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Chapter 7: Problem 4

An equation explaining chief executive officer salary is $$\begin{aligned}\widehat{\log (\text {salary})}=& 4.59+.257 \log (\text {sales})+.011 \text { roe }+.158 \text { finance} \\\&(.30)\quad(.032)\quad(.004) \quad(.089)\\\&+.181 \text { consprod }-.283 \text { utility} \\\&(.085) \quad(.099) \\\n=& 209, R^{2}=.357\end{aligned}$$ The data used are in CEOSAL1, where finance, consprod, and utility are binary variables indicating the financial, consumer products, and utilities industries. The omitted industry is transportation. i. Compute the approximate percentage difference in estimated salary between the utility and transportation industries, holding sales and roe fixed. Is the difference statistically significant at the \(1 \%\) level? ii. Use equation (7.10) to obtain the exact percentage difference in estimated salary between the utility and transportation industries and compare this with the answer obtained in part (i). iii. What is the approximate percentage difference in estimated salary between the consumer products and finance industries? Write an equation that would allow you to test whether the difference is statistically significant.

Short Answer

Expert verified
i. Approximately -24.7%, not statistically significant. ii. Exact difference is -24.63%. iii. Approximately 2.3%; use t-test for significance.

Step by step solution

01

Understanding the equation components

The equation given is a regression equation for estimating the CEO salary based on various predictors. The logarithm of salary is modeled as a function of logarithm of sales, return on equity (roe), and binary indicators for finance, consumer products (consprod), and utility industries. Each coefficient represents the change in the log of salary for a unit change in the variable.
02

Estimate percentage difference (utility vs. transportation) - Approximate

To find the approximate percentage difference in estimated salary between the utility and transportation industries, examine the coefficient for 'utility', which is -0.283. Convert this coefficient to a percentage difference using the approximation given by \(100 \times (\exp(-0.283) - 1)\).
03

Calculate exact percentage difference (utility vs. transportation)

To find the exact percentage difference, use the formula \((\exp(\beta) - 1) \times 100\) for the utility coefficient. Calculate \((\exp(-0.283) - 1) \times 100\).
04

Statistical significance test at 1% level (utility vs. transportation)

Determine if the difference is statistically significant by looking at the statistical output in parentheses. For 'utility', the standard error is 0.099. Use the t-statistic formula \(t = \frac{-0.283}{0.099}\) and compare with the critical value for a 1% significance level (approximately 2.58 for large sample sizes, two-tailed test).
05

Approximate percentage difference (consumer products vs. finance)

To find the approximate percentage difference in salary between consumer products and finance, compare their coefficients: 0.181 (consprod) and 0.158 (finance). Calculate the difference, \(0.181 - 0.158 = 0.023\), and convert this to a percentage \((0.023 \times 100)\).
06

Statistical significance test (consumer products vs. finance)

To test for statistical significance, write the hypothesis test equation. Let \((\beta_{consprod} - \beta_{finance})\) be the difference in coefficients, and test using \(t = \frac{0.023}{\sqrt{0.085^2 + 0.089^2}}\) to determine if zero is within the confidence interval at a 5% significance level using a z-test or t-test.

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Key Concepts

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

Regression Analysis
Regression analysis is a powerful statistical tool used to understand the relationship between one dependent variable and one or more independent variables. In this context, it helps in estimating CEO salaries based on different determinants like sales, return on equity (roe), and industry classification.

To perform regression analysis, we use a regression equation that models the dependent variable as a function of the independent variables. Each term in the equation consists of a coefficient that quantifies the relationship between the independent variable and the dependent variable. By examining these coefficients, we can determine how changes in the independent variables lead to changes in the dependent variable.
  • The dependent variable in our case is the natural logarithm of CEO salary, which allows us to interpret the coefficients in terms of percentage differences.
  • The independent variables include both continuous variables, like sales and roe, and categorical variables, like industry type, which are represented as binary indicators.
This analysis helps in quantifying how much of the variation in CEO salaries can be explained by these factors, as indicated by the R-squared value. A higher R-squared implies a better fit of the model to the observed data.
CEO Salary Determinants
Several factors influence the salary levels of CEOs, and regression equations can be used to quantify these determinants. These factors are either continuous or binary in nature.
  • Sales: Often, a company's size, measured by sales, is a primary determinant of CEO salary.
  • Return on Equity (roe): This financial metric indicates how efficiently a company is using its equity to generate profit, influencing CEO pay based on perceived performance.
  • Industry Type: The industry in which a company operates impacts CEO salaries, captured by binary variables such as finance, consumer products, and utility.
These elements allow us to dissect how much each factor contributes to the overall salary expectation. Recognizing these determinants aids companies and researchers in understanding the underlying dynamics of compensation.
Binary Variables
Binary variables are a vital part of econometric analysis when dealing with qualitative factors like industry type. In the CEO salary regression, binary variables are used to indicate which industry a company belongs to.
  • Binary Indicators: These take the value of 0 or 1. For instance, 'finance' equals 1 for financial companies and 0 otherwise.
  • Omitted Variable: When using binary variables, one category is often omitted to avoid multicollinearity. In this analysis, 'transportation' is omitted, meaning its effect is captured in the intercept.
Using binary variables lets us assess how belonging to a certain category affects the dependent variable compared to the base category. It simplifies complex qualitative data into manageable quantitative factors, enabling their inclusion in the analysis.
Logarithmic Transformation
Logarithmic transformation is a common technique in econometrics for managing skewed data and interpreting coefficients as elasticities. In the CEO salary equation, both the dependent variable (salary) and an independent variable (sales) are logarithmically transformed.
  • Interpreting Coefficients: The coefficients in a log-log model represent elasticity. They show how a 1% change in an independent variable affects the dependent variable in percentage terms.
  • Benefits of Transformation: It stabilizes variance and normalizes distributions, making the data more suitable for linear regression.
This transformation helps in presenting a more realistic view of the relationships, particularly in cases where the data exhibit exponential growth characteristics. Understanding and leveraging logarithmic transformations can enhance the depth and accuracy of econometric analyses.

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Most popular questions from this chapter

For a child \(i\) living in a particular school district, let voucher, be a dummy variable equal to one if a child is selected to participate in a school voucher program, and let \(s\)core\(_{i}\) be that child's score on a subsequent standardized exam. Suppose that the participation variable, voucher, is completely randomized in the sense that it is independent of both observed and unobserved factors that can affect the test score. i. If you run a simple regression score, on voucher, using a random sample of size \(n\), does the OLS estimator provide an unbiased estimator of the effect of the voucher program? ii. Suppose you can collect additional background information, such as family income, family structure (e.g. whether the child lives with both parents), and parents' education levels. Do you need to control for these factors to obtain an unbiased estimator of the effects of the voucher program? Explain. iii. Why should you include the family background variables in the regression? Is there a situation in which you would not include the background variables?

Suppose you collect data from a survey on wages, education, experience, and gender. In addition, you ask for information about marijuana usage. The original question is: "On how many separate occasions last month did you smoke marijuana?" i. Write an equation that would allow you to estimate the effects of marijuana usage on wage, while controlling for other factors. You should be able to make statements such as, "Smoking marijuana five more times per month is estimated to change wage by \(x \% .\) ii. Write a model that would allow you to test whether drug usage has different effects on wages for men and women. How would you test that there are no differences in the effects of drug usage for men and women? iii. Suppose you think it is better to measure marijuana usage by putting people into one of four categories: nonuser, light user (1 to 5 times per month), moderate user (6 to 10 times per month), and heavy user (more than 10 times per month). Now, write a model that allows you to estimate the effects of marijuana usage on wage. iv. Using the model in part (iii), explain in detail how to test the null hypothesis that marijuana usage has no effect on wage. Be very specific and include a careful listing of degrees of freedom. v. What are some potential problems with drawing causal inference using the survey data that you collected?

Let \(d\) be a dummy (binary) variable and let \(z\) be a quantitative variable. Consider the model $$y=\beta_{0}+\delta_{0} d+\beta_{1} z+\delta_{1} d \cdot z+u_{;}$$ this is a general version of a model with an interaction between a dummy variable and a quantitative variable. [An example is in equation \((7.17) .]\) i. Because it changes nothing important, set the error to zero, \(u=0 .\) Then, when \(d=0\) we can write the relationship between \(y\) and \(z\) as the function \(f_{0}(z)=\beta_{0}+\beta_{1} z\). Write the same relationship when \(d=1\), where you should use \(f_{1}(z)\) on the left-hand side to denote the linear function of \(z\). ii. Assuming that \(\delta_{1} \neq 0\) (which means the two lines are not parallel), show that the value of \(z^{*}\) such that \(f_{0}\left(z^{*}\right)=f 1\left(z^{*}\right)\) is \(z^{*}=-\delta_{0} / \delta_{1}\). This is the point at which the two lines intersect [as in Figure 7.2 (b)]. Argue that \(z^{*}\) is positive if and only if \(\delta_{0}\) and \(\delta_{1}\) have opposite signs. iii. Using the data in TWOYEAR, the following equation can be estimated: $$\begin{aligned}\widehat{\log (\text {wage})}=& 2.289-.357 \text { female }+.50 \text { totcoll}+.030 \text { female } \text { -totcoll} \\\&(0.011) \quad (.015) \quad (.003) \quad (.005) \\\n=& 6,763, R^{2}=.202\end{aligned}$$ where all coefficients and standard errors have been rounded to three decimal places. Using this equation, find the value of totcoll such that the predicted values of \(\log (w a g e)\) are the same for men and women. iv. Based on the equation in part (iii), can women realistically get enough years of college so that their earnings catch up to those of men? Explain.

Using the data in SLEEP75 (see also Problem 3 in Chapter 3), we obtain the estimated equation $$\begin{aligned}\widehat{\text {sleep}}=& 3,840.83-.163 \text {totwrk}-11.71 \text {educ}-8.70 \text { age} \\\&(235.11)\quad(.018) \quad (5.86) \quad (11.21)\\\&+.128 \text { age}^{2}+87.75 \text { male } \\ &(.134) \quad (34.33)\\\n=& 706, R^{2}=.123, \bar{R}^{2}=.117\end{aligned}$$ The variable sleep is total minutes per week spent sleeping at night, totwr \(k\) is total weekly minutes spent working, educ and age are measured in years, and male is a gender dummy. i. All other factors being equal, is there evidence that men sleep more than women? How strong is the evidence? ii. Is there a statistically significant tradeoff between working and sleeping? What is the estimated tradeoff? iii. What other regression do you need to run to test the null hypothesis that, holding other factors fixed, age has no effect on sleeping?
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