Exact Linear Restrictions on Parameters in the Classical Linear Regression Model

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This note will establish a simple method for incorporating exact linear restrictions on estimated parameters in the context of the classical linear regression model. The standard textbook approach to the problem consists of an application of the Lagrangian technique wherein the sum of squared residuals is minimized while constrained by the exact linear restrictions on the parameters. A typical discussion of the method is given in Johnston (1972). Although a large number of problems in applied econometrics are concerned with the estimation of regression coefficients subject to linear restrictions, the author's extensive search through the literature has failed to turn up any published version of the results which are to be presented in this paper.1 In the area of empirical demand analysis one can find specialized methods which have been developed for the particular problem at hand. M-Nost of the methods assume the restrictions to be that some subset of the elements are zero, or that the sum of the coefficients is zero (cf. Stone (1954)). In more elaborately conceived analyses of demand the linear restrictions apply to individual commodity demand functions which are embedded in a simultaneous system of demand functions (cf. Byron (1970)). Virtually all the approaches which have been employed in the applied econometric literature have consisted of optimization techniques, sometimes applied to maximum likelihood functions. The method presented here is different from the textbook approach inasmuch as the derivation of the estimators is more straightforward and does not entail the use of the Lagrangian technique. These features of the method seem to have computational as well as pedagogical advantages.