Abstract:
Conventional standard errors reflect the fact that the observed data is sampled from an infinite super-population, but this approach to uncertainty may be unnatural in settings where all units in the population are observed (e.g. all 50 U.S. states). In such settings, it may be more natural to view the uncertainty as design-based, i.e. arising from the stochastic assignment of treatment. This paper develops a design-based framework for uncertainty that is suitable for analyzing
quasi-experimental'' settings commonly studied in economics. A key feature of our framework is that each unit has an idiosyncratic probability of receiving treatment, but these idiosyncratic probabilities are unknown to the researcher. We derive conditions under which difference-in-differences (DiD) and related estimators are unbiased for an interpretable causal estimand. When the DiD estimator is unbiased, conventional confidence intervals are valid but potentially conservative in large populations. An interesting feature of our setting is that conventional standard errors tend to be more conservative when treatment probabilities differ across units, which helps to mitigate undercoverage from bias. As a result, conventional confidence intervals for DiD can potentially still have correct coverage even if the design-based analog to parallel trends does not hold exactly. Our results also have implications for the appropriate level to cluster standard errors and for the analysis of instrumental variables.