1. Stock and Watson, #12.2. Consider the regression model with a single regressor: Y = Bo + BX; +ui. Suppose that the least squares assumptions in Key Concept 4.3 are satisfied. a. Show that X; is a valid instrument. That is, show that Key Concept 12.3 is satisfied with Zi = Xi. b. Show that the IV regression assumptions in Key Concept 12.4 are satisfied with this choice of Zi. c. Show that the IV estimator constructed using Zi === X; is identical to the OLS estimator. 2. Stock and Watson, #12.5. Consider the instrumental variable regression model YiBo+B1X; + BW; +ui where X; is correlated with u, and Z; is an instrument. Suppose that the first three assumptions in Key Concept 12.4 are satisfied. Which IV assumption is not satisfied when: a. Z; is independent of (Yi, Xi, W;)? b. Zi = Wi? c. W; 1 for all i? = d. Z; = X;? 3. Stock and Watson, #12.10 Consider the instrumental variable regression model Y = Bo+ BX; + BW; +ui, where Zi is an instrument. Suppose that data on W; are not available and the model is estimated omitting Wi from the regression. a. Suppose that Z; and W; are uncorrelated. Is the IV estimator consistent? b. Suppose that Z; and W; are correlated. Is the IV estimator consistent?