EViews 4.0 User's Guide
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1. For example the same ARIMA 1 1 1 model can be estimated by the one line com mand ls d m1 c ar 1 ma 1 The latter method should generally be preferred for an important reason If you define a new variable such as DM1 above and use it in your estimation procedure then when you forecast from the estimated model EViews will make forecasts of the dependent variable DM1 That is you will get a forecast of the differenced series If you are really interested in forecasts of the level variable in this case M1 you will have to manually transform the Estimating ARIMA Models 307 forecasted value and adjust the computed standard errors accordingly Moreover if any other transformation or lags of M1 are included as regressors EViews will not know that they are related to DM1 If however you specify the model using the difference operator expression for the dependent variable d m1 the forecasting procedure will provide you with the option of forecasting the level variable in this case M1 The difference operator may also be used in specifying exogenous variables and can be used in equations without ARMA terms Simply include them in the list of regressors in addition to the endogenous variables For example d cs 2 c d gdp 2 d gdp 1 2 d gdp 2 2 time is a valid specification that employs the difference operator on both the left hand and right hand sides of the equation ARMA Terms The AR and MA parts of your model w2. or using lag operators 2 You can estimate this second order process by including both the MA 1 and MA 2 terms in your equation specification Estimating ARIMA Models 309 With quarterly data you might want to add sma 4 to take account of seasonality If you specify the equation as cs c ad ma 1 ma 2 sma 4 then the estimated model is CS bi BAD uy 5 4 13 25 Ut 1 Ea 0L 0L ya wL JEt The error process is equivalent to Up E OE1 HOE FWE 4 t WOE _5 WOE _ 13 26 The parameter wis associated with the seasonal part of the process This is just an MA 6 process with nonlinear restrictions on the coefficients You can also include both SAR and SMA terms Output from ARIMA Estimation The output from estimation with AR or MA specifications is the same as for ordinary least squares with the addition of a lower block that shows the reciprocal roots of the AR and MA polynomials If we write the general ARMA model using the lag polynomial p L and 6 L as p L u A L e 13 27 then the reported roots are the roots of the polynomials oa 0 and O a 0 13 28 The roots which may be imaginary should have modulus no greater than one The output will display a warning message if any of the roots violate this condition If p has a real root whose absolute value exceeds one or a pair of complex reciprocal roots outside the unit circle that is with modulus greater than one it means t
3. AR 1 specification EViews transforms the nonlinear model Y f t B u 13 11 Up pu 1 tE into the alternative nonlinear specification Yi PY 1 t fp P Pfi 9 amp 13 12 and estimates the coefficients using a Marquardt nonlinear least squares algorithm Higher order AR specifications are handled analogously For example a nonlinear AR 3 is estimated using nonlinear least squares on the equation Yt P1Yt 1 P2Yt 2 P3Yt 3 fp B Pfi B 13 13 paf 49 B P3 f i_3 b Et For details see Fair 1984 pp 210 214 and Davidson and MacKinnon 1996 pp 331 341 ARIMA Theory ARIMA autoregressive integrated moving average models are generalizations of the sim ple AR model that use three tools for modeling the serial correlation in the disturbance e The first tool is the autoregressive or AR term The AR 1 model introduced above uses only the first order term but in general you may use additional higher order AR terms Each AR term corresponds to the use of a lagged value of the residual in the forecasting equation for the unconditional residual An autoregressive model of order p AR p has the form Up Pyle PUat pPpUt pt Et 13 14 e The second tool is the integration order term Each integration order corresponds to differencing the series being forecast A first order integrated component means that the forecasting model is designed for the first difference of the original series A
4. assigns coefficients for the ARMA terms As with other estimation methods when you specify your equation as a list of variables EViews uses the built in C coefficient vector It assigns coefficient numbers to the variables in the following order e First are the coefficients of the variables in order of entry e Next come the AR terms in the order you typed them e The SAR MA and SMA coefficients follow in that order Thus the following two specifications will have their coefficients in the same order yc x ma 2 ma 1 sma 4 ar l1 y sma 4 c ar 1 ma 2 x ma 1 You may also assign values in the C vector using the param command param c 1 50 c 2 8 c 3 2 c 4 6 c 5 1 c 6 5 The starting values will be 50 for the constant 0 8 for X 0 2 for AR 1 0 6 for MA 2 0 1 for MA 1 and 0 5 for SMA 4 Following estimation you can always see the assignment of coefficients by looking at the Representations view of your equation You can also fill the C vector from any estimated equation without typing the numbers by choosing Procs Update Coefs from Equation in the equation toolbar Backcasting MA terms By default EViews backcasts MA terms Box and Jenkins 1976 Consider an MA q model of the form Y XB u 13 31 Up Ep Oye 1 HOE a o OE Given initial values B and Q EViews first computes the unconditional residuals for t 1 2 T and uses the backward recursion Ey b124417 Site Pat q 13 32
5. equation produces the following view Estimating AR Models 299 Equation EQ_CS Workfile UROOT OY x View Procs Objects Print Name Freeze Estimate Forecast Stats Resids Correlogram of Residuals Date 08 19 97 Time 13 20 Sample 1948 3 1988 4 Included observations 162 Autocorrelation Partial Correlation AC PAC Stat Prob 1 0163 0163 43653 0 037 2 0 202 0 180 11 134 0 004 3 0 2142 0 1665 18 631 0 000 4 0 040 0 044 18 904 0 001 5 0 018 0 092 18 956 0 002 6 0 001 0 027 18 956 0 004 7 0 040 0 017 19 226 0 008 8 0 211 0 194 26 873 0 001 9 0 063 0 005 27 563 0 001 10 0 024 0 072 27 667 0 002 11 0 050 0 039 28 111 0 003 0 004 The correlogram has spikes at lags up to three and at lag eight The Q statistics are signifi cant at all lags indicating significant serial correlation in the residuals Selecting View Residual Tests Serial Correlation LM Test and entering a lag of 4 yields the following result Breusch Godfrey Serial Correlation LM Test F statistic 3 654696 Probability 0 007109 Obs R squared 13 96215 Probability 0 007417 The test rejects the hypothesis of no serial correlation up to order four The Q statistic and the LM test both indicate that the residuals are serially correlated and the equation should be re specified before using it for hypothesis tests and forecasting Estimating AR Models Before you use the tools described in this
6. section you may first wish to examine your model for other signs of misspecification Serial correlation in the errors may be evidence of serious problems with your specification In particular you should be on guard for an excessively restrictive specification that you arrived at by experimenting with ordinary least squares Sometimes adding improperly excluded variables to your regression will eliminate the serial correlation For a discussion of the efficiency gains from the serial correlation correction and some Monte Carlo evidence see Rao and Griliches 1969 300 Chapter 13 Time Series Regression First Order Serial Correlation To estimate an AR 1 model in EViews open an equation by selecting Quick Estimate Equation and enter your specification as usual adding the expression AR 1 to the end of your list For example to estimate a simple consumption function with AR 1 errors CS Cy coG DP Ut Up Putt amp 13 5 you should specify your equation as cs c gdp ar 1 EViews automatically adjusts your sample to account for the lagged data used in estima tion estimates the model and reports the adjusted sample along with the remainder of the estimation output Higher Order Serial Correlation Estimating higher order AR models is only slightly more complicated To estimate an AR amp you should enter your specification followed by expressions for each AR term you wish to include If you wish
7. to estimate a model with autocorrelations from one to five CS cy oG DP uy 13 6 Up P1Ut 1 F Polya t P5le_5 Et you should enter cs c gdp ar 1 ar 2 ar 3 ar 4 ar 5 By requiring that you enter all of the autocorrelations you wish to include in your model EViews allows you great flexibility in restricting lower order correlations to be zero For example if you have quarterly data and want to include a single term to account for sea sonal autocorrelation you could enter cs c gdp ar 4 Nonlinear Models with Serial Correlation EViews can estimate nonlinear regression models with additive AR errors For example suppose you wish to estimate the following nonlinear specification with an AR 2 error CS GDP u Led A 13 7 My C3Ug_1 F Catena tE Estimating AR Models 301 Simply specify your model using EViews expressions followed by an additive term describing the AR correction enclosed in square brackets The AR term should contain a coefficient assignment for each AR lag separated by commas cs c 1 gdp c 2 ar 1 c 3 ar 2 c 4 EViews transforms this nonlinear model by differencing and estimates the transformed nonlinear specification using a Gauss Newton iterative procedure see How EViews Esti mates AR Models on page 302 Two Stage Regression Models with Serial Correlation By combining two stage least squares or two stage nonlinear least squares with AR terms you can e
8. 0 028048 0 413540 0 067825 0 9460 M1 10 0 030806 0 407523 0 075593 0 9398 M1 11 0 018509 0 389133 0 047564 0 9621 M1 12 0 057373 0 228826 0 250728 0 8022 R squared 0 852398 Mean dependent var 71 72679 Adjusted R squared 0 846852 S D dependent var 19 53063 S E of regression 7 643137 Akaike info criterion 6 943606 Sum squared resid 20212 47 Schwarz criterion 7 094732 Log likelihood 1235 849 F statistic 153 7030 Durbin Watson stat 0 008255 __ Prob F statistic 0 000000 Taken individually none of the coefficients on lagged M1 are statistically different from zero Yet the regression as a whole has a reasonable R with a very significant F statistic though with a very low Durbin Watson statistic This is a typical symptom of high col linearity among the regressors and suggests fitting a polynomial distributed lag model To estimate a fifth degree polynomial distributed lag model with no constraints enter the commands smpl 59 1 89 12 ls ip c pdl m1 12 5 The following result is reported at the top of the equation window Polynomial Distributed Lags PDLs 319 Dependent Variable IP Method Least Squares Date 08 15 97 Time 17 53 Sample adjusted 1960 01 1989 12 Included observations 360 after adjusting endpoints Variable Coefficient Std Error t Statistic Prob C 40 67311 0 815195 49 89374 0 0000 PDLO1 4 66E 05 0 055566 0 000839 0 9993 PDLO2 0 015625 0 062884 0 248479 0 8039 PDLO3 0 000160 0 013909 0
9. 011485 0 9908 PDL04 0 001862 0 007700 0 241788 0 8091 PDLO5 2 58E 05 0 000408 0 063211 0 9496 PDLO6 4 93E 05 0 000180 0 273611 0 7845 R squared 0 852371 Mean dependent var 71 72679 Adjusted R squared 0 849862 S D dependent var 19 53063 S E of regression 7 567664 Akaike info criterion 6 904899 Sum squared resid 20216 15 Schwarz criterion 6 980462 Log likelihood 1235 882 F statistic 339 6882 Durbin Watson stat 0 008026 __ Prob F statistic 0 000000 This portion of the view reports the estimated coefficients y of the polynomial in Equation 13 38 on page 315 The terms PDLO1 PDLO2 PDLO3 correspond to 24 o in Equation 13 40 The implied coefficients of interest 8 f in equation 1 are reported at the bottom of the table together with a plot of the estimated polynomial Lag Distribution of M1 i Coefficient Std Error T Statistic 0 0 10270 0 14677 0 69970 1 0 01159 0 10948 0 10587 2 0 00215 0 10139 0 02123 3 0 00920 0 06150 0 14955 4 0 01766 0 07435 0 23756 5 0 01363 0 06974 0 19547 6 4 7E 05 0 05557 0 00084 7 0 01399 0 07080 0 19764 8 0 01821 0 07537 0 24158 9 0 00798 0 06399 0 12475 1 10 0 01017 0 10454 0 09726 11 0 01260 0 11069 0 11386 1 12 0 04737 0 15693 0 30182 Sum of Lags 0 08780 0 00297 29 5534 F v The Sum of Lags reported at the bottom of the table is the sum of the estimated coefficients on the distributed lag and has the interpretation of the long run effect of M1
10. 040236 12 76402 0 0000 SMA 4 0 960399 0 000601 1598 423 0 0000 R squared 0 991557 Mean dependent var 6 978830 Adjusted R squared 0 991484 S D dependent var 2 919607 S E of regression 0 269420 Akaike info criterion 0 225489 Sum squared resid 33 75296 Schwarz criterion 0 269667 Log likelihood 47 98992 F statistic 13652 76 Durbin Watson stat 2 099958 __ Prob F statistic 0 000000 Inverted AR Roots 09 98 Inverted MA Roots 99 00 99i 00 99i 51 99 This estimation result corresponds to the following specification y 8 61 u 13 29 1 0 98L 1 0 94L u 1 0 51L 1 0 96L Je or equivalently to Y 0 0088 0 98y _ 1 0 94y _ 4 0 92y _5 E 13 30 0 5le _ 0 96e _4 0 49e _5 Note that the signs of the MA terms may be reversed from those in textbooks Note also that the inverted roots have moduli very close to one which is typical for many macro time series models Estimation Options ARMA estimation employs the same nonlinear estimation techniques described earlier for AR estimation These nonlinear estimation techniques are discussed further in Chapter 12 Additional Regression Methods on page 283 Estimating ARIMA Models 311 You may need to use the Estimation Options dialog box to control the iterative process EViews provides a number of options that allow you to control the iterative procedure of the estimation algorithm In general you can rely on the EViews choices but on occ
11. 155 484 79 0 000 0 198 0 119 497 40 0 000 0 184 0 078 508 30 0 000 0 193 0 114 520 29 0 000 0 286 0 380 546 80 0 000 10 0 436 0 225 608 54 0 000 11 0 686 0 141 720 38 0 000 12 0 648 0 115 857 70 0 000 13 0 529 0 476 949 28 0 000 14 0 324 0 256 983 78 0 000 15 0 110 0 109 987 76 0 000 16 0 046 0 062 988 45 0 000 17 0 111 0 121 992 52 0 000 18 0 164 0 101 1001 5 0 000 19 0 188 0 002 1013 3 0 000 0 000 O0 A WN e After fitting a candidate ARIMA specification you should verify that there are no remaining autocorrelations that your model has not accounted for Examine the autocorrelations and the partial autocorrelations of the innovations the residuals from the ARIMA model to see if any important forecast ing power has been overlooked EViews provides views for diagnostic checks after estima tion Estimating ARIMA Models EViews estimates general ARIMA specifications that allow for right hand side explanatory variables Despite the fact that these models are sometimes termed ARIMAX specifications we will refer to this general class of models as ARIMA To specify your ARIMA model you will e Difference your dependent variable if necessary to account for the order of integra tion e Describe your structural regression model dependent variables and regressors and add any AR or MA terms as described above 306 Chapter 13 Time Series Regression Differencing The d operator can be used to specif
12. Chapter 13 Time Series Regression In this section we discuss single equation regression techniques that are important for the analysis of time series data testing for serial correlation estimation of ARMA models using polynomial distributed lags and testing for unit roots in potentially nonstationary time series The chapter focuses on the specification and estimation of time series models A number of related topics are discussed elsewhere standard multiple regression techniques are dis cussed in Chapters 11 and 12 forecasting and inference are discussed extensively in Chapters 14 and 15 vector autoregressions are discussed in Chapter 20 and state space models and the Kalman filter are discussed in Chapter 22 Serial Correlation Theory A common finding in time series regressions is that the residuals are correlated with their own lagged values This serial correlation violates the standard assumption of regression theory that disturbances are not correlated with other disturbances The primary problems associated with serial correlation are e OLS is no longer efficient among linear estimators Furthermore since prior residu als help to predict current residuals we can take advantage of this information to form a better prediction of the dependent variable e Standard errors computed using the textbook OLS formula are not correct and are generally understated e If there are lagged dependent variables on the right hand side OL
13. S estimates are biased and inconsistent EViews provides tools for detecting serial correlation and estimation methods that take account of its presence In general we will be concerned with specifications of the form Ye 2 B U 13 1 Up 2 1 VT EY where x is a vector of explanatory variables observed at time t z _ is a vector of vari ables known in the previous period and y are vectors of parameters u is a disturbance term and is the innovation in the disturbance The vector z _ may contain lagged values of u lagged values of or both 296 Chapter 13 Time Series Regression The disturbance u is termed the unconditional residual It is the residual based on the structural component 2 but not using the information contained in z _ The innova tion is also known as the one period ahead forecast error or the prediction error It is the difference between the actual value of the dependent variable and a forecast made on the basis of the independent variables and the past forecast errors The First Order Autoregressive Model The simplest and most widely used model of serial correlation is the first order autoregres sive or AR 1 model The AR 1 model is specified as Y B uy Up PUy_1 E 13 2 The parameter p is the first order serial correlation coefficient In effect the AR 1 model incorporates the residual from the past observation into the regression model for the cur rent
14. asion you may wish to override the default settings Iteration Limits and Convergence Criterion Controlling the maximum number of iterations and convergence criterion are described in detail in Iteration and Convergence Options on page 651 Derivative Methods EViews always computes the derivatives of AR coefficients analytically and the derivatives of the MA coefficients using finite difference numeric derivative methods For other coeffi cients in the model EViews provides you with the option of computing analytic expres sions for derivatives of the regression equation if possible or computing finite difference numeric derivatives in cases where the derivative is not constant Furthermore you can choose whether to favor speed of computation fewer function evaluations or whether to favor accuracy more function evaluations in the numeric derivative computation Starting Values for ARMA Estimation As discussed above models with AR or MA terms are estimated by nonlinear least squares Nonlinear estimation techniques require starting values for all coefficient estimates Nor mally EViews determines its own starting values and for the most part this is an issue that you need not be concerned about However there are a few times when you may want to override the default starting values First estimation will sometimes halt when the maximum number of iterations is reached despite the fact that convergence is not achieved Resum
15. at all lags should be nearly zero and all Q statistics should be insignificant with large p values Note that the p values of the Q statistics will be computed with the degrees of freedom adjusted for the inclusion of ARMA terms in your regression There is evidence that some care should be taken in interpreting the results of a Ljung Box test applied to the residuals from an ARMAX specification see Dezhbaksh 1990 for simulation evidence on the finite sample performance of the test in this setting Details on the computation of correlograms and Q statistics are provided in greater detail in Chapter 7 Series on page 167 Serial Correlation LM Test Selecting View Residual Tests Serial Correlation LM Test carries out the Breusch God frey Lagrange multiplier test for general high order ARMA errors In the Lag Specification dialog box you should enter the highest order of serial correlation to be tested The null hypothesis of the test is that there is no serial correlation in the residuals up to the specified order EViews reports a statistic labeled F statistic and an Obs R squared 298 Chapter 13 Time Series Regression 2 re 2 te NR the number of observations times the R square statistic The N R statistic has an asymptotic X distribution under the null hypothesis The distribution of the F statistic is not known but is often used to conduct an informal test of the null See Serial Correlatio
16. autocorrelation properties You can use the correlogram view of a series for this purpose as outlined in Correlogram on page 165 This phase of the ARIMA modeling procedure is called identification not to be confused with the same term used in the simultaneous equations literature The nature of the corre lation between current values of residuals and their past values provides guidance in selecting an ARIMA specification The autocorrelations are easy to interpret each one is the correlation coefficient of the current value of the series with the series lagged a certain number of periods The partial autocorrelations are a bit more complicated they measure the correlation of the current and lagged series after taking into account the predictive power of all the values of the series with smaller lags The partial autocorrelation for lag 6 for example measures the added predictive power of u _g when u4 u _5 are already in the prediction model In fact the partial autocorrelation is precisely the regression coefficient of u _ ina regression where the earlier lags are also used as predictors of u If you suspect that there is a distributed lag relationship between your dependent left hand variable and some other predictor you may want to look at their cross correlations before carrying out estimation The next step is to decide what kind of ARIMA model to use If the autocorrelation func tion dies off smoothly at a geome
17. e is a pre specified constant given by 7 k 2 if p is even eae k 1 2 if p is odd The PDL is sometimes referred to as an Almon lag The constant is included only to avoid numerical problems that can arise from collinearity and does not affect the estimates of b 316 Chapter 13 Time Series Regression This specification allows you to estimate a model with amp lags of x using only p parameters if you choose p gt k EViews will return a Near Singular Matrix error If you specify a PDL EViews substitutes Equation 13 38 into Equation 13 37 yielding Y A V121 E V222 Vp 412 p41 t Et 13 40 where Zy Lit Li 1t H Tik Zo X 1 C a _ k 0 ti k 13 41 z p1 E r 1 2 Pa _y t K 28 ay Once we estimate y from Equation 13 40 we can recover the parameters of interest 3 and their standard errors using the relationship described in Equation 13 38 This proce dure is straightforward since is a linear transformation of y The specification of a polynomial distributed lag has three elements the length of the lag k the degree of the polynomial the highest power in the polynomial p and the con straints that you want to apply A near end constraint restricts the one period lead effect of x on y to be zero By M M 1 O 41 1 0 0 13 42 A far end constraint restricts the effect of x on y to die off beyond the number of specified lags Beer NE
18. econd set of residuals are the estimated one period ahead forecast errors As the name suggests these residuals represent the forecast errors you would make if you com puted forecasts using a prediction of the residuals based upon past values of your data in addition to the contemporaneous information In essence you improve upon the uncondi 302 Chapter 13 Time Series Regression tional forecasts and residuals by taking advantage of the predictive power of the lagged residuals nE 2 For AR models the residual based regression statistics such as the R the standard error of regression and the Durbin Watson statistic reported by EViews are based on the one period ahead forecast errors A set of statistics that is unique to AR models is the estimated AR parameters For the simple AR 1 model the estimated parameter is the serial correlation coefficient of the unconditional residuals For a stationary AR 1 model the true p lies between 1 extreme negative serial correlation and 1 extreme positive serial correlation The stationarity condition for general AR p processes is that the inverted roots of the lag polynomial lie inside the unit circle EViews reports these roots as Inverted AR Roots at the bottom of the regression output There is no particular problem if the roots are imaginary but a station ary AR model should have all roots with modulus less than one How EViews Estimates AR Models Textbooks
19. esidual correlogram looks as follows Polynomial Distributed Lags PDLs 315 E Equation EQ1 Workfile ARMA1 of x View Procs Objects Print Name Freeze Estimate Forecast Stats Resids Correlogram of Residuals Date 08 14 97 Time 18 30 Sample 1954 06 1993 07 Included observations 470 Q statistic probabilities adjusted for 4 ARMA term s Autocorrelation Partial Correlation AC PAC Stat Prob The correlogram has a significant spike at lag 5 and all subsequent Q statistics are highly significant This result clearly indicates the need for respecification of the model Polynomial Distributed Lags PDLs A distributed lag is a relation of the type Y w Box Byxp_y Bete pt Et 13 37 The coefficients 8 describe the lag in the effect of x on y In many cases the coefficients can be estimated directly using this specification In other cases the high collinearity of current and lagged values of x will defeat direct estimation You can reduce the number of parameters to be estimated by using polynomial distributed lags PDLs to impose a smoothness condition on the lag coefficients Smoothness is expressed as requiring that the coefficients lie on a polynomial of relatively low degree A polynomial distributed lag model with order p restricts the 8 coefficients to lie on a p th order polynomial of the form Bee so 2 MOREE Bi Were E y Fx y G20 13 38 for j 1 2 k wher
20. f ference stationary series since the first difference of y is stationary Y Y i 1 L y 13 45 A difference stationary series is said to be integrated and is denoted as I d where d is the order of integration The order of integration is the number of unit roots contained in the series or the number of differencing operations it takes to make the series stationary For the random walk above there is one unit root so it is an I 1 series Similarly a stationary series is I 0 Standard inference procedures do not apply to regressions which contain an integrated dependent variable or integrated regressors Therefore it is important to check whether a series is stationary or not before using it in a regression The formal method to test the sta tionarity of a series is the unit root test
21. f thumb with 50 or more observations and only a few independent variables a DW statistic below about 1 5 is a strong indication of positive first order serial correlation See Johnston and DiNardo 1997 Chapter 6 6 1 for a thorough discussion on the Durbin Watson test and a table of the significance points of the statistic There are three main limitations of the DW test as a test for serial correlation First the dis tribution of the DW statistic under the null hypothesis depends on the data matrix x The usual approach to handling this problem is to place bounds on the critical region creating a region where the test results are inconclusive Second if there are lagged dependent vari ables on the right hand side of the regression the DW test is no longer valid Lastly you may only test the null hypothesis of no serial correlation against the alternative hypothesis of first order serial correlation Two other tests of serial correlation the Q statistic and the Breusch Godfrey LM test overcome these limitations and are preferred in most applications Correlograms and Q statistics If you select View Residual Tests Correlogram Q statistics on the equation toolbar EViews will display the autocorrelation and partial autocorrelation functions of the residu als together with the Ljung Box Q statistics for high order serial correlation If there is no serial correlation in the residuals the autocorrelations and partial autocorrelations
22. f REV where the coefficients of REV lie on a 4th degree polynomial with a constraint at the far end The pd1 specification may also be used in two stage least squares If the series in the pdl is exogenous you should include the PDL of the series in the instruments as well For this purpose you may specify pdl as an instrument all pd1 variables will be used as instruments For example if you specify the TSLS equation as sales c inc pdl orders 1 12 4 with instruments fed fed 1 pdl the distributed lag of ORDERS will be used as instruments together with FED and FED 1 Polynomial distributed lags cannot be used in nonlinear specifications Example The distributed lag model of industrial production IP on money M1 yields the following results 318 Chapter 13 Time Series Regression Dependent Variable IP Method Least Squares Date 08 15 97 Time 17 09 Sample adjusted 1960 01 1989 12 Included observations 360 after adjusting endpoints Variable Coefficient Std Error t Statistic Prob Cc 40 67568 0 823866 49 37171 0 0000 M1 0 129699 0 214574 0 604449 0 5459 M1 1 0 045962 0 376907 0 121944 0 9030 M1 2 0 033183 0 397099 0 083563 0 9335 M1 3 0 010621 0 405861 0 026169 0 9791 M1 4 0 031425 0 418805 0 075035 0 9402 M1 5 0 048847 0 431728 0 113143 0 9100 M1 6 0 053880 0 440753 0 122245 0 9028 M1 7 0 015240 0 436123 0 034944 0 9721 M1 8 0 024902 0 423546 0 058795 0 9531 M1 9
23. fications with ARMA errors see Two stage Least Squares on page 275 Nonlinear Models with ARMA errors EViews will estimate nonlinear ordinary and two stage least squares models with autore gressive error terms For details see the extended discussion in Nonlinear Least Squares beginning on page 282 EViews does not currently estimate nonlinear models with MA errors You can however use the state space object to specify and estimate these models see ARMAX 2 3 with a Random Coefficient on page 568 Weighted Models with ARMA errors EViews does not have procedures to automatically estimate weighted models with ARMA error terms if you add AR terms to a weighted model the weighting series will be ignored You can of course always construct the weighted series and then perform estima tion using the weighted data and ARMA terms Diagnostic Evaluation If your ARMA model is correctly specified the residuals from the model should be nearly white noise This means that there should be no serial correlation left in the residuals The Durbin Watson statistic reported in the regression output is a test for AR 1 in the absence of lagged dependent variables on the right hand side As discussed above more general tests for serial correlation in the residuals can be carried out with View Residual Tests Correlogram Q statistic and View Residual Tests Serial Correlation LM Test For the example seasonal ARMA model the r
24. hat the autore gressive process is explosive If 0 has reciprocal roots outside the unit circle we say that the MA process is noninvertible which makes interpreting and using the MA results difficult However noninvertibility poses no substantive problem since as Hamilton 1994 p 65 notes there is always an equivalent representation for the MA model where the reciprocal roots lie inside the unit circle Accordingly you should re estimate your model with different starting values until you get a moving average process that satisfies invertibility Alternatively you may wish to turn off MA backcasting see Backcasting MA terms on page 312 310 Chapter 13 Time Series Regression If the estimated MA process has roots with modulus close to one it is a sign that you may have over differenced the data The process will be difficult to estimate and even more dif ficult to forecast If possible you should re estimate with one less round of differencing Consider the following example output from ARMA estimation Dependent Variable R Method Least Squares Date 08 14 97 Time 16 53 Sample adjusted 1954 06 1993 07 Included observations 470 after adjusting endpoints Convergence achieved after 25 iterations Backcast 1954 01 1954 05 Variable Coefficient Std Error t Statistic Prob C 8 614804 0 961559 8 959208 0 0000 AR 1 0 983011 0 009127 107 7077 0 0000 SAR 4 0 941898 0 018788 50 13275 0 0000 MA 1 0 513572 0
25. ill be specified using the keywords ar and ma as part of the equation We have already seen examples of this approach in our specification of the AR terms above and the concepts carry over directly to MA terms For example to estimate a second order autoregressive and first order moving average error process ARMA 2 1 you would include expressions for the AR 1 AR 2 and MA 1 terms along with your other regressors c gov ar 1 ar 2 ma 1 Once again you need not use the AR and MA terms consecutively For example if you want to fit a fourth order autoregressive model to take account of seasonal movements you could use AR 4 by itself c gov ar 4 You may also specify a pure moving average model by using only MA terms Thus c gov ma 1 ma 2 indicates an MA 2 model for the residuals The traditional Box Jenkins or ARMA models do not have any right hand side variables except for the constant In this case your list of regressors would just contain a C in addi tion to the AR and MA terms For example c ar 1 ar 2 ma 1 ma 2 is a standard Box Jenkins ARMA 2 2 308 Chapter 13 Time Series Regression Seasonal ARMA Terms Box and Jenkins 1976 recommend the use of seasonal autoregressive SAR and seasonal moving average SMA terms for monthly or quarterly data with systematic seasonal movements A SAR p term can be included in your equation specification for a seasonal autoregressive term with lag p The lag poly
26. ing the estimation with starting values from the previous step causes estimation to pick up where it left off instead of start ing over You may also want to try different starting values to ensure that the estimates are a global rather than a local minimum of the squared errors You might also want to supply starting values if you have a good idea of what the answers should be and want to speed up the estimation process To control the starting values for ARMA estimation click on the Options button in the Equation Specification dialog Among the options which EViews provides are several alter natives for setting starting values that you can see by accessing the drop down menu labeled Starting Coefficient Values for ARMA EViews default approach is OLS TSLS which runs a preliminary estimation without the ARMA terms and then starts nonlinear estimation from those values An alternative is to use fractions of the OLS or TSLS coefficients as starting values You can choose 8 5 3 or you can start with all coefficient values set equal to zero 312 Chapter 13 Time Series Regression The final starting value option is User Supplied Under this option EViews uses the coeffi cient values that are in the coefficient vector To set the starting values open a window for the coefficient vector C by double clicking on the icon and editing the values To properly set starting values you will need a little more information about how EViews
27. k 4 y 0 13 43 If you restrict either the near or far end of the lag the number of y parameters estimated is reduced by one to account for the restriction if you restrict both the near and far end of the lag the number of y parameters is reduced by two By default EViews does not impose constraints How to Estimate Models Containing PDLs You specify a polynomial distributed lag by the pd1 term with the following information in parentheses each separated by a comma in this order e The name of the series e The lag length the number of lagged values of the series to be included e The degree of the polynomial e A numerical code to constrain the lag polynomial optional Polynomial Distributed Lags PDLs 317 1 constrain the near end of the lag to zero 2 constrain the far end 3 constrain both ends You may omit the constraint code if you do not want to constrain the lag polynomial Any number of pd1 terms may be included in an equation Each one tells EViews to fit distrib uted lag coefficients to the series and to constrain the coefficients to lie on a polynomial For example ls sales c pdl orders 8 3 fits SALES to a constant and a distributed lag of current and eight lags of ORDERS where the lag coefficients of ORDERS lie on a third degree polynomial with no endpoint con straints Similarly ls div c pdl rev 12 4 2 fits DIV to a distributed lag of current and 12 lags o
28. n LM Test on page 297 for further discussion of the serial correlation LM test Example As an example of the application of these testing procedures consider the following results from estimating a simple consumption function by ordinary least squares Dependent Variable CS Method Least Squares Date 08 19 97 Time 13 03 Sample 1948 3 1988 4 Included observations 162 Variable Coefficient Std Error t Statistic Prob Cc 9 227624 5 898177 1 564487 0 1197 GDP 0 038732 0 017205 2 251193 0 0257 CS 1 0 952049 0 024484 38 88516 0 0000 R squared 0 999625 Mean dependent var 1781 675 Adjusted R squared 0 999621 S D dependent var 694 5419 S E of regression 13 53003 Akaike info criterion 8 066045 Sum squared resid 29106 82 Schwarz criterion 8 123223 Log likelihood 650 3497 F statistic 212047 1 Durbin Watson stat 1 672255 _ Prob F statistic 0 000000 A quick glance at the results reveals that the coefficients are statistically significant and the fit is very tight However if the error term is serially correlated the estimated OLS standard errors are invalid and the estimated coefficients will be biased and inconsistent due to the presence of a lagged dependent variable on the right hand side The Durbin Watson statis tic is not appropriate as a test for serial correlation in this case since there is a lagged dependent variable on the right hand side of the equation Selecting View Residual Tests Correlogram Q statistics from this
29. nomial used in estimation is the product of the one specified by the AR terms and the one specified by the SAR terms The purpose of the SAR is to allow you to form the product of lag polynomials Similarly SMA q can be included in your specification to specify a seasonal moving aver age term with lag q The lag polynomial used in estimation is the product of the one defined by the MA terms and the one specified by the SMA terms As with the SAR the SMA term allows you to build up a polynomial that is the product of underlying lag poly nomials For example a second order AR process without seasonality is given by Up P141 Pozo Ez 13 19 which can be represented using the lag operator L Lx 2 _ as 2 1 pyL pol Ju 13 20 You can estimate this process by including ar 1 and ar 2 terms in the list of regres sors With quarterly data you might want to add a sar 4 expression to take account of seasonality If you specify the equation as sales c inc ar 1 ar 2 sar 4 then the estimated error structure would be 1 p L pb L u c 13 21 The error process is equivalent to Up PyUy 1 t Pgty g OUt 4 OP1Uy 5 PPU et E 13 22 The parameter f is associated with the seasonal part of the process Note that this is an AR 6 process with nonlinear restrictions on the coefficients As another example a second order MA process without seasonality may be written Up E Oye 1 O0 amp _ 2 13 23
30. nsider that are specific to estimation of ARMA models First MA models are notoriously difficult to estimate In particular you should avoid high order MA terms unless absolutely required for your model as they are likely to cause esti mation difficulties For example a single large spike at lag 57 in the correlogram does not necessarily require you to include an MA 57 term in your model unless you know there is something special happening every 57 periods It is more likely that the spike in the corre logram is simply the product of one or more outliers in the series By including many MA terms in your model you lose degrees of freedom and may sacrifice stability and reliabil ity of your estimates If the underlying roots of the MA process have modulus close to one you may encounter estimation difficulties with EViews reporting that it cannot improve the sum of squares or that it failed to converge in the maximum number of iterations This behavior may be a sign that you have over differenced the data You should check the correlogram of the series to determine whether you can re estimate with one less round of differencing Lastly if you continue to have problems you may wish to turn off MA backcasting 314 Chapter 13 Time Series Regression TSLS with ARIMA errors Two stage least squares or instrumental variable estimation with ARIMA poses no particu lar difficulties For a detailed discussion of how to estimate TSLS speci
31. observation Higher Order Autoregressive Models More generally a regression with an autoregressive process of order p AR p error is given by Ye PP uy Up P1Ut 1 F PoU PpUt pt Et 13 3 The autocorrelations of a stationary AR p process gradually die out to zero while the par tial autocorrelations for lags larger than p are zero Testing for Serial Correlation Before you use an estimated equation for statistical inference e g hypothesis tests and forecasting you should generally examine the residuals for evidence of serial correlation EViews provides several methods of testing a specification for the presence of serial corre lation The Durbin Watson Statistic EViews reports the Durbin Watson DW statistic as a part of the standard regression out put The Durbin Watson statistic is a test for first order serial correlation More formally the DW statistic measures the linear association between adjacent residuals from a regres sion model The Durbin Watson is a test of the hypothesis p 0 in the specification Testing for Serial Correlation 297 If there is no serial correlation the DW statistic will be around 2 The DW statistic will fall below 2 if there is positive serial correlation in the worst case it will be near zero If there is negative correlation the statistic will lie somewhere between 2 and 4 Positive serial correlation is the most commonly observed form of dependence As a rule o
32. often describe techniques for estimating AR models The most widely discussed approaches the Cochrane Orcutt Prais Winsten Hatanaka and Hildreth Lu procedures are multi step approaches designed so that estimation can be performed using standard linear regression All of these approaches suffer from important drawbacks which occur when working with models containing lagged dependent variables as regressors or models using higher order AR specifications see Davidson and MacKinnon 1994 pp 329 341 Greene 1997 p 600 607 EViews estimates AR models using nonlinear regression techniques This approach has the advantage of being easy to understand generally applicable and easily extended to non linear specifications and models that contain endogenous right hand side variables Note that the nonlinear least squares estimates are asymptotically equivalent to maximum likeli hood estimates and are asymptotically efficient To estimate an AR 1 model EViews transforms the linear model Y 2 B u Up Pur tE 13 9 into the nonlinear model Yi PYe 1 44 p21 B 13 10 by substituting the second equation into the first and rearranging terms The coefficients p and are estimated simultaneously by applying a Marquardt nonlinear least squares algo rithm to the transformed equation See Appendix D Estimation Algorithms and Options on page 645 for details on nonlinear estimation ARIMA Theory 303 For a nonlinear
33. on IP assum ing stationarity Note that selecting View Coefficient Tests for an equation estimated with PDL terms tests the restrictions on y not on In this example the coefficients on the fourth PDL05 and fifth order PDL06 terms are individually insignificant and very close to zero To test the 320 Chapter 13 Time Series Regression joint significance of these two terms click View Coefficient Tests Wald Coefficient Restrictions and type c 6 0 c 7 0 in the Wald Test dialog box See Chapter 14 for an extensive discussion of Wald tests in EViews EViews displays the result of the joint test Wald Test Equation IP_PDL Null Hypothesis C 6 0 C 7 0 F statistic 0 039852 Probability 0 960936 Chi square 0 079704 Probability 0 960932 There is no evidence to reject the null hypothesis suggesting that you could have fit a lower order polynomial to your lag structure Nonstationary Time Series The theory behind ARMA estimation is based on stationary time series A series is said to be weakly or covariance stationary if the mean and autocovariances of the series do not depend on time Any series that is not stationary is said to be nonstationary A common example of a nonstationary series is the random walk Y Y t Ez 13 44 where is a stationary random disturbance term The series y has a constant forecast value conditional on t and the variance is increasing over time The random walk is a di
34. sec ond order component corresponds to using second differences and so on e The third tool is the MA or moving average term A moving average forecasting model uses lagged values of the forecast error to improve the current forecast A first order moving average term uses the most recent forecast error a second order term uses the forecast error from the two most recent periods and so on An MA q has the form U Ep 01E 1 HOE g OE 13 15 304 Chapter 13 Time Series Regression Please be aware that some authors and software packages use the opposite sign con vention for the 0 coefficients so that the signs of the MA coefficients may be reversed The autoregressive and moving average specifications can be combined to form an ARMA p q specification Up PyUz_y E PU F H PpUt p tE 13 16 01E 1 HOE a t o t OEt q Although econometricians typically use ARIMA models applied to the residuals from a regression model the specification can also be applied directly to a series This latter approach provides a univariate model specifying the conditional mean of the series as a constant and measuring the residuals as differences of the series from its mean Principles of ARIMA Modeling Box Jenkins 1976 In ARIMA forecasting you assemble a complete forecasting model by using combinations of the three building blocks described above The first step in forming an ARIMA model for a series of residuals is to look at its
35. stimate models where there is correlation between regressors and the innovations as well as serial correlation in the residuals If the original regression model is linear EViews uses the Marquardt algorithm to estimate the parameters of the transformed specification If the original model is nonlinear EViews uses Gauss Newton to estimate the AR corrected specification For further details on the algorithms and related issues associated with the choice of instruments see the discussion in TSLS with AR errors beginning on page 278 Output from AR Estimation When estimating an AR model some care must be taken in interpreting your results While the estimated coefficients coefficient standard errors and t statistics may be inter preted in the usual manner results involving residuals differ from those computed in OLS settings To understand these differences keep in mind that there are two different residuals associ ated with an AR model The first are the estimated unconditional residuals a y 2 b 13 8 which are computed using the original variables and the estimated coefficients b These residuals are the errors that you would observe if you made a prediction of the value of y using contemporaneous information but ignoring the information contained in the lagged residual Normally there is no strong reason to examine these residuals and EViews does not auto matically compute them following estimation The s
36. to compute backcast values of to _ _ 1 To start this recursion the q values for the innovations beyond the estimation sample are set to zero brag Spey Se epg 0 13 33 Next a forward recursion is used to estimate the values of the innovations Estimating ARIMA Models 313 0 18 _1 baby_g gt 13 34 using the backcasted values of the innovations to initialize the recursion and the actual residuals If your model also includes AR terms EViews will p difference the to elimi nate the serial correlation prior to performing the backcast Lastly the sum of squared residuals SSR is formed as a function of the 8 and using the fitted values of the lagged innovations T 2 ssr 8 9 Sy mX B t 1 P ta 13 35 t p 1 This expression is minimized with respect to 8 and The backcast step forward recursion and minimization procedures are repeated until the estimates of 8 and converge If backcasting is turned off the values of the pre sample are set to zero Pg p 0 13 36 and forward recursion is used to solve for the remaining values of the innovations Dealing with Estimation Problems Since EViews uses nonlinear least squares algorithms to estimate ARMA models all of the discussion in Chapter 12 Solving Estimation Problems on page 289 is applicable espe cially the advice to try alternative starting values There are a few other issues to co
37. tric rate and the partial autocorrelations were zero after one lag then a first order autoregressive model is appropriate Alternatively if the autocor relations were zero after one lag and the partial autocorrelations declined geometrically a first order moving average process would seem appropriate If the autocorrelations appear Estimating ARIMA Models 305 to have a seasonal pattern this would suggest the presence of a seasonal ARMA structure see Seasonal ARMA Terms on page 308 For example we can examine the correlogram of the DRI Basics housing series in the HS WF1 workfile by selecting View Correlogram from the HS series toolbar The wavy cyclical correlo gram with a seasonal frequency BBL ie el ema ze sel rel Stet sae eels a La CS O OO suggests fitting a seasonal Date 08 21 97 Time 18 03 ARMA model to HS Sample 1959 01 1984 12 Included observations 312 The goal of ARIMA analysis isa Autocorrelation Partial Correlation AC PAC Stat Prob parsimonious representation of the process governing the resid ual You should use only enough AR and MA terms to fit the properties of the residuals The Akaike information crite rion and Schwarz criterion pro vided with each set of estimates may also be used as a guide for the appropriate lag order selec tion 0 860 0 860 233 15 0 000 0 660 0 308 370 89 0 000 0 454 0 102 436 22 0 000 0 306 0 106 466 00 0 000 0 243 0
38. y differences of series To specify first differencing simply include the series name in parentheses after d For example d gdp specifies the first difference of GDP or GDP GDP 1 More complicated forms of differencing may be specified with two optional parameters n and s d x n specifies the n th order difference of the series X d x n 1 L 2 13 17 where L is the lag operator For example d gdp 2 specifies the second order difference of GDP d gdp 2 gdp 2 gdp 1 gdp 2 d x n s specifies n th order ordinary differencing of X with a seasonal difference at lag s d x n s 1 L Q L a 13 18 For example d gdp 0 4 specifies zero ordinary differencing with a seasonal difference at lag 4 or GDP GDP 4 If you need to work in logs you can also use the dlog operator which returns differences in the log values For example dlog gdp specifies the first difference of log GDP or log GDP log GDP 1 You may also specify the n and s options as described for the simple d operator dlog x n s There are two ways to estimate integrated models in EViews First you may generate a new series containing the differenced data and then estimate an ARMA model using the new data For example to estimate a Box Jenkins ARIMA 1 1 1 model for M1 you can enter series dmi d m1 ls dmi c ar 1 ma 1 Alternatively you may include the difference operator d directly in the estimation specifi cation
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