A Multiple-Index Model and Dimension Reduction

Abstract

Dimension reduction can be used as an initial step in statistical modeling. Further specification of model structure is imminent and important when the reduced dimension is still greater than 1. In this article we investigate one method of specification that involves separating the linear component from the nonlinear components, leading to further dimension reduction in the unknown link function and, thus, better estimation and easier interpretation of the model. The specified model includes the popular econometric multiple-index model and the partially linear single-index model as its special cases. A criterion is developed to validate the model specification. An algorithm is proposed to estimate the model directly. Asymptotic distributions for the estimators of the parameters and the nonparametric link function are derived. Air pollution data in Chicago are used to illustrate the modeling procedure and to demonstrate its advantages over the existing dimension reduction approaches.