Consider the one-variable regression model \(Y_{i}=\beta_{0}+\beta_{1} X_{i}+u_{i}\), and suppose it satisfies the least squares assumptions in Key Concept 4.3. Suppose \(Y_{i}\) is measured with error, so the data are \(\widetilde{Y}_{i}=Y_{i}+w_{i}\), where \(w_{i}\) is the measurement error, which is i.i.d. and independent of \(Y_{i}\) and \(X_{i}\). Consider the population regression \(\widetilde{Y}_{i}=\beta_{0}+\beta_{1} X_{i}+v_{i}\), where \(v_{i}\) is the regression error, using the mismeasured dependent variable, \(\widetilde{Y}_{i}\).

a. Show that \(v_{i}=u_{i}+w_{i}\).

b. Show that the regression \(\widetilde{Y}_{i}=\beta_{0}+\beta_{1} X_{i}+v_{i}\) satisfies the least squares 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."