Lecture 17

Lecture 17 - Econ 513, USC, Fall 2005 Lecture 17. Discrete...

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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 6 = 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. So, suppose that Spargo raises its prices. Fewer people will go there, and they will go
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Lecture 17 - Econ 513, USC, Fall 2005 Lecture 17. Discrete...

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