Springer Texts in Business and Economics

Theil’s minimum mean square estimator of «. » can be written as

n / Г n

a. " = EXiYi/ EX? + (a2/"2) i=i 7 Li=i

Substituting Yi = "Xi + Ui we get

ре Xi? + e XiUi

" = -=—=—

P Xi2 + (a2/"2)

i=1

and since E(XiUi) = 0, we get

b.

Therefore, Bias(") = E(") — " = " (yC—) — " = —[c/(l + c)]". This bias is positive (negative) when " is negative (positive). This also means that " is biased towards zero.

n

a2L Xi2

i=1

2

EXi2 c (o2/p2)

.i = 1

using var(ui) = o2 and cov(ui, uj) = 0 for i ф j. This can also be written as var(Q) = - j—— . Therefore,

P Xi2(1+c)2 i = 1

MSE(Q) = Bias2(Q) + var(Q)

n

But "2 Xi2c = o2 from the definition of c. Hence,

i=1

n

The Bias(°ois) = 0 and var(°ois) = o2/ J] Xi2. Hence

i= 1

n

MSE(0 ok) = var(0 ois) = o2/£ Xi2.

i=1

This is larger than MSE(") since the latter has a positive constant (o2/"2) in the denominator.