Wouldn't It Be Nice …? The Automatic Unbiasedness of OLS (and GLS)

In a recent issue of this journal, Larocca (2005) makes two notable claims about the best linear unbiasedness of ordinary least squares (OLS) estimation of the linear regression model. The first, drawn from McElroy (1967), is that OLS remains best linear unbiased in the face of a particular kind of autocorrelation (constant for all pairs of observations). The second, much larger and more heterodox, is that the disturbance need not be assumed uncorrelated with the regressors for OLS to be best linear unbiased. The assumption is unnecessary, Larocca says, because “orthogonality [of disturbance and regressors] is a property of all OLS estimates” (p. 192). Of course OLS's being best linear unbiased still requires that the disturbance be homoskedastic and (McElroy's loophole aside) nonautocorrelated, but Larocca also adds that the same automatic orthogonality obtains for generalized least squares (GLS), which is also therefore best linear unbiased, when the disturbance is heteroskedastic or autocorrelated.