Selection of Regressors

Selection of regressors is an old and important problem in econometrics as well as in any other field that uses regression analysis. Many criteria of selection have been proposed in the econometric and other literature, but I have not found a good unified treatment of the subject. This paper is such an attempt. Selection of regressors should be based on economic-theoretic considerations as well as on statistical evidence. Economic theory can ofteni indicate what regressors should be included in any equation, and sometimes, but less often, a class of functional forms that should be considered. In addition, it canl often tell the researcher the likely sign of a coefficient and sometimes even a shorter range within which the coefficient is likely to lie. These types of a priori information should be utilized either formally by means of the Bayesian method or informally by means of the classical method. Hence, mechanical reliance on goodness of fit should be avoided as much as possible. However, other things being equal, it would be convenient to be able to choose a regression equation on the basis of a single statistic such as the sum of squared residuals or its monotonic transformation, R2. However, R2 has an obvious weakness; i.e. it can be maximized by maximizing the number of regressors. Therefore, some kind of correction that accounts for the degrees of freedom is necessary. In the econometric literature, Theil 1961, p. 213 was the first to propose such a correction, and since then his R2 has become an important index with which an econometrician judges the merit of a regression equation. However, a theoretical weakness of Theil's R2 is that it is not based on an explicit consideration of a loss fuLnction. For a criticism of K2 on an empirical ground, see Mayer 1975. In this paper I reiterate my suggestion (Amemiya 1966 and 1972) to develop a selection criterion based on minimizing the unconditional mean square prediction error (UMSPE) as the risk functioni, which results in a modification of R2 that corrects for the degrees of freedom to a greater extent than Theil's R2. I call this Prediction Criterion (PC) and show that it can be used as a selection criterion in the nonlinear regression model and/or in the case where the error term has a general variancecovariance matrix. Thus it has a wide applicability.