Palgrave Handbook of Econometrics: Applied Econometrics

William Greene 545

for all observations, in each of the four choices. The right-hand side reports the
same statistics for the observations that made the particular choices. Thus, for
example, the average terminal time for all 210 observations for the air choice is
61.01 minutes. For the 58 individuals who chose air, the average terminal time for
air is 46.534 minutes. We note before beginning that the sample proportions for
the four travel modes in this sample are 0.27619, 0.30000, 0.14286 and 0.28095,
respectively. Long study of this market revealed that the population values of these
proportions should be closer to 0.14, 0.13, 0.09 and 0.64, respectively. The sample
observations were deliberately drawn so that the car alternative received fewer
observations than random sampling would predict. The sample ischoice based.A
general adjustment for that phenomenon is the Manski–Lerman (1977) weighted
endogenous sampling maximum likelihood (WESML) correction, which consists
of two parts. First, we would fit a weighted log-likelihood:

lnL(WESML)=

∑n

i= 1

∑J

j= 1

πj

pj

dijlnij,

wheredij=1 if individualichooses alternativejand 0 otherwise,πjis the true
population proportion,pjis the sample proportion, andijis the probability for
outcomejimplied by the model. The second aspect of the correction is to use
a sandwich style corrected estimator for the asymptotic covariance matrix of the
MLE:
V(WESML)=H−^1 (G′G)H−^1 ,

whereHis the inverse of the (weighted) Hessian and (G′G)−^1 would be the
BHHH estimator based on first derivatives. The results to follow do not include
this correction – the results in the example would change slightly if they were
incorporated.
We fit a variety of models. The same utility functions were specified for all:

Ui,AIR=αAIR+βttTTMEi,AIR+βitINVTi,AIR+βgcGCi,AIR+γAHINCi+εi,AIR,

Ui,TRAIN=αTRAIN+βttTTMEi,TRAIN+βitINVTi,TRAIN+βgcGCi,TRAIN+εi,TRAIN,

Ui,BUS=αBUS+βttTTMEi,BUS+βitINVTi,BUS+βgcGCi,BUS+εi,BUS,

Ui,CAR=βttTTMEi,CAR+βitINVTi,CAR+βgcGCi,CAR+εi,CAR.

The estimated parameters for the several specifications are given in Table 11.11.
Model MNL is the base case multinomial logit model. Model MNP is the multi-
nomial probit model. The three nested logit (NL) models are nested logit models
with different tree structures:

NL(1) = Private (air, car), Public (train, bus).

NL(2) = Fly (air), Ground (train, bus, car)

NL(3) = Fly (air), Rail (train), Drive (car), Autobus (bus).

For the third of these, one of the inclusive value parameters,μj, must be constrained
to equal one. Model HEV (heteroskedastic extreme value) is the extreme value