Bias reduction of autoregressive estimates in time series regression model through restricted maximum likelihood

This article considers maximum likelihood (ML) and restricted maximum likelihood (REML) estimation of time series regression models with autoregressive AR(p) noise. Approximate biases of the ML and REML estimators of the AR parameters, based on their approximate representations, are derived. In addition, a bias result for the ML estimator (MLE) of the error variance is established. Numerical results are presented to illustrate the biases of the MLE and REML estimator for the AR parameters, and simulation results are provided to assess the adequacy of our approximations. The impact of bias of the AR estimates on testing of linear trend in a regression trend model is also investigated. For a time series of short or moderate sample length, the REML estimator is generally much less biased than the MLE. Consequently, the REML approach leads to more accurate inferences for the regression parameters.