Econ 513, USC, Fall 2005
Lecture 14.
Discrete Response Models:
Ordered Multinomial Response
Models
Now we consider discrete response models with more than two possible responses. In this
lecture we limit ourselves to the case where the outcomes are ordered.
There are two important cases. In the first there is an underlying continuous variable
but we only observe an indicator for a particular range. An example is earnings data that
may come coded in intervals: 0, [0
,
10], [10
,
50], [50
,
∞
).
Another example is educational
choices where outcomes may be coded as less than high school, high school, some college,
more than college. Such data are referred to as intervalcoded data. Typically we model the
underlying continuous variable as linear and normal (or logistic) with a set of covariates:
y
*
i
=
x
i
β
+
ε
i
,
with the observed outcome an indicator for the interval for
j
= 0
, . . . , J
:
y
i
=
j
if
α
j
≤
y
*
< α
j
+1
,
with
α
J
+1
=
∞
,
α
0
=
∞
, and
α
j

1
< α
j
.
In this case the key assumption is that the
boundaries
α
j
are known a priori. In this case we are typically interested in the conditional
expectation of the latent outcome
E
[
y
*

x
]
,
rather than in the distribution of the observed outcome,
Pr(
y
=
j

x
)
.
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 Fall '07
 Rashidian
 Econometrics, Conditional Probability, Probability theory, Likelihood function, Yi, log likelihood function, discrete response

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