Quasi-likelihood regression models for Markov chains

We consider regression models in which covariates and responses jointly form a higher order Markov chain. A quasi-likelihood model speciies parametric models for the conditional means and variances of the responses given the past observations. A simple estimator for the parameter is the maximum quasi-likelihood estimator. We show that it does not use the information in the model for the conditional variances, and construct an eecient estimating function which involves estimators for the third and fourth centered conditional moments of the responses. In many applications one assumes that the innovations are not arbitrary martin-gale increments but independently and identically distributed. We determine how much additional information about the parameter such an assumption contains. To make the exposition more readable, we rst treat the case in which only the conditional mean is speciied.