Abstract Many marketing applications, including credit card incentive programs, offer rewards to customers who exceed specific spending thresholds to encourage increased consumption. Quantifying the causal effect of these thresholds on customers is crucial for effective marketing strategy design. Although regression discontinuity design is a standard method for such causal inference tasks, its assumptions can be violated when customers, aware of the thresholds, strategically manipulate their spending to qualify for the rewards. To address this issue, we propose a novel framework for estimating the causal effect under threshold manipulation. The main idea is to model the observed spending distribution as a mixture of two distributions: one representing customers strategically affected by the threshold, and the other representing those unaffected. To fit the mixture model, we adopt a two-step Bayesian approach consisting of modeling non-bunching customers and fitting a mixture model to a sample around the threshold. We show posterior contraction of the resulting posterior distribution of the causal effect under large samples. Furthermore, we extend this framework to a hierarchical Bayesian setting to estimate heterogeneous causal effects across customer subgroups, allowing for stable inference even with small subgroup sample sizes. We demonstrate the effectiveness of our proposed methods through simulation studies and illustrate their practical implications using a real-world marketing dataset.
Jan 20, 2026
Abstract In-store advertising, such as digital signage and in-store posters, is a crucial advertising method that influences customer purchasing behavior. While their effectiveness is typically evaluated by displaying ads on a store-by-store basis and comparing the purchasing behavior of those exposed to ads with those who are not, obtaining ad exposure data for individual customers is costly, making it challenging to conduct accurate causal inference with individual-level treatment variables. A common approach to address this issue is to perform causal inference considering non-compliance, setting visitors to stores implementing an ad campaign as the treatment group and similar customers who have visited comparable stores as the control group. In this setting, a popular estimator is the ratio of two Differences-in-Differences (DID) estimates: one for the outcome variable and another for the treatment variable. However, previous study assume that the DID estimate for the treatment variable is known from public data, which is not always the case. To overcome this limitation, we propose a method to estimate causal effects by utilizing the fact that, for binary treatment variables, the DID estimate of the treatment variable represents the change in the proportion of compliers in the treatment group. Our method leverages a Gaussian Mixture Model to estimate the proportion. This approach allows for the estimation of the treatment effect on the compliers even in advertising strategies where ad exposure data for individual customers is unavailable.
Mar 4, 2025