  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.
