Inferring causal impact using Bayesian structural time-series models

Fabian Gallusser
Jim Koehler
Nicolas Remy
Steven L. Scott
Annals of Applied Statistics, 9(2015), pp. 247-274
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An important problem in econometrics and marketing is to infer the causal impact that a designed market intervention has exerted on an outcome metric over time. In order to allocate a given budget optimally, for example, an advertiser must assess to what extent different campaigns have contributed to an incremental lift in web searches, product installs, or sales. This paper proposes to infer causal impact on the basis of a diffusion-regression state-space model that predicts the counterfactual market response that would have occurred had no intervention taken place. In contrast to classical difference-in-differences schemes, state-space models make it possible to (i) infer the temporal evolution of attributable impact, (ii) incorporate empirical priors on the parameters in a fully Bayesian treatment, and (iii) flexibly accommodate multiple sources of variation, including the time-varying influence of contemporaneous covariates, i.e., synthetic controls. Using a Markov chain Monte Carlo algorithm for model inversion, we illustrate the statistical properties of our approach on synthetic data. We then demonstrate its practical utility by evaluating the effect of an online advertising campaign on search-related site visits. We discuss the strengths and limitations of state-space models in enabling causal attribution in those settings where a randomised experiment is unavailable. The CausalImpact R package provides an implementation of our approach.