Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Causal Inference under Threshold Manipulation: Bayesian Mixture Modeling and Heterogeneous Treatment Effects

Kohsuke Kubota, Shonosuke Sugasawa

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.

Causal Inference under Threshold Manipulation: Bayesian Mixture Modeling and Heterogeneous Treatment Effects

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.

Paper Structure

This paper contains 22 sections, 1 theorem, 42 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Bayesian Modeling of Threshold Manipulation via Mixtures
  4. Causal Inference Under Threshold Manipulation
  5. Mixture Model for Bunching and Non-bunching Distributions
  6. Two-Step Bayesian Inference
  7. Theoretical Analysis
  8. Heterogeneous Treatment Effect via Hierarchical Formulation
  9. Simulation Studies
  10. Application to Promotion with Thresholds in Marketing
  11. Concluding Remarks
  12. Technical Details
  13. Regularity conditions
  14. Proof of Theorem 1
  15. Singh-Madala distribution and its adjusted likelihood
  16. ...and 7 more sections

Key Result

Theorem 1

Assume regularity conditions in Appendix A.1. Under $n \to \infty$, it holds that for a large constant $M$.

Figures (7)

  • Figure 1: Conceptual illustration of our proposed mixture model (red). Modeling the observed distribution (skyblue) as a mixture of a non-bunching distribution (green, unaffected by the threshold) and a strategic bunching distribution (blue, distorted near the threshold).
  • Figure 2: Point estimates and 90% credible intervals of the subgroup-specific causal effects $\Delta_g$ by BMTM and HBMTM.
  • Figure 3: Observed distribution of the total payment amount during the promotion (skyblue) and the three thresholds (red).
  • Figure 4: The non-bunching distribution for each subgroup, estimated using the Singh-Maddala distribution.
  • Figure 5: Point estimates and 90% credible intervals of each subgroup's causal effect $\Delta_g$ obtained by the proposed method HBMTM at each threshold.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Theorem 1