Multivariate flattening for better pre- dictions

Multivariate regression with p responses as opposed to p multiple regressions is getting increasingly more attention, especially in the context of prediction. Multivariate flattening methods are investigated as a way to obtain improved predictions over ordinary least-squares. With respect to sum of squares of prediction error, or SPE risk, an unbiased estimate of the risk is derived for two recent prediction methods, OPT and GCV, proposed by Breiman & Friedman (1997). Expressions for the exact SPE risk of OPT and GCV are derived generally for p ≥ 1 and evaluated numerically for p = 1.