Markov-Switching Vector Autoregressive Models: Monte Carlo Experiment, Impulse Response Analysis, and Granger-Causal Analysis

Transcription

1 Markov-Switching Vector Autoregressive Models: Monte Carlo Experiment, Impulse Response Analysis, and Granger-Causal Analysis Matthieu Droumaguet Thesis submitted for assessment with a view to obtaining the degree of Doctor of Economics of the European University Institute Florence, December 2012

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3 European University Institute Department of Economics Markov-Switching Vector Autoregressive Models: Monte Carlo Experiment, Impulse Response Analysis, and Granger-Causal Analysis Matthieu Droumaguet Thesis submitted for assessment with a view to obtaining the degree of Doctor of Economics of the European University Institute Examining Board Prof. Massimiliano Marcellino, European University Institute (Supervisor) Prof. Ana Beatriz Galvão, Queen Mary University of London Prof. Hans-Martin Krolzig, University of Kent Prof. Helmut Lütkepohl, DIW Berlin and Freie Universität Berlin Matthieu Droumaguet, 2012 No part of this thesis may be copied, reproduced or transmitted without prior permission of the author

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5 Contents 1 Monte Carlo characterization of MS-VARs Introduction VAR models with Markov-switching in regime Estimation Monte Carlo experiment Finite-sample evidence Summary and implications A Appendix 35 A.1 Monte Carlo experiment results for MS-VAR models A.2 Statistics ratios of MS-VAR models over VAR models Bibliography 65 2 Bayesian impulse responses for MS-VAR models Introduction Markov-switching vector autoregressive model Impulse responses Likelihood, prior, and posterior Gibbs sampler Nonlinearities in oil markets Conclusions B Appendix 107 B.1 Alternative classical approach, the rolling estimation B.2 Structural breaks, the Qu and Perron test iii

6 B.3 Kilian (2009) s impulse responses Bibliography Testing noncausality in MS-VAR models Introduction MS-VAR model Granger Causality - Following Warne (2000) Bayesian Testing The Block MH sampler for restricted MS-VAR models Granger causal analysis of US money-income data Conclusions C Appendix 157 C.1 Alternative restrictions for noncausality C.2 Summary of the posterior densities simulations C.3 Characterization of estimation efficiency Bibliography 163

7 Abstract This dissertation has for prime theme the exploration of nonlinear econometric models featuring a hidden Markov chain. Occasional and discrete shifts in regimes generate convenient nonlinear dynamics to econometric models, allowing for structural changes similar to the exogenous economic events occurring in reality. The first paper sets up a Monte Carlo experiment to explore the finite-sample properties of the estimates of vector autoregressive models subject to switches in regime governed by a hidden Markov chain. The main main finding of this article is that the accuracy with which regimes are determined by the expectation maximixation algorithm shows improvement when the dimension of the simulated series increases. However this gain comes at the cost of higher sample size requirements for models with more variables. The second paper advocates the use of Bayesian impulse responses for a Markovswitching vector autoregressive model. These responses are sensitive to the Markovswitching properties of the model and, based on densities, allow statistical inference to be conducted. Upon the premise of structural changes occurring on oil markets, the empirical results of Kilan (2009) are reinvestigated. The effects of the structural shocks are characterized over four estimated regimes. Over time, the regime dynamics are evolving into more competitive oil markets, with the collapse of the OPEC. Finally, the third paper proposes a method of testing restrictions for Granger noncausality in mean, variance and distribution in the framework of Markov-switching VAR models. Due to the nonlinearity of the restrictions derived by Warne (2000), classical tests have limited use. The computational tools for posterior inference consist of a novel Block Metropolis-Hastings sampling algorithm for estimation of the restricted models, and of standard methods of computing the Posterior Odds Ratio. The analysis may be applied to financial and macroeconomic time series with changes of parameter values over time and heteroskedasticity. v

8 Keywords: Markov-switching Vector Autoregressive models, Expectation Maximization algorithm, Monte Carlo experiment, Gibbs Sampling, Impulse Response Analysis, Granger Causality, Regime Inference, Posterior Odds Ratio, Block Metropolis-Hastings Sampling. JEL classification: C11, C15, C22, C32, C53, E32, Q43

9 Acknowledgements Professor Massimiliano Marcellino Professor Helmut Lütkepohl Tomasz Woźniak My family My dear friends Thank you vii

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11 Chapter 1 Characterization of the Estimates of Markov-Switching Vector Autoregressive Models Through Monte Carlo Simulations Abstract. Through a Monte Carlo experiment, this paper examines the finitesample properties of the estimates of vector autoregressive models subject to switches in regime governed by a hidden Markov chain. The main main finding of this article is that the accuracy with which regimes are determined by the EM algorithm shows improvement when the dimension of the simulated series increases. However this gain comes at the cost of higher sample size requirements for models with more variables. I thank Pierre Guérin, Helmut Lütkepohl, and Massimilliano Marcellino for their very useful comments on the paper. 1

12 2 CHAPTER 1. MONTE CARLO CHARACTERIZATION OF MS-VARS 1.1 Introduction The discipline of econometrics is devoted to the analysis and test of the empirical relationships between economic variables. Hendry (1996) chronicles the historical debate over the constancy of the parameters underlying economic models, whose essence lies in the essay by Robbins (1932), doubting about the existence of permanent and constant values for the formal categories of economic analysis. Or rephrased through Robbins s frugal metaphor: The demand for herrings, however, is not a simple derivative of needs. It is, as it were, a function of a great and many apparently independent variables. It is a function of fashion, and by fashion is meant something more than the ephemeral results of an Eat British Herrings campain; the demand for herrings might be substantially changed by a change in theological views of the economic subjects entering the market. It is a function of the availability of other foods. [...] Discoveries in the art of cooking may change their relative desirability. Is it possible reasonably to suppose that coefficients derived from the observation of a particular herring market at a particular time and place have any permanent significance - save as Economic History? Major exogenous events, such as the formation of the international monetary system at Bretton Woods after the second world war, are quite likely to redefine the economic landscape and arguably to change the predictive power of formerly insightful econometric models. Hendry (1996) futuristically illustrates this point: An analogy might be a spacecraft to a distant planet being exactly on course and forecast to land successfully, just before being destroyed by a meteorite. The huge literature on tests for structural breaks, surveyed in Hansen (2001)orPerron (2006) testifies to the whole attention that econometricians pay to this phenomena. Once agreed that doubt may be cast upon the stability of some data generating processes governing economic time series, one needs to find a methodology to deal with it. An appealing econometric framework taking into account such structural changes is the one including discrete regimes governed by a hidden Markov chain, modeling time-series as combination of data-generating processes, and popularized in Hamilton (1989). Occasional and discrete shifts in regimes generate the required nonlinear dynamics to econometric models, allowing for structural changes similar to the exogenous economic events occurring in reality. The unobservable characteristic of the Markov chain is also convenient for the econometrician who in practice has to draw probabilistic inference about what the current regime of the time series is. The growing popularity of models with regime switching and the large scope of investigated economic time series for which dramatic breaks in

13 1.1. INTRODUCTION 3 their behavior occur due to some event is surveyed in Hamilton (2008). Among the most famous applications of such models, is certainly Hamilton (1989), where the succession of expansionary and recessionary phases in business cycles is considered. Sims and Zha (2006) use switches in regimes within a structural vector autoregressive [VAR] model to assess the impact of changes in the U.S. monetary policy. Currency crises were also studied through the Markov-switching framework in Jeanne and Masson (2000), with the empirical example of speculative attacks against the French franc in The area of fiscal policy is examined by Davig (2004), with the U.S. tax reforms of 1964 and Markov-switching models are not restricted to economic time series, and applications to financial time series also have been considered, in for instance Dai et al. (2007), where the latent variables introduce regime-shift risks to dynamic term structure model used for U.S. Treasury zero-coupon bond yields. However, the finite-sample properties of vector autoregressive models with shifts in the regime have been scarcely studied. Besides Psaradakis and Sola (1998) who perform a Monte Carlo experiment on univariate autoregressive [AR] processes with Markov regimeswitching in the mean and in the variance, I am not aware of any attempt to characterize the estimates of such models by simulation. This is certainly due to the non-linearities present in the models, rendering their estimation problematic to program. Hence, while the estimation theory has already been formulated see Krolzig (1997) for full coverage of estimation few software packages for estimation are available to the practitioner. 1 The contribution of this paper is twofold. Firstly it extends Psaradakis and Sola (1998) to multivariate time series with up to 20 equations. Markov switching vector autoregressive [MS VAR] models considered for the staged Monte Carlo experiment are models with switches in intercepts. Three classes of models are scrutinized: models with regime switches in the intercepts only, models with regime switches in the variance only, and models with regime switches in all the model parameters, in other words the intercepts vector, the autoregressive coefficients matrix, and the variance-covariance matrix. Secondly, studies such as Ang and Bekaert (2002) show that incorporating incorporating international short-rate and term spread information to interest rate series provide better regime classification than in the univariate case. Hence I consider statistics for evaluating the accuracy in the estimation of the latent regime in the univariate and multivariate cases. This can be seen as an exercise of precision for dating the turning points of the time series. 1 An overview of existing software packages is provided in Section 1.3.

14 4 CHAPTER 1. MONTE CARLO CHARACTERIZATION OF MS-VARS The results of this Monte Carlo experiment pave the way for future applications making use of large dimensional MS VAR models. The gain of precision in regime estimation occurring when adding new variables to the model, justifies the use of large MS VAR models in applications involving regime changes in the economy, and the detection of turning points. The remainder of the present paper is structured as follows. Section 1.2 introduces multivariate models with switches in regime governed by Markov chains. Section 1.3 discusses the estimation of such processes and the algorithm used to estimate them, the expectation maximization [EM] algorithm. Next, the setup for a Monte Carlo experiment is devised in Section 1.4. Finally, Section 1.5 dissects the results of the experiment. 1.2 Vector autoregressive models with Markov-switching in regime MS VAR are non-linear models, confluent of the linear vector autoregressive models and of the hidden Markov chain models. Krolzig (1997) discusses them in depth, from their origin to their estimation. Krolzig (1997) established the taxonomy of models belonging to the MS VAR class. Models can be classified into two categories: models with switches in their intercept and models with switches in their mean. While the seminal application of Hamilton (1989) confronted a MS VAR model incorporating switches in mean to U.S. GDP series for the study of business cycles this class of models is more complex to estimate due to the dependence of the mean to history of the latent variable. 2 The class of models with switches in intercept, comparatively behaving nicely in terms of estimation, are more suited to a Monte Carlo experiment. Among those two categories, models can be further classified, depending on which of VAR parameters are allowed to vary across regimes. The three VAR parameters are the intercept (or mean), the autoregressive coefficients, and the variance-covariance matrix. The next section is dedicated to the three types of models estimated in this Monte Carlo experiment. 2 My experience in estimating models with switches in the mean was unfruitful, the EM algorithm suffering from convergence issues. Even after convergence, the estimates were heavily depending on the parameters initial values.

15 1.2. VAR MODELS WITH MARKOV-SWITCHING IN REGIME MSI(M) VAR(p) model In MSI(M) VAR(p) models, as defined in Krolzig (1997), only the intercepts vary across regimes. M stands for the number of regimes and p for the number of lags of autoregressive terms to take into account. If y t is a K dimensional time-series, the corresponding MSI VAR model is written as: A 01 + p i=1 A iy t i +Σ 1 2 e t y t =., (1.1) A 0M + p i=1 A iy t i +Σ 1 2 e t where e t NID(0, I K ). Each regime is characterized by an intercept A 0i. The autoregressive terms A 1,...,A p, and a variance-covariance matrix Σ are common across all regimes according to a hidden Markov chain. This model is based on the assumption of varying intercepts according to the state of the economy controlled by the unobserved variable s t. Traditionally, and abstracting the difference between switches in mean and switches in intercepts, MSI(M) VAR(p) models were used business cycle applications, the first of them being Hamilton (1989). To complete the description of the data-generating process, one introduces a model for the regime generating process, which then allows to infer the evolutions of regimes from the data. In Markov-switching models, the unobservable realization of the regime s t {1,...,M} is governed by a discrete time, discrete state Markov stochastic process, which is defined by the transition probabilities: M p ij = Pr(s t+1 = j s t = i), p ij = 1, for all i, j {1,, M}. j=1 The transition probabilities between the states are collected into the transition probability matrix P: p 11 p p 1M p 21 p p 2M P = p M1 p M2... p MM s t follows an ergodic M state Markov process. A Markov chain is irreducible in the sense

16 6 CHAPTER 1. MONTE CARLO CHARACTERIZATION OF MS-VARS that no state is absorbing, i.e. when occurring the Chain does not stay stuck into a state. Ergodicity of the chain refers to the fact that each states are aperiodic and recurrent. Under these two conditions the ergodic probability vector of the Markov chain can be interpreted as the unconditional probability distribution of the states MSH(M) VAR(p) model In MSH(M) VAR(p) models, only the variance covariance matrix varies across regime. They are written as: A 0 + p i=1 A iy t i +Σ 1 2 y t =. A 0 + p i=1 A iy t i +Σ e t M e t, (1.2) where e t NID(0, I K ). Each regime is characterized by its proper variance-covariance matrix Σ i. With Markovswitching heteroskedasticity, the variance of errors can also differ between the regimes. After the change in regime there is thus an immediate one-time jump in the variance of errors. The intercept A 0 and autoregressive terms A 1,...,A p remain constant over all regimes. This model is based on the assumption of varying heteroskedasticity according to the state of the economy, controlled by the latent variable s t of the same nature as in MSI VAR models. These models have recently been used in Lanne et al. (2010) where within the reduced form error covariance matrix varying across states context the Markov regime switching property is exploited to identify structural shocks MSIAH(M) VAR(p) model The less restrictive MS VAR specification is the one where all parameters of the process are conditioned on the state s t. MSIAH VAR model are written as: where e t NID(0, I K ). A 01 + p i=1 A i1y t i +Σ e t y t =. A 0M + p i=1 A imy t i +Σ 1 2 M e t, (1.3)

17 1.3. ESTIMATION 7 Each regime is characterized by an intercept A 0i, autoregressive parameter matrices A 1i,...,A pi, and a variance-covariance matrix Σ i. In this general specification all parameters are allowed to switch between regimes according to a hidden Markov chain. This model is also based on the assumption of varying model parameters according to the state of the economy controlled by the unobserved variable s t, similarly to the former two models. These models introducing switches in the autoregressive parameters, have thus been used for impulse response analysis. For instance Ehrmann et al. (2003) propose to study regime-dependent impulse responses in the context of such models, conditional on staying on the regime after the shock. 1.3 Estimation Estimation techniques Estimation of Markov-switching autoregressive models has been initiated in Hamilton (1989). This paper describes how to draw probabilistic inference about the latent state s t given observations on y t, giving birth to the so called Hamilton filter. It then relates this result to the sample likelihood, which can be estimated on series with the help of numerical optimization methods using gradient methods, such as the Newton-Raphson algorithm. 3 Despite being appropriate for the estimation of a restricted number of parameters, the use of numerical optimizers is prohibited for larger dimensional systems or when the number of lags increase. An answer to that is the expectation maximization algorithm, introduced to the Markov-switching models of time-series econometrics in Hamilton (1990). The EM algorithm by construction finds an analytic solution to the sample likelihood derivatives from the smoothed inference about the unobserved regime s t. Estimation of higher dimensional models is permitted by the EM algorithm. Krolzig (1997) provides analytical solution to the maximization step for the whole class of Markov-switching models. Another argument in favor of the EM algorithm over the maximization of the likelihood using the calculation of gradients is made by Mizrach and Watkins (1999), and is related to the existence of local maxima in the likelihood function associated with Markov switching models. Mixture distributions possibly have as many local maxima as there are regimes in the model, and likelihood functions derived from these densities inherit the same features. The EM algorithm however, not involving the hill-climbing of any likelihood surface but 3 The book Kim and Nelson (1999) also presents an estimation strategy based on the Hamilton filter associated to numerical optimization algorithms.

18 8 CHAPTER 1. MONTE CARLO CHARACTERIZATION OF MS-VARS rather providing an algebraic solution to the maximization problem, may perform better in avoid local maxima. Exhaustiveness requires to refer to the recent developments of Sims and Zha (2006), presenting an application of the estimation of MSVAR models within the Bayesian framework. The approach is flexible, allows to work with multivariate series, and additionally provides tools to compare model specifications, through their Marginal Data Densities. Software packages GAUSS source code replicating Hamilton (1989) or examples of the book Kim and Nelson (1999) are provided by the authors. Also, Bellone (2005) wrote the open-source MSVARlib package in GAUSS. However all of these programs use numerical optimizers, hence are not appropriate for higher dimensional estimation. Krolzig (1998) implemented the models described in Krolzig (1997) in the proprietary software package Ox. The closed nature of this program renders it impossible to use beyond the scope the authors allow us to. No modification of the algorithm, nor Monte Carlo experiment is possible through this program and its use has again to be discarded. Sims et al. (2008) provide the theoretical framework to Bayesian estimation of MSVAR models, as well as some Matlab and C++ programs for practitioners. While very promising, the code is not yet polished enough to be usable. Staying in the classical framework, the GAUSS programs developed by Warne (1999) make use of the EM algorithm. I preferred to use the open source software language R 4 to implement the EM algorithm described in Krolzig (1997). 5 R s openness makes it a fast evolving programming environment for which one can release packages that are likely to be used by other practitioners Implementation of the Expectation Maximization algorithm The implementation of the EM algorithm developed for this article is flexible because it estimates different type of models, with a flexibility in the following parameters: Number of regimes, M. Number of lags in the autoregressive part, p. 4 The R computing language is developed by the R Development Core Team (2009). 5 This paper focusing on the algorithm of Krolzig (1997), a rapid comparison of the results yielded by the package in Ox and this implementation was performed. Similarity in the estimates ensured that the present implementation was correct.

19 1.3. ESTIMATION 9 Number of equations in the series, either univariate (K = 1) or multivariate (K = 2, 5, 10, 20). Initialization The parameters to be initiated are the matrices of autoregressive coefficients, A 1i,, A pi for each regime i {1,...,M} in the case of MSIAH VAR models, the matrix of transition probabilities P, and the initial state ˆξ The procedure is automatized and the approach is similar to the one employed in Bellone (2005). For the intercepts and autoregressive terms, I compute the ordinary least squares [OLS] regression on the whole or split series, depending on which model is estimated. 7 From the OLS regression results I compute the residuals either for each regime or on the whole series, as well as their variance-covariance matrix, used later in the expectation step. The transition probability matrix P is initialized with arbitrarily diagonal values. 8 Offdiagonals columns of each row share the remaining probabilities, so that the transition probabilities for each state sum up to unity. Expectation step The BLHK filter, as described in Krolzig (1997), performs the filtering and smoothing operations on the regime probabilities ξ t. Pr (s t = 1) ξ t =. Pr (s t = M) 6 Hamilton (1994) defines P(s t = j Y t,θ) denotes the conditional probability that the analyst assigns the possibility that the tth observation was generated by regime j. Those probabilities are collected for j = 1,...,M in the (M 1) vector ˆξ t t. 7 Before splitting them, Bellone (2005) sorts the series by the values of the first column. While this approach seems reasonable for business cycle applications with univariate series, I do not proceed like that for MSH VAR and MSIAH VAR, in order to release the assumption that one series is more prominent than others for determining the value of the Markov chain. However for MSI VAR models series have to be sorted beforehand, otherwise the resulting initialization parameters are often too similar for the EM algorithm to differentiate between them, which results in poor convergence performance of the algorithm. 8 Diagonals of 0.7 are used for the simulations performed throughout this paper.

20 10 CHAPTER 1. MONTE CARLO CHARACTERIZATION OF MS-VARS Filtering The filter introduced by Hamilton (1989) is an iterative algorithm calculating the optimal forecast of the value of ξ t+1 on the basis of the information set in t consisting of the observed values of y t, namely Y t = (y t, y t 1,, y 1 p ). The initial state ˆξ 1 0 needs to be initialized with some value to start the iterations. As suggested in Hamilton (1994), I use the vector of ergodic regime probabilities ξ =Π, where Π satisfies the equation PΠ =Π. This step is a forward recursion, i.e. for t = 1,, T, written as: ( ) ˆξ t+1 t = P η t ˆξ t t 1 ( ), ηt F ˆξ t 1 t 1 1 M where F = P and η t is the collection of M densities and is defined as: p ( ) y t s t = 1, Y t 1 η t =. p ( ) y t s t = M, Y t 1 Smoothing Full-sample information is used to make an inference about the unobserved regimes by incorporating the previously neglected sample information Y t+1 T = (y t+1,, y T ) into the inference about ξ t. This step is a backward recursion, for j = 1,, T 1. The iteration consists in the following equation: ˆξ T j T = [ P ( ˆξ T j+1 T ˆξ T j+1 T j )] ˆξ T j T j Matrix of transition probabilities The transition probabilities matrix are estimated from the filtered, and the smoothed probabilities ˆξ t t and ˆξ t T calculated in the expectation step. The (M 2 1) vector of transition probabilities ρ obtained as follows: 1. Calculate the joint probabilities p ( ) s t+1 = j,,s t = i Y t for all st, s t+1 = 1,, M gives the (M 2 1) vector of regime probabilities: 9 ˆξ (2) t T = vec(p) [( ˆξ (1) t+1 T ) ] ˆξ (1) t+1 t ˆξ (1) t t 9 The transition probability matrix P from previous iterations is used during this step.

21 1.3. ESTIMATION Sum up over all T: 3. Write: ˆξ (2) = T 1 t=1 ˆξ (2) t T ˆξ (1) = ( 1 M I M) ˆξ (2) 4. The vector of transition probabilities ρ is obtained by: ρ = ˆξ (2) ( 1 M ˆξ (1)) 5. Finally, reshape ρ into a (M M) matrix to get the transitions probability matrix P. Normal equations The maximization step typically boils down to the computation of the maximum likelihood for the model: L (λ Y) := p (Y T Y 0 ; λ) T = p (Y t Y t 1 ; λ) = = t=1 T t=1 p ( y t ξ t, Y t 1,θ ) p (ξ t Y t 1 ; λ) ξ t T η ˆξ t t t 1 t=1 The conditional densities p ( y t ξ t, Y t 1,θ ) are composed of several normal distributions, see Krolzig (1997), hence rendering L non-normal. Directly maximizing log L thus requires the use of non-linear optimization algorithms, which are costly from the computational point of view and increasingly with the estimation of multivariate models. In the EM

22 12 CHAPTER 1. MONTE CARLO CHARACTERIZATION OF MS-VARS algorithm, the parameters (intercepts, autoregressive terms, and variance) are derived from the first-order condition of the maximum likelihood estimation. Krolzig (1997) shows that it is sufficient to do only one single generalized least squares [GLS] estimation within each maximization step to ensure convergence to a stationary point of the likelihood. MSI VAR models The regression equation for MSI VAR models is: 10 M y = (Ξ m 1 T I K ) A 0m + ( ) X I K A + u, u N (0, Ω), Ω=IT Σ m=1 Writing β as the collection of ν m and α in a (( M + Kp ) K ) matrix, the GLS estimates are : 11 ˆβ = ˆΞ X ˆΞˆΞˆΞ ˆΞˆΞˆΞ 1 X ˆΞˆΞˆΞ X X Ȳ X Ȳ ˆΣ = ˆT 1 Û ˆΞÛ MSH VAR models The regression equation in the MSH VAR case is the following: M M y = (1 T I K ) β + u, u N (0, Ω), Ω= Ξ m Σ m m=1 m=1 10 Defining y as y = ( ) y 1,...,y 11 T Notations are introduced: And for the indicators of smoothed probabilities: Y j = ( y 1 j,...,y T j ) X = ( Y 1,...,Y p ) Û = 1 M Y Z ˆβ Z = ( I M 1 T, 1 M X ) ˆΞˆΞˆΞ = ( ˆξ 1 T,..., ˆξ T T ) ˆΞ = diag ( 1 T ˆΞˆΞˆΞ )

23 1.3. ESTIMATION 13 The GLS estimates are written as 12 ˆβ = M ( X ˆΞ m X ) M ( ) ˆΣ 1 m 1 X ˆΞ m ˆΣ 1 m y m=1 ˆΣ m = ˆT 1 m Û ˆΞ m Û m=1 MSIAH VAR models The regression equation for the MSIAH VAR case is the following: M ( ) M y = Ξm X I K βm + u, u N (0, Ω), Ω= Ξ m Σ m m=1 m=1 In that case, the GLS estimates are written as: 13 ˆβ m = ( X ˆΞ m X ) 1 X ˆΞ m Y ˆΣ m = ˆT m 1 ˆΞ Û m m Û m 12 Some of the notations change: X = ( ) 1 T, Y 1,...,Y p Û m = Y ( ) X I K ˆβ And for the indicators of smoothed probabilities: ˆξ m = ( ˆξ m1 T,..., ) ˆξ mt T ˆΞ m = diag ( ) ˆξ m ˆT m = tr ( ) ˆΞ m = 1T ˆξ m 13 The residuals are now obtained by: Û m = Y X ˆβ m

24 14 CHAPTER 1. MONTE CARLO CHARACTERIZATION OF MS-VARS Convergence criteria After the initialization, the algorithm iterates on the expectation and maximization steps, until convergence. Two measures for convergence between j th and j + 1 th iterations are considered here. The first one is the absolute percentage change in the logarithm of the likelihood value, calculated as: ln L ( λ (j+1) ) ( ) Y T ln L λ (j) YT Δ 1 = 100 ln L ( λ (j) ) Y T The second one is the maximum change between iterations j and j + 1, among all parameters, formulated as follows: Δ 2 = max i { λ(j+1) i λ (j) i } Convergence is considered achieved when one of the criterion is judged small enough, i.e. Δ 1 δ or Δ 2 δ. 14 In order not to enter into infinite iterations, a parameter for the maximum number of allowed iterations before convergence is implemented. If the EM has not converged within it, the algorithm stops. 15 Error handling and information about convergence are also provided by the algorithm returns, facilitating simulation exercises such as bootstrap or Monte Carlo experiments. 1.4 Monte Carlo experiment The purpose of this Monte Carlo experiment is to study the properties of the EM algorithm for the estimation of simulated univariate and multivariate series from MSI VAR, MSH VAR, and MSIAH VAR models, paying particular attention to higher dimensional systems composed of many variables. First of all, we are interested in finding out whether these models are estimable at all, and under which circumstances. All Markov-switching VAR models may not be equally 14 Typically, a value of δ = 10 6 is used. 15 By default the maximum number of iterations allowed before convergence of the algorithm is 100.

25 1.4. MONTE CARLO EXPERIMENT 15 easy to estimate for the EM algorithm, and for some sets of parameters, the algorithm may not converge at all. Furthermore, we are interested in looking at the accuracy of the estimates, consisting of the mean square error in the estimated model parameters, i.e. the intercepts, autoregressive coefficients, and variance-covariance matrices. Finally, I take a closer look at the estimates of the realizations of the hidden Markov chain, to see how well the EM manages to estimate the regimes Specificities of the experiment: multiple dimensions and distance between regimes In time-series econometrics, a Monte Carlo experiment can usually be decomposed into four phases: 1. Choice of model parameters. 2. Simulation of N series using the parameter set. Each series is generated by drawing a new history of residuals from the appropriate random distribution. 3. Estimation of each individual series using the algorithm. 4. Aggregation of the results of the individual estimations into the final result. In a standard Monte Carlo experiment, parameters remain identical over all the simulations. While this is the proper way to proceed to gather finite-sample evidence on models where the number of parameters stays the same over all the simulations, what to do when for instance one wants to study the properties of an estimator over different dimensions of one model? 16 The number of parameters to estimate varies with the number of equations in the model and thus steps 1, 2, and 4 of the aforementioned list do not fit to such an experiment. The Monte Carlo experimental design has to be adapted to characterizes the properties of estimators over a different number of possible parameters. Regarding the dimensions of interest for a Monte Carlo experiment on Markov switching VAR model, consider the peculiarity in such models: the switch in regimes through the latent variable following a Markov process. Intuitively, the relative distance between the regimes should affect the estimation, processes composed of regimes that are very close from another should be more cumbersome to estimate than processes that have very 16 The number of equations in the model K is varying over the experiment.

26 16 CHAPTER 1. MONTE CARLO CHARACTERIZATION OF MS-VARS distant regimes. Also, a regime which is rarely occurring over the sample will also be more problematic to estimate than a regime that occurs frequently over the sample. 17. The parameters defining the distance between regimes are the intercepts, the autoregressive coefficients, and the variance-covariance matrix. For each of these, two cases will be considered in the experiment, one close and another distant case, see details in next section. Finally, two factors that may influence the estimation are also considered. The first is the stationarity in the simulated series as their distance to the unit root. The second one is the persistence of the regimes, expressed in the matrix of transition probabilities between regimes. The next section details the design of the experiment Experimental design Parameters The experiment is conducted on MSI(2) VAR(1), MSH(2) VAR(1), MSIAH(2) VAR(1) processes, i.e. with M = 2 regimes, and p = 1 lag in the autoregressive part. It is conducted over the following dimensions: the sample size (T) of the processes are the same as in Psaradakis and Sola (1998) and are the ones typically used in practice. Univariate and multivariate processes are simulated and estimated, up to series containing 20 equations simulations are repeated for every set of parameters. This is summarized as: M = 2 p = 1 T {100, 200, 400, 800} K {1, 2, 5, 10, 20} N = 1000 MSI VAR In MSI(2) VAR(1) models, the intercept coefficients are regime dependent (A 01 and A 02 ). We set up two distances between the two regimes. For close intercepts, the first regime contains values of -1 for all equations whereas the second regime is valued 17 This is a neglected dimension in this paper, for which only a minimum number of occurrences of a regime over the simulated series is required to be sufficient.

27 1.4. MONTE CARLO EXPERIMENT 17 to 1. In the more distant case, the first regime has intercepts of -5 and the second one of 5. the (K 1) vector of intercepts, is invariant across regimes and contains only ones for all equations. A 1, the (K K) matrix of autoregressive coefficients is a diagonal matrix. All the diagonal values are equal to 0.6 in the stationary case, all eigenvalues well inside the unit circle, or 0.9 which is closer from the non-stationarity region. Σ, the (K K) variance-covariance matrix is also diagonal. For the transition probabilities, two cases are considered. The more persistent case will have an average expected duration of a regime of 20 periods (0.95 in the diagonals of P). In the less persistent case, regimes will be expected to last for 5 periods only. All elements (A 01, A 02 ) {( 1, 1), ( 5, 5)} Diagonals of A 1 {0.6, 0.9}. Non-diagonal elements are 0 Diagonals of Σ=1. Non-diagonal elements are 0 ( ) p 11, p 22 {(0.8, 0.8), (0.95, 0.95)} Combining these cases to consider their separate and joint effect, we have to consider 8 experiments for MSI VAR models. Adding the sample size dimension and the number of equations ones to the experiment yields 160 experiments in total, each of them consisting of 1000 simulations plus estimations for MSI VAR models. MSH VAR A 0 the (K 1) vector of intercepts, is invariant across regimes and contains only ones for all equations. A 1, the (K K) matrix of autoregressive coefficients is a diagonal matrix. All the diagonal values are equal to 0.6 in the stationary case, all eigenvalues well in the safe zone of the unit circle, or 0.9 which is closer from the non-stationarity region. Σ, the (K K) variance-covariance matrix is also diagonal. In MSH(2) VAR(1) models, the variance is regime dependent and we consider either close regimes, where the diagonal elements of the first regime are 1 and of the second regimes are 5, or more distant regimes respectively with diagonal elements of 1 and 25. All elements A 0 = 1 Diagonals of A 1 {0.6, 0.9}. Non-diagonal elements are 0 Diagonals of (Σ 1, Σ 2 ) {(1, 5), (1, 25)}. Non-diagonal elements are 0

28 18 CHAPTER 1. MONTE CARLO CHARACTERIZATION OF MS-VARS ( p 11, p 22 ) {(0.8, 0.8), (0.95, 0.95)} 160 experiments are also conducted for MSH VAR model, each of them consisting of 1000 simulations plus estimations. MSIAH VAR In MSIAH(2) VAR(1) models, all parameters vary across regimes. Adding up to the variations of intercepts and variance-covariance of the same nature as for the precedent models, a variation of the autoregressive matrices is integrated. Closer matrices are diagonals of -0.6 in the first regime and 0.6 in the second. The numbers become -0.9 and 0.9 for the more distant case. 18 All elements (A 01, A 02 ) {( 1, 1), ( 5, 5)} Diagonals of (A 11, A 12 ) {( 0.6, 0, 6), ( 0.9, 0, 9)}. Non-diagonal elements are 0 Diagonals of (Σ 1, Σ 2 ) {(1, 5), (1, 25)}. Non-diagonal elements are 0 ( ) p 11, p 22 {(0.8, 0.8), (0.95, 0.95)} Again, 160 experiments are also conducted for MSIAH VAR model, each of them consisting of 1000 simulations plus estimations. Benchmark: VAR As a point of reference, experiments are run on K-dimensional VAR(p) models: p y t = A 0 + A i y t i +Σ 1 2 et, (1.4) where e t NID(0, I K ). The parameters varying over the experiments are: i=1 All elements A 0 {1, 5} Diagonals of A 1 {0.6, 0.9}. Non-diagonal elements are 0 Diagonals of Σ {1, 5, 25}. Non-diagonal elements are 0 18 Hence we can not discriminate if it is the distance to unit root or between the AR coefficients that affect the estimation.

29 1.5. FINITE-SAMPLE EVIDENCE 19 Criteria for successful experiment EM estimation For each call to the EM algorithm, the maximum number of iterations of the EM algorithm authorized before achieving convergence is 100. If convergence occurs, the algorithm is considered to be successful, provided there was a log-likelihood gain of at least 5% between the first and last iterations. 19 Each experiment Each experiment consists in 1000 simulated and estimated series. To reduce the computational burden of the whole procedure, the upper limit of 100 failed EM estimations (as defined above) per experiment is set. For each experiment, I report the rate of failed estimations. The following section analyzes the outcomes of the Monte Carlo experiment. 1.5 Finite-sample evidence The complete results for all experiments are detailed in the tables of the Appendix. Common and rather intuitive results arise among the three Markov-switching type of estimated models. First of all when the number of observations increase over the simulations (from 100 to 800 per simulation), the algorithms converge more often and the experiments are considered as more successful. Naturally a higher sample size facilitates the estimation. Also, the sample size necessary to estimate processes naturally increases with the dimension of the simulated series (univariate ones requiring less voluminous series to be estimated than higher dimensional ones). The number of parameters to estimate inflates with the number of variables in the time series, as illustrated below for the models studied in this paper: 19 5% was arbitrarily chosen, based on my own experience. It was chosen as low as possible, so that not too many estimations are discarded.

30 20 CHAPTER 1. MONTE CARLO CHARACTERIZATION OF MS-VARS MSI(M) VAR(p) : M (M 1) } {{ } Transition probabilities MSH(M) VAR(p) : M (M 1) } {{ } Transition probabilities MSIAH(M) VAR(p) : M (M 1) } {{ } Transition probabilities + K ( M + Kp ) + } {{ } Intercept + AR K (K + 1) 2 } {{ } Variance-covariance + K ( 1 + Kp ) KM (K + 1) + } {{ } 2 } {{ } Intercept + AR Variance-covariance + KM ( 1 + Kp ) KM (K + 1) + } {{ } 2 } {{ } Intercept + AR Variance-covariance Table 1.1: Number of model parameters for the models studied in the present Monte Carlo experiment. K stands for the number of equations in the model. K MSI(2) VAR(1) MSH(2) VAR(1) MSIAH(2) VAR(1) As illustrated in Table 1.1, the number of parameters to estimate grows fast with an increase of the number of equations in the model (K). MSI VAR have the least parameters among the three types of studied models. Due to the variance-covariance matrix and/or the autoregressive terms, the MSH VAR and MSIAH VAR models see their regime-dependent parameters grow to the order 2 of K, whereas this rate of growth for regime-dependent parameters is not squared for MSI VAR models. In any case, the three models possess a high number of parameters to estimate for systems with many equations. This explains the deterioration of the estimation performance of the algorithm for these models when the sample size stays small Percentage of successful estimations Tables A.1, A.6, and A.11 display the percentage of successful estimations for the thousand simulations of each experiment, respectively for MSI VAR, MSH VAR, and MSIAH VAR models. The next sections present the estimation successes specific to the three types of

31 1.5. FINITE-SAMPLE EVIDENCE 21 models. MSI VAR Being the models with the lowest number of parameters to estimate, models with regime change in intercept only are the most successfully estimated among the three studied ones. The distance between the intercepts is the only regime-varying parameter for MSI VAR models. This Monte Carlo experiment was designed with two cases for intercept distance between regimes, one with close regimes and another with more distant ones. The closer case has intercepts vectors of the first regime equal to -1 and intercepts vectors of the second regime equal to 1, whereas the in the more distant case, intercepts are -5 and 5 in each regime. The EM algorithm performs better for the more distant case, where it discriminates better between the two regimes. For the closer case, the worst rate of failures of the experiment is 24%, to compare with 12.4% for the more distant intercepts case. Series with smaller sample size are less successfully estimated, and this phenomena is more pronounced for series of higher dimensions. The persistence of the processes is also a varying dimension over the experiments. Results indicate that for MSI VAR, processes closer to unit root 20 are estimated better, since no experiment was considered successful for processes more distant to the unit root. The last source of variation between experiments is the persistence of the regimes, introduced in the form of the transition probabilities between regimes. They can be either less persistent with probabilities to remain in the same regime in next period of 0.8 for both regimes or more persistent with probabilities of Simulated series with lower regime persistence are consistently more successfully estimated by the EM in comparison to series with more persistent processes. MSH VAR Experiments on MSH VAR models display more failures than for MSI VAR models. The distance between the diagonals of the variance-covariance matrices in each regime is once again an important determinant of success in the EM estimation, with as should be expected better estimation performance for simulated processes having higher difference between regimes in their variance-covariance matrices. In both the close and distant 20 The diagonals of the autoregressive coefficients matrix A 1 were chosen to be equal either to 0.6 or 0.9.

32 22 CHAPTER 1. MONTE CARLO CHARACTERIZATION OF MS-VARS cases, the success rates of the experiments increase with the number of equations in the models. However, adding equations to the system only increases the efficiency of the algorithm provided the sample size for is large enough. Indeed, small sample size become a handicap as the number of parameters to estimate increase. Every equation of the simulated series being subject to switches at simultaneous times certainly eases the regime detection occurring during the expectation step in the EM algorithm. Different distances to unit root for the processes i.e. A 1 = 0.6 ora 1 = 0.9, does not noticeably modify the estimation performance. Yet, the persistence in the processes has an strong impact on the experiment successes, and more persistent regimes (P = 0.95) have more easily estimated than models with less persistent regimes. This is not surprising, as the only source of variation between the regimes is the variance, and intuitively prolonged periods of the same variance should be easier to detect than rapid switches between different regimes of variance. MSIAH VAR Among the three categories of models, the MSIAH VARwitness the most contrast between their regimes, with regime switches in intercepts, autoregressive terms, and covariances matrices. The EM algorithm would be expected to estimate these models with ease, in comparison to MSI VAR and MSI VAR models. However the number of parameters to estimate is much higher, as shows the Table 1.1. As expected, MSIAH VAR models are the successfully estimated in this Monte Carlo analysis, with only few failures for univariate models, K = 1, or in higher dimension when the number of observations is low. The EM algorithm has more latitude to distinguish between the regimes in its expectation step, which results in better convergence for these models. However, due to the high number of parameters to estimate, higher dimensional systems are less successfully estimated than models less parameters intensive such as the MSI VAR. One can clearly observe a decrease in the rate of success when jumping from 10 equations MSIAH VAR models to models with 20 equations. Higher distance between regimes in the three regime-varying parameters yields better convergence performance of the EM algorithm, for low dimensional MS VAR models with K = 1, 2. This tendency vanishes for higher dimensions for which closer regimes have more success rate in estimation. The persistence in the processes slightly improves on the successes.

33 1.5. FINITE-SAMPLE EVIDENCE 23 As for the regime persistence, it does not notably influence the experiments success rate. However, processes with more persistent regimes are subject to higher algorithm failure rate when the sample size is to small for the number of parameters to estimate, for example when the processes have 10 or 20 equations and when the simulated series have 100 or 200 observations Empirical distribution of the Maximum Likelihood Estimator Psaradakis and Sola (1998) considered the mean, skewness and kurtosis of the estimators. Here, due to the high number of experiments to summarize, only the second moment of the error mean squared error [MSE] incorporating bias and variance of the estimator, is considered. MSI VAR Intercepts The mean squared error for the first intercept coefficient for each regimes, A 01 and A 02, are summarized in Table A.2. Results indicate that the mean squared error decreases when the sample size gets larger. The precision in the intercept estimates remains about the same for different distances between regimes, provided the sample size is large enough, otherwise closer regimes are logically more precisely estimated, as an artefact of the experiment. The estimation of intercepts for models further away from the unit root constantly outperforms the estimation of processes closer from the unit root, but only moderately. Higher persistences in the Markov chain deteriorate the precision in the intercepts estimation. It is worth to notice that for a large enough sample size, the mean squared error does not suffer from a deterioration when the number of equations in the model increase. Also, when comparing the magnitude of the mean squared errors between MSI VAR models and the ones of standard VAR models with the comparable parameters, which is done through ratios exposed in Table A.16, in almost every experiment the intercepts are estimated with much more precision for MSI VAR models. The ratios have indeed values comprised between and 1.1. More distant regimes for the MSI VAR, where the EM estimates better, produce the best results. Autoregressive coefficients Table A.3 reports the mean squared error statistics of the Monte Carlo experiments for the upper-left element of A 1, the matrix of autoregressive

34 24 CHAPTER 1. MONTE CARLO CHARACTERIZATION OF MS-VARS coefficients. In comparison with the estimates of the intercepts for the same MSI VAR processes, estimates of the autoregressive terms have the same magnitude of mean squared errors. Nor the distance between the regimes nor the persistences in the regimes or in the processes influence the mean square error of the autoregressive coefficients. Neither an increase in the number of variables K results in worse estimates of the AR coefficients, except for K = 10 or K = 20 where smaller sample size are less precisely estimated. Comparing to the estimates of standard VAR models, as reported in the form of ratios in Table A.17, models with regime switches in their intercept show slightly better estimation precision the estimation of the AR coefficients, provided the sample size is large enough to estimate all the parameters. Variance-covariance matrix Table A.4 recapitulates the mean squared errors statistics of the Monte Carlo experiments for the first element of the variance-covariance matrix Σ. The results are very comparable to the ones concerning the intercepts of the same MSI VAR models, both with regards to the magnitude of the statistics and to the effects of the distance between regimes or persistence of the regimes. When increasing the number of variables in the system, the mean squared errors of the estimates of the variance-covariance matrix coefficients do remain fairly at the same levels that for lower dimensional MSI VAR models. Table A.18 indicates that the variance estimation of MSI VAR processes yields worse, though comparable precision to the simpler VAR processes. However, the ratios surge for higher dimensional systems with small sample size, with a maximum ratio of 1161 for K = 20. MSH VAR Intercepts Table A.7 shows the Monte Carlo experiments outcomes in the form of mean squared error statistics of the first element of the intercept vector A 0, constant over regimes for MSH VAR models. A higher distance between the regimes a higher variance difference between regimes in the MSH VAR case helps the EM algorithm to estimate the intercepts more precisely, as lower mean squared errors for the right-hand side panel witness. Roughly the statistics are at least two times higher for closer regimes. A higher persistence in the processes notably worsens the precision with which the