Shrinkage estimators of BLUE for time series regression models

Abstract

The least squares estimator (LSE) seems a natural estimator of linear regression models. Whereas, if the dimension of the vector of regression coefficients is greater than 1 and the residuals are dependent, the best linear unbiased estimator (BLUE), which includes the information of the covariance matrix $\Gamma$ of residual process has a better performance than LSE in the sense of mean square error. As we know the unbiased estimators are generally inadmissible, Senda and Taniguchi (2006) introduced a James–Stein type shrinkage estimator for the regression coefficients based on LSE, where the residual process is a Gaussian stationary process, and provides sufficient conditions such that the James–Stein type shrinkage estimator improves LSE. In this paper, we propose a shrinkage estimator based on BLUE. Sufficient conditions for this shrinkage estimator to improve BLUE are also given. Furthermore, since $\Gamma$ is infeasible, assuming that $\Gamma$ has a form of $\Gamma =\Gamma \left(\theta \right)$, we introduce a feasible version of that shrinkage estimator with replacing $\Gamma \left(\theta \right)$ by $\Gamma \left(\hat{\theta }\right)$ which is introduced in Toyooka (1986). Additionally, we give the sufficient conditions where the feasible version improves BLUE. Besides, the results of a numerical studies confirm our approach.