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

Discrete-choice models such as the multinomial-logit (MNL) model are increasingly being used to model customer purchase behaviour in hotels, airlines, fashion retail and e-commerce. Their estimation however has proved difficult in practice. One reason for this difficulty is well-known: most firms in retail do not, or unable to, record customers who were interested in purchasing but did not buy the product (``no-purchasers"). Estimating demand even with the simplest discrete-choice model such as the MNL becomes challenging then as we do not know the fraction that have chosen an outside option (not purchased). Indeed, the parameters of the MNL model may not be identifiable with such data. Some previous approaches have proposed using ``market-share" to pin down the parameter associated with the outside option. However, in many industries, market-share data is difficult to obtain, and for some, such as fashion products, has little meaning. In this paper we point out an additional difficulty that arises in practice: Many firms constantly monitor sales and optimize their prices and assortments based on partially observed demand. This leads to an optimization-induced endogeneity as the input used for estimation has been processed by optimization that takes both past data as well as future demand trends in setting controls. As we demonstrate, methods that work well on randomly generated assortments may do badly on optimized assortment data. In this paper we propose a robust method when the firm cannot observe no-purchases, has no market-share information, and the optimization-induced endogeneity exists in the data. The method is a two-step GMM (Generalized Method of Moments) procedure for which we show the estimates are consistent, and we give intuition for its robustness. In Monte-Carlo simulations the performance of our method matches existing methods on randomly generated controls, and is superior in accuracy and robustness when optimization-induced endogeneity is present. On a large real-world data set from the fashion industry --- subject to markdowns as well as stock-outs and unknown management controls --- our method provides very reasonable and robust estimates compared to existing methods.