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

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, $\tilde {\mathrm { X}} _ { i } ^ { 1 } = \mathrm { X } _ { i } ^ { 1 } + w _ { i } ^ { 1 }$ where $\bar { X }$ 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: $\tilde{\mathrm { X }} _ { i } ^ { 2 } = \mathrm { X } _ { i } ^ { 2 } + w _ { i } ^ { 2 }$ What would happen if you ran a regression of grades received by students in the final on midterm grades?