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

Exercise 12

Consider the one-variable regression model Yj = ß 0 + ß 1 X i + u i , and suppose that it satisfies the assumption in Key Concept 4.3. Suppose that Y i is measured with error, so the data are
where w i is the measurement error which is i.i.d. and independent of Y i and X i. Consider the population regression
where v i is the regression error using the mismeasured dependent variable,
i
a. Show that v i = u i + w i.
b. Show that the regression
satisfies the assumptions in Key Concept 4.3. (Assume that w i is independent of Y j and X j for all values of i and j and has a finite fourth moment.)
c. Are the OLS estimators consistent
d. Can confidence intervals be constructed in the usual way
e. Evaluate these statements: "Measurement error in the X 's is a serious problem. Measurement error in Y is not."