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Inference with "Difference in Differences" with a Small Number of Policy Changes

  • Timothy Conley
  • Christopher Taber

Difference in differences methods have become very popular in applied work. This paper provides a new method for inference in these models when there are a small number of policy changes. This situation occurs in many implementations of these estimators. Identification of the key parameter typically arises when a group "changes" some particular policy. The asymptotic approximations that are typically employed assume that the number of cross sectional groups, N, times the number of time periods, T, is large. However, even when N or T is large, the number of actual policy changes observed in the data is often very small. In this case, we argue that point estimators of treatment effects should not be thought of as being consistent and that the standard methods that researchers use to perform inference in these models are not appropriate. We develop an alternative approach to inference under the assumption that there are a finite number of policy changes in the data, using asymptotic approximations as the number of non-changing groups gets large. In this situation we cannot obtain a consistent point estimator for the key treatment effect parameter. However, we can consistently estimate the finite-sample distribution of the treatment effect estimator, up to the unknown parameter itself. This allows us to perform hypothesis tests and construct confidence intervals. For expositional and motivational purposes, we focus on the difference in differences case, but our approach should be appropriate more generally in treatment effect models which employ a large number of controls, but a small number of treatments. We demonstrate the use of the approach by analyzing the effect of college merit aide programs on college attendance. We show that in some cases the standard approach can give misleading results.

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Paper provided by National Bureau of Economic Research, Inc in its series NBER Technical Working Papers with number 0312.

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Date of creation: Jul 2005
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Handle: RePEc:nbr:nberte:0312
Note: TWP
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  1. David Card & Alan B. Krueger, 1993. "Minimum Wages and Employment: A Case Study of the Fast Food Industry in New Jersey and Pennsylvania," NBER Working Papers 4509, National Bureau of Economic Research, Inc.
  2. Dynarski, Susan, 2000. "Hope for Whom? Financial Aid for the Middle Class and Its Impact on College Attendance," National Tax Journal, National Tax Association, vol. 53(n. 3), pages 629-62, September.
  3. Dynarski, Susan, 2004. "The New Merit Aid," Working Paper Series rwp04-009, Harvard University, John F. Kennedy School of Government.
    • Susan Dynarski, 2004. "The New Merit Aid," NBER Chapters, in: College Choices: The Economics of Where to Go, When to Go, and How to Pay For It, pages 63-100 National Bureau of Economic Research, Inc.
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  7. Christopher M. Cornwell & David B. Mustard & Deepa Sridhar, 2005. "The Enrollment Effects of Merit-Based Financial Aid: Evidence from Georgia's HOPE Scholarship," HEW 0501002, EconWPA.
  8. Gary T. Henry & Ross Rubenstein, 2002. "Paying for grades: Impact of merit-based financial aid on educational quality," Journal of Policy Analysis and Management, John Wiley & Sons, Ltd., vol. 21(1), pages 93-109.
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  11. Matzkin, Rosa L., 1986. "Restrictions of economic theory in nonparametric methods," Handbook of Econometrics, in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 42, pages 2523-2558 Elsevier.
  12. Susan Dynarski, 2000. "Hope for Whom? Financial Aid for the Middle Class and Its Impact on College Attendance," NBER Working Papers 7756, National Bureau of Economic Research, Inc.
  13. Gallant, A Ronald & Nychka, Douglas W, 1987. "Semi-nonparametric Maximum Likelihood Estimation," Econometrica, Econometric Society, vol. 55(2), pages 363-90, March.
  14. Chamberlain, Gary, 1980. "Analysis of Covariance with Qualitative Data," Review of Economic Studies, Wiley Blackwell, vol. 47(1), pages 225-38, January.
  15. Marianne Bertrand & Esther Duflo & Sendhil Mullainathan, 2002. "How Much Should We Trust Differences-in-Differences Estimates?," NBER Working Papers 8841, National Bureau of Economic Research, Inc.
  16. Newey, Whitney K. & McFadden, Daniel, 1986. "Large sample estimation and hypothesis testing," Handbook of Econometrics, in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 36, pages 2111-2245 Elsevier.
  17. Moulton, Brent R, 1990. "An Illustration of a Pitfall in Estimating the Effects of Aggregate Variables on Micro Unit," The Review of Economics and Statistics, MIT Press, vol. 72(2), pages 334-38, May.
  18. Anderson, Patricia M. & Meyer, Bruce D., 2000. "The effects of the unemployment insurance payroll tax on wages, employment, claims and denials," Journal of Public Economics, Elsevier, vol. 78(1-2), pages 81-106, October.
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