Prediction Error Variances Under Heteroskedasticity. This is based on Termayne (1985). Consider the t-th observation of the linear regression model given in (9.1).

yt = x



tβ + ut t = 1, 2, . . . , T where yt is a scalar x

t is 1 × K and β is a K × 1 vector of unknown coefficients. ut is assumed to have zero mean, heteroskedastic variances E(u2t

) = (z

tγ)2 where z

t is a 1 × r vector of observed variables and γ is an r×1 vector of parameters. Furthermore, these ut’s are not serially correlated, so that E(utus) = 0 for t = s.

(a) Find the var(βOLS) and var(βGLS) for this model.

(b) Suppose we are forecasting y for period f in the future knowing xf , i.e., yf = x

fβ +uf with f > T. Let ef and ef be the forecast errors derived using OLS and GLS, respectively. Show that the prediction error variances of the point predictions of yf are given by var(ef) = x



f (

T t=1 xtx



t)−1[

T t=1 xtx



t(z



tγ)2](

T t=1 xtx



t)−1xf + (z



fγ)2 var(ef) = x



f [

T t=1 xtx



t(z



tγ)2]−1xf + (z



fγ)2

(c) Show that the variances of the two forecast errors of conditional mean E(yf/xf) based upon βOLS and βGLS and denoted by cf and cf , respectively are the first two terms of the corresponding expressions in part (b).

(d) Now assume that K = 1 and r = 1 so that there is only one single regressor xt and one zt variable determining the heteroskedasticity. Assume also for simplicity that the empirical moments of xt match the population moments of a Normal random variable with mean zero and variance θ. Show that the relative efficiency of the OLS to the GLS predictor of yf is equal to (T +1)/(T + 3), whereas the relative efficiency of the corresponding ratio involving the two predictions of the conditional mean is (1/3).