(i) Consider the simple regression model y 5 b0 1 b1x 1 u under the first four Gauss-Markov assumptions. For some function g(x), for example g1x2 5 x2 or g1x2 5 log11 1 x2 2, define zi 5 g1xi 2. Define a slope estimator as b

|

1 5 a a n

i51 1zi 2 z2yib^a a n

i51 1zi 2 z2xib.

Show that b

|

1 is linear and unbiased. Remember, because E(u|x) = 0, you can treat both xi and zi as nonrandom in your derivation.

(ii) Add the homoskedasticity assumption, MLR.5. Show that Var1b

|

1 2 5 s2 a a n

i51 1zi 2 z2 2 b^a a n

i51 1zi 2 z2xib 2

.

(iii) Show directly that, under the Gauss-Markov assumptions, Var1b^

1 2 # Var1b

|

1 2, where b^

1 is the OLS estimator. [Hint: The Cauchy-Schwartz inequality in Appendix B implies that an21 a

n i51 1zi 2 z2 1xi 2 x2 b 2

# an21 a

n i51 1zi 2 z2 2 b an21 a

n i51 1xi 2 x2 2 b;

notice that we can drop x from the sample covariance.]