Best Linear Prediction. (Problems 16 and 17 are based on Amemiya (1994)). Let X and Y be two random variables with means X and Y

Best Linear Prediction. (Problems 16 and 17 are based on Amemiya (1994)). Let X and Y be two random variables with means μX and μY and variances σ2 X and σ2 Y , respectively. Suppose that

ρ = correlation(X, Y ) = σXY /σXσY where σXY = cov(X, Y ). Consider the linear relationship Y = α+βX where α and β are scalars:

(a) Show that the best linear predictor of Y based on X, where best in this case means the minimum mean squared error predictor which minimizes E(Y −α−βX)2 with respect to α

and β is given by Y = α + βX where α = μY

− βμX and β = σXY /σ2 X = ρσY /σX.

(b) Show that the var(Y ) = ρ2σ2 Y and that u = Y − Y , the prediction error, has mean zero and variance equal to (1 − ρ2)σ2 Y . Therefore, ρ2 can be interpreted as the proportion of σ2 Y that is explained by the best linear predictor Y .

(c) Show that cov(Y , u) = 0.