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1997 | Buch

In the last decade time series econometrics has changed dramatically. One increasingly prominent field has become the treatment of regime shifts and non-linear mod- elling strategies. While the importance ofregime shifts, particularly in macroeconometric systems, seems to be generally accepted, there is no established theory suggesting a unique approach for specifying econometric models that embed changes in regime. Kapitel lesen Erstes Kapitel lesen

Markov-Switching Vector Autoregressions

Modelling, Statistical Inference, and Application to Business Cycle Analysis

verfasst von: Dr. Hans-Martin Krolzig

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Economics and Mathematical Systems

Enthalten in: Professional Book Archive

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Über dieses Buch

This book contributes to re cent developments on the statistical analysis of multiple time series in the presence of regime shifts. Markov-switching models have become popular for modelling non-linearities and regime shifts, mainly, in univariate eco­ nomic time series. This study is intended to provide a systematic and operational ap­ proach to the econometric modelling of dynamic systems subject to shifts in regime, based on the Markov-switching vector autoregressive model. The study presents a comprehensive analysis of the theoretical properties of Markov-switching vector autoregressive processes and the related statistical methods. The statistical concepts are illustrated with applications to empirical business cyde research. This monograph is a revised version of my dissertation which has been accepted by the Economics Department of the Humboldt-University of Berlin in 1996. It con­ sists mainly of unpublished material which has been presented during the last years at conferences and in seminars. The major parts of this study were written while I was supported by the Deutsche Forschungsgemeinschajt (DFG), Berliner Graduier­ tenkolleg Angewandte Mikroökonomik and Sondeiforschungsbereich 373 at the Free University and Humboldt-University of Berlin. Work was finally completed in the project The Econometrics of Macroeconomic Forecasting founded by the Economic and Social Research Council (ESRC) at the Institute of Economies and Statistics, University of Oxford. It is a pleasure to record my thanks to these institutions for their support of my research embodied in this study.

Inhaltsverzeichnis

Frontmatter
In the last decade time series econometrics has changed dramatically. One increasingly prominent field has become the treatment of regime shifts and non-linear mod- elling strategies. While the importance ofregime shifts, particularly in macroeconometric systems, seems to be generally accepted, there is no established theory suggesting a unique approach for specifying econometric models that embed changes in regime.
Abstract
In the last decade time series econometrics has changed dramatically. One increasingly prominent field has become the treatment of regime shifts and non-linear mod- elling strategies. While the importance ofregime shifts, particularly in macroeconometric systems, seems to be generally accepted, there is no established theory suggesting a unique approach for specifying econometric models that embed changes in regime.
Hans-Martin Krolzig
Chapter 1. The Markov-Switching Vector Autoregressive Model
Abstract
This first chapter is devoted to a general introduction into the Markov-switching vector autoregressive (MS-VAR) time series model. In Section 1.2 we present the fundamental assumptions constituting this class of models. The discussion of the two components of MS-VAR processes will clarify their on time invariant vector auto-regressive and Markov-chain models. Some basic stochastic properties of MS-VAR processes are presented in Section 1.3. Finally, MS-VAR models are compared to alternative non-normal and non-linear time series models proposed in the literature. As most non-linear models have been developed for univariate time series, this discussion is restricted to this case. However, generalizations to the vector case are also considered.
Hans-Martin Krolzig
Chapter 2. The State-Space Representation
Abstract
In the following chapters we will be concerned with the statistical analysis of MS(M)-VAR(p) models. As a formal framework for these investigations we employ the state-space model which has been proven useful for the study of time series with unobservable states. In order to motivate the introduction of state-space representations for MS(M)-VAR(p) models it might be helpful to sketch its use for the three main tasks of statistical inference:
1.
Filtering & smoothing of regime probabilities: Given the conditional density function p(yt|Yt-1, ξt), the discrete Markovian chain as regime generating process ξt, and some assumptions about the initial state \({y_0} = {\left( {{{y'}_0},...,{{y'}_{1 - p}}} \right)^\prime }\) of the observed variables and the unobservable initial state ξ0 of the Markov chain, the complete density function p(ξ, Y) is specified. The statistical tools to provide inference for ξt given a specified observation set Yτ, τ ≤ T are the filter and smoother recursions which reconstruct the time path of the regime, \(\left\{ {{\xi _t}} \right\}_{t = 1}^T\) under alternative information sets:
 
  • $${\hat \xi _{t\left| \tau \right.}},\quad \tau < t\quad predicted\quad regime\,probabilities.$$
  • $${\hat \xi _{t\left| \tau \right.}},\quad \tau = t\quad filtered\quad regime\,probabilities,$$
  • $${\hat \xi _{t\left| \tau \right.}},\quad t < \tau \leqslant T\quad smoothed\quad regime\,probabilities.$$
  • In the following, mainly the filtered regime probabilities, \({\hat \xi _{t\left| t \right.}}\) and full-sample smoothed regime probabilities, \({\hat \xi _{t\left| T \right.}}\), are considered. See Chapter 5.
2.
Parameter estimation & testing: If the parameters of the model are un known, classical Maximum Likelihood as well as Bayesian estimation methods are feasible. Here, the filter and smoother recursions provide the analytical tool to construct and evaluate the likelihood function. See Chapters 6–9.
 
3.
Forecasting: Given the state-space form, prediction of the system is a straightforward task. See Chapter 4 and Section 8.5.
 
Hans-Martin Krolzig
Chapter 3. VARMA-Representation of MSI-VAR and MSM-VAR Processes
Abstract
The previous chapter introduced the state-space representation as the basic tool for describing vector autoregressive processes with Markovian regime shifts. This chapter looks in greater depth at the relationship between Markov-switching vector autoregressions and linear time series models. We develop a finite order VARMA representations theorem for vector autoregressive processes with Markovian regime shifts in the mean or the intercept term of the multiple time series. This result generalizes concepts recently proposed by Poskitt & Chung [1994] for univariate hidden Markov-chains, and by Krolzig [1995] for univariate MSM(M)-AR(p) and MSI(M)-AR(p) processes.
Hans-Martin Krolzig
Chapter 4. Forecasting MS-VAR Processes
Abstract
One major objective of time series analysis is the creation of suitable models for prediction. It is convenient to choose the optimal predictor \({\hat y_{t + h\left| t \right.}}\) in the sense of a minimizer of the mean squared prediction error (MSPE),
$${\hat y_{t + h\left| t \right.}}: = \arg \,\mathop {\min }\limits_{\hat y} {\rm E}\left[ {{{\left( {{y_{t + h}} - \hat y} \right)}^2}\left| {{\lambda _t}} \right.} \right].$$
(4.1)
Hans-Martin Krolzig
Chapter 5. The BLHK Filter
Abstract
An important task associated with the statistical analysis of MS-VAR models is discussed in this chapter: the filtering and smoothing of regime probabilities. In the MS-VAR model the state vector ξ t is given a structural interpretation. Thus an inference on this unobserved variable is of interest for its own sake. However, the filtered and smoothed state probabilities provide not only information about the regime at time t, but also open the way for the computation of the likelihood function and consequently for maximum likelihood estimation and likelihood ratio tests.
Hans-Martin Krolzig
Chapter 6. Maximum Likelihood Estimation
Abstract
In the last chapter attention was given to the determination of the state vector1 ξ for given observations Y and known parameters A. In this chapter the maximum likelihood estimation of the parameters \(\lambda = (\theta \prime ,\rho \prime ,\xi {\prime _0})\prime \) of an MS-VAR model is considered. The aim of this chapter is (i.) to provide the reader with an introduction to the methodological issues of ML estimation of MS-VAR models in general, (ii.) to propose with the EM algorithm an estimation technique for all discussed types of the MS-VAR models, (iii.) to inform the reader about alternative techniques which can be used for special purposes or model extensions and (iv.) to give some basic asymptotic results.
Hans-Martin Krolzig
Chapter 7. Model Selection and Model Checking
Abstract
The last two chapters have demonstrated that the estimation methods and filtering techniques are now well established for MS-VAR processes. Most unresolved questions arising in empirical investigations with MS-VAR models concern the issue of model specification. In Section 6.6 we discussed the asymptotic distribution of the maximum likelihood estimator of MS-VAR models. In the literature (cf. e.g. Hamilton [1993]) it has been assumed that standard asymptotic theory holds:
$$\sqrt T (\bar \lambda - \lambda )\mathop \to \limits^d N(0,{I^{ - 1}})$$
The asymptotic normal distribution of the ML estimator ensures that most model diagnostics and tests known from the time-invariant VAR(p) model (cf. the discussion in Lütkepohl [1991, ch. 4]) can be applied generally with only some slight modifications.
Hans-Martin Krolzig
Chapter 8. Multi-Move Gibbs Sampling
Abstract
In this section we discuss the use of simulation techniques to estimate and forecast MS-VAR processes. A general feature of MS-VAR models is that they approximate non-linear processes as piecewise linear by restricting the processes to be linear in each regime. Since the distribution of the observed variable y t is assumed normal conditional on the unobserved regime vector ξt, the MS-VAR model is well suited for Gibbs sampling techniques.
Hans-Martin Krolzig
Chapter 9. Comparative Analysis of Parameter Estimation in Particular MS-VAR Models
Abstract
The general framework for ML estimation of the MS(M)-VAR(p) model was laid out in Chapter 6. In Chapter 8 the methodological issues of Gibbs sampling and its conceptional differences to the EM algorithm have been discussed. In this chapter, we will focus on the technical aspects of estimation of the VAR coefficients under the various types of restrictions.1
Hans-Martin Krolzig
Chapter 10. Extensions of the Basic MS-VAR Model
Abstract
In the preceding chapters we have made three essential assumptions with regard to the specification of MS-VAR processes: we have assumed that (i.) the system is autonomous, i.e. no exogenous variables enter into the system, (ii.) the regime-dependent parameters depend only on the actual regime but not on the former history, and (iii.) the hidden Markov chain is homogeneous, i.e. the transition probabilities are time-invariant. As we have seen in the foregoing discussion, these assumptions allow for various specifications. Modelling with MS-VAR processes is discussed extensively in the last part of this study for some empirical investigations related to business cycle analysis. However, there might be situations where the assumptions made about the MS-VAR model result in limitations for modelling.
Hans-Martin Krolzig
Chapter 11. Markov-Switching Models of the German Business Cycle
Abstract
The statistical measurement of business cycles has recently experienced a revival of interest. Empirical business cycle research has always been interested in the chronology of contraction and expansion epochs (cf. inter alia Quah [1994]). This view is expressed in the primary descriptive definition of the ‘business cycle’ proposed by Burns & Mitchell [1946, p. 3] which is however compatible with most business cycle theories (cf. Klein [1995]):
“Business cycles are a type of fluctuations found in the aggregate economic activity of nations that organize their work mainly in business enterprise: a cycle consists of expansions occurring at about the same time in many economic activities, followed by similarly general recessions, contractions, and revivals which merge into the expansion phase of the next cycle; this sequence of changes is recurrent but not periodic; in duration business cycles vary from more than one year to ten or twelve years; they are not divisible into shorter cycles of similar character with amplitudes approximating their own.”
Hans-Martin Krolzig
Chapter 12. Markov-Switching Models of Global and International Business Cycles
Abstract
Business cycle research focuses traditionally on (i.) the co-movement of macroeconomic time series and (ii.) the regime switching nature of macroeconomic activity. Recent theoretical and empirical research has revived interest in each issue separately as pointed out by Diebold & Rudebusch [1994]. A synthesis of the dynamic factor and the non-linear approach for the modelling of macroeconomic fluctuations associated with these different traditions in empirical macroeconomics is provided by the MS-VAR business cycle model, where the regime shift governing process generates dynamic factor structures. The purpose of this chapter therefore is not only to illustrate the MS-VAR model and the related methods developed in this study, but also to lend new insight into the common center of these two research strategies.
Hans-Martin Krolzig
Chapter 13. Cointegration Analysis of VAR Models with Markovian Shifts in Regime
Abstract
The following consideration proposes a new methodological approach to the analysis of cointegrated linear systems with shifts in regime. The main difference with the foregoing analysis, as well as the previous literature, is the application of the MS-VAR model and the associated statistical procedures to cointegrated time series. Whereas the relevance of shifts in regimes of cointegrated time series has recently found a growing audience, the current state-of-the-art in this increasingly important field is rudimentary.
Hans-Martin Krolzig
Epilogue
Abstract
A study like the present one can of course make no claims of encyclopedic completeness and it would be pointless to list all the concepts which are related to the MS-VAR model but which have not been discussed in this presentation. If this study may have intended to develop an operational econometric approach for the statistical analysis of economic time series with MS-VAR models, then we can conclude that some progress has been made. Concerning inter alia the flexibility of modelling and the computational effort of estimation, this study has put forward the MS-VAR model as an alternative to linear, normal systems. In some other respects our results are more preliminary, but realistically we could not have expected to resolve all problems.
Hans-Martin Krolzig
Backmatter
Metadaten
Titel
Markov-Switching Vector Autoregressions
verfasst von
Dr. Hans-Martin Krolzig
Copyright-Jahr
1997
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-51684-9
Print ISBN
978-3-540-63073-9
DOI
https://doi.org/10.1007/978-3-642-51684-9