In this Section we report on a brief Monte Carlo experiment designed
to evaluate the performance of the Falstaff estimator of location.
We limit the choice of error densities to the Student t family in order to
exploit a simple normal/independent variance reduction technique.
Instead of simply simply generating t-variates and directly computing
mean squared errors of the estimators we generate observations from
the model 
for 
and 
.
This enables us to compute the optimal (weighted least squares) estimator
conditional of the 
's. Since this estimator as a known
distribution we can remove this source of variability from the Monte
Carlo and focus on the departure of our estimators from this
idealized, but obviously unattainable estimator. This is most easily
accomplished by replacing the realized n-vector y by the standardized
vector 
where
and
.
See, e.g. Simon (1976) for further details.
We consider three choices of the degrees of freedom parameter of the
t-distribution: 
and six choices of the sample
size 
.
The coefficient of kurtosis is unbounded for the t(1) and t(3)
distributions, and equals 9 for the t(5).
For each configuration we consider
9 versions of the the Falstaff estimator with the q-dimension of the
augmentation matrix varying from 0 to 8. Obviously, q=0 provides the
benchmark, ordinary least squares estimator against which we will
compare the performance of the other estimators. For each configuration
the augmentation matrix, D, is generated as iid from the standard
normal distribution. And 10,000 replications are done for each
configuration.
Figure 1 presents the results of the simulation. Columns of the
array of figures correspond to the three t distributions, and rows to
the six sample sizes. The horizontal line in each figure represents
performance of the sample mean, corresponding to the Falstaff estimator
with q=0. The curve plotted in each panel represents the performance,
measured by mean squared error,
of the other Falstaff estimators relative to the sample mean for each
configuration.
The vertical bars represent 95 percent confidence intervals for the
point estimates represented by the curve.
When the line drops below the horizontal line it indicates
improvement in performance over that of the sample mean. Thus, for the
Cauchy, 
, cases there is dramatic improvement over the entire
range of q simulated in the experiment. For the Student on 3
degrees of freedom configurations, there is improvement for modest q
when the sample size is small, and improvement over the entire range
when the sample size is larger. In the last column of the array,
corresponding to the Student on 5 degrees of freedom, the results show
no improvement from the Falstaff estimator for the smallest sample sizes,
10 and 15, slight improvement at n=25 for q=2, and significant
improvement for the larger sample sizes, for moderate q.
These results confirm the theoretical conclusions of the second-order
asymptotics, and also indicate that the choice of an optimal q is
rather delicately tied to the degree of non-normality of the error
density and the sample size. See Koenker and Machado (1996) for a more
serious theoretical consideration of this aspect of the GMM problem.