13.10 Consider the regression model with heterogeneous regression coefficients Yi = b0 + b1iXi + vi, where (vi, Xi, b1i) are i.i.d. random variables with b1 = E1b1i2.

a. Show that the model can be written as Yi = b0 + b1Xi + ui, where ui = 1b1i - b12Xi + vi.

b. Suppose Xi is randomly assigned, so that 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 outliers are rare, so that 1ui, Xi2 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. Now suppose Xi is not randomly assigned, that E3vi Xi4 = 0, but that b1i and Xi are positively correlated, so that observations with largerthan-
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? Will the OLS estimator of b1 be unbiased for E1b1i2?