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On the residual empirical process based on the ALASSO in high dimensions and its functional oracle property

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  • Chatterjee, A.
  • Gupta, S.
  • Lahiri, S.N.

Abstract

This paper considers post variable-selection inference in a high dimensional penalized regression model based on the ALASSO method of Zou (2006). It is shown that under suitable sparsity conditions, the residual empirical process based on the ALASSO provides valid inference methodology in very high dimensional regression problems where conventional methods fail. It is also shown that the ALASSO based residual empirical process satisfies a functional oracle property, i.e., in addition to selecting the set of relevant variables with probability tending to one, the ALASSO based residual empirical process converges to the same limiting Gaussian process as the OLS based residual empirical process under the oracle. The functional oracle property is critically exploited to construct asymptotically valid confidence bands for the error distribution function and prediction intervals for unobserved values of the response variable in the high dimensional set up, where traditional non-penalized methods are known to fail. Simulation results are presented illustrating finite sample performance of the proposed methodology.

Suggested Citation

  • Chatterjee, A. & Gupta, S. & Lahiri, S.N., 2015. "On the residual empirical process based on the ALASSO in high dimensions and its functional oracle property," Journal of Econometrics, Elsevier, vol. 186(2), pages 317-324.
  • Handle: RePEc:eee:econom:v:186:y:2015:i:2:p:317-324
    DOI: 10.1016/j.jeconom.2015.02.012
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    References listed on IDEAS

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    1. Liao, Zhipeng & Phillips, Peter C. B., 2015. "Automated Estimation Of Vector Error Correction Models," Econometric Theory, Cambridge University Press, vol. 31(3), pages 581-646, June.
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    3. Eric Gautier & Alexandre Tsybakov, 2011. "High-Dimensional Instrumental Variables Regression and Confidence Sets," Working Papers 2011-13, Center for Research in Economics and Statistics.
    4. 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.
    5. Chatterjee, A. & Lahiri, S. N., 2011. "Bootstrapping Lasso Estimators," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 608-625.
    6. Alexandre Belloni & Victor Chernozhukov & Christian Hansen, 2011. "Inference for High-Dimensional Sparse Econometric Models," Papers 1201.0220, arXiv.org.
    7. Ming Yuan & Yi Lin, 2007. "Model selection and estimation in the Gaussian graphical model," Biometrika, Biometrika Trust, vol. 94(1), pages 19-35.
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    Cited by:

    1. Ning Xu & Jian Hong & Timothy C. G. Fisher, 2016. "Model selection consistency from the perspective of generalization ability and VC theory with an application to Lasso," Papers 1606.00142, arXiv.org.

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