Question

Log( salary )=β0+β1LSAT+β2GPA+β3log( libvol )+β4log(cost)+β5rank+u where LSAT is the median LSAT score for the graduating class, GPA is the median college GPA for the class, libvol is the number of volumes in the law school library, cost is the annual cost of attending law school, and rank is a law school ranking with rank=1 being the best.

(A) Explain why we expect β5<=0.
(B) What signs do you expect for the other slope parameters? Justify your answer.
(C) Using actual data, the estimated equation is: log( salary )=8.34+0.0047LSAT+0.248GPA+0.095log( libvol )+0.038log( cost )−0.0033rankn=136, R2=0.842 What is the predicted ceteris paribus difference in salary for schools with a median GPA different by one point?
(D) Interpret the coefficient of log( libvol ) 1
(E) Would you say it is better to attend a higher-ranked law school? How much is the difference in ranking of 20 worth in terms of predicted starting salary?

Answer 1

We expect β5 ≤ 0 because higher-ranked law schools tend to yield higher salaries. Other parameters are expected to be positive, with a one-point increase in GPA leading to a 24.8% salary increase and a 1% increase in library volumes leading to a 0.095% salary increase. A 20-rank decrease in law school rank would decrease salary by 0.066%.

Step-by-step explanation:

When analyzing the regression equation log(salary) = β0 + β1LSAT + β2GPA + β3log(libvol) + β4log(cost) + β5rank + u, we anticipate that β5 ≤ 0 because a higher rank (closer to 1) is generally associated with better law schools, which in turn tend to lead to higher salaries. Thus, as rank increases (indicating a lower rank), we expect salaries to decrease, signifying a negative relationship.

For the other slope parameters, positive signs are expected. Higher LSAT scores (β1 > 0) and GPAs (β2 > 0) generally indicate better academic performance, which could result in higher salaries. The volume of the library (libvol) is also expected to have a positive coefficient (β3 > 0), as larger libraries might indicate better resources for students, potentially leading to better career prospects. Lastly, the cost of attendance (β4 > 0) is often higher at prestigious schools, which may lead to higher salaries.

The estimated equation provided and the coefficient on GPA (0.248) indicate that a one-point increase in median GPA is predicted to increase the starting salary by 24.8% ceteris paribus.

The coefficient of log(libvol), which is 0.095, suggests that a 1% increase in the number of volumes in the law school library is associated with a 0.095% increase in the predicted salary.

As the coefficient of rank is -0.0033, we can infer that attending a higher-ranked law school correlates with a higher starting salary. A 20-rank increase, ceteris paribus, would decrease the predicted salary by approximately 0.066%, given the negative coefficient on rank.

Answer 2

The response explains the expected signs of the slope parameters as well as provides an interpretation of the coefficient of log(libvol) and the difference in predicted starting salary for different law school rankings.

Step-by-step explanation:

(A) We expect β5<=0 because a higher law school ranking (rank=1 being the best) is associated with a higher salary, so as the ranking decreases (higher rank numbers), the salary is expected to decrease. Therefore, β5, the coefficient for rank, should be negative.

(B) The signs we expect for the other slope parameters are:

• β0: This is the intercept and represents the predicted salary when all other variables are 0. It is not necessarily related to the signs of the other slope parameters.
• β1 (LSAT): We expect a positive sign because a higher LSAT score is associated with a higher salary.
• β2 (GPA): We expect a positive sign because a higher GPA is associated with a higher salary.
• β3 (log(libvol)): We expect a positive sign because a larger law school library (higher libvol) is associated with a higher salary.
• β4 (log(cost)): We expect a negative sign because a higher cost of attending law school is associated with a lower salary.

(C) To calculate the predicted ceteris paribus difference in salary for schools with a median GPA different by one point, we substitute the values into the equation and compare the resulting salaries. Since the equation is in logarithmic form, we exponentiate the predicted salaries to obtain actual salary values.

(D) The coefficient of log(libvol) (β3) represents the change in salary associated with a one-unit increase in log(libvol), while holding all other variables constant. It tells us the magnitude and direction of the effect of the number of volumes in the law school library on salary.

(E) To determine if it is better to attend a higher-ranked law school, we can compare the predicted starting salaries for different ranks. The difference in ranking of 20 would be worth β5 times the difference, which in this case is -0.0033 * 20.