Local Asymptotic Distribution Theory for Regime Switching Test in Bilinear Models

Since Hamilton’s seminal paper [1], Markov switching models have become increasingly useful in several areas, such as financial applications and dynamic econometrics. In our study, we are interested in a class of bilinear models with a Markov-switching regime (MS-BL). By treating the transition probabilities as nuisance parameters, the local asymptotic normality (LAN) of the log-likelihood ratio is established for a first-order bilinear two-state Markov-switched model with respect to the model coefficients using the approach of root mean square differentiability. This method provides excellent advantages and valuable alternatives for solving well-known difficult problems in hypothesis testing affected by regime-switching. The non-identification of specific nuisance parameters under the null hypothesis and the fact that the null hypothesis gives us a local optimum (which means that the score function is equal to zero) are the most challenging problems found. Under appropriate conditions and based on this framework, Wald tests asymptotic distributions, under the null and non-null hypothesis, are derived for first-order bilinear two-state Markov-switched models.