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Chapter 4: Problem 2

Consider an equation to explain salaries of CEOs in terms of annual firm sales, return on equity (roe, in percentage form), and return on the firm's stock (ros, in percentage form): $$\log (\text {salary})=\beta_{0}+\beta_{1} \log (\text {sales})+\beta_{2} \text {roe}+\beta_{3} r o s+u$$ i. In terms of the model parameters, state the null hypothesis that, after controlling for sales and roe, ros has no effect on CEO salary. State the alternative that better stock market performance increases a CEO's salary. ii. Using the data in CEOSAL1, the following equation was obtained by OLS: $$\begin{aligned} \widehat{\log (\text {salary})}=& 4.32+.280 \log (\text {sales})+.0174 \mathrm{roe}+.00024 \mathrm{ros} \\ &(.32)(.035) \\ n=& 209, R^{2}=.283 \end{aligned}$$ By what percentage is salary predicted to increase if ros increases by 50 points? Does ros have a practically large effect on salary? iii. Test the null hypothesis that ros has no effect on salary against the alternative that ros has a positive effect. Carry out the test at the \(10 \%\) significance level. iv. Would you include ros in a final model explaining CEO compensation in terms of firm performance? Explain.

Short Answer

Expert verified
1. Null: \(\beta_3=0\); Alternative: \(\beta_3>0\). 2. Salary increases by 1.21% if ros increases by 50 points. 3. \(\mathrm{ros}\) does not significantly affect salary; fail to reject null. 4. Do not include \(\mathrm{ros}\) in the final model.

Step by step solution

01

Define the Null and Alternative Hypotheses

To address the first part, we define the null and alternative hypotheses relating to the parameter \( \beta_3 \), which represents the effect of \( \mathrm{ros} \) (return on stock) on CEO salary. - Null Hypothesis \( (H_0): \beta_3 = 0 \). This implies that \( \mathrm{ros} \) has no effect on salary after controlling for sales and roe.- Alternative Hypothesis \( (H_1): \beta_3 > 0 \). This suggests that better stock market performance increases the CEO's salary.
02

Calculate the Predicted Increase in Salary for ros Increase

With the equation provided, the coefficient for \( \mathrm{ros} \) is \( 0.00024 \). If \( \mathrm{ros} \) increases by 50 points, the predicted change in \( \log(\text{salary}) \) is:\[ \text{Change in } \log(\text{salary}) = 0.00024 \times 50 = 0.012 \]To find the percentage change in salary, use \( 100 \times [e^{0.012} - 1] \). Calculating the exponential:\[ e^{0.012} \approx 1.0121 \]Thus, the percentage increase in salary is approximately:\[ 100 \times 0.0121 = 1.21\% \]
03

Evaluate the Practical Effect Size

The predicted impact of a 50-point increase in \( \mathrm{ros} \) is a 1.21% increase in salary. Considering the small magnitude of this change relative to typical CEO salaries, \( \mathrm{ros} \) does not have a practically large effect on salary.
04

Test the Null Hypothesis About the Effect of ros

To test if \( \mathrm{ros} \) has no effect on salary, we perform a hypothesis test for \( \beta_3 = 0 \). Given the standard error \( 0.00024 \), the t-statistic is calculated as:\[ t = \frac{0.00024}{0.00024} = 1 \]At a 10% significance level and with \( n = 209 \), compare the t-statistic to the critical value from a t-distribution table. Since \( t = 1 \) is likely below the critical value at this level (typically 1.645 for one-sided tests), we fail to reject the null hypothesis.
05

Decision on Including ros in the Model

Given that the significance test suggests \( \mathrm{ros} \) does not have a statistically significant effect and considering the practical significance, it would be reasonable to omit \( \mathrm{ros} \) from a final model of CEO compensation focused solely on firm performance metrics.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing
Hypothesis testing is a fundamental part of econometrics that helps us make informed decisions based on data analysis. In this context, we want to understand the effect of certain variables, like return on stock (ros), on CEO compensation. To test this, we establish two hypotheses:
- The **null hypothesis** (denoted as \( H_0 \)) posits that the variable of interest, ros, has no effect on CEO salary. This is written as \( \beta_3 = 0 \).
- The **alternative hypothesis** proposes that the ros does have an effect, increasing CEO salary. This is represented as \( \beta_3 > 0 \).
We use statistical tests to determine whether the data supports \( \beta_3 = 0 \) or suggests it's significantly different, indicating a meaningful effect of ros on CEO salary.
CEO Compensation
CEO compensation isn't just about what CEOs take home annually. It's influenced by various performance metrics that keep companies competitive and profitable, such as sales figures and stock market performance. Here, we try to determine how these metrics affect CEO salary using statistical models.
Understanding what influences a CEO's compensation helps in setting fair CEO pay structures that incentivize performance without unfairly inflating salaries. This ensures alignment between CEO efforts and company success, fostering greater accountability and motivation.
Linear Regression
Linear regression is a statistical tool used to model the relationship between a dependent variable and one or more independent variables. In our study, the regression equation \[\log ( ext {salary}) = \beta_{0} + \beta_{1} \log ( ext {sales}) + \beta_{2} \text {roe} + \beta_{3} \text{ros} + u\]is employed to explore how CEO salary (dependent variable) relates to firm sales, return on equity, and return on stock (independent variables).
Through linear regression, we analyze how changes in the independent variables might lead to changes in the dependent variable. Specifically, the coefficients \( \beta_1, \beta_2, \text{and} \beta_3 \) provide insight into the strength and direction of these relationships.
OLS Estimation
Ordinary Least Squares (OLS) estimation is a method used to find the best-fitting line through data points in linear regression. This method is critical in econometric analysis because it minimizes the sum of the squared differences (i.e., residuals) between observed and predicted values of the dependent variable.
In our example, OLS helps us estimate:
  • The constant term \(4.32\)
  • The impact of log(sales) with a coefficient of \(0.280\)
  • The effect of roe with a coefficient of \(0.0174\)
  • The influence of ros showcased by \(0.00024\)
These coefficients help explain how much CEO salary changes in response to changes in firm performance metrics. OLS thus provides clarity on whether our hypothesized relationships hold true and guide decisions about the inclusion of variables like ros in the final compensation model.

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Most popular questions from this chapter

Are rent rates influenced by the student population in a college town? Let rent be the average monthly rent paid on rental units in a college town in the United States. Let pop denote the total city population, avginc the average city income, and pctstu the student population as a percentage of the total population. One model to test for a relationship is $$\log (\text {rent})=\beta_{0}+\beta_{1} \log (p o p)+\beta_{2} \log (\text {avginc})+\beta_{3} p c t s t u+u$$ i. State the null hypothesis that size of the student body relative to the population has no ceteris paribus effect on monthly rents. State the alternative that there is an effect. ii. What signs do you expect for \(\beta_{1}\) and \(\beta_{2} ?\) iii. The equation estimated using 1990 data from RENTAL for 64 college towns is $$\begin{aligned} &\widehat{\log (\text {rent})}=.043+.066 \log (\text {pop})+.507 \log (\text {avginc})+.0056 \text { pctstu}\\\ &\begin{aligned} (.844)(.039) &(.081) \\ n=64, R^{2}=.458 \end{aligned} \end{aligned}$$ What is wrong with the statement: "A 10\% increase in population is associated with about a \(6.6 \%\) increase in rent"? iv. Test the hypothesis stated in part (i) at the \(1 \%\) level.

In Section \(4-5 .\) we used as an example testing the rationality of assessments of housing prices. There, we used a log-log model in price and assess [see equation ( \(4.47)\) ]. Here, we use a level-level formulation. i. In the simple regression model $$\text { price }=\beta_{0}+\beta_{1} \text { assess }+u$$ the assessment is rational if \(\beta_{1}=1\) and \(\beta_{0}=0 .\) The estimated equation is $$\begin{aligned} \widehat{\text {price}} &=-14.47+.976 \text { assess} \\ &(16.27)(.049) \\ n &=88, \mathrm{SSR}=165,644.51, R^{2}=.820 \end{aligned}$$ First, test the hypothesis that \(\mathrm{H}_{0}: \beta_{0}=0\) against the two- sided alternative. Then, test \(\mathrm{H}_{0}: \beta_{1}=1\) against the two- sided alternative. What do you conclude? ii. To test the joint hypothesis that \(\beta_{0}=0\) and \(\beta_{1}=1\), we need the SSR in the restricted model. This amounts to computing \(\sum_{i=1}^{n}\left(\text {price}_{i}-\text {assess}_{i}\right)^{2},\) where \(n=88,\) because the residuals in the restricted model are just price \(_{i}\) - assess \(_{i}\). (No estimation is needed for the restricted model because both parameters are specified under \(\mathrm{H}_{0}\).) This turns out to yield \(\mathrm{SSR}=209,448.99\) Carry out the \(F\) test for the joint hypothesis. iii. Now, test \(\mathrm{H}_{0}: \beta_{2}=0, \beta_{3}=0,\) and \(\beta_{4}=0\) in the model $$\text { price }=\beta_{0}+\beta_{1} \text { assess }+\beta_{2} \text { lotsize }+\beta_{3} \text { sqr } f t+\beta_{4} b d r m s+u$$ The \(R\) -squared from estimating this model using the same 88 houses is .829 iv. If the variance of price changes with assess, lotsize, sqrft, or bdrms, what can you say about the F test from part (iii)?

Consider the estimated equation from Example 4.3 , which can be used to study the effects of skipping class on college GPA: $$\begin{aligned} \widehat{\text {colGPA}} &=1.39+.412 \mathrm{hsGPA}+.015 \mathrm{ACT}-.083 \text { skipped} \\ &(.33)(.094) \\ n &=141, R^{2}=.234 \end{aligned}$$ i. Using the standard normal approximation, find the \(95 \%\) confidence interval for \(\beta_{h, G P A}\). ii. Can you reject the hypothesis \(\mathrm{H}_{0}: \beta_{h s, G P A}=.4\) against the two-sided alternative at the \(5 \%\) level? iii. Can you reject the hypothesis \(\mathrm{H}_{0}: \beta_{h_{h} G P A}=1\) against the two-sided alternative at the \(5 \%\) level?

The variable rdintens is expenditures on research and development (R\&D) as a percentage of sales. Sales are measured in millions of dollars. The variable profmarg is profits as a percentage of sales. Using the data in RDCHEM for 32 firms in the chemical industry, the following equation is estimated: $$\begin{aligned} \widehat{\text {rdintens}} &=.472+.321 \log (\text {sales})+.050 \text { profmarg} \\ &(1.369)(.216) \\ n &=32, R^{2}=.099 \end{aligned}$$ i. Interpret the coefficient on \(\log (\text {sales} \text { ). In particular, if sales increases by } 10 \%,\) what is the estimated percentage point change in \(r\)dintens? Is this an economically large effect? ii. Test the hypothesis that R\&D intensity does not change with sales against the alternative that it does increase with sales. Do the test at the \(5 \%\) and \(10 \%\) levels. iii. Interpret the coefficient on profmarg. Is it economically large? iv. Does profinarg have a statistically significant effect on rdintens?

The data in MEAPSINGLE were used to estimate the following equations relating school-level performance on a fourth-grade math test to socioeconomic characteristics of students attending school. The variable free, measured at the school level, is the percentage of students eligible for the federal free lunch program. The variable medinc is median income in the ZIP code, and \(p c t s g l e\) is percent of students not living with two parents (also measured at the ZIP code level). See also Computer Exercise \(\mathrm{C} 11\) in Chapter \(3 .\) $$\begin{aligned} \widehat{\text {math} 4} &=96.77-.833 \text { pctsgle} \\ &(1.60)(.071) \\ n &=299, R^{2}=.380 \end{aligned}$$ $$\begin{aligned} \widehat{\text {math} 4}=& 93.00-.275 \text { pctsgle }-.402 \text { free} \\ &(1.63)(.117) \\ n &=299, R^{2}=.459 \end{aligned}$$ $$\begin{aligned} \widehat{\text {math} 4}=& 24.49-.274 \text { pctsgle }-.422 \text { free}-.752 \text { lmedinc }+9.01 \text { lexppp} \\ &(59.24)(.161) \\ n=& 299, R^{2}=.472 \end{aligned}$$ $$\begin{aligned} \widehat{\text {math} 4} &=17.52-.259 \text { pctsgle }-.420 \text { free}+8.80 \text { lexppp} \\ &(32.25)(.117) \\ n &=299, R^{2}=.472 \end{aligned}$$ i. Interpret the coefficient on the variable \(p\) ctsgle in the first equation. Comment on what happens when free is added as an explanatory variable. ii. Does expenditure per pupil, entered in logarithmic form, have a statistically significant effect on performance? How big is the estimated effect? iii. If you had to choose among the four equations as your best estimate of the effect of \(p c t s g l e\) and obtain a \(95 \%\) confidence interval of \(\beta_{\text {ptegle}},\) which would you choose? Why?
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