Validating Markov Switching VAR Through Spectral Representations

Abstract

We develop a method to validate the use of Markov Switching models in modelling time series subject to structural changes. Particularly, we consider multivariate autoregressive models subject to Markov Switching and derive close-form formulae for the spectral density of such models, based on their autocovariance functions and stable representations. Within this framework, we check the capability of the model to capture the relative importance of high- and low-frequency variability of the series. Applications to U.S. macroeconomic and financial data illustrate the behaviour at different frequencies.

Appendix

Derivation of Formula ( 2.4 ):

The spectral density of the process $(\mathbf y_t)$ in (2.3) is given by

$$ \begin{aligned} F_\mathbf{y} (\omega ) = \sum _{h=-\infty }^{+ \infty } \varGamma _\mathbf{y} ( |h| ) e^{-i \omega h}&= \varGamma _\mathbf{y} (0) + \sum _{h=1}^{+ \infty } \varGamma _\mathbf{y} (h) e^{-i \omega h} + \sum _{k=-\infty }^{-1} \varGamma _\mathbf{y} (k) e^{-i \omega k} \\&= \varGamma _\mathbf{y} (0) + \sum _{h=1}^{+ \infty } \varGamma _\mathbf{y} (h) e^{-i \omega h} + \sum _{h=1}^{+\infty } \varGamma _\mathbf{y} (k) e^{i \omega h}. \end{aligned} $$

Note that

$$ \begin{aligned} \left( \sum _{h=1}^{n} A^h \right) \left( I - A \right)&= (A+A^2 +\dots + A^n)(I-A) = A - A^{n+1} \end{aligned} $$

which is equal to A when n goes to infinity with the spectral radius of A less than 1. Hence

$$ \left( \lim _{n \rightarrow + \infty } \sum _{h=1}^{n} A^h \right) \left( I - A \right) = A \qquad \text {and} \qquad \sum _{h=1}^{+ \infty } A^h =A(I-A)^{-1} . $$

It turns out that spectral density of the process in (2.3) is given by

$$ \begin{aligned} F_\mathbf{y} (\omega )&= {\widetilde{\varvec{\varLambda }}} {\widetilde{\mathbf{D}}} {\widetilde{\varvec{\varLambda }}}^{'}+ {\widetilde{\varvec{\varSigma }}} ( {\widetilde{\mathbf{D}}} \otimes \mathbf{I}_K) {\widetilde{\varvec{\varSigma }}}^{'}+ {\varvec{\varSigma }} (( \mathbf{D P}_{\infty }) \otimes \mathbf{I}_K) {\varvec{\varSigma }}^{'} \\&\quad + \sum _{h=1}^{+ \infty }{\widetilde{\varvec{\varLambda }}} \mathbf{F}^h {\widetilde{\mathbf{D}}}{\widetilde{\varvec{\varLambda }}}^{'}e^{-i \omega h} + \sum _{h=1}^{+\infty }{\widetilde{\varvec{\varLambda }}} \mathbf{F}^h {\widetilde{\mathbf{D}}}{\widetilde{\varvec{\varLambda }}}^{'}e^{i \omega h} \\&= Q +{\widetilde{\varvec{\varLambda }}}\sum _{h=1}^{+ \infty } (\mathbf{F} e^{-i \omega })^h {\widetilde{\mathbf{D}}}{\widetilde{\varvec{\varLambda }}}^{'}+ {\widetilde{\varvec{\varLambda }}}\sum _{h=1}^{+ \infty } (\mathbf{F} e^{i \omega })^h {\widetilde{\mathbf{D}}}{\widetilde{\varvec{\varLambda }}}^{'} \\&=Q +{\widetilde{\varvec{\varLambda }}}(\mathbf{F} e^{-i \omega })(\mathbf{I}-\mathbf{F} e^{-i \omega } )^{-1} {\widetilde{\mathbf{D}}}{\widetilde{\varvec{\varLambda }}}^{'}+ {\widetilde{\varvec{\varLambda }}}(\mathbf{F} e^{i \omega })(\mathbf{I}-\mathbf{F} e^{i \omega } )^{-1} {\widetilde{\mathbf{D}}}{\widetilde{\varvec{\varLambda }}}^{'} \\&= Q+ 2 {\widetilde{\varvec{\varLambda }}} \mathbf{F} \mathcal{R}e \{ ( \mathbf{I}_{M-1} e^{i \omega } -\mathbf{F})^{-1} \} {\widetilde{\mathbf{D}}} {\widetilde{\varvec{\varLambda }}}^{'} \end{aligned} $$

where $\mathcal{R}e$ denotes the real part of the complex matrix $( \mathbf{I}_{M-1} e^{i \omega } -\mathbf{F})^{-1}$, and

$$\begin{aligned} Q = {\widetilde{\varvec{\varLambda }}} {\widetilde{\mathbf{D}}} {\widetilde{\varvec{\varLambda }}}^{'}+ {\widetilde{\varvec{\varSigma }}} ( {\widetilde{\mathbf{D}}} \otimes \mathbf{I}_K) {\widetilde{\varvec{\varSigma }}}^{'}+ {\varvec{\varSigma }} (( \mathbf{D P}_{\infty }) \otimes \mathbf{I}_K) {\varvec{\varSigma }}^{'}. \end{aligned}$$

   $\square$