Palgrave Handbook of Econometrics: Applied Econometrics

William Greene 489

complication arises in interpretation of the model. The partial effects in this
augmented model are:

δi=

∂Prob(di= 1 |wi,vi)

∂

(

wi

vi

) =f

(

wi′θ

exp(vi′τ)

)(

θ

[−(w′iθ)/exp(v′iτ)]τ

)

.

Ifwiandvihave variables in common, then the two effects are added. Whether
they do or not, this calls into question the interpretation of the original coefficients
in the model. Ifwiandvido share variables, then the partial effect may have sign
and magnitude that both differ from those of the coefficients,θ. At a minimum, as
before, the scales of the partial effects are different from those of the coefficients.
For testing for homoskedasticity, the same three statistics as before are useable.
(This is a parametric restriction on the model;H 0 :τ= 0 .) The derivatives of the
log-likelihood function are presented in Greene (2008, Ch. 23). As usual, the LM
test is the simplest to carry out. The term necessary to compute the LM statistic
under the null hypothesis is:

gi=qifi

(

wi

(−w′iθ)vi

)

.

A second specification test of interest concerns the distribution. Silva (2001) has
suggested a score (LM) test that is based on adding a constructed variable to the
logit or probit model. An alternative way of testing the two competing models
could be based on Vuong’s (1989) statistic. Vuong’s test is computed using:

λVuong=

√

nm

sm

, where m=

1

n

in= 1 [lnLi(probit)−lnLi(logit)],

andsmis the sample standard deviation. Vuong shows that, under certain assump-
tions (likely to be met here for these two models),λVuonghas a limiting standard
normal distribution. Large positive values (larger than+1.96) favor the probit
model, while large negative values (less than−1.96) favor the logit model. The
power of these two statistics for this setting remains to be investigated. As with
all specification tests, the power depends crucially on the true but unknown
underlying model, which may be unlike either candidate model.

11.3.2.6 The fit of the model

As noted earlier, in modeling binary (or other discrete) choices, there is no direct

counterpart to theR^2 goodness-of-fit statistic. A common computation which,
unfortunately in spite of its name, does not provide such a measure is thelikelihood
ratio index, which is also called the

pseudoR^2 = 1 −lnL/lnL 0 ,

where lnLis the log-likelihood for the estimated model (which must include a
constant term) and lnL 0 is the log-likelihood function for a model that only has a