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Uniform post selection inference for LAD regression models

  • Alexandre Belloni
  • Victor Chernozhukov

    ()

    (Institute for Fiscal Studies and MIT)

  • Kengo Kato

We develop uniformly valid confidence regions for a regression coefficient in a high-dimensional sparse LAD (least absolute deviation or median) regression model. The setting is one where the number of regressors p could be large in comparison to the sample size n, but only s « n of them are needed to accurately describe the regression function. Our new methods are based on the instrumental LAD regression estimator that assembles the optimal estimating equation from either post l 1- penalised LAD regression or l 1- penalised LAD regression. The estimating equation is immunised against non-regular estimation of nuisance part of the regression function, in the sense of Neyman. We establish that in a homoscedastic regression model, under certain conditions, the instrumental LAD regression estimator of the regression coefficient is asymptotically root-n normal uniformly with respect to the underlying sparse model. The resulting confidence regions are vaild uniformly with respect to the underlying model. The new inference methods outperform the naive, 'oracle based' inference methods, which are known to be not uniformly valid- with coverage property failing to hold uniformly with respect the underlying model- even in the setting with p= 2. We also provide Monte-Carlo experiments which demonstrate that standard post-selection inference breaks down over large parts of the parameter space, and the proposed method does not.

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File URL: http://www.cemmap.ac.uk/wps/cwp241313.pdf
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Paper provided by Centre for Microdata Methods and Practice, Institute for Fiscal Studies in its series CeMMAP working papers with number CWP24/13.

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Date of creation: Jun 2013
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Handle: RePEc:ifs:cemmap:24/13
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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. He, Xuming & Shao, Qi-Man, 2000. "On Parameters of Increasing Dimensions," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 120-135, April.
  3. Lee, Sokbae, 2003. "Efficient Semiparametric Estimation Of A Partially Linear Quantile Regression Model," Econometric Theory, Cambridge University Press, vol. 19(01), pages 1-31, February.
  4. Chernozhukov, Victor & Hansen, Christian, 2008. "Instrumental variable quantile regression: A robust inference approach," Journal of Econometrics, Elsevier, vol. 142(1), pages 379-398, January.
  5. A. Belloni & D. Chen & V. Chernozhukov & C. Hansen, 2012. "Sparse Models and Methods for Optimal Instruments With an Application to Eminent Domain," Econometrica, Econometric Society, vol. 80(6), pages 2369-2429, November.
  6. Powell, James L., 1986. "Censored regression quantiles," Journal of Econometrics, Elsevier, vol. 32(1), pages 143-155, June.
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