3. (Stock and Watson #13.10) Consider the regression model with heterogenous regression coefficients Yi=0i+1iXi+vi where (vi,Xi,0i,1i) are i.i.d random variables with 0=E(0i) and 1=E(1i). a. Show that the model can be written as Yi=0+1Xi+ui where ui=(0i0)+ (1i1)Xi+vi b. Suppose that E(0iXi)=0,E(1iXi)=1, and E(viXi)=0. Show that E(uiXi)=0 c. Show that assumptions 1 and 2 of Key Concept 4.3 are satisfied. d. Suppose that outliers are rare, so that (ui,Xi) have finite fourth moments. Is it appropriate to use OLS and the methods of Chapters 4 and 5 to estimate and carry out inference about the average values of 0i and 1i ? e. Suppose that 1i and Xi, are positively correlated, so that observations with larger than average values of Xi tend to have larger than average values of 1i. . Are the assumptions in Key Concept 4.3 satisfied? If not, which assumption(s) is (are) violated? Is it appropriate to use OLS and the methods of Chapters 4 and 5 to estimate and carry out inference about the average value of 0i and 1i ? 3. (Stock and Watson #13.10) Consider the regression model with heterogenous regression coefficients Yi=0i+1iXi+vi where (vi,Xi,0i,1i) are i.i.d random variables with 0=E(0i) and 1=E(1i). a. Show that the model can be written as Yi=0+1Xi+ui where ui=(0i0)+ (1i1)Xi+vi b. Suppose that E(0iXi)=0,E(1iXi)=1, and E(viXi)=0. Show that E(uiXi)=0 c. Show that assumptions 1 and 2 of Key Concept 4.3 are satisfied. d. Suppose that outliers are rare, so that (ui,Xi) have finite fourth moments. Is it appropriate to use OLS and the methods of Chapters 4 and 5 to estimate and carry out inference about the average values of 0i and 1i ? e. Suppose that 1i and Xi, are positively correlated, so that observations with larger than average values of Xi tend to have larger than average values of 1i. . Are the assumptions in Key Concept 4.3 satisfied? If not, which assumption(s) is (are) violated? Is it appropriate to use OLS and the methods of Chapters 4 and 5 to estimate and carry out inference about the average value of 0i and 1i