Specification and misspecification in reduced rank regression

SUMMARY. In the classical linear regression model with p dependent variables con-stituting the vector Y and q independent variables constituting the vector X the rank k of the regression matrix B of Y on X may be less than p and q. In that case the estimator of B, called the “reduced rank regression estimator, ” is composed of the k canonical variables corresponding to the k largest canonical correlations (Anderson, 1951). The asymptotic distribution of this estimator has been found when the rank is correctly specified and X and the residual Y − BX are independent with finite variances (Anderson, 1999b). The reduced rank regression estimator is more efficient than the least squares estimator, markedly so if k is small. This paper considers the properties of the estimator when the rank is not correctly specified. When the specified rank of B is less than the true rank, biases occur; when the specified rank is greater, variances of estimators and predictors are unnecessarily large. These results are related to tests concerning the rank. 1