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 vertical bar X, but that of Y vertical bar theta(T)X' 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(T)X, 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.