Bayesian Method for Causal Inference in Spatially-Correlated Multivariate Time Series

Measuring the causal impact of an advertising campaign on sales is an important problem for advertising companies interested in modeling consumer demand at stores in different locations. This paper proposes a new causal inference method that uses a Bayesian multivariate time series model to capture the spatial correlation between stores. Control stores which are used to build counterfactuals over the causal period are chosen before running the advertising campaign. The novelty of this method is to estimate causal effects by comparing the posterior distributions of latent variables given by the observed data and its counterfactual data. We use one-sided Kolmogorov-Smirnov distance to quantify the difference between the two posterior distributions. We found that this method is able to detect smaller scale of causal impact as measurement errors are automatically filtered out in the causal analysis compared to a commonly used method. A two-stage algorithm is used to estimate the model. A G-Wishart prior with a given graphical structure on the precision matrix is used to impose sparsity in spatial correlation. The graphical structure needs not correspond to a decomposable graph. We model the local linear trend by a stationary multivariate autoregressive process to prevent the prediction intervals from being explosive. A detailed simulation study shows the effectiveness of the proposed approach to causal inference. We apply the proposed method to a real dataset to measure the effect of an advertising campaign for a consumer product sold at stores of a large national retail chain.