Chapter on Discrete Choice Models Linear Probability Model ...

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where V j denotes V (x j ,z) for simplicity. This can be shown as follows. Since yj ∗ =max(y∗ 1 , ···,y∗ m )is equivalent to ² k
The Hessian matrix is non-positive deÞnite and the log likelihood function is concave. The conditional logit model has a restrictive property known as the independence of irrelevant alternative property (IIA): The probability odds ratio for the jth and the kth choices is the same irrespective of the total number m of choices. This is so, because Prob(I j =1|{1, 2,...,m}) Prob(I k =1|{1, 2,...,m}) = eV j e V k = Prob(I j =1|{j, k}) Prob(I k =1|{j, k}) . The IIA property is inappropriate for some situations. An example is the so-called blue bus and red bus paradox. Suppose that each consumer has three alternatives in choosing the red bus, the blue bus or his/her own automobile to work. Let x 1 (red bus), x 2 (blue bus) and x 3 (car) be the attributes of the three transportation modes. Suppose that consumers treat the two buses indifferent and are also indifferent between the automobile mode and the bus mode. In such situation, Prob(I 1 =1|x 1 ,x 2 )=Prob(I 1 = 1|x 1 ,x 3 )=Prob(I 2 =1|x 2 ,x 3 )=1/2 andProb(I 1 =1|x 1 ,x 2 ,x 3 )=Prob(I 2 =1|x 1 ,x 2 ,x 3 )=1/4. It follows that Prob(I1=1|x1,x2,x3) Prob(I1=1|x1,x3) Prob(I 3=1|x 1,x 2,x 3) =1/2 and Prob(I 3=1|x 1,x 3) =1. TheIIApropertyisnotsatisÞed in this situation. The inconsistency occurs because the two bus modes are perceived as similar alternatives rather than independent alternatives by the individual. 3
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Chapter on Discrete Choice Models Linear Probability Model ...

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