(Instrumental variable estimation). Consider the model Yi = Xi + ui, where Xi and are K 1. Assuming that E(uiXi) = 0 so that Xi is endogenous. Further assume that there exists an instrumental variable (IV) Zi, of dimension K 1, such that E(uiZi) = 0 and E(XiZi) is non-singular. That is, the instrumental variable is uncorrelated with disturbance ui, and correlated with the regressor xi.

(a) Show that = [E(ZiXi)]1E(ZiYi) (show your derivations). (b) From the result from (a), we can estimate by IV = [n1 ni=1 ZiXi]1n1 ni=1 ZiYi =

(ZX)1ZY . Assuming iid data, show that IV p . (c) Derive the asymptotic distribution of n(IV ). (Hint: using LLN and CLT).

(d) Provide a consistent estimator for the asymptotic variance of n(IV ).

6. (Instrumental variable estimation). Consider the model Yi = X:B + ui, where Xi and B are K x 1. Assuming that E(u;Xi) * 0 so that X; is endogenous. Further assume that there exists an instrumental variable (IV) Zi, of dimension K x 1, such that E(uiZi) = 0 and E(X;Z':) is non-singular. That is, the instrumental variable is uncorrelated with disturbance us, and correlated with the regressor Ci. (a) Show that = [E(Z;X:)]-'E(ZY;) (show your derivations). (b) From the result from (a), we can estimate S by Biv = [n-'Et, Z;X:]-in 'Eh ZiYi = (Z'X) -1Z'Y. Assuming iid data, show that BIV " B. (c) Derive the asymptotic distribution of Vn(Biv - B). (Hint: using LLN and CLT). (d) Provide a consistent estimator for the asymptotic variance of Vn(BIV - B)