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.

翻译：本文将潜在结果框架推广至包含干预的时间序列，通过定义随机过程上的因果效应来实现。动态系统中的干预不仅改变结果水平，还改变演化动态——影响持续性与转移规律。我们的框架将潜在结果视为完整轨迹，使得因果估计量、识别条件和估计量可以直接在路径空间上构建。由此产生的动态平均处理效应（DATE）刻画了因果效应随时间演化的特征，并在单期情况下退化为经典平均处理效应。针对观测数据，我们推导出动态逆概率加权估计量，该估计量在动态可忽略性与正值性条件下具有无偏性。当处理单元稀缺时，我们证明DATE背后的条件均值轨迹具有线性状态空间表示，从而实现了动态线性模型的构建。仿真实验表明，将时间建模为因果机制的内在要素，能够揭示静态方法系统性误估的动态效应。一项关于COVID-19封锁政策的实证研究，展示了该框架在估计与分解处理效应方面的实用价值。