The Treatment of Linear Restrictions in Regression Analysis
John S. Chipman
M. M. Rao
This paper concerns itself with the following problems in univariate multiple regression analysis: (1) least squares estimation of the regression coefficients when these are assumed to be subject to a set of linear restrictions; (2) testing one set of linear restrictions when another (possibly vacuous) set of linear restrictions is assumed to hold. In (2) it is assumed that the true residuals are normally distributed, and in both cases it is assumed that the independent variables are fixed variates, and that the variance-covariance matrix of the dependent variable is known in any sample up to multiplication by an unknown scalar. The formulas for the estimators and test statistics are given in terms of the original variables, and a simple proof is provided of the unbiasedness of the F test. Extensive use is made of the properties of idempotent matrices, and geometric interpretations of the results are briefly described.