  In this paper we study the effective degrees of freedom of a general class of
reduced rank estimators for multivariate regression in the framework of Stein's
unbiased risk estimation (SURE). We derive a finite-sample exact unbiased
estimator that admits a closed-form expression in terms of the singular values
or thresholded singular values of the least squares solution and hence readily
computable. The results continue to hold in the high-dimensional scenario when
both the predictor and response dimensions are allowed to be larger than the
sample size. The derived analytical form facilitates the investigation of its
theoretical properties and provides new insights into the empirical behaviors
of the degrees of freedom. In particular, we examine the differences and
connections between the proposed estimator and a commonly-used naive estimator,
i.e., the number of free parameters. The use of the proposed estimator leads to
efficient and accurate prediction risk estimation and model selection, as
demonstrated by simulation studies and a data example.
