Autoregression , estimating functions , and optimality criteria

Abstract We consider d-order Markov chains satisfying a conditional constraint E(aθ(Xi−1, Xi) | Xi−1) = 0, where Xi−1 = (Xi−1, . . . , Xi−d). These comprise quasi-likelihood models and nonlinear and conditionally heteroscedastic autoregressive models with martingale innovations. Estimators for θ can be obtained from estimating equations ∑n i=1Wθ(Xi−1) aθ(Xi−1, Xi) = 0. We review different criteria for choosing good weights Wθ(Xi−1). They usually lead to weights that depend on unknown features of the transition distribution and must be estimated. We compare the approach via estimating functions with other ways of constructing estimators for θ, and discuss efficiency of the estimators in the sense of Hájek and LeCam. Analogous comparisons may be made for regression models.