Springer Texts in Business and Economics

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

Подпись: Dividing both sides by (і/ст?) one getsi=1

n n n n

Подпись: a =Yi/o? 1/* -" X,/ o2 1/o2)

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

= Y* - "x*.

Подпись: i=1

Подпись: nn Yp 1/4) i=1 i=1
Подпись: YiXi/of = Xi/of
Подпись: i=1

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

2

Подпись: i=1Xi/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

Подпись: 2Subtract 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=1 i=1 i=1 / i=1

Therefore, E(") = " as expected, and

/ n 2

I P wi* (Xi — X )u|

P wi* (Xi — X*)2

i=1

Подпись: w* = 0.
image204 Подпись: i=1

P = P + ^------------------------------------- = P + ^-------------

b. From problem 5.3 part (b) we get

var(" blue) = —----------------------

P wi* (Xi - X*)2

i=1

_ n / n

where w* = (1/cr?) = (1/ct2X®) and X = P w’Xjpw* =

i=i! i=i

p (Xi/Xi8)

i=1 v______

n ’

1/Xi8

i=1

For Xi = 1,2,.., 10 and 8 = 0.5, 1, 1.5 and 2, this variance is tabulated below.

c. The relative efficiency of OLS is given by the ratio of the two variances computed in parts (a) and (b). This is tabulated below for various values

of 8.

8

var(" ols)

var(" BLUE)

var(" BLUE)/var(" ols)

0.5

0.0262 c2

0.0237 c2

0.905

1

0.0667 c2

0.04794 c2

0.719

1.5

0.1860 c2

0.1017 c2

0.547

2

0.5442 c2

0.224 c2

0.412

As 8 increases from 0.5 to 2, the relative efficiency of OLS with respect to BLUE decreases from 0.9 for mild heteroskedasticity to 0.4 for more serious heteroskedasticity.

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Springer Texts in Business and Economics

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