Statistics & Probability Letters 8 (1989) 457-461 North-Holland October 1989 zyxwvuts APPLICATIONS OF A NECESSARY AND SUFFICIENT CONDITION FOR OLS TO BE BLUE zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Badi H. BALTAGI Department of Economics, Texas A&M University College Station, TX 77843-4228, USA Received September 1987 Revised October 1988 Abstract: This paper considers three examples from the statistics and econometrics literature where OLS is BLUE and demonstrates the easiness of verifying this result with the necessary and sufficient condition (NSC) derived by Zyskind (1967) and more recently Milliken and Albohali (1984). In particular, the third example is extended to a more general model where OLS is still BLUE. Once again this result is verified by the Milliken and Albohali NSC. Keywords: general linear model, seemingly unrelated regressions, pooled time-series of cross-sections, error components models, grouped data. 1. Introduction Statisticians have been concerned with the deriva- tion of necessary and sufficient conditions for OLS to be BLUE, especially when the variance- covariance matrix of the disturbances D is un- known. See for example Rao (1967), Zyskind (1967) Kruskal (1968) Balestra (1970), Graybill (1976), and more recently Milliken and Albohali (1984) to mention a few. It is the purpose of this paper to illustrate the usefulness of these condi- tions by focusing on the Milliken and Albohali condition and applying it to three well known areas in the statistics and econometrics literature. First, a set of Zellner’s (1962) seemingly unrelated regressions (SUR) is considered, and the two textbook cases (under which the Zellner GLS estimator reduces to OLS) are shown to satisfy Milliken’s and Albohali’s NSC. Next, a pooled time-series of cross-sections regression is consid- ered where the disturbances have an error compo- nents structure (see Dielman, 1983; and Hsiao, 1986; for recent surveys), and the within and between regressions are shown to satisfy this NSC. Third, a regression with grouped data is consid- ered where a microvariable is explained by aggre- gates, and the contemporaneous disturbances are equicorrelated, see Kloek (1981). Again, it is shown that such a model satisfies Milliken’s and Al- bohali’s NSC. In fact, this paper extends Kloek’s result to any two-way error component model where the microvariable is explained by a constant and aggregate variables that are invariant over time or across cross-sections. It is shown that under such a model OLS is BLUE. Once again this result is verified by Milliken’s and Albohali’s NSC. 2. Model and results Consider the general linear model y=xp+u 0) where X is an nxk matrix of rank k, E(u)=0 and var( u) = $2. Using the results of Zyskind (1967), Kruskal (1968) and more recently Milliken and Albohali (1984), it can be easily shown that li OLS is BLUE if and only if X’Q-‘~~=O wherefi,=I,-X(X’X)-‘X’ (2) 0167-7152/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland) 457