Stochastic Regressors

Abstract

To this point we have considered the linear statistical model y = Xβ + e, where the regressors X are treated as fixed in repeated samples. In many cases, however, this assumption is not tenable. The explanatory variables economists and other social scientists use are often generated by stochastic processes beyond their control. In this chapter we consider the consequences of relaxing the fixed Xassumption under two distinct circumstances. In the first instance multivariate normality is assumed. That is, (y_t, x’_t)’ is assumed to be distributed as a (K + l)-multivariate normal random vector with variance-covariance matrix Σ In the second instance, a more relaxed set of assumptions is considered; the case where X and e are independently distributed along with other mild conditions on their distributions. The major conclusion of this chapter’s study of these stochastic regressor models is that, under general conditions, the essential results of the fixed regressor models (i.e. the classical linear regression and classical normal linear regression models of Definitions 2.2.1 and 3.2.1, respectively) remain intact even if an experimenter does not have control of the settings of the regressor matrix. There is the problem of multicollinearity, however, but that is the subject of Chapter 13.