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On the Use of Two-Way Fixed Effects Regression Models for Causal Inference with Panel Data
Open accessAbstract
Abstract The two-way linear fixed effects regression ( 2FE ) has become a default method for estimating causal effects from panel data. Many applied researchers use the 2FE estimator to adjust for unobserved unit-specific and time-specific confounders at the same time. Unfortunately, we demonstrate that the ability of the 2FE model to simultaneously adjust for these two types of unobserved confounders critically relies upon the assumption of linear additive effects. Another common justification for the use of the 2FE estimator is based on its equivalence to the difference-in-differences estimator under the simplest setting with two groups and two time periods. We show that this equivalence does not hold under more general settings commonly encountered in applied research. Instead, we prove that the multi-period difference-in-differences estimator is equivalent to the weighted 2FE estimator with some observations having negative weights. These analytical results imply that in contrast to the popular belief, the 2FE estimator does not represent a design-based, nonparametric estimation strategy for causal inference. Instead, its validity fundamentally rests on the modeling assumptions.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Instead, we prove that the multi-period difference-in-differences estimator is equivalent to the weighted FE estimator with some observations having negative weights.”
- Also stated in: § . The Multi-period Difference-in-Differences Estimator · boosted
“Instead, the following theorem establishes that the general multi-period DiD estimator given in Equation ( ) is equivalent to a weighted two-way fixed effects regression estimator.”
- Also stated in: § T (Difference-in-Differences · plain
“Second, although the multi-period DiD estimator can be shown to be equivalent to the weighted FE estimator, some control observations will have negative regression weights.”
§ On the Use of Two-Way Fixed Effects Regression Models for Causal Inference with Panel Data > Abstract
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § . The Multi-period Difference-in-Differences Estimator · boosted
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The next proposition establishes that this two-way matching estimator, which eliminates mismatches within-unit and within-time dimension, can be written as a weighted FE estimator.”
§ On the Use of Two-Way Fixed Effects Regression Models for Causal Inference with Panel Data > Adjustment for Unobserved Confounders
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The treatment varies within each unit as well as within each time period.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Although it is generally impossible to eliminate all mismatches, in this section we show that we can do so under the difference-in-differences (DiD) design.”
- Also stated in: § . The Multi-period Difference-in-Differences Estimator · plain
“It is worth noting that all three sets only contain control observations, thereby eliminating all mismatches.”
§ On the Use of Two-Way Fixed Effects Regression Models for Causal Inference with Panel Data > The Difference-in-Differences Design
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The parallel trend assumption holds for the DiD design.
- Also stated in: § . The Multi-period Difference-in-Differences Estimator · plain
In the paper
“The two-way fixed effects estimator defined in Equation ( ) is equivalent to the following matching estimator,”
- Also stated in: § Adjustment for Unobserved Confounders · plain
“The proposition shows that the estimated counterfactual outcome of a given observation, that is,”
§ On the Use of Two-Way Fixed Effects Regression Models for Causal Inference with Panel Data > Adjustment for Unobserved Confounders
What the sources say
- Not matchedcites Imai and Kim
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § Adjustment for Unobserved Confounders · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Therefore, even the weighted FE estimator, which has fewer mismatches than the standard FE estimator, suffers from some mismatches.”
§ On the Use of Two-Way Fixed Effects Regression Models for Causal Inference with Panel Data > P
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The estimator also has an adjustment set whose observations belong to neither the same unit nor the same time period as the observation being matched with.”
§ On the Use of Two-Way Fixed Effects Regression Models for Causal Inference with Panel Data > P
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We show that this equivalence does not hold under more general settings commonly encountered in applied research.”
- Also stated in: § Introduction · plain
“However, we show that this equivalence does not hold under more general settings frequently encountered in applied research.”
§ On the Use of Two-Way Fixed Effects Regression Models for Causal Inference with Panel Data > Abstract
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § Introduction · plain
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Unfortunately, we demonstrate that the ability of the FE model to simultaneously adjust for these two types of unobserved confounders critically relies upon the assumption of linear additive effects.”
- Also stated in: § Introduction · boosted
“Unfortunately, we show that the FE's ability to simultaneously adjust for the two types of unobserved confounders critically hinges upon the assumption of linear additive effects.”
§ On the Use of Two-Way Fixed Effects Regression Models for Causal Inference with Panel Data > Abstract
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § Introduction · boosted
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“Another common justification is based on the fact that the FE estimator is equivalent to the difference-in-differences estimator under the simplest setting with two groups and two time periods (e.g., Bertrand, Duflo, and Mullainathan ; Angrist and Pischke ).”
- Also stated in: § . The Multi-period Difference-in-Differences Estimator · boosted
“It is well known that the standard nonparametric DiD estimator is numerically equivalent to the FE estimator in the simplest setting, in which there are only two time periods and the treatment is administered only to one group of units in the second time period.”
§ On the Use of Two-Way Fixed Effects Regression Models for Causal Inference with Panel Data > Introduction
What the sources say
- Not matchedcites Bertrand, Duflo, and Mullainathan
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites Angrist and Pischke
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Load-bearingStrength:boosted· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § . The Multi-period Difference-in-Differences Estimator · boosted
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Since the regression weights should generally positive, the results of this section shows that the justification of the FE estimator as the DiD estimator is not warranted unless the linearity assumption is imposed.”
§ On the Use of Two-Way Fixed Effects Regression Models for Causal Inference with Panel Data > T (Difference-in-Differences
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes
The paper relies on this without support.
The two-way FE regression is a default method for estimating causal effects from panel data among applied researchers.
“The two-way linear fixed effects regression ( FE) has become a default method for estimating causal effects from panel data.”
“Many social scientists use the two-way fixed effects ( FE) regression, or linear regression with unit and time fixed effects, as the default methodology for estimating causal effects from panel data.”
The paper relies on this without support.
Many applied researchers use the FE estimator to adjust for unobserved unit-specific and time-specific confounders at the same time.
“Many applied researchers use the FE estimator to adjust for unobserved unit-specific and time-specific confounders at the same time.”
“Applied researchers o en use the FE regression to adjust for unobserved unit-specific and time-specific confounders at the same time.”
The authors state this as an assumption.
The two types of unobserved confounders (unit-specific and time-specific) are additive and separable, with no interaction between them.
“Thus, although the interaction between these two types of unobserved confounders is assumed to be absent, there is no functional-form restriction on h(•) and f (•).”
“In other words, since the treatment is binary, the model makes no restriction other than the additivity and separability of the two types of unobserved confounders.”
The authors state this as an assumption.
The parallel trend assumption holds for the DiD design.
“To establish the relations between the FE and DiD estimators, we begin by considering the following parallel trend assumption, A (Parallel Trend).”
The authors state this as an assumption.
The treatment varies within each unit as well as within each time period.
“Assume that the treatment varies within each unit as well as within each time period, that is, 0 < T t =1 X i t < T for each i and 0 < N i =1 X i t < N for each t.”
The authors state this as an assumption.
There is at least one treated and control unit, and at least one unit with D_it = 1.
“Assume that there is at least one treated and control unit, that is, 0 < N i =1 T t =1 X i t < NT , and that there is at least one unit with D i t = 1, that is, 0 < N i =1 T t =1 D i t .”
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In the citing paper
“This approach yields an interpretable causal effect under generalizations of the parallel trends assumption and and an additional 'no carryover' assumption that imposes that the potential outcomes depend only on current treatment status and not on the full treatment history.”
From What's trending in difference-in-differences? A synthesis of the recent econometrics literature (2023) · cites this paper as Imai and Kim (2021)
Experimental — accuracy not yet measuredThe cited source says what the claim attributes to it.
In the citing paper
“de Chaisemartin and D'Haultfœuille (2020) and Imai and Kim (2021) consider settings where units are treated at different times, but do not necessarily require that treatment is an absorbing state.”
From What's trending in difference-in-differences? A synthesis of the recent econometrics literature (2023) · cites this paper as Imai and Kim (2021)
The cited source says what the claim attributes to it.
In the citing paper
“Their estimators intuitively compare changes in outcomes for units whose treatment status changed to other units whose treatment status remained constant over the same periods.”
From What's trending in difference-in-differences? A synthesis of the recent econometrics literature (2023) · cites this paper as Imai and Kim (2021)
Citations
Cites
How Much Should We Trust Differences-in-Differences Estimates?
Bertrand, M., Duflo, E., & Mullainathan, S. (2004) · The Quarterly Journal of Economics · doi:10.1162/003355304772839588 · Cited by 10,641 · cited 2×
“Another common justification is based on the fact that the 2FE estimator is equivalent to the difference-in-differences estimator under the simplest setting with two groups and two time periods (e.g., Bertrand, Duflo, and Mullainathan 2004; Angrist and Pischke 2009).”
MethodMostly Harmless Econometrics: An Empiricist's Companion
Angrist, J. D., & Pischke, J.-S. (2009) · doi:10.1515/9781400829828 · Cited by 9,467 · cited 2×
“Our result is also related to that of Arkhangelsky et al. (2018) who shows that the weighted two-way fixed effects estimator can improve the synthetic control method of Abadie et al.”
MethodResultAbadie, A., Diamond, A., & Hainmueller, J. (2010) · Journal of the American Statistical Association · doi:10.1198/jasa.2009.ap08746 · Cited by 5,513 · cited 1×
“Recently, a number of researchers have extended the synthetical control method of Abadie, Diamond, and Hainmueller (2010) to more general settings (e.g., Xu 2017; Ben-Michael, Feller, and Rothstein 2019).”
BackgroundTwo-Way Fixed Effects Estimators with Heterogeneous Treatment Effects
de Chaisemartin, C., & D’Haultfœuille, X. (2020) · American Economic Review · doi:10.1257/aer.20181169 · Cited by 5,031
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“Recently, a number of researchers have extended the synthetical control method of Abadie, Diamond, and Hainmueller (2010) to more general settings (e.g., Xu 2017; Ben-Michael, Feller, and Rothstein 2019).”
BackgroundDesign-based analysis in Difference-In-Differences settings with staggered adoption
Athey, S., & Imbens, G. W. (2022) · Journal of Econometrics · doi:10.1016/j.jeconom.2020.10.012 · Cited by 917 · cited 2×
“In addition, a number of scholars have recently considered causal interpretations of the standard 2FE estimator (see, e.g., Borusyak and Jaravel 2017; Abraham and Sun 2018; Athey and Imbens 2018; Chaisemartin and D’Haultfœuille 2018; Goodman-Bacon 2018).”
BackgroundDifference-in-Differences with Variation in Treatment Timing
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“Indeed, by extending the matching framework of Imai and Kim (2019), we show that the simultaneous adjustment for the two types of unobserved confounders cannot be done nonparametrically under the 2FE framework.”
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Liebl, D., & Schüwer, U. (2021) · SSRN Electronic Journal · doi:10.2139/ssrn.3924620
“Note that classic two-way fixed effects panel models, with separate unit-level and time-level fixed effects, would not be suitable for our analysis, because they are not able to capture the regionally- and time-varying dynamics and involved non-linear effects during a pandemic (see also the critique in De Chaisemartin…”
Resource Shocks, Corruption, and Local Business in AfricaPreprint
Pavlik, J. B., & Ross, A. (2020) · SSRN Electronic Journal · doi:10.2139/ssrn.3688541
Giebel, M. (2021) · SSRN Electronic Journal · doi:10.2139/ssrn.3992810
Carnegie, A. S., & Carson, A. (2020) · Cambridge University Press eBooks · doi:10.1017/9781108778114.008
Carnegie, A. S., & Carson, A. (2020) · Cambridge University Press eBooks · doi:10.1017/9781108778114.002
Carnegie, A. S., & Carson, A. (2020) · Cambridge University Press eBooks · doi:10.1017/9781108778114.006
Carnegie, A. S., & Carson, A. (2020) · Cambridge University Press eBooks · doi:10.1017/9781108778114.005
Sensitive Information in Global Governance: The Past and the Present
Carnegie, A. S., & Carson, A. (2020) · Cambridge University Press eBooks · doi:10.1017/9781108778114.003
Carnegie, A. S., & Carson, A. (2020) · Cambridge University Press eBooks · doi:10.1017/9781108778114.001
Carnegie, A. S., & Carson, A. (2020) · Cambridge University Press eBooks · doi:10.1017/9781108778114.004
(2020) · Cambridge University Press eBooks · doi:10.1017/9781108778114.009
Finnegan, J., Lipscy, P. Y., Meckling, J., & Metz, F. (2021) · doi:10.33774/apsa-2021-2f3gx
Carnegie, A. S., & Carson, A. (2020) · Cambridge University Press eBooks · doi:10.1017/9781108778114.010
The Legislative Legacy of Strict Voter Identification LawsPreprint
Campos, A., Harden, J. J., & Bussing, A. (2021) · SSRN Electronic Journal · doi:10.2139/ssrn.3904159
Understanding community level influences on the prevalence of SARS-CoV-2 infection in EnglandPreprint
Joshi, C., Ali, A., ÓConnor, T., et al. (2022) · medRxiv · doi:10.1101/2022.04.14.22273759
The Effect of Mandated CEO-Chairman Separation on Shareholder ValuePreprint
Guimarães, R. D. S., & Trevisan, G. (2022) · SSRN Electronic Journal · doi:10.2139/ssrn.4122646
Gniadkowska‐Szymańska, A., & Gubareva, M. (2022) · Finanse i Prawo Finansowe · doi:10.18778/2391-6478.3.35.03
Duvall, S. F., & Mason, J. R. (2022) · Journal of School Choice · doi:10.1080/15582159.2022.2135843
Revisiting the J-shape. Human development and fertility in the United States
Schubert, H.-A., Dudel, C., Kolobova, M., & Myrskylä, M. (2023) · doi:10.4054/mpidr-wp-2023-022
“We apply two-way fixed effects regression, which only removes the additive contribution of state and year effects (Imai and Kim, 2021).”
MethodSubtracting 420 from 922: Marijuana Legalization and the Gun Control Act After BruenPreprint
Goldrosen, N. (2023) · SSRN Electronic Journal · doi:10.2139/ssrn.4478216
Concentrating on Bailouts: Government Guarantees and Bank Asset CompositionPreprint
Eufinger, C., Gorostiaga, J. P., & Richter, B. (2023) · SSRN Electronic Journal · doi:10.2139/ssrn.4522039
Seeing is Believing: The Effects of Optometrist Scope of Practice ExpansionPreprint
Bae, K., Timmons, E. J., & Nandy, P. (2023) · SSRN Electronic Journal · doi:10.2139/ssrn.4556268
Generalized Synthetic Control for TestOps at ABI: Models, Algorithms, and Infrastructure
Costa, L. M., Farias, V. F., Foncea, P., Gan, J. (., Garg, A., Montenegro, I. R., Pathak, K., Peng, T., & Popovic, D. (2023) · INFORMS Journal on Applied Analytics · doi:10.1287/inte.2023.0028
Chen, H., Bayram, M. E., & Lee, C. (2023) · Finance research letters · doi:10.1016/j.frl.2023.104599
Ciaccio, A. (2023) · SSRN Electronic Journal · doi:10.2139/ssrn.4603295
Three essays on bank risk-taking from a portfolio perspective
Gorostiaga, J. P. J. (. P. J. (2023) · Deposito Adademico Digital Universidad De Navarra (University of Navarra) · doi:10.15581/10171/67259
Mühlenweg Andrea, M., & Maike, S. (2023) · Sozialer Fortschritt · doi:10.3790/sfo.2023.03.muehlenweg.schmitt
Peer Dismissal and Contagious Career ConcernsPreprint
Lee, J. (2023) · SSRN Electronic Journal · doi:10.2139/ssrn.4636780
Temporary agency employment: A supplement to regular jobs – under certain circumstances
Neumann, U. (2023) · Competition & Change · doi:10.1177/10245294231221884
“The ability of TWFE models to simultaneously adjust for within-and between-establishment effects critically relies upon the assumption of linear additive effects (Imai and Kim, 2021).”
Aja‐Eke, D., & Brazys, S. R. (2023) · SSRN Electronic Journal · doi:10.2139/ssrn.4669972
More Likely to Be Persuaded: The Impact of Air Pollution on Decisions MakingPreprint
Li, X., Meng, K., & Shi, X. (2023) · SSRN Electronic Journal · doi:10.2139/ssrn.4672383
Efeitos da experiência de orçamento participativo no Brasil
Ferreira, R. P. M. (2023) · Digital Library of Theses and Dissertations (Universidade de São Paulo) · doi:10.11606/d.12.2023.tde-14122023-173406
Three essays on health economics in china
Dong, Y. (2024) · Queensland University of Technology · doi:10.5204/thesis.eprints.246072
Mühlenweg, A., & Schmitt, M. (2024) · Sozialer Fortschritt · doi:10.3790/sfo.73.2.81
Monetary Policy Shocks and Bank Profitability: Evidence from the Euro AreaPreprint
Časta, M., & Švéda, J. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4770689
Trade and the Politics of Electoral ReformPreprint
Becher, M., & Gonzalez, I. M. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4788761
Are Individual and Corporate Social Responsibility Substitutes?Preprint
Neshat, N. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4836989
Wu, J., & Liew, C. Y. (2024) · Asian Academy of Management Journal of Accounting and Finance · doi:10.21315/aamjaf2024.20.1.9
“A bidirectional fixed effect was used for regression in this article (Wintoki et al., 2012; Imai & Kim, 2021; Liew & Devi, 2021; Liew et al., 2021).”
MethodDo Armed Drones Counter Terrorism, Or Are They Counterproductive? Evidence from Eighteen CountriesPreprint
Schwartz, J. A., Fuhrmann, M., & Horowitz, M. C. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4818417
The Differential Error Components Models under Treatment Effect HeterogeneityPreprint
Jin, Y. (2024) · Preprints.org · doi:10.20944/preprints202407.0569.v1
The Hidden Subsidy of the Affordable Care ActPreprint
Sigaud, L., Bjoerkheim, M., & Melo, V. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4889920
Essays on systemic risk and banking
Borges, W. A. D. F. (2024) · Digital Library of Theses and Dissertations (Universidade de São Paulo) · doi:10.11606/t.96.2024.tde-25042024-103038
Metadata
- Authors
- Imai, K., & Kim, I. S.
- Affiliations
- Harvard University (US)
- Massachusetts Institute of Technology (US)
- Corresponding author
- Kim, In Song
- Venue
- Political Analysis
- Publisher
- Cambridge University Press (CUP)
- Published
- 2020-11-12
- Volume
- 29
- Issue
- 3
- Pages
- 405-415
- ISSN
- 1047-1987 (ISSN-L)
- 1476-4989
- OpenAlex
- W3098174737
- Semantic Scholar
- 660ac5f400de68ad64a9f31370bc59a30ec98d22
- Cited by
- 623
- References
- 20
- Citations by year
- 2026: 105 · 2025: 142 · 2024: 126 · 2023: 92 · 2022: 68 · 2021: 54 · 2020: 30
- FWCI
- 42.96
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Political Analysis (publishedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- DSpace@MIT (Massachusetts Institute of Technology) (submittedVersion, cc-by-nc-sa)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Causal Inference Techniques
- Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Spatial and Panel Data Analysis
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Keywords
- causal inference, panel data, difference-in-differences, unmeasured confounding, negative weights, design-based estimation, model assumptions
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