Adaptive Estimators for Parameters of the Autoregression Function of a Markov Chain

Suppose we observe an ergodic Markov chain on the real line, with a parametric model for the autoregression function, i.e. the conditional mean of the transition distribution. If one speciies, in addition, a paramet-ric model for the conditional variance, one can deene a simple estimator for the parameter, the maximum quasi-likelihood estimator. It is robust against misspeciication of the conditional variance, but not eecient. We construct an estimator which is adaptive in the sense that it is eecient if the conditional variance is misspeciied, and asymptotically as good as the maximum quasi-likelihood estimator if the conditional variance is correctly speciied. The adaptive estimator is a weighted nonlinear least squares estimator, with weights given by predictors for the conditional variance.