Professor Hsieh decides to run an experiment to measure the effect of time press

Question

Professor Hsieh decides to run an experiment to measure the effect of time pressure on final exam scores. He gives each of the 50 students in his course the same final exam, but some students have 90 minutes to complete the exam, while the others have 120 minutes. Each student is randomly assigned one of the examination times based on the flip of a coin (25 students will be assigned to the 90 minutes group and vice versa). Let Y_i denote the test score of student i and let X_i denote the amount of time assigned to student i (X_i = 90 or 120). Consider the regression model Y_i = alpha + beta X_i + u_i;. Explain why E[u_i| X_i] = 0 for this regression model. Instead of flipping a coin, Prof. Hsieh decides to assign 90 minutes to junior and 120 minutes to senior. Will this cause any problem? It is reasonable to assume that senior students have higher math ability in general as they might have completed more math-related courses. If so, will the assignment in (B) lead to upward or downward bias of OLS estimation?

Explanation / Answer

Question 1: Why E[i/Xi] = 0

E[i/Xi] = 0 is one of the assumptions of Classical liear regression model.

It says for any given fixed value of X(Explanatory variable or dependent variable) , the expected value of the disturbance term i is zero.

Let us understand the terms i and E[Y/Xi] before we go to find why E[i/Xi] = 0.

E[Y/Xi] is nothing but the average marks obtained by all the students who had been assigned 90 minutes for the test.i.e., E[Y/Xi] is mean or expected value of Y (Independent variable) for a given X (which is 90 in this case)

Generally, if the given data follows linear regression assumptions, we can have the individual Y values at any given X is always distributed around its mean values  E[Y/Xi] .This is shown in the chart attached.The deviations or distances of each Yi around its expected values can be expressed as Yi - E[Y/Xi] which is nothing but our i..

If we calculate the i 's ( Yi - E[Y/Xi] ) for each individual values of Y and total them all it will result in zero.Please refer to the below table for example.

Simillarly if we do for X= 120 we get sum of all i = 0 for given X =120. Sinc ethe total is zero we have the expected value of i for any given value of X is zero . i.e  E[i/Xi] = 0.

Question 2:

Flipping a coin will definitely have some randomness in assigning the 90 min/120 min test for the students through whihc we can see the impact of time.

But if we assign 120mins to Seniors and 90 minutes to Juniors ,the results could be biased as with seniors we can expect higher ability levels and also with extra time given to them , we can expect higher score for seniors compared to Juniors. Obviously in this case we cannot exactly study the impact of time alone on the results as we can expect the impact of ability levels on the results along with time.Hence we will not be able to measure the effect of time pressure on final score with out an bias.

Question 3:

As explained above ,  if we assign 120mins to Seniors and 90 minutes to Juniors ,the results could be upward biased as with seniors we can expect higher ability levels and also with extra time given to them , we can expect higher score for seniors compared to Juniors.

We are actuallyhaving a case where seniority level leads to more ability and higher marks where >0 .This may cause upward bias of the ols estimation.

However we can add another categorical variable (taking values Junior or Senior) to the model and can study the time impact with respect to seniority level.

X Y u=Y-E[Y/X] 90 80 3 90 93 16 90 95 18 90 70 -7 90 60 -17 90 65 -12 Mean 77 Sum 0

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.