13.10 Consider the regression model with heterogeneous regression coefficients Yi = b0i + b1iXi + vi, where (vi, Xi, b0i, b1i) are i.i.d. random variables with b0 = E(b0i) and b1 = E(b1i).

a. Show that the model can be written as Yi = b0 + b1Xi + ui, where ui = (b0i - b0) + (b1i - b1)Xi + vi.

b. Suppose that E3b0i Xi4 = b0, E3b1i Xi4 = b1, and E3vi Xi4 = 0 Show that E3ui Xi4 = 0.

c. Show that Assumption #1 and Assumption #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 b0i and b1i?

e. Suppose that b1i and Xi are positively correlated so that observations with larger-than-average values of Xi tend to have larger-than-average values of b1i. 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 b0i and b1i?