Consider the regression setup where Y=b'0+E, b

(b, bp) being a vector of regressor variables, 0=(0,,....,)' being a vector of unknown regression coefficients, and a being a N(0, 2) random error (known, for simplicity). Some data, X, is available to estimate 0; let 6(x) denote the estimator. The goal of the investigation, however, is to predict future (independent) values of Y arising from this model. Indeed, such Y will be predicted (for each corresponding

b) by =b'6(X), and the loss in estimating Y by is squared-error prediction loss, (Y-).

(a) Show, for a given

b, that choice of 6(x) in the prediction problem is equivalent to choice of 6(x) in the problem of estimating under loss L(0,6)=(0-6) bb' (0-6).

(b) Suppose b N, (0, Q) (independent of future Y and past X). Show that the prediction problem is equivalent to the problem of estimating 0 under loss L(0,6) (0-6)'Q(0-8).