Same Root Different Leaves: Time Series and Cross-Sectional Methods in Panel Data

Abstract: A central goal in social science is to evaluate the causal effect of a policy. One dominant approach is through panel data analysis in which the behaviors of multiple units are observed over time. The information across time and space motivates two general approaches: (i) horizontal regression (i.e., unconfoundedness), which exploits time series patterns, and (ii) vertical regression (e.g., synthetic controls), which exploits cross-sectional patterns. Conventional wisdom states that the two approaches are fundamentally different. We establish this position to be partly false for estimation but generally true for inference. In particular, we prove that both approaches yield identical point estimates under several standard settings. For the same point estimate, however, each approach quantifies uncertainty with respect to a distinct estimand. In turn, the confidence interval developed for one estimand may have incorrect coverage for another. This emphasizes that the source of randomness that researchers assume has direct implications for the accuracy of inference.