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Can One Estimate the Conditional Distribution of Post-Model-Selection Estimators?

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We consider the problem of estimating the conditional distribution of a post-model-selection estimator where the conditioning is on the selected model. The notion of a post-model-selection estimator here refers to the combined procedure resulting from first selecting a model (e.g., by a model selection criterion like AIC or by a hypothesis testing procedure) and second estimating the parameters in the selected model (e.g., by least-squares or maximum likelihood), all based on the same data set. We show that it is impossible to estimate this distribution with reasonable accuracy even asymptotically. In particular, we show that no estimator for this distribution can be uniformly consistent (not even locally). This follows as a corollary to (local) minimax lower bounds on the performance of estimators for this distribution. Similar impossibility results are also obtained for the conditional distribution of linear functions (e.g., predictors) of the post-model-selection estimator.

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  • 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.
    • Handle: RePEc:cwl:cwldpp:1444
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    1. Diebold, F.X. & Kilian, L. & Nerlove, Marc, 2006. "Time Series Analysis," Working Papers 28556, University of Maryland, Department of Agricultural and Resource Economics.
    2. Brownstone, David, 1990. "Bootstrapping improved estimators for linear regression models," Journal of Econometrics, Elsevier, vol. 44(1-2), pages 171-187.
    3. repec:cup:etheor:v:11:y:1995:i:3:p:537-49 is not listed on IDEAS
    4. Leeb, Hannes & Pötscher, Benedikt M., 2008. "Can One Estimate The Unconditional Distribution Of Post-Model-Selection Estimators?," Econometric Theory, Cambridge University Press, vol. 24(2), pages 338-376, April.
    5. Hjort N.L. & Claeskens G., 2003. "Frequentist Model Average Estimators," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 879-899, January.
    6. Hannes Leeb, 2006. "The distribution of a linear predictor after model selection: Unconditional finite-sample distributions and asymptotic approximations," Papers math/0611186, arXiv.org.
    7. Kapetanios, George, 2001. "Incorporating lag order selection uncertainty in parameter inference for AR models," Economics Letters, Elsevier, vol. 72(2), pages 137-144, August.
    8. Kabaila, Paul, 1995. "The Effect of Model Selection on Confidence Regions and Prediction Regions," Econometric Theory, Cambridge University Press, vol. 11(3), pages 537-549, June.
    9. Pötscher, B.M., 1991. "Effects of Model Selection on Inference," Econometric Theory, Cambridge University Press, vol. 7(2), pages 163-185, June.
    10. Leeb, Hannes & Pötscher, Benedikt M., 2005. "Model Selection And Inference: Facts And Fiction," Econometric Theory, Cambridge University Press, vol. 21(1), pages 21-59, February.
    11. Danilov, Dmitry & Magnus, J.R.Jan R., 2004. "On the harm that ignoring pretesting can cause," Journal of Econometrics, Elsevier, vol. 122(1), pages 27-46, September.
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    More about this item

    Keywords

    Inference after model selection; Post-model-selection estimator; Pre-test estimator; Selection of regressors; Akaikeis information criterion AIC; Model uncertainty; Consistency; Uniform consistency; Lower risk bound;
    All these keywords.

    JEL classification:

    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
    • C50 - Mathematical and Quantitative Methods - - Econometric Modeling - - - General

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