When are time series predictions causal? The potential system and dynamic causal effects

1. Problem Statement

Time series analysis frequently relies on predictive quantities (e.g., impulse response functions, local projections, SVARs) to infer the causal impact of an assignment $A_t$ on a future outcome $Y_{t+h}$. However, a fundamental gap exists between standard time series prediction and rigorous causal inference.

• The Core Question: Under what conditions does the difference in conditional expectations, $E[Y_{t+h} | X_t, D_{1:t-1}, A_t = a_t] - E[Y_{t+h} | X_t, D_{1:t-1}, A_t = a'_t]$, actually measure the causal effect of changing the assignment from $a'_t$ to $a_t$?
• The Challenge: Time series data lacks the replication found in cross-sectional randomized experiments. Furthermore, standard time series models often conflate prediction with causality (e.g., Granger causality is predictive, not causal). Existing causal frameworks (like potential outcomes) are often designed for panel data or cross-sections and do not naturally accommodate the dynamic, sequential nature of time series assignments and confounders.

2. Methodology: The Potential System (PS)

The authors introduce a foundational nonparametric framework called the Potential System (PS) to bridge time series dynamics with nonparametric causal inference.

A. Core Definitions

The PS is defined by five assumptions linking a Data Generating Process (DGP) and a Counterfactual Process:

1. Data Generating Process (DGP): Defines observed features ($X_t$), assignments ($A_t$), and outcomes ($Y_t$) generated by a sequential assignment mechanism (SAM) $\alpha_t$ dependent on past history $D_{1:t-1}$ and current features.
2. Counterfactual Process: Defines "potential" features and outcomes $Z_t(a_{1:t})$ for any possible assignment path $a_{1:t}$.
   • Non-anticipation (CP.1a): Future assignments cannot affect current outcomes.
   • Triangularity (CP.1b): Current features depend only on past assignments, not the current one.
3. Consistency (LP): The observed data corresponds to the potential system under the realized assignment path.

B. Dynamic Causal Effects

The paper defines Dynamic Causal Effects as the difference between potential outcomes under different assignment paths at a specific horizon $h$:
$$\text{Effect} = Y_{t,h}(a_t) - Y_{t,h}(a'_t)$$
This framework unifies various causal measures:

• Total Dynamic Causal Effect: Includes the feedback loop where changing $A_t$ alters future assignments $A_{t+1}, \dots$.
• Direct Dynamic Causal Effect: Isolates the effect of $A_t$ on $Y_{t+h}$ holding future assignments fixed.
• Summaries: The authors define Average Treatment Effects (ATE), Conditional ATE (CATE), Filtered Treatment Effects (FTE), and Conditional Filtered Treatment Effects (CFTE) within this time-series context.

C. Identification via Sequential Assignment Mechanisms (SAM)

The paper establishes conditions under which these counterfactual quantities can be identified from observational data. It introduces four key constraints on the SAM:

1. Branch-Sequential Unconfoundedness (SAM.BSU): $A_t \perp\!\!\perp \text{Future Potential Branches} \mid X_t, D_{1:t-1}$.
2. Branch-Sequential Randomization (SAM.BSR): $A_t \perp\!\!\perp \text{Future Potential Branches} \mid D_{1:t-1}$.
3. Branch-Unconfoundedness (SAM.BU): $A_t \perp\!\!\perp \text{Future Potential Branches} \mid X_t$.
4. Branch-Randomization (SAM.BR): $A_t \perp\!\!\perp \text{Future Potential Branches}$ (unconditional).

Theorem 2 demonstrates that under these assumptions, causal summaries (like ATE or CATE) are exactly equal to differences in observable conditional expectations (predictions). For example, under SAM.BSU, the CFTE is identified as:
$$E[Y_{t+h} \mid X_t, D_{1:t-1}, A_t = a_t] - E[Y_{t+h} \mid X_t, D_{1:t-1}, A_t = a'_t]$$

3. Key Contributions and Results

A. Unification of Literature

The PS serves as a "primitive" framework that connects:

• Structural Vector Autoregressions (SVARs): The linear special case of the PS recovers SVARs and Impulse Response Functions (IRFs), providing a nonparametric justification for them.
• Potential Outcomes & Panel Data: It extends the work of Robins, Imbens, and others to pure time series, handling sequential assignments and dynamic confounders.
• Control Theory: The PS bridges observational models with Stochastic Dynamic Programming (SDP) and control literature, showing how optimal control policies can be viewed as specific assignment mechanisms.
• Granger Causality: It clarifies that while Granger causality is about predictability, the PS provides the conditions under which predictability implies causality.

B. Special Cases and Extensions

The paper details how the PS specializes to known models:

• Linear Markov PS: Recovers the homogeneous linear VAR(1) and SVAR(1) structures. It shows that under linearity, the dynamic causal effect is non-stochastic and equals the standard IRF.
• News Impact Models: Specializes to financial econometrics models where volatility responds to news shocks.
• Instrumental Variables (IV): Extends the PS to time series IVs (Definition 8), allowing for the identification of Local Average Treatment Effects (LATE) in dynamic settings.
• Design-Based Inference: The paper adapts the "design-based" approach (conditioning on potential outcomes) to time series. It proposes a method for exact finite-sample inference (similar to Fisher's randomization tests) by simulating assignment paths under a null hypothesis, even in the presence of confounders (if they are PS-exogenous).
• Stochastic Dynamic Programming: It formalizes the link between causal effects and optimal control, defining the "value" of a policy as the minimum expected future loss.

C. Linear Projections and Stationarity

The authors analyze the relationship between causal effects and linear projections (Local Projections). They show that under covariance stationarity and specific unconfoundedness assumptions, the coefficients of linear regressions of $Y_{t+h}$ on $A_t$ (and controls) consistently estimate the causal effects. They also address issues like measurement error (attenuation bias) in this context.

4. Significance

• Foundational Rigor: The paper provides the first rigorous nonparametric definition of a "potential system" specifically for time series, closing the gap between the prediction-heavy time series literature and the causal inference literature.
• Justification of Standard Tools: It offers a theoretical justification for widely used tools like Impulse Response Functions and Local Projections, specifying exactly when they can be interpreted causally.
• Flexibility: By being nonparametric, the PS allows researchers to move beyond linear assumptions while retaining the ability to define and identify dynamic causal effects.
• New Methodologies: It opens the door for new estimation and inference techniques, including doubly robust estimators for time series, exact design-based inference for time series experiments, and causal learning in control systems.
• Policy Relevance: The framework is directly applicable to macroeconomic policy analysis, financial risk management, and personalized medicine (N-of-1 trials), where understanding the dynamic, lagged effects of interventions is critical.

In summary, Carlson and Shephard provide a comprehensive theoretical architecture that transforms time series predictions into causal statements by explicitly modeling the counterfactual dynamics of the system, thereby unifying disparate strands of econometric and statistical theory.