Likelihood Ratio Based Tests for Markov Regime Switching

Markov regime switching models are widely considered in economics and \u85nance. Although there have been persistent interests (see e.g., Hansen, 1992, Garcia, 1998, and Cho and White, 2007), the asymptotic distributions of likelihood ratio based tests have remained unknown. This paper considers such tests and establishes their asymptotic distributions in the context of non-linear models allowing for multiple switching parameters. The analysis simultaneously addresses three di ¢ culties: (i) some nuisance parameters are unidenti\u85ed under the null hypothesis, (ii) the null hypothesis yields a local optimum, and (iii) conditional regime probabilities follow stochastic processes that can only be represented recursively. Addressing these issues permits substantial power gains in empirically relevant situations. Besides obtaining the testsasymp-totic distributions, this paper also obtains four sets of results that can be of independent interest: (1) a characterization of conditional regime probabilities and their high order derivatives with respect to the models parameters, (2) a high order approximation to the log likelihood ratio permitting multiple switching parameters, (3) a re\u85nement to the asymptotic distribution, and (4) a uni\u85ed algorithm for simulating the critical values. For models that are linear under the null hypothesis, the elements needed for the algorithm can all be computed analytically. The above results also shed light on why some bootstrap procedures can be inconsistent and why standard information criteria, such as the Bayesian information criterion (BIC), can be sensitive to the hypothesis and the models structure. When applied to the US quarterly real GDP growth rates, the methods suggest fairly strong evidence favoring the regime switching speci cation, which holds consistently over a range of sample periods