Markov regime switching models are widely considered in economics and finance. 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 difficulties: (i) some nuisance parameters are unidentified 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 tests' asymptotic 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 model's parameters, (2) a high order approximation to the log likelihood ratio permitting multiple switching parameters, (3) a refinement to the asymptotic distribution, and (4) a unified 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 model's structure. When applied to the US quarterly real GDP growth rates, the methods suggest fairly strong evidence favoring the regime switching specification, which holds consistently over a range of sample periods.

中文翻译：

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基于似然比的马尔可夫机制切换测试

马尔可夫政权转换模型在经济学和金融学中被广泛考虑。尽管一直有兴趣（例如，参见 Hansen，1992、Garcia，1998 和 Cho 和 White，2007），但基于似然比的检验的渐近分布仍然未知。本文考虑了此类测试，并在允许多个切换参数的非线性模型的背景下建立了它们的渐近分布。该分析同时解决了三个困难：(i) 在原假设下某些令人讨厌的参数无法识别，(ii) 原假设产生局部最优，以及 (iii) 条件状态概率遵循只能递归表示的随机过程。解决这些问题可以在经验相关的情况下获得显着的功率增益。除了获得测试 渐近分布，本文还获得了四组可以独立感兴趣的结果：（1）条件状态概率的表征及其相对于模型参数的高阶导数，（2）对数似然的高阶近似比率允许多个切换参数，(3) 对渐近分布的改进，以及 (4) 用于模拟临界值的统一算法。对于在零假设下为线性的模型，算法所需的元素都可以通过分析计算。上述结果还阐明了为什么某些引导程序可能不一致，以及为什么标准信息标准（例如贝叶斯信息标准 (BIC)）会对假设和模型结构敏感。