Optimality Properties of Theil's BLUS Residuals
Stanley I. Grossman, George P. H. Styan · Journal of the American Statistical Association · 1972
In the linear model y=Xβ+u, the error vector u is not observable. Following Theil [8] we approximate J'u, a vector of selected components of u, by A'y with A chosen so that ϵ(A'y) = 0 and U(A'y) = σ2I. We study Σa = U(A'y—J'u) and give a new proof that fr Σa is minimized at A = B, where B'y is the vector of BLUS residuals. The claim in [9] that Σa — Σb is positive semi-definite is shown not to hold in general; we prove, however, that the BLUS residuals minimize each individual characteristic root of Σa.