Design: Identification and Fixed Effects

Fixed effects are central to many econometric analyses. Here we discuss their nuts and bolts.

Objective

This session aims to provide a bird’s view of the conceptual framework used in causal inference analyses and to discuss how fixed effects work, under the hood.

Summary

Causal identification strategies are ways to form assumptions that may allow us to go from a quantity we can typically estimate in an applied economics study (the difference in average observed outcomes) to the quantity of interest, the Average Treatment effect on the Treated (ATT). Some of these strategies leverage repeated observations (DiD, event-study, etc) and make use of fixed effects. We discuss nuts and bolts of the use of fixed effects. In particular, they not only partial out variation but also change the interpretation of the coefficients being estimated and affect the variation used identifying variation. Additional challenges arise when using TWFEs.

Session Outline

1. Notes on the Potential Outcomes Framework
2. Overview of identification strategies based on repeated observations
3. Some nuts and bolts of fixed effects (interpretation, projections and identifying variation)
4. TWFE

Materials

Open slides in html

Open slides in pdf

Specific resources for this lecture

If you should read only one thing of FEs

Chapter 16 of The Effect of Huntington-Klein (2022)

Courses

• A large share of the slides are derived from notes by Claire Palandri and related slides by Anna Papp.
• Chapter 16 to 18 of The Effect of Huntington-Klein (2022)
• Chapter 8 to 10 of Cunningham (2021)
• Chapter 5 of Angrist and Pischke (2009)

Papers

• Regression weights: Aronow and Samii (2016) and Angrist and Pischke (2009) (section 3.3.1) interpret OLS regression as a weighted average of observations, depending on how well the endogenous variable of interest is correlated with other covariates.
• TWFE: de Chaisemartin and D’Haultfœuille (2023) and Roth et al. (2023) both review recent developments in the TWFE literature

R Pacakges

There is a breadth of packages to implement analyses with fixed-effects (plm, fixest, lfe for instance).

I have used the 3 mentioned above in various projects and recommend using fixest for its versatility. It allows to implement analysis with FE, IVs, limited outcomes and so on. Paired with modelsummary to compute standard errors, you should be able to run a lot of your analyses.

References

Angrist, Joshua D., and Jörn-Steffen Pischke. 2009. Mostly Harmless Econometrics: An Empiricist’s Companion. 1 edition. Princeton: Princeton University Press.

Aronow, Peter M., and Cyrus Samii. 2016. “Does Regression Produce Representative Estimates of Causal Effects?” American Journal of Political Science 60 (1): 250–67. https://doi.org/10.1111/ajps.12185.

Cunningham, Scott. 2021. Causal Inference: The Mixtape. Yale University Press. https://doi.org/10.2307/j.ctv1c29t27.

de Chaisemartin, Clément, and Xavier D’Haultfœuille. 2023. “Two-Way Fixed Effects and Differences-in-Differences with Heterogeneous Treatment Effects: A Survey.” The Econometrics Journal 26 (3): C1–30. https://doi.org/10.1093/ectj/utac017.

Huntington-Klein, Nick. 2022. The Effect: An Introduction to Research Design and Causality. 1st edition. Chapman and Hall/CRC.

Roth, Jonathan, Pedro H. C. Sant’Anna, Alyssa Bilinski, and John Poe. 2023. “What’s Trending in Difference-in-Differences? A Synthesis of the Recent Econometrics Literature.” Journal of Econometrics 235 (2): 2218–44. https://doi.org/10.1016/j.jeconom.2023.03.008.