Automatically Differentiable Random Coefficient Logistic Demand Estimation

The random coefficient logistic demand model of Berry et al. (1995) (henceforth BLP) has been a workhorse of the New Empirical Industrial Organization literature, allowing for varied substitution patterns across products, and accounting for endogeneity of price. The reliability of its estimation has been the subject of rigorous debate (Nevo, 2000; Conlon and Gortmaker, 2020; Knittel and Metaxoglou, 2014), and the estimator itself has been the study of many proposed advances in econometric techniques as a sophisticated yet widely used structural model (Hong et al., 2020; Forneron and Ng, 2020). The most common implementation of the BLP estimator involves the use of a nested fixed point (NFP) as an inner loop within an outer loop of GMM estimation, although we acknowledge the Mathematical Programming with Equilibrium Constraints (MPEC) approach of Dubé et al. (2012), which is beyond the scope of this paper. Dubé et al. (2012) and Conlon and Gortmaker (2020) find that derivative-free optimization algorithms such as the Nelder-Meade or simplex algorithms often fail to converge or converge to the wrong solution. As such, the literature has settled on the use of analytical derivatives with a derivative-based optimization algorithm such as L-BFGS. Nevo (2000) provides the analytical derivative for demand-only (DO) BLP in detail, and Conlon and Gortmaker (2020) indicate that the same is possible for demand-and-supply (DS) BLP, although it involves tensor products.