Abstract

Exact formulae are provided for the calculation of multivariate skewness and kurtosis of Markov-switching Vector Auto-Regressive (MS VAR) processes as well as for the general class of MS state space (MS SS) models. The use of the higher-order moments in non-linear modeling is illustrated with two examples. A Matlab code that implements the results is available from the authors.

Highlights

  • Markov-switching models are widespread in applied macroeconomics and finance

  • MS models have been introduced by Hamilton (1989) with the aim of capturing the asymmetry of the business cycle

  • Two examples below illustrate the use of the higher-order moments in multivariate MS models

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Summary

Introduction

Markov-switching models are widespread in applied macroeconomics and finance. By extending linear specifications with a discrete latent process that controls parameter switches, MS models have gained the ability to fit time series subject to non-linearities. Given the empirical evidence about the existence of policy regimes, the last generation of dynamic stochastic general equilibrium models includes Markov-switching policy reaction functions (see Davig and Leeper, 2007; Davig et al, 2004). In this context MS VAR models arise as fundamental solution of the forward-looking structural equations (Farmer et al, 2009, 2011). The statistical properties of MS VAR models have been analyzed, among others, by Yang (2000), Francq and Zakoian (2001, 2002), and Cavicchioli (2013, 2014) These studies focus on stationarity issues, on the first two unconditional moments, and on the determination of the number of regimes. All proofs are given as supplementary material (see Appendix A)

Model and assumptions
Multivariate measures of skewness and kurtosis
Markov-switching vector autoregressive models
Markov-switching state space models
Examples
Conclusion
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