Consider the one-variable regression model: Y, B+ BX; +u,, and sup pose that it satisfies the assumption in Key Concept 4.3. Suppose that Y, is measured with error, so that the data are Y, Y,+w, where w, is the mea- surement error which is i.i.d. and independent of Y, and X. Consider the population regression Y = B+ BX, + v,, where v, is the regression error using the mismeasured dependent variable. Y.

a. Show that vu + w

b. Show that the regression YB, BX, + v, satisfies the assumptions in Key Concept 4.3. (Assume that w, is independent of Y, and X, 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."