Palgrave Handbook of Econometrics: Applied Econometrics

488 Discrete Choice Modeling

mentioned earlier. UsingVBHHHproduces a particularly convenient computation,
as well as an interesting and surprisingly simple test of the null hypothesis that all
coefficients save the constant are zero. UsingVBHHHand expanding the terms, we
have:

λLM=

(∑n

i= 1 qiwif

0

i

)′(∑n

i= 1 q

2

i(f

0

i)

(^2) w
iw
′
i
)− 1 (∑n
i= 1 qiwif
0
i
)
,
and an immediate simplification occurs becauseq^2 i =1. The density is computed
at the restricted estimator, however obtained. If the null hypothesis is that all
coefficients are zero save for the constant, then, for the logit model,fi^0 =f^0 =
P 1 ( 1 −P 1 ). For the probit model, the estimator of the constant term will be#−^1 (P 1 )
andf^0 =φ[#−^1 (P 1 )]. Taking this constant outside the summation ingleaves
in= 1 qiwi=n[P 1 w 1 −P 0 w 0 ], wherew 1 is the sample mean of then 1 observations
withdiequal to one andw 0 is the mean of then 0 remaining observations. Note
that the constantf^0 falls out of the resulting statistic, and we are left with the LM
statistic for testing this null hypothesis:
λLM=n^2 [P 1 w 1 −P 0 w 0 ]′
(
W′W
)− 1
[P 1 w 1 −P 0 w 0 ],
whereWis the data matrix withith row equal tow′i. As in the case of the likelihood
ratio (LR) statistic, the same computation is used for both the probit and logit
models.
11.3.2.5 Specification tests
Two specification issues are typically addressed in the context of these parametric
models, heteroskedasticity and the distributional assumption. For the former, since
there are no useful “residuals” whose squares will reveal anything about scaling in
the model, general approaches such as the Breusch and Pagan (1979, 1980) LM test
or the White (1980) test are not available. Heteroskedasticity must be built into the
model and tested parametrically. Moreover, there is no robust approach to estima-
tion and inference that will accommodate heteroskedasticity without specifically
making it part of the model. (In linear regression, the ordinary least squares (OLS)
estimator and White’s (1980) heteroskedasticity robust covariance matrix serve that
purpose.) A common approach to modeling heteroskedasticity in parametric binary
choice models is based on Harvey’s (1976) exponential model:
d∗i=x′iβ+z′iγ+εi, E[εi|xi,zi,vi]=0, Var[εi|xi,zi,vi]=[exp(v′iτ)]^2
di=1ifd∗i>0, anddi=0 otherwise,
whereviis a known set of variables (that does not include a constant term) andτ
is a new parameter vector to be estimated. The adjustment of the log-likelihood is
fairly straightforward; the terms are changed to accommodate
Prob(di= 1 |xi,zi,vi)=F[w′iθ/exp(v′iτ)].
Maximization of the likelihood function with respect to all the parameters is
somewhat more complicated, as the function is no longer globally concave. The