Randomized trials and observational studies, more often than not, run over a certain period of time during which the treatment effect evolves. Many conventional methods for estimating treatment effects are limited to the i.i.d. setting and are not suited for inferring the time dynamics of the treatment effect. The time series encountered in these settings are highly informative but often nonstationary due to the changing effects of treatment. This increases the difficulty of the task, since stationarity, a common assumption in time series analysis, cannot be reasonably assumed. Another challenge is the heterogeneity of the treatment effect when the treatment affects units differently. The task of estimating heterogeneous treatment effects from nonstationary and, in particular, interventional time series is highly relevant but remains largely unexplored. We propose Causal Transfer, a method which fits state-space models to observational-interventional data in order to learn the effect of the treatment and how it evolves over time. Causal Transfer does not assume the data to be stationary and can be applied to randomized trials and observational studies in which treatment is confounded. Causal Transfer adjusts the effect for possible confounders and transfers the learned effect to other time series and, thereby, estimates various forms of treatment effects, such as the average treatment effect (ATE), the sample average treatment effect (SATE), or the conditional average treatment effect (CATE). By learning the time dynamics of the effect, Causal Transfer can also predict the treatment effect for unobserved future time points and determine the long-term consequences of treatment.
Causal inference from observational data is an ambitious but highly relevant task, with diverse applications ranging from natural to social sciences. Within the scope of nonparametric time series, causal inference defined through interventions is largely unexplored, although time order simplifies the problem substantially. A marginal integration scheme is considered for inferring causal effects from observational time series data, MINT-T (marginal integration in time series), which is an adaptation for time series of a previously proposed method for the case of independent data. This approach for stationary stochastic processes is fully nonparametric and, assuming no instantaneous effects consistently recovers the total causal effect of a single intervention with optimal one-dimensional nonparametric convergence rate n−2 5 assuming regularity conditions and twice differentiability of a certain corresponding regression function. Therefore, MINT-T remains largely unaffected by the curse of dimensionality as long as smoothness conditions hold in higher dimensions and it is feasible for a large class of stationary time series, including nonlinear and multivariate processes. For the case with instantaneous effects, we provide a procedure which guards against false positive causal statements.