SCAD-penalized regression for varying-coefficient models with autoregressive errors

Varying-coefficient models are useful extension of classical linear models. This paper is concerned with the statistical inference of varying-coefficient regression models with autoregressive errors. By combining the estimated residuals, the smoothly clipped absolute deviation (SCAD) penalty and Yule–Walker equations, a novel method is proposed to fit the error structure, including determining the order of the autoregressive error and efficiently estimating the autoregressive parameters. With appropriate selection of the tuning parameters, we establish the consistency of this method and the oracle property of the resulting regularized estimators. Based on the fitted autoregressive error structure, we further propose a two-stage local linear estimation for the unknown coefficient functions of the mean model to improve efficiency. The procedure is based on a pre-whitening transformation of the dependent variable. The resultant estimator of the unknown coefficient functions is asymptotically efficient and has the same asymptotic distribution as it would be if the autoregressive error structure were known with certainty. Simulation studies demonstrate that our asymptotic theory is applicable for finite samples, and the analysis of two real data sets illustrates the usefulness of our developed methodologies.