Linear fixed effects regression models are a primary workhorse for
causal inference among applied researchers. And yet, it has been
shown that even when the treatment is exogenous within each unit,
the linear regression models with unit-specific fixed effects do not
consistently estimate the average treatment effect in the presence
of heterogenous treatment effects and treatment assignment
probabilities across units. In this paper, we offer a simple
solution. Specifically, we show that weighted fixed effects
regression models consistently estimate the average treatment effect
under various identification strategies such as propensity score
weighting, first difference, stratified randomization,
post-treatment stratification, and difference-in-differences. We
prove the results by establishing various finite sample equivalence
relationships between fixed effects and matching estimators. At the
basic level, the results suggest that these estimators do not
fundamentally differ in their ability to cope with unobserved
heterogeneity. More importantly, our analysis identifies the
information implicitly used by fixed effects models to estimate
counterfactual outcomes necessary for causal inference, highlighting
the potential sources of their bias and inefficiency. In addition,
the proposed framework offers simple, model-based standard errors
for various matching estimators. Finally, we illustrate the
proposed methodology by revisiting the controversy concerning the
effects of the General Agreement on Tariffs and Trade (GATT)
membership on international trade. Open-source software is
available for fitting the proposed weighted linear fixed effects estimators.
(Last revised, May 2012) |