Efficient estimation in nonlinear autoregressive time-series models

This paper discusses efficient estimation for a class of nonlinear time-series models with unknown error densities. It establishes local asymptotic normality in this semi-parametric setting. This is then used to describe efficient estimates and to discuss the question of adaptation. Stein's necessary condition for adaptive estimation is satisfied if the error densities are symmetric, but is also satisfied in some models with asymmetric error densities. The paper gives several methods of constructing efficient estimates. These results are then applied to construct efficient estimators in SETAR(2; 1, 1), EXPAR(1) and ARMA(1, 1) models. We observe that adaptation is not possible in the SETAR(2; 1, 1) model with asymmetric errors while the efficient estimators in the ARMA(1, 1) model are adaptive even for asymmetric error densities. Section 8 contains a result that is useful in verifying the continuity of the stationary density with respect to the underlying parameters.