  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 (2012) 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 root n, in the spirit of the method introduced by
Robinson (1988) for partially linear models. When the errors in the model are
martingale differences, this approach can lead to more effcient estimates than
the method considered in Zhang and Wu (2012). 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.
