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Gretl User's Guide

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1. Figure 7 1 gretl s gnuplot controller 7 2 Boxplots Boxplots are not generated using gnuplot but rather by gretl itself These plots after Tukey and Chambers display the distribution of a variable The central box encloses the middle 50 percent of the data i e it is bounded by the first and third quartiles The whiskers extend to the minimum and maximum values A line is drawn across the box at the median Chapter 7 Graphs and plots 45 In the case of notched boxes the notch shows the limits of an approximate 90 percent confidence interval This is obtained by the bootstrap method which can take a while if the data series is very long Clicking the mouse in the boxplots window brings up a menu which enables you to save the plots as encapsulated postscript EPS or as a full page postscript file Under the X window system you can also save the window as an XPM file under MS Windows you can copy it to the clipboard as a bitmap The menu also gives you the option of opening a summary window which displays five number summaries minimum first quartile median third quartile maximum plus a confidence interval for the median if the notched option was chosen Some details of gretl s boxplots can be controlled via a plain text file named boxplotrc which is looked for in turn in the current working directory the user s home directory corresponding to the environment variable HOME and the gretl user directo
2. Hodrick Robert and Prescott Edward C 1997 Postwar U S Business Cycles An Empirical Inves tigation Journal of Money Credit and Banking 29 pp 1 16 Johansen S ren 1995 Likelihood Based Inference in Cointegrated Vector Autoregressive Models Oxford Oxford University Press Keane Michael P and Wolpin Kenneth I 1997 The Career Decisions of Young Men Journal of Political Economy 105 pp 473 522 Kiviet J F 1986 On the Rigour of Some Misspecification Tests for Modelling Dynamic Relation ships Review of Economic Studies 53 pp 241 61 Kwiatkowski D Phillips P C B Schmidt P and Shin Y 1992 Testing the Null of Stationarity Against the Alternative of a Unit Root How Sure Are We That Economic Time Series Have a Unit Root Journal of Econometrics 54 pp 159 78 Locke C 1976 A Test for the Composite Hypothesis that a Population has a Gamma Distribution Communications in Statistics Theory and Methods A5 4 pp 351 64 Lucchetti R Papi L and Zazzaro A 2001 Banks Inefficiency and Economic Growth A Micro Macro Approach Scottish Journal of Political Economy 48 pp 400 424 McCullough B D and Renfro Charles G 1998 Benchmarks and software standards A case study of GARCH procedures Journal of Economic and Social Measurement 25 pp 59 71 Mackinnon J G 1996 Numerical Distribution Functions for Unit Root and Cointegration Tests
3. Journal of Applied Econometrics 11 pp 601 18 Bibliography 177 MacKinnon J G and White H 1985 Some Heteroskedasticity Consistent Covariance Matrix Esti mators with Improved Finite Sample Properties Journal of Econometrics 29 pp 305 25 Maddala G S 1992 Introduction to Econometrics 2nd edition Englewood Cliffs NJ Prentice Hall Matsumoto M and Nishimura T 1998 Mersenne twister a 623 dimensionally equidistributed uniform pseudo random number generator ACM Transactions on Modeling and Computer Simulation 8 pp 3 30 Mroz T 1987 The Sensitivity of an Empirical Model of Married Women s Hours of Work to Eco nomic and Statistical Assumptions Econometrica 55 pp 765 99 Nerlove M 1999 Properties of Alternative Estimators of Dynamic Panel Models An Empirical Analysis of Cross Country Data for the Study of Economic Growth in Hsiao C Lahiri K Lee L F and Pesaran M H eds Analysis of Panels and Limited Dependent Variable Models Cambridge Cambridge University Press Neter J Wasserman W and Kutner M H 1990 Applied Linear Statistical Models 3rd edition Boston MA Irwin Newey W K and West K D 1987 A Simple Positive Semi Definite Heteroskedasticity and Auto correlation Consistent Covariance Matrix Econometrica 55 pp 703 8 Newey W K and West K D 1994 Automatic Lag Selection in Covariance Matrix Estimation Review of Eco
4. The best known such criterion for linear models estimated via least squares is the adjusted R SSR n k 52 d gt TSS n 1 where n is the number of observations in the sample k denotes the number of parameters esti mated and SSR and TSS denote the sum of squared residuals and the total sum of squares for the dependent variable respectively Compared to the ordinary coefficient of determination or unadjusted R the adjusted calculation penalizes the inclusion of additional parameters other things equal 19 2 Information criteria A more general criterion in a similar spirit is Akaike s 1974 Information Criterion AIC The original formulation of this measure is AIC 2 6 2k 19 1 where represents the maximum loglikelihood as a function of the vector of parameter esti mates 0 and k as above denotes the number of independently adjusted parameters within the model In this formulation with AIC negatively related to the likelihood and positively related to the number of parameters the researcher seeks the minimum AIC The AIC can be confusing in that several variants of the calculation are in circulation For exam ple Davidson and MacKinnon 2004 present a simplified version AIC 6 k which is just 2 times the original in this case obviously one wants to maximize AIC In the case of models estimated by least squares the loglikelihood can be
5. matrix y price matrix b inv X X X y printf estimated coefficient vector n b matrix u y X b scalar SSR u u scalar s2 SSR Crows X rows b matrix V s2 inv X X V matrix se sqrt diag V printf estimated standard errors n se compare with built in function ols price const sqft vcv 86 Chapter 12 Matrix manipulation abs cols eigengen I int mcov minc nul lspace qrdecomp sort tr zeros atan cos eigensym imaxc inv maxc minr onenorm rank sqrt transp Table 12 3 Alphabetical listing of matrix functions cdemean det exp imaxr Tdet maxr mlag ones rcond sumc unvech cholesky diag fft iminc Tngamma meanc mnormal princomp rows sumr values cmult dnorm ffti iminr log meanr mshape qform seq svd vec cnorm dsort gamma infnorm mcorr mexp muni form qnorm sin tan vech 87 Chapter 13 Cheat sheet This chapter explains how to perform some common and some not so common tasks in gretl s scripting language Some but not all of the techniques listed here are also available through the graphical interface Although the graphical interface may be more intuitive and less intimidating at first we encourage users to take advantage of the power of gretl s scripting language as soon as they feel comfortable with the program 13 1 Dataset handling Weird periodicities Problem You have data sampled each 3 m
6. If you are able to figure out the derivatives of the regression function with respect to the para meters it is advisable to supply those derivatives as shown in the examples above If that is not possible gretl will compute approximate numerical derivatives The properties of the NLS algo rithm may not be so good in this case see Section 16 7 If analytical derivatives are supplied they are checked for consistency with the given nonlinear function If the derivatives are clearly incorrect estimation is aborted with an error message If the derivatives are suspicious a warning message is issued but estimation proceeds This warning may sometimes be triggered by incorrect derivatives but it may also be triggered by a high degree of collinearity among the derivatives Note that you cannot mix analytical and numerical derivatives you should supply expressions for all of the derivatives or none Chapter 16 Nonlinear least squares 109 16 5 Controlling termination The NLS estimation procedure is an iterative process Iteration is terminated when the criterion for convergence is met or when the maximum number of iterations is reached whichever comes first Let k denote the number of parameters being estimated The maximum number of iterations is 100 x k 1 when analytical derivatives are given and 200 x k 1 when numerical derivatives are used Let denote a small number The iteration is deemed to have converged if at le
7. A simple example of a Monte Carlo loop in progressive mode is shown in Example 9 1 Example 9 1 Simple Monte Carlo loop nulldata 50 seed 547 genr x 100 uniform open a progressive loop to be repeated 100 times loop 100 progressive genr u 10 normal O construct the dependent variable genr y 10 x u run OLS regression ols y const x grab the coefficient estimates and R squared genr a coeff const genr b coeff x genr r2 rsq arrange for printing of stats on these print a b r2 and save the coefficients to file store coeffs gdt a b endloop This loop will print out summary statistics for the a and b estimates and R across the 100 rep etitions After running the loop coeffs gdt which contains the individual coefficient estimates from all the runs can be opened in gretl to examine the frequency distribution of the estimates in detail The command nulldata is useful for Monte Carlo work Instead of opening a real data set nulldata 50 for instance opens a dummy data set containing just a constant and an index variable with a series length of 50 Constructed variables can then be added using the genr com mand See the set command for information on generating repeatable pseudo random series Chapter 9 Loop constructs 55 Iterated least squares Example 9 2 uses a while loop to replicate the estimation of a nonlinear consumption function of the form C a BY
8. SSR SSR where n is the total number of observations used On this variant H cannot be negative since adding additional regressors to the RE model cannot raise the SSR By default gretl computes the Hausman test via the matrix difference method largely for compara bility with other software but it uses the regression method if you pass the option hausman reg to the panel command Robust standard errors For most estimators gretl offers the option of computing an estimate of the covariance matrix that is robust with respect to heteroskedasticity and or autocorrelation and hence also robust standard errors In the case of panel data robust covariance matrix estimators are available for the pooled and fixed effects model but not currently for random effects Please see section 14 4 for details 15 2 Dynamic panel models Special problems arise when a lag of the dependent variable is included among the regressors in a panel model Consider a dynamic variant of the pooled model 15 1 Vit XitB PVit 1 Uit 15 7 First if the error uj includes a group effect v then yjz_ is bound to be correlated with the error since the value of v affects y at all t That means that OLS applied to 15 7 will be inconsistent as well as inefficient The fixed effects model sweeps out the group effects and so overcomes this particular problem but a subtler issue remains which applies to both fixed and random effects estimation Consi
9. b2 0 280405 0 245828 1 141 0 25401 GMM criterion 0 0110046 Gretl computes the HAC covariance matrix by default when a GMM model is estimated on time series data You can control the kernel and the bandwidth that is the value of k in 18 10 using the set command See chapter 14 for further discussion of HAC estimation You can also ask gretl not to use the HAC version by saying set force_hc on 18 5 Areal example the Consumption Based Asset Pricing Model To illustrate gretl s implementation of GMM we will replicate the example given in chapter 3 of Hall 2005 The model to estimate is a classic application of GMM and provides an example of a case when orthogonality conditions do not stem from statistical considerations but rather from economic theory A rational individual who must allocate his income between consumption and investment in a financial asset must in fact choose the consumption path of his whole lifetime since investment translates into future consumption It can be shown that an optimal consumption path should satisfy the following condition pu er OFF ren cere Fels 18 11 where p is the asset price U is the individual s utility function 6 is the individual s subjective discount rate and 7 is the asset s rate of return between time t and time t k F is the infor mation set at time t equation 18 11 says that the utility lost at time t by purchasing the asset instead of consumptio
10. ll j and the observation rule is given by y for sf gt 0 21 11 dd f O for s lt 0 One of the most popular applications of this model in econometrics is a wage equation coupled with a labor force participation equation we only observe the wage for the employed If y and sf were conditionally independent there would be no reason not to use OLS for estimating equation 21 9 otherwise OLS does not yield consistent estimates of the parameters j A widely used estimator is the so called Heckit estimator named after Heckman 1979 The pro cedure can be briefly outlined as follows first a probit model is fit on equation 21 10 next the generalized residuals are inserted in equation 21 9 to correct for the effect of sample selection Example 21 5 shows two estimates from the dataset used in Mroz 1987 the first one replicates Table 22 7 in Greene 2003 while the second one replicates table 17 1 in Wooldridge 2002a Note that the heckit inp script provided with gretl as an example script is invoked Chapter 21 Discrete and censored dependent variables Example 21 5 Heckit model open mroz gdt include heckit inp genr EXP2 AXA2 genr WA2 WAA2 genr KIDS KL6 K618 gt 0 Greene s specification list X list Z const AX EXP2 WE CIT const WA WA2 FAMINC KIDS WE heckit WW X LFP Z Wooldridge s specification series NWINC FAMINC WW WHRS series lww log Ww list X const WE AX EXP2 list
11. We wish to select for analysis only the women If we have a gender dummy variable with value 1 for men and 0 for women we could do smp1 gender 0 restrict to this effect Or suppose we want to restrict the sample to respondents with incomes over 50 000 Then we could use smp1 income gt 50000 restrict A question arises here If we issue the two commands above in sequence what do we end up with in our sub sample all cases with income over 50000 or just women with income over 50000 By default in a gretl script the answer is the latter women with income over 50000 The second restriction augments the first or in other words the final restriction is the logical product of the new restriction and any restriction that is already in place If you want a new restriction to replace any existing restrictions you can first recreate the full dataset using smp1 full Alternatively you can add the replace option to the smpl command smp1 income gt 50000 restrict replace This option has the effect of automatically re establishing the full dataset before applying the new restriction Unlike a simple setting of the sample restricting the sample may result in selection of non contiguous observations from the full data set It may also change the structure of the data set This can be seen in the case of panel data Say we have a panel of five firms indexed by the variable firm observed in each of several years identified b
12. clicking on its row in the main data window Most options will be self explanatory Note that you can rename a variable and can edit its descriptive label under Edit attributes You can also Define a new variable via a formula e g involving some function of one or more existing variables For the syntax of such formulae look at the online help for Generate variable syntax or see the genr command in the Gretl Command Reference One simple example foo x1 x2 Chapter 2 Getting started 11 will create a new variable foo as the product of the existing variables x1 and x2 In these formulae variables must be referenced by name not number e Model menu For details on the various estimators offered under this menu please consult the Gretl Command Reference Also see Chapter 16 regarding the estimation of nonlinear models e Help menu Please use this as needed It gives details on the syntax required in various dialog entries 2 4 Keyboard shortcuts When working in the main gretl window some common operations may be performed using the keyboard as shown in the table below Return Opens a window displaying the values of the currently selected variables it is the same as selecting Data Display Values Delete Pressing this key has the effect of deleting the selected variables A confirma tion is required to prevent accidental deletions e Has the same effect as selecting Edit attributes from the V
13. depending on the specification of the model and the estimation method chosen 1 Estimate a pure AR model by Least Squares nonlinear least squares if the model requires it otherwise OLS Set the AR parameter values based on this regression and set the MA parameters to a small positive value 0 0001 Chapter 20 Time series models 137 2 The Hannan Rissanen method First estimate an autoregressive model by OLS and save the residuals Then in a second OLS pass add appropriate lags of the first round residuals to the model to obtain estimates of the MA parameters To see the details of the ARMA estimation procedure add the verbose option to the command This prints a notice of the initialization method used as well as the parameter values and log likelihood at each iteration Besides the build in initialization mechanisms the user has the option of specifying a set of starting values manually This is done via the set command the first argument should be the keyword initvals and the second should be the name of a pre specified matrix containing starting values For example matrix start 0 0 85 0 34 set initvals start arma 1 1 y The specified matrix should have just as many parameters as the model in the example above there are three parameters since the model implicitly includes a constant The constant if present is always given first otherwise the order in which the parameters are expected is the same as the order
14. e as presented in Greene 2000 Example 11 3 This script is included in the gretl distribution under the name greene11_3 inp you can find it in gretl under the menu item File Script files Practice file Greene The option print final for the ols command arranges matters so that the regression results will not be printed each time round the loop but the results from the regression on the last iteration will be printed when the loop terminates Example 9 2 Nonlinear consumption function open greenel1_3 gdt run initial OLS ols CO Y genr essbak ess genr essdiff 1 genr beta coeff Y genr gamma 1 iterate OLS till the error sum of squares converges loop while essdiff gt 00001 form the linearized variables genr CO C gamma beta YAgamma log Y genr x1 YAgamma genr x2 beta YAgamma log Y run OLS ols CO 0 x1 x2 print final no df corr vcv genr beta coeff x1 genr gamma coeff x2 genr ess ess genr essdiff abs ess essbak essbak genr essbak ess endloop print parameter estimates using their proper names noecho printf alpha printf beta printf gamma g n coeff 0 g n beta g n gamma Example 9 3 shows how a loop can be used to estimate an ARMA model exploiting the outer product of the gradient OPG regression discussed by Davidson and MacKinnon in their Estimation and Inference in Econometrics Indexed loop examples Example 9 4 shows an
15. followed by a cross section for the same variable in 1970 The following gretl script illustrates how we can accomplish the stacking for both x1 and x2 We assume that the original data file is called panel txt and that in this file the columns are headed with variable names p1 p2 p5 The columns are not really variables but in the first instance we pretend that they are open panel txt genr x1 stack pl p5 length 50 genr x2 stack pl p5 offset 50 length 50 setobs 50 1 1 stacked cross section store panel gdt x1 x2 The second line illustrates the syntax of the stack function The double dots within the parenthe ses indicate a range of variables to be stacked here we want to stack all 5 columns for all 5 years The full data set contains 100 rows in the stacking of variable x1 we wish to read only the first 50 rows from each column we achieve this by adding length 50 Note that if you want to stack a non contiguous set of columns you can put a comma separated list within the parentheses as in genr x stack p1 p3 p5 On line 3 we do the stacking for variable x2 Again we want a length of 50 for the components of the stacked series but this time we want gretl to start reading from the 50th row of the original data and we specify offset 50 Line 4 imposes a panel interpretation on the data finally we save the data in gretl format with the panel interpretation discarding the original varia
16. g g n xi mean Y xi sd Y end loop In this case we evaluate the conditional mean and standard deviation of the variable Y for each value of Z8 Chapter 9 Loop constructs 53 For each loop The fourth form of loop control also uses the internal variable i but in this case the variable ranges over a specified list of strings The loop is executed once for each string in the list This can be useful for performing repetitive operations on a list of variables Here is an example of the syntax loop foreach i peach pear plum print i endloop This loop will execute three times printing out peach pear and plum on the respective itera tions If you wish to loop across a list of variables that are contiguous in the dataset you can give the names of the first and last variables in the list separated by rather than having to type all the names For example say we have 50 variables AK AL WY containing income levels for the states of the US To run a regression of income on time for each of the states we could do genr time loop foreach i AL WY ols i const time endloop For loop The final form of loop control uses a simplified version of the for statement in the C programming language The expression is composed of three parts separated by semicolons The first part specifies an initial condition expressed in terms of a control variable the second part gives a continuation condi
17. gretl could also be built using GTK 1 2 Support for this was dropped at version 1 6 0 of gretl 168 Appendix B Building gretl 169 In this guided example we will build gretl complete with documentation This introduces a few more requirements but gives you the ability to modify the documentation files as well like the help files or the manuals We assume that the basic GNU utilities are already installed on the system together with these other programs e some TpX BTpXsystem tetex or tex ive will do beautifully e Gnuplot e ImageMagick We also assume that the user has administrative privileges and knows how to install packages The examples below are carried out using the apt get shell command but they can be performed with menu based utilities like aptitude dselect or the GUI based program synaptic Users of Linux distributions which employ rpm packages e g Red Hat Fedora Mandriva SuSE may want to refer to the dependencies page on the gretl website The first step is installing the C compiler and related utilities On a Debian system these are contained in a bunch of packages that can be installed via the command apt get install gcc autoconf automake1 9 libtool flex bison gcc doc libc6 dev libc dev libgfortranl libgfortranl dev gettext pkgconfig Then it is necessary to install the development dev packages for the libraries that gretl uses Library command GLIB apt get install libglib2 0 dev GTK 2 0 apt
18. handle this and we will not discuss it here Chapter 14 Robust covariance matrix estimation 94 common point of departure namely the idea that the squared OLS residuals are likely to be too small on average This point is quite intuitive The OLS parameter estimates satisfy by design the criterion that the sum of squared residuals Sa Y 01 XB is minimized for given X and y Suppose that B B This is almost certain to be the case even is OLS is not biased it would be a miracle if the 6 calculated from any finite sample were exactly equal to f But in that case the sum of squares of the true unobserved errors Su X y Xb is bound to be greater than gt a The elaborated variants on HC take this point on board as follows e HC Applies a degrees of freedom correction multiplying the HCy matrix by T T k e HC Instead of using e for the diagonal elements of Q uses 0 1 ht where hy X X X X the t diagonal element of the projection matrix P which has the property that P y The relevance of h is that if the variance of all the u is o the expectation of a is 02 1 hy or in other words the ratio a h has expectation o As Davidson and MacKinnon show 0 lt h lt 1 for all t so this adjustment cannot reduce the the diagonal elements of and in general revises them upward e HC3 Uses n2 1 h The additional factor of 1 h in the denomina
19. i 1 with m d i 1 d k 1 ki Wk 1 k Yk The same expansion can be used with the lag operator so that if we defined Y 1 ye X this could be considered shorthand for Yt Xt 0 5Xt 1 0 125Xt 2 0 0625Xt 3 In gretl this transformation can be accomplished by the syntax genr Y fracdiff X 0 5 The Hodrick Prescott filter This filter is accessed using the hpfiltO function which takes one argument the name of the variable to be processed A time series y may be decomposed into a trend or growth component g and a cyclical component Ct Yea Oot Cty t 1 2 T The Hodrick Prescott filter effects such a decomposition by minimizing the following T T 1 gt Or o A gt gra 9t 9 90 t 1 t 2 The first term above is the sum of squared cyclical components ct y gt The second term is a multiple A of the sum of squares of the trend component s second differences This second term penalizes variations in the growth rate of the trend component the larger the value of A the higher is the penalty and hence the smoother the trend series Note that the hpf11t function in gretl produces the cyclical component ct of the original series If you want the smoothed trend you can subtract the cycle from the original genr ct hpfilt yt genr gt yt ct Hodrick and Prescott 1997 suggest that a value of A 1600 is reasonable for quarterly data The default value in gretl is 100 time
20. in the first case containing the variances on the diagonal and covariances of the variables in the columns of A and in the second containing the correlations of the variables For an n x n matrix A mexp A returns an n x n matrix holding the matrix exponential Z Ak I A A r a G k 0 This series is sure to converge The cholesky function computes the Cholesky decomposition L of a symmetric positive definite matrix A A LL L is lower triangular has zeros above the diagonal The diag function returns the principal diagonal of an n x n matrix A as a column vector that is an n vector v such that vi ii The cdemean function applied to an m x n matrix A returns an m x n matrix B such that bij aij Aj where A denotes the mean of column j of A The vec function applied to an m x n matrix A returns a column vector of length mn formed by stacking the columns of A The vech function applied to an n x n matrix A returns a column vector of length n n 1 2 formed by stacking the elements of the lower triangle of A column by column Note that A must be square for the operation to make sense A should also be symmetric The unvech function performs the inverse operation producing a symmetric matrix Chapter 12 Matrix manipulation 79 The mlag function requires two arguments a matrix and a scalar lag order m Applied to an T x k matrix A this function returns a T x k matrix B such that b ai m j ls
21. muniform m n m xn matrix filled with uniform random values mnormal m n m x n matrix filled with normal random values seq a b row vector containing the numbers from a to b The dimensions m and n or in the case of seq the limits a and b may be given numerically by reference to pre existing scalar variables or as expressions that evaluate to scalars The muniform and mnormal matrix functions fill the matrix with drawings from the uniform 0 1 distribution and the standard normal distribution respectively The seq function generates a sequence of integers from a to b inclusive increasing if a lt b or decreasing if a gt b Matrix reshaping A matrix can also be created by re arranging the elements of a pre existing matrix This is accom plished via the mshape function It takes three arguments the input matrix A and the rows and columns of the target matrix r and c respectively Elements are read from A and written to the target in column major order If A contains fewer elements than n r x c they are repeated cyclically if A has more elements only the first n are used For example Chapter 12 Matrix manipulation 80 matrix a mnormal 2 3 a matrix b mshape a 3 1 b matrix b mshape a 5 2 b produces a a 1 2323 0 99714 0 39078 0 54363 0 43928 0 48467 matrix b mshape a 3 1 Generated matrix b b b 1 2323 0 54363 0 99714 matrix b mshape a 5 2 Replaced matrix b b b 1
22. source file containing the text or mathematics to be typset interspersed with mark up that defines how it should be formatted The second step is to run the source through a processing engine that does the actual formatting Typically this is either e a program called latex that generates so called DVI device independent output or e a program called pdflatex that generates PDF output For previewing one uses either a DVI viewer typically xdvi on GNU Linux systems or a PDF viewer for example Adobe s Acrobat Reader or xpdf depending on how the source was processed If the DVI route is taken there s then a third step to produce printable output typically using the program dvips to generate a PostScript file If the PDF route is taken the output is ready for printing without any further processing On the MS Windows and Mac OS X platforms gretl calls pdflatex to process the source file and expects the operating system to be able to find the default viewer for PDF output DVI is not supported On GNU Linux the default is to take the DVI route but if you prefer to use PDF you can do the following select the menu item Tools Preferences General then the Programs tab Find the item titled Command to compile TeX files and set this to pdflatex Make sure the Command to view PDF files is set to something appropriate 22 2 TpX related menu items The model window The fullest TfX support in gretl is found
23. 0 241287 Cross tabulation of Z1 rows against Z6 columns C 91 121 14 16 17 1 181 20 TOT 0 4 36 106 70 52 45 2 315 1 3 8 48 45 37 67 78 286 TOTAL 7 44 154 115 89 112 80 601 Pearson chi square test 123 177 6 df p value 3 50375e 24 Cross tabulation of Z4 rows against Z5 columns 1 E 2 JE 3JE 4 5 TOT 0 17 60 35 45 14 171 1 31 104 94 145 56 430 TOTAL 48 164 129 190 70 601 Pearson chi square test 11 1615 4 df p value 0 0248074 Cross tabulation of Z4 rows against Z6 columns C 91 121 14 16 1 17 181 20 TOT 0 1 8 39 47 30 32 14 171 1 6 36 115 68 59 80 66 430 TOTAL 7 44 154 115 89 112 80 601 Pearson chi square test 18 3426 6 df p value 0 0054306 Pearson s x test for independence is automatically displayed provided that all cells have expected frequencies under independence greater than 1077 However a common rule of thumb states that this statistic is valid only if the expected frequency is 5 or greater for at least 80 percent of the cells If this condition is not met a warning is printed Additionally the options row or column options can be given in this case the output displays row or column percentages respectively If you want to cut and paste the output of xtab to some other program e g a spreadsheet you may want to use the zeros option this option causes cells with zero frequency to display the number 0 instead of being empty Ch
24. 2 pp 216 231 Invited paper Special Issue on Program Generation Optimization and Platform Adaptation Goossens M Mittelbach F and Samarin A 2004 The BT X Companion 2nd edition Boston Addison Wesley Gourieroux C Monfort A Renault E and Trognon A 1987 Generalized Residuals Journal of Econometrics 34 pp 5 32 Greene William H 2000 Econometric Analysis 4th edition Upper Saddle River NJ Prentice Hall Greene William H 2003 Econometric Analysis 5th edition Upper Saddle River NJ Prentice Hall Gujarati Damodar N 2003 Basic Econometrics 4th edition Boston MA McGraw Hill Hall Alastair D 2005 Generalized Method of Moments Oxford Oxford University Press Hamilton James D 1994 Time Series Analysis Princeton NJ Princeton University Press Hannan E J and Quinn B G 1979 The Determination of the Order of an Autoregression Journal of the Royal Statistical Society B 41 pp 190 95 Hansen L P 1982 Large Sample Properties of Generalized Method of Moments Estimation Econometrica 50 pp 1029 1054 Hansen L P and Singleton K J 1982 Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models Econometrica 50 pp 1269 86 Hausman J A 1978 Specification Tests in Econometrics Econometrica 46 pp 1251 71 Heckman J 1979 Sample Selection Bias as a Specification Error Econometrica 47 ppi53 161
25. 2323 0 48467 0 54363 1 2323 0 99714 0 54363 0 43928 0 99714 0 39078 0 43928 Single return two matrix functions The function qform constructs a quadratic form in a matrix A and a conformable symmetric matrix X The command B qform A X calculates B AXA This is computed more efficiently than the alternative command B A X A In addition the result is symmetric by construction The function cmult computes the complex product of two input matrices A and B representing complex numbers These matrices must have the same number of rows n and either one or two columns The first column contains the real part and the second if present the imaginary part The return value is an n x 2 matrix or if the product has no imaginary part an n vector Multiple return matrix functions The functions that take one or more matrices as arguments and compute one or more matrices are qrdecomp QR decomposition eigensym Eigen analysis of symmetric matrix eigengen Eigen analysis of general matrix svd Singular value decomposition SVD Chapter 12 Matrix manipulation 81 The syntax for all but the last of these functions is of the form matrix B func A amp C while for svd it is matrix B func A amp C amp D The first argument A represents the input data that is the matrix whose decomposition or analysis is required The second argument and in the case of svd the third must be either the name of an existing mat
26. 9 Figure 14 1 Three HAC kernels ADNA Bartlett Parzen QS In gretl you select the kernel using the set command with the hac_kernel parameter set hac_kernel parzen set hac_kernel qs set hac_kernel bartlett Selecting the HAC bandwidth The asymptotic theory developed by Newey West and others tells us in general terms how the HAC bandwidth p should grow with the sample size T that is p should grow in proportion to some fractional power of T Unfortunately this is of little help to the applied econometrician working with a given dataset of fixed size Various rules of thumb have been suggested and gretl implements two such The default is p 0 75T 3 as recommended by Stock and Watson 2003 An alternative is p 4 T 100 9 as in Wooldridge 2002b In each case one takes the integer part of the result These variants are labeled nw1 and nw2 respectively in the context of the set command with the hac_lag parameter That is you can switch to the version given by Wooldridge with Chapter 14 Robust covariance matrix estimation 97 set hac_lag nw2 As shown in Table 14 1 the choice between nw1 and nw2 does not make a great deal of difference T p nwl p nw2 50 2 3 100 3 4 150 3 4 200 4 4 300 5 5 400 5 3 Table 14 1 HAC bandwidth two rules of thumb You also have the option of specifying a fixed numerical value for p as in set hac_lag 6 In addition you can set a distinct bandwidth for use with the Quadratic
27. CML method is nearly equivalent to maximum likelihood under the hypothesis of normality the difference is that the first t 1 observations are considered fixed and only enter the like lihood function as conditioning variables As a consequence the two methods are asymptotically equivalent under standard conditions except for the fact discussed above that our CML imple mentation treats the constant and exogenous variables as per equation 20 3 The two methods can be compared as in the following example open datal0 1 arma 1 1 r arma 1 1 r conditional which produces the estimates shown in Table 20 1 As you can see the estimates of p and O are quite similar The reported constants differ widely as expected see the discussion following equations 20 4 and 20 5 However dividing the CML constant by 1 q we get 7 38 which is not far from the ML estimate of 6 93 Table 20 1 ML and CML estimates Parameter ML CML H 6 93042 0 673202 1 07322 0 488661 0 855360 0 0512026 0 852772 0 0450252 0 0 588056 0 0809769 0 591838 0 0456662 Convergence and initialization The numerical methods used to maximize the likelihood for ARMA models are not guaranteed to converge Whether or not convergence is achieved and whether or not the true maximum of the likelihood function is attained may depend on the starting values for the parameters Gretl employs one of the following two initialization mechanisms
28. Check the numerical accuracy of gretl against the reference results for linear regression made available by the US National Institute of Standards and Technol ogy Preferences Set the paths to various files gretl needs to access Choose the font in which gretl displays text output Select or unselect expert mode If this mode is selected various warning messages are suppressed Activate or suppress gretl s messaging about the availability of program updates Configure or turn on off the main window toolbar See the Gretl Command Reference for further details e Data menu Select all Several menu items act upon those variables that are currently selected in the main window This item lets you select all the variables Display values Pops up a window with a simple not editable printout of the values of the selected variable or variables Edit values Opens a spreadsheet window where you can edit the values of the selected variables Add observations Gives a dialog box in which you can choose a number of observations to add at the end of the current dataset for use with forecasting Remove extra observations Active only if extra observations have been added automati cally in the process of forecasting deletes these extra observations Read info Edit info Read info just displays the summary information for the current data file Edit info allows you to make changes to it if you have permission
29. For example some series are monthly and some are quarterly Your first step is to formulate a strategy Do you want to end up with a quarterly or a monthly data set A basic point to note here is that compacting data from a higher frequency e g monthly to a lower frequency e g quarterly is usually unproblematic You lose information in doing so but in general it is perfectly legitimate to take say the average of three monthly observations to create a quarterly observation On the other hand expanding data from a lower to a higher frequency is not in general a valid operation In most cases then the best strategy is to start by creating a data set of the lower frequency and then to compact the higher frequency data to match When you import higher frequency data from a database into the current data set you are given a choice of compaction method average sum start of period or end of period In most instances average is likely to be appropriate You can also import lower frequency data into a high frequency data set but this is generally not recommended What gretl does in this case is simply replicate the values of the lower frequency series as many times as required For example suppose we have a quarterly series with the value 35 5 in 1990 1 the first quarter of 1990 On expansion to monthly the value 35 5 will be assigned to the observations for January February and March of 1990 The expanded variable is
30. For some purposes it may be useful to save a string that is a sequence of characters as a named variable that can be reused Versions of gretl higher than 1 6 0 offer this facility but some of the refinements noted below are available only in gretl 1 6 3 and higher To define a string variable you can use either of two commands string or sprintf The string command is simpler you just type for example string foo some stuff I want to save The first field after string is the name under which the string should be saved then comes an equals sign then comes the string to be saved enclosed in double quotes The latter can be repre sented as a sequence of sub strings if need be as in mow string bits first and second See below for further variants of the string command including use of getenv The sprintf command is more flexible It works exactly as gretl s printf command except that the format string must be preceded by the name of a string variable For example scalar x 8 sprintf foo var d x To retrieve the value of a string variable you give the name of the variable preceded by the at sign In most contexts the notation is treated as a macro That is if a sequence of characters in a gretl command following the symbol is recognized as the name of a string variable the value of that variable is sustituted literally into the command line before the regular parsing of the command i
31. Fourier Chapter 5 Special functions in genr 38 transforms and inverting the result If n Zk gt XjVk js j l then F z F x O Fly That is F Z r F MORF ye For computing the Fourier transform gretl uses the external library fftw3 see Frigo and Johnson 2003 This guarantees extreme speed and accuracy In fact the CPU time needed to perform the transform is O n logn for any n This is why the array of numerical techniques employed in fftw3 is commonly known as the Fast Fourier Transform Gretl provides two matrix functions for performing the Fourier transform and its inverse fft and ffti In fact gretl s implementation of the Fourier transform is somewhat more specialized the input to the fft function is understood to be real Conversely ffti takes a complex argument and delivers a real result For example xl1 1 2 3 perform the transform f fft a perform the inverse transform x2 ffticf yields 1 6 0 1 x 2 f 1 5 0 866 xXx2 2 3 1 5 0 866 3 where the first column of f holds the real part and the second holds the complex part In general if the input to fft has n columns the output has 2n columns where the real parts are stored in the odd columns and the complex parts in the even ones Should it be necessary to compute the Fourier transform on several vectors with the same number of elements it is numerically more efficient to group them into a matrix rather than invoking fft for
32. New script item or the new script toolbar button In either case a script window will open see Figure 3 1 gretl mrw2 inp gaart open mrw gdt genr lny log gdp85 genr ngd 0 05 popgrow 100 0 genr Ingd log ngd genr linv log inv 100 0 generate variable for testing Solow restriction genr x3 linv lngd restrict sample to non oil producing countries smpl o nonoil modell lt ols lny const linv lngd genr essu ess genr dful df ols lny const x3 q genr Fl ess essu essu dful restrict sample to the better data countries smpl o intermed model2 lt ols lny const linv lngd genr essu ess genr dfu2 df ols lny const x3 q genr F2 ess essu essu dfu2 PTE ee TA ee oe Figure 3 1 Script window editing a command file The toolbar at the top of the script window offers the following functions left to right 1 Save the file 2 Save the file under a specified name 3 Print the file ander Windows or the gnome desktop only 4 Execute the commands in the file 5 Copy selected text 6 Paste the selected text 7 Find and replace text 8 Undo the last Paste or Replace action 9 Help if you place the cursor in a command word and press the question mark you will get help on that command 10 Close the window When you click the Execute icon all output is directed to a single window where it can be edited saved or copied to the clipboard To
33. Section 4 3 for details Function files Handles function packages see Section 10 4 which allow you to access functions written by other users and share the ones written by you Exit Quit the program If expert mode is not selected you ll be prompted to save any unsaved work e Tools menu Statistical tables Look up critical values for commonly used distributions normal or Gaussian t chi square F and Durbin Watson P value finder Open a window which enables you to look up p values from the Gaussian t chi square F gamma or binomial distributions See also the pvalue command in the Gretl Command Reference Test statistic calculator Calculate test statistics and p values for a range of common hy pothesis tests population mean variance and proportion difference of means variances and proportions See also the item Bivariate tests under the Model menu Command log Open a window containing a record of the commands executed so far Gretl console Open a console window into which you can type commands as you would using the command line program gretlcli as opposed to using point and click Chapter 2 Getting started 9 Start Gnu R Start R if it is installed on your system and load a copy of the data set currently open in gretl See Appendix D Sort variables Rearrange the listing of variables in the main window either by ID number or alphabetically by name NIST test suite
34. Since Q E uu the matrix being estimated here can also be written as X E X uu X which expresses gt as the long run covariance of the random k vector X u From a computational point of view it is not necessary or desirable to store the potentially very large T x T matrix as such Rather one computes the sandwich filling by summation as p f 0 gt w f j 1 Chapter 14 Robust covariance matrix estimation 96 where the k x k sample autocovariance matrix j for j gt 0 is given by T A 1 ee MN gt UrUt j Xt Xt j t j 1 and w is the weight given to the autocovariance at lag j gt 0 This leaves two questions How exactly do we determine the maximum lag length or bandwidth p of the HAC estimator And how exactly are the weights w to be determined We will return to the difficult question of the bandwidth shortly As regards the weights Gretl offers three variants The default is the Bartlett kernel as used by Newey and West This sets ya t 23 j lt p 0 gt p so the weights decline linearly as j increases The other two options are the Parzen kernel and the Quadratic Spectral QS kernel For the Parzen kernel 1 6af 6a O0 lt aj lt 0 5 wj 2 1 aj 0 5 lt a lt 1 0 aj gt 1 where a j p 1 and for the OS kernel 25 Sam wj cosm 2 1245 mj y where dj j p and mj 6rrd 5 Figure 14 1 shows the weights generated by these kernels for p 4 and j 1 to
35. The size of such numbers in bytes depends on the computer platform but is typically eight To give a rough notion of magnitudes suppose we have a data set with 10 000 observations on 500 variables That s 5 million floating point numbers or 40 million bytes If we define the megabyte MB as 1024 x 1024 bytes as is standard in talking about RAM it s slightly over 38 MB The program needs additional memory for workspace but even so handling a data set of this size should be quite feasible on a current PC which at the time of writing is likely to have at least 256 MB of RAM If RAM is not an issue there is one further limitation on data size though it s very unlikely to be a binding constraint That is variables and observations are indexed by signed integers and on a typical PC these will be 32 bit values capable of representing a maximum positive value of 231 1 2 147 483 647 The limits mentioned above apply to gretl s native functionality There are tighter limits with regard to two third party programs that are available as add ons to gretl for certain sorts of time series analysis including seasonal adjustment namely TRAMO SEATS and X 12 ARIMA These pro grams employ a fixed size memory allocation and can t handle series of more than 600 observa tions 4 8 Data file collections If you re using gretl in a teaching context you may be interested in adding a collection of data files and or scripts that relate
36. a GARCH model is by no means a straightforward task Con sider equation 20 12 the conditional variance at any point in time o depends on the conditional variance in earlier periods but o is not observed and must be inferred by some sort of Maxi mum Likelihood procedure Gretl uses the method proposed by Fiorentini Calzolari and Panattoni 1996 which was adopted as a benchmark in the study of GARCH results by McCullough and Renfro 1998 It employs analytical first and second derivatives of the log likelihood and uses a mixed gradient algorithm exploiting the information matrix in the early iterations and then switch ing to the Hessian in the neighborhood of the maximum likelihood This progress can be observed if you append the verbose option to gretl s garch command Several options are available for computing the covariance matrix of the parameter estimates in connection with the garch command At a first level one can choose between a standard and a robust estimator By default the Hessian is used unless the robust option is given in which case the QML estimator is used A finer choice is available via the set command as shown in Table 20 2 It is not uncommon when one estimates a GARCH model for an arbitrary time series to find that the iterative calculation of the estimates fails to converge For the GARCH model to make sense there are strong restrictions on the admissible parameter values and it is no
37. a configuration dialog which can also be reached from the main menu bar Tools Preferences General HCCME There you can select from a set of possible robust estimation variants and can also choose to make robust estimation the default The specifics of the available options depend on the nature of the data under consideration cross sectional time series or panel and also to some extent the choice of estimator Although we introduced robust standard errors in the context of OLS above they may be used in conjunction with other estimators too The following three sections of this chapter deal with matters that are specific to the three sorts of data just mentioned Note that additional details regarding covariance matrix estimation in the context of GMM are given in chapter 18 We close this introduction with a brief statement of what robust standard errors can and cannot achieve They can provide for asymptotically valid statistical inference in models that are basically correctly specified but in which the errors are not iid The asymptotic part means that they may be of little use in small samples The correct specification part means that they are not a magic bullet if the error term is correlated with the regressors so that the parameter estimates themselves are biased and inconsistent robust standard errors will not save the day 14 2 Cross sectional data and the HCCME With cross sectional data the
38. alpha delta end gmm iterate gmm e delta ewr consratA alpha 1 orthog e inst weights V1 params alpha delta end gmm iterate 100000 I nelem inst const consrat 1 consrat 2 ewr 1 ewr 2 128 Chapter 18 GMM estimation 129 Example 18 5 Estimation of the Consumption Based Asset Pricing Model output Model 1 1 step GMM estimates using the 465 observations 1959 04 1997 12 e d ewr consratA alpha 1 1 PARAMETER ESTIMATE STDERROR T STAT P VALUE alpha 3 14475 6 84439 0 459 0 64590 d 0 999215 0 0121044 82 549 lt 0 00001 GMM criterion 2778 08 Model 2 1 step GMM estimates using the 465 observations 1959 04 1997 12 e d ewr consratA alpha 1 1 PARAMETER ESTIMATE STDERROR T STAT P VALUE alpha 0 398194 2 26359 0 176 0 86036 d 0 993180 0 00439367 226 048 lt 0 00001 GMM criterion 14 247 Model 3 Iterated GMM estimates using the 465 observations 1959 04 1997 12 e d ewr consratA alpha 1 1 PARAMETER ESTIMATE STDERROR T STAT P VALUE alpha 0 344325 2 21458 0 155 0 87644 d 0 991566 0 00423620 234 070 lt 0 00001 GMM criterion 5491 78 J test Chi square 3 11 8103 p value 0 0081 Model 4 Iterated GMM estimates using the 465 observations 1959 04 1997 12 e d ewr consratA alpha 1 1 PARAMETER ESTIMATE STDERROR T STAT P VALUE alpha 0 344315 2 21359 0 156 0 87639 d 0 991566 0 00423469 234 153 lt 0 00001 GMM criterion 5491 78 J test Chi square 3 11 8
39. are automatically interpreted as a cross section In the unlikely event that cross sectional data are wrongly interpreted as time series you can correct this by selecting the Data Dataset struc ture menu item Click the cross sectional radio button in the dialog box that appears then click Forward Click OK to confirm your selection Time series data When you import data from a spreadsheet or plain text file gretl will make fairly strenuous efforts to glean time series information from the first column of the data if it looks at all plausible that such information may be present If time series structure is present but not recognized again you can use the Data Dataset structure menu item Select Time series and click Forward select the appropriate data frequency and click Forward again then select or enter the starting observation 2See www estima com Chapter 4 Data files 22 and click Forward once more Finally click OK to confirm the time series interpretation if it is correct or click Back to make adjustments if need be Besides the basic business of getting a data set interpreted as time series further issues may arise relating to the frequency of time series data In a gretl time series data set all the series must have the same frequency Suppose you wish to make a combined dataset using series that in their original state are not all of the same frequency
40. browser 6 Open this manual in PDF format Chapter 2 Getting started 12 7 Open the help item for script commands syntax i e a listing with details of all available commands 8 Open the dialog box for defining a graph 9 Open the dialog box for estimating a model using ordinary least squares 10 Open a window listing the sample datasets supplied with gretl and any other data file collec tions that have been installed If you don t care to have the toolbar displayed you can turn it off under the Tools Preferences General menu Go o the Toolbar tab and uncheck the show gretl toolbar box Chapter 3 Modes of working 3 1 Command scripts As you execute commands in gretl using the GUI and filling in dialog entries those commands are recorded in the form of a script or batch file Such scripts can be edited and re run using either gretl or the command line client gretlcli To view the current state of the script at any point in a gretl session choose Command log under the Tools menu This log file is called session inp and it is overwritten whenever you start a new session To preserve it save the script under a different name Script files will be found most easily using the GUI file selector if you name them with the extension inp To open a script you have written independently use the File Script files menu item to create a script from scratch use the File Script files
41. classes the only files needed are as noted above the amsmath dcolumn and longtable packages These should be included in any reasonably full TeX implementation Chapter 22 Gretl and T X 161 22 4 Character encodings People using gretl in English speaking locales are unlikely to have a problem with this but if you re generating T X output in a locale where accented characters not in the ASCII character set are employed you may want to pay attention here Gretl generates TpX output using whatever character encoding is standard on the local system If the system encoding is in the ISO 8859 family this will probably be OK wihout any special effort on the part of the user Newer GNU Linux systems however typically use Unicode UTF 8 This is also OK so long as your TeX system can handle UTF 8 input which requires use of the latex ucs package So if you are using gretl to generate TX in a non English locale where the system encoding is UTF 8 you will need to ensure that the latex ucs package is installed This package may or may not be installed by default when you install Tx For reference if gretl detects a UTF 8 environment the following lines are used in the TeX preamble usepackage ucs usepackage ut f8x inputenc 22 5 Installing and learning Tex This is not the place for a detailed exposition of these matters but here are a few pointers So far as we know every GNU Linux distribution has a package or set of
42. column In this context the flattening of a panel data set can be done in either of two ways e Stacked time series the successive vertical blocks each comprise a time series for a given unit e Stacked cross sections the successive vertical blocks each comprise a cross section for a given period You may input data in whichever arrangement is more convenient Internally however gretl always stores panel data in the form of stacked time series Chapter 4 Data files 23 When you import panel data into gretl from a spreadsheet or comma separated format the panel nature of the data will not be recognized automatically most likely the data will be treated as undated A panel interpretation can be imposed on the data using the graphical interface or via the setobs command In the graphical interface use the menu item Data Dataset structure In the first dialog box that appears select Panel In the next dialog you have a three way choice The first two options Stacked time series and Stacked cross sections are applicable if the data set is already organized in one of these two ways If you select either of these options the next step is to specify the number of cross sectional units in the data set The third option Use index variables is applicable if the data set contains two variables that index the units and the time periods respectively the next step is then to select those variables For examp
43. each vector separately As an example consider the multiplication of two polynomals a x 1 0 5x b x 1 0 3x 0 8x c x a x b x 1 0 8x 0 65x 0 4x The coefficients of the polynomial c x are the convolution of the coefficents of a x and b x the following gretl code fragment illustrates how to compute the coefficients of c x define the two polynomials a 1 0 5 0 O b 1 0 3 0 8 0 perform the transforms fa fft a fb fft b complex multiply the two transforms fc cmult fa fb compute the coefficients of c via the inverse transform c ffti fc 1See chapter 12 Chapter 5 Special functions in genr 39 Maximum efficiency would have been achieved by grouping a and b into a matrix The computa tional advantage is so little in this case that the exercise is a bit silly but the following alternative may be preferable for a large number of rows columns define the two polynomials 1 0 5 0 0 1 0 3 0 8 0 perform the transforms jointly fft a b complex multiply the two transforms fc cmult f 1 2 f 3 41 compute the coefficients of c via the inverse transform c ffti fc h WM Traditionally the Fourier tranform in econometrics has been mostly used in time series analysis the periodogram being the best known example Example script 5 3 shows how to compute the periodogram of a time series via the fft function Example 5 3 Period
44. enables the author of a function to pre empt an ordinary execution error and or offer a more specific and helpful error message For example if nelem xlist 0 funcerr xlist must not be empty end if Error checking When gretl first reads and compiles a function definition there is minimal error checking the only checks are that the function name is acceptable and so far as the body is concerned that you are not trying to define a function inside a function see Section 10 1 Otherwise if the function body contains invalid commands this will become apparent only when the function is called and its commands are executed Printing of output The usual mechanism whereby gretl echos commands and reports on the creation of new variables is suppressed by default when a function is being executed If you want to turn this on for example Chapter 10 User defined functions 63 for the purpose of debugging function code you can issue one or both of the following commands inside the function set echo on set messages on 10 4 Function packages As of gretl 1 6 0 there is a mechanism to package functions and make them available to other users of gretl This is currently experimental but here is a walk through of the process Load a function in memory There are several ways to load a function e If you have a script file containing function definitions open that file and run it e Create a script file from scratch In
45. for the US and ewr is the return price ratio of a fictitious asset constructed by averaging all the stocks in the NYSE The instrument set contains the constant and two lags of each variable The command set force_hc on on the second line of the script has the sole purpose of replicating the given example as mentioned above it forces gretl to compute the long run variance of the orthogonality conditions according to equation 18 9 rather than 18 10 We run gmm four times one step estimation for each of two initial weights matrices then iterative estimation starting from each set of initial weights Since the number of orthogonality conditions 5 is greater than the number of estimated parameters 2 the choice of intial weights should make a difference and indeed we see fairly substantial differences between the one step estimates Models 1 and 2 On the other hand iteration reduces these differences almost to the vanishing point Models 3 and 4 Part of the output is given in 18 5 It should be noted that the J test leads to a rejection of the hypothesis of correct specification This is perhaps not surprising given the heroic assumptions required to move from the microeconomic principle in equation 18 11 to the aggregate system that is actually estimated 18 6 Caveats A few words of warning are in order despite its ingenuity GMM is possibly the most fragile esti mation method in econometrics The number of non obvious choices
46. get install libgtk2 0 dev PNG apt get install libpngl2 dev XSLT apt get install libxslt1 dev LAPACK apt get install lapack3 dev FFTW apt get install fftw3 dev GMP apt get install libgmp3 dev GMP is optional but recommended The dev packages for these libraries are necessary to compile gretl you ll also need the plain non dev library packages to run gretl but most of these should already be part of a standard installation In order to enable other optional features like audio support you may need to install more libraries At this point it is possible to build from the source 1 Download the latest gretl source package from gretl sourceforge net The released versions are guaranteed to build correctly However if you like to live dangerously you may download the CVS version which contains the work in progress towards the next release For instruc tions on how to download the CVS version please refer to the appropriate SourceForge page 2 Unzip and untar the package On a system with the GNU utilities available the command would be tar xvfz gretl N tar gz replace N with the specific version number of the file you downloaded at step 1 3 Change directory to the gretl source directory created at step 2 e g gret1 1 6 2 The next command you need is configure this is a complex script that detects which tools you have on your system and sets things up The configure command accepts many options you may want to run
47. import the file into the target program When you finish a gretl session you are given the option of saving all the output from the session to a single file Note that on the gnome desktop and under MS Windows the File menu includes a command to send the output directly to a printer t When pasting or importing plain text gretl output into a word processor select a monospaced or typewriter style font e g Courier to preserve the output s tabular formatting Select a small font 10 point Courier should do to prevent the output lines from being broken in the wrong place Note that when you copy as RTF under MS Windows Windows will only allow you to paste the material into appli cations that understand RTF Thus you will be able to paste into MS Word but not into notepad Note also that there appears to be a bug in some versions of Windows whereby the paste will not work properly unless the target application e g MS Word is already running prior to copying the material in question Chapter 2 Getting started 8 2 3 The main window menus Reading left to right along the main window s menu bar we find the File Tools Data View Add Sample Variable Model and Help menus File Tools Data View Add Sample Variable Model e File menu Open data Open a native gretl data file or import from other formats See Chapter 4 Append data Add data to the current working data set from a gretl data file a comma
48. in the GUI model window This has a menu item titled LaTeX with sub items View Copy Save and Equation options see Figure 22 1 Experts will be aware of something called plain TX which is processed using the program tex The great majority of TeX users however use the ATX macros initially developed by Leslie Lamport Gretl does not support plain Tex 156 Chapter 22 Gretl and TpX 157 Figure 22 1 BI X menu in model window File Edit Tests Save Graphs Analysis MEE View Model 1 OLS estimates using the 51 obser copy Dependent variable ENROLL FR Save VARIABLE COEFFICIENT S Equation options VALUE The first three sub items have branches titled Tabular and Equation By Tabular we mean that the model is represented in the form of a table this is the fullest and most explicit presentation of the results See Table 22 1 for an example this was pasted into the manual after using the Copy Tabular item in gretl a few lines were edited out for brevity Table 22 1 Example of BTX tabular output Model 1 OLS estimates using the 51 observations 1 51 Dependent variable ENROLL Variable Coefficient Std Error t statistic p value const 0 241105 0 0660225 3 6519 0 0007 CATHOL 0 223530 0 0459701 4 8625 0 0000 PUPIL 0 00338200 0 00271962 1 2436 0 2198 WHITE 0 152643 0 0407064 3 7499 0 0005 Mean of dependent variable 0 0955686 S D of dependent variable
49. including sales tax genr ravgprs avgprs cpi real avg cig specific tax genr rtax tax cpi real average total tax genr rtaxs taxs cpi real average sales tax genr rtaxso rtaxs rtax logs of consumption price income genr lIpackpc log packpc genr Travgprs log ravgprs genr perinc income pop cpi genr lperinc log perinc restrict sample to 1995 observations smpl restrict year 1995 Equation 10 16 by tsls list xlist const Travgprs lperinc list zlist const rtaxso rtax lperinc tsls Ipackpc xlist zlist robust setup for gmm matrix Z zlist matrix W inv Z Z series e 0 scalar b0 1 scalar b1 T scalar b2 1 gmm e lpackpc bO bl lravgprs b2 lperinc orthoge Z weights W params bO b1 b2 end gmm 125 Chapter 18 GMM estimation 126 Example 18 3 TSLS via GMM partial output Model 1 TSLS estimates using the 48 observations 1 48 Dependent variable lpackpc Instruments rtaxso rtax Heteroskedasticity robust standard errors variant HCO VARIABLE COEFFICIENT STDERROR T STAT P VALUE const 9 89496 0 928758 10 654 lt 0 00001 Travgprs 1 27742 0 241684 5 286 lt 0 00001 Tperinc 0 280405 0 245828 1 141 0 25401 Model 2 1 step GMM estimates using the 48 observations 1 48 e Ipackpc b0 b1 lravgprs b2 lperinc PARAMETER ESTIMATE STDERROR T STAT P VALUE bO 9 89496 0 928758 10 654 lt 0 00001 b1 1 27742 0 241684 5 286 lt 0 00001
50. is printed stating how many observations were dropped On the other hand the skipping of missing observations is not supported for all procedures exceptions include all autoregressive estimators system estimators such as SUR and nonlinear least squares In the case of panel data the skipping of missing observations is supported only if their omission leaves a balanced panel If missing observations are found in cases where they are not supported gretl gives an error message and refuses to produce estimates In case missing values in the middle of a dataset present a problem the misszero function use with care is provided under the genr command By doing genr foo misszero bar you can produce a series foo which is identical to bar except that any missing values become zeros Then Chapter 4 Data files 26 you can use carefully constructed dummy variables to in effect drop the missing observations from the regression while retaining the surrounding sample range 4 7 Maximum size of data sets Basically the size of data sets both the number of variables and the number of observations per variable is limited only by the characteristics of your computer Gretl allocates memory dynami cally and will ask the operating system for as much memory as your data require Obviously then you are ultimately limited by the size of RAM Aside from the multiple precision OLS option gretl uses double precision floating point numbers throughout
51. learn more about the possibilities of scripting take a look at the gretl Help item Command reference or start up the command line program gretlcli and consult its help or consult the Gretl Command Reference If you click on the Execute icon when part of a script is highlighted gretl will only run that portion 13 Chapter 3 Modes of working 14 Moreover if you want to run just the current line you can do so by pressing Ctrl Enter In addition the gretl package includes over 70 practice scripts Most of these relate to Ra manathan 2002 but they may also be used as a free standing introduction to scripting in gretl and to various points of econometric theory You can explore the practice files under File Script files Practice file There you will find a listing of the files along with a brief description of the points they illustrate and the data they employ Open any file and run it to see the output Note that long commands in a script can be broken over two or more lines using backslash as a continuation character You can if you wish use the GUI controls and the scripting approach in tandem exploiting each method where it offers greater convenience Here are two suggestions e Open a data file in the GUI Explore the data generate graphs run regressions perform tests Then open the Command log edit out any redundant commands and save it under a specific name Run the script to generate a single file
52. maximum lag order inside the parentheses before the list name and separated by a comma For example list xlist x1 x2 x3 list laglist lags 2 xlist or scalar order 4 list laglist lags order xlist These command will populate 1aglist with the specified number of lags of the variables in xlist As with the ordinary lags command you can omit the order in which case this is determined automatically based on the frequency of the data One further special feature is available when generating lags namely you can give the name of a single variable in place of a named list on the right hand side as in series 1x log x list laglist lags 4 1x Note that the ordinary syntax for e g logs is just Chapter 11 Named lists and strings 69 logs x1 x2 x3 If xlist is a named list you can also say logs xlist but this form will not save the logs as a named list for that you need the form list loglist logs xlist Checking for missing values Gretl offers several functions for recognizing and handling missing values see the Gretl Command Reference for details In this context it is worth remarking that the ok function can be used with a list argument For example list xlist x1 x2 x3 series xok ok xlist After these commands the series xok will have value 1 for observations where none of x1 x2 or x3 has a missing value and value 0 for any observations where this condition is not met 11 2 Named strings
53. mode If the progressive option is given for a command loop the effects of the commands ols print and store are modified as follows Chapter 9 Loop constructs 54 ols The results from each individual iteration of the regression are not printed Instead after the loop is completed you get a printout of a the mean value of each estimated coefficient across all the repetitions b the standard deviation of those coefficient estimates c the mean value of the estimated standard error for each coefficient and d the standard deviation of the estimated standard errors This makes sense only if there is some random input at each step print When this command is used to print the value of a variable you do not get a print each time round the loop Instead when the loop is terminated you get a printout of the mean and standard deviation of the variable across the repetitions of the loop This mode is intended for use with variables that have a single value at each iteration for example the error sum of squares from a regression store This command writes out the values of the specified variables from each time round the loop to a specified file Thus it keeps a complete record of the variables across the iterations For example coefficient estimates could be saved in this way so as to permit subsequent examination of their frequency distribution Only one such store can be used in a given loop 9 4 Loop examples Monte Carlo example
54. of specification in the arma or arima command In the example the constant is set to zero 1 to 0 85 and 0 to 0 34 You can get gretl to revert to automatic initialization via the command set initvals auto Estimation via X 12 ARIMA As an alternative to estimating ARMA models using native code gretl offers the option of using the external program X 12 ARIMA This is the seasonal adjustment software produced and main tained by the U S Census Bureau it is used for all official seasonal adjustments at the Bureau Gretl includes a module which interfaces with X 12 ARIMA it translates arma commands using the syntax outlined above into a form recognized by X 12 ARIMA executes the program and retrieves the results for viewing and further analysis within gretl To use this facility you have to install X 12 ARIMA separately Packages for both MS Windows and GNU Linux are available from the gretl website http gretl sourceforge net To invoke X 12 ARIMA as the estimation engine append the flag x 12 arima as in arma p q y x 12 arima As with native estimation the default is to use exact ML but there is the option of using conditional ML with the conditional flag However please note that when X 1 2 ARIMA is used in conditional ML mode the comments above regarding the variant treatments of the mean of the process y do not apply That is when you use X 12 ARIMA the model that is estimated is 20 2 regardless of whether esti
55. of the commands ols print and store in a manner that may be useful with Monte Carlo analyses see Section 9 3 The following sections explain the various forms of the loop control expression and provide some examples of use of loops t If you are carrying out a substantial Monte Carlo analysis with many thousands of repetitions memory capacity and processing time may be an issue To minimize the use of computer resources run your script using the command line program gretlcli with output redirected to a file 51 Chapter 9 Loop constructs 52 9 2 Loop control variants Count loop The simplest form of loop control is a direct specification of the number of times the loop should be repeated We refer to this as a count loop The number of repetitions may be a numerical constant as in loop 1000 or may be read from a variable as in loop replics In the case where the loop count is given by a variable say replics in concept replics is an integer scalar If it is in fact a series its first value is read If the value is not integral it is converted to an integer by truncation Note that replics is evaluated only once when the loop is initially compiled While loop A second sort of control expression takes the form of the keyword whi 1e followed by an inequality the left hand term should be the name of a predefined variable the right hand side may be either a numerical constant or the name of another predefined variable F
56. one gets used to it Some readers may find it helpful to note that the conditional assignment operator works exactly the same way as the IF function in spreadsheets Chapter 13 Cheat sheet 90 13 3 Neat tricks Interaction dummies Problem You want to estimate the model y X B1 Z B2 d B3 di zi B4 t where di is a dummy variable while x and z are vectors of explanatory variables Solution list X x1 x2 x3 list Z z1 z2 list dZ null loop foreach i Z series d i d i list dZ dZ d i end loop ols y X Z d dZ Comment It s amazing what string substitution can do for you isn t it Realized volatility Problem Given data by the minute you want to compute the realized volatility for the hour as RV 5 Sy ee Imagine your sample starts at time 1 1 Solution smp1 full genr time genr minute int time 60 1 genr second time 60 setobs minute second panel genr rv psd y A2 setobs 1 1 smpl second 1 restrict store foo rv Comment Here we trick gretl into thinking that our dataset is a panel dataset where the minutes are the units and the seconds are the time this way we can take advantage of the special function psd panel standard deviation Then we simply drop all observations but one per minute and save the resulting data store foo rv translates as store in the gretl datafile foo gdt the series rv Part II Econometric methods Chapte
57. p3 genr rp2 p2 p3 ols rcost const y rpl rp2 Cobb Douglas cost function with homogeneity restrictions and inefficiency scalar b0 coeff const scalar b1 coeff y scalar b2 coeff rp1 scalar b3 coeff rp2 scalar su 0 1 scalar sv 0 1 mle log InCcnorm e lambda ss In ss 0 5 e ss A2 scalar ss sqrt suA2 svA2 scalar lambda su sv series e rcost bO const bl y b2 rp1 b3 rp2 params b0 b1 b2 b3 su sv end mle Chapter 17 Maximum likelihood estimation 118 then the density of y conditional on F _ is normal MlFr 1 N p ht By means of the properties of conditional distributions the joint density can be factorized as follows T FO Ti Four f Yo t 1 If we treat yo as fixed then the term f yo does not depend on the unknown parameters and there fore the conditional log likelihood can then be written as the sum of the individual contributions as T lu w a B gt t 17 6 t 1 1 YH 1 Y my t tos 9 Ju J z oeo hi The following script shows a simple application of this technique which uses the data file djclose it is one of the example dataset supplied with gretl and contains daily data from the Dow Jones stock index where open djclose series y 100 1diff djclose scalar mu 0 0 scalar omega 1 scalar alpha 0 4 scalar beta 0 0 mle 11 0 5 log h eA2 h series e y mu series h var y series h omega alpha
58. please see the gretl data site this is a free download although it is not included in the main gretl package Example 15 2 opens pwt56_60_89 gdt a subset of the PWT containing data on 120 countries 1960 89 for 20 variables with no missing observations the full data set which is also supplied in the pwt package for gretl has many missing observations Total growth of real GDP 1960 89 is calculated for each country and regressed against the 1960 level of real GDP to see if there is evidence for convergence i e faster growth on the part of countries starting from a low base 3Also see Clint Cummins benchmarks page http www stanford edu clint bench Chapter 15 Panel data Example 15 2 Use of the Penn World Table open pwt56_60_89 gdt for 1989 the last obs lag 29 gives 1960 the first obs genr gdp60 RGDPL 29 find total growth of real GDP over 30 years genr gdpgro RGDPL gdp60 gdp60 restrict the sample to a 1989 cross section smpl restrict YEAR 1989 convergence did countries with a lower base grow faster ols gdpgro const gdp60 result No Try an inverse relationship genr gdp60inv 1 gdp60 ols gdpgro const gdp60inv no again Try treating Africa as special genr afdum CCODE 1 genr afslope afdum gdp60 ols gdpgro const afdum gdp60 afslope 106 Chapter 16 Nonlinear least squares 16 1 Introduction and examples Gretl supports nonlinear least squares NL
59. probably not have need to read or write such files other than via gretl itself but if you want to manipulate them using other software tools you should examine the DTD and also take a look at a few of the supplied practice data files data4 1 gdt gives a simple example data4 10 gdt is an example where observation labels are included A 2 Traditional ESL format For backward compatibility gretl can also handle data files in the traditional format inherited from Ramanathan s ESL program In this format which was the default in gretl prior to version 0 98 a data set is represented by two files One contains the actual data and the other information on how the data should be read To be more specific 1 Actual data A rectangular matrix of white space separated numbers Each column represents a variable each row an observation on each of the variables spreadsheet style Data columns can be separated by spaces or tabs The filename should have the suffix gdt By default the data file is ASCII plain text Optionally it can be gzip compressed to save disk space You can insert comments into a data file if a line begins with the hash mark the entire line is ignored This is consistent with gnuplot and octave data files 2 Header The data file must be accompanied by a header file which has the same basename as the data file plus the suffix hdr This file contains in order e Optional comments on the data set off by the opening s
60. run packages are available in rpm format suitable for Red Hat Linux and related systems on the gretl webpage http gret1l sourceforge net However we re hopeful that some users with coding skills may consider gretl sufficiently interest ing to be worth improving and extending The documentation of the libgretl API is by no means complete but you can find some details by following the link Libgretl API docs on the gretl home page People interested in the gretl development are welcome to subscribe to the gretl devel mailing list If you prefer to compile your own or are using a unix system for which pre built packages are not available instructions on building gretl can be found in Appendix B MS Windows The MS Windows version comes as a self extracting executable Installation is just a matter of downloading gretl_instal1 exe and running this program You will be prompted for a location to install the package the default is c 1userdatalgret1 ln this manual we use Linux as shorthand to refer to the GNU Linux operating system What is said herein about Linux mostly applies to other unix type systems too though some local modifications may be needed Chapter 1 Introduction 3 Updating If your computer is connected to the Internet then on start up gretl can query its home website at Wake Forest University to see if any program updates are available if so a window will open up informing you of that fact If you w
61. samples will enable us to estimate more of the true wts elements that is for t and s more widely separated in time but will not contribute ever increasing information regarding the maximally separated wrs pairs since the maximal separation itself grows with the sample size To ensure consistency we have to confine our attention to processes exhibiting temporally limited dependence or in other words cut off the computation of the ts values at some maximum value of p t s where p is treated as an increasing function of the sample size T although it cannot increase in proportion to T The simplest variant of this idea is to truncate the computation at some finite lag order p where p grows as say T The trouble with this is that the resulting may not be a positive definite matrix In practical terms we may end up with negative estimated variances One solution to this problem is offered by The Newey West estimator Newey and West 1987 which assigns declining weights to the sample autocovariances as the temporal separation increases To understand this point it is helpful to look more closely at the covariance matrix given in 14 5 namely OXY UXO OX This is known as a sandwich estimator The bread which appears on both sides is X X yA This is a k x k matrix and is also the key ingredient in the computation of the classical covariance matrix The filling in the sandwich is a Xx X kxk kxT TXT Txk
62. sc_a genr se_m stderr sc_m noecho print printf constant 8g se 6g t 4f n C se_c c se_c printf arl term 8g se 6g t 4f n a se_a a se_a printf mal term 8g se 6g t 4f n m se_m m se_m Chapter 9 Loop constructs 57 Example 9 4 Panel statistics open hospitals gdt loop i 1991 2000 smpl year i restrict replace summary 1 2 3 4 endloop Example 9 5 illustrates string substitution in an indexed loop Example 9 5 String substitution open bea dat loop i 1987 2001 genr V COMP i genr TC GOC i PBT i genr C TC V ols PBT i const TC V endloop The first time round this loop the variable V will be set to equal COMP1987 and the dependent variable for the ols will be PBT1987 The next time round V will be redefined as equal to COMP1988 and the dependent variable in the regression will be PBT1988 And so on Chapter 10 User defined functions 10 1 Defining a function Since version 1 3 3 gretl has contained a mechanism for defining functions in the context of a script This functionality has been through some changes in search of a stable and extensible framework We believe that the version present in gretl 1 6 1 should provide a good basis for future development Functions must be defined before they are called The syntax for defining a function looks like this function function name parameters function body end function function name is the
63. section The fixed and random effects models In gretl version 1 6 0 and higher the fixed and random effects models for panel data can be es timated in their own right In the graphical interface these options are found under the menu item Model Panel Fixed and random effects In the command line interface one uses the panel command with or without the random effects option This section explains the nature of these models and comments on their estimation via gretl The pooled OLS specification may be written as Vit XitB Uit 15 1 where Yit is the observation on the dependent variable for cross sectional unit i in period t Xit is a 1 x k vector of independent variables observed for unit i in period t 6 is a k x 1 vector of parameters and uit is an error or disturbance term specific to unit i in period t The fixed and random effects models have in common that they decompose the unitary pooled error term Uit For the fixed effects model we write Uit Qi Eit yielding Vit Xith Qi Eit 15 2 That is we decompose uj into a unit specific and time invariant component and an observation specific error r The s are then treated as fixed parameters in effect unit specific y intercepts which are to be estimated This can be done by including a dummy variable for each cross sectional unit and suppressing the global constant This is sometimes called the Least Squares Dummy Vari ables LSDV method
64. separated values file or a spreadsheet file Save data Save the currently open native gretl data file Save data as Write out the current data set in native format with the option of using gzip data compression See Chapter 4 Export data Write out the current data set in Comma Separated Values CSV format or the formats of GNU R or GNU Octave See Chapter 4 and also Appendix D Send to Send the current data set as an e mail attachment New data set Allows you to create a blank data set ready for typing in values or for importing series from a database See below for more on databases Clear data set Clear the current data set out of memory Generally you don t have to do this since opening a new data file automatically clears the old one but sometimes it s useful Script files A script is a file containing a sequence of gretl commands This item contains entries that let you open a script you have created previously User file open a sample script or open an editor window in which you can create a new script Session files A session file contains a snapshot of a previous gretl session including the data set used and any models or graphs that you saved Under this item you can open a saved session or save the current session Databases Allows you to browse various large databases either on your own computer or if you are connected to the internet on the gretl database server See
65. specifically to your course in such a way that students can browse and access them easily There are three ways to access such collections of files e For data files select the menu item File Open data Sample file or click on the folder icon on the gretl toolbar e For script files select the menu item File Script files Practice file When a user selects one of the items e The data or script files included in the gretl distribution are automatically shown this includes files relating to Ramanathan s Introductory Econometrics and Greene s Econometric Analysis e The program looks for certain known collections of data files available as optional extras for instance the datafiles from various econometrics textbooks Davidson and MacKinnon Gujarati Stock and Watson Verbeek Wooldridge and the Penn World Table PWT 5 6 See the data page at the gretl website for information on these collections If the additional files are found they are added to the selection windows 4genr also offers the inverse function to misszero namely zeromiss which replaces zeros in a given series with the missing observation code Chapter 4 Data files 27 e The program then searches for valid file collections not necessarily known in advance in these places the system data directory the system script directory the user directory and all first level subdirectories of these For reference typical values for these dir
66. t ratios displayed in parentheses under the parameter estimates The default is to show standard errors if you want t ratios select that item Other windows Several other sorts of output windows also have TeX preview copy and save enabled In the case of windows having a graphical toolbar look for the TeX button Figure 22 2 shows this icon second from the right on the toolbar along with the dialog that appears when you press the button Figure 22 2 TeX icon and dialog Variable ENROLL CATHOL PUPIL One aspect of gretl s TeX support that is likely to be particularly useful for publication purposes is the ability to produce a typeset version of the model table see section 3 4 An example of this is shown in Table 22 2 22 3 Fine tuning typeset output There are three aspects to this adjusting the appearance of the output produced by gretl in BTE preview mode adjusting the formatting of gretl s tabular output for models when using the tabprint command and incorporating gretl s output into your own T X files Previewing in the GUI As regards preview mode you can control the appearance of gretl s output using a file named gretlpre tex which should be placed in your gretl user directory see the Gretl Command Ref erence If such a file is found its contents will be used as the preamble to the TX source The default value of the preamble is as follows documentclass 11pt article usepackage
67. term may differ across the cross sectional units e The covariance of the errors across the units may be non zero in each time period e If the between variation is not removed the errors may exhibit autocorrelation not in the usual time series sense but in the sense that the mean error for unit i may differ from that of unit j This is particularly relevant when estimation is by pooled OLS Gretl currently offers two robust covariance matrix estimators specifically for panel data These are available for models estimated via fixed effects pooled OLS and pooled two stage least squares The default robust estimator is that suggested by Arellano 2003 which is HAC provided the panel is of the large n small T variety that is many units are observed in relatively few periods The Arellano estimator is Sa XX aes Xajti x X X i 1 where X is the matrix of regressors with the group means subtracted in the case of fixed effects u denotes the vector of residuals for unit i and n is the number of cross sectional units Cameron and Trivedi 2005 make a strong case for using this estimator they note that the ordinary White HCCME can produce misleadingly small standard errors in the panel context because it fails to take autocorrelation into account In cases where autocorrelation is not an issue however the estimator proposed by Beck and Katz 1995 and discussed by Greene 2003 chapter 13 may be appropriate This
68. tex where xx is replaced by the first two letters of the current setting of the LANG environment variable For example if you are running the program in Polish using LANG p1_PL then gretl will do the following when writing the preamble for a TeX source file 1 Look for a file named gret1pre_p1 tex in the gretl user directory If this is not found then 2 look for a file named gret1pre tex in the gretl user directory If this is not found then 3 use the default preamble Conversely suppose you usually run gretl in a language other than English and have a suitable gretlpre tex file in place for your native language If on some occasions you want to produce TEX output in English then you could create an additional file gretlpre_en tex this file will be used for the preamble when gretl is run with a language setting of say en_US Command line options After estimating a model via a script or interactively via the gretl console or using the command line program gretlcli you can use the commands tabprint or eqnprint to print the model to file in tabular format or equation format respectively These options are explained in the Gretl Command Reference If you wish alter the appearance of gretl s tabular output for models in the context of the tabprint command you can specify a custom row format using the format flag The format string must be enclosed in double quotes and must be tied to the flag with an equals sign The patt
69. the Octave manual for details As for R the exported data file preserves any time series structure that is apparent to gretl The series are saved as individual structures The data should be brought into R using the source command In addition gretl has a convenience function for moving data quickly into R Under gretl s Tools menu you will find the entry Start GNU R This writes out an R version of the current gretl data set in the user s gretl directory and sources it into a new R session The particular way R is invoked depends on the internal gretl variable Rcommand whose value may be set under the Tools Preferences menu The default command is RGui exe under MS Windows Under X it is xterm e R Please note that at most three space separated elements in this command string will be processed any extra elements are ignored 173 Appendix E Listing of URLs Below is a listing of the full URLs of websites mentioned in the text Estima RATS http www estima com FFTW3 http ww fftw org Gnome desktop homepage http www gnome org GNU Multiple Precision GMP library http swox com gmp GNU Octave homepage http www octave org GNU R homepage http ww r project org GNU R manual http cran r project org doc manuals R intro pdf Gnuplot homepage http www gnuplot info Gnuplot manual http ricardo ecn wfu edu gnuplot html Gretl data page http gretl sourceforge net gretl_data html G
70. the file np gdt supplied with gretl among the sample datasets Suppose you want to compute the rate of change for the variable iprod via your new function and store the result in a series named foo Go to File Function files On local machine You will be shown a list of the installed packages including the one you have just created If you select it and click on Execute or double click on the name of the function package a window similar to the one shown in figure 10 3 will appear Click Ok and the series foo will be generated see figure 10 4 You may have to go to Data Refresh data in order to have your new variable show up in the main window variable list or just press the r key Alternatively the same could have been accomplished by the script include pc gfn open np foo pc iprod Chapter 10 User defined functions gretl function call Call to function pc Arguments name type selection y series liprod Assign return value optional type selection or new variable series foo ly Figure 10 3 Using your package gretl gnuplot graph 1880 1900 1920 click on graph for pop up menu Figure 10 4 Percent change in industrial production Chapter 11 Named lists and strings 11 1 Named lists Many gretl commands take one or more lists of variables as arguments To make this easier to handle in the context of command scripts
71. the version at Wake Forest Assuming you have managed to open a database you can import selected series into gretl s workspace by using the Series Import menu item in the database window or via the popup menu that ap pears if you click the right mouse button or by dragging the series into the program s main window Creating a gretl data file independently It is possible to create a data file in one or other of gretl s own formats using a text editor or software tools such as awk sed or perl This may be a good choice if you have large amounts of data already in machine readable form You will of course need to study the gretl data formats XML format or traditional format as described in Appendix A 4 5 Structuring a dataset Once your data are read by gretl it may be necessary to supply some information on the nature of the data We distinguish between three kinds of datasets 1 Cross section 2 Time series 3 Panel data The primary tool for doing this is the Data Dataset structure menu entry in the graphical inter face or the setobs command for scripts and the command line interface Cross sectional data By a cross section we mean observations on a set of units which may be firms countries in dividuals or whatever at a common point in time This is the default interpretation for a data file if gretl does not have sufficient information to interpret data as time series or panel data they
72. tmp loop for i 1 n quiet series x tmp i ret y 1 x 0 end loop return series ret end function open Keane gdt status status 1 dep var must be O based smpl year 87 ok status restrict matrix X educ exper expersq black const scalar k cols X matrix theta zeros 2 k 1 mle loglik mlogitlogprobs status X theta params theta end mle verbose hessian 152 Chapter 21 Discrete and censored dependent variables 153 In this case regressing y on the xis does not yield consistent estimates of the parameters because the conditional mean E y x is not equal to wii xi Bj It can be shown that restricting the sample to non zero observations would not yield consistent estimates either The solution is to estimate the parameters via maximum likelihood The syntax is simply tobit depvar indvars As usual progress of the maximization algorithm can be tracked via the verbose switch while uhat returns the generalized residuals An important difference between the Tobit estimator and OLS is that the consequences of non normality of the disturbance term are much more severe non normality implies inconsistency for the Tobit estimator For this reason the output for the tobit model includes the Chesher Trish 1987 test for normality by default Generalized Tobit model In the so called Tobit II model there are two latent variables k j l Mrs ZijYj Ni 21 10
73. undertake the compilation proper this is done by running the make command which takes care of compiling all the necessary source files in the correct order All you need to do is type make This step will likely take several minutes to complete a lot of output will be produced on screen Once this is done you can install your freshly baked copy of gretl on your system via make install Appendix B Building gretl 171 On most systems the make install command requires you to have administrative privileges Hence either you log in as root before launching make install or you may want to use the sudo utility sudo make install Appendix C Numerical accuracy Gretl uses double precision arithmetic throughout except for the multiple precision plugin in voked by the menu item Model Other linear models High precision OLS which represents floating point values using a number of bits given by the environment variable GRETL_MP_BITS default value 256 The normal equations of Least Squares are by default solved via Cholesky decompo sition which is accurate enough for most purposes with the option of using QR decomposition instead The program has been tested rather thoroughly on the statistical reference datasets pro vided by NIST the U S National Institute of Standards and Technology and a full account of the results may be found on the gretl website follow the link Numerical accuracy Giovanni Baiocchi and Walte
74. unique identifier for the function Names must start with a letter They have a maximum length of 31 characters if you type a longer name it will be truncated Function names cannot contain spaces You will get an error if you try to define a function having the same name as an existing gretl command The parameters for a function are given in the form of a comma separated list Parameters can be of any of the types shown below Type Description bool scalar variable acting as a Boolean switch int scalar variable acting as an integer scalar scalar variable series data series list named list of series matrix named matrix or vector Each element in the listing of parameters must include two terms a type specifier and the name by which the parameter shall be known within the function An example follows function myfunc series y list xvars bool verbose Each of the type specifiers with the exception of list may be modified by prepending an asterisk to the associated parameter name as in function myfunc series y scalar b The meaning of this modification is explained below see section 10 3 it is related to the use of pointer arguments in the C programming language In addition parameters may be modified by the tag const again see 10 3 Besides these required elements the specification of a scalar parameter may include up to three pieces of additional information a minimum value a maximum and a default These addition
75. used with monthly or quarterly data to extract the business cycle component namely the component between 6 and 36 quarters Usual choices for k are 8 or 12 maybe higher for monthly series The default values for the frequency bounds are 8 and 32 and the default value for the approximation order k is 8 You can adjust these values using the set command The keyword for setting the frequency limits is bkbp_limits and the keyword for k is bkbp_k Thus for example if you were using monthly data and wanted to adjust the frequency bounds to 18 and 96 and k to 24 you could do set bkbp_limits 18 96 set bkbp_k 24 These values would then remain in force for calls to the bkfi 1t function until changed by a further use of set 5 4 Panel data specifics Dummy variables In a panel study you may wish to construct dummy variables of one or both of the following sorts a dummies as unique identifiers for the units or groups and b dummies as unique identifiers for the time periods The former may be used to allow the intercept of the regression to differ across the units the latter to allow the intercept to differ across periods Two special functions are available to create such dummies These are found under the Add menu in the GUI or under the genr command in script mode or gretlcli 1 unit dummies script command genr unitdum This command creates a set of dummy variables identifying the cross sectional units The variable
76. v is also assumed then it is possible to show that the density function of e has the form 2 1 Fei qe E oP 5 17 4 where and are respectively the distribution and density function of the standard normal o Joi oy and Chapter 17 Maximum likelihood estimation 116 As a consequence the log likelihood for one observation takes the form apart form an irrelevant constant Agi Li log 9 lt logo E Therefore a Cobb Douglas cost function with stochastic frontier is the model described by the following equations logC logCf ej m n logC c gt Bjlogyij gt aylog pij j l j l Ei Uit Vi ui N 0 0 vi N 0 0 In most cases one wants to ensure that the homogeneity of the cost function with respect to the prices holds by construction Since this requirement is equivalent to 4 aj 1 the above equation for C i can be rewritten as m n log Ci log Pin c gt Bjlogyij gt aj log pi log Pin i 17 5 j 1 j 2 The above equation could be estimated by OLS but it would suffer from two drawbacks first the OLS estimator for the intercept c is inconsistent because the disturbance term has a non zero expected value second the OLS estimators for the other parameters are consistent but inefficient in view of the non normality of i Both issues can be addressed by estimating 17 5 by maximum likelihood Nevertheless OLS estimation is a q
77. with q degrees of freedom and if the p value is small enough reject the null hypothesis of homoskedasticity in favor of the alternative of ARCH q This test is implemented in gretl via the arch command This command may be issued following the estimation of a time series model by OLS or by selection from the Tests menu in the model window again following OLS estimation The result of the test is reported and if the TR from the auxiliary regression has a p value less than 0 10 ARCH estimates are also reported These estimates take the form of Generalized Least Squares GLS specifically weighted least squares using weights that are inversely proportional to the predicted variances of the disturbances 6 derived from the auxiliary regression In addition the ARCH test is available after estimating a vector autoregression VAR In this case however there is no provision to re estimate the model via GLS GARCH The simple ARCH q process is useful for introducing the general concept of conditional het eroskedasticity in time series but it has been found to be insufficient in empirical work The dynamics of the error variance permitted by ARCH q are not rich enough to represent the patterns found in financial data The generalized ARCH or GARCH model is now more widely used Chapter 20 Time series models 142 The representation of the variance of a process in the GARCH model is somewhat but not exactly analogous to th
78. x gt xT Let 09 indicate the true value of 0 Theo retical considerations either of statistical or economic nature may suggest that a relationship like the following holds Elx g 0 0 lt 0 0pb 18 1 with g a continuous and invertible function That is to say there exists a function of the data and the parameter with the property that it has expectation zero if and only if itis evaluated at the true parameter value For example economic models with rational expectations lead to expressions like 18 1 quite naturally If the sampling model for the x s is such that some version of the Law of Large Numbers holds then r xt gt g o Mns t 1 hence since g is invertible the statistic 6 g x 0o so is a consistent estimator of 0 A different way to obtain the same outcome is to choose as an estimator of 0 the value that minimizes the objective function T 2 F 0 SUD X g 0 1 18 2 t 1 the minimum is trivially reached at 6 g X since the expression in square brackets equals 0 The above reasoning can be generalized as follows suppose O is an n vector and we have m relations like E fi x1 0 0 fori 1 m 18 3 where E is a conditional expectation on a set of p variables z called the instruments In the above simple example m 1 and f x 0 x g 0 and the only instrument used is z 1 Then it must also be true that E filxe 0 zje El f 0
79. 0 0522150 Sum of squared residuals 0 0709594 Standard error of residuals 0 0 0388558 Unadjusted R 0 479466 Adjusted R 0 446241 F 3 47 14 4306 The Equation option is fairly self explanatory the results are written across the page in equa tion format as below ENROLL 0 241105 0 223530 CATHOL 0 00338200 PUPIL 0 152643 WHITE 0 066022 0 04597 0 0027196 0 040706 51 R 0 4462 F 3 47 14 431 6 0 038856 standard errors in parentheses The distinction between the Copy and Save options for both tabular and equation is twofold First Copy puts the TeX source on the clipboard while with Save you are prompted for the name of a file into which the source should be saved Second with Copy the material is copied as a fragment while with Save it is written as a complete file The point is that a well formed TeX source file must have a header that defines the documentc lass article report book or whatever and tags that say begin document and end document This material is included when you do Chapter 22 Gretl and T X 158 Save but not when you do Copy since in the latter case the expectation is that you will paste the data into an existing TX source file that already has the relevant apparatus in place The items under Equation options should be self explanatory when printing the model in equa tion form do you want standard errors or
80. 0 fori 1 m and j 1 p 18 4 equation 18 4 is known as an orthogonality condition or moment condition The GMM estimator is defined as the minimum of the quadratic form F 0 W f wf 18 5 121 Chapter 18 GMM estimation 122 where fis a 1 x m p vector holding the average of the orthogonality conditions and W is some symmetric positive definite matrix known as the weights matrix A necessary condition for the minimum to exist is the order condition n lt m p The statistic 0 Argmin F 0 W 18 6 0 is a consistent estimator of O whatever the choice of W However to achieve maximum asymp totic efficiency W must be proportional to the inverse of the long run covariance matrix of the orthogonality conditions if W is not known a consistent estimator will suffice These considerations lead to the following empirical strategy 1 Choose a positive definite W and compute the one step GMM estimator 01 Customary choices for W are Im p Or Im 8 Z Z 7 2 Use 6 to estimate V Fi 0 and use its inverse as the weights matrix The resulting esti mator 0 is called the two step estimator 3 Re estimate V fi 0 by means of 02 and obtain 63 iterate until convergence Asymp totically these extra steps are unnecessary since the two step estimator is consistent and efficient however the iterated estimator often has better small sample properties and should be independent of the choice of W made at step 1
81. 103 p value 0 0081 Chapter 18 GMM estimation 130 4 With time series data there is no hard rule on the appropriate number of lags to use when computing the long run covariance matrix see section 18 4 Our advice is to go by trial and error since results may be greatly influenced by a poor choice Future versions of gretl will include more options on covariance matrix estimation One of the consequences of this state of things is that replicating various well known published studies may be extremely difficult Any non trivial result is virtually impossible to reproduce unless all details of the estimation procedure are carefully recorded Chapter 19 Model selection criteria 19 1 Introduction In some contexts the econometrician chooses between alternative models based on a formal hy pothesis test For example one might choose a more general model over a more restricted one if the restriction in question can be formulated as a testable null hypothesis and the null is rejected on an appropriate test In other contexts one sometimes seeks a criterion for model selection that somehow measures the balance between goodness of fit or likelihood on the one hand and parsimony on the other The balancing is necessary because the addition of extra variables to a model cannot reduce the degree of fit or likelihood and is very likely to increase it somewhat even if the additional variables are not truly relevant to the data generating process
82. 17 described as of Higher difficulty and Average difficulty respectively The additional failure in numerical approximation mode was on MGH10 Higher difficulty maximum number of iterations reached The table gives information on several aspects of the tests the number of outright failures the average number of iterations taken to produce a solution and two sorts of measure of the accuracy of the estimates for both the parameters and the standard errors of the parameters For each of the 54 runs in each mode if the run produced a solution the parameter estimates obtained by gretl were compared with the NIST certified values We define the minimum correct figures for a given run as the number of significant figures to which the least accurate gretl esti mate agreed with the certified value for that run The table shows both the average and the worst case value of this variable across all the runs that produced a solution The same information is shown for the estimated standard errors The second measure of accuracy shown is the percentage of cases taking into account all parame ters from all successful runs in which the gretl estimate agreed with the certified value to at least the 6 significant figures which are printed by default in the gretl regression output Table 16 1 Nonlinear regression the NIST tests Analytical derivatives Numerical derivatives Failures in 54 tests 4 5 Average iterations 32 127 Me
83. 8 There is at present no mechanism for specifying an order other than 1 for the initial VAR A further refinement is available in this context namely data based bandwidth selection It makes intuitive sense that the HAC bandwidth should not simply be based on the size of the sample but should somehow take into account the time series properties of the data and also the kernel chosen A nonparametric method for doing this was proposed by Newey and West 1994 a good concise account of the method is given in Hall 2005 This option can be invoked in gretl via set hac_lag nw3 This option is the default when prewhitening is selected but you can override it by giving a specific numerical value for hac_lag Even the Newey West data based method does not fully pin down the bandwidth for any particular sample The first step involves calculating a series of residual covariances The length of this series is given as a function of the sample size but only up to a scalar multiple for example it is given as O T for the Bartlett kernel Gretl uses an implied multiple of 1 14 4 Special issues with panel data Since panel data have both a time series and a cross sectional dimension one might expect that in general robust estimation of the covariance matrix would require handling both heteroskedasticity and autocorrelation the HAC approach In addition some special features of panel data require attention e The variance of the error
84. Alternatively one can subtract the group mean from each of variables and estimate a model without a constant In the latter case the dependent variable may be written as Yu Vit Vi The group mean is defined as T 1 Ji T 2 9 lit is possible to break a third component out of uit namely wt a shock that is time specific but common to all the units in a given period In the interest of simplicity we do not pursue that option here 100 Chapter 15 Panel data 101 where T is the number of observations for unit i An exactly analogous formulation applies to the independent variables Given parameter estimates obtained using such de meaned data we can recover estimates of the as using 12 A i T vit XitB These two methods LSDV and using de meaned data are numerically equivalent Gretl takes the approach of de meaning the data If you have a small number of cross sectional units a large num ber of time series observations per unit and a large number of regressors it is more economical in terms of computer memory to use LSDV If need be you can easily implement this manually For example genr unitdum ols y x du_ See Chapter 5 for details on uni tdum The amp estimates are not printed as part of the standard model output in gretl there may be a large number of these and typically they are not of much inherent interest However you can retrieve them after estimation of the fixed effe
85. Appendix B Building gretl 170 configure help first to see what options are available One option you way wish to tweak is prefix By default the installation goes under usr local but you can change this For example configure prefix usr will put everything under the usr tree Another useful option refers to the fact that by default gretl offers support for the gnome desktop If you want to suppress the gnome specific features you can pass the option without gnome to configure In order to have the documentation built we need to pass the relevant option to configure as in configure enable build doc You will see a number of checks being run and if everything goes according to plan you should see a summary similar to that displayed in Example B 1 Example B 1 Output from configure enable build doc Configuration Installation path usr local Use readline library yes Use gnuplot for graphs yes Use PNG for gnuplot graphs yes Use LaTeX for typesetting output yes Gnu Multiple Precision support yes MPFR support no LAPACK support yes FFTW3 support yes Build with GTK version 2 0 Script syntax highlighting yes Use installed gtksourceview yes Build with gnome support no Build gret documentation yes Build message catalogs yes Gnome installation prefix NA X 12 ARIMA support yes TRAMO SEATS support yes Experimental audio support no Now type make to build gretl We are now ready to
86. BYOBS keyword is replaced by BYVAR and followed by the keyword BINARY this indi cates that the corresponding data file is in binary format Such data files can be written from greticli using the store command with the s flag single precision or the o flag double precision 2 If BYOBS is followed by the keyword MARKERS gretl expects a data file in which the first column contains strings 8 characters maximum used to identify the observations This may be handy in the case of cross sectional data where the units of observation are identifiable countries states cities or whatever It can also be useful for irregular time series data such as daily stock price data where some days are not trading days in this case the observations can be marked with a date string such as 10 01 98 Remember the 8 character maximum Note that BINARY and MARKERS are mutually exclusive flags Also note that the markers are not considered to be a variable this column does not have a corresponding entry in the list of variable names in the header file 3 If a file with the same base name as the data file and header files but with the suffix 1b1 is found it is read to fill out the descriptive labels for the data series The format of the label file is simple each line contains the name of one variable as found in the header file followed by one or more spaces followed by the descriptive label Here is an example price New car price index 1982
87. Calling a function 10 3 Function programming details lt 5 04 005 eee we ee eee 10 4 Function packages 11 Named lists and strings 11 1 Named lists 11 2 Named strings 12 Matrix manipulation 12 1 Introduction 12 2 Creating matrices 12 3 Selecting sub matrices 33 33 34 35 36 37 40 40 40 41 42 42 43 43 44 46 46 47 51 51 52 53 54 58 58 59 60 63 67 67 69 Contents 12 4 Matrix operators 125 Matti Sc alar OBRAS lt 3 cc edocs ese ee BOE a SOR ee Be ea 12 6 Matrix functions 12 7 Matrix accessors 12 8 Namespace issues 12 9 Creating a data series from a matrix 12 10Matrices and lists 12 11 Deleting a matrix 12 12Printing a matrix 12 13 Example OLS using matrices 13 Cheat sheet 13 1 Dataset handling 13 2 Creating modifying variables 13 3 Neat tricks II Econometric methods 14 Robust covariance matrix estimation 14 1 Introduction 14 2 Cross sectional data and the HCCME 14 3 Time series data and HAC covariance matrices 00000 eee eee 14 4 Special issues with panel data 15 Panel data 15 1 Estimation of panel models 5 25 2465 554 eed Ss 4644944085 4 ES eR ASS Re 152 Dynamic panel Models o scone eS er Ra ROE ERED EER REE TEER Ee AR 15 3 Illustration the Penn World Table 16 Nonlinear least squares 16 1 Introduction and examples 16 2 Initializing the parameters 16 3 NLS d
88. Gretl User s Guide Gnu Regression Econometrics and Time series Allin Cottrell Department of Economics Wake Forest university Riccardo Jack Lucchetti Dipartimento di Economia Universita Politecnica delle Marche May 2007 Permission is granted to copy distribute and or modify this document under the terms of the GNU Free Documentation License Version 1 1 or any later version published by the Free Software Foundation see http ww gnu org licenses fdl html 1 Contents Introduction 1 1 Features at aglan oo a Ee A EHR Dake RR ade S 1 2 Acknowledgements oo pocan be ee ee ee ae a 13 Installg the programs oa cde kee De e a ee a a a E a Running the program Getting started 2 LES PU FESTESSION eh Kae ew Ae ee EP ee ee 22 Estimation OUND eese kek ore eee eee OEE Cee RR ERE ERE Ree EES 2 4 The main window Menus lt lt lt carers ee ee a 24 Keyboard shortens 0 4 acca ewe Ka ada a a a ee ae a ome UG rel O Modes of working 3 1 Command SCTIPIS socs ce baw eee ee eee EA ewe Ga aw eee Sf Savi Seri OSOS su a ee ES ee wee ROS EE PRE EEE EEE ESE Oe OE ds Whe ret COSO sa seo ee e A Mew ee We a Re E 34 The SESSION Concept a he we ER ae ee A ae Ge eK Data files A1 NAVE TONAL so a hee Ree RA Ra AS ee wa aes 42 Uther iata tile Lora cc ek ia Go we wee ee Rw O a aw GY Meet a a 4 3 Binary databases cc ee 4 4 Creating a data file from scratch 0 0 cee ee eee 45 Structuring a dataset s
89. In addition for square A and integer k gt 0 B AAk produces a matrix B which is A raised to the power k Note that the operator cannot be used in place of A for this purpose because in a matrix context it is reserved for the Kronecker product Chapter 12 Matrix manipulation 76 Expression Effect matrix B A k by kajj matrix B A k_ bij aij k matrix B k A bij k aij matrix B A k bij aij k matrix B A k by ajj k matrix B k A by k ajj matrix B A k bj aij modulo k Table 12 1 Matrix scalar operators 12 6 Matrix functions Creation I mnormal muniform ones seq zeros Shape size cols diag dsort mlag mshape rows sort transp unvech vec vech Element by element abs atan cnorm cos dnorm exp gamma int Tngamma log qnorm sin sqrt tan Matrix algebra cholesky det eigengen eigensym fft ffti infnorm inv Tdet mexp nullspace onenorm qform qrdecomp rank rcond svd tr cmult Statistical cdemean imaxc imaxr iminc iminr mcorr mcov maxc maxr meanc meanr minc minr princomp sumc sumr values Table 12 2 Table of matrix functions by category Table 12 2 lists the matrix functions that gretl provides an alphabetized version of the table is provided at the end of this chapter as Table 12 3 The following functions are available for element by element transformations of matrices log exp sin cos tan atan int abs sqrt dnorm cnorm qnorm gamma and Ingamma These functions have the effects documented in re
90. In the special case when the number of parameters n is equal to the total number of orthogonality conditions m p the GMM estimator 0 is the same for any choice of the weights matrix W so the first step is sufficient in this case the objective function is 0 at the minimum If on the contrary n lt m p the second step or successive iterations is needed to achieve efficiency and the estimator so obtained can be very different in finite samples from the one step estimator Moreover the value of the objective function at the minimum suitably scaled by the number of observations yields Hansen s J statistic this statistic can be interpreted as a test statistic that has a x distribution with m p n degrees of freedom under the null hypothesis of correct specification See Davidson and MacKinnon 1993 section 17 6 for details In the following sections we will show how these ideas are implemented in gretl through some examples 18 2 OLS as GMM It is instructive to start with a somewhat contrived example consider the linear model y x B ur Although most of us are used to read it as the sum of a hazily defined systematic part plus an equally hazy disturbance a more rigorous interpretation of this familiar expression comes from the hypothesis that the conditional mean E y x is linear and the definition of ut as y E y x From the definition of ut it follows that E uf x O The following orthogona
91. S using a variant of the Levenberg Marquandt algorithm The user must supply a specification of the regression function prior to giving this specification the parameters to be estimated must be declared and given initial values Optionally the user may supply analytical derivatives of the regression function with respect to each of the parameters The tolerance criterion for terminating the iterative estimation procedure can be adjusted using the set command The syntax for specifying the function to be estimated is the same as for the genr command Here are two examples with accompanying derivatives Example 16 1 Consumption function from Greene nls C alpha beta YAgamma deriv alpha 1 deriv beta YAgamma deriv gamma beta YAgamma log Y end nis Example 16 2 Nonlinear function from Russell Davidson nls y alpha beta x1 1 beta x2 deriv alpha 1 deriv beta x1 x2 beta beta end nis Note the command words n1s which introduces the regression function deriv which introduces the specification of a derivative and end nls which terminates the specification and calls for estimation If the vcv flag is appended to the last line the covariance matrix of the parameter estimates is printed 16 2 Initializing the parameters The parameters of the regression function must be given initial values prior to the nls command This can be done using the genr command or in the GUI program via the me
92. Spectral kernel since this need not be an integer For example set qs_bandwidth 3 5 Prewhitening and data based bandwidth selection An alternative approach is to deal with residual autocorrelation by attacking the problem from two sides The intuition behind the technique known as VAR prewhitening Andrews and Monahan 1992 can be illustrated by a simple example Let x be a sequence of first order autocorrelated random variables Xt PXt 1 t Ut The long run variance of x can be shown to be Vir ut a p In most cases ur is likely to be less autocorrelated than x so a smaller bandwidth should suffice Estimation of Vr xt can therefore proceed in three steps 1 estimate p 2 obtain a HAC estimate of t xt PX1 1 and 3 divide the result by 1 p Vir Xt The application of the above concept to our problem implies estimating a finite order Vector Au toregression VAR on the vector variables amp X i In general the VAR can be of any order but in most cases 1 is sufficient the aim is not to build a watertight model for amp but just to mop up a substantial part of the autocorrelation Hence the following VAR is estimated Er A r 1 Et Then an estimate of the matrix X QX can be recovered via TA sat ANA where is any HAC estimator applied to the VAR residuals You can ask for prewhitening in gretl using set hac_prewhiten on Chapter 14 Robust covariance matrix estimation 9
93. Z X NWINC WA KL6 K618 heckitClww X LFP Z 154 Part III Technical details 155 Chapter 22 Gretl and TpX 22 1 Introduction TeX initially developed by Donald Knuth of Stanford University and since enhanced by hundreds of contributors around the world is the gold standard of scientific typesetting Gretl provides various hooks that enable you to preview and print econometric results using the TEX engine and to save output in a form suitable for further processing with Tex This chapter explains the finer points of gretl s TpX related functionality The next section describes the relevant menu items section 22 3 discusses ways of fine tuning TeX output section 22 4 ex plains how to handle the encoding of characters not found in English and section 22 5 gives some pointers on installing and learning TeX if you do not already have it on your computer Just to be clear TEX is not included with the gretl distribution it is a separate package including several programs and a large number of supporting files Before proceeding however it may be useful to set out briefly the stages of production of a final document using TEX For the most part you don t have to worry about these details since in regard to previewing at any rate gretl handles them for you But having some grasp of what is going on behind the scences will enable you to understand your options better The first step is the creation of a plain text
94. a Rosenbrock amp theta print theta Computing the Jacobian To construct a covariance matrix for estimates produced via BFGS_max you may wish to calculate a numerical approximation to the relevant Jacobian The fdjac function again takes two arguments an nx 1 matrix holding initial parameter values and a string specifying a call to a function that calculates and returns an m x 1 matrix n lt m given the current parameter values and any other relevant data On successful completion it returns an m x n matrix holding the Jacobian For example matrix Jac fdjac theta SumOC amp theta amp X where we assume that SumOC is a user defined function with the following structure function SumOC matrix theta matrix X matrix V do some computation return matrix V end function 5 10 The discrete Fourier transform The discrete Fourier transform can be best thought of as a linear invertible transform of a complex vector Hence if x is an n dimensional vector whose k th element is xx ax iby then the output of the discrete Fourier transform is a vector f F x whose k th element is Se is where wjk ani Since the transformation is invertible the vector x can be recovered from f via the so called inverse transform 7 T GRES The Fourier transform is used in many diverse situations on account of this key property the convolution of two vectors can be performed efficiently by multiplying the elements of their
95. abase supplied with gretl to replicate Johansen s example found in his 1995 book open denmark coint2 2 LRM LRY IBO IDE rc seasonal In this case the vector 7 in equation 20 14 comprises the four variables LRM LRY IBO IDE The number of lags equals p in 20 14 plus one Part of the output is reported below Johansen test Number of equations 4 Lag order 2 Estimation period 1974 3 1987 3 CT 53 Case 2 Restricted constant Rank Eigenvalue Trace test p value Lmax test p value 0 0 43317 49 144 0 1284 30 087 0 0286 1 0 17758 19 057 0 7833 10 362 0 8017 2 0 11279 8 6950 0 7645 6 3427 0 7483 3 0 043411 2 3522 0 7088 2 3522 0 7076 Since both the trace and A max accept the null hypothesis that the smallest eigenvalue is in fact 0 we may conclude that the series are in fact non stationary However some linear combination may be I 0 as indicated by the rejection of the A max of the hypothesis that the rank of IT is 0 the trace test gives less clear cut evidence for this The Johansen cointegration test The Johansen test for cointegration is used to establish the rank of f in other words how many cointegration vectors the system has This test has to take into account what hypotheses one is Chapter 20 Time series models 145 willing to make on the deterministic terms which leads to the famous five cases A full and general illustration of the five cases requires a fair amount of matrix
96. above the main sorts of statements that are permitted within an mle block namely e auxiliary commands to generate helper variables e deriv statements to specify the gradient with respect to each of the parameters and e a params statement to identify the parameters in case analytical derivatives are not given For the purpose of debugging ML estimators one additional sort of statement is allowed you can print the value of a relevant variable at each step of the iteration This facility is more restricted then the regular print command The command word print should be followed by the name of just one variable a scalar series or matrix In the last example above a key variable named tmp was generated forming the basis for the analytical derivatives To track the progress of this variable one could add a print statement within the ML block as in series tmp dnorm ndx y P 1 y 1 P print tmp Chapter 18 GMM estimation 18 1 Introduction and terminology The Generalized Method of Moments GMM is a very powerful and general estimation method which encompasses practically all the parametric estimation techniques used in econometrics It was introduced in Hansen 1982 and Hansen and Singleton 1982 an excellent and thorough treatment is given in Davidson and MacKinnon 1993 chapter 17 The basic principle on which GMM is built is rather straightforward Suppose we wish to estimate a scalar parameter 0 based on a sample x1
97. ach variable occupies a column and there can only be one variable per column You cannot use the semi colon as a row separator in this case if you want the series arranged in rows append the transpose symbol The range of data values included in the matrix depends on the current setting of the sample range Please note while gretl s built in statistical functions are capable of handling missing values the matrix arithmetic functions are not When you build a matrix from series that include missing values observations for which at least one series has a missing value are skipped Instead of giving an explicit list of variables you may instead provide the name of a saved list see Chapter 11 as in 72 Chapter 12 Matrix manipulation 73 list xlist x1 x2 x3 matrix A xlist When you provide a named list the data series are by default placed in columns as is natural in an econometric context if you want them in rows append the transpose symbol As a special case of constructing a matrix from a list of variables you can say matrix A dataset This builds a matrix using all the series in the current dataset apart from the constant variable 0 When this dummy list is used it must be the sole element in the matrix definition You can however create a matrix that includes the constant along with all other variables using column wise concatenation see below as in matrix A const dataset You can creat
98. adsheet is quite basic and has no support for functions or formulas Data transformations are done via the Add or Variable menus in the main gretl window Selecting from a database Another alternative is to establish your dataset by selecting variables from a database Gretl comes with a database of US macroeconomic time series and as mentioned above the program will reads RATS 4 databases Begin with gretl s File Databases menu item This has three forks Gretl native RATS 4 and On database server You should be able to find the file fedstl1 bin in the file selector that Chapter 4 Data files 21 opens if you choose the Gretl native option this file which contains a large collection of US macroeconomic time series is supplied with the distribution You won t find anything under RATS 4 unless you have purchased RATS data If you do possess RATS data you should go into gretl s Tools Preferences General dialog select the Databases tab and fill in the correct path to your RATS files If your computer is connected to the internet you should find several databases at Wake Forest University under On database server You can browse these remotely you also have the option of installing them onto your own computer The initial remote databases window has an item showing for each file whether it is already installed locally and if so if the local version is up to date with
99. al 58 Chapter 10 User defined functions 59 values should directly follow the name of the parameter enclosed in square brackets and with the individual elements separated by colons For example suppose we have an integer parameter order for which we wish to specify a minimum of 1 a maximum of 12 and a default of 4 We can write int order 1 12 4 If you wish to omit any of the three specifiers leave the corresponding field empty For example 1 4 would specify a minimum of 1 and a default of 4 while leaving the maximum unlimited For a parameter of type bool you can specify a default of 1 true or 0 false as in bool verbose 0 You may define a function that has no parameters these are called routines in some programming languages In this case use the keyword void in place of the listing of parameters function myfunc2 void When a function is called the parameters are instantiated by arguments given by the caller There are automatic checks in place to ensure that the number of arguments given in a function call matches the number of parameters and that the types of the given arguments match the types specified in the definition of the function An error is flagged if either of these conditions is violated One qualification allowance is made for omitting arguments at the end of the list provided that default values are specified in the function definition To be precise the check is that the number of argumen
100. algebra but an intuitive understanding of the issue can be gained by means of a simple example Consider a series x which behaves as follows Xt M Xt 1 Et where m is a real number and e is a white noise process As is easy to show x is a random walk which fluctuates around a deterministic trend with slope m In the special case m 0 the deterministic trend disappears and x is a pure random walk Consider now another process y defined by v k Xt Ut where again k is a real number and u is a white noise process Since u is stationary by definition xt and y cointegrate that is their difference Zt Vt Xt k ut is a stationary process For k 0 Z is simple zero mean white noise whereas for k 0 the process Zt is white noise with a non zero mean After some simple substitutions the two equations above can be represented jointly as a VAR 1 system ve k m n O 1 Yt 1 a Ut Et Xt m 0 1 Xt 1 Et or in VECM form el lea e od A a ol less e 1 Yt 1 Ho AB x Nt Ho amp Zt 1 Nt where is the cointegration vector and is the loadings or adjustments vector We are now in a position to consider three possible cases 1 m 0 In this case x is trended as we just saw it follows that y also follows a linear trend because on average it keeps at a distance k from x The vector uy is unrestricted This case is the default for gretl s vecm command 2 m 0 and k 4 0 In
101. amma and binomial The syntax of these functions is set out in the Gretl Command Reference here we expand on some subtleties The cumulative density function or CDF for a random variable is the integral of the variable s density from its lower limit typically either or 0 to any specified value x The p value at Chapter 5 Special functions in genr 34 least the one tailed right hand p value as returned by the pvalue function is the complementary probability the integral from x to the upper limit of the distribution typically 0 In principle therefore there is no need for two distinct functions given a CDF value pp you could easily find the corresponding p value as 1 po or vice versa In practice with finite precision computer arithmetic the two functions are not redundant This requires a little explanation In gretl as in most statistical programs floating point numbers are represented as doubles double precision values that typically have a storage size of eight bytes or 64 bits Since there are only so many bits available only so many floating point numbers can be represented doubles do not model the real line Typically doubles can represent numbers over the range roughly 1 7977x 10208 but only to about 15 digits of precision Suppose you re interested in the left tail of the x distribution with 50 degrees of freedom you d like to know the CDF value for x 0 9 Take a look at the following interactiv
102. an be written as y H ut where ur is some zero mean stationary process then not only does the sample average of the y s provide a consistent estimator of u but the long run variance of ur is a well defined finite number Neither of these properties hold under the alternative The test itself is based on the following statistic Dini St where S Sa es and G is an estimate of the long run variance of e y Y Under the null this statistic has a well defined nonstandard asymptotic distribution which is free of nuisance parameters and has been tabulated by simulation Under the alternative the statistic diverges As a consequence it is possible to construct a one sided test based on n where Ho is rejected if n is bigger than the appropriate critical value gretl provides the 90 95 97 5 and 99 quantiles Usage example kpss m y where mis an integer representing the bandwidth or window size used in the formula for estimating the long run variance m e 7 lil E g 2 1 m 1 i m The y terms denote the empirical autocovariances of e from order m through m For this estimator to be consistent m must be large enough to accommodate the short run persistence of et but not too large compared to the sample size T In the GUI interface of gretl this value defaults q 1 4 to the integer part of 4 55 The above concept can be generalized to the case where y is thought to be stationary around a de
103. an of min correct figures 8 120 6 980 parameters Worst of min correct figures 4 3 parameters Mean of min correct figures 8 000 5 673 standard errors Worst of min correct figures 5 2 standard errors Percent correct to at least 6 figures 96 5 91 9 parameters Percent correct to at least 6 figures 97 7 77 3 standard errors Using analytical derivatives the worst case values for both parameters and standard errors were improved to 6 correct figures on the test machine when the tolerance was tightened to 1 0e 14 3The data shown in the table were gathered from a pre release build of gretl version 1 0 9 compiled with gcc 3 3 linked against glibc 2 3 2 and run under Linux on an i686 PC IBM ThinkPad A21m 4For the standard errors I excluded one outlier from the statistics shown in the table namely Lanczos1 This is an odd case using generated data with an almost exact fit the standard errors are 9 or 10 orders of magnitude smaller than the coefficients In this instance gretl could reproduce the certified standard errors to only 3 figures analytical derivatives and 2 figures numerical derivatives Chapter 16 Nonlinear least squares 111 Using numerical derivatives the same tightening of the tolerance raised the worst values to 5 correct figures for the parameters and 3 figures for standard errors at a cost of one additional failure of convergence Note the overall superiority of analytical derivatives on average so
104. and in particular within user defined functions gretl offers the possibility of named lists Creating and modifying named lists A named list is created using the keyword list followed by the name of the list an equals sign and either null to create an empty list or one or more variables to be placed on the list For example list xlist 1234 list reglist income price list empty_list null The name of the list must start with a letter and must be composed entirely of letters numbers or the underscore character The maximum length of the name is 15 characters list names cannot contain spaces When adding variables to a list you can refer to them either by name or by their ID numbers Once a named list has been created it will be remembered for the duration of the gretl session and can be used in the context of any gretl command where a list of variables is expected One simple example is the specification of a list of regressors list xlist x1 x2 x3 x4 ols y O xlist Lists can be modified in two ways To redefine an existing list altogether use the same syntax as for creating a list For example list xlist list xlist 12 45 3 6 After the second assignment xlist contains just variables 4 5 and 6 To append or prepend variables to an existing list we simply make use of the fact that a named list can stand in for a longhand list For example we can do list xlist xlist 5 6 7 list xlist 9 10 xli
105. and q represent the non seasonal AR and MA orders and P and Q the seasonal orders For example arma 1 1 11 y would be used to estimate the following model 1 PL 1 PL yt H 1 OL 1 OL er If y is a quarterly series and therefore s 4 the above equation can be written more explicitly as Yi H P Mt 1 H BOY H P P VYt 5 H Et OEt 1 O r 4 0 O r 5 Such a model is known as a multiplicative seasonal ARMA model Differencing and ARIMA The above discussion presupposes that the time series y has already been subjected to all the transformations deemed necessary for ensuring stationarity see also section 20 2 Differencing is the most common of these transformations and gretl provides a mechanism to include this step into the arma command the syntax arma pdq y would estimate an ARMA p q model on A y It is functionally equivalent to series tmp y loop for i 1 d tmp diff tmp end loop arma p q tmp except with regard to forecasting after estimation see below When the series y is differenced before performing the analysis the model is known as ARIMA T for Integrated for this reason gretl provides the arima command as an alias for arma Seasonal differencing is handled similarly with the syntax arm pdq PDQ y where D is the order for seasonal differencing Thus the command arma 100 111 y would produce the same parameter estim
106. ant to activate this feature check the box marked Tell me about gretl updates under gretl s Tools Preferences General menu The MS Windows version of the program goes a step further it tells you that you can update gretl automatically if you wish To do this follow the instructions in the popup window close gretl then run the program titled gretl updater you should find this along with the main gretl program item under the Programs heading in the Windows Start menu Once the updater has completed its work you may restart gretl Part I Running the program Chapter 2 Getting started 2 1 Let s run a regression This introduction is mostly angled towards the graphical client program please see Chapter 24 below and the Gretl Command Reference for details on the command line program gretlcli You can supply the name of a data file to open as an argument to gretl but for the moment let s not do that just fire up the program You should see a main window which will hold information on the data set but which is at first blank and various menus some of them disabled at first What can you do at this point You can browse the supplied data files or databases open a data file create a new data file read the help items or open a command script For now let s browse the supplied data files Under the File menu choose Open data Sample file A second notebook type window will open presenting the sets of da
107. apter 9 Loop constructs 9 1 Introduction The command loop opens a special mode in which gretl accepts a block of commands to be re peated one or more times This feature may be useful for among other things Monte Carlo simu lations bootstrapping of test statistics and iterative estimation procedures The general form of a loop is loop control expression progressive verbose quiet loop body endloop Five forms of control expression are available as explained in section 9 2 Not all gretl commands are available within loops The commands that are accepted in this context are shown in Table 9 1 Table 9 1 Commands usable in loops add adf append arima break coint coint2 corc corr criteria diff else end endif endloop freq garch genr gmm heem hilu hsk hurst if kpss labels lad lags ldiff logs loop meantest mle mpols multiply nls ols omit outfile pca pergm print printf pvalue pwe rename rhodiff runs set setinfo shell smpl spearman square store string summary testuhat tobit tsls var varlist vartest vecm wls xtab By default the genr command operates quietly in the context of a loop without printing informa tion on the variable generated To force the printing of feedback from genr you may specify the verbose option to loop The quiet option suppresses the usual printout of the number of iterations performed which may be desirable when loops are nested The progressive option to loop modifies the behavior
108. are complex conjugates print 1 print v column 1 contains the first eigenvector real B A v 1 c ll Sd B should equal c print B print c columns 2 3 contain the real and imaginary parts of eigenvector 2 B A v 2 3 cmult ones 3 1 1 2 v 2 3 B should equal c print B print c Chapter 12 Matrix manipulation 83 the singular values of A they are real and non negative and are returned in descending order The first min m n columns of U and V are the left and right singular vectors of A The svd function returns the singular values in a vector of length k The left and or right singu lar vectors may be obtained by supplying non null values for the second and or third arguments respectively For example matrix s svd A amp U amp V matrix s svd A null null matrix s svd A null amp V In the first case both sets of singular vectors are obtained in the second case only the singular values are obtained and in the third the right singular vectors are obtained but U is not computed Please note when the third argument is non null it is actually V that is provided 12 7 Matrix accessors In addition to the matrix functions discussed above various accessor strings allow you to create copies of internal matrices associated with models previously estimated coeff vector of estimated coefficients stderr vector of estimated standard errors uh
109. ariable menu F2 Same as e Included for compatibility with other programs g Has the same effect as selecting Define new variable from the Variable menu which maps onto the genr command h Opens a help window for gretl commands F1 Same as h Included for compatibility with other programs r Refreshes the variable list in the main window has the same effect as selecting Refresh window from the Data menu t Graphs the selected variable a line graph is used for time series datasets whereas a distribution plot is used for cross sectional data 2 5 The gretl toolbar At the bottom left of the main window sits the toolbar The icons have the following functions reading from left to right 1 Launch a calculator program A convenience function in case you want quick access to a calculator when you re working in gretl The default program is calc exe under MS Win dows or xcalc under the X window system You can change the program under the Tools Preferences General menu Programs tab 2 Start a new script Opens an editor window in which you can type a series of commands to be sent to the program as a batch 3 Open the gretl console A shortcut to the Gretl console menu item Section 2 3 above 4 Open the gretl session icon window 5 Open the gretl website in your web browser This will work only if you are connected to the Internet and have a properly configured
110. ast one of the following conditions is satisfied e Both the actual and predicted relative reductions in the error sum of squares are at most e e The relative error between two consecutive iterates is at most e This default value of e is the machine precision to the power 3 4 but it can be adjusted using the set command with the parameter nls_toler For example set nls_toler 0001 will relax the value of e to 0 0001 16 6 Details on the code The underlying engine for NLS estimation is based on the minpack suite of functions available from netlib org Specifically the following minpack functions are called Imder Levenberg Marquandt algorithm with analytical derivatives chkder Check the supplied analytical derivatives Imdif Levenberg Marquandt algorithm with numerical derivatives fdjac2 Compute final approximate Jacobian when using numerical derivatives dpmpar Determine the machine precision On successful completion of the Levenberg Marquandt iteration a Gauss Newton regression is used to calculate the covariance matrix for the parameter estimates If the robust flag is given a robust variant is computed The documentation for the set command explains the specific options available in this regard Since NLS results are asymptotic there is room for debate over whether or not a correction for degrees of freedom should be applied when calculating the standard error of the regression and the standard errors of the paramete
111. at vector of residuals yhat vector of fitted values vcv covariance matrix see below rho autoregressive coefficients for error process jalpha matrix a loadings from Johansen s procedure jbeta matrix f cointegration vectors from Johansen s procedure jvbeta covariance matrix for the unrestricted elements of f from Johansen s procedure If these accessors are given without any prefix they retrieve results from the last model estimated if any Alternatively they may be prefixed with the name of a saved model plus a period in which case they retrieve results from the specified model Here are some examples matrix u uhat matrix b m1 coeff matrix v2 m1 vcv 1 2 1 2 The first command grabs the residuals from the last model the second grabs the coefficient vector from model m1 and the third which uses the mechanism of sub matrix selection described above grabs a portion of the covariance matrix from model m1 If the model in question is actually a system a VAR or VECM or system of simultaneous equa tions uhat retrieves the matrix of residuals one column per equation and vcv gets the cross equation covariance matrix in the special case of a VAR or a VECM coeff returns the companion matrix At present the other accessors are not available for equation systems After a vector error correction model is estimated via Johansen s procedure the matrices jalpha and jbeta are also available Thes
112. ates as genr dsy sdiff y arma 1 O 1 1 dsy where we use the sdiff function to create a seasonal difference e g for quarterly data Yt Y1 4 Chapter 20 Time series models 136 Estimation The default estimation method for ARMA models is exact maximum likelihood estimation under the assumption that the error term is normally distributed using the Kalman filter in conjunc tion with the BFGS maximization algorithm The gradient of the log likelihood with respect to the parameter estimates is approximated numerically This method produces results that are directly comparable with many other software packages The constant and any exogenous variables are treated as in equation 20 2 The covariance matrix for the parameters is computed using a nu merical approximation to the Hessian at convergence The alternative method invoked with the conditional switch is conditional maximum like lihood CML also known as conditional sum of squares see Hamilton 1994 p 132 This method was exemplified in the script 9 3 and only a brief description will be given here Given a sample of size T the CML method minimizes the sum of squared one step ahead prediction errors generated by the model for the observations to T The starting point to depends on the orders of the AR polynomials in the model The numerical maximization method used is BHHH and the covariance matrix is computed using a Gauss Newton regression The
113. ay or delete the object This popup menu also gives you the option of editing graphs The model table In econometric research it is common to estimate several models with a common dependent vari able the models differing in respect of which independent variables are included or perhaps in respect of the estimator used In this situation it is convenient to present the regression results in the form of a table where each column contains the results coefficient estimates and standard errors for a given model and each row contains the estimates for a given variable across the models In the Icon view window gretl provides a means of constructing such a table and copying it in plain text BTX or Rich Text Format Here is how to do it 2 1 Estimate a model which you wish to include in the table and in the model display window under the File menu select Save to session as icon or Save as icon and close 2 Repeat step 1 for the other models to be included in the table up to a total of six models 3 When you are done estimating the models open the icon view of your gretl session by se lecting Icon view under the View menu in the main gretl window or by clicking the session icon view icon on the gretl toolbar 4 In the Icon view there is an icon labeled Model table Decide which model you wish to appear in the left most column of the model table and add it to the table either by dragging its icon
114. base year If you want to save data in traditional format use the t flag with the store command either in the command line program or in the console window of the GUI program A 3 Binary database details A gretl database consists of two parts an ASCII index file with filename suffix idx containing information on the series and a binary file suffix bin containing the actual data Two examples of the format for an entry in the idx file are shown below GOM910 Composite index of 11 leading indicators 1987 100 M 1948 01 1995 11 n 575 currbal Balance of Payments Balance on Current Account SA Q 1960 1 1999 4 n 160 The first field is the series name The second is a description of the series maximum 128 charac ters On the second line the first field is a frequency code M for monthly Q for quarterly A for annual B for business daily daily with five days per week and D for daily seven days per week No other frequencies are accepted at present Then comes the starting date N B with two digits following the point for monthly data one for quarterly data none for annual a space a hyphen Appendix A Data file details 167 another space the ending date the string n and the integer number of observations In the case of daily data the starting and ending dates should be given in the form YYYY MM DD This format must be respected exactly Optionally the first line of the index file may contain a short c
115. be satisfied e The first row must contain valid variable names A valid variable name is of 15 characters maximum starts with a letter and contains nothing but letters numbers and the underscore character _ Longer variable names will be truncated to 15 characters Qualifications to the above First in the case of an ASCII or CSV import if the file contains no row with variable names the program will automatically add names v1 v2 and so on Second by the first row is meant the first relevant row In the case of ASCII and CSV imports blank rows and rows beginning with a hash mark are ignored In the case of Excel and Gnumeric imports you are presented with a dialog box where you can select an offset into the spreadsheet so that gretl will ignore a specified number of rows and or columns e Data values these should constitute a rectangular block with one variable per column and one observation per row The number of variables data columns must match the number of variable names given See also section 4 6 Numeric data are expected but in the case of importing from ASCII CSV the program offers limited handling of character string data if Chapter 4 Data files 20 a given column contains character data only consecutive numeric codes are substituted for the strings and once the import is complete a table is printed showing the correspondence between the strings and the codes Dates or observation labels Optionall
116. ble addition test Example 21 2 Variable addition test in a probit model open greene19_1 probit GRADE const GPA PSI series u uhat ols u const GPA PSI TUCE q printf Variable addition test for TUCE n printf Rsq T g p val g n trsq pvalue X 1 trsq Ordered models These models are simple variations of ordinary logit probit models and are usually applied in case the dependent variable is a discrete and ordered measurement not necessarily quantitative For example this sort of model can be applied when the dependent variable is a qualitative assessment like Good Average and Bad Assuming we have p categories the probability that individual i falls in the j th category is given by F Z Uo for j 0 P yi JIXi 4 F zit uy F zit uj 1 for0 lt j lt p 21 7 1 F Z Up 1 for j p The unknown parameters u are called the cutoff points and are estimated together with the Bs For identification purposes Ho is assumed to be O In terms of the unobserved variable y the model can be equivalently cast as P y j xi P Mj 1 lt Y lt Hj In order to apply these models the dependent variable must be marked as discrete and its lowest value must be 0 Example 21 3 reproduces the estimation given in chap 15 of Wooldridge 2002a Note that gretl does not provide a separate command for ordered models the logit and probit commands automatically estimate the ordered versi
117. bles p1 through p5 The illustrative script above is appropriate when the number of variable to be processed is small When then are many variables in the data set it will be more efficient to use a command loop to accomplish the stacking as shown in the following script The setup is presumed to be the same as in the previous section 50 units 5 periods but with 20 variables rather than 2 open panel txt loop for i 1 20 genr k i 1 50 genr x i stack p1 p5 offset k length 50 endloop setobs 50 1 01 stacked cross section store panel gdt x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 Panel data marker strings It can be helpful with panel data to have the observations identified by mnemonic markers A special function in the genr command is available for this purpose In the example above suppose all the states are identified by two letter codes in the left most column of the original datafile When the stacking operation is performed these codes will be stacked along with the data values If the first row is marked AR for Arkansas then the marker AR will end up being shown on each row containing an observation for Arkansas That s all very well but these markers don t tell us anything about the date of the observation To rectify this we could do Chapter 4 Data files 25 genr time genr year 1960 5 time genr markers s d marker year The first line generates a 1 bas
118. ch is likely to be at least 10 to the power 300 and so should not be confused with legitimate data values In a native format data file they should be represented as NA When importing CSV data gretl accepts several common representations of missing values in cluding 999 the string NA in upper or lower case a single dot or simply a blank cell Blank cells should of course be properly delimited e g 120 6 5 38 in which the middle value is presumed missing As for handling of missing values in the course of statistical analysis gretl does the following e In calculating descriptive statistics mean standard deviation etc under the summary com mand missing values are simply skipped and the sample size adjusted appropriately e In running regressions gretl first adjusts the beginning and end of the sample range trun cating the sample if need be Missing values at the beginning of the sample are common in time series work due to the inclusion of lags first differences and so on missing values at the end of the range are not uncommon due to differential updating of series and possibly the inclusion of leads If gretl detects any missing values inside the possibly truncated sample range for a regression the result depends on the character of the dataset and the estimator chosen In many cases the program will automatically skip the missing observations when calculating the regression results In this situation a message
119. clude at least one function definition and run the script e Open the GUI console and type a function definition interactively This method is not partic ularly recommended you are probably better composing a function non interactively For example suppose you decide to package a function that returns the percentage change of a time series Open a script file and type function pc series y series foo diff y y 1 return series foo end function aiii OL CONSDIE ASAS gretl console type help for a list of commands 7 genr x uniform Replaced series x ID 2 7 genr dpcx pcx Generated series dpcx ID 3 print x dpcx byobs Obs x 8024579 6330880 4343051 1391594 3507580 5807393 2800963 8875319 4181660 5418941 0 05245 5216106 9040799 1171185 5927462 4445166 WONAUBWNHE 00000 0000000000 i i i i raa OROOMOOCOONOGOOPrPFOOO Figure 10 1 Output of function check Now run your function You may want to make sure your function works properly by running a few tests For example you may open the console and type Chapter 10 User defined functions 64 genr x uniform genr dpcx pc x print x dpcx byobs You should see something similar to figure 10 1 The function seems to work ok Once your function is debugged you may proceed to the next stage Create a package Start the GUI program and take a look at the File Function files menu This m
120. columns corresponding to the years This would be followed vertically by a block with the same structure for variable x2 A fragment of such a data file is shown below with quinquennial observations 1965 1985 Imagine the table continued for 48 more states followed by another 50 rows for variable x2 x1 1965 1970 1975 1980 1985 AR 100 0 110 5 118 7 131 2 160 4 AZ 100 0 104 3 113 8 120 9 140 6 If a datafile with this sort of structure is read into gretl the program will interpret the columns as distinct variables so the data will not be usable as is But there is a mechanism for correcting the situation namely the stack function within the genr command Consider the first data column in the fragment above the first 50 rows of this column constitute a cross section for the variable x1 in the year 1965 If we could create a new variable by stacking the 3Note that you will have to modify such a datafile slightly before it can be read at all The line containing the variable name in this example x1 will have to be removed and so will the initial row containing the years otherwise they will be taken as numerical data Chapter 4 Data files 24 first 50 entries in the second column underneath the first 50 entries in the first we would be on the way to making a data set by observation in the first of the two forms mentioned above stacked cross sections That is we d have a column comprising a cross section for x1 in 1965
121. containing a concise record of your work e Start by establishing a new script file Type in any commands that may be required to set up transformations of the data see the genr command in the Gretl Command Reference Typically this sort of thing can be accomplished more efficiently via commands assembled with forethought rather than point and click Then save and run the script the GUI data window will be updated accordingly Now you can carry out further exploration of the data via the GUI To revisit the data at a later point open and rerun the preparatory script first 3 2 Saving script objects When you estimate a model using point and click the model results are displayed in a separate window offering menus which let you perform tests draw graphs save data from the model and so on Ordinarily when you estimate a model using a script you just get a non interactive printout of the results You can however arrange for models estimated in a script to be captured so that you can examine them interactively when the script is finished Here is an example of the syntax for achieving this effect Modell lt ols Ct 0 Yt That is you type a name for the model to be saved under then a back pointing assignment arrow then the model command You may use names that have embedded spaces if you like but such names must be wrapped in double quotes Model 1 lt ols Ct O Yt Models saved in this way will appear as icons
122. cts a scalar and the fourth selects rows 1 2 and 6 from matrix A In addition there is a pre defined index specification diag which selects the principal diagonal of a square matrix as in B diag where B is square Chapter 12 Matrix manipulation 74 You can use selections of this sort on either the right hand side of a matrix generating formula or the left Here is an example of use of a selection on the right to extract a 2 x 2 submatrix B froma 3 x 3 matrix A matrix A matrix B A 1 2 2 3 Il aren rR N w S 5 6 7 8 9 And here are examples of selection on the left The second line below writes a 2 x 2 identity matrix into the bottom right corner of the 3 x 3 matrix A The fourth line replaces the diagonal of A with 1s matrix A 1 2 3 4 5 6 7 8 9 matrix A 2 3 2 3 I 2 matrix d 1 1 1 matrix A diag d 12 4 Matrix operators The following binary operators are available for matrices addition subtraction ordinary matrix multiplication pre multiplication by transpose matrix division see below element wise multiplication element wise division A element wise exponentiation column wise concatenation row wise concatenation Kronecker product test for equality Here are explanations of the less obvious cases For matrix addition and subtraction in general the two matrices have to be of the same dimensions but an exception to this rule is granted if
123. cts model if you wish In the graphical interface go to the Save menu in the model window and select per unit constants In command line mode you can do genr newname ahat where newname is the name you want to give the series For the random effects model we write Uit Vi it SO the model becomes Vit Xi B Vi Eit 15 3 In contrast to the fixed effects model the v s are not treated as fixed parameters but as random drawings from a given probability distribution The celebrated Gauss Markov theorem according to which OLS is the best linear unbiased esti mator BLUE depends on the assumption that the error term is independently and identically distributed IID In the panel context the IID assumption means that E uz in relation to equa tion 15 1 equals a constant d for all i and t while the covariance E u su equals zero for all s t and the covariance E u ru r equals zero for all j i If these assumptions are not met and they are unlikely to be met in the context of panel data OLS is not the most efficient estimator Greater efficiency may be gained using generalized least squares GLS taking into account the covariance structure of the error term Consider observations on a given unit i at two different times s and t From the hypotheses above it can be worked out that Var u s Var uit a o while the covariance between uis and Uit is given by E Uisuit o In matr
124. d by 14 4 do not provide valid means of statistical inference In the recent history of econometrics there are broadly two approaches to the problem of non iid errors The traditional approach is to use an FGLS estimator For example if the departure from the iid condition takes the form of time series dependence and if one believes that this could be modeled as a case of first order autocorrelation one might employ an AR 1 estimation method such as Cochrane Orcutt Hildreth Lu or Prais Winsten If the problem is that the error variance is non constant across observations one might estimate the variance as a function of the independent variables and then perform weighted least squares using as weights the reciprocals of the estimated variances While these methods are still in use an alternative approach has found increasing favor that is use OLS but compute standard errors or more generally covariance matrices that are robust with respect to deviations from the iid assumption This is typically combined with an emphasis on using large datasets large enough that the researcher can place some reliance on the asymptotic consistency property of OLS This approach has been enabled by the availability of cheap computing power The computation of robust standard errors and the handling of very large datasets were daunting tasks at one time but now they are unproblematic The other point favoring the newer 92 Chapter 14 Rob
125. d provides a means of retrieving various values calculated by the program in the course of estimating models or testing hypotheses The variables that can be retrieved in this way are listed in the Gretl Command Reference here we say a bit more about the special variables test and pvalue These variables hold respectively the value of the last test statistic calculated using an explicit testing command and the p value for that test statistic If no such test has been performed at the time when these variables are referenced they will produce the missing value code The explicit testing commands that work in this way are as follows add joint test for the significance of vari ables added to a model adf Augmented Dickey Fuller test see below arch test for ARCH chow Chow test for a structural break coeffsum test for the sum of specified coefficients cusum the Harvey Collier t statistic kpss KPSS stationarity test no p value available Imtest see below meantest test for difference of means omit joint test for the significance of variables omitted from a model reset Ramsey s RESET restrict general linear restriction runs runs test for randomness testuhat test for normality of residual and vartest test for difference of vari ances In most cases both a test and a pvalue are stored the exception is the KPSS test for which a p value is not currently available An important point to notice about this
126. dadsa eee ke A Se ee a 4 6 Missing data Values ce 4 7 Maximum size of datasets 0 o o de Data He colleenons ecca 44 aaa a Ra a Special functions in genr Dl is A NO 52 Long Tum VA rCIANCE oi RA E a 5 3 Timie series filters wk a a AA 54 Panel data specifics asosi ca ce Re baiki e Re Re ee 11 11 13 13 14 15 15 18 18 18 18 19 21 25 26 26 Contents 55 Resampling and bootstrapping coseno a a a eee SO Cumulative densities and pales 2 coasa ea 2000 na pa ee RE e S7 Handling missing Valles oi ccas Sak EAS OE RR ERERM ERS EE RE RE RR 58 Retrieving internal Variables 6s ee a eee 58 Numerical maximization sc pik ss Sw wok E A A RG Ee OS RRR a 5 10 The discrete Fourier transform 6 6 ace ea Bae ea eRe Ee ew 6 Sub sampling a dataset G1 TWGCONION msta ce sk eA eo Als ee ale RAR Be ho Ree Be ee ee eo O 2 Seting Te SAMD E oc ck SA Lene ee RAS Re eRe RRA Ee a 6 3 Restricting the sample 24 6 4 de Ge ae ea RG be ee be 6 4 Random sampling 2 0 02 G5 The Sample menu HS os ee a ee wee a Ra eRe Pee a MA 7 Graphs and plots 7 1 Gnuplot graphs 7 2 BOXPIOTS 2 224444 8 Discrete variables 8 1 Declaring variables as discrete 2 es 8 2 Commands for discrete variables 000000 cee ee es 9 Loop constructs 9 1 Introduction 9 2 Loop control variants 9 3 Progressive mode 9 4 Loop examples 10 User defined functions 10 1 Defining a function 10 2
127. der the de meaned representation of fixed effects as applied to the dynamic model Vit XitB PVit 1 Eit where Vit Vit Vi and Eit Uit Uj or Uit Qi using the notation of equation 15 2 The trouble is that 1 1 will be correlated with via the group mean yi The disturbance influences Vit directly which influences yi which by construction affects the value of V for all t The same issue arises in relation to the quasi demeaning used for random effects Estimators which ignore this correlation will be consistent only as T in which case the marginal effect of it on the group mean of y tends to vanish One strategy for handling this problem and producing consistent estimates of and p was pro posed by Anderson and Hsiao 1981 Instead of de meaning the data they suggest taking the first difference of 15 7 an alternative tactic for sweeping out the group effects Avit AXitB PAYit 1 Nit 15 8 and is inefficient Chapter 15 Panel data 105 where nit AUit A Vi Eit Eit Eit 1 We re not in the clear yet given the structure of the error nit the disturbance e 1 is an influence on both nit and Ayit 1 Vit Vit 1 The next step is then to find an instrument for the contaminated AV t 1 Anderson and Hsiao suggest using either 1 2 or AYVi t 2 both of which will be uncorrelated with nit provided that the underlying errors Eit are not themselves serially corr
128. du_1 will have value 1 in each row corresponding to a unit 1 observation 0 otherwise du_2 will have value 1 in each row corresponding to a unit 2 observation 0 otherwise and so on 2 time dummies script command genr timedum This command creates a set of dummy variables identifying the periods The variable dt_1 will have value 1 in each row correspond ing to a period 1 observation 0 otherwise dt_2 will have value 1 in each row corresponding to a period 2 observation 0 otherwise and so on Chapter 5 Special functions in genr 32 If a panel data set has the YEAR of the observation entered as one of the variables you can create a periodic dummy to pick out a particular year e g genr dum CYEAR 1960 You can also create periodic dummy variables using the modulus operator For instance to create a dummy with value 1 for the first observation and every thirtieth observation thereafter 0 otherwise do genr index genr dum Cindex 1 30 0 Lags differences trends If the time periods are evenly spaced you may want to use lagged values of variables in a panel regression but see section 15 2 below you may also wish to construct first differences of variables of interest Once a dataset is identified as a panel gretl will handle the generation of such variables correctly For example the command genr x1_1 x1 1 will create a variable that contains the first lag of x1 where available and the missing value code wh
129. e 1 A2 beta h 1 params mu omega alpha beta end mle 17 5 Analytical derivatives Computation of the score vector is essential for the working of the BFGS method In all the previous examples no explicit formula for the computation of the score was given so the algorithm was fed numerically evaluated gradients Numerical computation of the score for the i th parameter is performed via a finite approximation of the derivative namely 90 01 0n POr 0 01 Ny ca On E01 055 01 h On 00 2h where h is a small number In many situations this is rather efficient and accurate However one might want to avoid the approximation and specify an exact function for the derivatives As an example consider the following script nulldata 1000 Chapter 17 Maximum likelihood estimation 119 genr x1 normal O genr x2 normal genr x3 normal O genr ystar x1 x2 x3 normal genr y ystar gt 0 scalar b0 scalar b1 scalar b2 scalar b3 oooo mle logl y In P 1 yY In 1 P series ndx b0 b1 x1 b2 x2 b3 x3 series P cnorm ndx params b0 b1 b2 b3 end mle verbose Here 1000 data points are artificially generated for an ordinary probit model y is a binary variable which takes the value 1 if y B1x11 B2xX21 B3X3t amp gt O and 0 otherwise Therefore y 1 with probability B 1x1 B2x2 63x32 Tt The probability function for one observation can be wr
130. e ARMA representation of the level of a time series The variance at time t is allowed to depend on both past values of the variance and past values of the realized squared disturbance as shown in the following system of equations Y XipB u 20 10 Ut Or et 20 11 q Pp Of a gt amit gt 607 20 12 i 1 j l As above e is an iid sequence with unit variance X is a matrix of regressors or in the simplest case just a vector of 1s allowing for a non zero mean of y Note that if p 0 GARCH collapses to ARCH q the generalization is embodied in the 6 terms that multiply previous values of the error variance In principle the underlying innovation e could follow any suitable probability distribution and besides the obvious candidate of the normal or Gaussian distribution the t distribution has been used in this context Currently gretl only handles the case where e is assumed to be Gaussian However when the robust option to the garch command is given the estimator gretl uses for the covariance matrix can be considered Quasi Maximum Likelihood even with non normal distur bances See below for more on the options regarding the GARCH covariance matrix Example garch p q y const x where p gt 0 and q gt O denote the respective lag orders as shown in equation 20 12 These values can be supplied in numerical form or as the names of pre defined scalar variables GARCH estimation Estimation of the parameters of
131. e graph page The graph page icon in the session window offers a means of putting together several graphs for printing on a single page This facility will work only if you have the BIEX typesetting system installed and are able to generate and view either PDF or PostScript output In the Icon view window you can drag up to eight graphs onto the graph page icon When you double click on the icon or right click and select Display a page containing the selected graphs in PDF or EPS format will be composed and opened in your viewer From there you should be able to print the page To clear the graph page right click on its icon and select Clear On systems other than MS Windows you may have to adjust the setting for the program used to view postscript Find that under the Programs tab in the Preferences dialog box under the Tools menu in the main window On Windows you may need to adjust your file associations so that the appropriate viewer is called for the Open action on files with the ps extension FIXME discuss PDF here Saving and re opening sessions If you create models or graphs that you think you may wish to re examine later then before quitting gretl select Session files Save session from the File menu and give a name under which to save the session To re open the session later either e Start gretl then re open the session file by going to the File Session files Open sessio
132. e have a number of columns equal to the chosen cointegration rank therefore the product matrix Pi jalpha jbeta returns the reduced rank estimate of A 1 Since B is automatically identified via the Phillips nor malization see section 20 4 its unrestricted elements do have a proper covariance matrix which can be retrieved through the jvbeta accessor Chapter 12 Matrix manipulation 84 12 8 Namespace issues Matrices share a common namespace with data series and scalar variables In other words no two objects of any of these types can have the same name It is an error to attempt to change the type of an existing variable for example 3 ones 2 2 wrong scalar x matrix x It is possible however to delete or rename an existing variable then reuse the name for a variable of a different type scalar x 3 delete x matrix x ones 2 2 OK 12 9 Creating a data series from a matrix Section 12 2 above describes how to create a matrix from a data series or set of series You may sometimes wish to go in the opposite direction that is to copy values from a matrix into a regular data series The syntax for this operation is series sname mspec where sname is the name of the series to create and mspec is the name of the matrix to copy from possibly followed by a matrix selection expression Here are two examples series S X series ul U 1 It is assumed that x and U are pre existing matrices In the s
133. e is the invocation of gnuplot under MS Windows where a full path to the program is given 162 Chapter 24 The command line interface 24 1 Gretl at the console The gretl package includes the command line program gretlcli On Linux it can be run from a terminal window xterm rxvt or similar or at the text console Under MS Windows it can be run in a console window sometimes inaccurately called a DOS box greticli has its own help file which may be accessed by typing help at the prompt It can be run in batch mode sending output directly to a file see also the Gretl Command Reference If gretlcli is linked to the readline library this is automatically the case in the MS Windows version also see Appendix B the command line is recallable and editable and offers command completion You can use the Up and Down arrow keys to cycle through previously typed commands On a given command line you can use the arrow keys to move around in conjunction with Emacs editing keystokes The most common of these are Keystroke Effect Ctrl a goto start of line Ctrl e go to end of line Ctrl d delete character to right where Ctr1 a means press the a key while the Ctr1 key is also depressed Thus if you want to change something at the beginning of a command you don t have to backspace over the whole line erasing as you go Just hop to the start and add or delete characters If you type the first letters of a comma
134. e likely to be associated with higher variability in absolute terms The obvious fix employed in many macroeconometric studies is to use the logs of such series rather than the raw levels Provided the proportional variability of such series remains roughly constant over time the log transformation is effective Other forms of heteroskedasticity may resist the log transformation but may demand a special treatment distinct from the calculation of robust standard errors We have in mind here autore gressive conditional heteroskedasticity for example in the behavior of asset prices where large disturbances to the market may usher in periods of increased volatility Such phenomena call for specific estimation strategies such as GARCH see chapter 20 Chapter 14 Robust covariance matrix estimation 95 Despite the points made above some residual degree of heteroskedasticity may be present in time series data the key point is that in most cases it is likely to be combined with serial correlation autocorrelation hence demanding a special treatment In White s approach the estimated covariance matrix of the u remains conveniently diagonal the variances E ue may differ by t but the covariances E u u are all zero Autocorrelation in time series data means that at least some of the the off diagonal elements of should be non zero This introduces a substantial complication and requires another piece of terminology estimates o
135. e new matrices or replace existing matrices by means of various transformations just as with scalars and data series The relevant mechanisms are discussed in the next several sections t Names of matrices must satisfy the same requirements as names of gretl variables in general the name can be no longer than 15 characters must start with a letter and must be composed of nothing but letters numbers and the underscore character 12 3 Selecting sub matrices You can select sub matrices of a given matrix using the syntax A rows cols where rows can take any of these forms empty selects all rows a single integer selects the single specified row two integers separated by a colon selects a range of rows the name of a matrix selects the specified rows With regard to the second option the integer value can be given numerically as the name of an existing scalar variable or as an expression that evaluates to a scalar With the last option the index matrix given in the rows field must be either p x 1 or 1 x p and should contain integer values in the range 1 to n where n is the number of rows in the matrix from which the selection is to be made The cols specification works in the same way mutatis mutandis Here are some examples matrix B A 1 matrix B A 2 3 3 5 matrix B A 2 2 matrix idx 1 2 6 matrix B A idx The first example selects row 1 from matrix A the second selects a 2 x 3 submatrix the third sele
136. e session genr pl cdf X 50 0 9 Generated scalar pl ID 2 8 94977e 35 genr p2 pvalue X 50 0 9 Generated scalar p2 ID 3 1 genr test 1 p2 Generated scalar test ID 4 0 The cdf function has produced an accurate value but the pvalue function gives an answer of 1 from which it is not possible to retrieve the answer to the CDF question This may seem surprising at first but consider if the value of p1 above is correct then the correct value for p2 is 1 8 94977x 10735 But there s no way that value can be represented as a double that would require over 30 digits of precision Of course this is an extreme example If the x in question is not too far off into one or other tail of the distribution the cdf and pvalue functions will in fact produce complementary answers as shown below genr pl cdf X 50 30 Generated scalar p1 ID 2 0 0111648 genr p2 pvalue X 50 30 Generated scalar p2 CID 3 0 988835 genr test 1 p2 Generated scalar test ID 4 0 0111648 But the moral is that if you want to examine extreme values you should be careful in selecting the function you need in the knowledge that values very close to zero can be represented as doubles while values very close to 1 cannot 5 7 Handling missing values Four special functions are available for the handling of missing values The boolean function missingO takes the name of a variable as its single argument it returns a se
137. e tests one expects to see for models estimated on straight time series or cross sectional data Nonetheless various panel specific tests are printed along with the parameter estimates as a matter of course as follows When you estimate a model using fixed effects you automatically get an F test for the null hy pothesis that the cross sectional units all have a common intercept That is to say that all the as are equal in which case the pooled model 15 1 with a column of 1s included in the X matrix is adequate When you estimate using random effects the Breusch Pagan and Hausman tests are presented automatically The Breusch Pagan test is the counterpart to the F test mentioned above The null hypothesis is that the variance of v i equals zero if this hypothesis is not rejected then again we conclude that the simple pooled model is adequate The Hausman test probes the consistency of the GLS estimates The null hypothesis is that these estimates are consistent that is that the requirement of orthogonality of the v and the X is satisfied The test is based on a measure H of the distance between the fixed effects and random effects estimates constructed such that under the null it follows the x distribution with degrees of freedom equal to the number of time varying regressors in the matrix X If the value of H is large this suggests that the random effects estimator is not consistent and the fixed effects model is
138. econd example the series u1 is formed from the first column of the matrix U For this operation to work the matrix or matrix selection must be a vector with length equal to either the full length of the current dataset n or the length of the current sample range n If n lt n then only n elements are drawn from the matrix if the matrix or selection comprises n elements the n values starting at element t are used where t represents the starting observation of the sample range Any values in the series that are not assigned from the matrix are set to the missing code 12 10 Matrices and lists To facilitate the manipulation of named lists of variables see Chapter 11 it is possible to convert between matrices and lists In section 12 2 above we mentioned the facility for creating a matrix from a list of variables as in matrix M listname That formulation with the name of the list enclosed in braces builds a matrix whose columns hold the variables referenced in the list What we are now describing is a different matter if we say matrix M listname without the braces we get a row vector whose elements are the ID numbers of the variables in the list This special case of matrix generation cannot be embedded in a compound expression The syntax must be as shown above namely simple assignment of a list to a matrix To go in the other direction you can include a matrix on the right hand side of an expression t
139. ectories are shown in Table 4 1 Linux MS Windows system data dir usr share gretl data c userdata gret data system script dir usr share gretl scripts c userdata gretl scripts user dir home me gret1 c userdatalgretlluser Table 4 1 Typical locations for file collections Any valid collections will be added to the selection windows So what constitutes a valid file collec tion This comprises either a set of data files in gretl XML format with the gdt suffix or a set of script files containing gretl commands with inp suffix in each case accompanied by a master file or catalog The gretl distribution contains several example catalog files for instance the file descriptions in the misc sub directory of the gretl data directory and ps_descriptions in the misc sub directory of the scripts directory If you are adding your own collection data catalogs should be named descriptions and script cat alogs should be be named ps_descriptions In each case the catalog should be placed along with the associated data or script files in its own specific sub directory e g usr share gret1 data mydata or c userdata gret1 data mydata The syntax of the plain text description files is straightforward Here for example are the first few lines of gretl s misc data catalog Gretl various illustrative datafiles arma artificial data for ARMA script example ects_nls Nonlinear least squares example hamilton Pric
140. ed A convenient way of formalizing this situation is to consider the variable y as a Bernoulli random variable and analyze its distribution conditional on the explanatory variables x That is 1 P 21 1 zf 0 1 P where Pi P y 1 x is a given function of the explanatory variables xi In most cases the function P is a cumulative distribution function F applied to a linear combi nation of the x s In the probit model the normal cdf is used while the logit model employs the logistic function A Therefore we have probit Pi F z zi 21 2 i 1 logit Pi F zi A zi Tae 21 3 k Zi xijPj 21 4 j 1 where z is commonly known as the index function Note that in this case the coefficients 6 cannot be interpreted as the partial derivatives of E yi x with respect to xij However for a given value of x it is possible to compute the vector of slopes that is OF z Ox slope x z Z Gretl automatically computes the slopes setting each explanatory variable at its sample mean Another equivalent way of thinking about this model is in terms of an unobserved variable y which can be described thus g vi Y Xij j i Zi i Gt j l We observe y 1 whenever y gt 0 and y 0 otherwise If e is assumed to be normal then we have the probit model The logit model arises if we assume that the density function of e is oA Ei ei dei 1 e amp 2 A Ei B
141. ed index representing the period of each observation and the second line uses the time variable to generate a variable representing the year of the observation The third line contains this special feature if and only if the name of the new variable to generate is markers the portion of the command following the equals sign is taken as C style format string which must be wrapped in double quotes followed by a comma separated list of arguments The arguments will be printed according to the given format to create a new set of observation markers Valid arguments are either the names of variables in the dataset or the string marker which denotes the pre existing observation marker The format specifiers which are likely to be useful in this context are s for a string and d for an integer Strings can be truncated for example 3s will use just the first three characters of the string To chop initial characters off an existing observation marker when constructing a new one you can use the syntax marker n where n is a positive integer in the case the first n characters will be skipped After the commands above are processed then the observation markers will look like for example AR 1965 where the two letter state code and the year of the observation are spliced together with a colon 4 6 Missing data values These are represented internally as DBL_MAX the largest floating point number that can be repre sented on the system whi
142. elated The Anderson Hsiao estimator is not provided as a built in function in gretl since gretl s sensible handling of lags and differences for panel data makes it a simple application of regression with instrumental variables see Example 15 1 which is based on a study of country growth rates by Nerlove 1999 3 Example 15 1 The Anderson Hsiao estimator for a dynamic panel model Penn World Table data as used by Nerlove open penngrow gdt Fixed effects for comparison panel Y 0 Y 1 X Random effects for comparison panel Y 0 Y 1 X random effects take differences of all variables diff Y X Anderson Hsiao using Y 2 as instrument tsls d_Y d_Y 1 d_X 0 d_X Y 2 Anderson Hsiao using d_Y 2 as instrument tsls d_Y d_Y 1 d_X 0 d_X d_Y 2 Although the Anderson Hsiao estimator is consistent it is not most efficient it does not make the fullest use of the available instruments for Ay 1 nor does it take into account the differenced structure of the error nit It is improved upon by the methods of Arellano and Bond 1991 and Blundell and Bond 1998 There is provisional support for the Arellano Bond method in current gretl please see the documentation for the arbond command 15 3 Illustration the Penn World Table The Penn World Table homepage at pwt econ upenn edu is a rich macroeconomic panel dataset spanning 152 countries over the years 1950 1992 The data are available in gretl format
143. enu contains four items On local machine On server Edit package New package Select New package the command will return an error message unless at least one user defined function is currently loaded in memory see the previous point in the first dialog you get to select e A public function to package e Zero or more private helper functions Public functions are directly available to users private functions are part of the behind the scenes mechanism in a function package gretl function package editor iio jji IOO Author Version 1 0 Date YYYY MM DD 2007 01 12 Package description Percentage change a E Minimum gretl version 1 a 5 Si O Data requirement No special requirement q p q Help text for pc Usage pc x x must be a time series Returns x t x t 1 x t 1 Edit function code Upload package to server on save save as Y Close Figure 10 2 The package editor window On clicking OK a second dialog should appear see figure 10 2 where you get to enter the package information currently author version date and a short description You also get to enter help text for the public interface You have a further chance to edit the code of the functions to be packaged by selecting them from the drop down selector and clicking on Edit function code Finally you can choose to upload the package on gretl
144. equirements offers GTK version 2 4 0 or higher see gtk org Gretl calls gnuplot for graphing You can find gnuplot at gnuplot info As of this writing the most recent official release is 4 2 of March 2007 The MS Windows version of gretl comes with a Windows version gnuplot 4 2 the gretl website also offers an rpm of gnuplot 3 8j0 for x86 Linux systems Some features of gretl make use of portions of Adrian Feguin s gtkextra library The relevant parts of this package are included in slightly modified form with the gretl source distribution A binary version of the program is available for the Microsoft Windows platform Windows 98 or higher This version was cross compiled under Linux using mingw the GNU C compiler gcc ported for use with win32 and linked against the Microsoft C library msvcrt d11 It uses Tor Lillqvist s port of GTK 2 0 to win32 The free open source Windows installer program is courtesy of Jordan Russell jrsoftware org B 2 Build instructions a step by step example In this section we give instructions detailed enough to allow a user with only a basic knowledge of a Unix system to build gretl These steps were tested on a fresh installation of Debian Etch For other Linux distributions especially Debian based ones like Ubuntu and its derivatives little should change Other Unix like operating systems such as MacOSX and BSD would probably require more substantial adjustments lUp till version 1 5 1
145. ere are a few changes in this respect in version 1 6 1 12 2 Creating matrices Matrices can be created using any of these methods 1 By direct specification of the scalar values that compose the matrix in numerical form by reference to pre existing scalar variables or using computed values 2 By providing a list of data series 3 By providing a named list of series 4 Using a formula of the same general type that is used with the genr command whereby a new matrix is defined in terms of existing matrices and or scalars or via some special functions To specify a matrix directly in terms of scalars the syntax is for example matrix A 1 2 3 4 5 6 The matrix is defined by rows the elements on each row are separated by commas and the rows are separated by semi colons The whole expression must be wrapped in braces Spaces within the braces are not significant The above expression defines a 2 x 3 matrix Each element should be a numerical value the name of a scalar variable or an expression that evaluates to a scalar Directly after the closing brace you can append a single quote to obtain the transpose To specify a matrix in terms of data series the syntax is for example matrix A x1 x2 x3 where the names of the variables are separated by commas Besides names of existing variables you can use expressions that evaluate to a series For example given a series x you could do matrix A x xA2 E
146. ere the lag is not available e g at the start of the time series for each group When you run a regression using such variables the program will automatically skip the missing observations When a panel data set has a fairly substantial time dimension you may wish to include a trend in the analysis The command genr time creates a variable named time which runs from 1 to T for each unit where T is the length of the time series dimension of the panel If you want to create an index that runs consecutively from 1 to m x T where m is the number of units in the panel use genr index Basic statistics by unit The functions pmean Q and psd can be used to generate basic descriptive statistics mean and standard deviation for a given variable on a per group basis Suppose we have a panel data set comprising 8 time series observations on each of N units or groups Then the command genr pmx pmean x creates a series of this form the first 8 values corresponding to unit 1 contain the mean of x for unit 1 the next 8 values contain the mean for unit 2 and so on The psd function works in a similar manner The sample standard deviation for group i is computed as x xi S i T where T denotes the number of valid observations on x for the given unit x denotes the group mean and the summation is across valid observations for the group If T lt 2 however the standard deviation is recorded as 0 One particular use of
147. erion HQC 35 7227 26 81 2 0 189 Predicted 0 1 Actual 0 18 3 1 3 8 Model 2 Probit estimates using the 32 observations 1 32 Dependent variable GRADE VARIABLE COEFFICIENT STDERROR T STAT SLOPE at mean const 7 45232 2 54247 2 931 GPA 1 62581 0 693883 2 343 0 533347 TUCE 0 0517288 0 0838903 0 617 0 0169697 PSI 1 42633 0 595038 2 397 0 467908 Mean of GRADE 0 344 Number of cases correctly predicted 26 81 2 f beta x at mean of independent vars 0 328 McFadden s pseudo R squared 0 377478 Log likelihood 12 8188 Likelihood ratio test Chi square 3 15 5459 p value 0 001405 Akaike information criterion AIC 33 6376 Schwarz Bayesian criterion BIC 39 5006 Chapter 21 Discrete and censored dependent variables 150 Hannan Quinn criterion HQC 35 581 Predicted 0 1 Actual 0 18 3 1 3 8 In this context the uhat accessor function takes a special meaning it returns generalized resid uals as defined in Gourieroux et al 1987 which can be interpreted as unbiased estimators of the latent disturbances These are defined as 21 6 yi Pi for the logit model Uji 5 5 Yi SES 1 yi EN for the probit model Among other uses generalized residuals are often used for diagnostic purposes For example it is very easy to set up an omitted variables test equivalent to the familiar LM test in the context of a linear regression example 21 2 shows how to perform a varia
148. ern for the format string is as follows There are four fields representing the coefficient standard error t ratio and p value respectively These fields should be separated by vertical bars they may contain a printf type specification for the formatting of the numeric value in question or may be left blank to suppress the printing of that column subject to the constraint that you can t leave all the columns blank Here are a few examples format 4F 4F 4F 4F format 4F 4F 3f format 5F 4F 4f format 8g 8g 4f The first of these specifications prints the values in all columns using 4 decimal places The second suppresses the p value and prints the t ratio to 3 places The third omits the t ratio The last one again omits the t and prints both coefficient and standard error to 8 significant figures Once you set a custom format in this way it is remembered and used for the duration of the gretl session To revert to the default formatting you can use the special variant format default Further editing Once you have pasted gretl s TeX output into your own document or saved it to file and opened it in an editor you can of course modify the material in any wish you wish In some cases machine generated TpX is hard to understand but gretl s output is intended to be human readable and editable In addition it does not use any non standard style packages Besides the standard BIFX document
149. es and exchange rate U S and Italy The first line which must start with a hash mark contains a short name here Gretl which will appear as the label for this collection s tab in the data browser window followed by a colon followed by an optional short description of the collection Subsequent lines contain two elements separated by a comma and wrapped in double quotation marks The first is a datafile name leave off the gdt suffix here and the second is a short de scription of the content of that datafile There should be one such line for each datafile in the collection A script catalog file looks very similar except that there are three fields in the file lines a filename without its inp suffix a brief description of the econometric point illustrated in the script and a brief indication of the nature of the data used Again here are the first few lines of the supplied misc script catalog Gretl various sample scripts mow arma ARMA modeling artificial data ects_nls Nonlinear least squares Davidson artificial data leverage Influential observations artificial data n o longley Multicollinearity US employment If you want to make your own data collection available to users these are the steps 1 Assemble the data in whatever format is convenient Chapter 4 Data files 28 2 Convert the data to gretl format and save as gdt files It is probably easiest to con
150. estimator which takes into account contemporaneous correlation across the units and heteroskedasticity by unit is nn Spx XX 230 uxi xa The covariances 6 are estimated via Chapter 14 Robust covariance matrix estimation 99 where T is the length of the time series for each unit Beck and Katz call the associated standard errors Panel Corrected Standard Errors PCSE This estimator can be invoked in gretl via the command set pcse on The Arellano default can be re established via set pcse off Note that regardless of the pcse setting the robust estimator is not used unless the robust flag is given or the Robust box is checked in the GUI program Chapter 15 Panel data 15 1 Estimation of panel models Pooled Ordinary Least Squares The simplest estimator for panel data is pooled OLS In most cases this is unlikely to be adequate but it provides a baseline for comparison with more complex estimators If you estimate a model on panel data using OLS an additional test item becomes available In the GUI model window this is the item panel diagnostics under the Tests menu the script counterpart is the hausman command To take advantage of this test you should specify a model without any dummy variables represent ing cross sectional units The test compares pooled OLS against the principal alternatives the fixed effects and random effects models These alternatives are explained in the following
151. etl specify model OLS Dependent variable Choose gt ct Y Set as default Independent variables Add gt const lt lt Remove I Robust standard errors Ys Gear X cancel 0 Heip Figure 2 3 Model specification dialog To select the dependent variable highlight the variable you want in the list on the left and click the Choose button that points to the Dependent variable slot If you check the Set as default box this variable will be pre selected as dependent when you next open the model dialog box Shortcut double clicking on a variable on the left selects it as dependent and also sets it as the default To select independent variables highlight them on the left and click the Add button or click the right mouse button over the highlighted variable To select several variable in the list box drag the mouse over them to select several non contiguous variables hold down the Ctr1 key and click Chapter 2 Getting started 7 on the variables you want To run a regression with consumption as the dependent variable and income as independent click Ct into the Dependent slot and add Yt to the Independent variables list 2 2 Estimation output Once you ve specified a model a window displaying the regression output will appear The output is reasonably comprehensive and in a standard format Figure 2 4 gretl model 1 File Edit Tests Graphs Model data LaTex Model 1 OLS estimates usin
152. f Y If this is not possible or too difficult the gradient may be evaluated numerically The choice of the starting value 90 is crucial in some contexts and inconsequential in others In general however it is advisable to start the algorithm from sensible values whenever possible If a consistent estimator is available this is usually a safe and efficient choice this ensures that in large samples the starting point will be likely close to O and convergence can be achieved in few iterations The maxmimum number of iterations allowed for the BFGS procedure and the relative tolerance for assessing convergence can be adjusted using the set command the relevant variables are bfgs_maxiter default value 500 and bfgs_toler default value the machine precision to the power 3 4 Covariance matrix and standard errors By default the covariance matrix of the parameter estimates is based on the Outer Product of the Gradient That is Varors 0 G 6 G 6 where G 6 is the T x k matrix of contributions to the gradient Two other options are available If the hessian flag is given the covariance matrix is computed from a numerical approximation to the Hessian at convergence If the robust option is selected the quasi ML sandwich estimator is used A r A R Varom H 0 1G 0 G 0 H 0 where H denotes the numerical approximation to the Hessian Chapter 17 Maximum likelihood estimation 114 17 2 Gamma estimat
153. f the covariance matrix that are asymptotically valid in face of both heteroskedasticity and autocorrelation of the error process are termed HAC heteroskedasticity and autocorrelation consistent The issue of HAC estimation is treated in more technical terms in chapter 18 Here we try to convey some of the intuition at a more basic level We begin with a general comment residual autocorrelation is not so much a property of the data as a symptom of an inadequate model Data may be persistent though time and if we fit a model that does not take this aspect into account properly we end up with a model with autocorrelated disturbances Conversely it is often possible to mitigate or even eliminate the problem of autocorrelation by including relevant lagged variables in a time series model or in other words by specifying the dynamics of the model more fully HAC estimation should not be seen as the first resort in dealing with an autocorrelated error process That said the obvious extension of White s HCCME to the case of autocorrelated errors would seem to be this estimate the off diagonal elements of Q that is the autocovariances E u u using once again the appropriate OLS residuals ts 4 s This is basically right but demands an important amendment We seek a consistent estimator one that converges towards the true Q as the sample size tends towards infinity This can t work if we allow unbounded serial depen dence Bigger
154. ffer so far try the next phase of the walk through Close gretl then re open it Now go to File Function files On local machine If the previous stage above has gone OK you should see the file you packaged and saved with its short description If you click on Info you get a window with all the information gretl has gleaned from the function package If you click on the View code icon in the toolbar of this new window you get a script view window showing the actual function code Now back to the Function packages window if you click on the package s name the functions are loaded into gretl ready to be called by clicking on the Call button After loading the function s from the package open the GUI console Try typing help foo replac ing foo with the name of the public interface from the loaded function package if any help text was provided for the function it should be presented In a similar way you can browse and load the function packages available on the gretl server by selecting File Function files On server Once your package is installed on your local machine you can use the function it contains via the graphical interface as described above or by using the CLI namely in a script or through the console In the latter case you load the function via the include command specifying the package file as the argument complete with the gfn extension To continue with our example load
155. fined scalars The parameter u can be dropped if necessary by appending the option nc no constant to the command If estimation of 20 5 is needed the switch conditional must be appended to the command as in arma p q y conditional Generalizing this principle to the estimation of 20 2 or 20 3 you get that arma p q y const x1 x2 would estimate the following model Vt XB p t 1 Xt 18 Pp Y p x pB Er O1Et 1 OqEt q where in this instance xtp Bo x1 181 Xt 262 Appending the conditional switch as in arma p q y const x1 x2 conditional would estimate the following model Vt XtyY P YVt 1 PpYt p Et OJEr 1 OqEt q Ideally the issue broached above could be made moot by writing a more general specification that nests the alternatives that is P L yt XB zty 0 L er 20 6 we would like to generalize the arma command so that the user could specify for any estimation method whether certain exogenous variables should be treated as x s or zts but we re not yet at that point and neither are most other software packages Seasonal models A more flexible lag structure is desirable when analyzing time series that display strong seasonal patterns Model 20 1 can be expanded to p L L yt O L O L et 20 7 For such cases a fuller form of the syntax is available namely Chapter 20 Time series models 135 armapq PQ y where p
156. first case E holds the eigenvalues of M and V holds the eigenvectors In the second E holds the eigenvalues but the eigenvectors are not computed The function eigengen computes the eigenvalues and optionally the eigenvectors of a general n xn matrix The eigenvalues are returned directly in an n x 2 matrix the first column holding the real components and the second column the imaginary components If the eigenvectors are required that is if the second argument to eigengen is not null they are returned in an n x n matrix The column arrangement of this matrix is somewhat non trivial the eigenvectors are stored in the same order as the eigenvalues but the real eigenvectors occupy one column whereas complex eigenvectors take two the real part comes first the total num ber of columns is still n because the conjugate eigenvector is skipped Example 12 1 provides a hopefully clarifying example The function svd computes all or part of the singular value decomposition of the real m x n matrix A The decomposition is A UZV where gt is an m x n matrix which is zero except for its k min m n diagonal elements U is an m x m orthogonal matrix and V is an n x n orthogonal matrix The diagonal elements of are Chapter 12 Matrix manipulation 82 Example 12 1 Complex eigenvalues and eigenvectors set seed 34756 matrix v A mnormal 3 3 do the eigen analysis 1 eigengen A v eigenvalue 1 is real 2 and 3
157. g the 36 observations 1959 1994 Dependent variable Ct VARIABLE COEFFICIENT STDERROR T STAT 2Prob t gt T 0 const 384 105 151 330 2 538 0 015892 2 Yt 0 932738 0 0106966 87 199 lt 0 00001 Mean of dependent variable 12490 9 Standard deviation of dep var 2940 03 Sum of squared residuals 1 34675e 06 Standard error of residuals 199 023 Unadjusted R squared 0 995548 Adjusted R squared 0 995417 Degrees of freedom 34 Durbin Watson statistic 0 513696 First order autocorrelation coeff 0 768301 MODEL SELECTION STATISTICS SGMASQ 39610 3 AIC 41806 1 FPE 41810 9 43109 7 SCHWARZ 45650 5 SHIBATA 41566 4 41940 3 RICE 42085 9 ly Close Figure 2 4 Model output window The output window contains menus that allow you to inspect or graph the residuals and fitted values and to run various diagnostic tests on the model For most models there is also an option to print the regression output in BIEX format See Chap ter 22 for details To import gretl output into a word processor you may copy and paste from an output window using its Edit menu or Copy button in some contexts to the target program Many not all gretl windows offer the option of copying in RTF Microsoft s Rich Text Format or as Tx If you are pasting into a word processor RTF may be a good option because the tabular formatting of the output is preserved Alternatively you can save the output to a plain text file then
158. greene greenel19_1 gdt periodicity 1 maxobs 32 observations range 1 32 Listing 5 variables 1 GPA 0 const freq TUCE 2 TUCE Frequency distribution for TUCE obs 1 32 number of bins 7 mean 21 9375 sd 3 90151 interval lt 13 13 417 16 16 250 19 19 083 21 21 917 24 24 750 27 gt 27 417 250 083 917 750 583 583 midpt 12 14 29 000 833 17 20 23 26 000 667 500 333 167 frequency NN ODODE F 3 PSI 4 GRADE rel cum 12 3 12 12 6 25 75 25 75 43 75 12 71 88 88 93 75 25 100 Test for null hypothesis of normal distribution Chi square 2 1 872 with p value 0 39211 discrete TUCE mark TUCE as discrete freq TUCE Frequency distribution for TUCE obs 1 32 freque 12 1 14 1 ncy rel 3 12 3 12 cum 3 12 6 25 00 i 00 we Chapter 8 Discrete variables 49 17 3 9 38 15 62 19 3 9 38 25 00 20 2 6 25 31 25 21 4 12 50 43 75 22 2 6 25 50 00 23 4 12 50 62 50 24 3 9 38 71 88 25 4 12 50 84 38 26 2 6 25 90 62 27 1 3 12 93 75 28 1 3 12 96 88 29 1 3 12 100 00 Test for null hypothesis of normal distribution Chi square 2 1 872 with p value 0 39211 As can be seen from the sample output a Doornik Hansen test for normality is computed auto matically This test is suppressed for discrete variables whe
159. hat defines a list as in Chapter 12 Matrix manipulation 85 list Xl M where M is a matrix The matrix must be suitable for conversion that is it must be a row or column vector containing non negative whole number values none of which exceeds the highest ID number of a variable series or scalar in the current dataset Example 12 2 illustrates the use of this sort of conversion to normalize a list moving the constant variable 0 to first position Example 12 2 Manipulating a list function normalize_list matrix x If the matrix representing a list contains var 0 but not in first position move it to first position if x 1 0 scalar k cols x loop for i 2 i lt k i quiet if x i 0 x i x 1 EL 0 break endif end loop end if end function open data9 7 list Xl 2304 matrix x Xl normalize_list amp x list Xl x 12 11 Deleting a matrix To delete a matrix just write delete M where M is the name of the matrix to be deleted 12 12 Printing a matrix To print a matrix you can simply give the name of the matrix in question on a line by itself or you can use the print command matrix M mnormal 100 2 M print M 12 13 Example OLS using matrices Example 12 3 shows how matrix methods can be used to replicate gretl s built in OLS functionality Chapter 12 Matrix manipulation Example 12 3 OLS via matrix methods open data4 1 matrix X const sqft
160. he name of a variable as in the listing of parameters There can be only one such statement An example of a valid return statement is shown below return scalar SSR Having a function return a list is one way of permitting the return of more than one variable That is you can define several variable inside a function and package them as a list in this case they are not destroyed when the function exits Here is a simple example which also illustrates the possibility of setting the descriptive labels for variables generated in a function function make_cubes list xlist list cubes null loop foreach i xlist quiet series 13 iA3 setinfo 13 d cube of i list cubes 13 end loop return list cubes end function open data4 1 list xlist price sqft list cubelist make_cubes xlist print xlist cubelist byobs labels Note that the return statement does not cause the function to return exit at the point where it appears within the body of the function Rather it specifies which variable is available for assign ment when the function exits and a function exits only when a the end of the function code is reached b a gretl error occurs or c a funcerr statement is reached The funcerr keyword which may be followed by a string enclosed in double quotes causes a function to exit with an error flagged If a string is provided this is printed on exit otherwise a generic error message is printed This mechanism
161. here we mean unchanged relative to the last smp1 setting or relative to the full dataset if no sub sample has been defined up to this point For example after smp1 1970 1 2003 4 smp1 2000 4 the sample range will be 1970 1 to 2000 4 An incremental or relative form of setting the sample range is also supported In this case a relative offset should be given in the form of a signed integer or a semicolon to indicate no change for both the starting and ending point For example smpl 1 will advance the starting observation by one while preserving the ending observation and smpl 2 1 will both advance the starting observation by two and retard the ending observation by one An important feature of setting the sample as described above is that it necessarily results in the selection of a subset of observations that are contiguous in the full dataset The structure of the dataset is therefore unaffected for example if it is a quarterly time series before setting the sample it remains a quarterly time series afterwards 40 Chapter 6 Sub sampling a dataset 41 6 3 Restricting the sample By restricting the sample we mean selecting observations on the basis of some Boolean logical criterion or by means of a random number generator This is likely to be most relevant for cross sectional or panel data Suppose we have data on a cross section of individuals recording their gender income and other characteristics
162. hesher A and Irish M 1987 Residual Analysis in the Grouped and Censored Normal Linear Model Journal of Econometrics 34 pp 33 61 Cureton E 1967 The Normal Approximation to the Signed Rank Sampling Distribution when Zero Differences are Present Journal of the American Statistical Association 62 pp 1068 1069 Davidson R and MacKinnon J G 1993 Estimation and Inference in Econometrics New York Oxford University Press Davidson R and MacKinnon J G 2004 Econometric Theory and Methods New York Oxford University Press Doornik J A and Hansen H 1994 An Omnibus Test for Univariate and Multivariate Normality working paper Nuffield College Oxford 175 Bibliography 176 Doornik J A 1998 Approximations to the Asymptotic Distribution of Cointegration Tests Jour nal of Economic Surveys 12 pp 573 93 Reprinted with corrections in M McAleer and L Oxley Practical Issues in Cointegration Analysis Oxford Blackwell 1999 Edgerton D and Wells C 1994 Critical Values for the Cusumsq Statistic in Medium and Large Sized Samples Oxford Bulletin of Economics and Statistics 56 pp 355 65 Fiorentini G Calzolari G and Panattoni L 1996 Analytic Derivatives and the Computation of GARCH Estimates Journal of Applied Econometrics 11 pp 399 417 Frigo M and Johnson S G 2005 The Design and Implementation of FFTW3 Proceedings of the IEEE 93
163. ialog window 16 4 Analytical and numerical derivatives 16 5 Controlling termination cia we Pe a Re we A 16 6 Details on the code 16 7 Numerical accuracy 17 Maximum likelihood estimation 17 1 Generic ML estimation with gretl 17 2 Gamma estimation 74 75 76 83 84 84 84 85 85 85 88 88 89 90 91 92 92 93 94 98 100 100 104 105 107 107 107 108 108 109 109 109 Contents 17 3 Stochastic frontier cost function 22 sss vk ee de e el ee ee ee 17 4 GARCH models 17 5 Analytical derivatives 2 06 0 4 66 be we a e a Re 17 6 Debugging ML scrip 18 GMM estimation AS ese Se ee es Be ee Pe Bee a ee a ae Ge a 18 1 Introduction and terminology 2 6 eee deo ee wR eG Se wR 18 2 OLS as GMM 18 3 TSLS as GMM 18 4 Covariance MatriX options se co be be loa rad REBEL A Ee ee ee 18 5 A real example the 18 6 Caveats 19 Model selection criteria 19 1 Introduction 19 2 Information criteria 20 Time series models 20 1 ARIMA models 20 2 Unit root tests 20 3 ARCH and GARCH Consumption Based Asset Pricing Model 20 4 Cointegration and Vector Error Correction Models o ooo 21 Discrete and censored dependent variables 21 1 Logitand probit models gt scr dla a a e ee ee bo 21 2 The Tobit model III Technical details 22 Gretl and Tex 22 1 Introduction 222 TEX Pelaved menu items lt oso i ees a ee a
164. ich will detect when a new version is available and offer the option of auto updating International Gretl will produce its output in English French Italian Spanish Polish or German depending on your computer s native language setting 1 2 Acknowledgements The gretl code base originally derived from the program ESL Econometrics Software Library written by Professor Ramu Ramanathan of the University of California San Diego We are much in debt to Professor Ramanathan for making this code available under the GNU General Public Licence and for helping to steer gretl s early development Chapter 1 Introduction 2 We are also grateful to the authors of several econometrics textbooks for permission to package for gretl various datasets associated with their texts This list currently includes William Greene au thor of Econometric Analysis Jeffrey Wooldridge Introductory Econometrics A Modern Approach James Stock and Mark Watson Introduction to Econometrics Damodar Gujarati Basic Economet rics and Russell Davidson and James MacKinnon Econometric Theory and Methods GARCH estimation in gretl is based on code deposited in the archive of the Journal of Applied Econometrics by Professors Fiorentini Calzolari and Panattoni and the code to generate p values for Dickey Fuller tests is due to James MacKinnon In each case we are grateful to the authors for permission to use their work With regard to the internationalization
165. id with mean zero and variance 1 and where 0 is taken to be the positive square root of Of Q 1 denotes the information set as of time t 1 and of is the conditional variance that is the variance conditional on information dated t 1 and earlier It is important to notice the difference between ARCH and an ordinary autoregressive error process The simplest first order case of the latter can be written as Ut Put 1 Et l lt p lt l where the es are independently and identically distributed with mean zero and variance o With an AR 1 error if p is positive then a positive value of ut will tend to be followed with probability greater than 0 5 by a positive us 1 With an ARCH error process a disturbance ut of large absolute value will tend to be followed by further large absolute values but with no presumption that the successive values will be of the same sign ARCH in asset prices is a stylized fact and is consistent with market efficiency on the other hand autoregressive behavior of asset prices would violate market efficiency One can test for ARCH of order q in the following way 1 Estimate the model of interest via OLS and save the squared residuals ti 2 Perform an auxiliary regression in which the current squared residual is regressed on a con stant and q lags of itself 3 Find the TR value sample size times unadjusted R for the auxiliary regression 4 Refer the TR value to the x distribution
166. idual effects are representable as a legitimate part of the disturbance term that is zero mean random variables uncorrelated with the regressors As a consequence the fixed effects estimator always works but at the cost of not being able to estimate the effect of time invariant regressors The richer hypothesis set of the random effects estimator ensures that parameters for time invariant regressors can be estimated and that esti mation of the parameters for time varying regressors is carried out more efficiently These advan tages though are tied to the validity of the additional hypotheses If for example there is reason to think that individual effects may be correlated with some of the explanatory variables then the random effects estimator would be inconsistent while fixed effects estimates would still be valid It is precisely on this principle that the Hausman test is built see below if the fixed and random effects estimates agree to within the usual statistical margin of error there is no reason to think the additional hypotheses invalid and as a consequence no reason not to use the more efficient RE estimator Testing panel models If you estimate a fixed effects or random effects model in the graphical interface you may notice that the number of items available under the Tests menu in the model window is relatively limited Panel models carry certain complications that make it difficult to implement all of th
167. ied by whatever goes on inside the function There is however a mechanism for allowing a function and its caller to cooperate such that an outer variable can be modified by the function In effect this allows a function to return more than one value although only one variable can be returned directly see below The method is at least superficially similar to the passing of the address of a variable in the C programming language The parameter in question is marked with a prefix of in the function definition and the corresponding argument is marked with the complementary prefix amp in the caller For example function get_uhat_and_ess series y list xvars scalar ess ols y 0 xvars quiet ess ess series uh uhat return series uh end function main script open data4 1 list xlist 2 3 4 function call scalar SSR series resid get_uhat_and_ess price xlist amp SSR In the above we may say that the function is given the address of the scalar variable SSR and it assigns a value to that variable under the local name ess For anyone used to programming in C Chapter 10 User defined functions 61 note that it is not necessary or even possible to dereference the variable in question within the function using the operator Unembellished use of the name of the variable is sufficient to access the variable in outer scope An address parameter of this sort can be used as a means of offering o
168. ights matrix will influence the solution Partial output from this script is shown in 18 3 The estimated standard errors from GMM are robust by default if we supply the robust option to the ts1s command we get identical results 18 4 Covariance matrix options The covariance matrix of the estimated parameters depends on the choice of W through PWD ty wawsi wy 18 8 where J is a Jacobian term E nh an 00 and Q is the long run covariance matrix of the orthogonality conditions Gretl computes J by numeric differentiation there is no provision for specifying a user supplied analytical expression for J at the moment As for Q a consistent estimate is needed The simplest choice is the sample covariance matrix of the fts T O fel Fel 18 9 t 1 This estimator is robust with respect to heteroskedasticity but not with respect to autocorrela tion A heteroskedasticity and autocorrelation consistent HAC variant can be obtained using the Bartlett kernel or similar A univariate version of this is used in the context of the 1rvar function see equation 5 1 The multivariate version is set out in equation 18 10 1 T k k 0 7 gt 5 wf t0r 18 10 t k i k lThe data file used in this example is available in the Stock and Watson package for gretl See http gretl sourceforge net gretl_data html Chapter 18 GMM estimation Example 18 2 TSLS via GMM open cig_ch10 gdt real avg price
169. in the gretl icon view window see Section 3 4 after the script is executed In addition you can arrange to have a named model displayed in its own window automatically as follows Mode11 show Again if the name contains spaces it must be quoted Model 1 show lThis feature is not unique to gretl other econometric packages offer the same facility However experience shows that while this can be remarkably useful it can also lead to writing dinosaur scripts that are never meant to be executed all at once but rather used as a chaotic repository to cherry pick snippets from Since gretl allows you to have several script windows open at the same time you may want to keep your scripts tidy and reasonably small Chapter 3 Modes of working 15 The same facility can be used for graphs For example the following will create a plot of Ct against Yt save it under the name CrossPlot it will appear under this name in the icon view window and have it displayed CrossPlot lt gnuplot Ct Yt CrossPlot show You can also save the output from selected commands as named pieces of text again these will appear in the session icon window from where you can open them later For example this com mand sends the output from an augmented Dickey Fuller test to a text object named ADF1 and displays it in a window ADF1 lt adf 2 x1 ADF1 show Objects saved in this way whether models graphs or pieces of text output can be des
170. indexed loop in which the smp1 is keyed to the index variable i Suppose we have a panel dataset with observations on a number of hospitals for the years 1991 to 2000 where the year of the observation is indicated by a variable named year We restrict the sample to each of these years in turn and print cross sectional summary statistics for variables 1 through 4 Chapter 9 Loop constructs 56 Example 9 3 ARMA 1 1 open armaloop gdt genr c genr a genr m Il ooo he series e 1 0 genr de_c e genr de_a e genr de_m e genr crit 1 loop while crit gt 1 0e 9 one step forecast errors genr e y c a y 1 m e 1 log likelihood genr loglik 0 5 sum eA2 print loglik partials of forecast errors wrt c a and m genr de_c 1 m de_c 1 genr de_a y 1 m de_a 1 genr de_m e 1 m de_m 1 partials of 1 wrt c a and m genr sc_c de_c e genr sc_a de_a e genr sc_m de_m e OPG regression ols const sc_c sc_a sc_m print final no df corr vcv Update the parameters genr dc coeff sc_c genr c c dc genr da coeff sc_a genr a a da genr dm coeff sc_m genr m m dm printf constant 8g gradient 6g n c dc printf arl coefficient 8g gradient 6g n a da printf mal coefficient 8g gradient 6g n m dm genr crit T ess print crit endloop genr se_c stderr sc_c genr se_a stderr
171. ing form so if you have a cross section sample you may just use d t 123 of course if the dataset has data labels use the corresponding label For example if you open the dataset mrw gdt supplied with gretl among the examples a dummy variable for Italy could be generated via genr Dita t Italy Note that this method does not require scripting at all In fact you might as well use the GUI Menu Add Define new variable for the same purpose with the same syntax Generating an ARMA 1 1 Problem Generate y 0 9y 1 0 5 1 With e NIID O 1 Solution alpha 0 9 theta 0 5 series e normal O series y 0 series y alpha y 1 e theta e 1 Comment The statement series y 0 is necessary because the next statement evaluates y re cursively so y 1 must be set Note that you must use the keyword series here instead of writing genr y Oor simply y 0 to ensure that y is a series and not a scalar Conditional assignment Problem Generate y via the following rule lt for dt 1 vt zt for di 0 Solution series y d x Z Comment There are several alternatives to the one presented above One is a brute force solution using loops Another one more efficient but still suboptimal would be y d x 1 d z The ternary conditional assignment operator is not only the most numerically efficient way to accom plish what we want it is also remarkably transparent to read when
172. interesting statistical properties in its own right it will be denoted here as g 0 Itis a lWe are supposing here that our data are a realization of continuous random variables For discrete random variables everything continues to apply by referring to the probability function instead of the density In both cases the distribution may be conditional on some exogenous variables 112 Chapter 17 Maximum likelihood estimation 113 k vector with typical element 0010 y 00 0 00 00 gi 0 t 1 Gradient based methods can be shortly illustrated as follows 1 pick a point 09 O evaluate g 0p if g 0p is small stop Otherwise compute a direction vector d g 0p Bo uN evaluate 01 09 d g 00 yI substitute Oy with 01 6 restart from 2 Many algorithms of this kind exist they basically differ from one another in the way they compute the direction vector d g 0o to ensure that 01 gt 00 so that we eventually end up on the maximum The method gretl uses to maximize the log likelihood is a gradient based algorithm known as the BFGS Broyden Fletcher Goldfarb and Shanno method This technique is used in most econometric and statistical packages as it is well established and remarkably powerful Clearly in order to make this technique operational it must be possible to compute the vector g for any value of 0 In some cases this vector can be written explicitly as a function o
173. inutes from 9am onwards you ll probably want to specify the hour as 20 periods Solution setobs 20 9 1 special Comment Now functions like sdiffQ seasonal difference or estimation methods like seasonal ARIMA will work as expected Dropping missing observations selectively Problem You have a dataset with many variables and want to restrict the sample to those observa tions for which there are no missing observations for the variables x1 x2 and x3 Solution list X x1 x2 x3 genr sel ok X smpl sel restrict Comment You can now save the file via a store command to preserve a subsampled version of the dataset By operations Problem You have a discrete variable d and you want to run some commands for example estimate a model by splitting the sample according to the values of d Solution matrix vd values d m rows vd loop for i 1 m scalar sel vd i smpl d sel restrict replace ols y const x end loop smp1 full 88 Chapter 13 Cheat sheet 89 Comment The main ingredient here is a loop You can have gretl perform as many instructions as you want for each value of d as long as they are allowed inside a loop 13 2 Creating modifying variables Generating a dummy variable for a specific observation Problem Generate dt 0 for all observation but one for which d 1 Solution genr d t 1984 2 Comment The internal variable t is used to refer to observations in str
174. ion Suppose we have a sample of T independent and identically distributed observations from a Gamma distribution The density function for each observation x is f x 7 x xP exp ax 17 2 The log likelihood for the entire sample can be written as the logarithm of the joint density of all the observations Since these are independent and identical the joint density is the product of the individual densities and hence its log is T L a p tog P exp caw S4 17 3 rip where Li p log axt y p log xt Axt and y is the log of the gamma function In order to estimate the parameters and p via ML we need to maximize 17 3 with respect to them The corresponding gretl code snippet is scalar alpha 1 scalar p 1 mle logl p InCalpha x Ingamma p ln x alpha x end mle The two statements alpha 1 p 1 are necessary to ensure that the variables p and alpha exist before the computation of logl is attempted The values of these variables will be changed by the execution of the mle command upon successful completion they will be replaced by the ML estimates The starting value is 1 for both this is arbitrary and does not matter much in this example more on this later The above code can be made more readable and marginally more efficient by defining a variable to hold x This command can be embedded into the mle block as follows scalar alpha 1 scalar p 1 m
175. is anomaly was fixed in gretl version 1 0pre3 released October 10 2002 The current version of gretl includes a plugin that runs the NIST linear regression test suite You can find this under the Tools menu in the main window When you run this test the introductory text explains the expected result If you run this test and see anything other than the expected result please send a bug report to cottrel1 wfu edu As mentioned above all regression statistics are printed to 6 significant figures in the current version of gretl except when the multiple precision plugin is used then results are given to 12 figures If you want to examine a particular value more closely first save it for example using the genr command then print it using print long see the Gretl Command Reference This will show the value to 10 digits or more if you set the internal variable longdigits to a higher value via the set command 172 Appendix D Related free software Gretl s capabilities are substantial and are expanding Nonetheless you may find there are some things you can t do in gretl or you may wish to compare results with other programs If you are looking for complementary functionality in the realm of free open source software we recommend the following programs The self description of each program is taken from its website e GNU R r project org R is a system for statistical computation and graphics It consists of a
176. itten as P yi T 1 mu oo Since the observations are independent and identically distributed the log likelihood is simply the sum of the individual contributions Hence T f 3 y log m 1 yt log 1 Tr t 1 The verbose switch at the end of the end mle statement produces a detailed account of the iterations done by the BFGS algorithm In this case numerical differentiation works rather well nevertheless computation of the analytical 1 score is straightforward since the derivative A can be written as ot ot OTT OBi Ome OB via the chain rule and it is easy to see that of Mu 1 y OTT Tit 1 Tr OTT 3B P Bixit P2X2t B3X3t Xit L The mle block in the above script can therefore be modified as follows mle logl y In P 1 yY In 1 P series ndx bO b1 x1 b2 x2 b3 x3 series P cnorm ndx series tmp dnorm ndx y P 1 y 1 P deriv bO tmp deriv b1 tmp x1 deriv b2 tmp x2 deriv b3 tmp x3 end mle verbose 2 Again gretl does provide a native probit command see section 21 1 but a probit model makes for a nice example here Chapter 17 Maximum likelihood estimation 120 Note that the params statement has been replaced by a series of deriv statements these have the double function of identifying the parameters over which to optimize and providing an analytical expression for their respective score elements 17 6 Debugging ML scripts We have discussed
177. ix notation we may group all the T observations for unit i into the vector y and write it as yi X B ui 15 4 The vector u which includes all the disturbances for individual i has a variance covariance matrix given by Var u i gt el o 15 5 where J is a square matrix with all elements equal to 1 It can be shown that the matrix 0 Kiel 2 where 0 1 oe has the property KK 671 Chapter 15 Panel data 102 It follows that the transformed system Kiy KiXiB K u 15 6 satisfies the Gauss Markov conditions and OLS estimation of 15 6 provides efficient inference But since Kiyi yi 09 GLS estimation is equivalent to OLS using quasi demeaned variables that is variables from which we subtract a fraction 0 of their average Notice that for o gt 0 0 1 while for o 0 0 0 This means that if all the variance is attributable to the individual effects then the fixed effects estimator is optimal if on the other hand individual effects are negligible then pooled OLS turns out unsurprisingly to be the optimal estimator To implement the GLS approach we need to calculate 0 which in turn requires estimates of the variances o and 0 These are often referred to as the within and between variances respec tively since the former refers to variation within each cross sectional unit and the latter to variation between the units Several means of estimating these magnitude
178. ized boolean expression may be added to limit the sample for the variable in question A space must be inserted between the variable name or number and the expression Suppose you have salary figures for men and women and you have a dummy variable GENDER with value 1 for men and O for women In that case you could draw comparative boxplots with the following line in the boxplots dialog salary GENDER 1 salary GENDER 0 Chapter 8 Discrete variables When a variable can take only a finite typically small number of values then the variable is said to be discrete Some gretl commands act in a slightly different way when applied to discrete variables moreover gretl provides a few commands that only apply to discrete variables Specifically the dummify and xtab commands see below are available only for discrete variables while the freq frequency distribution command produces different output for discrete variables 8 1 Declaring variables as discrete Gretl uses a simple heuristic to judge whether a given variable should be treated as discrete but you also have the option of explicitly marking a variable as discrete in which case the heuristic check is bypassed The heuristic is as follows First are all the values of the variable reasonably round where this is taken to mean that they are all integer multiples of 0 25 If this criterion is met we then ask whether the variable takes on a fairly small set of distinct val
179. lFor convenience I will refer to the graphical client program simply as gretl in this manual Note however that the specific name of the program differs according to the computer platform On Linux it is called gret1_x11 while on MS Windows it is gretlw32 exe On Linux systems a wrapper script named gretl is also installed see also the Gretl Command Reference Chapter 2 Getting started 6 This file contains data pertaining to a classic econometric chestnut the consumption function The data window should now display the name of the current data file the overall data range and sample range and the names of the variables along with brief descriptive tags see Figure 2 2 File Utilities Session Data Sample Variable Model data3 6 gdt Descriptive label auto generated constant Personal consumption expenditures 1992 dollars Per capita disposable personal income 1992 dollars Annual Full range 1959 1 current sample 1959 1994 Figure 2 2 Main window with a practice data file open OK what can we do now Hopefully the various menu options should be fairly self explanatory For now we ll dip into the Model menu a brief tour of all the main window menus is given in Section 2 3 below gretl s Model menu offers numerous various econometric estimation routines The simplest and most standard is Ordinary Least Squares OLS Selecting OLS pops up a dialog box calling for a model specification see Figure 2 3 gr
180. language plus a run time environment with graphics a debugger access to certain system functions and the ability to run programs stored in script files It compiles and runs on a wide variety of UNIX platforms Windows and MacOS Comment There are numerous add on packages for R covering most areas of statistical work e GNU Octave www octave org GNU Octave is a high level language primarily intended for numerical computations It provides a convenient command line interface for solving linear and nonlinear problems numerically and for performing other numerical experiments using a language that is mostly compatible with Matlab It may also be used as a batch oriented language e JMulTi www jmulti de JMulTi was originally designed as a tool for certain econometric pro cedures in time series analysis that are especially difficult to use and that are not available in other packages like Impulse Response Analysis with bootstrapped confidence intervals for VAR VEC modelling Now many other features have been integrated as well to make it possi ble to convey a comprehensive analysis Comment JMulTi is a java GUI program you need a java run time environment to make use of it As mentioned above gretl offers the facility of exporting data in the formats of both Octave and R In the case of Octave the gretl data set is saved as a single matrix X You can pull the X matrix apart if you wish once the data are loaded in Octave see
181. latin1 inputenc usepackage amsmath usepackage dcolumn longtable begin document thispagestylef empty Note that the amsmath and dcolumn packages are required For some sorts of output the longtable package is also needed Beyond that you can for instance change the type size or the font by al tering the documentclass declaration or including an alternative font package Chapter 22 Gretl and T X 159 Table 22 2 Example of model table output OLS estimates Dependent variable ENROLL Model 1 const 0 2907 0 07853 CATHOL 0 2216 0 04584 PUPIL 0 003035 0 002727 WHITE 0 1482 0 04074 ADMEXP 0 1551 0 1342 n 51 R 0 4502 96 09 Model 2 0 2411 0 06602 0 2235 0 04597 0 003382 0 002720 0 1526 0 04071 51 0 4462 95 36 Model 3 0 08557 0 05794 0 2065 0 05160 0 001697 0 003025 51 0 2956 88 69 Standard errors in parentheses indicates significance at the 10 percent level indicates significance at the 5 percent level Chapter 22 Gretl and T X 160 The line usepackage latin1 inputenc is automatically modified if gretl finds itself running on a system where UTF 8 is the default character encoding see section 22 4 below In addition if you should wish to typeset gretl output in more than one language you can set up per language preamble files A localized preamble file is identified by a name of the form gretlpre_xx
182. lation to the genr command For example if a matrix A is already defined then matrix B sqrt A generates a matrix such that b a All of these functions require a single matrix as argument or an expression which evaluates to a single matrix Note that to find the matrix square root you need the cholesky function see below moreover the exp function computes the exponential element by element and therefore does not return the matrix exponential unless the matrix is diagonal to get the matrix exponential use mexp The functions sort dsort and values are available for matrices as well as data series In the matrix case the argument to these functions must be a vector p x 1 or 1 x p For sort and dsort the return value is a vector containing the elements of the input vector sorted in ascending sort Chapter 12 Matrix manipulation 77 or descending dsort order of magnitude For values the return is a vector containing the distinct values in the input vector sorted in ascending order Several matrix specific functions are available These functions fall into five categories 1 Those taking a single matrix as argument and returning a scalar 2 Those taking a single matrix as argument plus in some cases an additional parameter and returning a matrix 3 Those taking one or two dimensions as arguments and returning a matrix 4 Those taking two matrices as arguments and returning a matrix 5 Those taking
183. lds the parameter values which produce the maximum Here is an example matrix X dataset matrix theta 1 100 scalar J BFGSmax theta ObjFunc amp theta amp X It is assumed here that ObjFunc is a user defined function see Chapter 10 with the following general set up function ObjFunc matrix theta matrix X scalar val do some computation return scalar val end function The operation of the BFGS maximizer can be adjusted using the set variables bfgs_maxiter and bfgs_toler see Chapter 17 In addition you can provoke verbose output from the maximizer by assigning a positive value to max_verbose again via the set command The Rosenbrock function is often used as a test problem for optimization algorithms It is also known as Rosenbrock s Valley or Rosenbrock s Banana Function on account of the fact that its contour lines are banana shaped It is defined by f x y 1 x 100 y x The function has a global minimum at x y 1 1 where f x y 0 Example 5 2 shows a gretl script that discovers the minimum using BFGSmax giving a verbose account of progress Chapter 5 Special functions in genr 37 Example 5 2 Finding the minimum of the Rosenbrock function function Rosenbrock matrix param scalar x param 1 scalar y param 2 scalar f 1 x A2 100 y xA2 A2 return scalar f end function nulldata 10 matrix theta 0 0 set max_verbose 1 BFGSmax thet
184. le a data file might contain a country code variable and a variable representing the year of the observation In that case gretl can reconstruct the panel structure of the data regardless of how the observation rows are organized The setobs command has options that parallel those in the graphical interface If suitable index variables are available you can do for example setobs unitvar timevar panel vars where unitvar is a variable that indexes the units and timevar is a variable indexing the periods Alternatively you can use the form setobs freq 1 1 structure where freq is replaced by the block size of the data that is the number of periods in the case of stacked time series or the number of units in the case of stacked cross sections and structure is either stacked time series or stacked cross section Two examples are given below the first is suitable for a panel in the form of stacked time series with observations from 20 periods the second for stacked cross sections with 5 units setobs 20 1 1 stacked time series setobs 5 1 1 stacked cross section Panel data arranged by variable Publicly available panel data sometimes come arranged by variable Suppose we have data on two variables x1 and x2 for each of 50 states in each of 5 years giving a total of 250 observations per variable One textual representation of such a data set would start with a block for x1 with 50 rows corresponding to the states and 5
185. le log p In ax Ingamma p ln x ax series ax alpha x params alpha p end mle In this case it is necessary to include the line params alpha p to set the symbols p and alpha apart from ax which is a temporarily generated variable and not a parameter to be estimated In a simple example like this the choice of the starting values is almost inconsequential the algo rithm is likely to converge no matter what the starting values are However consistent method of moments estimators of p and can be simply recovered from the sample mean m and variance V since it can be shown that E x pl V x p o Chapter 17 Maximum likelihood estimation 115 it follows that the following estimators amp mV m a v ll are consistent and therefore suitable to be used as starting point for the algorithm The gretl script code then becomes scalar m mean x scalar alpha m var x scalar p m alpha mle logl p In ax Ingamma p ln x ax series ax alpha x params alpha p end mle Another thing to note is that sometimes parameters are constrained within certain boundaries in this case for example both and p must be positive numbers Gretl does not check for this it is the user s responsibility to ensure that the function is always evaluated at an admissible point in the parameter space during the iterative search for the maximum An effective technique is to define a variable for checking that the parameter
186. les for quarterly data e Sample menu Set range Select a different starting and or ending point for the current sample within the range of data available Restore full range self explanatory Define based on dummy Given a dummy indicator variable with values O or 1 this drops from the current sample all observations for which the dummy variable has value 0 Restrict based on criterion Similar to the item above except that you don t need a pre defined variable you supply a Boolean expression e g sqft gt 1400 and the sample is restricted to observations satisfying that condition See the entry for genr in the Gretl Command Reference for details on the Boolean operators that can be used Random sub sample Draw a random sample from the full dataset Drop all obs with missing values Drop from the current sample all observations for which at least one variable has a missing value see Section 4 6 Count missing values Give a report on observations where data values are missing May be useful in examining a panel data set where it s quite common to encounter missing values Set missing value code Set a numerical value that will be interpreted as missing or not available This is intended for use with imported data when gretl has not recognized the missing value code used e Variable menu Most items under here operate on a single variable at a time The active variable is set by highlighting it
187. lity condition is therefore available E f B 9 18 7 where f B y x B x The definitions given in the previous section therefore specialize here to e Gis B e the instrument is xt e fijt 0 is y x1B x1 utxt the orthogonality condition is interpretable as the requirement that the regressors should be uncorrelated with the disturbances Chapter 18 GMM estimation 123 e W can be any symmetric positive definite matrix since the number of parameters equals the number of orthogonality conditions Let s say we choose I e The function F 0 W is in this case LE F 0 W E Saro t 1 and it is easy to see why OLS and GMM coincide here the GMM objective function has the same minimizer as the objective function of OLS the residual sum of squares Note however that the two functions are not equal to one another at the minimum F 0 W O while the minimized sum of squared residuals is zero only in the special case of a perfect linear fit The code snippet contained in Example 18 1 uses gretl s gmm command to make the above opera tional Example 18 1 OLS via GMM initialize stuff series e 0 scalar beta 0 matrix V I 1 proceed with estimation gmm series e y x beta orthog e x weights V params beta end gmm We feed gretl the necessary ingredients for GMM estimation in a command block starting with gmm and ending with end gmm Three elements are compulsory within a g
188. lity in the data If the mean of the series in question is not constant or if the error process is not only heteroskedastic but also autoregressive it is necessary to take this into account when formulating an appropriate model For example it may be necessary to take the first difference of the variable in question and or to add suitable regressors X as in 20 10 20 4 Cointegration and Vector Error Correction Models Vector Error Correction Models as representation of a cointegrated system Consider a VAR of order p with a deterministic part given by ut typically a polynomial in time Then it is possible to write the n variate process y as Yi Ht Ar Mt 1 A2Vt 2 ApYt p Et however this model can be re cast in a form more suitable to analyze the phenomenon of cointe gration Since Y yz 1 Ay and Vt i Ve 1 AVt 1 AVYt 2 AVt i 1 the Vector Error Correction form of the previous model is given by p Ayt Ht Hyt 1 gt TiAyt i Et 20 14 i 1 where IT 5 Ai and kk Si Aj If the rank of IT is O the processes are all 1 1 if the rank of IT is full the processes are all I 0 in between II can be written as wf and you have cointegration The rank of II is investigated by computing the eigenvalues of a closely related matrix call it M whose rank is the same as II however M is by construction symmetric and positive semidefinite Chapter 20 Time series models 144 As a conseque
189. ls that have been affixed in either of these ways select Edit from the graph pop up and uncheck Show all data labels Advanced options If you know something about gnuplot and wish to get finer control over the appearance of a graph than is available via the graphical controller Edit option you have two further options e Once the graph is saved as a session icon you can right click on its icon for a further pop up menu One of the options here is Edit plot commands which opens an editing window with the actual gnuplot commands displayed You can edit these commands and either save them for future processing or send them to gnuplot with the Execute icon on the toolbar in the plot commands editing window e Another way to save the plot commands or to save the displayed plot in formats other than EPS or PNG is to use Edit item on a graph s pop up menu to invoke the graphical controller then click on the Output to file tab in the controller You are then presented with a drop down menu of formats in which to save the graph To find out more about gnuplot see the online manual or www gnuplot info See also the entry for gnuplot in the Gretl Command Reference and the graph and plot com mands for quick and dirty ASCII graphs Y axis Lines Output to file Title for axis Yt auto axis range manual range minimum 8399 02 maximum 18562 Y Apply E save X Close 0 Heip
190. lt si m lt T uf 0 otherwise That is the columns of B are lagged versions of the columns of A with missing values replaced by zeros The order m may be negative to generate leads instead of lags The nullspace function yields X the right null space of a matrix A it is assumed that A has full row rank X satisfies A X 0 The function princomp requires two arguments a T x k matrix X and a scalar p 0 lt p lt k Itis assumed that X contains T observations on each of k variables series The return value is a T x p matrix P containing the first p principal components of X The elements of P are computed as k G Pj gt Ziv i l where Z is the standardized value of variable i at observation t Zti Xti Xi G and v is the jth eigenvector of the correlation matrix of the X s with the eigenvectors ordered by decreasing value of the corresponding eigenvalues The functions fft and ffti return the real discrete Fourier transform and its inverse respectively if Xis an n x k matrix then fft OO is an n x 2k matrix containing the real part of the transform in the odd columns and the complex part in the even ones Conversely ffti takes a n x 2k argument and yields an n x k result See section 5 10 for some examples Matrix filling functions The functions taking one or two integers as arguments and returning a matrix are I n n x n identity matrix zeros m n m xn zero matrix ones m n m x n matrix filled with 1s
191. lutions to the test problems were obtained with substantially fewer iterations and the results were more accurate most notably for the estimated standard errors Note also that the six digit results printed by gretl are not 100 percent reliable for difficult nonlinear problems in particular when using numerical derivatives Having registered this caveat the percentage of cases where the results were good to six digits or better seems high enough to justify their printing in this form Chapter 17 Maximum likelihood estimation 17 1 Generic ML estimation with gretl Maximum likelihood estimation is a cornerstone of modern inferential procedures Gretl provides a way to implement this method for a wide range of estimation problems by use of the mle com mand We give here a few examples To give a foundation for the examples that follow we start from a brief reminder on the basics of ML estimation Given a sample of size T it is possible to define the density function for the whole sample namely the joint distribution of all the observations f Y 0 where Y 71 YT Its shape is determined by a k vector of unknown parameters 0 which we assume is contained in a set O and which can be used to evaluate the probability of observing a sample with any given characteristics After observing the data the values Y are given and this function can be evaluated for any legiti mate value of 6 In this case we prefer to call it the likelih
192. ly thing to do since a closed form solution is easily given by OLS 18 3 TSLS as GMM Moving closer to the proper domain of GMM we now consider two stage least squares TSLS as a case of GMM TSLS is employed in the case where one wishes to estimate a linear model of the form y Xb ut but where one or more of the variables in the matrix X are potentially endogenous correlated with the error term u We proceed by identifying a set of instruments Z which are explanatory for the endogenous variables in X but which are plausibly uncorrelated with u The classic two stage procedure is 1 regress the endogenous elements of X on Z then 2 estimate the equation of interest with the endogenous elements of X replaced by their fitted values from 1 An alternative perspective is given by GMM We define the residual t as y XB as usual But instead of relying on E u X 0 as in OLS we base estimation on the condition E u Z 0 In this case it is natural to base the initial weighting matrix on the covariance matrix of the instruments Example 18 2 presents a model from Stock and Watson s Introduction to Econometrics The demand for cigarettes is modeled as a linear function of the logs of price and income income is treated as exogenous while price is taken to be endogenous and two measures of tax are used as instruments Since we have two instruments and one endogenous variable the model is over identified and there fore the we
193. m lelsek i e Otherwise if p 1 and n q or q 1 and m p then C is m xn with cij aij X bx where k jifp 1elsek i e If none of the above conditions are satisfied the product is undefined and an error is flagged For example if A is a row vector with the same number of columns of B then the columns of C are the columns of B multiplied by the corresponding element of A Note that this convention makes it unnecessary in most cases to use diagonal matrices to perform transformations by means of ordinary matrix multiplication if Y XV where V is diagonal it is computationally much more convenient to obtain Y via the instruction o matrix Y X v where v is a row vector containing the diagonal of V Element wise division and element wise exponentiation work in a manner exactly analogous to element wise multiplication simply replace x by or the exponentation operation in the account given for multiplication In column wise concatenation of an mxn matrix A and an mxp matrix B the result is an mx n p matrix That is matrix C A B produces C A B Row wise concatenation of an m x n matrix A and an p x n matrix B produces an m p xn matrix That is matrix C A B A produces C B 12 5 Matrix scalar operators For matrix A and scalar k the operators shown in Table 12 1 are available Addition and subtrac tion were discussed in section 12 4 but we include them in the table for completeness
194. mation is by exact ML or conditional ML Forecasting ARMA models are often used for forecasting purposes The autoregressive component in particu lar offers the possibility of forecasting a process out of sample over a substantial time horizon Gretl supports forecasting on the basis of ARMA models using the method set out by Box and Jenkins 1976 The Box and Jenkins algorithm produces a set of integrated AR coefficients which take into account any differencing of the dependent variable seasonal and or non seasonal in the ARIMA context thus making it possible to generate a forecast for the level of the original variable 1See in particular their Program 4 on p 505ff Chapter 20 Time series models 138 By contrast if you first difference a series manually and then apply ARMA to the differenced series forecasts will be for the differenced series not the level This point is illustrated in Example 20 1 The parameter estimates are identical for the two models The forecasts differ but are mutually consistent the variable fcdiff emulates the ARMA forecast static one step ahead within the sample range and dynamic out of sample Example 20 1 ARIMA forecasting open greenel18_2 gdt log of quarterly U S nominal GNP 1950 1 to 1983 4 genr y log Y and its first difference genr dy diff y reserve 2 years for out of sample forecast smp1 1981 4 Estimate using ARIMA arima 111 y forecast ove
195. me further options are available if the dataset includes case markers that is labels identifying each observa tion With a scatter plot displayed when you move the mouse pointer over a data point its label is shown on the graph By default these labels are transient they do not appear in the printed or copied version of the graph They can be removed by selecting Clear data labels from the graph pop up menu If you want the labels to be affixed permanently so they will show up when the graph is printed or copied you have two options e To affix the labels currently shown on the graph select Freeze data labels from the graph pop up menu lFor best results when pasting graphs into MS Office applications choose the application s Edit Paste Special menu item and select the option Picture Enhanced Metafile 2For an example of such a dataset see the Ramanathan file data4 10 this contains data on private school enrollment for the 50 states of the USA plus Washington DC the case markers are the two letter codes for the states 43 Chapter 7 Graphs and plots 44 e To affix labels for all points in the graph select Edit from the graph pop up and check the box titled Show all data labels This option is available only if there are less than 55 data points and it is unlikely to produce good results if the points are tightly clustered since the labels will tend to overlap To remove labe
196. mechanism is that the internal variables test and pvalue are over written each time one of the tests listed above is performed If you want to reference these values you must do so at the correct point in the sequence of gretl commands A related point is that some of the test commands generate by default more than one test statistic and p value in these cases only the last values are stored To get proper control over the retrieval of values via test and pvalue you should formulate the test command in such a way that the result is unambiguous This comment applies in particular to the adf and Imtest commands e By default the adf command generates three variants of the Dickey Fuller test one based on a regression including a constant one using a constant and linear trend and one using a constant and a quadratic trend When you wish to reference test or pvalue in connection with this command you can control the variant that is recorded by using one of the flags nc C Ct or ctt with adf e By default the Imtest command which must follow an OLS regression performs several diagnostic tests on the regression in question To control what is recorded in test and pvalue you should limit the test using one of the flags logs autocorr squares or white As an aid in working with values retrieved using test and pvalue the nature of the test to which these values relate is written into the descriptive label for the genera
197. metric software for the GNU generation Journal of Applied Econometrics 18 pp 105 10 Baltagi B H 1995 Econometric Analysis of Panel Data New York Wiley Baxter M and King R G 1995 Measuring Business Cycles Approximate Band Pass Filters for Economic Time Series National Bureau of Economic Research Working Paper No 5022 Beck N and Katz J N 1995 What to do and not to do with Time Series Cross Section Data The American Political Science Review 89 pp 634 47 Belsley D Kuh E and Welsch R 1980 Regression Diagnostics New York Wiley Berndt E Hall B Hall R and Hausman J 1974 Estimation and Inference in Nonlinear Structural Models Annals of Economic and Social Measurement 3 4 pp 653 65 Blundell R and Bond S 1998 Initial Conditions and Moment Restrictions in Dynamic Panel Data Models Journal of Econometrics 87 pp 115 43 Bollerslev T and Ghysels E 1996 Periodic Autoregressive Conditional Heteroscedasticity Jour nal of Business and Economic Statistics 14 pp 139 51 Box G E P and Jenkins G 1976 Time Series Analysis Forecasting and Control San Franciso Holden Day Box G E P and Muller M E 1958 A Note on the Generation of Random Normal Deviates Annals of Mathematical Statistics 29 pp 610 11 Cameron A C and Trivedi P K 2005 Microeconometrics Methods and Applications Cambridge Cambridge University Press C
198. mies When using the approach of subtracting the group means the issue is that after de meaning these variables are nothing but zeros 2 A somewhat analogous prohibition applies to the random effects estimator This estimator is in effect a matrix weighted average of pooled OLS and the between estimator Suppose we have observations on n units or individuals and there are k independent variables of interest If k gt n the between estimator is undefined since we have only n effective observations and hence so is the random effects estimator If one does not fall foul of one or other of the prohibitions mentioned above the choice between fixed effects and random effects may be expressed in terms of the two econometric desiderata efficiency and consistency From a purely statistical viewpoint we could say that there is a tradeoff between robustness and efficiency In the fixed effects approach we do not make any hypotheses on the group effects that is the time invariant differences in mean between the groups beyond the fact that they exist Chapter 15 Panel data 103 and that can be tested see below As a consequence once these effects are swept out by taking deviations from the group means the remaining parameters can be estimated On the other hand the random effects approach attempts to model the group effects as drawings from a probability distribution instead of removing them This requires that indiv
199. mm block 1 one or more orthog statements 2 one weights statement 3 one params statement The three elements should be given in the stated order The orthog statements are used to specify the orthogonality conditions They must follow the syntax orthog x Z where x may be a series matrix or list of series and Z may also be a series matrix or list In example 18 1 the series e holds the residuals and the series x holds the regressor If x had been a list a matrix the orthog statement would have generated one orthogonality condition for each element column of x Note the structure of the orthogonality condition it is assumed that the term to the left of the semicolon represents a quantity that depends on the estimated parameters and so must be updated in the process of iterative estimation while the term on the right is a constant function of the data The weights statement is used to specify the initial weighting matrix and its syntax is straight forward The params statement specifies the parameters with respect to which the GMM criterion should be minimized it follows the same logic and rules as in the mle and nls commands Chapter 18 GMM estimation 124 The minimum is found through numerical minimization via BFGS see section 5 9 and chapter 17 The progress of the optimization procedure can be observed by appending the verbose switch to the end gmm line In this example GMM estimation is clearly a rather sil
200. most likely departure from iid errors is heteroskedasticity non constant variance In some cases one may be able to arrive at a judgment regarding the likely form of the heteroskedasticity and hence to apply a specific correction The more common case however is where the heteroskedasticity is of unknown form We seek an estimator of the covari ance matrix of the parameter estimates that retains its validity at least asymptotically in face of unspecified heteroskedasticity It is not obvious a priori that this should be possible but White 1980 showed that Varn XX I X X X X 14 5 does the trick As usual in statistics we need to say under certain conditions but the conditions are not very restrictive is in this context a diagonal matrix whose non zero elements may be estimated using squared OLS residuals White referred to 14 5 as a heteroskedasticity consistent covariance matrix estimator HCCME Davidson and MacKinnon 2004 chapter 5 offer a useful discussion of several variants on White s HCCME theme They refer to the original variant of 14 5 in which the diagonal elements of are estimated directly by the squared OLS residuals e as HCo The associated standard errors are often called White s standard errors The various refinements of White s proposal share a ln some specialized contexts spatial autocorrelation may be an issue Gretl does not have any built in methods to
201. n or e From the command line type gret1 r sessionfile where sessionfile is the name under which the session was saved 3For PDF output you need pdflatex and either Adobe s PDF reader or xpdf on X11 For PostScript you must have dvips and ghostscript installed along with a viewer such as gv ggv or kghostview The default viewer for systems other than MS Windows is gv Chapter 4 Data files 4 1 Native format gretl has its own format for data files Most users will probably not want to read or write such files outside of gretl itself but occasionally this may be useful and full details on the file formats are given in Appendix A 4 2 Other data file formats gretl will read various other data formats Plain text ASCH files These can be brought in using gretl s File Open Data Import ASCII menu item or the import script command For details on what gretl expects of such files see Section 4 4 e Comma Separated Values CSV files These can be imported using gretl s File Open Data Import CSV menu item or the import script command See also Section 4 4 e Worksheets in the format of either MS Excel or Gnumeric These are also brought in using gretl s File Open Data Import menu The requirements for such files are given in Sec tion 4 4 Stata data files dta Eviews workfiles wf1 When you import data from the ASCII or CSV formats gretl opens a diagnostic window repor
202. n goods must be matched by a corresponding increase in the discounted future utility of the consumption financed by the asset s return Since the future is uncertain the individual considers his expectation conditional on what is known at the time when the choice is made We have said nothing about the nature of the asset so equation 18 11 should hold whatever asset we consider hence it is possible to build a system of equations like 18 11 for each asset whose price we observe Chapter 18 GMM estimation 127 If we are willing to believe that e the economy as a whole can be represented as a single gigantic and immortal representative individual and e the function U x Y a is a faithful representation of the individual s preferences then setting k 1 equation 18 11 implies the following for any asset j E ER ES ide Pit Ct where Cr is aggregate consumption and and 6 are the risk aversion and discount rate of the representative individual In this case it is easy to see that the deep parameters and 6 can be estimated via GMM by using Y C a 1 ey 6 tet e 1 Pit Ct as the moment condition while any variable known at time t may serve as an instrument F 1 18 12 In the example code given in 18 4 we replicate selected portions of table 3 7 in Hall 2005 The variable consrat is defined as the ratio of monthly consecutive real per capita consumption ser vices and nondurables
203. n of AutoRegressive Integrated Moving Average ARIMA models These are models that can be written in the form p L y OL et 20 1 where L and L are polynomials in the lag operator L defined such that L x Xt n and t is a white noise process The exact content of yt of the AR polynomial and of the MA polynomial 0 will be explained in the following Mean terms The process y as written in equation 20 1 has without further qualifications mean zero If the model is to be applied to real data it is necessary to include some term to handle the possibility that y has non zero mean There are two possible ways to represent processes with nonzero mean one is to define u as the unconditional mean of yt namely the central value of its marginal distribution Therefore the series y ut has mean 0 and the model 20 1 applies to i In practice assuming that ur is a linear function of some observable variables x the model becomes P L yt xtp O L et 20 2 This is sometimes known as a regression model with ARMA errors its structure may be more apparent if we represent it using two equations Y Xxip ur p Lju O L e The model just presented is also sometimes known as ARMAX ARMA eXogenous variables It seems to us however that this label is more appropriately applied to a different model another way to include a mean term in 20 1 is to base the representation on the conditi
204. name and the second is a named list of regressors Note that while the function offers the variable myess as a return value it is ignored by the caller in this instance As a side note here if you want a function to calculate some value having to do with a regression but are not interested in the full results of the regression you may wish to use the quiet flag with the estimation command as shown above A second example shows how to write a function call that assigns a return value to a variable in the caller function definition function get_uhat series y list xvars ols y 0 xvars quiet series uh uhat return series uh end function main script open data4 1 list xlist 2 3 4 function call series resid get_uhat price xlist 10 3 Function programming details Scope of variables All variables created within a function are local to that function and are destroyed when the func tion exits unless they are made available as return values and these values are picked up or assigned by the caller Functions do not have access to variables in outer scope that is variables that exist in the script from which the function is called except insofar as these are explicitly passed to the function as arguments By default when a variable is passed to a function as an argument what the function actually gets is a copy of the outer variable which means that the value of the outer variable is not modif
205. nce all its eigenvalues are real and non negative tests on the rank of II can therefore be carried out by testing how many eigenvalues of M are 0 If all the eigenvalues are significantly different from 0 then all the processes are stationary If on the contrary there is at least one zero eigenvalue then the y process is integrated although some linear combination B 7 might be stationary On the other extreme if no eigenvalues are significantly different from 0 then not only the process y is non stationary but the same holds for any linear combination f y in other words no cointegration occurs The two Johansen tests are the A max test for hypotheses on individual eigenvalues and the trace test for joint hypotheses The gretl command coint2 performs these two tests As in the ADF test the asymptotic distribution of the tests varies with the deterministic kernel ut one includes in the VAR gretl provides the following options for a short discussion of the meaning of the five options see section 20 4 below Ht command option 0 nc Ho 0 Uo 0 rc Ho default Ho pit 1 0 crt Ho Mit ct Note that for this command the above options are mutually exclusive In addition you have the option of using the seasonal options for augmenting u with centered seasonal dummies In each case p values are computed via the approximations by Doornik 1998 The following code uses the denmark dat
206. nd arma 0 0 013 y const y 2 on a quarterly series would estimate the parameters of the model Yi H P Yi 2 H t OEt 4 However this workaround is not really recommended it should deliver correct estimates but will break the existing mechanism for forecasting 20 2 Unit root tests The ADF test The ADF Augmented Dickey Fuller test is as implemented in gretl the t statistic on in the following regression p AVe Me PYt 1 y yiAYVt i Et 20 8 i l This test statistic is probably the best known and most widely used unit root test It is a one sided test whose null hypothesis is p O versus the alternative y lt O Under the null y must be differenced at least once to achieve stationarity under the alternative y is already stationary and no differencing is required Hence large negative values of the test statistic lead to the rejection of the null One peculiar aspect of this test is that its limit distribution is non standard under the null hy pothesis moreover the shape of the distribution and consequently the critical values for the test depends on the form of the up term A full analysis of the various cases is inappropriate here Hamilton 1994 contains an excellent discussion but any recent time series textbook covers this topic Suffice it to say that gretl allows the user to choose the specification for us among four different alternatives Ht command option 0 nc Ho C H
207. nd name then press the Tab key readline will attempt to complete the command name for you If there s a unique completion it will be put in place automatically If there s more than one completion pressing Tab a second time brings up a list 24 2 CLI syntax Probably the most useful mode for heavy duty work with gretlcli is batch non interactive mode in which the program reads and processes a script and sends the output to file For example greticli b scriptfile gt outputfile The scriptfile is treated as a program argument it should specify a data file to use internally using the syntax open datafile Don t forget the b batch switch otherwise the program will wait for user input after executing the script l Actually the key bindings shown below are only the defaults they can be customized See the readline manual 163 Part IV Appendices 164 Appendix A Data file details A 1 Basic native format In gretl s native data format a data set is stored in XML extensible mark up language Data files correspond to the simple DTD document type definition given in gretldata dtd which is supplied with the gretl distribution and is installed in the system data directory e g usr share gret1 data on Linux Data files may be plain text or gzipped They contain the actual data values plus additional information such as the names and descriptions of variables the frequency of the data and so on Most users will
208. nomic Studies 61 pp 631 53 R Core Development Team 2000 An Introduction to R version 1 1 1 Ramanathan Ramu 2002 Introductory Econometrics with Applications 5th edition Fort Worth Harcourt Schwarz G 1978 Estimating the dimension of a model Annals of Statistics 6 pp 461 64 Shapiro S and Chen L 2001 Composite Tests for the Gamma Distribution Journal of Quality Technology 33 pp 47 59 Silverman B W 1986 Density Estimation for Statistics and Data Analysis London Chapman and Hall Stock James H and Watson Mark W 2003 Introduction to Econometrics Boston MA Addison Wesley Swamy P A V B and Arora S S 1972 The Exact Finite Sample Properties of the Estimators of Coefficients in the Error Components Regression Models Econometrica 40 pp 261 75 White H 1980 A Heteroskedasticity Consistent Covariance Matrix Astimator and a Direct Test for Heteroskedasticity Econometrica 48 pp 817 38 Windmeijer F 2005 A Finite Sample Correction for the Variance of Linear Efficient Two step GMM Estimators Journal of Econometrics 126 pp 25 51 Wooldridge Jeffrey M 2002a Econometric Analysis of Cross Section and Panel Data Cambridge Mass MIT Press Wooldridge Jeffrey M 2002b Introductory Econometrics A Modern Approach 2nd edition Mason Ohio South Western
209. ntinuous variable into classes you can use the genr command and its arithmetic functions as in the following example 46 Chapter 8 Discrete variables 47 nulldata 100 generate a variable with mean 2 and variance 1 genr x normalQ 2 split into 4 classes genr z x gt 0 x gt 2 004 now declare z as discrete discrete z Once a variable is marked as discrete this setting is remembered when you save the file 8 2 Commands for discrete variables The dummi fy command The dummi fy command takes as argument a series x and creates dummy variables for each distinct value present in x which must have already been declared as discrete Example open greene22_2 discrete Z5 mark Z5 as discrete dummify Z5 The effect of the above command is to generate 5 new dummy variables labeled DZ5_1 through DZ5_5 which correspond to the different values in Z5 Hence the variable DZ5_4 is 1 if Z5 equals 4 and 0 otherwise This functionality is also available through the graphical interface by selecting the menu item Add Dummies for selected discrete variables The dummify command can also be used with the following syntax list dlist dummify x This not only creates the dummy variables but also a named list see section 11 1 that can be used afterwards The following example computes summary statistics for the variable Y for each value of Z5 open greene22_2 discrete Z5 mark Z5 as discrete list foo dummif
210. nu item Variable Define new variable In some cases where the nonlinear function is a generalization of or a restricted form of a linear model it may be convenient to run an ols and initialize the parameters from the OLS coefficient 107 Chapter 16 Nonlinear least squares 108 estimates In relation to the first example above one might do ols CO Y genr alpha coeff 0 genr beta coeff Y genr gamma 1 And in relation to the second example one might do ols y O x1 x2 genr alpha coeff 0 genr beta coeff x1 16 3 NLS dialog window It is probably most convenient to compose the commands for NLS estimation in the form of a gretl script but you can also do so interactively by selecting the item Nonlinear Least Squares under the Model Nonlinear models menu This opens a dialog box where you can type the function specification possibly prefaced by genr lines to set the initial parameter values and the derivatives if available An example of this is shown in Figure 16 1 Note that in this context you do not have to supply the nls and end nls tags NLS Specify function and derivatives if possible Please refer to Help for guidance genr alpha 10 genr beta 5 genr gamma 1 C alpha beta Y gamma deriv alpha 1 0 deriv beta Y gamma deriv gamma beta Y gamma log Y 0K X cancel 5 Help Figure 16 1 NLS dialog box 16 4 Analytical and numerical derivatives
211. o Hit ct Ho pat yt ctt These options are not mutually exclusive when they are used together the statistic will be reported separately for each case By default gretl uses by default the combination c ct ctt For each case approximate p values are calculated by means of the algorithm developed in MacKinnon 1996 The gretl command used to perform the test is adf for example adf 4 x1 c ct would compute the test statistic as the t statistic for y in equation 20 8 with p 4 in the two cases Ht Mo and Hr Uo pit The number of lags p in equation 20 8 should be chosen as to ensure that 20 8 is a parame trization flexible enough to represent adequately the short run persistence of Ay Setting p too low results in size distortions in the test whereas setting p too high would lead to low power As a convenience to the user the parameter p can be automatically determined Setting p toa negative number triggers a sequential procedure that starts with p lags and decrements p until the t statistic for the parameter y exceeds 1 645 in absolute value Chapter 20 Time series models 140 The KPSS test The KPSS test Kwiatkowski Phillips Schmidt and Shin 1992 is a unit root test in which the null hypothesis is opposite to that in the ADF test under the null the series in question is stationary the alternative is that the series is 1 1 The basic intuition behind this test statistic is very simple if y c
212. of gretl thanks go to Ignacio D az Emparanza Spanish Michel Robitaille and Florent Bresson French Cristian Rigamonti Italian Tadeusz Kufel and Pawel Kufel Polish and Markus Hahn and Sven Schreiber German Gretl has benefitted greatly from the work of numerous developers of free open source software for specifics please see Appendix B Our thanks are due to Richard Stallman of the Free Software Foundation for his support of free software in general and for agreeing to adopt gretl as a GNU program in particular Many users of gretl have submitted useful suggestions and bug reports In this connection par ticular thanks are due to Ignacio Diaz Emparanza Tadeusz Kufel Pawel Kufel Alan Isaac Cri Rigamonti Sven Schreiber Talha Yalta and Dirk Eddelbuettel who maintains the gretl package for Debian GNU Linux 1 3 Installing the programs Linux On the Linux platform you have the choice of compiling the gretl code yourself or making use of a pre built package Building gretl from the source is necessary if you want to access the development version or customize gretl to your needs but this takes quite a few skills most users will want to go for a pre built package Some Linux distributions feature gretl as part of their standard offering Debian for example or Ubuntu in the universe repository If this is the case all you need to do is install gretl through your package manager of choice eg synaptic Ready to
213. ogram via the Fourier transform nulldata 50 generate an AR 1 process series e normal O series x 0 x 0 9 x 1 e compute the periodogram scale 2 pi nobs X x F fft X S sumr F A2 S S 2 nobs 2 1 scale omega seq 1 nobs 2 2 pi nobs omega omega S compare the built in command pergm x print omega Chapter 6 Sub sampling a dataset 6 1 Introduction Some subtle issues can arise here This chapter attempts to explain the issues A sub sample may be defined in relation to a full data set in two different ways we will refer to these as setting the sample and restricting the sample respectively 6 2 Setting the sample By setting the sample we mean defining a sub sample simply by means of adjusting the starting and or ending point of the current sample range This is likely to be most relevant for time series data For example one has quarterly data from 1960 1 to 2003 4 and one wants to run a regression using only data from the 1970s A suitable command is then smp1 1970 1 1979 4 Or one wishes to set aside a block of observations at the end of the data period for out of sample forecasting In that case one might do smpl 2000 4 where the semicolon is shorthand for leave the starting observation unchanged The semicolon may also be used in place of the second parameter to mean that the ending observation should be unchanged By unchanged
214. olpin 1997 The dependent variable is the occupational status of an individual 0 in school 1 not in school and not working 2 working and the explanatory variables are education and work experience linear and square plus a black binary variable The full data set is a panel here the analysis is confined to a cross section for 1987 For explanations of the matrix methods employed in the script see chapter 12 21 2 The Tobit model The Tobit model is used when the dependent variable of a model is censored Assume a latent variable y can be described as k YES xij bj i j l where e N 0 o If yi were observable the model s parameters could be estimated via ordinary least squares On the contrary suppose that we observe y defined as Sof for yf gt 0 y 0 for y lt 0 21 8 lWe assume here that censoring occurs from below at 0 Censoring from above or at a point different from zero can be rather easily handled by re defining the dependent variable appropriately The more general case of two sided censoring is not handled by gretl via a native command yet but it is possible to estimate such models using the mle command see chapter 17 Chapter 21 Discrete and censored dependent variables Example 21 4 Multinomial logit function mlogitlogprobs series y matrix X matrix theta scalar n max y scalar k cols X matrix b mshape theta k n matrix tmp X b series ret In 1 sumr exp
215. ommand under gretl s Help menu RATS 4 databases Thanks to Thomas Doan of Estima who made available the specification of the database format used by RATS 4 Regression Analysis of Time Series gretl can also handle such databases Well actually a subset of same I have only worked on time series databases containing monthly and quarterly series My university has the RATS G7 database containing data for the seven largest OECD economies and gretl will read that OK t Visit the gretl data page for details and updates on available data 4 4 Creating a data file from scratch There are several ways of doing this 1 Find or create using a text editor a plain text data file and open it with gretl s Import ASCIT option 2 Use your favorite spreadsheet to establish the data file save it in Comma Separated Values format if necessary this should not be necessary if the spreadsheet program is MS Excel or Gnumeric then use one of gretl s Import options CSV Excel or Gnumeric as the case may be 3 Use gretl s built in spreadsheet 4 Select data series from a suitable database 5 Use your favorite text editor or other software tools to a create data file in gretl format inde pendently Here are a few comments and details on these methods Common points on imported data Options 1 and 2 involve using gretl s import mechanism For gretl to read such data success fully certain general conditions must
216. omment up to 64 characters on the source and nature of the data following a hash mark For example Federal Reserve Board interest rates The corresponding binary database file holds the data values represented as floats that is single precision floating point numbers typically taking four bytes apiece The numbers are packed by variable so that the first n numbers are the observations of variable 1 the next m the observations on variable 2 and so on Appendix B Building gretl B 1 Requirements Gretl is written in the C programming language abiding as far as possible by the ISO ANSI C Standard C90 although the graphical user interface and some other components necessarily make use of platform specific extensions The program was developed under Linux The shared library and command line client should compile and run on any platform that supports ISO ANSI C and has the libraries listed in Table B 1 If the GNU readline library is found on the host system this will be used for gretcli providing a much enhanced editable command line See the readline homepage Library purpose website Zlib data compression info zip org libxml2 XML manipulation xmlsoft org LAPACK linear algebra netlib org FFTW3 Fast Fourier Transform fftw org glib 2 0 Numerous utilities gtk org Table B 1 Libraries required for building gretl The graphical client program should compile and run on any system that in addition to the above r
217. on if the dependent variable is not binary pro vided it has already been marked as discrete After estimating ordered models the uhat accessor yields generalized residuals as in binary mod els additionally the yhat accessor function returns 2 so it is possible to compute an unbiased estimator of the latent variable y simply by adding the two together Chapter 21 Discrete and censored dependent variables 151 Example 21 3 Ordered probit model open pension gdt series pctstck pctstck 50 discrete pctstck probit pctstck const choice age educ female black married finc25 finc35 finc50 finc75 finc100 finc101 wealth89 prftshr Multinomial logit When the dependent variable is not binary and does not have a natural ordering multinomial models are used Gretl does not provide a native implementation of these yet but simple models can be handled via the mle command see chapter 17 We give here an example of a multinomial logit model Let the dependent variable yi take on integer values 0 1 p The probability that yi k is given by exp x Bx P Yi klxi p poaki Py exp xib For the purpose of identification one of the outcomes must be taken as the baseline it is usually assumed that Bo 0 in which case exp xi Bx P yi klx Vi kid TSP pb and i P yi 0 x 1 5 exp xib Example 21 4 reproduces Table 15 2 in Wooldridge 2002a based on data on career choice from Keane and W
218. onal mean of yt that is the central value of the distribution of y given its own past Assuming again that this can be represented as a linear combination of some observable variables z the model would expand to P L y zty O L es 20 3 The formulation 20 3 has the advantage that y can be immediately interpreted as the vector of marginal effects of the z variables on the conditional mean of y And by adding lags of z to this specification one can estimate Transfer Function models which generalize ARMA by adding the effects of exogenous variable distributed across time Gretl provides a way to estimate both forms Models written as in 20 2 are estimated by maximum likelihood models written as in 20 3 are estimated by conditional maximum likelihood For more on these options see the section on Estimation below 133 Chapter 20 Time series models 134 In the special case when x Z 1 that is the models include a constant but no exogenous variables the two specifications discussed above reduce to PLD yt u O L et 20 4 and P L yt amp 0 L er 20 5 respectively These formulations are essentially equivalent but if they represent one and the same process u and ox are fairly obviously not numerically identical rather a 1 di bp u The gretl syntax for estimating 20 4 is simply arma p q y The AR and MA lag orders p and q can be given either as numbers or as pre de
219. one has to make when using GMM is high and in finite samples each of these can have dramatic consequences on the eventual output Some of the factors that may affect the results are 1 Orthogonality conditions can be written in more than one way for example if E x u 0 then E x u 1 0 holds too It is possible that a different specification of the moment conditions leads to different results 2 As with all other numerical optimization algorithms weird things may happen when the ob jective function is nearly flat in some directions or has multiple minima BFGS is usually quite good but there is no guarantee that it always delivers a sensible solution if one at all 3 The 1 step and to a lesser extent the 2 step estimators may be sensitive to apparently trivial details like the re scaling of the instruments Different choices for the initial weights matrix can also have noticeable consequences Chapter 18 GMM estimation Example 18 4 Estimation of the Consumption Based Asset Pricing Model open hall gdt set force_hc on scalar alpha scalar delta series e 0 oo uw list inst matrix VO matrix Z inst matrix V1 nobs inv Z Z gmm e delta ewr consratA alpha 1 orthog e inst weights VO params alpha delta end gmm gmm e delta ewr consratA alpha 1 orthog e inst weights V1 params alpha delta end gmm gmm e delta ewr consratA alpha 1 orthog e inst weights VO params
220. one of the operands is a 1 x 1 matrix or scalar The scalar is implicitly promoted to the status of a matrix of the correct dimensions all of whose elements are equal to the given scalar value For example if A is an m x n matrix and k a scalar then the commands A k A k matrix C matrix D both produce m x n matrices with elements cij aij k and dij aij k respectively By pre multiplication by transpose we mean for example that matrix C X Y produces the product of X transpose and Y In effect the expression X Y is shorthand for X Y which is also valid In matrix division A B is algebraically equivalent to B A pre multiplication by the inverse of the divisor Therefore the following two expressions are equivalent in principle Chapter 12 Matrix manipulation 75 matrix C A B matrix C inv B A where inv is the matrix inversion function see below for more on matrix functions The first form however may be more accurate than the second the solution is obtained via LU decomposition without the explicit calculation of the inverse In element wise multiplication if we write matrix C A B then the result depends on the dimensions of A and B Let A be an m x n matrix and let B be p x q e If m p andn q then C is mx n with cij aij X bij This is known as the Hadamard product e Otherwise if m 1 and n q orn 1 and m p then C is p x q with cij ax X bij where k jif
221. one or more matrices as arguments and returning one or more matrices These sets of functions are discussed in turn below Matrix to scalar functions The functions which take a single matrix as argument and return a scalar are rows number of rows cols number of columns rank rank det determinant Tdet log determinant tr trace onenorm 1 norm infnorm infinity norm rcond reciprocal condition number The single matrix argument to these functions may be given as the name of an existing matrix or as an expression that evaluates to a single matrix Note that the functions det Idet and tr require a square matrix as input The rank function is computed via the QR decomposition The functions onenorm and infnorm return respectively the 1 norm and the infinity norm of a matrix The former is the maximum across the columns of the matrix of the sums of the absolute values of the column elements while the latter is the maximum across the rows of the matrix of the sums of the absolute values of the row elements The function rcond returns the reciprocal condition number for a symmetric positive definite matrix Matrix to matrix functions The functions which take a single matrix as argument and return a matrix are Chapter 12 Matrix manipulation 78 sumc sum by column sumr sum by row meanc mean by column meanr mean by row mcov covariance matrix mcorr correlation matrix mexp matrix exponential inv inverse cholesky Cholesky decom
222. onto the Model table icon or by right clicking on the model icon and selecting Add to model table from the pop up menu 5 Repeat step 4 for the other models you wish to include in the table The second model selected will appear in the second column from the left and so on 6 When you are finished composing the model table display it by double clicking on its icon Under the Edit menu in the window which appears you have the option of copying the table to the clipboard in various formats 7 If the ordering of the models in the table is not what you wanted right click on the model table icon and select Clear table Then go back to step 4 above and try again 2The model table can also be built non interactively in script mode For details on how to do this see the entry for modeltab in the Gretl Command Reference Chapter 3 Modes of working 17 A simple instance of gretl s model table is shown in Figure 3 3 gretl model table OSs w x OLS estimates Dependent variable price Model 1 Model 2 Model 3 const 129 1 121 2 52 35 88 30 80 18 37 29 sqft 0 1548 0 1483 0 1388 0 03194 0 02121 0 01873 bedrms 21 59 23 91 27 03 24 64 baths 12 19 43 25 n 14 14 14 Adj R 2 0 7868 0 8046 0 8056 Standard errors in parentheses indicates significance at the 10 percent level indicates significance at the 5 percent level Close Figure 3 3 Example of model table Th
223. ood function the need for another name stems from the fact that this function works as a density when we use the yrs as arguments and 0 as parameters whereas in this context 0 is taken as the function s argument and the data Y only have the role of determining its shape In standard cases this function has a unique maximum The location of the maximum is unaffected if we consider the logarithm of the likelihood or log likelihood for short this function will be denoted as 0 log f Y 0 The log likelihood functions that gretl can handle are those where 0 can be written as T 0 gt 0 t 1 which is true in most cases of interest The functions 4 0 are called the log likelihood contribu tions Moreover the location of the maximum is obviously determined by the data Y This means that the value 0 Y Argmax 4 0 17 1 050 is some function of the observed data a statistic which has the property under mild conditions of being a consistent asymptotically normal and asymptotically efficient estimator of 0 Sometimes it is possible to write down explicitly the function Y in general it need not be so In these circumstances the maximum can be found by means of numerical techniques These often rely on the fact that the log likelihood is a smooth function of 0 and therefore on the maximum its partial derivatives should all be 0 The gradient vector or score vector is a function that enjoys many
224. or example loop while essdiff gt 00001 Execution of the commands within the loop will continue so long as the specified condition evalu ates as true If the right hand term of the inequality is a variable it is evaluated at the top of the loop at each iteration Index loop A third form of loop control uses the special internal index variable i In this case you specify starting and ending values for i which is incremented by one each time round the loop The syntax looks like this loop i 1 20 The index variable may be used within the loop body in one or both of two ways you can access the value of 1 see Example 9 4 or you can use its string representation i see Example 9 5 The starting and ending values for the index can be given in numerical form or by reference to predefined variables In the latter case the variables are evaluated once when the loop is set up In addition with time series data you can give the starting and ending values in the form of dates as in loop 1 1950 1 1999 4 This form of loop is particularly useful in conjunction with the values matrix function when some operation must be carried out for each value of some discrete variable see chapter 8 Con sider the following example open greene22_2 open greene22_2 discrete Z8 v8 values Z8 n rows v8 n rows v8 loop for i 1 n scalar xi v8 i smpl Z8 xi restrict replace printf mean Y Z8 g 8 5f sd Y Z8
225. oth the probit and logit model are estimated in gretl via maximum likelihood since the score equations do not have a closed form solution numerical optimization is used However in most cases this is totally transparent to the user since usually only a few iterations are needed to ensure convergence The verbose switch can be used to track the maximization algorithm 148 Chapter 21 Discrete and censored dependent variables 149 Example 21 1 Estimation of simple logit and probit models open greenel19_1 logit GRADE const GPA TUCE PSI probit GRADE const GPA TUCE PSI As an example we reproduce the results given in Greene 2000 chapter 21 where the effective ness of a program for teaching economics is evaluated by the improvements of students grades Running the code in example 21 1 gives the following output Model 1 Logit estimates using the 32 observations 1 32 Dependent variable GRADE VARIABLE COEFFICIENT STDERROR T STAT SLOPE at mean const 13 0213 4 93132 2 641 GPA 2 82611 1 26294 2 238 0 533859 TUCE 0 0951577 0 141554 0 672 0 0179755 PSI 2 37869 1 06456 2 234 0 449339 Mean of GRADE 0 344 Number of cases correctly predicted f beta x at mean of independent vars McFadden s pseudo R squared 0 374038 Log likelihood 12 8896 Likelihood ratio test Chi square 3 15 4042 p value 0 001502 Akaike information criterion AIC 33 7793 Schwarz Bayesian criterion BIC 39 6422 Hannan Quinn crit
226. ources This is supported via the File Append data menu items gretl will check the new data for conformability with the existing dataset and if everything seems OK will merge the data You can add new variables in this way provided the data frequency matches that of the existing dataset Or you can append new observations for data series that are already present in this case the variable names must match up correctly Note that by default that is if you choose Open data rather than Append data opening a new data file closes the current one Using the built in spreadsheet Under gretl s File New data set menu you can choose the sort of dataset you want to establish e g quarterly time series cross sectional You will then be prompted for starting and ending dates or observation numbers and the name of the first variable to add to the dataset After supplying this information you will be faced with a simple spreadsheet into which you can type data values In the spreadsheet window clicking the right mouse button will invoke a popup menu which enables you to add a new variable column to add an observation append a row at the foot of the sheet or to insert an observation at the selected point move the data down and insert a blank row Once you have entered data into the spreadsheet you import these into gretl s workspace using the spreadsheet s Apply changes button Please note that gretl s spre
227. packages for TEX and in fact these are likely to be installed by default Check the documentation for your distribution For MS Windows several packaged versions of TX are available one of the most popular is MiKTpx at http www miktex org For Mac OS X a nice implementation is TfXMac at http itexmac sourceforge net An essential starting point for online TeX resources is the Comprehensive TX Archive Network CTAN at http www ctan org As for learning TeX many useful resources are available both online and in print Among online guides Tony Roberts KTgx from quick and dirty to style and finesse is very helpful at http ww sci usq edu au staff robertsa LaTeX latexintro html An excellent source for advanced material is The BTX Companion Goossens et al 2004 Chapter 23 Troubleshooting gretl 23 1 Bug reports Bug reports are welcome Hopefully you are unlikely to find bugs in the actual calculations done by gretl although this statement does not constitute any sort of warranty You may however come across bugs or oddities in the behavior of the graphical interface Please remember that the usefulness of bug reports is greatly enhanced if you can be as specific as possible what exactly went wrong under what conditions and on what operating system If you saw an error message what precisely did it say 23 2 Auxiliary programs As mentioned above gretl calls some other programs to accomplish certain
228. pe parameter equals zero is then 0 0166 using the t 40 distribution Depending on the context however we may doubt whether the ratio of coefficient to standard error truly follows the t 40 distribution In that case we could derive a bootstrap p value as shown in Example 5 1 Under the null hypothesis that the slope with respect to x is zero y is simply equal to its mean plus an error term We simulate y by resampling the residuals from the initial OLS and re estimate the model We repeat this procedure a large number of times and record the number of cases where the absolute value of the t ratio is greater than 2 5 the proportion of such cases is our bootstrap p value For a good discussion of simulation based tests and bootstrapping see Davidson and MacKinnon 2004 chapter 4 Example 5 1 Calculation of bootstrap p value ols y 0 x save the residuals genr ui uhat scalar ybar mean y number of replications for bootstrap scalar replics 10000 scalar tcount 0 series ysim 0 loop replics quiet generate simulated y by resampling ysim ybar resample ui ols ysim 0 x scalar tsim abs coeff x stderr x tcount tsim gt 2 5 endloop printf proportion of cases with t gt 2 5 g n tcount replics 5 6 Cumulative densities and p values The two functions cdf and pvalue provide complementary means of examining values from several probability distributions the standard normal Student s t x F g
229. phs and plots 7 1 Gnuplot graphs A separate program gnuplot is called to generate graphs Gnuplot is a very full featured graphing program with myriad options It is available from www gnuplot info but note that a copy of gnuplot is bundled with the MS Windows version of gretl gretl gives you direct access via a graphical interface to a subset of gnuplot s options and it tries to choose sensible values for you it also allows you to take complete control over graph details if you wish With a graph displayed you can click on the graph window for a pop up menu with the following options Save as PNG Save the graph in Portable Network Graphics format e Save as postscript Save in encapsulated postscript EPS format e Save as Windows metafile Save in Enhanced Metafile EMF format e Save to session as icon The graph will appear in iconic form when you select Icon view from the View menu e Zoom Lets you select an area within the graph for closer inspection not available for all graphs e Print Gnome desktop or MS Windows only lets you print the graph directly e Copy to clipboard MS Windows only lets you paste the graph into Windows applications such as MS Word e Edit Opens a controller for the plot which lets you adjust various aspects of its appearance e Close Closes the graph window Displaying data labels In the case of a simple X Y scatterplot with or without a line of best fit displayed so
230. position diag extract principal diagonal transp transpose cdemean subtract column means vec elements as column vector vech vectorize lower triangle unvech undo vech mlag matrix lag or lead nullspace right nullspace princomp principal components maxc column maxima values maxr row maxima values 1imaxc column maxima indices imaxr row maxima indices minc column minima values minr row minima values iminc column minima indices iminr row minima indices fft discrete Fourier transform ffti discrete inverse Fourier transform As with the previous set of functions the argument may be given as the name of an existing matrix or as an expression that evaluates to a single matrix For an m x n matrix A sumc A returns a row vector holding the n column sums and sumr A returns a column vector with the m row sums meanc A returns a row vector with the n column means and meanr A a column vector with the m row means Also for an m x n matrix A the max and min family of functions return either an m x 1 matrix the r variants which select the extremum of each row ora 1 x n matrix the c variants which select the column extrema The max vectors contain the values of the row or column maxima while the min ones hold the row or column minima The variants with an i prefix e g imaxc return not the values but the 1 based indices of the respective maxima or minima For a T x k matrix A mcov A and mcorr A both return k x k symmetric matrices
231. preferable There are two ways of calculating H the matrix difference method and the regression method The procedure for the matrix difference method is this e Collect the fixed effects estimates in a vector B and the corresponding random effects esti mates in f then form the difference vector f f e Form the covariance matrix of the difference vector as Var B B Var B Var B Y where Var fB and Var fB are estimated by the sample variance matrices of the fixed and random effects models respectively 2 Hausman 1978 showed that the covariance of the difference takes this simple form when B is an efficient estimator Chapter 15 Panel data 104 e Compute H BY B Given the relative efficiencies of and B the matrix Y should be positive definite in which case H is positive but in finite samples this is not guaranteed and of course a negative x value is not admissible The regression method avoids this potential problem The procedure is e Treat the random effects model as the restricted model and record its sum of squared resid uals as SSR Estimate via OLS an unrestricted model in which the dependent variable is quasi demeaned y and the regressors include both quasi demeaned X as in the RE model and the de meaned variants of all the time varying variables i e the fixed effects regressors record the sum of squared residuals from this model as SSRy e Compute H n SSR
232. psd may be worth noting If you want to form a sub sample of a panel that contains only those units for which the variable x is time varying you can do smpl psd x gt 0 restrict Special functions for data manipulation Besides the functions discussed above there are some facilities in genr designed specifically for manipulating panel data in particular for the case where the data have been read into the program from a third party source and they are not in the correct form for panel analysis These facilities are explained in Chapter 4 Chapter 5 Special functions in genr 33 5 5 Resampling and bootstrapping Another specialized function is the resampling with replacement of a series Given an original data series x the command genr xr resample x creates a new series each of whose elements is drawn at random from the elements of x If the original series has 100 observations each element of x is selected with probability 1 100 at each drawing Thus the effect is to shuffle the elements of x with the twist that each element of x may appear more than once or not at all in xr The primary use of this function is in the construction of bootstrap confidence intervals or p values Here is a simple example Suppose we estimate a simple regression of y on x via OLS and find that the slope coefficient has a reported t ratio of 2 5 with 40 degrees of freedom The two tailed p value for the null hypothesis that the slo
233. ptional information to the caller That is the corresponding argument is not strictly needed but will be used if present In that case the parameter should be given a default value of nu11 and the the function should test to see if the caller supplied a corresponding argument or not using the built in function isnul1Q For example here is the simple function shown above modified to make the filling out of the ess value optional function get_uhat_and_ess series y list xvars scalar ess null1 ols y 0 xvars quiet if lisnull ess ess ess endif series uh uhat return series uh end function If the caller does not care to get the ess value it should use nu11 in place of a real argument series resid get_uhat_and_ess price xlist null List arguments The use of a named list as an argument to a function gives a means of supplying a function with a set of variables whose number is unknown when the function is written for example sets of regressors or instruments Within the function the list can be passed on to commands such as ols or it can be unpacked using a foreach loop construct For example suppose you have a list X and want to calculate the standard deviation of each variable in the list loop foreach i X scalar sd_ i sd i end loop When a named list of variables is passed to a function the function is provided with a copy of the list The variables referenced in the list are however made directl
234. r 14 Robust covariance matrix estimation 14 1 Introduction Consider once again the linear regression model y XBtu 14 1 where y and u are T vectors X is a T x k matrix of regressors and f is a k vector of parameters As is well known the estimator of given by Ordinary Least Squares OLS is B XXIX y 14 2 If the condition E u X 0 is satisfied this is an unbiased estimator under somewhat weaker conditions the estimator is biased but consistent It is straightforward to show that when the OLS estimator is unbiased that is when E B B 0 its variance is Var B E B B B B X X 71X 0X X X 1 14 3 where Q E uu is the covariance matrix of the error terms Under the assumption that the error terms are independently and identically distributed iid we can write Q o I where 0 is the common variance of the errors and the covariances are zero In that case 14 3 simplifies to the classical formula Var B o X X 14 4 If the iid assumption is not satisfied two things follow First it is possible in principle to construct a more efficient estimator than OLS for instance some sort of Feasible Generalized Least Squares FGLS Second the simple classical formula for the variance of the least squares estimator is no longer correct and hence the conventional OLS standard errors which are just the square roots of the diagonal elements of the matrix define
235. r Distaso published a review of gretl in the Journal of Applied Economet rics 2003 We are grateful to Baiocchi and Distaso for their careful examination of the program which prompted the following modifications 1 The reviewers pointed out that there was a bug in gretl s p value finder whereby the pro gram printed the complement of the correct probability for negative values of z This was fixed in version 0 998 of the program released July 9 2002 2 They also noted that the p value finder produced inaccurate results for extreme values of x e g values of around 8 to 10 in the t distribution with 100 degrees of freedom This too was fixed in gretl version 0 998 with a switch to more accurate probability distribution code 3 The reviewers noted a flaw in the presentation of regression coefficients in gretl whereby some coefficients could be printed to an unacceptably small number of significant figures This was fixed in version 0 999 released August 25 2002 now all the statistics associated with a regression are printed to 6 significant figures 4 It transpired from the reviewer s tests that the numerical accuracy of gretl on MS Windows was less than on Linux For example on the Longley data a well known ill conditioned dataset often used for testing econometrics programs the Windows version of gretl was getting some coefficients wrong at the 7th digit while the same coefficients were correct on Linux Th
236. r estimates For comparability with OLS and in light of the reasoning given in Davidson and MacKinnon 1993 the estimates shown in gretl do use a degrees of freedom correction 16 7 Numerical accuracy Table 16 1 shows the results of running the gretl NLS procedure on the 27 Statistical Reference Datasets made available by the U S National Institute of Standards and Technology NIST for test ing nonlinear regression software For each dataset two sets of starting values for the parameters are given in the test files so the full test comprises 54 runs Two full tests were performed one 10n a 32 bit Intel Pentium machine a likely value for this parameter is 1 82 x 10722 2For a discussion of gretl s accuracy in the estimation of linear models see Appendix C Chapter 16 Nonlinear least squares 110 using all analytical derivatives and one using all numerical approximations In each case the default tolerance was used Out of the 54 runs gretl failed to produce a solution in 4 cases when using analytical derivatives and in 5 cases when using numeric approximation Of the four failures in analytical derivatives mode two were due to non convergence of the Levenberg Marquandt algorithm after the maxi mum number of iterations on MGHO9 and Bennett5 both described by NIST as of Higher diffi culty and two were due to generation of range errors out of bounds floating point values when computing the Jacobian on BoxBOD and MGH
237. r full period smpl full fcast fel Return to sub sample and run ARMA on the first difference of y smpl 1981 4 arma 1 1 dy smpl full fcast fc2 genr fcdiff t lt 1982 1 fc1 yC 1 t gt 1982 1 fc1 fc1 D compare the forecasts over the later period smp1 1981 1 1983 4 print y fcl fc2 fcdiff byobs The output from the last command is y fcl fc2 fcdiff 1981 1 7 964086 7 940930 0 02668 0 02668 1981 2 7 978654 7 997576 0 03349 0 03349 1981 3 8 009463 7 997503 0 01885 0 01885 1981 4 8 015625 8 033695 0 02423 0 02423 1982 1 8 014997 8 029698 0 01407 0 01407 1982 2 8 026562 8 046037 0 01634 0 01634 1982 3 8 032717 8 063636 0 01760 0 01760 1982 4 8 042249 8 081935 0 01830 0 01830 1983 1 8 062685 8 100623 0 01869 0 01869 1983 2 8 091627 8 119528 0 01891 0 01891 1983 3 8 115700 8 138554 0 01903 0 01903 1983 4 8 140811 8 157646 0 01909 0 01909 Limitations The structure of gretl s arma command does not allow you to specify models with gaps in the lag structure other than via the seasonal specification discussed above For example if you have a monthly time series you cannot estimate an ARMA model with AR terms or MA terms at just lags 1 3 and 5 At a pinch you could circumvent this limitation in respect of the AR part of the specification by the trick of including lags of the dependent variable in the list of exogenous variables For example Chapter 20 Time series models 139 the following comma
238. re the number of distinct values is less than 10 This command accepts two options quiet to avoid generation of the histogram when invoked from the command line and gamma for replacing the normality test with Locke s nonparametric test whose null hypothesis is that the data follow a Gamma distribution If the distinct values of a discrete variable need to be saved the values matrix construct can be used see chapter 12 The xtab command The xtab command cab be invoked in either of the following ways First xtab ylist xlist where ylist and xlist are lists of discrete variables This produces cross tabulations two way frequencies of each of the variables in ylist by row against each of the variables in xlist by column Or second xtab xlist In the second case a full set of cross tabulations is generated that is each variable in xlist is tabu lated against each other variable in the list In the graphical interface this command is represented by the Cross Tabulation item under the View menu which is active if at least two variables are selected Here is an example of use open greene22_2 discrete Z mark Z1 Z8 as discrete xtab Z1 Z4 Z5 Z6 which produces Cross tabulation of Z1 rows against Z5 columns 111 2 0 3J 4 5 TOT 0 20 91 75 93 36 315 1 28 73 54 97 34 286 Chapter 8 Discrete variables 50 TOTAL 48 164 129 190 70 601 Pearson chi square test 5 48233 4 df p value
239. retl homepage http gretl sourceforge net GTK homepage http w gtk org GTK port for win32 http ww gimp org tml gimp win32 Gtkextra homepage http gtkextra sourceforge net InfoZip homepage http ww info zip org pub infozip zlib JMulTi homepage http www jmulti de JRSoftware http ww jrsoftware org Mingw gcc for win32 homepage http ww mingw org Minpack http ww netlib org minpack Penn World Table http pwt econ upenn edu Readline homepage http cnsww cns cwru edu chet readline rltop html Readline manual http cnsww cns cwru edu chet readline readline html Xmlsoft homepage http xmlsoft org 174 Bibliography Akaike H 1974 A New Look at the Statistical Model Identification IEEE Transactions on Auto matic Control AC 19 pp 716 23 Anderson T W and Hsiao C 1981 Estimation of Dynamic Models with Error Components Journal of the American Statistical Association 76 pp 598 606 Andrews D W K and Monahan J C 1992 An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator Econometrica 60 pp 953 66 Arellano M 2003 Panel Data Econometrics Oxford Oxford University Press Arellano M and Bond S 1991 Some Tests of Specification for Panel Data Monte Carlo Evidence and an Application to Employment Equations The Review of Economic Studies 58 pp 277 97 Baiocchi G and Distaso W 2003 GRETL Econo
240. riance of X T7 py xt must be estimated differently One of the most widely used statistics for this purpose is a nonparametric kernel estimator with the Bartlett kernel defined as T k k DAS TO gt unn Di 5 1 t k Li k where the integer k is known as the window size and the w terms are the so called Bartlett weights defined as w 1 i It can be shown that for k large enough k T yields a consistent estimator of the variance of X Gretl implements this estimator by means of the function Irvar which takes two arguments the series whose long run variance must be estimated and the scalar k If k is negative the popular choice T 3 is used 5 3 Time series filters One sort of specialized function in genr is time series filtering In addition to the usual application of lags and differences gretl provides fractional differencing and two filters commonly used in macroeconomics for trend cycle decomposition the Hodrick Prescott filter and the Baxter King bandpass filter Fractional differencing The concept of differencing a time series d times is pretty obvious when d is an integer it may seem odd when d is fractional However this idea has a well defined mathematical content consider the function fe shez where z and d are real numbers By taking a Taylor series expansion around z 0 we see that fz 1 dz 20224 29 Chapter 5 Special functions in genr 30 or more compactly f z 1 gt wiz
241. ries with value 1 for each observation at which the given variable has a missing value and value 0 otherwise that is if the given variable has a valid value at that observation The function ok is complementary to missing it is just a shorthand for missing where is the boolean NOT operator For example one can count the missing values for variable x using genr nmiss_x sum missing x The function zeromiss Q which again takes a single series as its argument returns a series where all zero values are set to the missing code This should be used with caution one does not want to confuse missing values and zeros but it can be useful in some contexts For example one can determine the first valid observation for a variable x using Chapter 5 Special functions in genr 35 genr time genr xO min zeromiss time ok x The function misszero does the opposite of zeromiss that is it converts all missing values to zero It may be worth commenting on the propagation of missing values within genr formulae The general rule is that in arithmetical operations involving two variables if either of the variables has a missing value at observation t then the resulting series will also have a missing value at t The one exception to this rule is multiplication by zero zero times a missing value produces zero since this is mathematically valid regardless of the unknown value 5 8 Retrieving internal variables The genr comman
242. rix preceded by amp to indicate the address of the matrix in question in which case an auxiliary result is written to that matrix or the keyword nul11 in which case the auxiliary result is not produced or is discarded In case a non null second argument is given the specified matrix will be over written with the auxiliary result It is not required that the existing matrix be of the right dimensions to receive the result The qrdecomp function computes the OR decomposition of an m x n matrix A A QR where Q is an m x n orthogonal matrix and R is an n x n upper triangular matrix The matrix Q is returned directly while R can be retrieved via the second argument Here are two examples matrix R matrix Q qrdecomp M amp R matrix Q qrdecomp M null In the first example the triangular R is saved as R in the second R is discarded The first line above shows an example of a simple declaration of a matrix R is declared to be a matrix variable but is not given any explicit value In this case the variable is initialized as a 1 x 1 matrix whose single element equals zero The function eigensym computes the eigenvalues and optionally the right eigenvectors of a sym metric n x n matrix The eigenvalues are returned directly in a column vector of length n if the eigenvectors are required they are returned in an n x n matrix For example matrix V matrix E eigensym M amp V matrix E eigensym M null In the
243. ry which is displayed and may be changed under the Tools Preferences General menu Options that can be set in this way are the font to use when producing postscript output must be a valid generic postscript font name the default is Helvetica the size of the font in points also for postscript output default is 12 the minimum and maximum for the y axis range the width and height of the plot in pixels default 560 x 448 whether numerical values should be printed for the quartiles and median default don t print them and whether outliers points lying beyond 1 5 times the interquartile range from the central box should be indicated separately default no Here is an example font Times Roman fontsize 16 max 4 0 min 0 width 400 height 448 numbers 3 2f outliers true On the second to last line the value associated with numbers is a printf format string as in the C programming language if specified this controls the printing of the median and quartiles next to the boxplot if no numbers entry is given these values are not printed In the example the values will be printed to a width of 3 digits with 2 digits of precision following the decimal point Not all of the options need be specified and the order doesn t matter Lines not matching the pattern key value are ignored as are lines that begin with the hash mark After each variable specified in the boxplot command a parenthes
244. s see section 3 4 Chapter 2 Getting started 10 Graph specified vars Gives a choice between a time series plot a regular X Y scatter plot an X Y plot using impulses vertical bars an X Y plot with factor separation i e with the points colored differently depending to the value of a given dummy variable boxplots and a 3 D graph Serves up a dialog box where you specify the variables to graph See Chapter 7 for details Multiple graphs Allows you to compose a set of up to six small graphs either pairwise scatter plots or time series graphs These are displayed together in a single window Summary statistics Shows a full set of descriptive statistics for the variables selected in the main window Correlation matrix Active only if two or more variables are selected shows the pairwise correlation coefficients for the selected variables Principal components Active only if two or more variables are selected produces a Prin cipal Components Analysis of the selected variables Mahalonobis distances Active only if two or more variables are selected computes the Mahalonobis distance of each observation from the centroid of the selected set of vari ables e Add menu Offers various standard transformations of variables logs lags squares etc that you may wish to add to the data set Also gives the option of adding random variables and for time series data adding seasonal dummy variables e g quarterly dummy variab
245. s carried out This is illustrated in the following interactive session Chapter 11 Named lists and strings 70 scalar x 8 scalar x 8 Generated scalar x ID 2 8 sprintf foo var d x Saved string as foo print foo var8 Note the effect of the quotation marks in the line print foo The line print foo would not print a literal var8 as above After pre processing the line would read print var8 It would therefore print the value s of the variable var8 if such a variable exists or would generate an error otherwise In certain specific contexts however it is natural to treat variables as variables in their own right and gretl does so These contexts are e When they appear among the arguments to the commands printf and sprintf e On the right hand side of a string assignment e When they appear as an argument to the function isstring see below Here is an illustration of the use of named string arguments with printf string vstr variance Saved string as vstr printf vstr 12s n vstr vstr variance Note that vstr should not be put in quotes in this context Similarly with string copy vstr Built in strings Apart from any strings that the user may define some string variables are defined by gretl itself These may be useful for people writing functions that include shell commands The built in strings are as shown in Table 11 1 gretidir the gretl installa
246. s are admissible and setting the log likelihood as undefined if the check fails An example which uses the conditional assignment operator follows scalar m mean x scalar alpha m var x scalar p m alpha mle logl check p InC ax Ingamma p ln x ax NA series ax alpha x scalar check alpha gt 0 p gt 0 params alpha p end mle 17 3 Stochastic frontier cost function When modeling a cost function it is sometimes worthwhile to incorporate explicitly into the sta tistical model the notion that firms may be inefficient so that the observed cost deviates from the theoretical figure not only because of unobserved heterogeneity between firms but also because two firms could be operating at a different efficiency level despite being identical under all other respects In this case we may write C C uit Vi where C is some variable cost indicator C i is its theoretical value u is a zero mean disturbance term and v is the inefficiency term which is supposed to be nonnegative by its very nature A linear specification for Cf is often chosen For example the Cobb Douglas cost function arises when Cf is a linear function of the logarithms of the input prices and the output quantities The stochastic frontier model is a linear model of the form y x B e in which the error term i is the sum of u and v A common postulate is that u N 0 07 and v N 0 af If independence between u and
247. s have been suggested in the liter ature see Baltagi 1995 gretl uses the method of Swamy and Arora 1972 a is estimated by the residual variance from the fixed effects model and the sum a T 0 is estimated as T times the residual variance from the between estimator Yi XiB ei The latter regression is implemented by constructing a data set consisting of the group means of all the relevant variables Choice of estimator Which panel method should one use fixed effects or random effects One way of answering this question is in relation to the nature of the data set If the panel comprises observations on a fixed and relatively small set of units of interest say the member states of the European Union there is a presumption in favor of fixed effects If it comprises observations on a large number of randomly selected individuals as in many epidemiological and other longitudinal studies there is a presumption in favor of random effects Besides this general heuristic however various statistical issues must be taken into account 1 Some panel data sets contain variables whose values are specific to the cross sectional unit but which do not vary over time If you want to include such variables in the model the fixed effects option is simply not available When the fixed effects approach is implemented using dummy variables the problem is that the time invariant variables are perfectly collinear with the per unit dum
248. s is but a stub Maximum likelihood estimation of 6 can be shown to be equivalent to solving an eigenvector prob lem In practice if the cointegration rank is known to be r the n x r matrix B is estimated as the solution to the following matrix equation MB BLA where A is a diagonal matrix containing the eigenvalues Notice however that if all columns of are cointegration vectors then any arbitrary linear combinations of those is a cointegration vector too This means that the matrix f is under identified As a consequence its elements do not have a proper covariance matrix but this difficulty can be circumvented by imposing an appropriate number of restrictions The method gretl uses is known as the Phillips normalization The starting point is writing f in partitioned form as in Bi sfa Bo where f is an r xr matrix and B2 is n 1 x r Assuming that f has full rank can be post multiplied by Bis giving a I I B pep l Chapter 20 Time series models 147 The coefficients that gretl produces are B with 2 known as the matrix of unrestricted coefficients Chapter 21 Discrete and censored dependent variables 21 1 Logit and probit models It often happens that one wants to specify and estimate a model in which the dependent variable is not continuous but discrete A typical example is a model in which the dependent variable is the occupational status of an individual 1 employed 0 unemploy
249. s server as soon as it is saved by checking the relevant checkbox Clicking OK in this dialog leads you to a File Save dialog All being well this should be pointing towards a directory named functions either under the gretl system directory if you have write Chapter 10 User defined functions 65 permission on that or the gretl user directory This is the recommended place to save function package files since that is where the program will look in the special routine for opening such files see below Needless to say the menu command File Function files Edit package allows you to edit again a local function package A word on the file you just saved By default it will have a gfn extension This is a function package file unlike an ordinary gretl script file it is an XML file containing both the function code and the extra information entered in the packager Hackers might wish to write such a file from scratch rather than using the GUI packager but most people are likely to find it awkward Note that XML special characters in the function code have to be escaped e g amp must be represented as amp amp Also some elements of the function syntax differ from the standard script representation the parameters and return values if any are represented in XML Basically the function is pre parsed and ready for fast loading using libxml Load a package Why package functions in this way To see what s on o
250. s the square of the data frequency which of course yields 1600 for quarterly data The value can be adjusted using the set command with a parameter of hp_lambda For example set hp_lambda 1200 The Baxter and King filter This filter is accessed using the bkfi1t function which again takes the name of the variable to be processed as its single argument Consider the spectral representation of a time series yr TU V e dZ w TU Chapter 5 Special functions in genr 31 To extract the component of y that lies between the frequencies w and w one could apply a bandpass filter TU En F w e dZ w T where F w 1 for w lt w lt w and O elsewhere This would imply in the time domain applying to the series a filter with an infinite number of coefficients which is undesirable The Baxter and King bandpass filter applies to y a finite polynomial in the lag operator A L ct A L yt where A L is defined as k A L gt ail i k The coefficients a are chosen such that F w A e A e7 is the best approximation to F w for a given k Clearly the higher k the better the approximation is but since 2k observations have to be discarded a compromise is usually sought Moreover the filter has also other appealing theoretical properties among which the property that A 1 0 so a series with a single unit root is made stationary by application of the filter In practice the filter is normally
251. select Icon view from the View menu you should see the basic default set of icons these give you quick access to information on the data set if any correlation matrix Correlations and descriptive summary statistics Summary All of these are activated by double clicking the relevant icon The Data set icon is a little more complex double clicking opens up the data in the built in spreadsheet but you can also right click on the icon for a menu of other actions To add a model to the Icon view first estimate it using the Model menu Then pull down the File menu in the model window and select Save to session as icon or Save as icon and close Simply hitting the S key over the model window is a shortcut to the latter action To add a graph first create it under the View menu Graph specified vars or via one of gretl s other graph generating commands Click on the graph window to bring up the graph menu and select Save to session as icon Chapter 3 Modes of working 16 gretl current session gt D bb Data info Data set Notes Summary P Correlations Model table Graph page Session ye Model 1 Graph 1 Figure 3 2 Icon view one model and one graph have been added to the default icons Once a model or graph is added its icon will appear in the Icon view window Double clicking on the icon redisplays the object while right clicking brings up a menu which lets you displ
252. set is very large say several thousands of observations With such a dataset in memory the creation of a copy may lead to a situation where the computer runs low on memory for calculating regression results You can work around this as follows 1 Open the full data set and impose the sample restriction 2 Save a copy of the reduced data set to disk 3 Close the full dataset and open the reduced one 4 Proceed with your analysis 6 4 Random sampling With very large datasets or perhaps to study the properties of an estimator you may wish to draw arandom sample from the full dataset This can be done using for example smp1 100 random to select 100 cases If you want the sample to be reproducible you should set the seed for the random number generator first using set This sort of sampling falls under the restriction category a reduced copy of the dataset is made 6 5 The Sample menu items The discussion above has focused on the script command smp1 You can also use the items under the Sample menu in the GUI program to select a sub sample The menu items work in the same way as the corresponding smp1 variants When you use the item Sample Restrict based on criterion and the dataset is already sub sampled you are given the option of preserving or replacing the current restriction Replacing the current restriction means in effect invoking the replace option described above Section 6 3 Chapter 7 Gra
253. sible Two other useful resources for gretl users are the available documentation and the gretl users mailing list Flexible You can choose your preferred point on the spectrum from interactive point and click to batch processing and can easily combine these approaches Cross platform Gretl s home platform is Linux but it is also available for MS Windows and Mac OS X and should work on any unix like system that has the appropriate basic libraries see Appendix B Open source The full source code for gretl is available to anyone who wants to critique it patch it or extend it See Appendix B Sophisticated Gretl offers a full range of least squares based estimators either for single equations and for systems including vector autoregressions and vector error correction models Sev eral specific maximum likelihood estimators e g probit ARIMA GARCH are also provided natively more advanced estimation methods can be implemented by the user via generic maximum likelihood or nonlinear GMM Extendible Users can enhance gretl by writing their own functions and procedures in gretl s script ing language which includes a reasonably wide range of matrix functions Accurate Gretl has been thoroughly tested on several benchmarks among which the NIST refer ence datasets See Appendix C Internet ready Gretl can access and fetch databases from a server at Wake Forest University The MS Windows version comes with an updater program wh
254. st 11 12 Querying a list You can determine whether an unknown variable actually represents a list using the function islistQ 67 Chapter 11 Named lists and strings 68 series x11 log x1 series x12 log x2 list xlogs x11 x12 genr isl islist xlogs genr is2 islist xl1 The first genr command above will assign a value of 1 to isl since xlogs is in fact a named list The second genr will assign O to 152 since x11 is a data series not a list You can also determine the number of variables or elements in a list using the function nelem list xlist 1 2 3 genr nl nelem xlist The scalar n1 will be assigned a value of 3 since xlist contains 3 members You can display the membership of a named list as illustrated in this interactive session list xlist x1 x2 x3 Added list xlist list xlist print xlist x1 x2 x3 Note that print xlist will do something different namely print the values of all the variables in xlist as should be expected Generating lists of transformed variables Given a named list of variables you are able to generate lists of transformations of these variables using a special form of the commands logs lags diff 1diff sdiff or square In this context these keywords must be followed directly by a named list in parentheses For example list xlist x1 x2 x3 list Ixlist logs xlist list difflist diff xlist When generating a list of lags in this way you can specify the
255. t ing on its progress in reading the data If you encounter a problem with ill formatted data the messages in this wndow should give you a handle on fixing the problem For the convenience of anyone wanting to carry out more complex data analysis gretl has a facility for writing out data in the native formats of GNU R and GNU Octave see Appendix D In the GUI client this option is found under the File Export data menu in the command line client use the store command with the flag r R or m Octave 4 3 Binary databases For working with large amounts of data gretl is supplied with a database handling routine A database as opposed to a data file is not read directly into the program s workspace A database can contain series of mixed frequencies and sample ranges You open the database and select series to import into the working dataset You can then save those series in a native format data file if you wish Databases can be accessed via gretl s menu item File Databases For details on the format of gretl databases see Appendix A 1This is somewhat experimental See http ww ecn wfu edu eviews_format 18 Chapter 4 Data files 19 Online access to databases As of version 0 40 gretl is able to access databases via the internet Several databases are available from Wake Forest University Your computer must be connected to the internet for this option to work Please see the description of the data c
256. t always the case that there exists a set of values inside the admissible parameter space for which the likelihood is maximized The restrictions in question can be explained by reference to the simplest and much the most common instance of the GARCH model where p q 1 In the GARCH 1 1 model the conditional 2The algorithm is based on Fortran code deposited in the archive of the Journal of Applied Econometrics by the authors and is used by kind permission of Professor Fiorentini Chapter 20 Time series models 143 Table 20 2 Options for the GARCH covariance matrix command effect set garch_vcv hessian Use the Hessian set garch_vcv im Use the Information Matrix set garch_vcv op Use the Outer Product of the Gradient set garch_vcv qm QML estimator set garch_vcv bw Bollerslev Wooldridge sandwich estimator variance is Of Ap Qu 51071 20 13 Taking the unconditional expectation of 20 13 we get 0 a9 0107 610 so that 2_ Xo 1 OX 1 For this unconditional variance to exist we require that xx 6 lt 1 and for it to be positive we require that a gt 0 Oo A common reason for non convergence of GARCH estimates that is a common reason for the non existence of ax and 6 values that satisfy the above requirements and at the same time maximize the likelihood of the data is misspecification of the model It is important to realize that GARCH in itself allows only for time varying volati
257. t the VAR might still have an intercept because the difference between the two the interest rate spread might be stationary around a non zero mean for example because of a risk or liquidity premium The previous example can be generalized in three directions 1 If a VAR of order greater than 1 is considered the algebra gets more convoluted but the conclusions are identical 2 If the VAR includes more than two endogenous variables the cointegration rank r can be greater than 1 In this case is a matrix with r columns and the case with restricted constant entails the restriction that uy should be some linear combination of the columns of a 3 If a linear trend is included in the model the deterministic part of the VAR becomes uo p t The reasoning is practically the same as above except that the focus now centers on uy rather than uo The counterpart to the restricted constant case discussed above is a restricted trend case such that the cointegration relationships include a trend but the first differences of the variables in question do not In the case of an unrestricted trend the trend appears in both the cointegration relationships and the first differences which corresponds to the presence of a quadratic trend in the variables themselves in levels These two cases are specified by the option flags crt and ct respectively with the vecm command Identification of the cointegration vectors FIXME thi
258. ta files supplied with the package see Figure 2 1 Select the first tab Ramanathan The numbering of the files in this section corresponds to the chapter organization of Ramanathan 2002 which contains discussion of the analysis of these data The data will be useful for practice purposes even without the text gretl data files Ramanathan Greene Wooldridge Penn World Table 5 ata3 2 Incom ea ing Patents and R amp D expenditures Gross Income and Taxes by States Sealing compound shipment data Disposable income and consumption Toyota station wagon repairs Tuition and salary gain for MBAs Return on equity and assets Profits and sales 27 German companies Professors salaries at 7 universities Population of the United Kingdom Population of the USA Personal income and travel expenditures U S Population and GDP Prices of sinale familv hames Figure 2 1 Practice data files window If you select a row in this window and click on Info this opens a window showing information on the data set in question for example on the sources and definitions of the variables If you find a file that is of interest you may open it by clicking on Open or just double clicking on the file name For the moment let s open data3 6 t In gretl windows containing lists double clicking on a line launches a default action for the associated list entry e g displaying the values of a data series opening a file
259. tasks gnuplot for graphing BIEX for high quality typesetting of regression output GNU R If something goes wrong with such external links it is not always easy for gretl to produce an informative error message If such a link fails when accessed from the gretl graphical interface you may be able to get more information by starting gretl from the command prompt rather than via a desktop menu entry or icon On the X window system start gretl from the shell prompt in an xterm on MS Windows start the program gret1w32 exe from a console window or DOS box using the g or debug option flag Additional error messages may be displayed on the terminal window Also please note that for most external calls gretl assumes that the programs in question are available in your path that is that they can be invoked simply via the name of the program without supplying the program s full location Thus if a given program fails try the experiment of typing the program name at the command prompt as shown below Graphing Typesetting GNUR X window system gnuplot latex xdvi R MS Windows wgnuplot exe pdflatex RGui exe If the program fails to start from the prompt it s not a gretl issue but rather that the program s home directory is not in your path or the program is not installed properly For details on modifying your path please see the documentation or online help for your operating system or shell 1The exception to this rul
260. te 22 3 Fine tuning typeset output 2 ee eee ee ewe ee ee ee 22A Character EMOS sc aoaie ae Bee Ge he ee RR A OE ae A 22 5 Installing and learning TeX ooo soosi picoi a Ka Ea euk ee 23 Troubleshooting gretl 23 1 Bug reports 23 2 Auxiliary programs 24 The command line interface 24 1 Gretl at the console 115 116 118 120 133 133 139 140 143 148 148 151 155 156 156 156 158 161 161 162 162 162 163 Contents 24 2 CLI Syntax sce eda bee ee ee A IV Appendices A Data file details A l Basic native format A 2 Traditional ESLformat A 3 Binary database details B Building gretl El Requirements osea saa cn 0545 B 2 Build instructions a step by step example C Numerical accuracy D Related free software E Listing of URLs Bibliography 163 164 165 165 165 166 168 168 168 172 173 174 175 Chapter 1 Introduction 1 1 Features at a glance Gretl is an econometrics package including a shared library a command line client program and a graphical user interface User friendly Gretl offers an intuitive user interface it is very easy to get up and running with econometric analysis Thanks to its association with the econometrics textbooks by Ramu Ramanathan Jeffrey Wooldridge and James Stock and Mark Watson the package offers many practice data files and command scripts These are well annotated and acces
261. ted variable You can read the label for the variable using the label command with just one argument the name of the variable to check that you have retrieved the right value The following interactive session illustrates this point Chapter 5 Special functions in genr 36 adf 4 x1 c Augmented Dickey Fuller tests order 4 for x1 sample size 59 unit root null hypothesis a 1 test with constant model 1 L y bO a 1 y 1 e estimated value of a 1 0 216889 test statistic t 1 83491 asymptotic p value 0 3638 P values based on MacKinnon JAE 1996 genr pv pvalue Generated scalar pv CID 13 0 363844 label pv pv Dickey Fuller pvalue scalar 5 9 Numerical maximization Two special functions are available to aid in the construction of special purpose estimators namely BFGSmax the BFGS maximizer discussed in Chapter 17 and fdjac which produces a forward difference approximation to the Jacobian The BFGS maximizer The BFGSmax function takes two arguments a vector holding the initial values of a set of parame ters and a string specifying a call to a function that calculates the scalar criterion to be maximized given the current parameter values and any other relevant data If the object is in fact minimiza tion this function should return the negative of the criterion On successful completion BFGSmax returns the maximized value of the criterion and the matrix given via the first argument ho
262. terministic trend In this case formula 20 9 remains unchanged but the series e is defined as the residuals from an OLS regression of y on a constant and a linear trend This second form of the test is obtained by appending the trend option to the kpss command kpss n y trend Note that in this case the asymptotic distribution of the test is different and the critical values reported by gretl differ accordingly The Johansen tests Strictly speaking these are tests for cointegration However they can be used as multivariate unit root tests since they are the multivariate generalization of the ADF test See section 20 4 for more details 20 3 ARCH and GARCH Heteroskedasticity means a non constant variance of the error term in a regression model Autore gressive Conditional Heteroskedasticity ARCH is a phenomenon specific to time series models Chapter 20 Time series models 141 whereby the variance of the error displays autoregressive behavior for instance the time series ex hibits successive periods where the error variance is relatively large and successive periods where it is relatively small This sort of behavior is reckoned to be quite common in asset markets an unsettling piece of news can lead to a period of increased volatility in the market An ARCH error process of order q can be represented as q Ut Ot Et of E uz Q4 1 9 ou i 1 where the s are independently and identically distributed i
263. therefore useless for fine grained time series analysis outside of the special case where you know that the variable in question does in fact remain constant over the sub periods When the current data frequency is appropriate gretl offers both Compact data and Expand data options under the Data menu These options operate on the whole data set compacting or exanding all series They should be considered expert options and should be used with caution Panel data Panel data are inherently three dimensional the dimensions being variable cross sectional unit and time period For example a particular number in a panel data set might be identified as the observation on capital stock for General Motors in 1980 A note on terminology we use the terms cross sectional unit unit and group interchangeably below to refer to the entities that compose the cross sectional dimension of the panel These might for instance be firms countries or persons For representation in a textual computer file and also for gretl s internal calculations the three dimensions must somehow be flattened into two This flattening involves taking layers of the data that would naturally stack in a third dimension and stacking them in the vertical dimension Gretl always expects data to be arranged by observation that is such that each row represents an observation and each variable occupies one and only one
264. this case x is not trended and as a consequence neither is yt However the mean distance between y and x is non zero The vector uy is given by 5 which is not null and therefore the VECM shown above does have a constant term The constant however is subject to the restriction that its second element must be 0 More generally Uo is a multiple of the vector Note that the VECM could also be written as Yt 1 Ayt 1 Ut Et 1 Chapter 20 Time series models 146 which incorporates the intercept into the cointegration vector This is known as the restricted constant case it may be specified in gretl s vecm command using the option flag rc 3 m 0 and k 0 This case is the most restrictive clearly neither x nor y are trended and the mean distance between them is zero The vector uy is also O which explains why this case is referred to as no constant This case is specified using the option flag nc with vecm In most cases the choice between the three possibilities is based on a mix of empirical observation and economic reasoning If the variables under consideration seem to follow a linear trend then we should not place any restriction on the intercept Otherwise the question arises of whether it makes sense to specify a cointegration relationship which includes a non zero intercept One example where this is appropriate is the relationship between two interest rates generally these are not trended bu
265. tion in terms of the same control variable and the third part specifies an increment or decrement for the control variable to be applied each time round the loop The entire expression is enclosed in parentheses For example loop for r 0 01 r lt 991 r 01 In this example the variable r will take on the values 0 01 0 02 0 99 across the 99 iterations Note that due to the finite precision of floating point arithmetic on computers it may be necessary to use a continuation condition such as the above r lt 991 rather than the more natural r lt 99 Using double precision numbers on an x86 processor at the point where you would expect r to equal 0 99 it may in fact have value 0 990000000000001 To expand on the rules for the three components of the control expression 1 The initial condition must take the form LHS1 RHS1 RHS1 must be a numeric constant or a predefined variable If the LHS1 variable does not exist already it is automatically created 2 The continuation condition must be of the form LHS1 op RHS2 where op can be lt gt lt or gt and RHS2 is a numeric constant or a predefined variable If RHS2 is a variable it is evaluated each time round the loop 3 The increment or decrement expression must be of the form LHS1 DELTA or LHS1 DELTA where DELTA is a numeric constant or a predefined variable If DELTA is a variable it is evaluated only once when the loop is set up 9 3 Progressive
266. tion directory Quserdir user s gretl directory gnuplot path to or name of the gnuplot executable tramo path to or name of the tramo executable x12a path to or name of the x 12 arima executable tramodir tramo data directory x12adir x 12 arima data directory Table 11 1 Built in string variables Chapter 11 Named lists and strings 71 Reading strings from the environment In addition it is possible to read into gretl s named strings values that are defined in the external environment To do this you use the function getenv which takes the name of an environment variable as its argument For example string user getenv USER Saved string as user string home getenv HOME Saved string as home print user s home directory is Ahome cottrell s home directory is home cottrel1 To check whether you got a non empty value from a given call to getenv you can use the function isstring as in string temp getenv TEMP Saved empty string as temp scalar x isstring temp Generated scalar x ID 2 0 Note that isstring is really shorthand for is a string that actually contains something At present the getenv function can only be used on the right hand side of a string assignment as in the above illustrations Chapter 12 Matrix manipulation 12 1 Introduction Since version 1 5 1 gretl has offered the facility of creating and manipulating user defined matrices Th
267. to do so Print description Opens a window containing a full account of the current dataset in cluding the summary information and any specific information on each of the variables Add case markers Prompts for the name of a text file containing case markers short strings identifying the individual observations and adds this information to the data set See Chapter 4 Remove case markers Active only if the dataset has case markers identifying the obser vations removes these case markers Dataset structure invokes a series of dialog boxes which allow you to change the struc tural interpretation of the current dataset For example if data were read in as a cross section you can get the program to interpret them as time series or as a panel See also section 4 5 Compact data For time series data of higher than annual frequency gives you the option of compacting the data to a lower frequency using one of four compaction methods average sum start of period or end of period Expand data For time series data gives you the option of expanding the data to a higher frequency Transpose data Turn each observation into a variable and vice versa or in other words each row of the data matrix becomes a column in the modified data matrix can be useful with imported data that have been read in sideways e View menu Icon view Opens a window showing the content of the current session as a set of icon
268. tor relative to HC2 may be justified on the grounds that observations with large variances tend to exert a lot of influence on the OLS estimates so that the corresponding residuals tend to be under estimated See Davidson and MacKinnon for a fuller explanation The relative merits of these variants have been explored by means of both simulations and the oretical analysis Unfortunately there is not a clear consensus on which is best Davidson and MacKinnon argue that the original HCo is likely to perform worse than the others nonetheless White s standard errors are reported more often than the more sophisticated variants and there fore for reasons of comparability HCo is the default HCCME in gretl If you wish to use HC HC or HC you can arrange for this in either of two ways In script mode you can do for example set hc_version 2 In the GUI program you can go to the HCCME configuration dialog as noted above and choose any of these variants to be the default 14 3 Time series data and HAC covariance matrices Heteroskedasticity may be an issue with time series data too but it is unlikely to be the only or even the primary concern One form of heteroskedasticity is common in macroeconomic time series but is fairly easily dealt with That is in the case of strongly trending series such as Gross Domestic Product aggregate consumption aggregate investment and so on higher levels of the variable in question ar
269. tring and the closing string each of these strings to occur on lines by themselves Required list of white space separated names of the variables in the data file Names are limited to 8 characters must start with a letter and are limited to alphanumeric characters plus the underscore The list may continue over more than one line it is terminated with a semicolon Required observations line of the form 1 1 85 The first element gives the data fre quency 1 for undated or annual data 4 for quarterly 12 for monthly The second and third elements give the starting and ending observations Generally these will be 1 and the number of observations respectively for undated data For time series data one can use dates of the form 1959 1 quarterly one digit after the point or 1967 03 monthly two digits after the point See Chapter 15 for special use of this line in the case of panel data e The keyword BYOBS 165 Appendix A Data file details 166 Here is an example of a well formed data header file DATA9 6 Data on log money log income and interest rate from US Source Stock and Watson 1993 Econometrica Cunsmoothed data Period is 1900 1989 annual data Data compiled by Graham Elliott Tmoney lincome intrate 1 1900 1989 BYOBS The corresponding data file contains three columns of data each having 90 entries Three further features of the traditional data format may be noted 1 If the
270. troyed using the command free appended to the name of the object as in ADF1 free 3 3 The gretl console A further option is available for your computing convenience Under gretl s Tools menu you will find the item Gretl console there is also an open gretl console button on the toolbar in the main window This opens up a window in which you can type commands and execute them one by one by pressing the Enter key interactively This is essentially the same as gretlcli s mode of operation except that the GUI is updated based on commands executed from the console enabling you to work back and forth as you wish In the console you have command history that is you can use the up and down arrow keys to navigate the list of command you have entered to date You can retrieve edit and then re enter a previous command In console mode you can create display and free objects models graphs or text aa described above for script mode 3 4 The Session concept gretl offers the idea of a session as a way of keeping track of your work and revisiting it later The basic idea is to provide an iconic space containing various objects pertaining to your current working session see Figure 3 2 You can add objects represented by icons to this space as you go along If you save the session these added objects should be available again if you re open the session later If you start gretl and open a data set then
271. ts is at least equal to the number of required parameters and is no greater than the total number of parameters A scalar series or matrix argument to a function may be given either as the name of a pre existing variable or as an expression which evaluates to a variable of the appropriate type Scalar arguments may also be given as numerical values List arguments must be specified by name The function body is composed of gretl commands or calls to user defined functions that is functions may be nested A function may call itself that is functions may be recursive While the function body may contain function calls it may not contain function definitions That is you cannot define a function inside another function 10 2 Calling a function A user function is called or invoked by typing its name followed by zero or more arguments en closed in parentheses If there are two or more arguments these should be separated by commas The following trivial example illustrates a function call that correctly matches the function defini tion function definition function ols_ess series y list xvars ols y 0 xvars quiet scalar myess ess printf ESS g n myess return scalar myess end function main script open data4 1 list xlist 2 3 4 function call the return value is ignored here ols_ess price xlist Chapter 10 User defined functions 60 The function call gives two arguments the first is a data series specified by
272. ues where fairly small is defined as less than or equal to 8 If both conditions are satisfied the variable is automatically considered discrete To mark a variable as discrete you have two options 1 From the graphical interface select Variable Edit Attributes from the menu A dialog box will appear and if the variable seems suitable you will see a tick box labeled Treat this variable as discrete This dialog box can also be invoked via the context menu right click on a variable or by pressing the F2 key 2 From the command line interface via the discrete command The command takes one or more arguments which can be either variables or list of variables For example list xlist x1 x2 x3 discrete z1 xlist z2 This syntax makes it possible to declare as discrete many variables at once which cannot presently be done via the graphical interface The switch reverse reverses the declaration of a variable as discrete or in other words marks it as continuous For example discrete foo now foo is discrete discrete foo reverse now foo is continuous The command line variant is more powerful in that you can mark a variable as discrete even if it does not seem to be suitable for this treatment Note that marking a variable as discrete does not affect its content It is the user s responsibility to make sure that marking a variable as discrete is a sensible thing to do Note that if you want to recode a co
273. uick and convenient way to provide starting values for the MLE algorithm Example 17 1 shows how to implement the model described so far The banks91 file contains part of the data used in Lucchetti Papi and Zazzaro 2001 17 4 GARCH models GARCH models are handled by gretl via a native function However it is instructive to see how they can be estimated through the mle command The following equations provide the simplest example of a GARCH 1 1 model Yt UTE Ee Ur Ot ut N 0 1 he w ae Phi Since the variance of y depends on past values writing down the log likelihood function is not simply a matter of summing the log densities for individual observations As is common in time series models y cannot be considered independent of the other observations in our sample and consequently the density function for the whole sample the joint density for all observations is not just the product of the marginal densities Maximum likelihood estimation in these cases is achieved by considering conditional densities so what we maximize is a conditional likelihood function If we define the information set at time t as Fi Vt Yt 1 Chapter 17 Maximum likelihood estimation 117 Example 17 1 Estimation of stochastic frontier cost function open banks91 Cobb Douglas cost function ols cost const y pl p2 p3 Cobb Douglas cost function with homogeneity restrictions genr rcost cost p3 genr rpl pl
274. ust covariance matrix estimation 93 methodology is that while FGLS offers an efficiency advantage in principle it often involves making additional statistical assumptions which may or may not be justified which may not be easy to test rigorously and which may threaten the consistency of the estimator for example the common factor restriction that is implied by traditional FGLS corrections for autocorrelated errors James Stock and Mark Watson s Introduction to Econometrics illustrates this approach at the level of undergraduate instruction many of the datasets they use comprise thousands or tens of thousands of observations FGLS is downplayed and robust standard errors are reported as a matter of course In fact the discussion of the classical standard errors labeled homoskedasticity only is confined to an Appendix Against this background it may be useful to set out and discuss all the various options offered by gretl in respect of robust covariance matrix estimation The first point to notice is that gretl produces classical standard errors by default in all cases apart from GMM estimation In script mode you can get robust standard errors by appending the robust flag to estimation commands In the GUI program the model specification dialog usually contains a Robust standard errors check box along with a configure button that is activated when the box is checked The configure button takes you to
275. vert the data by importing them into the program from plain text CSV or a spreadsheet format MS Excel or Gnumeric then saving them You may wish to add descriptions of the individual variables the Variable Edit attributes menu item and add information on the source of the data the Data Edit info menu item 3 Write a descriptions file for the collection using a text editor 4 Put the datafiles plus the descriptions file in a subdirectory of the gretl data directory or user directory 5 If the collection is to be distributed to other people package the data files and catalog in some suitable manner e g as a zipfile If you assemble such a collection and the data are not proprietary I would encourage you to submit the collection for packaging as a gretl optional extra Chapter 5 Special functions in genr 5 1 Introduction The genr command provides a flexible means of defining new variables It is documented in the Gretl Command Reference This chapter offers a more expansive discussion of some of the special functions available via genr and some of the finer points of the command 5 2 Long run variance As is well known the variance of the average of T random variables x1 X2 Xr with equal vari ance g equals o2 T if the data are uncorrelated In this case the sample variance of x over the sample size provides a consistent estimator If however there is serial correlation among the xs the va
276. written as 6 50 log 27 logn gt log ssR 19 2 131 Chapter 19 Model selection criteria 132 Substituting 19 2 into 19 1 we get AIC n 1 log 27 logn nlogSSR 2k which can also be written as AIC n log 2k n 1 log 271 19 3 Some authors simplify the formula for the case of models estimated via least squares For instance William Greene writes 19 4 AIC log oh n n This variant can be derived from 19 3 by dividing through by n and subtracting the constant 1 log 27r That is writing AIC for the version given by Greene we have AICg Alc 1 log 271 Finally Ramanathan gives a further variant AICR em n which is the exponential of the one given by Greene Gretl began by using the Ramanathan variant but since version 1 3 1 the program has used the original Akaike formula 19 1 and more specifically 19 3 for models estimated via least squares Although the Akaike criterion is designed to favor parsimony arguably it does not go far enough in that direction For instance if we have two nested models with k 1 and k parameters respec tively and if the null hypothesis that parameter k equals O is true in large samples the AIC will nonetheless tend to select the less parsimonious model about 16 percent of the time see Davidson and MacKinnon 2004 chapter 15 An alternative to the AIC which avoids this problem is the Schwarz 1978 Ba
277. y the first column may contain strings such as dates or labels for cross sectional observations Such strings have a maximum of 8 characters as with variable names longer strings will be truncated A column of this sort should be headed with the string obs or date or the first row entry may be left blank For dates to be recognized as such the date strings must adhere to one or other of a set of specific formats as follows For annual data 4 digit years For quarterly data a 4 digit year followed by a separator either a period a colon or the letter Q followed by a 1 digit quarter Examples 1997 1 2002 3 1947Q1 For monthly data a 4 digit year followed by a period or a colon followed by a two digit month Examples 1997 01 2002 10 CSV files can use comma space or tab as the column separator When you use the Import CSV menu item you are prompted to specify the separator In the case of Import ASCII the program attempts to auto detect the separator that was used If you use a spreadsheet to prepare your data you are able to carry out various transformations of the raw data with ease adding things up taking percentages or whatever note however that you can also do this sort of thing easily perhaps more easily within gretl by using the tools under the Add menu Appending imported data You may wish to establish a gretl dataset piece by piece by incremental importation of data from other s
278. y Z5 loop foreach i foo smpl i restrict replace summary Y end loop smp1 full Since dummi fy generates a list it can be used directly in commands that call for a list as input such as ols For example open greene22_2 discrete Z5 mark Z5 as discrete ols Y O dummify Z5 The freq command The freq command displays absolute and relative frequencies for a given variable The way fre quencies are counted depends on whether the variable is continuous or discrete This command is also available via the graphical interface by selecting the Variable Frequency distribution menu entry Chapter 8 Discrete variables For discrete variables frequencies are counted for each distinct value that the variable takes For continuous variables values are grouped into bins and then the frequencies are counted for each bin The number of bins by default is computed as a function of the number of valid observations in the currently selected sample via the rule shown in Table 8 1 However when the command is invoked through the menu item Variable Frequency Plot this default can be overridden by the user For example the following code open greene19_1 freq TUCE Table 8 1 Number of bins for various sample sizes Observations Bins 8 lt n lt l6 5 16 lt n lt 50 7 50 lt n lt 850 vr n gt 850 29 discrete TUCE mark TUCE as discrete freq TUCE yields Read datafile usr local share gret1 data
279. y accessible to the function in a similar manner to the case where a scalar or series argument is passed in pointer form as discussed above Passing a list is therefore another means of allowing a function to do more in the way of modifying data at the level of the caller than simply offering a return value If the variables will not be modified inside the function it is a good idea to flag this fact using the const modifier in the listing of parameters function myfunc scalar y const list X When a list is marked const any attempt to rename delete or overwrite the original values of the variables in the list will generate an error If a list argument to a function is optional this should be indicated by appending a default value of null as in function myfunc scalar y list X nul1 In that case if the caller gives null as the list argument then the named list X inside the function will be empty This possibility can be detected using the nelem function which returns O for an empty list This mechanism can also be used to check whether a named but empty list was supplied as an argument Chapter 10 User defined functions 62 Return values Functions can return nothing just printing a result perhaps or they can return a single variable a scalar series list or matrix The return value if any is specified via a statement within the function body beginning with the keyword return followed by a type specifier and t
280. y the variable year Then the restriction smp1 year 1995 restrict produces a dataset that is not a panel but a cross section for the year 1995 Similarly smp1 firm 3 restrict produces a time series dataset for firm number 3 For these reasons possible non contiguity in the observations possible change in the structure of the data gretl acts differently when you restrict the sample as opposed to simply setting it In the case of setting the program merely records the starting and ending observations and uses these as parameters to the various commands calling for the estimation of models the computation of statistics and so on In the case of restriction the program makes a reduced copy of the dataset and by default treats this reduced copy as a simple undated cross section If you wish to re impose a time series or panel interpretation of the reduced dataset you can do so using the setobs command or the GUI menu item Data Dataset structure lWith one exception if you start with a balanced panel dataset and the restriction is such that it preserves a balanced panel for example it results in the deletion of all the observations for one cross sectional unit then the reduced dataset is still by default treated as a panel Chapter 6 Sub sampling a dataset 42 The fact that restricting the sample results in the creation of a reduced copy of the original dataset may raise an issue when the data
281. yesian information criterion BIC The BIC can be written in line with Akaike s formulation of the AIC as BIC 26 0 klogn The multiplication of k by logn in the BIC means that the penalty for adding extra parameters grows with the sample size This ensures that asymptotically one will not select a larger model over a correctly specified parsimonious model A further alternative to AIC which again tends to select more parsimonious models than AIC is the Hannan Quinn criterion or HOC Hannan and Quinn 1979 Written consistently with the formulations above this is HQC 24 0 2kloglogn The Hannan Quinn calculation is based on the law of the iterated logarithm note that the last term is the log of the log of the sample size The authors argue that their procedure provides a strongly consistent estimation procedure for the order of an autoregression and that compared to other strongly consistent procedures this procedure will underestimate the order to a lesser degree Gretl reports the AIC BIC and HOC calculated as explained above for most sorts of models The key point in interpreting these values is to know whether they are calculated such that smaller values are better or such that larger values are better In gretl smaller values are better one wants to minimize the chosen criterion Chapter 20 Time series models 20 1 ARIMA models Representation and syntax The arma command performs estimatio

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