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.

中文翻译：

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基于似然比的马尔可夫体制转换检验

马尔可夫政权转换模型在经济学和金融学中非常普遍。尽管对它们的兴趣持续存在，但基于似然比的用于检测状态切换的测试的渐近分布仍然未知。这项研究检查了此类测试，并在非线性模型的背景下建立了它们的渐近分布，从而允许多个参数受到状态切换的影响。该分析解决了三个难题：（i）在原假设下无法识别某些令人讨厌的参数，（ii）原假设产生局部最优，并且（iii）条件体制概率遵循只能以递归方式表示的随机过程。解决这些问题可在经验上相关的设置中获得可观的功率增益。这项研究还提出了以下结果：（1）关于模型参数的条件状态概率及其导数的表征；（2）对数似然比的高阶近似；（3）渐近分布的细化；以及（4）a统一算法模拟临界值。对于在原假设下是线性的模型，算法所需的元素都可以通过解析来计算。此外，以上结果说明了为什么某些引导过程可能不一致，以及为什么标准信息标准可能对假设和模型结构敏感。当应用于美国季度实际国内生产总值（GDP）增长率数据时，这些方法会发现相对有力的证据，这些证据有利于政权转换规范。最后，