Palgrave Handbook of Econometrics: Applied Econometrics

540 Discrete Choice Modeling

11.7.2 Nested logit models

The nested logit model allows for the grouping of alternatives into “nests” with
correlation across elements in a group. The natural analogy is to a “tree structure”;
e.g., Figure 11.2 suggests an elaborate, three-level treatment of an eight-alternative-
choice set.

Commute TRUNK

LIMBS

Plane Helicopter Car_Drv Car_Ride Train Bus Ferry Raft TWIGS

Fly Drive Land Water BRANCHES

Private Public

Figure 11.2 Nested choice set

The specific choice probabilities are redefined to be the conditional probability
of alternative twigjin branchb, limbl, and trunkr,j|b,l,r, At the next level up
the tree, we define the conditional probability of choosing a particular branch in
limbl, trunkr,b|l,r, the conditional probability of choosing a limb in trunkr,l|r,
and, finally, the probability of choosing a trunkr. By the laws of probability, the
unconditional probability of the observed choices made by an individual is:

P(j,b,l,r)=P(j|b,l,r)×P(b|l,r)×P(l|r)×P(r).

This is the contribution of an individual observation to the likelihood function for
the sample. (Note that in our example, there is only one trunk, soP(r)=1.)
The two-level nested logit model is the leading case, and occupies most of
the received applications. In this instance, a common specification places the
individual specific characteristics, such as demographic variables, in the branch
probabilities. For this basic model, then:

P(j|b)=

exp(x′j|bβ)

∑

q|bexp(x

′

q|bβ)

=

exp(x′j|bβ)

exp(Jb)

,

whereJbis the inclusive value for branchb,

Jb=log

∑

q|b

exp(x′q|bβ).

At the next level up the tree, we define the conditional probability of choosing a
particular branch:

P(b)=

exp

[

λb(z′iγb+Jb)

]

∑

sexp

[

λs(z′iγs+Js)

]=

exp

[

λb(z′iγb+Jb)

]

exp(I)

,