Inference in partially linear models with correlated errors

Abstract

We study the problem of performing statistical inference on the linear effects in partially linear models with correlated errors. To estimate these effects, we introduce usual, modified and estimated modified backfitting estimators, relying on locally linear regression. We obtain explicit expressions for the conditional asymptotic bias and variance of the usual backfitting estimators under the assumption that the model errors follow a mean zero, covariance-stationary process. We derive similar results for the modified backfitting estimators under the more restrictive assumption that the model errors follow a mean zero, stationary autoregressive process of finite order. Our results assume that the width of the smoothing window used in locally linear regression decreases at a specified rate, and the number of data points in this window increases. These results indicate that the squared bias of the considered estimators can dominate their variance in the presence of correlation between the linear and non-linear variables in the model, therefore compromising their i/n-consistency. We suggest that this problem can be remedied by selecting an appropriate rate of convergence for the smoothing parameter of the-estimators. We argue that this rate is slower than the rate that is optimal for estimating the non-linear effect, and as such it 'undersmooths' the estimated non-linear effect. For this reason, data-driven methods devised for accurate estimation of the non-linear effect may fail to yield a satisfactory choice of smoothing for estimating the linear effects. We introduce three data-driven methods for accurate estimation of the linear effects. Two of these methods are modifications of the Empirical Bias Bandwidth Selection method of Opsomer and Ruppert (1999). The third method is a non-asymptotic plug-in method. We use the data-driven choices of smoothing supplied by these methods as a basis for constructing approximate confidence intervals and tests of hypotheses for the linear effects. Our inferential procedures do not account for the uncertainty associated with the fact that the choices of smoothing are data-dependent and the error correlation structure is estimated from the data. We investigate the finite sample properties of our procedures via a simulation study. We also apply these procedures to the analysis of data collected in a time-series air pollution study.