Errors of Prediction in Multiple Regression with Stochastic Regressor Variables

This paper investigates the errors of prediction of multiple regression equations, in which the regressor variables are regarded as drawn at random from a multivariate normal population. In most theoretical treatments of multiple regression it is assumed that one or more dependent variables are to be predicted, given the observed values of a number of variables. The independent or regressor variables are treated as constants. However, in many applications, it is more reasonable to regard them as random variables, drawn, for example, from a multivariate normal population. In this case the term variables is confusing, and it seems better to call them regressor variables. It has been suggested by Ehrenberg (1963) that regression or stochastic regressor variables is useless. This paper shows that although such regression has it limitations, especially if the number of regressor variables is large, it may be useful if the limitations are understood. The particular case to be discussed is that in which the regressor variables are drawn from a multivariate normal population. The k-dimensional vectors X1 , X2, ... X *... X, are given as a random