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Unformatted text preview: Imbens, Lecture Notes 14, ARE213 Spring ’06 1 ARE213 Econometrics Spring 2006 UC Berkeley Department of Agricultural and Resource Economics Discrete Response Models IV: Nested Logit Models Here we focus again on models for discrete choice with more than two choices. We assume that the outcome of interest, the choice Y takes on nonnegative integer values between zero and J ; Y ∈ { , 1 ,...,J } . Unlike the ordered case there is no particular meaning to the ordering. Examples are travel modes (bus/train/car), employment status (employed/unemployed/outofthelaborforce), housing choice: apartment, townhouse, sin gle family house, and many others. One way to induce correlation between the choices is through nesting them. We do this in three steps. I. Nested Logit with a Threevalued Outcome The utility associated with choice j is U ij = μ ij + ij = X ij β + ij . Before in the multinomial logit model we assumed that three residuals were independent with extreme value distributions, implying that the joint distribution function of the three residuals has the form F ( , 1 , 2 ) = exp( exp( )) exp( exp( 1 )) exp( exp( 2 )) . Now suppose the set of choices { , 1 , 2 } can be partitioned into 2 sets, the first consisting of only the first choice 0, and the second consisting of the pair of choices { 1 , 2 } . We assume the joint distribution function has the form F ( , 1 , 2 ) = exp( exp( )) exp( (exp( 1 /ρ + exp( 2 /ρ )) ρ ) . Imbens, Lecture Notes 14, ARE213 Spring ’06 2 So, is still independent of ( 1 , 2 ), but ( 1 , 2 ) are no longer independent, with joint distri bution function F ( 1 , 2 ) = exp( (exp( 1 /ρ + exp( 2 /ρ )) ρ ) . The correlation coefficient for ( 1 , 2 ) is 1 ρ 2 . If ρ = 1, we are back in the multinomial logit case. The marginal distribution function for 1 is F ( 1 ) = exp( exp( 1 )) , so that is just an extreme value distribution, with marginal probability density function...
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This note was uploaded on 08/01/2008 for the course ARE 213 taught by Professor Imbens during the Spring '06 term at Berkeley.
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