Index: Time Series Causal Inference

Time Series Causal Inference

Routing Summary

This folder covers Bayesian structural time-series models (BSTS) for inferring causal impact of interventions on time-series outcomes — the CausalImpact framework (Brodersen et al., Ann. Applied Stats. 2015). Contains 7 notes plus a State-Space and Kalman Filter sub-topic (the general filtering/smoothing machinery underlying BSTS).

• Need the core state-space model specification? → Bayesian Structural Time-Series Model
• Need the general Kalman filter / RTS smoother machinery (predict-update, smoothing, likelihood)? → State-Space and Kalman Filter
• Need trend and seasonality components? → Local Linear Trend and Seasonality
• Need automatic covariate selection (spike-and-slab)? → Spike-and-Slab Prior for Covariate Selection
• Need the MCMC / Gibbs sampler details? → MCMC Inference for CausalImpact
• Need pointwise / cumulative / running average impact formulas? → Counterfactual Impact Estimation
• Need the advertising application with placebo test? → CausalImpact Empirical Application

Sub-topics

Sub-topic	Notes	Covers
State-Space and Kalman Filter	5	Linear-Gaussian state-space models, the Kalman filter (predict-update recursion), the RTS smoother, and the marginal likelihood via the prediction-error decomposition — Särkkä (2013). The general machinery underlying BSTS.

Concept Map

Concept	Note	Type	Depends On	Key Result
BSTS state-space model	Bayesian Structural Time-Series Model	definition	—	Eqs (2.1), (2.2): $y_t=Z_t^{\top }{\alpha }_t+{\epsilon }_t$; modular state components
Local linear trend + seasonality	Local Linear Trend and Seasonality	definition	Bayesian Structural Time-Series Model	Eqs (2.3)-(2.5): stochastic level + slope; sum-to-zero seasonal
Spike-and-slab prior	Spike-and-Slab Prior for Covariate Selection	definition	Bayesian Structural Time-Series Model	Eqs (2.8)-(2.12); automatic selection from $J$ controls
MCMC inference	MCMC Inference for CausalImpact	concept	Local Linear Trend and Seasonality, Spike-and-Slab Prior for Covariate Selection	Gibbs + Durbin-Koopman smoother; linear in $m$, < 30s
Counterfactual impact	Counterfactual Impact Estimation	definition	MCMC Inference for CausalImpact	Eqs (2.15)-(2.17): ${\varphi }_t=y_t-{\tilde{y}}_t$; cumulative; running average
Empirical application	CausalImpact Empirical Application	example	Counterfactual Impact Estimation	22% lift [13%, 30%]; observational ≈ randomized; placebo = null

Concept Dependency Chain

Bayesian Structural Time-Series Model
  ├── Local Linear Trend and Seasonality (state components)
  │     └── trend: (2.3)-(2.4), seasonality: (2.5)
  ├── Spike-and-Slab Prior for Covariate Selection (variable selection)
  │     └── spike: (2.8)-(2.9), slab: (2.10)-(2.12), Zellner g-prior
  └── MCMC Inference for CausalImpact (Gibbs sampler)
        ├── Durbin-Koopman simulation smoother (state step)
        ├── Conjugate Gamma draws (variance step)
        └── Spike-slab Gibbs (variable selection step)
              └── Counterfactual Impact Estimation
                    ├── Pointwise impact: φ_t = y_t - ỹ_t (Eq 2.15)
                    ├── Cumulative impact: Σ φ_t (Eq 2.16)
                    └── Running average: (1/t-n) Σ φ_t (Eq 2.17)
                          └── CausalImpact Empirical Application
                                (advertising: 22% lift; placebo: null)

Notes

• Brodersen 2015 - Overview — CONTAINS: paper overview; 28-page Ann. Applied Stats.; comparison table (DiD, SC, ARIMA); key simulation and empirical results
• Bayesian Structural Time-Series Model — CONTAINS: observation equation (2.1), state equation (2.2); all dimensions defined; static/dynamic regression components
• Local Linear Trend and Seasonality — CONTAINS: local linear trend (2.3), AR(1) slope variant (2.4), seasonal model (2.5); block-diagonal assembly
• Spike-and-Slab Prior for Covariate Selection — CONTAINS: spike-slab factorization (2.8)-(2.9), Gaussian slab (2.10), Zellner g-prior (2.12), sufficient statistics (2.13)
• MCMC Inference for CausalImpact — CONTAINS: two Gibbs steps (state simulation + parameter simulation); Durbin-Koopman smoother; posterior predictive distribution (2.14)
• Counterfactual Impact Estimation — CONTAINS: pointwise impact (2.15), cumulative (2.16), running average (2.17); calibration properties; DiD connection
• CausalImpact Empirical Application — CONTAINS: Google advertising campaign; Analysis 1 (randomized: 22%), Analysis 2 (observational: 21%), Analysis 3 (placebo: null); model priors

Sources

• Brodersen - 2015 - Inferring causal impact using Bayesian structural time-series models.pdf — Brodersen KH, Gallusser F, Koehler J, Remy N, Scott SL. 2015. Ann. Appl. Stat. 9(1): 247–274.

Cross-Links to Existing Vault Notes

• Differences-in-Differences — CausalImpact generalizes DiD to time series (DD = special case with zero-variance local level, static regression, OLS)
• Counterfactual Inference — existing note on BART-based counterfactuals; CausalImpact is the time-series analog
• MCMC Basics — foundational MCMC concepts (Gibbs sampler)
• Synthetic Control — CausalImpact also generalizes SC by adding trend/seasonality and allowing negative weights