  Testing for regime switching when the regime switching probabilities are
specified either as constants (`mixture models') or are governed by a
finite-state Markov chain (`Markov switching models') are long-standing
problems that have also attracted recent interest. This paper considers testing
for regime switching when the regime switching probabilities are time-varying
and depend on observed data (`observation-dependent regime switching').
Specifically, we consider the likelihood ratio test for observation-dependent
regime switching in mixture autoregressive models. The testing problem is
highly nonstandard, involving unidentified nuisance parameters under the null,
parameters on the boundary, singular information matrices, and higher-order
approximations of the log-likelihood. We derive the asymptotic null
distribution of the likelihood ratio test statistic in a general mixture
autoregressive setting using high-level conditions that allow for various forms
of dependence of the regime switching probabilities on past observations, and
we illustrate the theory using two particular mixture autoregressive models.
The likelihood ratio test has a nonstandard asymptotic distribution that can
easily be simulated, and Monte Carlo studies show the test to have satisfactory
finite sample size and power properties.
