Simultaneous prediction using target function based on principal components estimator with correlated errors

Abstract

Prediction is pivotal in linear regression analysis, especially in applied sciences. Before target function was defined, the predictions of actual values and/or average values of the dependent variable were obtained individually rather than simultaneously. However, in many applied studies, obtaining simultaneous predictions of both the average values and the actual values is more appropriate. In this paper, the simultaneous prediction based on the principal components estimator with correlated errors in the linear regression model under the problem of multicollinearity, which has negative effects on the prediction, is considered by utilizing the target function. We define three new predictors and make theoretical comparisons of proposed predictors by using the mean squared error of predictions. Also, we support theoretical findings with a comprehensive simulation study and two numerical examples.

Appendix

Lemma A.1

(Farebrother 1976) Let $A$ be a pd matrix and let $\alpha$ be any vector then $A - \alpha \alpha^{\prime} \ge 0$ if and only if $\alpha^{\prime}A^{ - 1} \alpha \le 1$.

Lemma A.2

(Rao et al. 2008) Let $M > 0$ and $N \ge 0$ be $n \times n$ matrices, then $M > N$ if and only if maximum eigenvalue of $NM^{ - 1}$ is smaller than 1.