Limit theorems for estimating the parameters of differentiated product demand systems

We provide an asymptotic distribution theory for a class of generalized method of moments estimators that arise in the study of differentiated product markets when the number of observations is associated with the number of products within a given market. We allow for three sources of error: sampling error in estimating market shares, simulation error in approximating the shares predicted by the model, and the underlying model error. It is shown that the estimators are CAN provided the size of the consumer sample and the number of simulation draws grow at a large enough rate relative to the number of products. We consider the implications of the results for the Berry, Levinsohn and Pakes (1995) random coefficient logit model and the pure characteristics model analysed in Berry and Pakes (2002). The required rates differ for these two frequently used demand models. A small Monte Carlo study shows that the diffrences in asymptotic properties of the two models are reflected, in quite a striking way, in the models' small sample proper-ties. Moreover the limit distributions provide a good approximation to the actual Monte Carlo distribution of the parameter estimates. The results have important implications for the computational burden of the two models.