## Weighted Least Squares. This is based on Kmenta (1986)

a. From the first equation in (5.11), one could solve for a

n n n

5 ід2) = V°?) -" Xi/°2).

i=l i=1 i=1

i=1

n n n n

Yi/o? 1/* -" X,/ o2 1/o2)

Li=1 i=1 i=1 i=1

= Y* – "x*.

Substituting a in the second equation of (5.11) one gets

2

Xi/o? 1/o?) C" X2/o?).

i= 1 i=1

n

Multiplying both sides by (1/o?) and solving for " one gets (5.12b)

i= 1

pOA? P(Y’X’/o? – Lj(4o? PAA)

£ wi* (Xi — x*)2 i=1 |

Subtract this equation from the original regression equation to get Yi—Y* = "(Xi — X*) + (ui — it*). Substitute this in the expression for p in (5.12b), we get

n n

£ Wi* (Xi — X ) (ui — u*) £ Wi* (Xi — X )ui

£ Wi* (Xi — X*)[1] i=1 |

Ewi * (Xi — x*)2 i=1 |

where the second equality uses the fact that n n /n/n Ew* (Xi—X*) = E w*Xi— (E w* E w*Xi /Ew? i=... |