Bayesian Analysis

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

Bo Ning, Subhashis Ghosal, and Jewell Thomas

Full-text: Open access

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 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.

Article information

Source
Bayesian Anal., Volume 14, Number 1 (2019), 1-28.

Dates
First available in Project Euclid: 28 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.ba/1522202634

Digital Object Identifier
doi:10.1214/18-BA1102

Subjects
Primary: 62F15: Bayesian inference

Keywords
advertising campaign Bayesian variable selection causal inference graphical model stationarity time series

Rights
Creative Commons Attribution 4.0 International License.

Citation

Ning, Bo; Ghosal, Subhashis; Thomas, Jewell. Bayesian Method for Causal Inference in Spatially-Correlated Multivariate Time Series. Bayesian Anal. 14 (2019), no. 1, 1--28. doi:10.1214/18-BA1102. https://projecteuclid.org/euclid.ba/1522202634


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Supplemental materials

  • Supplement to “Bayesian method for causal inference in spatially-correlated multivariate time series”. This supplementary material contains five sections. Sections 1 and 2 provide the details on deriving the two-stage algorithm and the revised EMVS algorithm. Section 3 provides graphical and tabular representations of the results of the new method to infer causality using the univariate model. Section 4 provides model checking results. Section 5 describes the Kalman filter and backward smoothing algorithm.
  • R code for “Bayesian method for causal inference in spatially-correlated multivariate time series”. This supplementary material includes the original Bayesian multivariate time series model code written in R (Ning et al., 2019b). The code is also available on the website: https://github.com/Bo-Ning/Bayesian-multivariate-time-series-causal-inference.