Dynamic causal inference with time series data

TL;DR

This work develops a unified dynamic causal inference framework by extending the Rubin potential outcomes to stochastic trajectories, introducing the Dynamic Average Treatment Effect ($DATE$) as a time-varying measure of how an intervention alters the evolution of a process. It provides two identification regimes: Dynamic Inverse-Probability Weighting ($DIPW$) for settings with many comparable units under dynamic ignorability, and a state-space ($DLM$) representation for scarce-treated scenarios that imputes counterfactual trajectories. The paper establishes theoretical results (unbiasedness, consistency, and a representation theorem) and connects to practical estimation via $DIPW$ and $DLM$; it also decomposes the DATE into spot, persistent, and trend components for interpretability. Empirically, simulations show the $DLM$-based approach often outperforms alternatives in dynamic settings, and an application to the UK unemployment response to the COVID-19 lockdown demonstrates actionable decomposition of causal effects over time and through horizons.

Abstract

We generalize the potential outcome framework to time series with an intervention by defining causal effects on stochastic processes. Interventions in dynamic systems alter not only outcome levels but also evolutionary dynamics -- changing persistence and transition laws. Our framework treats potential outcomes as entire trajectories, enabling causal estimands, identification conditions, and estimators to be formulated directly on path space. The resulting Dynamic Average Treatment Effect (DATE) characterizes how causal effects evolve through time and reduces to the classical average treatment effect under one period of time. For observational data, we derive a dynamic inverse-probability weighting estimator that is unbiased under dynamic ignorability and positivity. When treated units are scarce, we show that conditional mean trajectories underlying the DATE admit a linear state-space representation, yielding a dynamic linear model implementation. Simulations demonstrate that modeling time as intrinsic to the causal mechanism exposes dynamic effects that static methods systematically misestimate. An empirical study of COVID-19 lockdowns illustrates the framework's practical value for estimating and decomposing treatment effects.

Figures (13)

• Figure 1: Illustration of the sampling mechanism in time series. Unlike the i.i.d. case, even after each unit is selected, each unit will have numerous paths it can take, from which one path is sampled or observed. The lighter color paths are unobserved potential paths, whereas the bold color paths are the observed paths. The collection of the latter is what is observed.
• Figure 2: Illustration of the proposed DATE. The light paths are the paths the stochastic process can take, and the bold color paths are their expectations. The DATE is defined as the difference between these expectations, per $t$, over the entire post-treatment period.
• Figure 3: Illustration of the DATE when there is only one treated and one control series. Although both series are observed, since there is only one path, the expectations of the paths are estimated. The difference between the two estimated expectations of paths is the DATE.
• Figure 4: True DATE for $T=72,120,240$.
• Figure 5: Coverage probability for each quantile from 5% to 95%. The x-axis is the estimated quantile, while the y-axis is the nominal quantile. The best case is the 45-degree dotted line. Shaded areas represent 95% intervals of the quantile estimates.

Theorems & Definitions (19)

• Definition 3.1: Dynamic Potential Outcomes
• Definition 3.2: Dynamic Average Treatment Effect
• Theorem 4.1
• Theorem 5.1
• Lemma S3.1
• Theorem S3.1
• Theorem S3.2
• Theorem S3.3
• proof
• Theorem S4.1