Inference for linear models with dependent errors

The paper is concerned with inference for linear models with fixed regressors and weakly dependent stationary time series errors. Theoretically, we obtain asymptotic normality for the M-estimator of the regression parameter under mild conditions and establish a uniform Bahadur representation for recursive M-estimators. Methodologically, we extend the recently proposed self-normalized approach of Shao from stationary time series to the regression set-up, where the sequence of response variables is typically non-stationary in mean. Since the limiting distribution of the self-normalized statistic depends on the design matrix and its corresponding critical values are case dependent, we develop a simulation-based approach to approximate the critical values consistently. Through a simulation study, we demonstrate favourable finite sample performance of our method in comparison with a block-bootstrap-based approach. Empirical illustrations using two real data sets are also provided.