Estimating Demand with Unobserved No-Purchases on Revenue-Managed Data

Problem definition : This paper studies the joint estimation of the consumer arrival rate and choice model parameters when “no-purchasers” (customers who considered the product but did not purchase) are not observable. Estimating this unconstrained demand even with the simplest discrete-choice model such as the multinomial logit (MNL) becomes challenging as we do not know the fraction that have chosen the outside option (i.e., not purchased). Methods have been proposed to use market share to pin down the parameter associated with the outside option. However, market share data are difficult to obtain in many situations, and in some industries, such as fashion retail, have little meaning as the items are difficult to compare. In this paper, we point out an additional difficulty that can arise in practice: Many firms monitor sales and optimize their prices and assortments within the sale period as part of their revenue management (RM) process, based on partially observed demand. This can potentially cause a revenue management induced endogeneity as the data used for estimation is the result of optimization (in turn based on prior data) to set controls. As we demonstrate, methods that work well on randomly generated assortments may do badly on optimized assortment data. Methodology/results : In this paper, we propose a robust method when the firm cannot observe no-purchases and has no market share information, and the data have been revenue-managed. We develop a two-step generalized method-of-moments (GMM) procedure that is based on a modified moment condition, and importantly, does not require instrumental variables (IVs), a significant advantage in practice. Managerial implications : In Monte Carlo simulations, the performance of our method matches existing methods when the controls are generated randomly, and is robust under all conditions, whether RM-induced endogeneity is present or not. On a large real-world data set from the fashion industry, subject to stock-outs and markdown pricing along with unknown management controls, our method provides robust estimates compared with existing methods without requiring any input on market shares, which is especially difficult to pin down at a category and season/collection level.