Further consequences of viewing LIML as an iterated Aitken estimator

Following an approach originally suggested by Pagan (1979. Economics Letters 3, 369–372) we have explored some additional consequences of viewing LIML as an iterated Aitken estimator. A curious result we derive in this context is that the Aitken estimator based on 2SLS residuals produces 2SLS estimate of the structural parameters. We derive a simple expression for the difference between 2SLS and LIML in finite samples, which turns out to be very similar to an expression for the approximate bias of 2SLS in small samples (Nagar, 1959. Econometrica 27, 575–595; Buse, 1992. Econometrica 60, 173–180; Bound et al., 1995. Journal of American Statistical Association 90, 443–450). The expression pinpoints the gain from iteration. Our analytical formula yields many of the results on the difference between 2SLS and LIML that Anderson and Sawa (1979. Econometrica 47, 163–182) and Anderson et al. (1982. Econometrica 50, 1009–1027) have obtained through extensive numerical simulations in a model with two endogenous variables. The robustness of 2SLS and LIML with respect to certain deviations from normality is also examined.