Estimating demand for differentiated products with zeroes in market share data

Abstract

In this paper, we introduce a new approach to estimating differentiated product demand systems that allows for products with zero sales in the data. Zeroes in demand are a common problem in differentiated product markets, but fall outside the scope of existing demand estimation techniques. We show that with a lower bound imposed on the expected sales quantities, we can construct upper and lower bounds for the conditional expectation of the inverse demand. These bounds can be translated into moment inequalities that are shown to yield consistent and asymptotically normal point estimators for demand parameters under natural conditions. In Monte Carlo simulations, we demonstrate that the new approach works well even when the fraction of zeroes is as high as 95%. We apply our estimator to supermarket scanner data and find that correcting the bias caused by zeroes has important empirical implications, for example, price elasticities become twice as large when zeroes are properly controlled.

Demand estimation with infrequent purchases and small market sizes

“…As in our previous simulations, implementing ad hoc fixes for BLP (1995) do not allow us to recover sensible estimates and typically return price parameters, which understate price sensitivity significantly. Similarly, we find that the estimator of Gandhi, Lu, and Shi (2023) results in significant bias due to the lack of "safe products. "…”

“…Similarly, dropping zeros lead to price elasticity estimates very close to zero, since observations with positive shares feature higher willingness to pay. We find that low arrival rates prevent any product from having consistently well-measured market shares, which causes the estimator of Gandhi, Lu, and Shi (2023) to yield biased demand estimates as well.…”

“…6 For comparison, Dubé, Hortaçsu, and Joo (2021) test their model on data with 42% zero shares, and Gandhi, Lu, and Shi (2023) test their estimator on synthetic data with 52% zero shares. Their bounds estimator is also tested on data with 96% zero observations.…”

“…We contrast our estimation method (Poisson RC) with Berry, Levinsohn, and Pakes (1995) and Gandhi, Lu, and Shi (2023) for the moderate (λ t = 25) and small (λ t = 5) arrivals data generating process. To implement these alternative approaches, we need to provide the estimators with a measure of empirical shares.…”

“…This allows us to rationalize zero-sale observations and account for the sampling distribution of sales driven by small market sizes. Relative to other proposed solutions to the zeros problem, including Quan and Williams (2018), Li (2019), Adam, He, and Zheng (2020), Dubé, Hortaçsu, and Joo (2021), Lima (2021), and Gandhi, Lu, and Shi (2023), we explicitly leverage market-size variation as a source of zero sales. An alternative approach would be to fix the market size and model the multinomial distribution of sales, similar to Conlon and Mortimer (2013).…”