the error term distribution are taken from McDonald and White: (a) Normal {N(0,1)}; (b) Mixture of
normals or variance-contaminated normal {0.9*N(0,1/9) +0.1*N(0,9)}; and (c) Lognormal. The
distributions were re-scaled and shifted, when necessary, to be drawn from a parent population with unitary
variance and zero mean. One thousand samples of size 50 were generated for each simulation experiment,
using the same X’s for each sample. For the second phase of the simulation, first order autocorrelation was
induced by multiplying each of the simulated error terms by the inverse of the P matrix defined above, for
two different ρ values of 0.5 and 0.8. Heteroskedasticity was induced by multiplying the errors by I+0.5,
where I is a binomial index variable taking values of zero or one with equal probability. For modeling
purposes I is assumed to be a known qualitative factor that shifts the variance of the dependent variable
from 0.5 to 1.5. In both cases the error term distributions maintain a zero mean and unit variance.
Simulation Results and Estimator Performance
As in previous studies (McDonald and White; Newey; Hsieh and Manski) the root mean squared
error (RMSE) of the slope estimators is used as the criteria for comparing the relative performance of
different estimators. The results of McDonald and White for three underlying error term distributions:
normal, kutotic-only and kurtotic and skewed (Table 1), are used as a basis for comparison. Under OLS,
the RMSE of the slope estimators is always around 0.28. When the true error term is i.i.d. normal, the
proposed estimator, yields maximum likelihood parameter estimates for Θ and μ that are zero or not
statistically different from zero, and slope estimates that are identical to those from OLS. Most of the
estimators explored in McDonald and White perform similarly or slightly worse than OLS, as expected,
since OLS is the most efficient estimator under i.i.d. normal error-term conditions.
The variance-contaminated normal implies a symmetric unimodal but thick-tailed error-term
distribution with a kurtosis coefficient of approximately 20 in this case. The proposed estimator performs
relatively well in estimating the slope coefficient (β2), with a RMSE of 0.115 (Table 2). Nine other
estimating techniques provide comparable efficiency levels, but none produces a RMSE lower than 0.11
(Table 1).