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: the sampling error in estimating market shares, the simulation error in approximating the shares predicted by the model, and the underlying model error. The limiting distribution of the parameter estimator is normal 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 specialise our distribution theory to the Berry, Levinsohn, and Pakes (1995) random coefficient logit model and a pure characteristic model. The required rates differ for these two frequently used demand models. A small Monte Carlo study shows that the difference in asymptotic properties of the two models are reflected in the models' small sample properties. These differences impact directly on the computational burden of the two models.