Palgrave Handbook of Econometrics: Applied Econometrics

William Greene 479

even in this form, it will generally bear little resemblance to the linear regression
model. As in other nonlinear cases, the interpretation of the model coefficients
is ambiguous. Partial effects on the probabilities associated with the choices for
individualiat timetare defined as:

δit(j,X′itβ,z′itγ)=∂Fit(j,X′itβ,z′itγ)/∂

(

xit,j

zit

)

=Fit′(j,Xitβ,z′itγ)

(

β

γ

)

.

These are likely to be of interest for particular individuals, or averaged across indi-
viduals in the sample. A crucial implication for use of the model is that these
partial effects may be quite different from the coefficients themselves. Since there
is no “regression” model at work, this calls into question the interpretation of the
model and its parts. No generality is possible at this point. We will return to the
issue below.
A related exercise in marginal analysis of the sample and estimated model is to
examine the aggregate outcomes predicted by the model:

nˆt,j=

∑n

i= 1

Fˆit(j,Xitβ,z′itγ)=

∑n

i= 1

dˆit,j.

where the “ˆ” indicates the estimate from the model. For example, ifxit,j,kdenotes
a policy variable, a price or a tax, say, we might be interested in:

nˆt,j=

∑n

i= 1

Fˆit(j,Xitβ,z′itγ|x^1 it,j,k)−

∑n

i= 1

Fˆit(j,Xitβ,zit′γ|x^0 it,j,k).

Although the subject of the impact in the partial effect is already scaled – it is
a probability between zero and one – it is still common for researchers to report
elasticities of probabilities rather than partial effects. These are:

ηit,j(variableit,j,k)=

F′it(j,Xitβ,z′itγ)

Fit(j,Xitβ,z′itγ)

×variableit,j,k×coefficientk.

This is prominently the case in the analysis of multinomial choice models, as we
will explore in section 11.7.
Finally, again because the model does not correspond to a regression except
in a very loose sense, the concept of fit measures is also ambiguous. There is no
counterpart to “explained variation” or “total variation” in this class of models, so
the idea behind the coefficient of determination(R^2 )in linear regression has no
meaning here. What is required to assess the fit of the model is, first, a specification
of how the model will be used to predict the outcome (choice), then an assessment
of how well the estimated model does in that regard.

11.2.3 Application

It will be helpful in the exposition below to illustrate the computations with a few
concrete examples based on “live” data. We will use two familiar datasets. The RWM