lect13 - Imbens, Lecture Notes 13, ARE213 Spring '06 ARE213...

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Imbens, Lecture Notes 13, ARE213 Spring ’06 1 ARE213 Econometrics Spring 2006 UC Berkeley Department of Agricultural and Resource Economics Discrete Response Models III: Multinomial, Conditional and 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 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 Frms), and covariates that vary by choice (and possibly individual), X ij . Examples of the Frst type include in- dividual 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 scientiFc) substance of the problem. Mc±adden 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 ± i =1 J ± j =0 1 { Y i = j ln P j ( X i ; θ ) . I. Multinomial Logit
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Imbens, Lecture Notes 13, ARE213 Spring ’06 2 Suppose we only have individual specifc covariates. Then we can model the response probability as Pr( Y = j | X )= exp( X ± β j ) 1+ J l =1 exp( X ± β l ) , For choices j =1 ,...,J and Pr( Y =0 | X 1 J l =1 exp( X ± β l ) , For the frst 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.
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lect13 - Imbens, Lecture Notes 13, ARE213 Spring '06 ARE213...

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