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On the distribution of the adaptive LASSO estimator

Author

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  • Pötscher, Benedikt M.
  • Schneider, Ulrike

Abstract

We study the distribution of the adaptive LASSO estimator (Zou (2006)) in finite samples as well as in the large-sample limit. The large-sample distributions are derived both for the case where the adaptive LASSO estimator is tuned to perform conservative model selection as well as for the case where tuning results in consistent model selection. We show that the finite-sample as well as the large-sample distributions are typically highly non-normal, regardless of the choice of the tuning parameter. The uniform convergence rate is also obtained, and is shown to be slower than n^{-1/2} in case the estimator is tuned to perform consistent model selection. In particular, these results question the statistical relevance of the `oracle' property of the adaptive LASSO estimator established in Zou 2006). Moreover, we also provide an impossibility result regarding the estimation of the distribution function of the adaptive LASSO estimator.

Suggested Citation

  • Pötscher, Benedikt M. & Schneider, Ulrike, 2007. "On the distribution of the adaptive LASSO estimator," MPRA Paper 6913, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:6913
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    References listed on IDEAS

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    1. Leeb, Hannes & Potscher, Benedikt M., 2008. "Sparse estimators and the oracle property, or the return of Hodges' estimator," Journal of Econometrics, Elsevier, vol. 142(1), pages 201-211, January.
    2. Jianqing Fan & Runze Li, 2004. "New Estimation and Model Selection Procedures for Semiparametric Modeling in Longitudinal Data Analysis," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 710-723, January.
    3. Wang, Hansheng & Li, Guodong & Jiang, Guohua, 2007. "Robust Regression Shrinkage and Consistent Variable Selection Through the LAD-Lasso," Journal of Business & Economic Statistics, American Statistical Association, vol. 25, pages 347-355, July.
    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. Pötscher, Benedikt M., 2007. "Confidence Sets Based on Sparse Estimators Are Necessarily Large," MPRA Paper 5677, University Library of Munich, Germany.
    6. 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.
    7. Leeb, Hannes & Pötscher, Benedikt M., 2003. "The Finite-Sample Distribution Of Post-Model-Selection Estimators And Uniform Versus Nonuniform Approximations," Econometric Theory, Cambridge University Press, vol. 19(1), pages 100-142, February.
    8. 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.
    9. Ming Yuan & Yi Lin, 2007. "Model selection and estimation in the Gaussian graphical model," Biometrika, Biometrika Trust, vol. 94(1), pages 19-35.
    10. Pötscher, Benedikt M. & Leeb, Hannes, 2009. "On the distribution of penalized maximum likelihood estimators: The LASSO, SCAD, and thresholding," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 2065-2082, October.
    11. 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.
    12. Wang, Hansheng & Leng, Chenlei, 2007. "Unified LASSO Estimation by Least Squares Approximation," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 1039-1048, September.
    13. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    14. Hao Helen Zhang & Wenbin Lu, 2007. "Adaptive Lasso for Cox's proportional hazards model," Biometrika, Biometrika Trust, vol. 94(3), pages 691-703.
    15. Hansheng Wang & Guodong Li & Chih‐Ling Tsai, 2007. "Regression coefficient and autoregressive order shrinkage and selection via the lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(1), pages 63-78, February.
    16. Hansheng Wang & Runze Li & Chih-Ling Tsai, 2007. "Tuning parameter selectors for the smoothly clipped absolute deviation method," Biometrika, Biometrika Trust, vol. 94(3), pages 553-568.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Ulrike Schneider & Martin Wagner, 2012. "Catching Growth Determinants with the Adaptive Lasso," German Economic Review, Verein für Socialpolitik, vol. 13(1), pages 71-85, February.
    2. Hui, Francis K.C. & Müller, Samuel & Welsh, A.H., 2020. "The LASSO on latent indices for regression modeling with ordinal categorical predictors," Computational Statistics & Data Analysis, Elsevier, vol. 149(C).
    3. Leeb, Hannes & Pötscher, Benedikt M. & Ewald, Karl, 2014. "On various confidence intervals post-model-selection," MPRA Paper 52858, University Library of Munich, Germany.
    4. Hui Xiao & Yiguo Sun, 2019. "On Tuning Parameter Selection in Model Selection and Model Averaging: A Monte Carlo Study," JRFM, MDPI, vol. 12(3), pages 1-16, June.
    5. Ruth M. Pfeiffer & Andrew Redd & Raymond J. Carroll, 2017. "On the impact of model selection on predictor identification and parameter inference," Computational Statistics, Springer, vol. 32(2), pages 667-690, June.
    6. Ulrike Schneider, 2016. "Confidence Sets Based on Thresholding Estimators in High-Dimensional Gaussian Regression Models," Econometric Reviews, Taylor & Francis Journals, vol. 35(8-10), pages 1412-1455, December.
    7. Leeb, Hannes & Pötscher, Benedikt M. & Kivaranovic, Danijel, 2018. "Comment on "Model Confidence Bounds for Variable Selection" by Yang Li, Yuetian Luo, Davide Ferrari, Xiaonan Hu, and Yichen Qin," MPRA Paper 90655, University Library of Munich, Germany.
    8. Bruce E. Hansen, 2016. "The Risk of James--Stein and Lasso Shrinkage," Econometric Reviews, Taylor & Francis Journals, vol. 35(8-10), pages 1456-1470, December.
    9. Xianyi Wu & Xian Zhou, 2019. "On Hodges’ superefficiency and merits of oracle property in model selection," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(5), pages 1093-1119, October.
    10. Audrino, Francesco & Camponovo, Lorenzo & Roth, Constantin, 2015. "Testing the lag structure of assets’ realized volatility dynamics," Economics Working Paper Series 1501, University of St. Gallen, School of Economics and Political Science.
    11. David Cheng & Abhishek Chakrabortty & Ashwin N. Ananthakrishnan & Tianxi Cai, 2020. "Estimating average treatment effects with a double‐index propensity score," Biometrics, The International Biometric Society, vol. 76(3), pages 767-777, September.
    12. Kun Chen & Kung-Sik Chan & Nils Chr. Stenseth, 2014. "Source-Sink Reconstruction Through Regularized Multicomponent Regression Analysis-With Application to Assessing Whether North Sea Cod Larvae Contributed to Local Fjord Cod in Skagerrak," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(506), pages 560-573, June.
    13. Pötscher, Benedikt M. & Schneider, Ulrike, 2008. "Confidence sets based on penalized maximum likelihood estimators," MPRA Paper 9062, University Library of Munich, Germany.
    14. Gabriela Ciuperca, 2014. "Model selection by LASSO methods in a change-point model," Statistical Papers, Springer, vol. 55(2), pages 349-374, May.

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    More about this item

    Keywords

    Penalized maximum likelihood; LASSO; adaptive LASSO; nonnegative garotte; finite-sample distribution; asymptotic distribution; oracle property; estimation of distribution; uniform consistency;
    All these keywords.

    JEL classification:

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

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