A Monte Carlo Study of Alternative Simultaneous Equation Estimators

We study the small sample properties of the simultaneous equation estimators
by a Monte Carlo approach. The four methods of estimation considered
are: least squares, two-stage least squares, unbiased and minimumsecond-
moment. The last of these four methods possesses the smallest secondorder
sampling moments about the true parameter value in a majority of cases,
while two-stage least squares shows the smallest bias in all cases. It is also
found that the usual asymptotic standard errors of two-stage least squares
give a rather satisfactory picture of the variability of the estimates about the
true value. This is not true for the least squares method in all cases considered.
Instead, it seems that the classical least squares standard errors measure
the variability of the estimates about the biased expectation, not about the
true value. In some cases this makes a very large difference.

1. INTRODUCTION AND SUMMARY
IN A RECENT article Wagner [4] examined certain small-sample properties
of limited-information maximum-likelihood, least squares, and instrumentalvariables
estimates for two models by a Monte Carlo approach. Although
these models are very simple-which is natural enough for a sampling
experiment-it seems appropriate for a variety of reasons to consider them
somewhat further. First, there are now several alternative estimation procedures
available, and it is worth-while to analyse these too. Secondly, by
using Wagner's models we can disregard certain methods of estimation
for the simple reason that they were already considered by him. Thirdly, it
appears that the two equations of both models are in a certain sense of
extreme types, so that we may hope that a Monte Carlo approach will
shed some light on the particular problems raised by such extremes. Wagner
considered only one equation in each model, and one which is over identified.
\Ve shall consider also the second equation, which is just-identified.
Just-identification implies that the two-stage least squares estimator is
identical with the limited information maximum likelihood estimator.