Using the regression given in equation (3.1): (a) Prove that OLS = n i=1 iYi where i = (1/n) Xwi and wi = xi/

Using the regression given in equation (3.1):

(a) Prove that αOLS =

n i=1 λiYi where λi = (1/n) − X¯wi and wi = xi/

n i=1 x2i

.

(b) Show that

n i=1 λi = 1 and

n i=1 λiXi = 0.

(c) Prove that any other linear estimator of α, say α =

n i=1 biYi must satisfy

n i=1  bi = 1 and n i=1 biXi = 0 for α to be unbiased for α.

(d) Let bi = λi + fi; show that

n i=1 fi = 0 and

n i=1 fiXi = 0.

(e) Prove that var(α) = σ2n i=1 b2i

= σ2n i=1 λ2i

+ σ2n i=1 f2 i =var(αOLS) + σ2n i=1 f2 i .