Using the regression given in equation (3.1): (a) Show that OLS = + ( OLS)X + u; and deduce that E(OLS) = .

Using the regression given in equation (3.1):

(a) Show that αOLS = α + (β − βOLS)¯X + ¯u; and deduce that E(αOLS) = α.

(b) Using the fact that βOLS

−β =

n i=1 xiui/

n i=1 x2i

; use the results in part

(a) to show that var(αOLS) = σ2[(1/n) + (¯X 2/

n i=1 x2i

)] = σ2n i=1 X2 i /n

n i=1 x2i

.

(c) Show that αOLS is consistent for α.

(d) Show that cov(αOLS, βOLS) = −¯Xvar(βOLS) = −σ2X¯ /

n i=1 x2i

. This means that the sign of the covariance is determined by the sign of ¯X . If ¯X is positive, this covariance will be negative. This also means that if αOLS is over-estimated, βOLS will be under-estimated.