William Greene 519Tobias and Li extended their analysis to this case simply by “transforming” the
dependent variable in Butler, Finegan and Siegfried’s second equation. Computing
the log-likelihood using sets of bivariate normal probabilities is fairly straightfor-
ward for the bivariate ordered probit model (see Greene, 2007b). However, the
classical study of these data using the bivariate ordered approach remains to be
done, so a side-by-side comparison to Tobias and Li’s Bayesian alternative estimator
is not possible. The endogeneity of the calculus dummy variables remains a feature
of the model, so both the MLE and the Bayesian posterior are less straightforward
than they might appear.
The bivariate ordered probit model has been applied in a number of settings
in the recent empirical literature, including husband and wife’s education levels
(Magee, Burbidge and Robb, 2000), family size (Calhoun, 1991) and many others.
In two early contributions to the field of pet econometrics, Butler and Chatterjee
analyze ownership of cats and dogs (1995) and dogs and televisions (1997).
11.5.3 Panel data applications
11.5.3.1 Fixed effects
D’Addio, Eriksson and Frijters (2003), using methodology developed by Frijters,
Haisken-DeNew and Shields (2004) and Ferrer-i-Carbonel and Frijters (2004), ana-
lyzed survey data on job satisfaction using the Danish component of the European
Community Household Panel. Their estimator for an ordered logit model is built
around the logic of Chamberlain’s estimator for the binary logit model. The
approach is robust to individual specific threshold parameters and allows time
invariant variables, so it differs sharply from the fixed effects models we have
considered thus far, as well as from the ordered probit model.^24 Unlike Chamber-
lain’s estimator for the binary logit model, however, their conditional estimator is
not a function of minimal sufficient statistics. As such, the incidental parameters
problem remains an issue.
Das and van Soest (2000) proposed a somewhat simpler approach. (See, as well,
Long’s 1997 discussion of the “parallel regressions assumption,” which employs
this device in a cross-section framework.) Consider the base case ordered logit
model with fixed effects:
yit∗=αi+x′itβ+εit,εit|Xi∼N[0, 1]
yit=jifμj− 1 <yit∗<μj,j=0, 1,...,Jandμ− 1 =−∞,μ 0 =0,μJ=+∞.The model assumptions imply that:
Prob(yit=j|Xi)=%(μj−αi−x′itβ)−%(μj− 1 −αi−x′itβ),where%(t)is the c.d.f. of the logistic distribution. Now, define a binary variable:
wit,j=1ifyit>j, j=0,...,J−1.It follows that:
Prob[wit,j= 1 |Xi]=%(αi−μj+x′itβ)=%(θi+x′itβ).