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Valid Confidence Intervals In Regression After Variable Selection

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  • Kabaila, Paul
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    Abstract

    We consider a linear regression model with regression parameters ( 1,..., p) and error variance parameter 2. Our aim is to find a confidence interval with minimum coverage probability 1 for a parameter of interest 1 in the presence of nuisance parameters ( 2,..., p, 2). We consider two confidence intervals, the first of which is the standard confidence interval for 1 with coverage probability 1 . The second confidence interval for 1 is obtained after a variable selection procedure has been applied to p. This interval is chosen to be as short as possible subject to the constraint that it has minimum coverage probability 1 . 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 p / is small. This comparison of confidence intervals is shown to be mathematically equivalent to a corresponding comparison of prediction intervals.

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    Bibliographic Info

    Article provided by Cambridge University Press in its journal Econometric Theory.

    Volume (Year): 14 (1998)
    Issue (Month): 04 (August)
    Pages: 463-482

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    Handle: RePEc:cup:etheor:v:14:y:1998:i:04:p:463-482_14

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    Cited by:
    1. Leeb, Hannes & Pötscher, Benedikt M. & Ewald, Karl, 2014. "On various confidence intervals post-model-selection," MPRA Paper 52858, University Library of Munich, Germany.
    2. Adam McCloskey, 2012. "Bonferroni-Based Size-Correction for Nonstandard Testing Problems," Working Papers 2012-16, Brown University, Department of Economics.
    3. 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.
    4. 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.
    5. 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.
    6. Pötscher, Benedikt M., 2007. "Confidence Sets Based on Sparse Estimators Are Necessarily Large," MPRA Paper 5677, University Library of Munich, Germany.
    7. Liu, Chu-An, 2012. "A plug-in averaging estimator for regressions with heteroskedastic errors," MPRA Paper 41414, University Library of Munich, Germany.

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