IDEAS home Printed from https://ideas.repec.org/p/fgv/eesptd/406.html
   My bibliography  Save this paper

Inference in differences-in-differences with few treated groups and heteroskedasticity

Author

Listed:
  • Ferman, Bruno
  • Pinto, Cristine Campos de Xavier

Abstract

Differences-in-Differences (DID) is one of the most widely used identification strategies in applied economics. However, how to draw inferences in DID models when there are few treated groups remains an open question. We show that the usual inference methods used in DID models might not perform well when there are few treated groups and errors are heteroskedastic. In particular, we show that when there is variation in the number of observations per group, inference methods designed to work when there are few treated groups tend to (under-) over-reject the null hypothesis when the treated groups are (large) small relative to the control groups. This happens because larger groups tend to have lower variance, generating heteroskedasticity in the group x time aggregate DID model. We provide evidence from Monte Carlo simulations and from placebo DID regressions with the American Community Survey (ACS) and the Current Population Survey (CPS) datasets to show that this problem is relevant even in datasets with large numbers of observations per group. We then derive an alternative inference method that provides accurate hypothesis testing in situations where there are few treated groups (or even just one) and many control groups in the presence of heteroskedasticity. Our method assumes that we can model the heteroskedasticity of a linear combination of the errors. We show that this assumption can be satisfied without imposing strong assumptions on the errors in common DID applications. With many pre-treatment periods, we show that this assumption can be relaxed. Instead, we provide an alternative inference method that relies on strict stationarity and ergodicity of the time series. Finally, we consider two recent alternatives to DID when there are many pre-treatment periods. We extend our inference methods to linear factor models when there are few treated groups. We also derive conditions under which a permutation test for the synthetic control estimator proposed by Abadie et al. (2010) is robust to heteroskedasticity and propose a modification on the test statistic that provided a better heteroskedasticity correction in our simulations.

Suggested Citation

  • Ferman, Bruno & Pinto, Cristine Campos de Xavier, 2015. "Inference in differences-in-differences with few treated groups and heteroskedasticity," Textos para discussão 406, FGV EESP - Escola de Economia de São Paulo, Fundação Getulio Vargas (Brazil).
  • Handle: RePEc:fgv:eesptd:406
    as

    Download full text from publisher

    File URL: http://bibliotecadigital.fgv.br/dspace/bitstream/10438/14170/8/TD%20411%20-%20BrunoFerman%20e%20CristinePinto-Maio.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. MacKinnon, James G. & Webb, Matthew D., 2020. "Randomization inference for difference-in-differences with few treated clusters," Journal of Econometrics, Elsevier, vol. 218(2), pages 435-450.
    2. Guido W. Imbens & Michal Kolesár, 2016. "Robust Standard Errors in Small Samples: Some Practical Advice," The Review of Economics and Statistics, MIT Press, vol. 98(4), pages 701-712, October.
    3. A. Colin Cameron & Jonah B. Gelbach & Douglas L. Miller, 2008. "Bootstrap-Based Improvements for Inference with Clustered Errors," The Review of Economics and Statistics, MIT Press, vol. 90(3), pages 414-427, August.
    4. Timothy G. Conley & Christopher R. Taber, 2011. "Inference with "Difference in Differences" with a Small Number of Policy Changes," The Review of Economics and Statistics, MIT Press, vol. 93(1), pages 113-125, February.
    5. White, Halbert, 1980. "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity," Econometrica, Econometric Society, vol. 48(4), pages 817-838, May.
    6. Brewer Mike & Crossley Thomas F. & Joyce Robert, 2018. "Inference with Difference-in-Differences Revisited," Journal of Econometric Methods, De Gruyter, vol. 7(1), pages 1-16, January.
    7. Alberto Abadie & Javier Gardeazabal, 2003. "The Economic Costs of Conflict: A Case Study of the Basque Country," American Economic Review, American Economic Association, vol. 93(1), pages 113-132, March.
    8. Pettersson-Lidbom, Per, 2012. "Does the size of the legislature affect the size of government? Evidence from two natural experiments," Journal of Public Economics, Elsevier, vol. 96(3), pages 269-278.
    9. Jeffrey M. Wooldridge, 2003. "Cluster-Sample Methods in Applied Econometrics," American Economic Review, American Economic Association, vol. 93(2), pages 133-138, May.
    10. Choi, Moonkyung Kate, 2011. "The impact of Medicaid insurance coverage on dental service use," Journal of Health Economics, Elsevier, vol. 30(5), pages 1020-1031.
    11. Stephen G. Donald & Kevin Lang, 2007. "Inference with Difference-in-Differences and Other Panel Data," The Review of Economics and Statistics, MIT Press, vol. 89(2), pages 221-233, May.
    12. Joshua D. Angrist & Jörn-Steffen Pischke, 2009. "Mostly Harmless Econometrics: An Empiricist's Companion," Economics Books, Princeton University Press, edition 1, number 8769, April.
    13. Marianne Bertrand & Esther Duflo & Sendhil Mullainathan, 2004. "How Much Should We Trust Differences-In-Differences Estimates?," The Quarterly Journal of Economics, Oxford University Press, vol. 119(1), pages 249-275.
    14. Kelly Bedard & Chau Do, 2005. "Are Middle Schools More Effective?: The Impact of School Structure on Student Outcomes," Journal of Human Resources, University of Wisconsin Press, vol. 40(3).
    15. James G. MacKinnon & Matthew D. Webb, 2017. "Wild Bootstrap Inference for Wildly Different Cluster Sizes," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 32(2), pages 233-254, March.
    16. Abadie, Alberto & Diamond, Alexis & Hainmueller, Jens, 2010. "Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California’s Tobacco Control Program," Journal of the American Statistical Association, American Statistical Association, vol. 105(490), pages 493-505.
    17. A. Colin Cameron & Douglas L. Miller, 2015. "A Practitioner’s Guide to Cluster-Robust Inference," Journal of Human Resources, University of Wisconsin Press, vol. 50(2), pages 317-372.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. MacKinnon, James G. & Webb, Matthew D., 2020. "Randomization inference for difference-in-differences with few treated clusters," Journal of Econometrics, Elsevier, vol. 218(2), pages 435-450.
    2. James G. MacKinnon & Matthew D. Webb, 2020. "When and How to Deal with Clustered Errors in Regression Models," Working Paper 1421, Economics Department, Queen's University.
    3. Baron, E. Jason & Goldstein, Ezra G. & Wallace, Cullen T., 2020. "Suffering in silence: How COVID-19 school closures inhibit the reporting of child maltreatment," Journal of Public Economics, Elsevier, vol. 190(C).
    4. James G. MacKinnon & Matthew D. Webb, 2017. "Pitfalls When Estimating Treatment Effects Using Clustered Data," Working Paper 1387, Economics Department, Queen's University.
    5. Carvalho, Carlos & Masini, Ricardo & Medeiros, Marcelo C., 2018. "ArCo: An artificial counterfactual approach for high-dimensional panel time-series data," Journal of Econometrics, Elsevier, vol. 207(2), pages 352-380.
    6. Gonzalez, Felipe & Prem, Mounu, 2020. "Police Repression and Protest Behavior: Evidence from Student Protests in Chile," SocArXiv 3xk5r, Center for Open Science.
    7. Marina Dias & Demian Pouzo, 2021. "Inference for multi-valued heterogeneous treatment effects when the number of treated units is small," Papers 2105.10965, arXiv.org.
    8. Cragun, Randy, 2019. "Effects of lower ages of majority on oral contraceptive use: Evidence on the validity of The Power of the Pill," MPRA Paper 100871, University Library of Munich, Germany, revised 03 Jun 2020.
    9. Martin Huber, 2019. "An introduction to flexible methods for policy evaluation," Papers 1910.00641, arXiv.org.
    10. Giovanni Peri & Vasil Yasenov, 2019. "The Labor Market Effects of a Refugee Wave: Synthetic Control Method Meets the Mariel Boatlift," Journal of Human Resources, University of Wisconsin Press, vol. 54(2), pages 267-309.
    11. Victor Chernozhukov & Kaspar Wuthrich & Yinchu Zhu, 2017. "An Exact and Robust Conformal Inference Method for Counterfactual and Synthetic Controls," Papers 1712.09089, arXiv.org, revised May 2021.
    12. Han, Luyi & Winters, John V., 2020. "Industry Fluctuations and College Major Choices: Evidence from an Energy Boom and Bust," Economics of Education Review, Elsevier, vol. 77(C).
    13. Bruno Ferman, 2019. "Assessing Inference Methods," Papers 1912.08772, arXiv.org, revised Aug 2021.
    14. Stearns, Jenna & White, Corey, 2018. "Can paid sick leave mandates reduce leave-taking?," Labour Economics, Elsevier, vol. 51(C), pages 227-246.
    15. Bittschi, Benjamin & Dwenger, Nadja & Rincke, Johannes, 2021. "Water the flowers you want to grow? Evidence on private recognition and donor loyalty," European Economic Review, Elsevier, vol. 131(C).
    16. Bruno Ferman, 2019. "Inference in Differences-in-Differences: How Much Should We Trust in Independent Clusters?," Papers 1909.01782, arXiv.org, revised Sep 2020.
    17. James G. MacKinnon & Morten Ørregaard Nielsen & Matthew D. Webb, 2021. "Cluster-Robust Inference: A Guide to Empirical Practice," Working Paper 1456, Economics Department, Queen's University.
    18. Bobonis, Gustavo J. & Stabile, Mark & Tovar, Leonardo, 2020. "Military training exercises, pollution, and their consequences for health," Journal of Health Economics, Elsevier, vol. 73(C).
    19. Bruno Ferman, 2020. "Inference in Differences-in-Differences with Few Treated Units and Spatial Correlation," Papers 2006.16997, arXiv.org, revised May 2021.
    20. Ferman, Bruno & Pinto, Cristine, 2017. "Placebo Tests for Synthetic Controls," MPRA Paper 78079, University Library of Munich, Germany.
    21. Wenjie Wang & Yichong Zhang, 2021. "Wild Bootstrap for Instrumental Variables Regressions with Weak and Few Clusters," Papers 2108.13707, arXiv.org.
    22. Ferman, Bruno, 2017. "Matching Estimators with Few Treated and Many Control Observations," MPRA Paper 78940, University Library of Munich, Germany.
    23. Andreas Hagemann, 2020. "Inference with a single treated cluster," Papers 2010.04076, arXiv.org.
    24. Shooshan Danagoulian & Derek Jenkins, 2021. "Rolling back the gains: Maternal stress undermines pregnancy health after Flint's water switch," Health Economics, John Wiley & Sons, Ltd., vol. 30(3), pages 564-584, March.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Hansen, Bruce E. & Lee, Seojeong, 2019. "Asymptotic theory for clustered samples," Journal of Econometrics, Elsevier, vol. 210(2), pages 268-290.
    2. repec:fgv:eesptd:411 is not listed on IDEAS
    3. James G. MacKinnon & Morten Ørregaard Nielsen & Matthew D. Webb, 2021. "Cluster-Robust Inference: A Guide to Empirical Practice," Working Paper 1456, Economics Department, Queen's University.
    4. James G. MacKinnon & Matthew D. Webb, 2020. "When and How to Deal with Clustered Errors in Regression Models," Working Paper 1421, Economics Department, Queen's University.
    5. James G. MacKinnon, 2019. "How cluster-robust inference is changing applied econometrics," Canadian Journal of Economics, Canadian Economics Association, vol. 52(3), pages 851-881, August.
    6. James G. MacKinnon & Matthew D. Webb, 2017. "Pitfalls When Estimating Treatment Effects Using Clustered Data," Working Paper 1387, Economics Department, Queen's University.
    7. Slawa Rokicki & Jessica Cohen & Gunther Fink & Joshua Salomon & Mary Beth Landrum, 2018. "Inference with difference-in-differences with a small number of groups: a review, simulation study and empirical application using SHARE data," CHaRMS Working Papers 18-01, Centre for HeAlth Research at the Management School (CHaRMS).
    8. Andreas Hagemann, 2019. "Permutation inference with a finite number of heterogeneous clusters," Papers 1907.01049, arXiv.org.
    9. A. Colin Cameron & Douglas L. Miller, 2010. "Robust Inference with Clustered Data," Working Papers 106, University of California, Davis, Department of Economics.
    10. Chor, Elise & Andresen, Martin Eckhoff & Kalil, Ariel, 2016. "The impact of universal prekindergarten on family behavior and child outcomes," Economics of Education Review, Elsevier, vol. 55(C), pages 168-181.
    11. Hernæs, Øystein, 2018. "Activation against absenteeism – Evidence from a sickness insurance reform in Norway," Journal of Health Economics, Elsevier, vol. 62(C), pages 60-68.
    12. Hwang, Jungbin, 2021. "Simple and trustworthy cluster-robust GMM inference," Journal of Econometrics, Elsevier, vol. 222(2), pages 993-1023.
    13. James G. MacKinnon & Morten Ørregaard Nielsen & Matthew D. Webb, 2020. "Testing for the appropriate level of clustering in linear regression models," Working Paper 1428, Economics Department, Queen's University.
    14. Heiko T. Burret & Lars P. Feld, 2018. "Vertical effects of fiscal rules: the Swiss experience," International Tax and Public Finance, Springer;International Institute of Public Finance, vol. 25(3), pages 673-721, June.
    15. Aleksey Oshchepkov & Anna Shirokanova, 2020. "Multilevel Modeling For Economists: Why, When And How," HSE Working papers WP BRP 233/EC/2020, National Research University Higher School of Economics.
    16. Jeffrey D. Michler & Anna Josephson, 2021. "Recent Developments in Inference: Practicalities for Applied Economics," Papers 2107.09736, arXiv.org.
    17. Entorf, Horst & Sattarova, Liliya, 2016. "The Analysis of Prison-Prisoner Data Using Cluster-Sample Econometrics: Prison Conditions and Prisoners' Assessments of the Future," IZA Discussion Papers 10209, Institute of Labor Economics (IZA).
    18. Matthew D. Webb, 2014. "Reworking Wild Bootstrap Based Inference For Clustered Errors," Working Paper 1315, Economics Department, Queen's University.
    19. Bullinger, Lindsey Rose, 2019. "The Effect of Paid Family Leave on Infant and Parental Health in the United States," Journal of Health Economics, Elsevier, vol. 66(C), pages 101-116.
    20. Congdon Fors, Heather & Houngbedji, Kenneth & Lindskog, Annika, 2019. "Land certification and schooling in rural Ethiopia," World Development, Elsevier, vol. 115(C), pages 190-208.
    21. Bruno Ferman, 2020. "Inference in Differences-in-Differences with Few Treated Units and Spatial Correlation," Papers 2006.16997, arXiv.org, revised May 2021.

    More about this item

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
    • C33 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Models with Panel Data; Spatio-temporal Models

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:fgv:eesptd:406. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: . General contact details of provider: https://edirc.repec.org/data/eegvfbr.html .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Núcleo de Computação da FGV EPGE (email available below). General contact details of provider: https://edirc.repec.org/data/eegvfbr.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.