Imbens, Lecture Notes 13, ARE213 Spring ’06
1
ARE213
Econometrics
Spring 2006 UC Berkeley Department of Agricultural and Resource Economics
Discrete Response Models III:
Multinomial, Conditional and Nested Logit Models
Here we focus again on models for discrete choice with more than two choices.
We
assume that the outcome of interest, the choice
Y
takes on nonnegative integer values
between zero and
J
;
Y
∈ {
0
,
1
, . . . , J
}
.
Unlike the ordered case there is no particular
meaning to the ordering.
Examples are travel modes (bus/train/car), employment status
(employed/unemployed/outofthelaborforce), marital status (single/married/divorced/widowed)
and many others.
We wish to model the distribution of
Y
in terms of covariates.
In some cases we will
distinguish between covariates
X
i
that vary by units (individuals or firms), and covariates
that vary by choice (and possibly individual),
X
ij
.
Examples of the first type include in
dividual characteristics such as age, or education.
An example of the second type is the
cost associated with the choice, for example the cost of commuting by bus/train/car. This
distinction only arises from the economics (or general scientific) substance of the problem.
McFadden developed the interpretation of these models through utility maximizing choice
behavior.
In that case we may be willing to put restrictions on the way covariates affect
choices: costs of a particular choice affect the utility of that choice, but not the utilities of
other choices.
The strategy is to develop a model for the conditional probability of choice
j
given the
covariates. Suppose the model is Pr(
Y
=
j

X
) =
P
j
(
X
;
θ
). Then the log likelihood function
is
L
(
θ
) =
N
i
=1
J
j
=0
1
{
Y
i
=
j
} ·
ln
P
j
(
X
i
;
θ
)
.
I. Multinomial Logit
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Imbens, Lecture Notes 13, ARE213 Spring ’06
2
Suppose we only have individual specific covariates.
Then we can model the response
probability as
Pr(
Y
=
j

X
) =
exp(
X β
j
)
1 +
∑
J
l
=1
exp(
X β
l
)
,
for choices
j
= 1
, . . . , J
and
Pr(
Y
= 0

X
) =
1
1 +
∑
J
l
=1
exp(
X β
l
)
,
for the first choice. This is a direct extension of the binary response logit model. It leads to
a very wellbehaved likelihood function and is easy to estimate. More interestingly it can be
viewed as a special case of the following conditional logit.
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 Spring '06
 IMBENS
 Likelihood function, Yi, xij, conditional logit model

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