496 Discrete Choice Modeling
Therefore:
Prob[di= 1 |xi,zi]=#[
x′iβ+γzi+(ρ/σ )ui
√
1 −ρ^2]
.Inserting the expression forui=(zi−w′iα), and using the normal density for
the marginal distribution ofziin the second equation of (11.6), we obtain the
log-likelihood function for the sample:
lnL=∑n
i= 1 ln#[
( 2 di− 1 )(
x′iβ+γwi+(ρ/σu)(zi−w′iα)
√
1 −ρ^2)]
+ln[
1
σu
φ(
zi−w′iα
σu)]
.11.3.6 Panel data models
The ongoing development of large, rich panel data sets on individual and fam-
ily market experiences, such as the GSOEP data we are using here, has brought
attention to panel data approaches for discrete choice modeling. The extensions
of familiar fixed and random effects models are not direct and bring statistical
and computational issues that are not present in linear regression modeling. This
section will detail the most widely used techniques. This area of research is one
of the most active theoretical arenas as well. We will only have space to note the
theoretical frontiers briefly in the conclusions.
11.3.6.1 Panel data modeling frameworks
The natural departure point for panel data analysis of binary choice is the extension
of the familiar fixed and random effects linear regression models. Since the models
considered here are nonlinear, however, the convenient least squares and feasible
generalized least squares methods are unavailable. This proves to be more than an
inconvenience in this setting, as it mandates consideration of some specification
issues. We will begin by considering extensions of the fixed and random effects
models, then turn to more general models of individual heterogeneity, the random
parameters and latent class models. The various models described here all carry over
to a range of specifications. However, in the applied literature, the binary choice
model is the leading case.
11.3.6.2 Fixed effects model
The fixed effects model is:
d∗it=αi+x′itβ+z′iγ+εit, t=1,...,Ti,i=1,...,n
dit=1ifd∗it>0, anddit=0 otherwise.We have made the distinction between time varying attributes and characteris-
tics,xit, and time invariant characteristics,zi. The common effects,αi, may be