The model is Yi X0 iei E ei j Xi 0 E e2 i j Xi 2i diag 21 , ,2 n The parameter is estimated by OLS b X 0X 1 X 0Y and GLS e X 01X 1 X 01Y Let be Y X b and e Y X e denote the residuals Let b R2 1be0be (Y 0Y ) and e R2 1ee0ee (Y 0Y ) denote the equation R2 where Y Y Y If the error ei is truly heteroskedastic will b R2 or e R2 be smaller

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Question: The model is Yi X0 iei E[ei j Xi ] 0 E e2 i j Xi 2i diag{21

The model is Yi Æ X0 i¯Åei E[ei j Xi ] Æ 0 E

e2 i

j Xi

Æ ¾2i

§ Æ diag{¾21

, ...,¾2 n}.

The parameter ¯ is estimated by OLS b¯ Æ

X 0X

¢¡1 X 0Y and GLS e¯ Æ

X 0§¡1X

¢¡1 X 0§¡1Y . Let be Æ Y ¡X b¯

and e Æ Y ¡ X e¯ denote the residuals. Let b R2 Æ 1¡be0be/(Y ¤0Y ¤) and e R2 Æ 1¡ee0ee/(Y ¤0Y ¤) denote the equation R2 where Y ¤ Æ Y ¡Y . If the error ei is truly heteroskedastic will b R2 or e R2 be smaller?

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