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Bonferroni-Based Size-Correction for Nonstandard Testing Problems

We develop powerful new size-correction procedures for nonstandard hypothesis testing environments in which the asymptotic distribution of a test statistic is discontinuous in a parameter under the null hypothesis. Examples of this form of testing problem are pervasive in econometrics and complicate inference by making size di- cult to control. This paper introduces two sets of new size-correction methods that correspond to two di erent general hypothesis testing frameworks. The new methods are designed to maximize the power of the underlying test while maintaining correct asymptotic size uniformly over the parameter space speci ed by the null hypothesis. They involve the construction of critical values that make use of reasoning derived from Bonferroni bounds. The rst set of new methods provides complementary alternatives to existing size-correction methods, entailing substantially higher power for many testing problems. The second set of new methods provides the rst available asymptotically size-correct tests for the general class of testing problems to which it applies. This class includes hypothesis tests on parameters after consistent model selection and tests on super-ecient/hard-thresholding estimators. We detail the construction and performance of the new tests in three speci c examples: testing after conservative model selection, testing when a nuisance parameter may be on a boundary and testing after consistent model selection.

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File URL: http://www.brown.edu/Departments/Economics/Papers/2012/2012-16_paper.pdf
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Paper provided by Brown University, Department of Economics in its series Working Papers with number 2012-16.

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Date of creation: 2012
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Handle: RePEc:bro:econwp:2012-16
Contact details of provider: Postal: Department of Economics, Brown University, Providence, RI 02912

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  1. Donald W.K. Andrews & Panle Jia, 2008. "Inference for Parameters Defined by Moment Inequalities: A Recommended Moment Selection Procedure," Cowles Foundation Discussion Papers 1676, Cowles Foundation for Research in Economics, Yale University.
  2. Chaudhuri, Saraswata & Zivot, Eric, 2011. "A new method of projection-based inference in GMM with weakly identified nuisance parameters," Journal of Econometrics, Elsevier, vol. 164(2), pages 239-251, October.
  3. Jorg Stoye, 2009. "More on Confidence Intervals for Partially Identified Parameters," Econometrica, Econometric Society, vol. 77(4), pages 1299-1315, 07.
  4. Kabaila, Paul, 1998. "Valid Confidence Intervals In Regression After Variable Selection," Econometric Theory, Cambridge University Press, vol. 14(04), pages 463-482, August.
  5. Donald W.K. Andrews & Xu Cheng & Patrik Guggenberger, 2011. "Generic Results for Establishing the Asymptotic Size of Confidence Sets and Tests," Cowles Foundation Discussion Papers 1813, Cowles Foundation for Research in Economics, Yale University.
  6. Pötscher, Benedikt M. & Leeb, Hannes, 2009. "On the distribution of penalized maximum likelihood estimators: The LASSO, SCAD, and thresholding," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 2065-2082, October.
  7. Andrews, Donald W.K. & Guggenberger, Patrik, 2009. "Incorrect asymptotic size of subsampling procedures based on post-consistent model selection estimators," Journal of Econometrics, Elsevier, vol. 152(1), pages 19-27, September.
  8. Leeb, Hannes & Pötscher, Benedikt M., 2012. "Testing in the Presence of Nuisance Parameters: Some Comments on Tests Post-Model-Selection and Random Critical Values," MPRA Paper 41459, University Library of Munich, Germany.
  9. Guggenberger, Patrik, 2010. "The Impact Of A Hausman Pretest On The Asymptotic Size Of A Hypothesis Test," Econometric Theory, Cambridge University Press, vol. 26(02), pages 369-382, April.
  10. Donald W. K. Andrews & Patrik Guggenberger, 2009. "Hybrid and Size-Corrected Subsampling Methods," Econometrica, Econometric Society, vol. 77(3), pages 721-762, 05.
  11. Moon, Hyungsik Roger & Schorfheide, Frank, 2009. "Estimation with overidentifying inequality moment conditions," Journal of Econometrics, Elsevier, vol. 153(2), pages 136-154, December.
  12. Andrews, Donald W.K. & Guggenberger, Patrik, 2010. "ASYMPTOTIC SIZE AND A PROBLEM WITH SUBSAMPLING AND WITH THE m OUT OF n BOOTSTRAP," Econometric Theory, Cambridge University Press, vol. 26(02), pages 426-468, April.
  13. Donald W. K. Andrews, 1999. "Estimation When a Parameter Is on a Boundary," Econometrica, Econometric Society, vol. 67(6), pages 1341-1384, November.
  14. Joseph P. Romano & Azeem M. Shaikh & Michael Wolf, 2014. "A Practical Two‐Step Method for Testing Moment Inequalities," Econometrica, Econometric Society, vol. 82, pages 1979-2002, 09.
  15. Andrews, Donald W.K. & Guggenberger, Patrik, 2010. "Applications of subsampling, hybrid, and size-correction methods," Journal of Econometrics, Elsevier, vol. 158(2), pages 285-305, October.
  16. Bruce E. Hansen, 2007. "Least Squares Model Averaging," Econometrica, Econometric Society, vol. 75(4), pages 1175-1189, 07.
  17. Donald W.K. Andrews & Xu Cheng, 2010. "Estimation and Inference with Weak, Semi-strong, and Strong Identification," Cowles Foundation Discussion Papers 1773, Cowles Foundation for Research in Economics, Yale University.
  18. Hannes Leeb & Benedikt M. Potscher, 2003. "Can One Estimate the Conditional Distribution of Post-Model-Selection Estimators?," Cowles Foundation Discussion Papers 1444, Cowles Foundation for Research in Economics, Yale University.
  19. Andrews, Donald W K, 2001. "Testing When a Parameter Is on the Boundary of the Maintained Hypothesis," Econometrica, Econometric Society, vol. 69(3), pages 683-734, May.
  20. Patrik Guggenberger & Frank Kleibergen & Sophocles Mavroeidis & Linchun Chen, 2012. "On the Asymptotic Sizes of Subset Anderson–Rubin and Lagrange Multiplier Tests in Linear Instrumental Variables Regression," Econometrica, Econometric Society, vol. 80(6), pages 2649-2666, November.
  21. Leeb, Hannes & P tscher, Benedikt M., 2005. "Model Selection And Inference: Facts And Fiction," Econometric Theory, Cambridge University Press, vol. 21(01), pages 21-59, February.
  22. Douglas Staiger & James H. Stock, 1997. "Instrumental Variables Regression with Weak Instruments," Econometrica, Econometric Society, vol. 65(3), pages 557-586, May.
  23. Anna Mikusheva, 2007. "Uniform Inference in Autoregressive Models," Econometrica, Econometric Society, vol. 75(5), pages 1411-1452, 09.
  24. Hansen, Peter Reinhard, 2005. "A Test for Superior Predictive Ability," Journal of Business & Economic Statistics, American Statistical Association, vol. 23, pages 365-380, October.
  25. Hansen, Bruce E., 2005. "Challenges For Econometric Model Selection," Econometric Theory, Cambridge University Press, vol. 21(01), pages 60-68, February.
  26. Jean-Marie Dufour, 1997. "Some Impossibility Theorems in Econometrics with Applications to Structural and Dynamic Models," Econometrica, Econometric Society, vol. 65(6), pages 1365-1388, November.
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