16-14. Prove the Gauss-Markov Theorem. Hint: (1) First express 0, 1 as linear functions of...

16-14. Prove the Gauss-Markov Theorem. Hint:

(1) First express ˆ β0, ˆ β1 as linear functions of the Yi, i = 1, . . . , n; that is, write

ˆ β1 =

n

i=1 wiYi, where wi = xi n



i=1 x2 i

;

ˆ β0 =

n

i=1 viYi, where vi = 1 n

Xwi.

(2) Write ˆ β1 = ni

=1 xiyi>ni

=1 x2 i

= β1 + wiεi and ˆ β0 = β0 +

(β1 − ˆ β1)X + ¯ε. Then find E( ˆ β1), E( ˆ β0).

(3) Determine the variances of ˆ β, ˆ β1 as V( ˆ β1) = E ( ˆ β1 − β1)2 = σ2

ε @

n

i=1 x2 i ;

V( ˆ β0) = E ( ˆ β0 − β0)2 = σ2

ε

1 n

+

X2

ni

=1 x2 i



(given that E[( ˆ β1 − β1¯ε ] = 0 and E(¯ε 2) = σ2

ε /n).

(4) Let β



1

= ni

=1 aiYi be an alternative linear estimator of β1 and verify that E(β



1) = β1 under the restrictionsni

=1 ai = 0 andni

=1 aiXi = 1.

(5) Given these restrictions, β



1

= β1 +ni

=1 aiεi and V(β



1) = σ2

ε ni

=1 a2i

.

(6) To compare β



1 with ˆ β1, set ai = wi + di, di constant for all i. Then verify that V(β



1) = V( ˆ β1) + σ2

ε ni

=1 d2 i

≥ V( ˆ β1). (A similar line of argumentation holds for ˆ β0.)