Trajectory Balancing: A Kernel Method for Causal Inference with TSCS Data

• In Event: Research in Causal Inference and Applications

Abstract

This paper proposes a new weighting method based on kernel balancing (Hazlett 2016) for estimating causal effects of treatments with time-series cross-sectional (TSCS) data. It uses a kernel transformation of the pre-treatment outcomes and covariates, and selects weights to obtain equal means for the control and treated on this higher dimensional description of each unit’s pre-treatment period. It can be used as a pre-processing method for panel models, such as difference-in-differences, or for a simple (weighted) difference in means estimator of the treatment effect on the treated in each post-treatment period. Our method extends the synthetic control approach (Abadie et al 2010), which weights control units to achieve equal means on pre-treatment outcomes and covariates for the treated unit and a control group. We show the “equal means” approach employed by traditional synthetic control has important limitations. Most critically, it obtains unbiased effect estimates only when the non-treatment potential outcomes at later timepoints for a given unit are linear in the pre-treatment outcomes and covariates of that unit. In a number of simple examples, such as exponential and logarithmic growth, this assumption does not hold. Our approach, by contrast, produces unbiased treatment effect estimates when the non-treatment potential outcomes fall in a large space of smooth, non-linear functions of the pre-treatment outcomes and covariates. We provide simulation evidence to show that our method overcomes the problematic linearity restriction inherent in the mean balance approach. Multiple applications are provided and suggest not only the desired estimation behavior but also that it is far less sensitivity to investigator decisions than existing approaches.

Authors

• Chad Hazlett

• Yiqing Xu, University of California, San Diego