Econ 513, USC, Fall 2005
Lecture 17. Discrete Response Models :
Random Coefficient or Mixed Multinomial Logit Models
Let us first recall some of the properties of the conditional logit. We consider a case with
3 choices, dinner at Spargo, Watergrill, or McDonalds (
y
∈ {
S, W, M
}
). There is only one
characteristic of the choice that matters, price.
To make the comparisons simpler, let us
suppose that the prices for the first two are equal and much higher than for the other one,
P
S
=
P
W
>> P
M
.
The coefficient on this characteristic in the utility function is
β <
0.
(We leave out the intercept in the utility function for simplicity. These would capture taste
preferences for the three restaurants.) So, the utilities for the three choices are
U
iS
=
β
·
P
S
+
iS
,
U
iW
=
β
·
P
W
+
iW
,
and
U
iM
=
β
·
P
M
+
iM
.
The probability of dinner at Spargo is
Pr(
y
i
=
S
) = Pr(
U
iS
= max(
U
iS
, U
iW
, U
iM
)) =
exp(
βP
S
)
exp(
βP
S
) + exp(
βP
W
) + exp(
βP
M
)
.
It follows from the IIA (independence of irrelevant alternatives) property of the conditional
logit that
Pr(
y
i
=
W

y
i
=
S
) = Pr(
U
iW
> U
iM

U
iS
<
max(
U
iS
, U
iW
, U
iM
)) =
exp(
βP
W
)
exp(
βP
W
) + exp(
βP
M
)
.
It is also clear that the probability that
U
iW
> U
iM
is
Pr(
U
iW
> U
iM
) =
exp(
βP
W
)
exp(
βP
W
) + exp(
βP
M
)
.
Thus it follows that
Pr(
U
iW
> U
iM

y
i
=
S
) = Pr(
U
iW
> U
iM

U
iS
= max(
U
iS
, U
iW
, U
iM
)) =
exp(
βP
W
)
exp(
βP
W
) + exp(
βP
M
)
.
1
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So an implication of the IIA property is that the probability that the second choice is
Watergrill given that the first choice is Spargo is the same as the conditional probability
that you choose Watergrill to begin with. Again very unappealing.
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 Fall '07
 Rashidian
 Econometrics, Conditional Probability, Likelihood function, Spargo

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