William Greene 509of thousands of samples of potentially thousands of observations. We used only
500 replications to produce the results in Table 11.3. The computations took about
five minutes. Using Newton’s method to maximize the log-likelihood directly took
less than five seconds. Unless one is wedded to the Bayesian paradigm then, on
strictly practical grounds, the MLE would be the preferred estimator.
Table 11.3 also lists the probit and logit random and fixed effects estimators. The
random effects estimators produce a reasonably large estimate ofρ^2 , roughly 0.44.
The high correlation across observations does cast some doubt on the validity of
the pooled estimator. The pooled estimator is inconsistent in either the fixed or
random effects cases. The logit results include two fixed effects estimators. The
model marked “U” is the unconditional (inconsistent) estimator. The one marked
“C” is Chamberlain’s consistent estimator. Note that, for all three fixed effects
estimators, it is necessary to drop from the sample any groups that have Doctorit
equal to zero or one for every period. There were 3,046 such groups, which is about
42% of the sample. We also computed the probit random effects model in two ways:
first by using the Butler and Moffitt method, then by using MSL estimation. In this
case, the estimators are very similar, as might be expected. The estimated squared
correlation coefficient is computed asρ^2 =σu^2 /(σε^2 +σu^2 ). For the probit model,
σε^2 =1. The MSL estimator computessu=0.9088376, from which we obtained
ρ^2. The estimated partial effects for the models are also shown in Table 11.3. The
average of the fixed effects constant terms is used to obtain a constant term for the
fixed effects case. Once again there is a considerable amount of variation across the
different estimators. On average, the fixed effects models tend to produce much
larger values than the pooled or random effects models.
Finally, we carried out two tests of the stability of the model. All of the estimators
listed in Table 11.3 derive from a model in which it is assumed that the same
coefficient vector applies in every period. To examine this assumption, we carried
out a homogeneity test of the hypothesis:
H 0 :β 1 =β 2 =...=βT,for theT=7 periods in the sample. The likelihood ratio statistic is:
λ= 2[(
tT= 1 lnLt)
−lnLPOOLED]
.The first part of the statistic is obtained by dividing the sample into the seven years
of data – the number of observations varies (3,874; 3,794; 3,792; 3,661; 4,483;
4,340; 3,377) – and then estimating the model separately for each year. The cal-
culated statistic is 202.97. The 5% critical value from the chi squared distribution
with(T− 1 ) 6 =36 degrees of freedom is 50.998, so the homogeneity assumption is
rejected by the data. As a second test, we separated the sample into men and women
and once again tested for homogeneity. The likelihood ratio test statistic is:
λ= 2 [lnLFEMALE+lnLMALE−lnLPOOLED]
= 2 [(−7855.219377)+(−9541.065897)−(−18019.55)]
=1246.529452.