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Oracle inequalities for high-dimensional panel data models

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  • Anders Bredahl Kock

    ()
    (Aarhus University and CREATES)

Abstract

This paper is concerned with high-dimensional panel data models where the number of regressors can be much larger than the sample size. Under the assumption that the true parameter vector is sparse we establish finite sample upper bounds on the estimation error of the Lasso under two different sets of conditions on the covariates as well as the error terms. Upper bounds on the estimation error of the unobserved heterogeneity are also provided under the assumption of sparsity. Next, we show that our upper bounds are essentially optimal in the sense that they can only be improved by multiplicative constants. These results are then used to show that the Lasso can be consistent in even very large models where the number of regressors increases at an exponential rate in the sample size. Conditions under which the Lasso does not discard any relevant variables asymptotically are also provided. In the second part of the paper we give lower bounds on the probability with which the adaptive Lasso selects the correct sparsity pattern in finite samples. These results are then used to give conditions under which the adaptive Lasso can detect the correct sparsity pattern asymptotically. We illustrate our finite sample results by simulations and apply the methods to search for covariates explaining growth in the G8 countries.

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File URL: ftp://ftp.econ.au.dk/creates/rp/13/rp13_20.pdf
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Bibliographic Info

Paper provided by School of Economics and Management, University of Aarhus in its series CREATES Research Papers with number 2013-20.

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Length: 37
Date of creation: 06 Dec 2013
Date of revision:
Handle: RePEc:aah:create:2013-20

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Web page: http://www.econ.au.dk/afn/

Related research

Keywords: Panel data; Lasso; Adaptive Lasso; Oracle inequality; Nonasymptotic bounds; High-dimensional models; Sparse models; Consistency; Variable selection; Asymptotic sign consistency.;

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  1. 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.
  2. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
  3. Robert J. Barro, 1989. "Economic Growth in a Cross Section of Countries," NBER Working Papers 3120, National Bureau of Economic Research, Inc.
  4. Jianqing Fan & Jinchi Lv, 2008. "Sure independence screening for ultrahigh dimensional feature space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(5), pages 849-911.
  5. Zou, Hui, 2006. "The Adaptive Lasso and Its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, American Statistical Association, vol. 101, pages 1418-1429, December.
  6. A. Belloni & V. Chernozhukov & L. Wang, 2011. "Square-root lasso: pivotal recovery of sparse signals via conic programming," Biometrika, Biometrika Trust, Biometrika Trust, vol. 98(4), pages 791-806.
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Cited by:
  1. Kaufmann, Hendrik & Kruse, Robinson & Sibbertsen, Philipp, 2012. "On tests for linearity against STAR models with deterministic trends," Economics Letters, Elsevier, vol. 117(1), pages 268-271.
  2. Bent Jesper Christensen & Morten Ørregaard Nielsen & Jie Zhu, 2012. "The impact of financial crises on the risk-return tradeoff and the leverage effect," CREATES Research Papers 2012-19, School of Economics and Management, University of Aarhus.

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