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Testing covariates in high-dimensional regression

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  • Wei Lan
  • Hansheng Wang
  • Chih-Ling Tsai

Abstract

In a high-dimensional linear regression model, we propose a new procedure for testing statistical significance of a subset of regression coefficients. Specifically, we employ the partial covariances between the response variable and the tested covariates to obtain a test statistic. The resulting test is applicable even if the predictor dimension is much larger than the sample size. Under the null hypothesis, together with boundedness and moment conditions on the predictors, we show that the proposed test statistic is asymptotically standard normal, which is further supported by Monte Carlo experiments. A similar test can be extended to generalized linear models. The practical usefulness of the test is illustrated via an empirical example on paid search advertising. Copyright The Institute of Statistical Mathematics, Tokyo 2014

Suggested Citation

  • Wei Lan & Hansheng Wang & Chih-Ling Tsai, 2014. "Testing covariates in high-dimensional regression," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(2), pages 279-301, April.
    • Handle: RePEc:spr:aistmt:v:66:y:2014:i:2:p:279-301
    DOI: 10.1007/s10463-013-0414-0
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    References listed on IDEAS

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    1. Zhong, Ping-Shou & Chen, Song Xi, 2011. "Tests for High-Dimensional Regression Coefficients With Factorial Designs," Journal of the American Statistical Association, American Statistical Association, vol. 106(493), pages 260-274.
    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, vol. 96, pages 1348-1360, December.
    3. Chen, Song Xi & Zhang, Li-Xin & Zhong, Ping-Shou, 2010. "Tests for High-Dimensional Covariance Matrices," Journal of the American Statistical Association, American Statistical Association, vol. 105(490), pages 810-819.
    4. Wang, Hansheng, 2009. "Forward Regression for Ultra-High Dimensional Variable Screening," Journal of the American Statistical Association, American Statistical Association, vol. 104(488), pages 1512-1524.
    5. Fan, Jianqing & Fan, Yingying & Lv, Jinchi, 2008. "High dimensional covariance matrix estimation using a factor model," Journal of Econometrics, Elsevier, vol. 147(1), pages 186-197, November.
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    Cited by:

    1. Ma, Yingying & Lan, Wei & Wang, Hansheng, 2015. "Testing predictor significance with ultra high dimensional multivariate responses," Computational Statistics & Data Analysis, Elsevier, vol. 83(C), pages 275-286.
    2. Lan, Wei & Ding, Yue & Fang, Zheng & Fang, Kuangnan, 2016. "Testing covariates in high dimension linear regression with latent factors," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 25-37.
    3. Yata, Kazuyoshi & Aoshima, Makoto, 2016. "High-dimensional inference on covariance structures via the extended cross-data-matrix methodology," Journal of Multivariate Analysis, Elsevier, vol. 151(C), pages 151-166.
    4. Rui Wang & Xingzhong Xu, 2021. "A Bayesian-motivated test for high-dimensional linear regression models with fixed design matrix," Statistical Papers, Springer, vol. 62(4), pages 1821-1852, August.
    5. Bin Guo & Song Xi Chen, 2016. "Tests for high dimensional generalized linear models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(5), pages 1079-1102, November.
    6. Ma, Yingying & Lan, Wei & Wang, Hansheng, 2015. "A high dimensional two-sample test under a low dimensional factor structure," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 162-170.

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