Optimal QR-based estimation in partially linear regression models with correlated errors using GCV criterion

Multicollinearity among the predictor variables is a serious problem in regression analysis. There are some classes of biased estimators for solving the problem in statistical literature. In these biased classes, estimation of the shrinkage parameter plays an important role in data analyzing. Using eigenvalue analysis, efforts have been made to develop skills and methods for computing risk function of the estimators in regression models. A modified estimator based on the QR decomposition to combat the multicollinearity problem of design matrix is proposed in partially linear regression model which makes the data to be less distorted than the other methods. The necessary and sufficient condition for the superiority of the partially generalized QR-based estimator over partially generalized least-squares estimator for selecting the shrinkage parameter is obtained. Under appropriate assumptions, the asymptotic bias and variance of the proposed estimators are obtained. Also, a generalized cross validation (GCV) criterion is proposed for selecting the optimal shrinkage parameter and the bandwidth of the kernel smoother and then, an extension of the GCV theorem is established to prove the convergence of the GCV mean. Finally, the Monté-Carlo simulation studies and a real application related to electricity consumption data are conducted to support our theoretical discussion.