Suppose that the multiple linear regression model yi=xiT+ei holds for i=1,,n, where the ei are independent errors, with E(eiX=xi)=0 but, unlike the standard setting, with different variances Var( eiX=xi)=i2. a) Show that the OLS estimate ^=(XTX)1XTY is still an unbiased estimate of . That is, show that E[^X]= under the model assumptions. b) Write the variance matrix of the errors ei as a diagonal matrix, say , with diagonal elements (12,,n2). Show that Var(^)=(XTX)1XTX(XTX)1..

1. Suppose that the multiple linear regression model yi=xiT+ei holds for i=1,,n, where the ei are
independent errors, with E(eiX=xi)=0 but, unlike the standard setting, with different variances Var(
eiX=xi)=i2. a) Show that the OLS estimate ^=(XTX)1XTY is still an unbiased estimate of . That is,
show that E[^X]= under the model assumptions. b) Write the variance matrix of the errors ei as a
diagonal matrix, say , with diagonal elements (12,,n2). Show that Var(^)=(XTX)1XTX(XTX)1.