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Predictive power of principal components for single-index model and sufficient dimension reduction

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  • Artemiou, Andreas
  • Li, Bing

Abstract

In this paper we demonstrate that a higher-ranking principal component of the predictor tends to have a stronger correlation with the response in single index models and sufficient dimension reduction. This tendency holds even though the orientation of the predictor is not designed in any way to be related to the response. This provides a probabilistic explanation of why it is often beneficial to perform regression on principal components—a practice commonly known as principal component regression but whose validity has long been debated. This result is a generalization of earlier results by Li (2007) [19], Artemiou and Li (2009) [2], and Ni (2011) [24], where the same phenomenon was conjectured and rigorously demonstrated for linear regression.

Suggested Citation

  • Artemiou, Andreas & Li, Bing, 2013. "Predictive power of principal components for single-index model and sufficient dimension reduction," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 176-184.
    • Handle: RePEc:eee:jmvana:v:119:y:2013:i:c:p:176-184
    DOI: 10.1016/j.jmva.2013.04.015
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    References listed on IDEAS

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    1. Peter Hall & You‐Jun Yang, 2010. "Ordering and selecting components in multivariate or functional data linear prediction," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(1), pages 93-110, January.
    2. Powell, James L & Stock, James H & Stoker, Thomas M, 1989. "Semiparametric Estimation of Index Coefficients," Econometrica, Econometric Society, vol. 57(6), pages 1403-1430, November.
    3. Li, Bing & Wang, Shaoli, 2007. "On Directional Regression for Dimension Reduction," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 997-1008, September.
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    1. Qian Chen & Ghulam Rasool Madni & Adnan Ali Shahzad, 2023. "The usage of spatial econometric approach to explore the determinants of ecological footprint in BRI countries," PLOS ONE, Public Library of Science, vol. 18(10), pages 1-14, October.
    2. Cook, R. Dennis, 2022. "A slice of multivariate dimension reduction," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    3. Jones, Ben & Artemiou, Andreas, 2021. "Revisiting the predictive power of kernel principal components," Statistics & Probability Letters, Elsevier, vol. 171(C).
    4. Mohammadreza Ramezani & Leili Abolhassani & Naser Shahnoushi Foroushani & Diane Burgess & Milad Aminizadeh, 2022. "Ecological Footprint and Its Determinants in MENA Countries: A Spatial Econometric Approach," Sustainability, MDPI, vol. 14(18), pages 1-20, September.
    5. Ben Jones & Andreas Artemiou, 2020. "On principal components regression with Hilbertian predictors," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(2), pages 627-644, April.

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