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Valid Confidence Intervals In Regression After Variable Selection

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  • Kabaila, Paul

Abstract

We consider a linear regression model with regression parameters (θ1,...,θp) and error variance parameter σ2. Our aim is to find a confidence interval with minimum coverage probability 1 − α for a parameter of interest θ1 in the presence of nuisance parameters (θ2,...,θp,σ2). We consider two confidence intervals, the first of which is the standard confidence interval for θ1 with coverage probability 1 − α. The second confidence interval for θ1 is obtained after a variable selection procedure has been applied to θp. This interval is chosen to be as short as possible subject to the constraint that it has minimum coverage probability 1 − α. The confidence intervals are compared using a risk function that is defined as a scaled version of the expected length of the confidence interval. We show that, subject to certain conditions including that [(dimension of response vector) − p] is small, the second confidence interval is preferable to the first when we anticipate (without being certain) that |θp|/σ is small. This comparison of confidence intervals is shown to be mathematically equivalent to a corresponding comparison of prediction intervals.

Suggested Citation

  • Kabaila, Paul, 1998. "Valid Confidence Intervals In Regression After Variable Selection," Econometric Theory, Cambridge University Press, vol. 14(4), pages 463-482, August.
  • Handle: RePEc:cup:etheor:v:14:y:1998:i:04:p:463-482_14
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    Cited by:

    1. Farchione, David & Kabaila, Paul, 2008. "Confidence intervals for the normal mean utilizing prior information," Statistics & Probability Letters, Elsevier, vol. 78(9), pages 1094-1100, July.
    2. McCloskey, Adam, 2017. "Bonferroni-based size-correction for nonstandard testing problems," Journal of Econometrics, Elsevier, vol. 200(1), pages 17-35.
    3. DiTraglia, Francis J., 2016. "Using invalid instruments on purpose: Focused moment selection and averaging for GMM," Journal of Econometrics, Elsevier, vol. 195(2), pages 187-208.
    4. Pötscher, Benedikt M., 2007. "Confidence Sets Based on Sparse Estimators Are Necessarily Large," MPRA Paper 5677, University Library of Munich, Germany.
    5. Paul Kabaila, 2009. "The Coverage Properties of Confidence Regions After Model Selection," International Statistical Review, International Statistical Institute, vol. 77(3), pages 405-414, December.
    6. Liu, Chu-An, 2012. "A plug-in averaging estimator for regressions with heteroskedastic errors," MPRA Paper 41414, University Library of Munich, Germany.
    7. Kabaila, Paul & Giri, Khageswor, 2009. "Large-sample confidence intervals for the treatment difference in a two-period crossover trial, utilizing prior information," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 652-658, March.
    8. Leeb, Hannes & Pötscher, Benedikt M. & Ewald, Karl, 2014. "On various confidence intervals post-model-selection," MPRA Paper 58326, University Library of Munich, Germany, revised 2014.
    9. Ruoyao Shi & Zhipeng Liao, 2018. "An Averaging GMM Estimator Robust to Misspecification," Working Papers 201803, University of California at Riverside, Department of Economics.
    10. Liu, Chu-An, 2015. "Distribution theory of the least squares averaging estimator," Journal of Econometrics, Elsevier, vol. 186(1), pages 142-159.
    11. Francis J. DiTraglia, 2011. "Using Invalid Instruments on Purpose: Focused Moment Selection and Averaging for GMM, Second Version," PIER Working Paper Archive 14-045, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania, revised 09 Dec 2014.
    12. Francis DiTraglia, 2011. "Using Invalid Instruments on Purpose: Focused Moment Selection and Averaging for GMM, Second Version," PIER Working Paper Archive 15-027, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania, revised 10 Aug 2015.
    13. Donald W.K. Andrews & Patrik Guggenberger, 2007. "Hybrid and Size-Corrected Subsample Methods," Cowles Foundation Discussion Papers 1606, Cowles Foundation for Research in Economics, Yale University.

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