586 Panel Data Methods
the distribution of the individual heterogeneity, which is conditioned out of the
model.
The conditional logit can be applied to ordered data by choosing a particular
threshold value and collapsing the data into a binary measure. A recent extension
of Chamberlain’s model, the conditional ordered fixed effects logit, proposed by
Ferrer-i-Carbonel and Frijters (2004), and applied to data on SAH by Frijterset al.
(2005), suggests a method to reduce the drastic loss in the number of observations
by identifying individual-specific threshold values to collapse the ordered depen-
dent variable into a binary format. Das and van Soest (1999) combine adjacent
categories so that the dependent variable is summarized as a binary variable, and
then use conditional logits. They repeat this for all the possible combinations of
adjacent categories to get a set of estimates of the parameters of interest. They
then define a linear combination of these estimates, with the optimal weighting
matrix used to compute the final estimate obtained from a minimum distance
approach. Ferrer-i-Carbonel and Frijters (2004) also propose an estimator that
collapses the ordered variable into a binary format, but they use an individual
specific threshold value. To find this individual threshold, the authors maximize
a weighted sum of log-likelihood functions, similar to Das and van Soest (1999),
subject to the constraint that the sum of squared weights across all possible thresh-
old values across all individuals must be equal to the number of individuals in
the sample. The threshold is selected for which the analytical expected Hessian
is minimized. However, this formulation of the estimator is highly computation
intensive, a fact which makes its wider application less attractive. In a simplifica-
tion of this estimator, one can simply use the within-individual means as a cut-off
criterion.
12.4.1.1 Dynamic models
Even for linear models the within-groups estimator breaks down in dynamic
models, such as:
yit=αyit− 1 +ui+εit. (12.17)
This is because the group mean is a function ofεitandεit− 1. An alternative is to
use the differenced equation:
yit=αyit− 1 +εit, (12.18)in which case bothyit− 2 andyit− 2 are valid instruments foryit− 1 as long as
the error term (εit)does not exhibit autocorrelation. Arellano and Bond (1991)
proposed generalized method of moments (GMM) estimators for dynamic panel
data models: linear models that can include leads and lags of the dependent variable
as well as a fixed effect. Instruments are created within the model by first taking
differences of the equation to sweep out the individual effect and then using lagged
levels or differences of the regressors as instruments.
Bover and Arellano (1997) extend the use of GMM to dynamic specifications for
categorical and limited dependent variable models, where it is not possible to take
first differences or orthogonal deviations as the latent variabley∗is unobserved.