Random Coefficient Demand Laplace

Random Coefficient Demand Laplace - Using a Laplace...

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Unformatted text preview: Using a Laplace Approximation to Estimate the Random Coefficients Logit Model by Non-linear Least Squares 1 Matthew C. Harding 2 Jerry Hausman 3 March 14, 2007 1 This paper is forthcoming in the International Economic Review . We thank Ketan Patel for excellent research assistance. We thank Ronald Butler, Kenneth Train, Joan Walker and partici- pants at the MIT Econometrics Lunch Seminar and at the Harvard Applied Statistics Workshop for comments. 2 Department of Economics, MIT; and Institute for Quantitative Social Science, Harvard Uni- versity. Email: [email protected] 3 Department of Economics, MIT. E-mail: [email protected] Abstract Current methods of estimating the random coefficients logit model employ simulations of the distribution of the taste parameters through pseudo-random sequences. These methods suffer from difficulties in estimating correlations between parameters and computational limitations such as the curse of dimensionality. This paper provides a solution to these problems by approximating the integral expression of the expected choice probability using a multivariate extension of the Laplace approximation. Simulation results reveal that our method performs very well, both in terms of accuracy and computational time. 1 Introduction Understanding discrete economic choices is an important aspect of modern economics. McFadden (1974) introduced the multinomial logit model as a model of choice behavior derived from a random utility framework. An individual i faces the choice between K different goods i = 1 ..K . The utility to individual i from consuming good j is given by U ij = x ij β + ² ij , where x ij corresponds to a set of choice relevant characteristics specific to the consumer-good pair ( i,j ). The error component ² ij is assumed to be independently identi- cally distributed with an extreme value distribution f ( ² ij ) = exp(- ² ij )exp(- exp(- ² ij )) . If individual i is constrained to choose a single good within the available set, utility maximization implies that some good j will be chosen over all other goods l 6 = j such that U ij > U il , for all l 6 = j . We are interested in deriving the probability that consumer i chooses good j , which is P ij = Pr[ x ij β + ² ij > x il β + ² il , for all l 6 = j ] . (1) McFadden (1974) shows that the resulting integral can be solved in closed form implying the familiar expression: P ij = exp( x ij β ) K ∑ k =1 exp( x ik β ) (= s ij ) . (2) In some analyses it is also useful to think of the market shares of different firms. Without loss of generality we can also consider the choice probability described above to be the share of the total market demand which goes to good j in market i and we will denote this by s ij ....
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Random Coefficient Demand Laplace - Using a Laplace...

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