What have we learned from this pathological parable? Our first moral is that we should be wary of reliance on conventional first-order asymptotics. It is tempting to adopt the view that our ability to consistently estimate V implies that we can safely ignore the consequences of this estimation. This is true when n is sufficiently large relative to q, but it may seriously misrepresent the situation commonly faced for moderate n. Our second moral is that it is possible to substantially improve upon the performance of the ordinary least squares estimator of the linear model when the error distribution is non-Gaussian. The Falstaff estimator is only one of many nonlinear estimators which stand ready to challenge the putative superiority of the Gauss-Markov estimator. If pressed, we would not choose Falstaff to lead us into battle, but like Jack Falstaff this portly estimator contains a little wisdom.
Finally, we must confess that the Falstaff estimator is really only a variant of the estimator proposed by Cragg (1983) designed to deal with heteroscedasticity of unknown form. While Cragg's estimator was intended to deal with real heteroscedasticity and exploited the existence of overidentifying restrictions based on measurable functions of the regressors already appearing in the model, Falstaff makes something similar out of thin air.