Econ 513, USC, Fall 2005
Lecture 15. Discrete Response Models:
Multinomial, Conditional and Nested Logit Models
Here we focus again on models for discrete choice with more than two outcomes.
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 individual
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
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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 Fall '07
 Rashidian
 Econometrics, Normal Distribution, Probability theory, probability density function, Likelihood function, McFadden

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