Brodersen 2015 - Overview

Summary

Brodersen, Gallusser, Koehler, Remy & Scott (2015) propose a Bayesian state-space model for inferring the causal impact of a market intervention on a time series outcome. By modeling a counterfactual prediction (what would have happened without the intervention), the approach generalizes DiD to handle temporal correlation, multiple covariates, local trends, and seasonality. Implemented in the CausalImpact R package.

Research Question and Contribution

Problem: Estimating causal effects of interventions (e.g., ad campaigns) on time-series outcomes. Classical DiD assumes i.i.d. data and only two time points; it is too restrictive for time-series settings.

Contribution:

1. A diffusion-regression state-space model combining local trend, seasonality, and a synthetic control component
2. Fully Bayesian inference via MCMC (Gibbs sampler), producing credible intervals for the causal effect
3. Spike-and-slab prior for automatic covariate selection among many potential control series
4. Three quantities of interest: pointwise impact ${\varphi }_t$, cumulative impact $\sum {\varphi }_t$, running average impact
5. CausalImpact R package for practical application

Published: Annals of Applied Statistics, 2015, Vol. 9, No. 1, pp. 247–274. DOI: 10.1214/14-AOAS788

Paper Structure

Section	Content
§1 Introduction	Problem, related work (DiD, synthetic control), approach overview
§2 Model	State-space equations; local linear trend; seasonality; static/dynamic regression; priors; inference (MCMC)
§3 Simulated data	Sensitivity, specificity, power, interval coverage; estimation accuracy
§4 Empirical application	Google advertising campaign; 3 analyses (randomized controls, observational, placebo)
§5 Discussion	Comparison to DiD, synthetic control, AR/MA; limitations

Key Results

• Advertising campaign (empirical): 22% cumulative lift [13%, 30%] using randomized controls; 21% [12%, 30%] using observational keyword searches as controls — nearly identical, validating the observational approach
• Placebo test: Causal effect of 2% [-6%, 10%] in untreated regions — not significant, as expected
• Power (simulation): 25% true effect detected correctly in >90% of cases; <1% effect missed in ~90% of cases
• Interval coverage: Central 95% PI contains truth in ~95% of simulations across campaign lengths

Relationship to Other Methods

Method	Relationship
Difference-in-Differences	Special case: DiD = state-space model with zero-variance local level, static regression, OLS
Synthetic Control	Special case: CausalImpact generalizes SC by adding trend, seasonality, and allowing negative weights
ARIMA/AR/MA	Special case: CausalImpact = state-space model with specific parameter restrictions
Multiple linear regression	Special case: zero-variance local level, no local trend, static coefficients

See Also

• Bayesian Structural Time-Series Model — the core model specification
• Local Linear Trend and Seasonality — state components
• Spike-and-Slab Prior for Covariate Selection — variable selection mechanism
• MCMC Inference for CausalImpact — posterior inference algorithm
• Counterfactual Impact Estimation — how causal effects are computed
• CausalImpact Empirical Application — advertising campaign case study
• Differences-in-Differences — classical approach that CausalImpact generalizes
• Synthetic Control — Abadie-style synthetic control that CausalImpact extends with trend, seasonality, and Bayesian inference
• Advertising and Promotion Effects — the primary empirical application domain for CausalImpact in marketing