Testing the Number of Regimes in Markov Regime Switching Models

TL;DR

This paper derives the asymptotic distribution of the likelihood ratio test for determining the number of regimes in Markov regime switching models, addressing a longstanding theoretical gap in econometrics and finance.

Contribution

It provides the first derivation of the asymptotic distribution for the likelihood ratio test in Markov regime switching models with multiple regimes.

Findings

Asymptotic distribution derived for testing $M_0$ vs. $M_0+1$ regimes.

Contiguous alternatives converge at rate $n^{-1/8}$ in models with normal density.

Bootstrap method validated for inference in these models.

Abstract

Markov regime switching models have been used in numerous empirical studies in economics and finance. However, the asymptotic distribution of the likelihood ratio test statistic for testing the number of regimes in Markov regime switching models has been an unresolved problem. This paper derives the asymptotic distribution of the likelihood ratio test statistic for testing the null hypothesis of $M_0$ regimes against the alternative hypothesis of $M_0 + 1$ regimes for any $M_0 \geq 1$ both under the null hypothesis and under local alternatives. We show that the contiguous alternatives converge to the null hypothesis at a rate of $n^{-1/8}$ in regime switching models with normal density. The asymptotic validity of the parametric bootstrap is also established.