542 Discrete Choice Modeling
whereτB|LR,σL|RandφRare parameters in the MNL probabilities. The marginal
effect is:
∂P(j,b,l,r)/∂x(k)|J,B,L,R=P(j,b,l,r)(k)F.
A marginal effect has four components: an effect on the probability of the particular
trunk, one on the probability for the limb, one for the branch, and one for the
probability for the twig. (Note that with one trunk,P(l)=P(1) = 1, and likewise
for limbs and branches.) For continuous variables, such as cost, it is common to
report, instead:
Elasticity=x(k)|J,B,L,R×(k|J,B,L,R)×F.The formulation of the nested logit model imposes no restrictions on the inclu-
sive value parameters. However, the assumption of utility maximization and the
stochastic underpinnings of the model do imply certain restrictions. For the former,
in principle, the inclusive value parameters must lie between zero and one. For the
latter, the restrictions are implied by the way the random terms in the utility func-
tions are constructed. In particular, the nesting aspect of the model is obtained by
writing:
εj|b,l,r=uj|b,l,r+vb|l,r.
That is, within a branch, the random terms are viewed as the sum of a unique com-
ponent and a common component. This has certain implications for the structure
of the scale parameters in the model. In particular, it is the source of the oft cited
(and oft violated) constraint that the IV parameters must lie between zero and one.
These are explored in Hunt (2000) and Hensher and Greene (2002).
11.7.3 Mixed logit and error component models
This model is somewhat similar to the random coefficients model for linear regres-
sions (see Bhat, 1996; Jain, Vilcassim and Chintagunta, 1994; Revelt and Train,
1998; Train, 2003; Berry, Levinsohn and Pakes, 1995). The model formulation is a
one-level MNL for individualsi=1,...,nin choice settingt. We begin with the
basic form of the MNL model, with alternative specific constantsαjiand attributes
xji:
Prob(yit=j|Xit)=exp(
αji+ix′it,jβi)∑Jit
q= 1 exp(
αqi+ix′it,qβi).The random parameters model emerges as the form of the individual specific
parameter vector,βi, is developed. The most familiar, simplest version of the model
specifies:
βki=βk+σkvki,
αji=αj+σjvji,whereβkis the population mean,vkiis the individual specific heterogeneity, with
mean zero and standard deviation one, andσkis the standard deviation of the dis-
tribution of theβkis aroundβk. The term “mixed logit” is often used in the literature