Efficient estimators for alternating quasi-likelihood models

We consider time series described by Markov chains that alternate periodically between different transition distributions, with conditional constraints involving unknown parameters. We obtain variance bounds and characterize efficient estimators for these parameters. Efficient estimators can be obtained as solutions of randomly weighted martingale estimating equations. Our model includes alternating heteroskedastic nonlinear autoregressive models whose innovations are martingale increments, in other words, alternating quasi-likelihood models. We consider in particular submodels of these in which the transition distributions do not alternate except for the conditional means and variances, and show that this information leads to better estimators for the parameters.