## Bayesian Analysis

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

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 28 March 2018

https://projecteuclid.org/euclid.ba/1522202634

Digital Object Identifier
doi:10.1214/18-BA1102

Mathematical Reviews number (MathSciNet)
MR3910036

Zentralblatt MATH identifier
07001973

Subjects
Primary: 62F15: Bayesian inference

#### 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

#### References

• Abadie, A. (2005). Semiparametric difference-in-differences estimators. The Review of Economic Studies 72, 1–19.
• Abadie, A. and J. Gardeazabal (2003). The economic costs of conflict: A case study of the basque country. American Economics Review 105, 113–132.
• Atay-Kayis, A. and H. Massam (2005). A Monte Carlo method for computing the marginal likelihood in nondecomposable Gaussian graphical models. Biometrika 92, 317–335.
• Barber, R. F. and M. Drton (2015). High-dimensional Ising model selection with Bayesian information criteria. Electronic Journal of Statistics 9, 249–275.
• Bojinov, I. and N. Shephard (2017). Time series experiments and causal estimands: exact randomization tests and trading. Arxiv:1706.07840v2.
• Bonhomme, S. and U. Sauder (2011). Recovering distributions in difference-in-differences models: A comparison of selective and comprehensive schooling. The Review of Economics and Statistics 93(2), 479–494.
• Brodersen, K. H., F. Gaullusser, J. Koehler, N. Remy, and S. L. Scott (2015). Inferring causal impact using Bayesian structural time-series models. The Annals of Applied Statistics 9, 247–274.
• de Jong, P. (1991). The diffuse Kalman filter. The Annals of Statistics 19, 1073–1083.
• de Jong, P. and S. Chu-Chun-Lin (1994). Stationary and non-stationary state space models. Journal of Time Series Analysis 15, 151–166.
• Ding, P. and F. Li (2017). Causal inference: A missing data perspective. Statistical Science (to appear).
• Donald, S. G. and K. Lang (2007). Inference with difference-in-differences and other panel data. The Review of Economics and Statistics 89, 221–233.
• Doudchenko, N. and G. W. Imbens (2016). Balancing, regression, difference-in-differences and synthetic control method: A synthesis. NBER Working Paper No. 22791.
• Durbin, J. and S. J. Koopman (2002). A simple and efficient simulation smoother for state space time series analysis. Biometrika 89, 603–615.
• Durbin, J. and S. J. Koopman (2012). Time Series Analysis by State Space Methods: Second Edition. Great Clarendon Street, Oxford OX2 6DP: Oxford University Press.
• Galindo-Garre, F. and J. K. Vermunt (2006). Avoiding boundary estimates in latent class analysis by Bayesian posterior mode estimation. Behaviormetrika 33, 43–59.
• Gelfand, A. E., A. F. M. Smith, and T.-M. Lee (1992). Bayesian analysis of constrained parameter and truncated data problems using Gibbs sampling. Journal of the American Statistics Association 87, 523–532.
• George, E. I. and R. E. McCulloch (1993). Variable selection via Gibbs sampling. Journal of the American Statistical Association 88(423), 881–889.
• Gunn, L. H. and D. B. Dunson (2005). A transformation approach for incorporating monotone or unimodal constraints. Biostatistics 6, 434–449.
• Harvey, A. C. and S. Peters (1990). Estimation procedures for structural time series models. Journal of Forecasting 9, 89–108.
• Holland, P. W. (1986). Statistics and causal inference. Journal of the American Statistical Association 81, 945–960.
• Imbens, G. W. and D. B. Rubin (2015). Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction. New York, NY, USA: Cambridge University Press.
• Khare, K., B. Rajaratnam, and A. Saha (2015). Bayesian inference for Gaussian graphical models beyond decomposable graphs. Arxiv:1505.00703.
• Koopman, S. J. (1997). Exact initial Kalman filtering and smoothing for nonstationary time series models. Journal of the American Statistical Association 92, 1630–1638.
• Lauritzen, S. L. (1996). Graphical Models. New York, USA: Oxford University Press Inc.
• Li, F., T. Zhang, Q. Wang, M. Z. Gonzalez, E. L. Maresh, and J. A. Coan (2015). Spatial Bayesian variable selection and grouping for high-dimensional scalar-on-image regression. The Annals of Applied Statistics 9, 687–713.
• Mitsakakis, N., H. Massam, and M. D. Escobar (2011). A Metropolis-Hastings based method for sampling from the G-Wishart distribution in Gaussian graphical models. Electronic Journal of Statistics 5, 18–30.
• Mohammadi, A. and E. C. Wit (2015). Bayesian structure learning in sparse Gaussian graphical models. Bayesian Analysis 10, 109–138.
• Ning, B., S. Ghosal, and J. Thomas (2019a). Supplement to “Bayesian method for causal inference in spatially-correlated multivariate time series”. Bayesian Analysis.
• Ning, B., S. Ghosal, and J. Thomas (2019b). R code for “Bayesian method for causal inference in spatially-correlated multivariate time series”. Bayesian Analysis.
• Ročková, V. and E. I. George (2014). EMVS: The EM approach to Bayesian variable selection. Journal of the American Statistical Association 109(506), 828–846.
• Rosenbaum, P. R. (2007). Interference between units in randomized experiments. Journal of the American Statistical Association 102, 191–200.
• Roverato, A. (2000). Cholesky decomposition of a hyper inverse Wishart matrix. Biometrika 87, 99–112.
• Roverato, A. (2002). Hyper inverse Wishart distribution for non-decomposable graphs and its application to Bayesian inference for Gaussian graphical models. Scandinavian Journal of Statistics 29, 391–411.
• Roy, A., T. S. McElroy, and P. Linton (2016). Constrained estimation of causal invertible VARMA models. Statistica Sinica (to appear).
• Rubin, D. B. (1974). Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of Educational Psychology 66, 688–701.
• Rubin, D. B. (1977). Assignment to treatment group on the basis of a covariate. Journal of Educational Statistics 2, 1–26.
• Rubin, D. B. (2005). Causal inference using potential outcomes: design, modeling, decisions. Journal of the American Statistical Association 100, 322–331.
• Smith, M. and L. Fahrmeir (2007). Spatial Bayesian variable selection with application to functional magnetic resonance imaging. Journal of the American Statistical Association 102, 417–431.
• Stein, P. (1952). Some general theorems on iterants. Journal of Research of the National Bureau of Standards 48, 82–83.
• Stuart, E. A. (2010). Matching methods for causal inference: A review and a look forward. Statistical Science 25, 1–21.
• Ueda, N. and R. Nakano (1998). Deterministic annealing EM algorithm. Neural Networks 11, 271–282.
• Uhler, C., A. Lenkoskiy, and D. Richardsz (2017). Exact formulas for the normalizing constants of Wishart distributions for graphical models. The Annals of Statistics 46, 90–118.
• Wang, H. and S. Z. Li (2012). Efficient Gaussian graphical model determination under G-Wishart prior distributions. Electronic Journal of Statistics 6, 168–198.

#### 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.