Reconciling Conflicting Gauss-Markov Conditions in the Classical Linear Regression Model

This article reconciles conflicting accounts of Gauss-Markov conditions, which specify when ordinary least squares (OLS) estimators are also best linear unbiased (BLU) estimators. We show that exogeneity constraints that are commonly assumed in econometric treatments of the Gauss-Markov theorem are unnecessary for OLS estimates of the classical linear regression model to be BLU. We also generalize a set of necessary and sufficient conditions first established by McElroy (1967, Journal of the American Statistical Association 62:1302–1304), but not yet generally recognized in the econometric literature, that are appropriate for many political science applications. McElroy's conditions relax the traditional Gauss-Markov restriction on autocorrelation in the errors to allow a type of correlation, exchangeability, that has two desirable characteristics: (1) exchangeable data occur in a potentially important class of political science models, and (2) the form of autocorrelation that occurs in exchangeable data has a ready intuition. We thus show that a common class of sample selection models that does not satisfy the Gauss-Markov conditions specified in econometrics textbooks is, in fact, BLU under OLS estimation.