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

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  • Leeb, Hannes
  • Pötscher, Benedikt M.

Abstract

We consider the problem of estimating the unconditional distribution of a post-model-selection estimator. 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 then 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 the unconditional 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 the distribution. These lower bounds are shown to approach 1/2 or even 1 in large samples, depending on the situation considered. Similar impossibility results are also obtained for the distribution of linear functions (e.g., predictors) of the post-model-selection estimator.

Suggested Citation

  • Leeb, Hannes & Pötscher, Benedikt M., 2005. "Can One Estimate the Unconditional Distribution of Post-Model-Selection Estimators ?," MPRA Paper 72, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:72
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    References listed on IDEAS

    as
    1. 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.
    2. repec:cup:etheor:v:11:y:1995:i:3:p:537-49 is not listed on IDEAS
    3. Hannes Leeb, 2006. "The distribution of a linear predictor after model selection: Unconditional finite-sample distributions and asymptotic approximations," Papers math/0611186, arXiv.org.
    4. 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.
    5. Kapetanios, George, 2001. "Incorporating lag order selection uncertainty in parameter inference for AR models," Economics Letters, Elsevier, vol. 72(2), pages 137-144, August.
    6. Brownstone, David, 1990. "Bootstrapping improved estimators for linear regression models," Journal of Econometrics, Elsevier, vol. 44(1-2), pages 171-187.
    7. 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.
    8. Francis X. Diebold & Lutz Kilian & Marc Nerlove, 2006. "Time Series Analysis," PIER Working Paper Archive 06-019, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
      • Diebold, F.X. & Kilian, L. & Nerlove, Marc, 2006. "Time Series Analysis," Working Papers 28556, University of Maryland, Department of Agricultural and Resource Economics.
    9. 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.
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    11. Lutz Kilian, 1998. "Accounting for Lag Order Uncertainty in Autoregressions: the Endogenous Lag Order Bootstrap Algorithm," Journal of Time Series Analysis, Wiley Blackwell, vol. 19(5), pages 531-548, September.
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    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

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

    JEL classification:

    • C20 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation

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