520 Discrete Choice Modeling
The “j” specific constant, which is the same for all individuals, is absorbed inθi.
Thus a fixed effects binary logit model applies to each of theJ−1 binary random
variables,wit,j. The method in section 11.3 can now be applied to each of the
J−1 random samples. This providesJ−1 estimators of the parameter vectorβ(but
no estimator of the threshold parameters). The authors propose to reconcile these
different estimators by using a minimum distance estimator of the common true
β. The minimum distance estimator at the second step is chosen to minimize:
q=∑J− 1
j= 0∑J− 1
m= 0(
βˆj−β)′[
V−jm^1](
βˆm−β)
,where
[
V−jm^1]
is thej,mblock of the inverse of the(J− 1 )K×(J− 1 )KpartitionedmatrixVthat contains Asy.Cov
[
βˆj,ˆβm]. The appropriate form of this matrix for a
set of cross-section estimators is given in Brant (1990). Das and van Soest (2000)
used the counterpart of Chamberlain’s fixed effects estimator, but do not provide
the specifics for computing the off diagonal blocks inV.
The full ordered probit model with fixed effects, including the individual spe-
cific constants, can be estimated by unconditional maximum likelihood using the
results in Greene (2008a, sec. 16.9.6.c). The likelihood function is concave (see
Pratt, 1981), so despite its superficial complexity, the estimation is straightforward.
(In the application below, with over 27,000 observations and 7,293 individual
effects, estimation of the full model required roughly five seconds of computa-
tion.) No theoretical counterpart to the Hsiao (1986, 2003) and Abrevaya (1997)
results on the smallTbias (incidental parameters problem) of the MLE in the pres-
ence of fixed effects has been derived for the ordered probit model. The Monte
Carlo results in Table 11.1 suggest that biases comparable to those in the binary
choice models persist in the ordered probit model as well. As in the binary choice
case, the complication of the fixed effects model is the small sample bias, not the
computation. The Das and van Soest (2000) approach finesses this problem, as
their estimator is consistent, but at the cost of losing the information needed to
compute partial effects or predicted probabilities.
11.5.3.2 Random effects
The random effects ordered probit model has been much more widely used than
the fixed effects model. Applications include Groot and van den Brink (2003),
who studied training levels of employees, with firm effects and gains to mar-
riage, Winkelmann (2004), who examined subjective measures of well-being with
individual and family effects, Contoyannis, Jones and Rice (2004), who analyzed
self-reported measures of health status, and numerous others. In the simplest case,
the quadrature method of Butler and Moffitt (1982) can be used.
11.5.4 Application
The GSOEP data that we have used earlier includes a self-reported measure of
health satisfaction,HSAT, that takes values 0, 1,..., 10. This is a typical application