Markov-Switching Vector Autoregressions Modelling, Statistical Infer
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 nom
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Hans-Martin Krolzig
Markov-Switching Vector Autoregressions Modelling, Statistical Inference, and Application to Business Cycle Analysis
Lecture N otes in Economies and Mathematical Systems Founding Editors: M. Beckmann H. P. Künzi Editorial Board:
H. Albach, M. Beckmann, G. Feichtinger, W Güth, W Hildenbrand, W Krelle, H. P. Künzi, K. Ritter, U. Schittko, P. Schönfeld, R. Selten Managing Editors: Prof. Dr. G. Fandei Fachbereich Wirtschaftswissenschaften Fernuniversität Hagen Feithstr. 140/AVZ II, D-58084 Hagen, Germany Prof. Dr. W. Trockel Institut für Mathematische Wirtschaftsforschung (IMW) Universität Bielefeld Universitäts,str. 25, D-33615 Bielefeld, Germany
454
Springer-Verlag Berlin Heidelberg GmbH
Hans-Martin Krolzig
Markov-Switching Vector Autoregressions Modelling, Statistical Inference, and Application to Business Cycle Analysis
Springer
Author Dr. Hans-Martin Krolzig University of Oxford Institute of Economics and Statistics St. Cross Building, Manor Road Oxford OX1 3UL, Great Britain
Lfbrary of Congress Cataloging-fn-Publication Data
Krolzig, Hans-Martin, 1964Markov-swltching vector autoregressions : modelling, statistical inference, and application to business cycle analysis / Hans-Martin Krolzig. p. cm. -- 0, i
= 1, ... , M, the process is called irreducible.
The assumptions of
ergodicity and irreducibility are essential for the theoretical properties of MS-VAR models, e.g. its property of being stationary. The estimation procedures, which will be introduced in Chapter 6 and Chapter 8 are flexible enough to capture even these degenerated cases, e.g. when there is a single jump ("structural break") into the absorbing state that prevails until the end of the observation period.
1.3
The Data Generating Process
After this introduction ofthe two components ofMS-VAR models, (i.) the Gaussian VAR model as the conditional data generating process and (ii.) the Markov chain as the regime generating process, we will briefly discuss their main implications for the data generating process. For given states ~t and lagged endogenous variables Yt-l = (Y~-l' Y~-2' ... ,Y~, Y~,
... , Y~ _p)' the conditional probability density function of Yt is denoted by
p(Yt I~t, Yt-l). It is convenient to assume in (1.5) and (1.7) a normal distribution of the error term Ut, so that
p(Ytl~t
= t m , Yt-d In(21l')-1/2In IEI- 1/ 2 exp{(Yt - Ymt)/E-;;/(Yt - Ymt)},
(1.10)
18
The Markov-Switching Vector Autoregressive Model
where Ymt = E[Ytl~t, Yt-l] is the conditional expectation of Yt in regime m. Thus the conditional density of Yt for a given regime
~t
is normal as in the VAR model
defined in equation (1.2). Thus:
NID (Ymt, ~m), ,....,
NID (Y~~t, I:(~t ® I K ))
,
(1.11)
where the conditional means Ymt are summarized in the vector Yt which is e.g. in MSI specifications of the form
v, + 2::=,'
Yt =
VM
AljYt-j
].
+ 2: j =1 AMjYt-j
Assuming that the information set available at time t - 1 consists only of the sampie observations and the pre-sample values collected in Yt-l and the states of the M
