Quasi-likelihood models and optimal inference

Consider an ergodic Markov chain on the real line, with parametric models for the conditional mean and variance of the transition distribution. Such a setting is an instance of a quasi-likelihood model. The customary estimator for the parameter is the maximum quasi-likelihood estimator. It is not eecient, but as good as the best estimator that ignores the parametric model for the conditional variance. We construct two eecient estimators. One is a convex combination of solutions of two estimating equations, the other a weighted nonlinear one-step least squares estimator, with weights involving predictors for the third and fourth centered conditional moments of the transition distribution. Additional restrictions on the model can lead to further improvement. We illustrate this with an autoregressive model whose error variance is related to the autoregression parameter.