PRELIMINARY TEST PREDICTOR IN THE LINEAR REGRESSION MODEL INCLUDING A PROXY VARIABLE
Journal of the Japan Statistical Society, Japanese Issue
In this paper, we show the mean square error of the pre-test predictor in the linear regression model including a proxy variable, and compare the mean square errors of the weighted average predictor and the pre-test predictor. where where c is the critical value in the pre-test. Since we can derive the MSE of the pre-test predictor by the way parallel to Toyoda [14, Appendix], we omit the detailed derivation. Performing some tedious calculations, we obtain where P(a,b)(01, 02; T) is given in
... ; T) is given in (11) in the text. (A-12) is very similar to the MSE given in Toyoda  and the only difference of our MSE and Toyoda's one is that our MSE has two noncentral parameters but Toyoda's one has only one noncentral parameter. Then, we should note that if the true variable is used (P=Z), 02 reduces to 0 and our MSE reduces to the one given in Toyoda  . Putting H'=(0, 1) and h=0, and noting (A-9) and (A-10), we get the MSE, (11), in the text.