Determinant Efficiencies in Ill-Conditioned Models

Abstract

The canonical correlations between subsets of OLS estimators are identified with design linkage parameters between their regressors. Known collinearity indices are 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 canonical Vector Alienation Coefficients between subset estimators—by duality, the Alienation Coefficients between subsets of regressors. These identities figure in the study of D and 𝐷𝑠 as determinant efficiencies for estimators and their subsets, specifically, 𝐷𝑠-efficiencies for the constant, linear, pure quadratic, and interactive coefficients in eight known small second-order designs. Studies on D- and 𝐷𝑠-efficiencies confirm that designs are seldom efficient for both. Determinant identities demonstrate the propensity for 𝐷𝑠-inefficient subsets to be masked through near collinearities in overall D-efficient designs.