Introduction to Econometrics 3rd Edition by James Stock, Mark Watson

Exercise 13

Consider the regression model with heterogenous regression coefficients
Y i = ß oi + ß 1i X i + V i where (v i -, X i , ß 0i , ß 1i ) are i.i.d. random variables with ß 0 = E ( ß 0i ) and ßi = E(ß 1 i ).
a. Show that the model can be written as Y i = ß 0 + ß 1 X i + u i where u i = ( ßoi - ßo ) + ( ß 1i - ß l ) X i + V i
b. Suppose that
. Show that
.
c. Show that assumptions 1 and 2 of Key Concept 4.3 are satisfied.
d. Suppose that outliers are rare so that (u i X i ) 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 ß 0i and ß 1i
e. Suppose that ß 1i and X i are positively correlated so that observations with larger than average values of X i , tend to have larger than average values of ß 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 ß 0i and ß 1i