2. Consider the regression model y=+x+u where x is an endogenous variable. a) Write xi=x^i+e^i and show that x^=x. b) Suppose there are k valid instruments: z1,z2,, Z k. Let x^i=^0+^1zi1+^2zi2++^kzik. Show that the 2SLS estimator is equivalent to an IV estimator using x^ as an instrument: i=1n(x^ix^)2i=1n(x^ix)(yiy)=i=1n(x^ix^)(xix)i=1n(x^ix^)(yiy). [Hint: OLS first order conditions imply i=1Ne^ix^i=0 and i=1Ne^i=0.] c) Suppose there is only one instrument z and x^i=^0+^1zi. Show that the 2 SLS estimator is numerically identical to the IV estimator: i=1n(x^ix)2i=1n(x^ix^)(yiy)=i=1n(ziz)(xix)i=1n(ziz)(yiy). 2. Consider the regression model y=+x+u where x is an endogenous variable. a) Write xi=x^i+e^i and show that x^=x. b) Suppose there are k valid instruments: z1,z2,, Z k. Let x^i=^0+^1zi1+^2zi2++^kzik. Show that the 2SLS estimator is equivalent to an IV estimator using x^ as an instrument: i=1n(x^ix^)2i=1n(x^ix)(yiy)=i=1n(x^ix^)(xix)i=1n(x^ix^)(yiy). [Hint: OLS first order conditions imply i=1Ne^ix^i=0 and i=1Ne^i=0.] c) Suppose there is only one instrument z and x^i=^0+^1zi. Show that the 2 SLS estimator is numerically identical to the IV estimator: i=1n(x^ix)2i=1n(x^ix^)(yiy)=i=1n(ziz)(xix)i=1n(ziz)(yiy)