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

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.