Let us do the twist again

摘要

Kr?mer (Sankhyā 42:130-131, 1980) posed the following problem: "Which are the y, given X and V, such that OLS and Gauss-Markov are equal?". In other words, the problem aimed at identifying those vectors y for which the ordinary least squares (OLS) and Gauss-Markov estimates of the parameter vector β coincide under the general Gauss-Markov model y=Xβ + u. The problem was later called a "twist" to Kruskal's Theorem, which provides conditions necessary and sufficient for the OLS and Gauss-Markov estimates of β to be equal. The present paper focuses on a similar problem to the one posed by Kr?mer in the aforementioned paper. However, instead of the estimation of β, we consider the estimation of the systematic part β, which is a natural consequence of relaxing the assumption that X and Y are of full (column) rank made by Kr?mer. Further results, dealing with the Euclidean distance between the best linear unbiased estimator (BLUE) and the ordinary least squares estimator (OLSE) of Xβ, as well as with an equality between BLUE and OLSE are also provided. The calculations are mostly based on a joint partitioned representation of a pair of orthogonal projectors.