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Pivotal estimation via square-root lasso in nonparametric regression

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  • Alexandre Belloni
  • Victor Chernozhukov

    ()
    (Institute for Fiscal Studies and MIT)

  • Lie Wang

Abstract

We propose a self-tuning √ Lasso method that simultaneiously resolves three important practical problems in high-dimensional regression analysis, namely it handles the unknown scale, heteroscedasticity, and (drastic) non-Gaussianity of the noise. In addition, our analysis allows for badly behaved designs, for example perfectly collinear regressors, and generates sharp bounds even in extreme cases, such as the infinite variance case and the noiseless case, in contrast to Lasso. We establish various non-asymptotic bounds for √ Lasso including prediction norm rate and sharp sparcity bound. Our analysis is based on new impact factors that are tailored to establish prediction rates. In order to cover heteroscedastic non-Gaussian noise, we rely on moderate deviation theory for self-normalized sums to achieve Gaussian-like results under weak conditions. Moreover, we derive bounds on the performance of ordinary least square (ols) applied to the model selected by √ Lasso accounting for possible misspecification of the selected model. Under mild conditions the rate of convergence of ols post √ Lasso is no worse than √ Lasso even with a misspecified selected model and possibly better otherwise. As an application, we consider the use of √ Lasso and post √ Lasso as estimators of nuisance parameters in a generic semi-parametric problem (nonlinear instrumental/moment condition or Z-estimation problem).

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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 CWP62/13.

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Date of creation: Dec 2013
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Handle: RePEc:ifs:cemmap:62/13

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  1. 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.
  2. A. Belloni & D. Chen & Victor Chernozhukov & Christian Hansen, 2010. "Sparse models and methods for optimal instruments with an application to eminent domain," CeMMAP working papers CWP31/10, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  3. Tingni Sun & Cun-Hui Zhang, 2012. "Scaled sparse linear regression," Biometrika, Biometrika Trust, vol. 99(4), pages 879-898.
  4. Amemiya, Takeshi, 1977. "The Maximum Likelihood and the Nonlinear Three-Stage Least Squares Estimator in the General Nonlinear Simultaneous Equation Model," Econometrica, Econometric Society, vol. 45(4), pages 955-68, May.
  5. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-54, July.
  6. Eric Gautier & Alexandre Tsybakov, 2011. "High-Dimensional Instrumental Variables Regression and Confidence Sets," Working Papers 2011-13, Centre de Recherche en Economie et Statistique.
  7. A. Belloni & V. Chernozhukov & L. Wang, 2011. "Square-root lasso: pivotal recovery of sparse signals via conic programming," Biometrika, Biometrika Trust, vol. 98(4), pages 791-806.
  8. repec:cup:cbooks:9780521496032 is not listed on IDEAS
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