Palgrave Handbook of Econometrics: Applied Econometrics

William Greene 521

of a scale variable that reflects an underlying continuous variable, “health.” The
frequencies and sample proportions for the reported values are as follows:

NEWHSAT 01234567 8 910

Frequency 447 255 642 1173 1390 4233 2530 4231 6172 3061 3192

Proportion 1.6% 0.9% 2.3% 4.2% 5.0% 15.4% 9.2% 15.4% 22.5% 11.2% 11.6%

We have fitted pooled and panel data versions of the ordered probit model to these
data. The model used is:

Uit∗=β 1 +β 2 Ageit+β 3 Incomeit+β 4 Educationit+β 5 Marriedit

+β 6 Workingit+εit+ci,

whereciwill be the common fixed or random effect. (We are interested in com-
paring the fixed and random effects estimators, so we have not included any time
invariant variables such as gender in the equation.) Table 11.5 lists five estimated
models. (Standard errors for the estimated threshold parameters are omitted.) The
first is the pooled ordered probit model. The second and third are fixed effects. Col-
umn 2 shows the unconditional fixed effects estimates using the results in Greene
(2008). Column 3 shows the Das and van Soest (2000) estimator. For the minimum
distance estimator, we used an inefficient weighting matrix, the block diagonal
matrix in which thejth block is the inverse of thejth asymptotic covariance matrix
for the individual logit estimators. With this weighting matrix, the estimator is:

βˆMDE=

[∑

9

j= 0

V−j^1

]− (^1) ∑
9
j= 0
V−j^1 βˆj,
and the estimator of the asymptotic covariance matrix is approximately equal to the
bracketed inverse matrix. The fourth set of results is the random effects estimator
computed using the MSL method. This model can be estimated using Butler and
Moffitt’s quadrature method: however, we found that, even with a large number of
nodes, the quadrature estimator converged to a point where the log-likelihood was
far lower than the MSL estimator, and at parameter values that were implausibly
different from the other estimates. Using different starting values and different
numbers of quadrature points did not change this outcome. The MSL estimator
for a random constant term is considerably slower, but produces more reasonable
results. The fifth set of results is the Mundlak form of the random effects model,
which includes the group means in the models as controls to accommodate possible
correlation between the latent heterogeneity and the included variables. As noted
earlier, the components of the ordered choice model must be interpreted with some
care. By construction, the partial effects of the variables on the probabilities of the
outcomes must change sign, so the simple coefficients do not show the complete
picture implied by the estimated model. Table 11.6 shows the partial effects for the
pooled model to illustrate the computations.