Monte Carlo Likelihood-Ratio Tests for Markov Switching Models

Markov switching models are widely used to capture nonlinearities arising from regime shifts. Most existing tests for the number of regimes focus on one versus two regimes. Even in such simple cases, this type of problem raises issues of non-standard asymptotic distributions, identification failure, and nuisance parameters. We address these difficulties by applying the technique of Monte Carlo tests, which yields both finite-sample and asymptotically valid procedures, without the need to establish an asymptotic distributional theory, nor the existence of an asymptotic distribution. Monte Carlo likelihood-ratio tests are developed for testing \(M_0\) regimes against \(M_0+m\) regimes, for any \(M_0≥1\) and \(m≥1\). The proposed tests apply to nonstationary processes, non-Gaussian errors, and multivariate models. A key contribution is the Maximized Monte Carlo likelihood-ratio test (MMC-LRT), an identification-robust procedure with both finite-sample and asymptotic validity. The framework also accommodates tests for regime synchronization and Markov switching GARCH models. Simulations show accurate size control and strong power. An empirical application using Markov switching VAR models finds weakened U.S.-Canada business cycle synchronization when COVID-period data are included, while applications to U.S. output growth support a three-regime specification consistent with previous empirical studies.