## Related questions with answers

Executive compensation has risen dramatically beyond the rising levels of an average worker's wage over the years. The government is even considering a cap on high-flying salaries for executives (The New York Times, February 9 , 2009). Consider the following portion of data which links total compensation (in dollar millions) of the $455$ highest-paid CEOs in $2006$ with three measures: industry-adjusted return on assets (ROA), industry-adjusted stock return (Return) and the firm's size (Total Assets in dollar millions).

$\begin{array}{|c|c|c|c|} \hline \text { Compensation } & \text { ROA } & \text { Return } & \text { Total Assets } \\ \hline 16.58 & 2.53 & -0.15 & 20917.5 \\ \hline 26.92 & 1.27 & 0.57 & 32659.5 \\ \hline \vdots & \vdots & \vdots & \vdots \\ \hline 2.3 & 0.45 & 0.75 & 44875.0 \\ \hline \end{array}$

a. Estimate three simple linear regression models that use Compensation as the response variable with $\mathrm{ROA}$, Return, or Total Assets as the explanatory variable. Which model do you select? Explain.

b. Estimate multiple linear regression models that use various combinations of two, or all three, explanatory variables. Which model do you select? Explain.

Solution

VerifiedWe are given a sample of compensations, ROA , returns and total assets of $455$ companies.

Our goal is to estimate three simple linear regression models with compensation as the response variable and ROA, return and total assets as explanatory variables, respectively.

We can write these models as:

$\begin{aligned} (i)\quad \text{Compensation}&=\beta_{1,0}+\beta_{1,1} \text{ ROA}+\epsilon\\ (ii)\quad\text{Compensation}&=\beta_{2,0}+\beta_{2,1} \text{ Return}+\epsilon\\ (iii)\quad \text{Compensation}&=\beta_{3,0}+\beta_{3,1} \text{ Total Assets}+\epsilon.\\ \end{aligned}$

We also want to decide which one of these three models is the best.

*How do we estimate these models?*

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