Bayesian method for causal inference in spatially-correlated multivariate time series

TL;DR

This paper introduces a Bayesian multivariate time series approach with spatial correlation modeling to detect weak causal effects of advertising campaigns across stores, improving impact detection in noisy, spatially-linked sales data.

Contribution

A novel Bayesian method with a spatially-aware multivariate model and counterfactual estimation for causal inference in store sales data.

Findings

Effective detection of weaker causal impacts demonstrated in simulations.

Application to real retail data shows improved impact measurement.

Spatial correlation modeling enhances causal inference accuracy.

Abstract

Measuring the causal impact of an advertising campaign on sales is an essential task for advertising companies. Challenges arise when companies run advertising campaigns in multiple stores which are spatially correlated, and when the sales data have a low signal-to-noise ratio which makes the advertising effects hard to detect. This paper proposes a solution to address both of these challenges. A novel Bayesian method is proposed to detect weaker impacts and a multivariate structural time series model is used to capture the spatial correlation between stores through placing a $\mathcal{G}$-Wishart prior on the precision matrix. The new method is to compare two posterior distributions of a latent variable---one obtained by using the observed data from the test stores and the other one obtained by using the data from their counterfactual potential outcomes. The counterfactual potential outcomes are estimated from the data of synthetic controls, each of which is a linear combination of sales figures at many control stores over the causal period. Control stores are selected using a revised Expectation-Maximization variable selection (EMVS) method. A two-stage algorithm is proposed to estimate the parameters of the model. To prevent the prediction intervals from being explosive, a stationarity constraint is imposed on the local linear trend of the model through a recently proposed prior. The benefit of using this prior is discussed in this paper. A detailed simulation study shows the effectiveness of using our proposed method to detect weaker causal impact. The new method is applied to measure the causal effect of an advertising campaign for a consumer product sold at stores of a large national retail chain.