On the Performance of Least Squares in Linear Regression with Undefined Error Means

This paper considers the linear regression model with multiple stochastic regressors, intercept, and errors that have undefined means. This model is of interest from a robustness perspective as a polar case. Generally, least squares estimators are inconsistent in this context. It is shown, however, that this inconsistency is restricted to the estimation of the intercept, if the regressors are highly variable. Rates of convergence of the least squares slope estimators are determined, and are shown to exceed the standard rate, n^{-1/2}, in certain contexts. The results place no restrictions on the temporal dependence of the errors, and require an unusually weak exogeneity condition between the regressors and errors. Implications of the results for robustness theory are discussed.