Trajectory Balancing : A General Reweighting Approach to Causal Inference With Time-Series Cross-Sectional Data

We introduce trajectory balancing, a general reweighting approach to causal inference with time-series cross-sectional (TSCS) data. We focus on settings in which one or more units is exposed to treatment at a given time, while a set of control units remain untreated throughout a time window of interest. First, we show that many commonly used TSCS methods imply an assumption that a unit's non-treatment potential outcomes in the post-treatment period are linear in that unit's pre-treatment outcomes as well as time-invariant covariates. Under this assumption, we introduce the mean balancing method that reweights the control units such that the averages of the pre-treatment outcomes and covariates are approximately equal between the treatment and (reweighted) control groups. Second, we relax the linearity assumption and propose the kernel balancing method that seeks an approximate balance on a kernel-based feature expansion of the pre-treatment outcomes and covariates. The resulting approach inherits the property of handling time-vary confounders as in synthetic control and latent factor models, but has the advantages of: (1) improving feasibility and stability with reduced user discretion compared to existing approaches; (2) accommodating both short and long pre-treatment time periods with many or few treated units; and (3) achieving balance on the high-order "trajectory" of pre-treatment outcomes rather than their simple average at each time period. We illustrate this method with simulations and two empirical examples