486 Discrete Choice Modeling
11.3.2.3 Marginal effects
Partial effects in the binary choice model are computed for continuous variables
using the general result:
δi=
∂Prob(di= 1 |wi)
∂wi
=f(w′iθ)θ.For a binary variable, such as gender or degree attained, the counterpart would be:
i=F(w′iθ+γk)−F(w′iθ)whereγkis the coefficient on the dummy variable of interest (assumed to be a
characteristic of the individual). These are typically evaluated for the average indi-
vidual in the sample, though current practice somewhat favors theaverage partial
effect:
δ=^1 n∑n
i= 1∂Prob(di= 1 |wi)
∂wi
=^1 n∑n
i= 1 f(w′
iθ)θ
=(
1
n∑n
i= 1 f(w′
iθ))
θ.(The two estimators will typically not differ substantively.) Standard errors for par-
tial effects are usually computed using the delta method. LetVdenote the estimator
of the asymptotic covariance matrix of the MLE ofθ. For a particular vector,wi:
Γi=
∂δi
∂θ′
=[
f′(w′iθ)]
I+[f(w′iθ)]θwi′.For a binary variable in the model, in addition to (or in) thewi, the corresponding
row ofΓiwould be:
Γi,k=∂i,k/∂(θ′,γk)=f(w′iθ+γk)[wi,1]−f(w′iθ)[wi,0].For the particular choice ofwi, then, the estimator of the asymptotic covariance
matrix forδiwould beiV′i, computed at the maximum likelihood estimates. It
is common to do this computation at the means of the data,w=^1 nin= 1 wi. For the
average partial effect, the computation is complicated a bit because the terms inδ
are correlated – they use the same estimator of the parameters – so the variance of
the mean is not (1/n)times the sum of the variances. It can be shown (see Greene,
2008, Ch. 23) that the appropriate computation reduces to:
Est.Asy.Var[δ]=V′, whereΓ=1
n∑n
i= 1 Γi.An alternative approach to computing standard errors for the marginal effects is
the method of Krinsky and Robb (1986). A set ofRrandom draws is taken from the
estimated (asymptotic) normal population with meanθˆMLEand varianceVand
the empirical mean squared deviation of the estimated partial effects is computed
using the MLE:
Est.Asy.Var[δ]=
1
R∑R
r= 1(
δr−δ)(
δr−δ)′
,whereδris computed at the random draw andδis computed atθˆMLE.