QUASI-LIKELIHOOD MODELS AND OPTIMAL INFERENCE

Consider an ergodic Markov chain on the real line, with parametric mod els for the conditional mean and variance of the transition distributi on. Such a setting is an instance of a quasi-likelihood model. The cus tomary estimator for the parameter is the maximum quasi-likelihood est imator. It is not efficient, but as good as the best estimator that ig nores the parametric model for the conditional variance. We construct two efficient estimators. One is a convex combination of solutions of two estimating equations, the other a weighted nonlinear one-step leas t squares estimator, with weights involving predictors for the third a nd 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.