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By Oakley Haldeman, Al Trace, Jimmy Lee

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As in section 1, let yi be the vector of dependent variables for observation i, iϭ 1,2,…,N, where yij ϭ1 iff choice j is chosen by i. The probability of i choosing j conditional on Xi is given in equation (8), and its frequency simulator is given in equation (10). The frequency simulator should be replaced by one of the simulators discussed in section 2, but for now we will use the frequency simulator for ease of presentation. As was discussed earlier, E[yij |Xi]ϭ Pr[yij ϭ 1|Xi]. Let Pi be a J-element vector with Pr[yij ϭ1|Xi] in the jth element of Pi, and let ␧i ϭyi ϪPi.

Then E[␧i |Xi]ϭ0 and E ͚QЈ␧ˆ ϭ0 (35) i i i for any set of exogenous instruments Qi. Thus, conditional on a chosen Qϭ(Q1,Q2,…,QN), the ␪ ϭ(␤,⍀) that satisfies ⌺iQЈi ␧i ϭ0 is the MOM estimator of ␪. Let Pˆi be an unbiased simulator of Pi, and let ␧ˆ i ϭyi ϪPˆi. Then the ␪ that solves ͚QЈ␧ˆ ϭ0 (36) i i i is the MSM estimator of ␪. To find a reasonable Q, consider the log likelihood contribution for the multinomial probit model Li ϭ ͚y logP . ij ij j 9 Extra conditions are found in McFadden (1989) and Pakes and Pollard (1989).

The authors consider a class of model where agents choose from a set of mutually exclusive alternatives, and must choose between them over a number of time periods. The general intractability of this type of model is again linked to the issue of information content, and specifically that the mapping from the observed data (in this instance choices and payoffs) to the posterior distribution of the structural parameters is in most cases intractable. It is worth emphasizing that the coupling of Bayesian inference in a discrete, dynamic framework gives rise to a number of additional problems which are not present in a pure discrete choice model.

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