Consider the one-variable regression model Yi = β0 + β1Xi + ui, and suppose it satisfies the least squares assumptions in Key Concept 4.3. Suppose Yi is measured with error, so the data are ?"i = Yi + wi, where wi is the measurement error, which is i.i.d. and independent of Yi. Consider the population regression ?"i = β0 + β1Xi + vi, where vi is the regression error, using the mismeasured dependent variable ?"i. a) Showthatvi =ui +wi. b) Show that the regression ?"i = β0 + β1Xi + 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.”
Please solve d-e, first parts asked in diffirent question.
Consider the one-variable regression model Yi = β0 + β1Xi + ui, and suppose it satisfies the least squares assumptions in Key Concept 4.3. Suppose Yi is measured with error, so the data are ?"i = Yi + wi, where wi is the measurement error, which is i.i.d. and independent of Yi. Consider the population regression ?"i = β0 + β1Xi + vi, where vi is the regression error, using the mismeasured dependent variable ?"i.
a) Showthatvi =ui +wi.
b) Show that the regression ?"i = β0 + β1Xi + 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.”
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