REGIME-SWITCHING MODELS
Contents
1. The Lakel hood Function ai e egte e EEEE OR ERTE ROEE 11 3 1 2 The Smoothed Probabilities ss 12 3 1 3 Differences Between Hamilton s and Our Code 0 ee eceeseseeeecneceeeeesecaeenees 12 3 1 4 Analytic Gradients ss 14 3 2 A Sample Program for Markov Switching Models cccesssssccescseeeeeeeceecseeeeeseceeeneeaes 15 3 3 Speeding up Estimation with C procedures 16 3 4 EM algorithms i ss hisser en dis A tA a Menthe Rene ms AR a EEA 17 3 5 Future Developments iriseoir ini svents ses EPSE EEEE Es EE SEEEN aria 17 4 0 Specification Tests for Regime Switching and Other Models 17 4 1 The Theory of Score Based Specification Tests 18 4 1 1 General Procedure for Score Based Tests wmisspec 18 4 2 Misspecification Tests in Switching Regressions 0 ceeeecceeeeceseeeeeeseeeeecneecaeceesaeenaes 19 4 2 1 Testing Simple Switching Models swmisspc 19 4 2 2 Testing Markov Switching Models mkvspec 20 5 0 Bibliography EEE eee Re eater a nee 21 1 0 Introduction This paper is a manual for using a set of Gauss procedures developed at the Bank of Canada to estimate regime switching models In these models the observable dependent variable s behavior is state dependent The state which itself is unobservable determines the process that generates2. Ricketts and D Rose J Gable S van Norden and R Vigfusson Exchange Rates and Oil Prices R A Amano and S van Norden Selection of the Truncation Lag in Structural VARs or VECMS with Long Run Restrictions A DeSerres and A Guay The Canadian Experience with Weighted Monetary Aggregates Long Run Demand for M1 The Empirical Performance of Alternative Monetary and Liquidity Aggregates Single copies of Bank of Canada papers may be obtained from D Longworth and J Atta Mensah S Hendry J Atta Mensah Publications Distribution Bank of Canada 234 Wellington Street Ottawa Ontario K1A 0G9 E mail publications bank banque canada ca WWW _ http www bank banque canada ca FTP ftp bank banque canada ca login anonymous to subdirectory pub publications working papers
3. SWEMMLE G incorporates both of these fea tures Furthermore it is structured to return the same information as the Gauss maxlik pro cedure which allows existing MLE programs to be easily adapted to joint EM ML estimation These procedures are part of the library file SWITCH LCG The output produced is con trolled by the same global variable __output used to control maxlik There are four other global variables that control other features of the procedure One _emprint controls whether the final MLE results are printed Another _emecm controls whether the unre stricted or restricted B B versions of the model are estimated The other two con trol when estimation is passed from the EM to the MLE procedures This occurs as soon as either a maximum number _emmaxit of EM iterations have been completed or when the relative change in each component of the parameter vector is less than some fraction _emconvg 6 Even more generally if the MLE gets stuck one might want to change to the EM procedure for several iterations before returning to the MLE Unfortunately we could find no way of aborting the Gauss maxlik procedure to do this in a flexible way However we are talking to Aptech Systems about ways to accomplish this and may incorporate such features in a future version of SWEMMLE G 17 2 3 A Sample Program For Simple Switching Models To give the user an understanding of how these procedures can be used le
4. converges to the maximum likelihood estimates of B4 B B 01 65 2 2 2 Gauss Programs A single iteration of the EM algorithm can be found in SWEM G It takes a data matrix and a set of parameter estimates calculates equations 1 13 and returns the new param eter estimates This may be used in a simple loop for model estimation The MLE procedures for estimating switching regressions also consider the special case where we impose the restriction B B P in estimation Imposing this restriction in the EM algorithm requires only slight modifications to the algorithm and procedure described above In particular we would now replace the 2 WLS regressions with a single WLS regression using weights W o W o2 to estimate B The unweighted residu als from this regression are then weighted by w and w to generate estimates of and respectively These modifications are embodied in the Gauss procedure SWEC MEM G which is otherwise analogous to SWEM G In order to use the above procedures all one needs is a simple loop that updates the param eter estimates at each pass and tests them for convergence To make the most of the rela tive strengths of the EM and MLE procedures however one might wish to begin estimation with the EM procedures to take advantage of their greater robustness As these estimates begin to converge one might then want to change to MLE because of its faster convergence The procedure listed below
5. X now have the usual linear relationship aside from a particular kind of heteroscedasticity some errors are generated by a high variance regime and some from a low The main use of this model is in tests of non switching models against switching alternatives For example suppose we wanted to test the hypothesis that Y XP e ae N 0 against the alternative of EQ 1 Econometrically this is difficult since some of the parameters of EQ 1 are unidentified under the null 4 Typically however what we care about is whether there is switching in means i e whether P B since evidence switching due solely to some form of heteroscedasticity may be cured by some other model of heteroscedasticity By estimating EQ 4 we have a model where we allow for heteroscedasticity and where all parameters are identified under the null Therefore we can use standard likelihood ratio tests of EQ 1 versus EQ 4 SWCONTAM G contains the procedures swecli and swecgrad which are analogues of swli and swgrad in SWLI_NEW G The only functional differences are that they assume the data matrix is of the form Y X X and the parameter vector is of the form B B3 0 0 EQ 6 Accordingly only _n and _n3 need be set before the procedure is called and _n2 is ignored The same applies to the procedure becinit in SWINTCON G which is analogous to binitial in SWINIT2 G However becinit uses an iterative weighted least squa
6. properties of the gradient vectors at a given set of parameter esti mates These tests can be adapted to the case of simple switching regressions They allow one to construct simple tests for serial correlation ARCH and Markov switching Section 4 1 reviews the econometric basis of these tests Section 4 1 1 lists a Gauss proce dure that can be used to calculate such tests for virtually any model whose gradient can be calculated Section 4 2 discusses the application and interpretation of the misspecification 26 tests in the context of switching regressions while Section 4 2 1 lists a Gauss procedure that applies a series of such tests to a standard switching regression 4 1 The Theory of Score Based Specification Tests Let L y e 9 denote the likelihood function of a given observation y conditional on some information set x and a choice of parameters The score h 8 is simply the gradi ent of Lx 6 with respedi to 0 Subject to a few regularity conditions such as no cor ner solutions and ergodicity it can be shown that at the true parameter estimates 0p h 8o should be unforecastable on the basis of any information available at t 1 which includes h _ 8 Since h 8 is a vector with the same dimensions as 8 say mx1 this implies that the absence of any first order serial correlation in A 0 alone gives us mxm testable restrictions While in theory we could test all of these conditions or test even higher o
7. switching model SWINIT2 G produces starting parameter values for the above SWCONTAM G contains likelihood function and gradients for restricted model with identical means in both regimes SWINTCON G produces starting parameter values for the above SWLI_NEW G and SWINIT2 G are for the most general model which has the form Yi Xi Bite Yo Xap By Ex Y3 X3 Bates peli Taps EQ 1 0 Gp 0 Elp op E3 N 0 O19 O 0 0 0 I Y is our observable dependent variable which is generated by a mixture of different regimes captured by the unobservables Y Y5 Y3 is a latent variable that perfectly classifies Y into the two regimes and X represents whatever extra information we have to make this classification Conditioning only on X the probability of being in regime 1 at tis P X ae B3 1 whereas if we also condition on Y it becomes 1 Alternatively one could use logit equations to determine the probability of being in the different regimes See Section 2 1 1 for more information 11 Y Xi By X B3 fe 1 pot X By se ph ara Ba aa AE P X Ba p where is the standard normal p d f and is the corresponding c d f In the simplest case X X X 31 1 and we just have a mixture of two normal distributions N B O1 and N B 0 SWLI_NEW G contains the likelihood and gradient procedures swli and swgrad It fol lows the standard syntax required by maxlik and assumes th
8. that this means we are now dealing with the conditional likelihood func tion instead of the usual unconditional function As Hamilton 1992 notes the difference is that the former treats the initial probability Pr S 0 as given a constant independent of the other parameters of the model If we endogenize it by imposing EQ 20 however then we have the usual unconditional likelihood function 20 supply another parameter p Pr S 0 In the absence of strong priors about which state we start in this can be set equal to the unconditional probability of being in state 0 which is l q a EQ 20 2 p q i Alternatively we could set p arbitrarily to zero or one implying that we know with cer tainty the regime at time zero or we could have p be a parameter of the model to be esti mated 3 1 2 The Smoothed Probabilities Hamilton 1989 also mentions a related but more complex function he calls a full sam ple smoother This gives the probability that S 0 conditioning on all the available data and not just that available up to time t In other words the smoother gives Pr S O Y 7 Yh EQ 21 instead of Pr S O Y Yh EQ 22 Previously calculating the smoothed probability at any time f requires that we work out the conditional p d f f Y ARE ere Y for each observation T from t to T and then go back and re evaluate it for each T under the assumption that S 0 This means that if the number
9. 22 3 1 4 Analytic Gradients The availability of the smoothed probabilities makes it feasible to calculate the likelihood function s analytic gradients These gradients further increase the speed of estimation of the models To estimate by maximum likelihood methods requires calculating the derivatives of the likelihood function with respect to the parameter vector This vector is also referred to as the score or the gradient vector This is usually done using numerical techniques that approximate the derivative by the change in the likelihood function for small changes in the parameter vector This is not especially efficient however as such techniques typically require N evaluations of the likelihood function to calculate the N elements of the score and N 1 to calculate the Hessian the matrix of second derivatives The use of analyti cal gradients can greatly reduce the number of calculations required to evaluate either of these objects This in turn considerably speeds up maximum likelihood estimation of such models with no loss in accuracy These gradients are further detailed in Gable van Norden and Vigfusson 1995 The gradi ents are derived and comparisons are made between the analytical and numerical methods In general the analytic derivatives can improve calculation efficiency by roughly a factor of four The relative speed of the analytic derivatives increases with sample size but the improvement seems to slow as sam
10. 34 ISBN 0 662 24293 9 Printed in Canada on recycled paper Abstract This paper is a user s guide to a set of Gauss procedures developed at the Bank of Canada for estimating regime switching models The procedures can estimate relatively quickly a wide variety of switching models and so should prove useful to the applied researcher Sample program listings are included R sum La pr sente tude constitue un guide d utilisation d un ensemble de proc dures de Gauss mises au point la Banque du Canada en vue de l estimation des mod les changement de r gime Ces proc dures permettent d estimer de fagon assez rapide une vaste gamme de mod les changement de r gime et devraient s av rer utiles pour la recherche appliqu e Des chantillons de programmes sont inclus dans l tude Contents 1 0 MANO CU sa Gen in Ne ae nt wutina elogederasonecauee 1 1 1 What Is N win This VErsiOM pes ninian Er Rae E cous E een opii i i eei i 1 2 0 Simple STUD ATID i Sines SR tn ST Te ta e at 2 2 1 Maximum Likelihood Methods 2 ZA SE Logit Probabilities ereere ts ne ennemie sinus 5 2 2 PM al eorithin n s mins a a a ER RE se 6 2 2 1 The FEM ATS Orit i sis in antenne tn 6 222 L Gauss Proprams nes tn a E R an AR alana ges 8 2 3 An Example Program For Simple Switching Models 00 cece eeeseeceseeeceeceseceeeeseeeeeeeeees 9 3 0 Markov Regime Switching fist etnedneanertilain 10 3 1 Maximum Likelihood Estimation ss 10 ZLE
11. Time Series Analysis 1994 devotes a chapter to Markov regime switching models Anyone interested in working with Markov switching models should consult it or another textbook for more details on the Markov switching models 3 1 Maximum Likelihood Estimation The general regime switching model that we consider describes the generation of a single variable y by 2 distinct states In any given state i y is generated by a linear regression model Y x Bit E E iid N 0 6 EQ 12 where x is a vector that is exogenous with respect to The states are assumed to follow a first order Markov process with transition probabilities that may vary over time accord ing to the formula P s ils 1 D De Yy EQ 13 where is the standard normal cumulative distribution The probability of being in the given state is dependent on last period s state This is what distinguishes this from sim ple mixtures of distributions In simple mixture models the probability of being in a given state is independent of last period s state In the two state model this simple switching model is nested into the Markov switching model when Pr S 0 S _ 0 1 Pr S 1 S _ 1 EQ 14 t t 1 7 We assume that Z is also exogenous with respect to 19 3 1 1 The Likelihood Function The dependence of this period s state on the previous period s state has a major effect on the difficulty of estimating the para
12. Working Paper 96 3 Document de travail 96 3 REG IME SWITCHING MODELS A Guide to the Bank of Canada Gauss Procedures by Simon van Norden and Robert Vigfusson Bank of Canada M Banque du Canada Regime Switching Models Guide to the Bank of Canada Gauss Procedures Simon van Norden E mail svannorden bank banque canada ca Robert Vigfusson E mail rvigfusson bank banque canada ca International Department Bank of Canada Ottawa Ontario Canada K1A 0G9 This paper is intended to make the results of Bank research available in preliminary form to other economists to encourage discussion and suggestions for revision The views expressed are those of the authors No responsibility for them should be attributed to the Bank of Canada Authors Note The Gauss procedures described in this paper are available from the Bank of Canada s Library Website http www bank banque canada ca library under working papers or by anonymous FTP to ftp bank banque canada ca in the directory pub vigfuss The code in this paper is being released free of charge In exchange we ask that its use be acknowledged in any published work including working papers and paid research papers The computer programs documentation and all other information in this working paper are provided for your information only and for use entirely at your own risk We do not warrant their accuracy nor their fitness for any purpose whatsoever ISSN 1192 54
13. at the data in the data matrix are of the form ly Xi X X and that the parameter vector is of the form B By Bs 5 03 EQ 3 The user must set the global variables _n _n2 and _n3 to tell the procedure the dimen sions of the elements in the data set and SWINIT2 G has the procedure binitial for generating initial values for the parameters and requires the same three global variables to be set before it is called It divides the sample into positive and negative valuts of Y then regresses the positive on X and the negative on X to generate B Bay B3 comes oni the regression of a scaled sion of Y on X3 This eens to work well enough if Y has a mean very close to zero Note that it omits ee in the data set containing missing values The procedure is called using the syntax theta binitial Xdata where Xdata has the same form as the data matrix for swli and swgrad SWCONTAM G and SWINTCON G estimate a special case of the general model described above where X X and we wish to restrict B P This means the model reduces to 2 The ambitious user would want to program Goldfeld and Quandt 1973 s method of moments estimator for this kind of thing but that is definitely non trivial However this procedure seems to work well for data with a mean close to zero 12 Y XP Pe Y3 X3 B3 3 N 0 0 3 lt 0 EQ 4 N 0 3 20 E E 3 0 This says that Y and
14. ated For the Markov switching case testing the gradient of the constant term in the transition equation gives a test for neglected higher order Markov switching effects Finally tests of the gradients of 6 and 6 amount to tests for first order regime specific ARCH effects since persistence here implies that the volatility in each regime seems to vary over time in a way captured by a first order autoregression 4 2 1 Testing Simple Switching Models swmisspc The Gauss program swmisspc calculates and prints the test statistics discussed in the previous section for the simple switching model It is designed to be used immediately after the estimation of a switching regression by maxlik It accepts as arguments the final parameter estimates and the data set used by swli and swgrad and prints the test statis tics and their marginal significance levels The procedure assumes that swgrad has been defined as have the global variables _n _n2 and _n3 Finally the procedure also assumes that the first regressor in each equation is a constant 15 More precisely the model must satisfy certain regularity conditions such as ergodicity See White 1987 16 In the notation of the previous section this means that c can include only diagonal elements of h h 8 However these are the ones that are most likely to be of interest 28 4 2 2 Testing Markov Switching Models mkvspec The Gauss program mkvspec calcul
15. ates and prints the test statistics or the Markov switching model It is designed to be used immediately after estimation using the same data set as and final estimates from maxlik The procedure assumes that the global vari ables discussed in subsection 3 2 have been defined and that the first regressor in each equations is a constant 29 5 0 Bibliography Diebold Francis X Joon Haeng Lee and Gretchen C Weinbach 1994 Regime Switching with Time Varying Transition Probabilities in C Hargreaves Nonstationary Time Series Analysis and Cointegration Oxford Oxford University Press Durland J Michael and Thomas H McCurdy 1994 Duration Dependent Transition in a Markov Model of U S GNP Growth Journal of Business amp Economic Statistics July 12 3 279 288 Elliot Robert J Lakhdar Aggoun and John B Moore 1995 Hidden Markov Models Estimation and Control Springer Verlag New York Engel Charles 1994 Can the Markov Switching Model forecast exchange rates Journal of International Economics 36 pp 151 165 Engel Charles and James D Hamilton 1990 Long Swings in the Dollar Are They in the Data and Do Markets Know It American Economic Review Sept 80 4 689 713 Gable Jeff Simon van Norden and Robert Vigfusson 1995 Analytical Derivatives for Markov Switching Models Bank of Canada Working Paper 95 7 Goldfeld S M and R E Quandt 1973 A Markov model for switching regressions Journal of E
16. conometrics 1 3 16 Hamilton James D 1989 A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle Econometrica 57 357 84 1992 Estimation Inference and Forecasting of Time Series Subject to Changes in Regime in C R Rao and G S Maddala Handbook of Statis tics Volume 10 1994 Time Series Analysis Princeton Princeton University Press 1996 Specification testing in Markov switching time series mod els Journal of Econometrics January 70 1 127 157 Hansen B E 1992 The Likelihood Ratio Test under Nonstandard Conditions Testing the Markov Switching Model Of GNP Journal of Applied Econometrics Vol 7 Supplement Hartley Michael J 1978 Comment on Quandt and Ramsey Journal of the American Statistical Association December Kim Chaing Jin 1994 Dynamic Linear Models with Markov Switching Journal of Econometric 60 1 22 30 Lam Pok sang 1990 The Hamilton Model with a General Autoregressive Component Estimation and Comparison with Other Models of Economic Time Series Journal of Monetary Economics 26 409 32 van Norden Simon 1996 Regime Switching as a Test for Exchange Rate Bubbles Journal of Applied Econometrics forthcoming van Norden Simon and H Schaller 1993 Speculative Behaviour Regime Switching and Stock Market Fundamentals Bank of Canada Working Paper 93 2 Vigfusson Robert 1996 Switching Between C
17. e At the start of the pro gram include a line with the Dlibrary function referencing the compiled library For example with our compiled library libmkz so the line would be DLIBRARY home int vigf regswt ccode libmkz so 12 The compiler to be used will depend on the computer where the programs will be run We used the GNU C compiler GCC 13 The dynamic library interface is currently as of 5 25 95 supported on the SunOS 4 1 3 Solaris 2 x AIX HPUX IRIX and OSF1 platforms Aptech README file In DOS instead of the dynamic libraries one uses the foreign language interface See the Gauss user manual for details 14 The Gauss on line help system gives further details on using the DLIBRARY command 25 To access the C procedures set the global variable Newalg in your main program equal to one No other changes are necessary 3 4 EM algorithm We have written the Gauss code for the EM algorithm for the case of constant transition probabilities Hamilton 1994 The algorithm is useful for getting close to a maximum in spite of potentially bad starting values The algorithm does have difficulties in final con vergence to the maximum Therefore we recommend using the EM algorithm first to come close to the maximum and then passing this coefficient vector to the standard maxlik approach for final convergence If Newalg equals one then the EM algorithm will use a C procedure augmented version of Kim s smoother Something h
18. hartists and Fundamentalists Bank of Canada Working Paper 96 1 White Halbert 1987 Specification Testing in Dynamic Models in Advances in Econometrics Fifth World Congress Volume II Truman F Bewley ed Cambridge Cambridge Press 1994 Estimation Inference and Specification Analysis Econometric Society Monographs Cambridge University Press 1996 96 1 96 2 96 3 95 10 95 11 95 12 Bank of Canada Working Papers Switching Between Chartists and Fundamentalists A Markov Regime Switching Approach Decomposing U S Nominal Interest Rates into Expected Inflation and Ex Ante Real Interest Rates Using Structural VAR Methodology Regime Switching Models A Guide to the Bank of Canada Gauss Procedures Deriving Agents Inflation Forecasts from the Term Structure of Interest Rates Estimating and Projecting Potential Output Using Structural VAR Methodology The Case of the Mexican Economy Empirical Evidence on the Cost of Adjustment and Dynamic Labour Demand Government Debt and Deficits in Canada A Macro Simulation Analysis Changes in the Inflation Process in Canada Evidence and Implications Inflation Learning and Monetary Policy Regimes in the G 7 Economies Analytical Derivatives for Markov Switching Models R Vigfusson P St Amant S van Norden and R Vigfusson C Ragan A DeSerres A Guay and P St Amant R A Amano T Macklem D Rose and R Tetlow D Hostland N
19. hese procedures have been developed and used on Unix workstations Some conversion may be necessary to use them on other operating systems For example several of the file names are longer than the DOS name length limit The rest of the paper is organized along the following lines Section 2 describes the proce dures used for estimating simple switching models Section 3 describes the procedures used for estimating Markov switching models Section 4 describes specification testing to be used after estimation of a switching model Section 5 contains a bibliography 1 1 What Is New in This Version There are some large differences between this and earlier releases of the switching proce dures A library file has been written making the procedures easier to use The Markov switching code has seen dramatic changes Most are related to the development of a fast algorithm for calculating full sample or smoothed probabilities Kim 1994 These prob abilities are necessary to make practical use of the analytic gradients that are discussed in Section 3 10 2 0 Simple Switching 2 1 Maximum Likelihood Methods This section documents the procedures used for maximum likelihood estimation of sim ple switching and normal mixture models These procedures calculate the likelihood function and its as well as starting parameter estimates There are four files containing procedures SWLI_NEW G contains likelihood function and gradients for the general
20. inear recursive formula any proce dure returning the likelihood function must have a loop that calculates the likelihood func tion of observation t given that for t 7 However like all interpreted computer languages Gauss is extremely inefficient at processing instructions inside a loop Therefore moving as many instructions as possible outside the loop greatly speeds execution Table 1 Speed Comparisons of Likelihood Functions and Maximum Likelihood Estimation Hamilton s New Evaluation of Likelihood Function Elapsed Time in Seconds 10 38 1 15 Time Savings 0 00 88 92 Speed up Factor 1 00 9 03 Maximum Likelihood Estimation Elapsed Time in Seconds 1246 3 297 09 Time Savings 0 0 76 16 Speed up Factor 1 0 4 19 The above table shows the results of an experiment to compare the speeds of the new like lihood function procedure with that of Hamilton s with Gauss version 2 0 It uses the GNP data from Hamilton 1989 and estimates a Markov mixture model with unequal variances in the two states and EQ 20 imposed The evaluation of the likelihood function with the new procedure is over 9 times faster than with Hamilton s procedures and the time required to estimate this simple model is cut from over 20 to less than 5 minutes 9 These results have not been updated to the current version However there is no reason that the newer ver sion should produce very different results
21. ings for ix and iy Value ix sameness iy 0 Variances Equal Across States Different explanatory variables in ee l q all equations ae ren 1 Variances Differ Across States Same explanatory variables in all rho is a parameter level equations Different in tran sition equations 2 Different explanatory variables in rho 0 all level equations Same explana tory variables in transition equa tions 3 Same variables for level equations rho 1 and same variables for transition equations 4 Same variables for level equations and transition equations Each row of terms an nstat by two matrix specifies what data are used by a particular equation The first nstat rows are for the level equations The rest are for the transition equations The first column of terms specifies the first column of data used in an equation The second specifies the last The dependent variable is assumed to occupy the first col umn of the data matrix The elements in terms may be specified such that the same vari ables are used in more than one equation 24 To use analytic derivatives in the maximum likelihood estimation one must add the line _mlgdprc amp ANGRAD Next make a call to the procedure mkvstart which returns the value of terms the vector that specifies which columns of the data matrix to use for each equation and nrows the number of observations It is assumed that the data are formatted such that the dependent variable is in the fir
22. maxgrd 2 2 EM algorithm Hartley 1978 described an EM algorithm to estimate the parameters of a switching regression The algorithm is an alternative to maximum likelihood estimation MLE and simply uses a combination of ordinary least squares OLS and WLS The EM algorithm converges more slowly than MLE near the final estimates but 1s thought to be more robust to poor starting values This should make it a natural complement to existing MLE proce dures This algorithm is now available as a set of Gauss procedures We will first describe the application of the EM algorithm to the estimation of switching regression systems The remainder of the section lists three Gauss procedures that imple ment this estimation technique The first procedure SWEM G performs one iteration of the EM algorithm The second procedure SWECMEM G is the analogue of SWEM G for the case where we restrict the coefficients in the two regimes to being identical The final procedure SWEMMLE G both controls the EM estimation and integrates it with the existing MLE procedures 2 2 1 The EM Algorithm Suppose an observable variable y is generated by two different latent unobservable vari ables Y4 Yy Yg is also a latent variable and classifies y in either regime Ya Xa Bit Es N 0 07 1 Ya Xp Bat Ep E 2 N 0 07 2 V3 Lis B3 83 amp 3 NO 1 3 Y Ya if Vs lt 0 15 y 9p if Y 3 gt 0 The EM algorithm is an iterative method f
23. maxlik variables Consult the user s man ual for information on them 4 Choose Settings for the SWEM procs if desired _emmaxit 100 _emconvg 0 002 5 Choose settings for the MAXLIK proc __row 0 _mlalgr 5 _mlstep 2 _mlcovp 2 _mlndmth NEWTON _output 0 _mlgtol 1E 5 _mlmiter 500 _n1i 2 _n2 2 _n3 2 _emprint 1 _emecm 0 __title A Switching Regression Model miparnm B23S0 BssSb MBLEO 8 Behe r Mig Sp MB gb27 i SES N i SSG ee a The next step is to provide some initial starting values let beta0 0 0074 0 0146 0 0298 0 0533 1 3723 2 4154 0 0405 0 1334 Finally there is a call to the swemmle procedure to estimate the model xswp llfswp g h retcode swemmle beta0 swd 18 3 0 Markov Regime Switching The section on Markov regime switching models consists of two parts First the proce dures used for maximum likelihood are explained Second an example program is given The Markov switching code has had four substantial improvements First Hamilton s original code was generalized and made faster Second analytic gradients were developed that made estimation even faster Third a C procedure was written to further speed up esti mation Finally Hamilton s EM algorithm 1994 has been programmed in Gauss This algorithm can reduce greatly the time consumed by bad starting values in comparison with the other methods Hamilton s textbook
24. mes faster than Hamilton s original code 23 With the above analysis we can predict that the analytic gradient procedure will be faster as either the sample size or the number of parameters increases In the unlikely case that the number of states increases with the number of parameters held constant there may be a relative improvement of the numeric technique More likely an increase in the number of states will cause a substantial increase in the number of parameters 3 2 A Sample Program for Markov Switching Models There is now a library file for the procedures switch lcg Hence it is no longer necessary to have a number of include statements Rather one merely needs to have a line such as library maxlik switch lcg The next required step is to clear the globals that will be used in the estimation These include nstat vei ix ty nrows sameness and terms The variable nstat is the number of states At present nstat must be equal to two The variable vei is an nsta by 1 vector The first nstat entries specify the number of variables in the corresponding level equation The next nstat nstat 1 specify the number of variables in the corresponding transition equa tion The variable ix controls whether variances differ across states or are equal The vari able iy sets the starting condition for the probability of being in the different states at time 0 TABLE 2 gives the possible settings for ix sameness and iy TABLE 2 Sett
25. meters of the mixture model To understand why note that the p d f of Y given Up Uy Oo O1 P in the simple mixture model can be written as Pr S 0 pdf Y CS 0 Pr S 1 pdf Y CS 1 EQ 15 p paf Y CS 0 1 p pdf Y CS 1 EQ 16 Y Ho Y H So 91 pe 1 p EQ 17 0 ST where is the standard normal p d f The key is that the probability of observing Y depends only on the parameters Ug H1 Og O1 P and not on other observations of Y In the Markov switching case however we are unable to duplicate the step from EQ 15 to EQ 16 This is because the probability of being in a given state now depends on the state the period before which is not observed Since the p d f of Y is necessary to calcu late the model s likelihood function this greatly complicates maximum likelihood estima tion Hamilton gets around this problem by replacing Pr S 0 in EQ 15 with PES SONT pee YD P PSs a IHY D la ECTS and Pr S 1 with Pr S _ 4 O Y _ 55 yd A p Pr S _ 1 Y _ Y qg EQ19 He then shows that Pr S _ O Y _1 can be calculated from Pr S _5 O0 Y _ gt 5 using a simple updating formula that depends only on Y _ and Mo Hy Og 01 P q In effect this means we can recursively calculate the likelihood function of Y given that of Y ea ie However to start the process off we must 8 The observant reader will notice
26. of calculations required to evaluate the likelihood function is a linear function of the number of observations say b T then the number of calculations required 2 to calculate the full sample smoother from 1 to T will be roughly b g 1 Kim 1994 provides a simpler method Based upon the fact that for the last observation the smoothed and ex post probabilities are equal he develops a recursive formula for the smoothed probabilities Pr S 0 Yr Pr S 1 Yr S Yr S rar POP 117 1 This reduces the number of calculation from almost T to about 2T 21 3 1 3 Differences Between Hamilton s and Our Code There are some features in Hamilton s procedures that are not present in our code As pre viously noted Hamilton s allow for autocorrelation within states while the new proce dures do not His EM procedures also handle vector Markov switching systems that is where Y is a vector of variables governed by a single state variable S Hamilton s pro grams also contain code for imposing a Bayesian prior on the likelihood function to help ensure convergence as described in Engel and Hamilton 1990 The new likelihood routines are much more efficiently coded than those available from Hamilton Much of this stems from dropping the serial correlation option However the biggest difference comes from minimizing the amount of calculation inside the loop Since the likelihood function is evaluated with a non l
27. or estimating such systems that relies on simple OLS and WLS regressions Given staining values for B4 B gt B3 G1 O2 we begin by calculating weights for a WLS regression 5 These weights are computed as wO A OAOD 4 wO A 1 OAOD 5 where f y is the normal probability density function for each regime 1 1 ss gap s een S Pi 6 fiO ne at A 6 is the cumulative density function of which represents the probability that Yy Yyy It is calculated as P x B3 7 where is the standard normal cumulative density function Finally h y is the proba bility density function of the observed variable y and is calculated as A y A fota fap 8 Hartley 1979 suggests the following WLS regression to obtain estimates of B i 1 2 B X Wi XTX W yl 9 W diag w es Waits 1 1 2 10 B is computed using OLS B3 X3 X3 X 6 11 where 513 is the conditional expectation of 3 given Yz f3 0 0 63 Elyga yd x3 B3 l 10 gt 7 90 F a 12 Finally to calculate the variance estimates 073i 1 2 the following WLS regression is employed Ew O X B W y X B 13 5 The aforementioned procedure SWINIT2 G could be used to find starting values 16 These new estimates Bi B gt Ba 0 z can be used as new starting values and the calcu lations of equations 1 13 repeated It can be shown that this iterative estimation process
28. owever is not right with our programmed version of the EM algorithm A slight discrepancy exists between the EM algorithm s results and the maxlik package s results In these cases if the maxlik estimate is used as a starting value for the EM algo rithm then the EM algorithm will move away from this estimate to its own even if the maxlik estimate has a higher likelihood The difference between these two estimates is slight so the EM algorithm continues to be useful for coming close to a maximum We are looking for a reason for the difference and would appreciate feedback on possible solu tions 3 5 Future Developments The analytic gradients and likelihood function currently can be used only for a two state system The code has been written such that the change to an n state system while non trivial is certainly feasible The code for the derivatives of EQ 12 is already capable of handling n states The global variables vei terms and nstat should facilitate the transition to the n state system The EM algorithm currently estimates only models with constant transition probabilities Diebold Lee and Weinbach 1994 however have an EM algorithm for the non constant transition probabilities model that we hope to include in a latter release of this software 4 0 Specification Tests for Regime Switching and Other Models White 1987 1994 and Hamilton 1990 1996 suggested a set of specification tests based on the serial correlation
29. ples become large An analysis of the code illustrates why the analytic derivatives are typically faster than the numerically calculated derivatives The numerically calculated derivatives make P the number of parameters calls to the likelihood function Most of the time the likelihood function is in turn spent in a loop over the sample size N One therefore could approxi mate the amount of calculations in the numeric derivatives as P N The analytic derivatives make one call to the smoother which in turn calculates the likeli hood function and then performs an additional loop over the sample size Hence the smoother requires approximately 2 N calculations Calculating the gradient then requires a loop over the number of states S for a set number of calculations L In total therefore this requires approximately 2 N S L calculations 10 Alternatives to gradient based maximum likelihood estimation include the EM algorithm and simulated annealing However simulated annealing is also inefficient and the EM algorithm has a lower rate of con vergence near the optimum than do most popular gradient based maximization methods such as the Newton algorithm The score is also useful for the calculation of standard errors and for diagnostic tests see White 1994 11 Our likelihood function was used for both methods of calculating the gradients So using our likelihood function with analytic gradients should result in its being about 15 20 ti
30. rder restrictions on its serial correlation in practice most of these restrictions are very difficult to interpret In what follows attention will limited to those restrictions that are most easily understood White 1987 constructs the general test by listing those elements of the mxm matrix h 0 h that we wish to test in the x1 vector c 8 He then lets 6 denote our maxi mum likelihood estimate of and lets be the 2 2 subblock of the inverse of the parti tioned matrix ae i Y h 8 h y Y a 6 Pal z pu 14 gt 8 2 0 c c 6 Lt 1 t 1 where T is the sample size In this case he shows that if the model is correctly specified the matrix product D T T 7 ze E TOODO 15 t 1 t 1 ca will have a x2 J asymptotic distribution 4 1 1 General Procedure for Score Based Tests wmisspec The Gauss procedure wmisspec may be used to calculate the test statistics defined by 15 Since the theory is completely general this can be applied to almost any econometric model for which we can calculate the gradient vector l For any of these models we can select any parameter or combination of parameters to use in our test for misspecification 27 Accordingly this procedure may be useful in a wide variety of situations The procedure does not require that the model be estimated by maximum likelihood All that is required is the actual gradient matrix For
31. res WLS method of approximating the robust estimator of B and then regresses the final weights on X to estimate p 3 In the heteroscedasticity literature this kind of model is formally known as the error contamination model hence the names of the files and procedures 4 Recent work has been done on this issue for the Markov regime switching case Hansen 1992 13 2 1 1 Logit Probabilities There is an alternative version of the above programs where the above probit weights are replaced with logit weights For the switching regression model of EQ 1 this means Pr Y Y2 P X3 B EQ 7 is replaced with 1 Pr Y Y 1 e XP EQ 8 and similarly for the error contamination model of EQ 4 Pre N 0 6 P X3 B3 EQ 9 is replaced with 1 a EQ 10 e Pr e N 0 Note that in both cases the probability is still bounded asymptotically between 0 and 1 and that the probability of observing regime 2 increases with X B3 This slight change allows one to examine the sensitivity of estimation results to minor changes in the weights accorded to each regime as the logit weights converge to their lim its more slowly than do the probit so they discount unlikely regimes less heavily In addi tion the exponential expressions given in EQ 8 and EQ 10 can be calculated more quickly and accurately than the cumulative normal distribution function which does not allow a closed form sol
32. some models such as the switching regression case con sidered below this can simply be done using a procedure that calculates gradients analyti cally Otherwise it will be necessary to supply a procedure that calculates the likelihood function and use the gradp function in Gauss to provide numerical gradient estimates As written the procedure requires as arguments only the 7xm matrix of gradients evalu ated at the parameter estimates and a column vector that lists the numbers of the columns in the gradient vector that we wish to include in the test It then returns the test statistic and its x2 marginal significance level 4 2 Misspecification Tests in Switching Regressions The specification tests are similar whether one is testing simple switching or Markov switching If we assume that the level equations each contain a constant then testing whether each of these gradients are autocorrelated amounts to testing whether there is omitted serial correlation in or The intuition here is that serial correlation in these gradients implies that we tend to find runs where the constant should be higher or lower which in this context implies persistence in the residuals or serial correlation For the sim ple switching case testing the gradient of the probability of being in a regime gives a test for neglected Markov like switching effects since serial correlation here implies that regimes seem to be more persistent than indic
33. st column then the explanatory variables for the level equations and then the explanatory variables for the transition equations terms mkvstart DATA vei nstat After loading in the data and setting all the necessary globals for maximum likelihood estimation including starting values make a call to the maximum likelihood procedure See Aptech s Maximum Likelihood manual for more information a b c d e maxlik DATA 0 amp swmkv StartValues 3 3 Speeding up Estimation with C procedures Besides the procedures already mentioned code exists to make estimating Markov switch ing models faster We speed up the calculations of the likelihood function and the smoothed probabilities by replacing the main loop with a C language procedure The C code actually only replaces a small portion of the code Because C handles do loops much more efficiently than Gauss the C augmented estimation is several times faster than the same procedure that uses the Gauss do loops To use the C procedures ones needs a C compiler and a version of Gauss no older than version 3 1 14 Before the C procedures can be used their source code has to be com piled Using these procedures on a computer running Unix requires the Gauss dynamic libraries feature that was introduced in version 3 1 14 The dynamic libraries commands allow Gauss to use functions written in other computer languages After the C procedures are compiled using them is fairly simpl
34. t s look at sam ple program swbasic prg The first thing that is required is to have a library statement that references the maxlik and switch lcg libraries After this library statement the program can then call all the proce dures mentioned above library maxlik switch lcg _swclass PROBIT The next step is to load in the data The exact method of loading in the data is dependent on whether the data are in ASCII format or in the form of a Gauss data set After the data have been read in they must be in the form of dependent variable explanatory variables regime 1 explanatory variables regime 2 and then the probability of being in regime 1 s variables 3 Load the data set The dataset SWREVBUB DAT has the format VWRETD EXCESS BUBBLE_A BUBBLE3A BUBBLE _B BUBBLE C BUBBLE3C open fdata dataset nobs rowsf fdata nvars colsf fdata Loading nobs observations and nvars variables from dataset crsp readr fdata nobs Using EXCESS and BUBBLE_A swd crsp 2 3 nobs rows swd This leaves nobs observations swd swd 1 ones nobs 1 swd 2 ones nobs 1 swd 2 ones nobs 1 swd 2 2 meanc swd 2 2 The global variables now need to be set In particular the globals _n1 _n2 and _n3 must be set There are others that are optional like the
35. the dependent variable The relationship between this period s state and the previous periods states determines what set of procedures one uses In a first order Markov model hereafter referred to as the Markov model the current state is dependent on only the last period s state In a simple switching model the current state is independent of the previous periods states The Bank of Canada procedures are for two state models with a single dependent variable The model can be either a Markov model or a simple switching model The procedures allow switching in the parameters for any number of explanatory variables including non constant transition probabilities Several Bank of Canada working papers have used earlier versions of this code Van Nor den 1996 and van Norden and Schaller 1993 both use the simple switching code while Vigfusson 1996 uses the Markov switching code A Bank of Canada working paper by Gable van Norden and Vigfusson 1995 provides more detail on the analytic gradients for the Markov switching model The Gauss procedures described in this paper are available from the Bank of Canada s Library Website http www bank banque canada ca library under working papers or by anonymous FTP to ftp bank banque canada ca in the directory pub vigfuss Use of this code requires Gauss version 3 1 The maximum likelihood estimation also requires Max lik version 3 While Gauss itself is available on many platforms t
36. ution and usually requires a series of iterative approximations Finally the two weighting systems are similar enough that the existing procedures for gen erating initial parameter estimates for B4 Ba B3 O1 O2 may be used without modifica tion for either probit or logit models Procedures for use with these new models may be found in the files SWLILOGT G and SWCONLGT G Each file contains procedures returning the log likelihood function and its gradient The names and formats of the procedures are identical to those used in the original probit procedures swli and swgrad for SWLILOGT swecli and swecgrad for SWCONLGT Only the library file SWITCH LCG needs to be modified for a user to switch from using the probit model to the logit model One comments out the library refer ences to the original probit procedures and removes the comments from the logit proce dures 14 The only lines that differ from the probit version are those calculating the weights given to each regime phi3 and those calculating the derivative with respect to B3 which is given by the formula O B a B X3 So 9_ _ EQ 11 LOB 0 B e 2B5 1 eX Bs d ap Pp B gt Ba O where B4 0 B are the probability density functions of the residuals from regime 1 and 2 respectively Note that the accuracy of the gradient procedures has been checked against the numerical derivatives of the likelihood function using the Gauss procedure
Download Pdf Manuals
Related Search