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Estimation and Inference of Discontinuity in Density


  • Taisuke Otsu
  • Ke-Li Xu
  • Yukitoshi Matsushita


Continuity or discontinuity of probability density functions of data often plays a fundamental role in empirical economic analysis. For example, for identification and inference of causal effects in regression discontinuity designs it is typically assumed that the density function of a conditioning variable is continuous at a cutoff point that determines assignment of a treatment. Also, discontinuity in density functions can be a parameter of economic interest, such as in analysis of bunching behaviors of taxpayers. To facilitate researchers to conduct valid inference for these problems, this article extends the binning and local likelihood approaches to estimate discontinuity of density functions and proposes empirical likelihood-based tests and confidence sets for the discontinuity. In contrast to the conventional Wald-type test and confidence set using the binning estimator, our empirical likelihood-based methods (i) circumvent asymptotic variance estimation to construct the test statistics and confidence sets; (ii) are invariant to nonlinear transformations of the parameters of interest; (iii) offer confidence sets whose shapes are automatically determined by data; and (iv) admit higher-order refinements, so-called Bartlett corrections. First- and second-order asymptotic theories are developed. Simulations demonstrate the superior finite sample behaviors of the proposed methods. In an empirical application, we assess the identifying assumption of no manipulation of class sizes in the regression discontinuity design studied by Angrist and Lavy (1999).

Suggested Citation

  • Taisuke Otsu & Ke-Li Xu & Yukitoshi Matsushita, 2013. "Estimation and Inference of Discontinuity in Density," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 31(4), pages 507-524, October.
  • Handle: RePEc:taf:jnlbes:v:31:y:2013:i:4:p:507-524 DOI: 10.1080/07350015.2013.818007

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    References listed on IDEAS

    1. Ke-Li Xu & Peter C. B. Phillips, 2011. "Tilted Nonparametric Estimation of Volatility Functions With Empirical Applications," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 29(4), pages 518-528, October.
    2. Whitney K. Newey & Richard J. Smith, 2004. "Higher Order Properties of Gmm and Generalized Empirical Likelihood Estimators," Econometrica, Econometric Society, vol. 72(1), pages 219-255, January.
    3. Emmanuel Saez, 2010. "Do Taxpayers Bunch at Kink Points?," American Economic Journal: Economic Policy, American Economic Association, vol. 2(3), pages 180-212, August.
    4. Smith, Richard J, 1997. "Alternative Semi-parametric Likelihood Approaches to Generalised Method of Moments Estimation," Economic Journal, Royal Economic Society, vol. 107(441), pages 503-519, March.
    5. Song Xi Chen & Hengjian Cui, 2006. "On Bartlett correction of empirical likelihood in the presence of nuisance parameters," Biometrika, Biometrika Trust, vol. 93(1), pages 215-220, March.
    6. Lee, David S., 2008. "Randomized experiments from non-random selection in U.S. House elections," Journal of Econometrics, Elsevier, vol. 142(2), pages 675-697, February.
    7. Imbens, Guido W. & Lemieux, Thomas, 2008. "Regression discontinuity designs: A guide to practice," Journal of Econometrics, Elsevier, vol. 142(2), pages 615-635, February.
    8. Davidson, Russell & MacKinnon, James G, 1998. "Graphical Methods for Investigating the Size and Power of Hypothesis Tests," The Manchester School of Economic & Social Studies, University of Manchester, vol. 66(1), pages 1-26, January.
    9. McCrary, Justin, 2008. "Manipulation of the running variable in the regression discontinuity design: A density test," Journal of Econometrics, Elsevier, vol. 142(2), pages 698-714, February.
    10. Gregory, Allan W & Veall, Michael R, 1985. "Formulating Wald Tests of Nonlinear Restrictions," Econometrica, Econometric Society, vol. 53(6), pages 1465-1468, November.
    11. Joshua D. Angrist & Victor Lavy, 1999. "Using Maimonides' Rule to Estimate the Effect of Class Size on Scholastic Achievement," The Quarterly Journal of Economics, Oxford University Press, vol. 114(2), pages 533-575.
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    Cited by:

    1. Ulrich Matter & Michaela Slotwinski, 2016. "Precise Control over Legislative Vote Outcomes: A Forensic Approach to Political Economics," CESifo Working Paper Series 6007, CESifo Group Munich.
    2. Jales, Hugo & Ma, Jun & Yu, Zhengfei, 2017. "Optimal bandwidth selection for local linear estimation of discontinuity in density," Economics Letters, Elsevier, vol. 153(C), pages 23-27.
    3. Slotwinski, Michaela & Schmidheiny, Kurt, 2014. "Behavioral Responses to Local Tax Rates: Quasi-Experimental Evidence from a Foreigners Tax Scheme in Switzerland," Annual Conference 2014 (Hamburg): Evidence-based Economic Policy 100292, Verein für Socialpolitik / German Economic Association.
    4. Andrew C. Eggers & Ronny Freier & Veronica Grembi & Tommaso Nannicini, 2015. "Regression Discontinuity Designs Based on Population Thresholds: Pitfalls and Solutions," Discussion Papers of DIW Berlin 1503, DIW Berlin, German Institute for Economic Research.
    5. Joshua D. Angrist & Victor Lavy & Jetson Leder-Luis & Adi Shany, 2017. "Maimonides Rule Redux," NBER Working Papers 23486, National Bureau of Economic Research, Inc.
    6. repec:eee:econom:v:201:y:2017:i:1:p:1-18 is not listed on IDEAS
    7. repec:spr:stpapr:v:58:y:2017:i:4:d:10.1007_s00362-016-0745-z is not listed on IDEAS
    8. repec:aea:jecper:v:31:y:2017:i:2:p:3-32 is not listed on IDEAS

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    JEL classification:

    • J1 - Labor and Demographic Economics - - Demographic Economics


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