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

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Author Info
Leeb, Hannes
Pötscher, Benedikt M.

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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; performance is here measured by the probability that the estimation error exceeds a given threshold. 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.

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Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 72.

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Date of creation: Apr 2005
Date of revision: Feb 2007
Handle: RePEc:pra:mprapa:72

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Related research
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.

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Find related papers by JEL classification:
C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
C20 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - General
C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Hypothesis Testing
C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Estimation
C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation and Testing

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References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. 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. [Downloadable!]
  2. Brownstone, David, 1990. "Bootstrapping improved estimators for linear regression models," Journal of Econometrics, Elsevier, vol. 44(1-2), pages 171-187. [Downloadable!] (restricted)
  3. 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. [Downloadable!]
  4. repec:cup:etheor:v:11:y:1995:i:3:p:537-49 is not listed on IDEAS
  5. Hannes Leeb & Benedikt M. Potscher, 2003. "Can One Estimate the Conditional Distribution of Post-Model-Selection Estimators?," Cowles Foundation Discussion Papers 1444, Cowles Foundation, Yale University. [Downloadable!]
    Other versions:
  6. Kapetanios, George, 2001. "Incorporating lag order selection uncertainty in parameter inference for AR models," Economics Letters, Elsevier, vol. 72(2), pages 137-144, August. [Downloadable!] (restricted)
  7. 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. [Downloadable!] (restricted)
  8. 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. [Downloadable!] (restricted)
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Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Pötscher, Benedikt M. & Schneider, Ulrike, 2007. "On the distribution of the adaptive LASSO estimator," MPRA Paper 6913, University Library of Munich, Germany. [Downloadable!]
  2. 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, revised Feb 2007. [Downloadable!]
    Other versions:
  3. Pötscher, Benedikt M. & Leeb, Hannes, 2007. "On the distribution of penalized maximum likelihood estimators: The LASSO, SCAD, and thresholding," MPRA Paper 5615, University Library of Munich, Germany. [Downloadable!]
  4. Peter C.B. Phillips, 2004. "Automated Discovery in Econometrics," Cowles Foundation Discussion Papers 1469, Cowles Foundation, Yale University. [Downloadable!]
    Other versions:
  5. Pötscher, Benedikt M., 2006. "The Distribution of Model Averaging Estimators and an Impossibility Result Regarding Its Estimation," MPRA Paper 73, University Library of Munich, Germany, revised Jul 2006. [Downloadable!]
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