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[1] You want to investigate the cause of differences in salaries of CEOs, and ask whether the differences in salaries can be explained by differences in their performances measured by the return on the equities of their firms. You specified three models and estimated them:

(1) Yi=963+18Xi
(2) ln(Yi)=6.7+0.01Xi (3) ln(Yi)=6.5+0.17ln(Xi)

where Yi represents the salary of CEO i measured in 1,000 dollars, and Xi represents the percentage return on the equities of their firms. Compare three firms: The returns on equities of firm A, B and C are 2%, 3% and 3.03%, respectively. Your model (1) predicts that the salary of A’s CEO is (lower, higher) than the salary of B’s CEO by $( ). On the other hand, your model (2) predicts that the salary of A’s CEO is (lower, higher) than the salary of B’s CEO by ( )%. Your model (3) predicts that the salary of B’s CEO is (lower, higher) than the salary of C’s CEO by ( )%.

Solution

Solution:

1) Model 1 => Y_i = 963+18X_i

where: X_i for A and B are 2 and 3 percent respectively.

Clubbing the X_i values in model 1 we get,

For A: Y_i =963 + 18 (0.02) = $963.36

For B: Y_i =963 + 18 (0.03) = $963.54

The difference in salary = $(963.54 - 963.36) = $0.18

Hence the model 1 predicts the salary of A\'s CEO is lower than the salary of B\'s CEO by $0.18

2) Model 2 => Ln(Y_i) = 6.7+0.01X_i

where: X_i for A and B are 2 and 3 percent respectively.

Clubbing the X_i values in model 2 we get,

For A: Ln(Y_i)=6.7 + 0.01 (0.02) = $6.7002

For B: Ln(Y_i)=6.7 + 0.01 (0.03) = $6.7003

The percentage difference in salary = ((6.7003 - 6.7002)/6.7003)*100 = 0.001%

Hence the model 1 predicts the salary of A\'s CEO is lower than the salary of B\'s CEO by 0.001%

3) Model 3 => Ln(Y_i) = 6.5+0.17Ln(X_i)

where: X_i for B and C are 3 and 3.03 percent respectively.

Clubbing the X_i values in model 3 we get,

For B: Ln(Y_i)=6.5 + 0.17 (1.10) = $6.69

For C: Ln(Y_i)=6.5 + 0.17 (1.11) = $6.69

The percentage difference in salary = 0

The Model 3 predicts no difference in salary of CEO of B and CEO of C