A Monte Carlo Study of Estimates of Simultaneous Linear Structural Equations

1. INTRODUCTION
CONSIDERABTLHEE OIZETIC~Lre search has been carried out by the Cowles Commission
[6],[7], on methods of estimating structural parameters in a system of
simultaneous linear stochastic equations. The estimates are derived from maximum
likelihood considerations and have been proved to be asymptotically unbiased
and efficient. But in many real economics studies, the amount of data is
limited, and samples often consist of only twenty to thirty observations. This
paper examines certain small sample properties of "limited-information-singleequation"
maximum-likelihood estimatesvor two models by a Monte Carlo
approach. That is, 100 sets of observatio~lso ver 20 time periods are generated
from the models, and then with these observations and various statistical techniques,
estimates of the parameters of an over-identified equation are obtained
and compared.
The models, differing only in the variance-covariance matrix of the disturbances,
consist of three equations, one of which is an identity. For each set of
data we find the L.I.S.E., least squares, and instrumental variables estimates of
the parameters; in addition, we con~putea sample estimate both of the variance
of the disturbance in the equation and of the variance of the L.I.S.E. parameter
estimates. An analysis of the results of the 100 sets of data produced for each
model indicates that least squares generally gives more biased but less variable
estimates than the L.I.S.E. method, that the sample estimate of the variance of
an L.I.S.E. statistic is reliable on the average, and that the t distribution may
be used in constructing confidence intervals with L.I.S.E. estimates of the parameters
and the sampling variance of the estimates.