William Greene 541whereIis the inclusive value for the model:
I=log∑sexp[λs(z′iδs+Js)].The original MNL results if the inclusive parameters,λs, are all equal to one.
Alternative normalizations and aspects of the nested logit model are discussed
in Hensher and Greene (2002) and Hunt (2000). A second form moves the scaling
down to the twig level, rather than at the branch level. Here it is made explicit
that, within a branch, the scaling must be the same for all alternatives, but it can
differ between the branches:
P(j|b)=exp[
μb(x′j|bβ)]∑
q|bexp[
μb(x′q|bβ)]=exp[
μb(x′j|bβ)]exp(Jb)
.Note that, in the summation in the inclusive valueI, the scaling parameter is not
varying with the summation index. It is the same for all twigs in the branch.
At the next level up the tree, we define the conditional probability of choosing
the particular branch:
P(b|l)=exp[(
z′iγs+( 1 /μb)Jb)]∑
sexp[(
z′iγs+( 1 /μs)Js)]=exp[(
z′iγs+( 1 /μb)Jb)]exp(I),whereIlis the inclusive value for limbl:
Il=log∑
s|l
exp[
γl(
α′ys|l+( 1 /μs|l)Js|l)]
.In the nested logit model withP(j,b,l,r)=P(j|b,l,r)×P(b|l,r)×P(l|r)×P(r), the
marginal effect of a change in attribute “k” in the utility function for alternative “J”
in branch “B” of limb “L” of trunk “R” on the probability of choice “j” in branch “b”
of limb “l” of trunk “r” is computed using the following result: lower-case letters
indicate the twig, branch, limb and trunk of the outcome upon which the effect is
being exerted. Upper-case letters indicate the twig, branch, limb and trunk which
contain the outcome whose attribute is being changed:
∂logP(alt=j,limb=l,branch=b,trunk=r)
∂x(k)|alt=J,limb=L,branch=B,trunk=r
=D(k|J,B,L,R)=(k)×F,where(k)= coefficient onx(k)inU(J|B,L,R) and:
F= 1 (r=R)× 1 (l=L)× 1 (b=B)×[ 1 (j=J)P(J|BLR)] (trunk effect),
= 1 (r=R)× 1 (l=L)×[ 1 (b=B)−P(B|LR)]×P(J|BLR)]×τB|LR (limb effect),
= 1 (r=R)×[ 1 (l=L)−P(L|R)]×P(B|LR)×P(J|BLR)×τB|LR×σL|R (branch effect),
=[ 1 (r=R)P(R)]×P(L|R)×P(B|LR)×P(J|BLR)×τB|LR×σL|R×φR (twig effect),