Large N, Small T, Multiple P: A Causal Matrix Completion Method for CRM Panel Data

Abstract

The prototypical customer relationship management (CRM) panel structure is composed of many customers (large N), with short histories (small T), and multiple outcome metrics (multiple P). Our paper aims to tackle the challenges of causal inference that firms face in such CRM settings, which are additionally characterized by unobserved heterogeneity, time dynamics, and staggered adoption. Despite the success of synthetic control methods (SCM) in contemporary marketing applications, extant variants typically necessitate "small N, large T" data regimes to be performant - e.g., a handful of firm- or jurisdiction-level donor units, each with long time series. To extend to the "large N, small T, multiple P" setting, we bridge SCM to the broader causal matrix completion (MC) paradigm and leverage the "multiple P" ubiquitous to contemporary CRM: the presence of multiple outcomes enables a shared matrix singular value decomposition (cf. SCM's factorization), which helps jointly identify individual-level latent factors to establish conditional ignorability, compensating for overall short time series at the customer level. We employ a Bayesian causal inference approach, specifying a joint posterior of the nonrandom missingness of potential outcomes, together with the likelihood of the observed outcomes. We introduce two distinct variants of Bayesian causal MC models, each estimated independently through the implementation of the Gibbs sampling (independent multiple P's) and the Hamiltonian Monte Carlo (concurrent multiple P's) -based data augmentation procedure. We empirically illustrate our approach through a comprehensive customer-level database of gift card purchases and redemptions from a U.S. hospitality startup. We compare the effectiveness with extant SCM under the German reunification empirical study and devise a generalized framework for marketing and statistics researchers applicable to a wide range of CRM panel structures.