Determinant Efficiencies in Ill-Conditioned Models

Abstract

Thecanonical correlationsbetween subsets ofOLSestimators are identified withdesign linkage parametersbetween their regressors. Knowncollinearity indicesare extended to encompass angles between each regressor vector and remaining vectors. One such angle quantifies the collinearity of regressors with the intercept, of concern in the corruption of all estimates due to ill-conditioning. Matrix identities factorize a determinant in terms of principal subdeterminants and the canonicalVector Alienation Coefficientsbetween subset estimators—by duality, theAlienation Coefficientsbetween subsets of regressors. These identities figure in the study ofDand as determinant efficiencies for estimators and their subsets, specifically, -efficiencies for the constant, linear, pure quadratic, and interactive coefficients in eight known small second-order designs. Studies onD- and -efficiencies confirm that designs are seldom efficient for both. Determinant identities demonstrate the propensity for -inefficient subsets to be masked through near collinearities in overallD-efficient designs.