Lecture 15

Lecture 15 - Econ 513, USC, Fall 2005 Lecture 15. Discrete...

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Econ 513, USC, Fall 2005 Lecture 15. Discrete Response Models: Multinomial, Conditional and Nested Logit Models Here we focus again on models for discrete choice with more than two outcomes. We assume that the outcome of interest, the choice y takes on non-negative integer values between zero and J ; y ∈ { 0 , 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/out-of-the-laborforce), marital status (single/married/divorced/widowed) and many others. We wish to model the distribution of y in terms of covariates. In some cases we will distinguish between covariates x i that vary by units (individuals or firms), and covariates that vary by choice (and possibly individual), x ij . Examples of the first type include individual characteristics such as age, or education. An example of the second type is the cost associated with the choice, for example the cost of commuting by bus/train/car. This distinction only arises from the economics (or general scientific) substance of the problem. McFadden developed the interpretation of these models through utility maximizing choice behavior. In that case we may be willing to put restrictions on the way covariates affect choices: costs of a particular choice affect the utility of that choice, but not the utilities of other choices. The strategy is to develop a model for the conditional probability of choice j given the covariates. Suppose the model is Pr( y = j | x ) = P j ( x ; θ ). Then the log likelihood function is L ( θ ) = N X i =1 J X j =0 1 { y i = j } · ln P j ( x i ; θ ) . I. Multinomial Logit Suppose we only have individual specific covariates. Then we can model the response probability as Pr( y = j | x ) = exp( x 0 β j ) 1 + J l =1 exp( x 0 β l ) , for choices j = 1 , . . . , J and Pr( y = 0 | x ) = 1 1 + J l =1 exp( x 0 β l ) , for the first choice. This is a direct extension of the binary response logit model. It leads to a very well-behaved likelihood function and is easy to estimate. More interestingly it can be viewed as a special case of the following conditional logit. 1
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II. Conditional Logit Suppose all covariates vary by choice (and possibly also by individual, but that is not essential here). Then McFadden proposed the conditional logit model: Pr( y i = j | x i 0 , . . . , x iJ ) = exp( x 0 ij β ) J l =0 exp( x 0 il β ) , for j = 0 , . . . , J .
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Lecture 15 - Econ 513, USC, Fall 2005 Lecture 15. Discrete...

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