Fixed Point Theorems by D. R. Smart

By D. R. Smart

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P1: GEM/IKJ P2: GEM/IKJ QC: GEM/ABE CB495-03Drv CB495/Train KEY BOARDED 48 T1: GEM August 20, 2002 12:14 Char Count= 0 Behavioral Models The limitation of the logit model arises when we attempt to allow tastes to vary with respect to unobserved variables or purely randomly.

This probability is a J-dimensional integral over the density of the J error terms in εn = εn1 , . . , εn J . The dimension can be reduced, however, through recognizing that only differences in utility matter. With J errors (one for each alternative), there are J − 1 error differences. The choice probability can be expressed as a (J − 1)-dimensional integral over the density of these error differences: Pni = Prob(Uni > Un j ∀ j = i) = Prob(εn j − εni < Vni − Vn j ∀ j = i) = Prob(ε˜n ji < Vni − Vn j ∀ j = i) = I (ε˜n ji < Vni − Vn j ∀ j = i)g(ε˜ni ) d ε˜ni P1: GEM/IKJ P2: GEM/IKJ QC: GEM/ABE CB495-02Drv CB495/Train KEY BOARDED T1: GEM September 18, 2002 11:15 Char Count= 0 Properties of Discrete Choice Models 27 where ε˜n ji = εn j − εni is the difference in errors for alternatives i and j; ε˜ni = ε˜n1i , .

The variance of each error difference depends on the variances and covariances of the original errors. For example, the variance of the difference between the first and second errors is Var(ε˜n21 ) = Var(εn2 − εn1 ) = Var(εn1 ) + Var(εn2 ) − 2 Cov(εn1 , εn2 ) = σ11 + σ22 − 2σ12 . We can similarly calculate the covariance between ε˜n21 , which is the difference between the first and second errors, and ε˜n31 , which is the difference between the first and third errors: Cov(ε˜n21 , ε˜n31 ) = E(εn2 − εn1 )(εn3 − εn1 ) = E(εn2 εn3 − εn2 εn1 − εn3 εn1 + εn1 εn1 ) = σ23 − σ21 − σ31 + σ11 .

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