'Weighted-least-squares estimation: Suppose that the errors from the linear regression model y X fl are independent and normally distributed, but with different

'Weighted-least-squares estimation: Suppose that the errors from the linear regression model y ¼ X fl þ " are independent and normally distributed, but with different variances, εi ; Nð0; σ2 i Þ, and that σ2 i ¼ σ2

ε=w2 i . Show that:

(a) The likelihood for the model is Lðfl; σ2

ε Þ ¼ 1

ð2πÞ

n=2 jSj 1=2 exp " 1 2

ðy " X flÞ

0 Sðy " X flÞ

" #

where S ¼ σ2

ε · diagf1=w2 1; ... ; 1=w2 ng [ σ2

εW"1

(b) The maximum-likelihood estimators of fl and σ2

ε are flb ¼ ðX0 WXÞ

"1 X0 Wy

σb2

ε ¼

PðEi=wiÞ

2 n

where e ¼ fEig ¼ y " Xflb.

(c) The MLE is equivalent to minimizing the weighted sum of squares Pw2 i E2 i .

(d) The estimated asymptotic covariance matrix of flb is given by VbðβbÞ ¼ σb2

ε ðX0 WXÞ

"1