Wald-Difference-in-Differences Estimation without Individual-level Treatment Data

In-store advertising, such as digital signage and posters, is a crucial method that influences customer behavior. While effectiveness is often evaluated by displaying ads on a store-by-store basis and comparing outcomes for those exposed to ads and those not, obtaining individual ad exposure data is costly, making it difficult to conduct causal inference with individual-level treatment variables. A common approach to address this issue is to perform causal inference considering non-compliance, treating visitors to stores with advertising as the treatment group and similar customers who visited comparable stores as the control. In this setting, a popular estimator is the ratio of two Difference-in-Differences (DID) estimates: one for the outcome and one for the treatment variable. However, prior studies assumed the DID estimate for the treatment variable is known, which is not always true. To address this, we propose a causal inference method using the fact that, for binary treatment, the DID estimate of the treatment variable represents the change in the proportion of compliers in the treatment group. Our method uses a Gaussian Mixture Model to estimate this proportion. This approach allows estimation of the treatment effect on compliers even when individual ad exposure data is unobserved.