19
other parameters of the duration model. Note that the selection equation parameters do
not suffer from this bias since they are estimated on all the data and are not affected by .
[Insert Figure 5 here.]
Besides the presence of bias in the parameter estimates, an additional problem
that occurred in our previous Monte Carlo analyses was biased standard errors. This
problem is also corrected by accounting for the selection process. A comparison of the
estimated standard errors to the standard deviation of the sampling distribution (not
shown) indicates that the average standard error provided by the FIML estimation is the
same, on average, as the standard deviation of the parameter estimate, even when is
outside the allowable range.
Given that the corrected model produces accurate estimates for the permitted
values of and biased estimates for other values, the question remains as to whether it is
preferable to naïve duration models that ignore the selection process entirely. As
demonstrated earlier, these models produce biased coefficients and biased standard errors
for all non-zero values of . Since there are concerns about both the coefficients and their
variation, we want to take both of these into account when we compare models.
Additionally, the increased complexity of the selection-corrected may lead to less precise
standard errors. Our comparison of the two types of models thus focuses on their
respective Root Mean Squared Errors (RMSE), which adds the standard deviation to the
squared bias.
24
Note that this comparison uses only the coefficient estimates and not the
estimated standard errors, which are generally incorrect for the naïve models.
24
2
))
ˆ
(bias(
)
ˆ
Var(
)
ˆ
MSE(
'
'
'
+
=
, where the variance is taken from the sampling distribution of the
parameter and the bias is calculated by taking the average estimate in the 1000 draws and subtracting the
true value.
Convention