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Uncorrelated residuals from linear models
Authors:Warren T Dent  George PH Styan
Institution:University of Iowa, Iowa City, IA 52242, USA;McGill University, Montreal, Quebec, Canada
Abstract:In the usual linear model y = +u, the error vector u is not observable and the vector r of least squares residuals has a singular covariance matrix that depends on the design matrix X. We approximate u by a vectorr1 = G(JA'y+Kz) of uncorrelated ‘residuals’, where G and (J, K) are orthogonal matrices, A'X = 0 and A'A = I, while z is either 0 or a random vector uncorrelated with u satisfying E(z) = E(J'u) = 0, V(z) = V(J'u) = σ2I. We prove that r1-r is uncorrelated with r-u, for any such r1, extending the results of Neudecker (1969). Building on results of Hildreth (1971) and Tiao and Guttman (1967), we show that the BAUS residual vector rh = r+P1z, where P1 is an orthonormal basis for X, minimizes each characteristic root of V(r1-u), while the vector rb of Theil's BLUS residuals minimizes each characteristic root of V(Jra-r), cf. Grossman and Styan (1972). We find that tr V(rh-u) < tr V(Jrb-u) if and only if the average of the singular values of P1K is less than 12, and give examples to show that BAUS is often better than BLUS in this sense.
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