Question

We are interested in the relationship between the compensation of Chief Executive Officers (CEO) of firms and the return on equity of their respective firm, using the dataset salary.xlsx. The variable salary shows the annual salary of a CEO in thousands of dollars, so that y = 150 indicates a salary of $150,000. Similarly, the variable ROE represents the average return on equity (ROE) for the CEO’s firm for the previous three years. A ROE of 20 indicates an average return of 20%.
a) Draw a boxplot and a histogram of the salary of CEO. Are there any apparent outliers in the data? Are there high leverage points?
b) Use your software to estimate the relationship and report your results.
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c) Looking at a plot of the residuals against predicted values and at the normal probability plot of residuals, does the estimated model appear satisfactory?
d) Use your software to estimate the model, this time by using the database salaryalt.xlsx which excludes all the data points for which the salary of the CEO appears extraordinarily large considering the ROE of their firm. Report your results.
e) Produce a histogram and a normal probability plot of the residuals of this regression. Does this regression appear to meet the conditions of absence of outliers and near normality?
f) What are the units of the slope coefficient b1 in this equation? What is the impact on the salary of the CEO of firm i if the ROE increases by 1%?
g) Use your results to calculate a 95% interval to estimate the mean salary of CEOs whose firms have an ROE of 20 per cent.

Answer 1

a) Draw a boxplot and a histogram of the salary of CEO. Are there any apparent outliers in the data? Are there high leverage points?

data<-read.csv("salary.csv",sep = ",")
boxplot(data$salary)
hist(data$salary)
summary(data$salary)

Yes, it has some outliers.

mod<-lm(salary~roe,data=data)
mod
summary(mod)
summary.aov(mod)

c)

plot(mod$fitted.values,mod$residuals)

Yes, the residuals are showing outliers on the positive side of y-axis which means that the model is not a perfect fit.

d) Use your software to estimate the model, this time by using the database salaryalt.xlsx which excludes all the data points for which the salary of the CEO appears extraordinarily large considering the ROE of their firm. Report your results.

> data<-read.csv("salaryalt.csv",sep = ",")
> boxplot(data$salary)
> hist(data$salary)
> summary(data$salary)
Min. 1st Qu. Median Mean 3rd Qu. Max.
223.0 729.8 998.5 1071.2 1345.8 2408.0
>
> mod<-lm(salary~roe,data=data)

e) Produce a histogram and a normal probability plot of the residuals of this regression. Does this regression appear to meet the conditions of absence of outliers and near normality?

qqnorm(mod$residuals)

Yes this data seems to adhere to the normality conditions.

f) What are the units of the slope coefficient b1 in this equation?

The unit is thousands of dollars

What is the impact on the salary of the CEO of firm if the ROE increases by 1%?

The salary will increase by15.56*1000 =15560

g) Use your results to calculate a 95% interval to estimate the mean salary of CEOs whose firms have an ROE of 20 per cent.

> Input<-data.frame(roe=c(20))
> predict(mod,newdata =Input,interval = "confidence",level = 0.95)
fit lwr   upr
1 1115.634 1050.878 1180.39

(1050.878 ,1180.39) in thousands of dollars or

(1050878, 1180390)