Testing for Regime Switching: A Comment - Carter, Andrew V.; Steigerwald, Douglas G. (Econometrica, 2012)
For such a model, we show that consistency of the quasi-maximum likelihood estimator for the population parameter values, on which consistency of the test is based, does not hold. We describe a condition that ensures consistency of the estimator and discuss the consistency of the test in the absence of consistency of the estimator. In Cho and White (2007), Testing for Regime Switching, the authors stud ied the asymptotic behavior of a statistic that tests the null hypothesis of one regime against the alternative of Markov switching between two regimes. A key insight is that a consistent test can be based on a quasi-likelihood that ignores the Markov structure of regime switching and treats the state variables that indicate regimes as a sequence of independent and identically distributed ran dom variables. Consistency of the test follows from consistency of the quasi maximum likelihood estimator (QMLE) under the alternative, which appears as Theorem 1(b) in Cho and White. Consistency of the QMLE requires that the expected quasi-log-likelihood attain a global maximum at the population parameter values. We show that this requirement does not hold for the au toregressive process analyzed in Cho and White. Thus, for models of regime switching in which the conditional mean contains autoregressive components, consistency of the test proposed by Cho and White has not been established. For the observable random variables {X, e Md}=1, d e N, the Markov regime-switching autoregressive process analyzed by Cho and White (Sec tion 3, p. 1697) is