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Inference in high-dimensional linear regression models

Listed author(s):
  • Tom Boot

    ()

    (University of Groningen, The Netherlands)

  • Didier Nibbering

    ()

    (Erasmus University Rotterdam, The Netherlands)

Registered author(s):

    We introduce an asymptotically unbiased estimator for the full high-dimensional parameter vector in linear regression models where the number of variables exceeds the number of available observations. The estimator is accompanied by a closed-form expression for the covariance matrix of the estimates that is free of tuning parameters. This enables the construction of confidence intervals that are valid uniformly over the parameter vector. Estimates are obtained by using a scaled Moore-Penrose pseudoinverse as an approximate inverse of the singular empirical covariance matrix of the regressors. The approximation induces a bias, which is then corrected for using the lasso. Regularization of the pseudoinverse is shown to yield narrower confidence intervals under a suitable choice of the regularization parameter. The methods are illustrated in Monte Carlo experiments and in an empirical example where gross domestic product is explained by a large number of macroeconomic and financial indicators.

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    File URL: http://papers.tinbergen.nl/17032.pdf
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    Paper provided by Tinbergen Institute in its series Tinbergen Institute Discussion Papers with number 17-032/III.

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    Date of creation: 14 Mar 2017
    Date of revision: 05 Jul 2017
    Handle: RePEc:tin:wpaper:20170032
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    1. Carmen Fernandez & Eduardo Ley & Mark F. J. Steel, 2001. "Model uncertainty in cross-country growth regressions," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 16(5), pages 563-576.
    2. Barro, Robert J. & Lee, Jong-Wha, 1993. "International comparisons of educational attainment," Journal of Monetary Economics, Elsevier, vol. 32(3), pages 363-394, December.
    3. Chikuse, Yasuko, 1990. "The matrix angular central Gaussian distribution," Journal of Multivariate Analysis, Elsevier, vol. 33(2), pages 265-274, May.
    4. Victor Chernozhukov & Christian Hansen & Martin Spindler, 2015. "Valid Post-Selection and Post-Regularization Inference: An Elementary, General Approach," Annual Review of Economics, Annual Reviews, vol. 7(1), pages 649-688, August.
    5. Mehmet Caner & Anders Bredahl Kock, 2014. "Asymptotically Honest Confidence Regions for High Dimensional Parameters by the Desparsified Conservative Lasso," CREATES Research Papers 2014-36, Department of Economics and Business Economics, Aarhus University.
    6. Tingni Sun & Cun-Hui Zhang, 2012. "Scaled sparse linear regression," Biometrika, Biometrika Trust, vol. 99(4), pages 879-898.
    7. 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.
    8. Stock J.H. & Watson M.W., 2002. "Forecasting Using Principal Components From a Large Number of Predictors," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 1167-1179, December.
    9. Cun-Hui Zhang & Stephanie S. Zhang, 2014. "Confidence intervals for low dimensional parameters in high dimensional linear models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 76(1), pages 217-242, January.
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