The information matrix test for Markov switching autoregressive models with covariate-dependent transition probabilities

The EM principle implies the moments underlying the information matrix test for multivariate Markov switching autoregressive models with covariate-dependent transition probabilities are the smoothed values of the moments we would test were the latent Markov chain observed. Thus, we identify components related to the heteroskedasticity, skewness and kurtosis of the multivariate regression residuals for each of the regimes, the neglected multivariate heteroskedasticity of the generalised residuals for each of the columns of the transition matrix, and a final component that assesses the conditional independence of these generalised residuals and the regression residuals, their squares and cross-products given the observed variables.