Robust Spectral Regression

Abstract

This paper addresses the problem of linear regression estimation when the disturbances follow a stationary process with its spectral density known only to be in a neighborhood of some specified spectral density, for instance, that of white noise. Rather than trying to adapt to a small unspecified autocorrelation, we follow here the robustness approach, and establish the extent of the regressors and disturbance spectra interaction which require serial correlation correction. We consider a class of generalized least-squares estimates, and find the estimator in this class which optimally robustifies the least-squares estimator against serial correlation. The estimator, when considered in the frequency domain, is of a form of weighted least squares with the most prominent frequencies of the regression spectrum being downweighted in a way similar to Huber's robust regression estimator.