Efficient semiparametric estimation in time-varying regression models

We study semiparametric inference in some linear regression models with time-varying coefficients, dependent regressors and dependent errors. This problem, which has been considered recently by Zhang and Wu [Inference of time-varying regression models. Ann Statist. 2012;40:1376-1402] under the functional dependence measure, is interesting for parsimony reasons or for testing stability of some coefficients in a linear regression model. In this paper, we propose a different procedure for estimating non-time-varying parameters at the rate, in the spirit of the method introduced by Robinson [Root-n-consistent semiparametric regression. Econometrica. 1988;56:931-954] for partially linear models. When the errors in the model are martingale differences, this approach can lead to more efficient estimates than the method considered in Zhang and Wu [Inference of time-varying regression models. Ann Statist. 2012;40:1376-1402]. For a time-varying AR process with exogenous covariates and conditionally Gaussian errors, we derive a notion of efficient information matrix from a convolution theorem adapted to triangular arrays. For independent but non-identically distributed Gaussian errors, we construct an asymptotically efficient estimator in a semiparametric sense.