516 Discrete Choice Modeling
By the laws of probability:Prob(yi=j|xi,,zi)=Prob(Ui∗≤μj)−Prob(Ui∗≤μj− 1 )=F(μj−x′iβ−z′iγ)−F(μj− 1 −x′iβ−z′iγ),whereF(t)is the assumed c.d.f., either normal or logistic. These are the terms that
enter the log-likelihood for a sample ofnobservations. The standard conditions for
maximum likelihood estimation apply here. The results in Table 11.1 suggest that
the force of the incidental parameters problem in the fixed effects case is similar to
that for the binomial probit model.
As usual in discrete choice models, partial effects in this model differ substan-
tively from the coefficients. Note, first, that there is no obvious regression at work.
Sinceyiis merely a labeling with no implicit scale, there is no conditional mean
function to analyze. In order to analyze the impact of changes in a variable, say
income, one can decompose the set of probabilities. For a continuous variable
xi,k, e.g.:
δi,k(j)=∂Prob(yi=j|xi,zi)/∂xi,k=−βk[f(μj−x′iβ−z′iγ)−f(μj− 1 −x′iβ−z′iγ)], j=0,...,J,wheref(t)is the density, dF(t)/dt. The sign of the partial effect is ambiguous,
since the difference of the two densities can have either sign. Moreover, since
jJ= 0 Prob(yi= 1 |xi,zi)=1, it follows thatjJ= 1 δi,k(j)=0. Since the c.d.f. is mono-
tonic, there is one sign change in the set of partial effects as the example below
demonstrates. For purposes of using and interpreting the model, it seems that
the coefficients are of relatively little utility – neither the sign nor the magnitude
directly indicates the effect of changes in a variable on the observed outcome.
Terza (1985) and Pudney and Shields (2000) suggested an extension of the
ordered choice model that would accommodate heterogeneity in the threshold
parameters. The extended model is:
Prob(yi=j|xi,zi)=F(μi,j−x′iβ−z′iγ)−F(μi,j− 1 −x′iβ−z′iγ),where:
μi,j=v′iπj, whereπ 0 =0,for a set of variablesvi. The model as shown has two complications First, it is
straightforward to constrain the fixed threshold parameters to preserve the ordering
needed to ensure that all probabilities are positive.^22 When there are variablesvi
in the construction, it is no longer possible to produce this result parametrically.
The authors (apparently) did not find it necessary to confront this constraint. A
second feature of the model (which was examined at length by the authors) is the
unidentifiability of elements ofπjwhenviand (xi,zi) contain the same variables.
This is a result of the linear functional form assumed forμi,j. Greene (2007a) and