Approximating conditional distribution functions using dimension   reduction
  Motivated by applications to prediction and forecasting, we suggest methods for approximating the conditional distribution function of a random variable Y given a dependent random d-vector X. The idea is to estimate not the distribution of Y|X, but that of Y|\theta^TX, where the unit vector \theta is selected so that the approximation is optimal under a least-squares criterion. We show that \theta may be estimated root-n consistently. Furthermore, estimation of the conditional distribution function of Y, given \theta^TX, has the same first-order asymptotic properties that it would enjoy if \theta were known. The proposed method is illustrated using both simulated and real-data examples, showing its effectiveness for both independent datasets and data from time series. Numerical work corroborates the theoretical result that \theta can be estimated particularly accurately. 
Peter Hall and Qiwei Yao