The likelihood ratio test under nonstandard conditions: Testing the markov switching model of gnp

A theory of testing under non-standard conditions is developed. By viewing the likelihood as a function of the unknown parameters, empirical process theory enables us to bound the asymptotic distribution of standardized likelihood ratio statistics, even when conventional regularity conditions (such as unidentified nuisance parameters and identically zero scores) are violated. This testing methodology is applied to the Markov switching model of GNP proposed by Hamilton (1989). The standardized likelihood ratio test is unable to reject the hypothesis of an AR(4) in favour of the Markov switching model. Instead, we find strong evidence for an alternative model. This model, like Hamilton's, is characterized by parameters which switch between states, but the states arrive independently over time, rather than following an unrestricted Markov process. The primary difference, however, is that the second autoregressive parameter, in addition to the intercept, switches between states.