Selecting Data Granularity Using the Power Likelihood
ERIC T. BRADLOW – UNIVERSITY OF PENNSYLVANIA
Imagine a firm that records sales over time, yielding data that can be used for making marketing decisions. When using such data, a firm wants to select the interval of analysis (e.g., weekly, monthly, quarterly) that provides an estimate of the effectiveness of their marketing mix with minimum bias and variance. (We refer to this granularity as the minimum discrepancy (“MD”) granularity.) However, in most empirical studies, the MD granularity is unknown. While extant literature has shown that empirical results depend on the chosen level of the granularity, it has not proposed a structured way to select it. Our research fills this gap by proposing a granularity “selection tool.” We first propose the desired properties a tool should have and then prove that the power likelihood, which raises each standard likelihood term to latent weights, satisfies these properties under certain boundary conditions. An extensive set of simulations shows that the power likelihood has high statistical accuracy in selecting the MD granularity, and more effectively than other approaches. Finally, we apply our framework to a Nielsen store-level scanner data set, and compare how selecting the granularity with the power likelihood and extant methods affect the empirical results and optimal marketing decisions.
This is joint work with Mingyung Kim and Raghuram Iyengar.
Keywords: data granularity, granularity selection, robust inference, power likelihood, Bayesian weighted likelihood, bias-variance tradeoff, aggregation bias, Fisher information, entropy