  Prediction, in regression and classification, is one of the main aims in
modern data science. When the number of predictors is large, a common first
step is to reduce the dimension of the data. Sufficient dimension reduction
(SDR) is a well established paradigm of reduction that keeps all the relevant
information in the covariates X that is necessary for the prediction of Y . In
practice, SDR has been successfully used as an exploratory tool for modelling
after estimation of the sufficient reduction. Nevertheless, even if the
estimated reduction is a consistent estimator of the population, there is no
theory that supports this step when non-parametric regression is used in the
imputed estimator. In this paper, we show that the asymptotic distribution of
the non-parametric regression estimator is the same regardless if the true SDR
or its estimator is used. This result allows making inferences, for example,
computing confidence intervals for the regression function avoiding the curse
of dimensionality.
