Aggregation Bias in Estimating Log‐Log Demand Function

Abstract

In this study, we consider the problem of estimating demand functions using historical sales data. In particular, we study how using aggregate data may result in bias in such estimations. We consider a log‐log demand model, and two ways of how the price in the aggregate data is calculated, simple average price or weighted average price. We study whether there exists estimation bias in each of the two cases as well as the direction of the bias. We show that in general, the correlation between “average price” and “price dispersion” in each time period has an effect on the direction of the bias. We then propose ways to reduce the aggregation bias. We prove that when the price processes satisfy certain mild conditions, our debiasing procedure can recover the true parameters asymptotically. We also obtain statistical properties of the debiased parameters. We perform numerical experiments on both synthetic and real data to demonstrate the effectiveness of our approach. The numerical results show that our proposed bias mitigation approach has the potential of effectively reducing the bias in both synthetic and practical settings.