Likelihood Ratio-Based Tests for Markov Regime Switching

Markov regime-switching models are very common in economics and finance. Despite persisting interest in them, the asymptotic distributions of likelihood ratio-based tests for detecting regime switching remain unknown. This study examines such tests and establishes their asymptotic distributions in the context of nonlinear models, allowing multiple parameters to be affected by regime switching. The analysis addresses three difficulties: (i) some nuisance parameters are unidentified under the null hypothesis, (ii) the null hypothesis yields a local optimum, and (iii) the conditional regime probabilities follow stochastic processes that can only be represented recursively. Addressing these issues permits substantial power gains in empirically relevant settings. This study also presents the following results: (1) a characterization of the conditional regime probabilities and their derivatives with respect to the model’s parameters, (2) a high-order approximation to the log-likelihood ratio, (3) a refinement of the asymptotic distribution, and (4) a unified algorithm to simulate the critical values. For models that are linear under the null hypothesis, the elements needed for the algorithm can all be computed analytically. Furthermore, the above results explain why some bootstrap procedures can be inconsistent, and why standard information criteria can be sensitive to the hypothesis and the model structure. When applied to US quarterly real gross domestic product (GDP) growth rate data, the methods detect relatively strong evidence favouring the regime-switching specification. Lastly, we apply the methods in the context of dynamic stochastic equilibrium models and obtain similar results as the GDP case.