9.2 Consider the one-variable regression model Yi = b0 + b1Xi + ui and suppose that it satisfies the least squares assumptions in Key Concept 4.3. Suppose that Yi is measured with error, so the data are Y 

i = Yi + wi, where wi is the measurement error, which is i.i.d. and independent of Yi and Xi. Consider the population regression Y 

i = b0 + b1Xi + vi, where vi is the regression error, using the mismeasured dependent variable, Y  i.

a. Show that vi = ui + wi.

b. Show that the regression Y 
i = b0 + b1Xi + vi satisfies the least squares assumptions in Key Concept 4.3. (Assume that wi is independent of Yj and Xj 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.”