Estimation of BLP models with high-dimensional controls

Abstract

This study proposes a framework for estimating demand in differentiated product markets with high dimensional product characteristics, building upon the seminal Berry, Levinsohn, and Pakes (1995) model, using market level data. We allow for a very large set of potential product characteristics, where the number of characteristics may exceed the number of market observations. Our contributions are twofold. First, we establish a general estimation theory for BLP models featuring high-dimensional nuisance parameters. We propose a Neyman orthogonal estimator specifically adapted to this framework, utilizing machine learning techniques, such as Lasso, to construct nuisance parameter estimators that are plugged into the Neyman orthogonal estimator. This approach offers a significant advantage: it achieves $\sqrt{T}$-asymptotic normality for parameters of interest--such as the price coefficient and price heterogeneity--even when nuisance parameters are estimated at slower rates due to their high dimensionality. Second, we apply this theory to a specialized BLP model under approximate sparsity, developing an estimation strategy for the high-dimensional nuisance parameters. The approximate sparsity condition posits that nuisance parameters can be controlled, up to a small approximation error, by a small and unknown subset of variables. In an economic context, this implies that while products have a vast array of characteristics, consumers focus on only a small subset of these due to bounded rationality. This condition makes the recovery of parameters of interest feasible by enabling nuisance parameter estimators to converge at the required rates. The practical performance of the method is evaluated through comprehensive Monte Carlo simulations, which demonstrate its efficacy in finite samples.