Your Professor Wants to Measure the Class's Knowledge of Econometrics $$\widetilde { X _ { i } ^ { 1 } } = X _ { i } ^ { 1 } + w _ { i } ^ { 1 } ,$$

Essay

Your professor wants to measure the class's knowledge of econometrics twice during the
semester, once in a midterm and once in a final.Assume that your performance, and that
of your peers, on the day of your midterm exam only measure knowledge imperfectly and
with an error, $$\widetilde { X _ { i } ^ { 1 } } = X _ { i } ^ { 1 } + w _ { i } ^ { 1 } ,$$
where $\widetilde { X }$ is your exam grade, $X$ is underlying econometrics knowledge, and $w$ is a random error with mean zero and variance $\sigma _ { w } ^ { 2 } . w$ may depend on whether you have a headache that day, whether or not the questions you had prepared for appeared on the exam, your mood, etc. A similar situation holds for the final, which is exam two: $\widetilde { X _ { i } ^ { 2 } } = X _ { i } ^ { 2 } + w _ { i } ^ { 2 }$ . What would happen if you ran a regression of grades received by students in the final on midterm grades?