For the artificial regression given in (11.55): (a) Show that OLS on this model yields 1,OLS =1,IV = (Z 1PWZ1)1Z 1PWy1. Hint: Y1 Y1 =

For the artificial regression given in (11.55):

(a) Show that OLS on this model yieldsδ 1,OLS =δ1,IV = (Z

1PWZ1)−1Z

1PWy1. Hint: Y1− ˆY1 =

¯ PWY1. Use the FWL Theorem to residual out these variables in (11.55) and use the fact that

¯ PWZ1 = [ ¯ PWY1, 0].

(b) Show that the var(δ 1,OLS) = s11(Z

1PWZ1)−1 where s11 is the mean squared error of the OLS regression in (11.55). Note that when η = 0 in (11.55), IV estimation is necessary and

s11 underestimates σ11 and will have to be replaced by (y1 − Z1δ 1,IV )(y1 − Z1δ1,IV )/T .