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Chapter 3: Problem 12

The following equation represents the effects of tax revenue mix on subsequent employment growth for the population of counties in the United States: $$ \text { growth }=\beta_{0}+\beta_{1} \text { sharep }+\beta_{2} \text { share_{I} }+\beta_{3} \text { shares }+\text { other factors, } $$ where growth is the percentage change in employment from 1980 to \(1990,\) sharep is the share of property taxes in total tax revenue, share_ is the share of income tax revenues, and shares is the share of sales tax revenues. All of these variables are measured in \(1980 .\) The omitted share, shareg, includes fees and miscellaneous taxes. By definition, the four shares add up to one. Other factors would include expenditures on education, infrastructure, and so on (all measured in 1980 ). i. Why must we omit one of the tax share variables from the equation? ii. Give a careful interpretation of \(\beta_{1}\)

Short Answer

Expert verified
One tax share must be omitted to avoid multicollinearity. \(\beta_{1}\) indicates how employment growth changes when the share of property taxes increases, replacing other omitted taxes.

Step by step solution

01

Understanding Zero-Sum Constraint

We have four different tax shares: property (\(\text{sharep}\)), income (\(\text{share_{I}}\)), sales (\(\text{shares}\)), and other (\(\text{shareg}\)). These shares must add up to 1, as they are parts of the total tax revenue distribution. If one variable is omitted, the remaining ones can describe the full variation because the omitted variable can be deduced by subtraction from the total sum of 1.
02

Avoiding Multicollinearity

The reason to omit one of the variables, like \(\text{shareg}\), is to avoid multicollinearity, which occurs when independent variables in a regression model are highly correlated. If all shares are included, the equation would be perfectly collinear because their sum is always 1, rendering regression coefficients unstable and interpretation difficult.
03

Interpreting \(\beta_{1}\)

\(\beta_{1}\) measures the change in employment growth rate for each unit increase in \(\text{sharep}\), holding other variables constant. A unit increase in \(\text{sharep}\) implies that funds are reallocated from the omitted share (\(\text{shareg}\) in our equation) to property taxes, and \(\beta_{1}\) quantifies the impact of such a reallocation on employment growth.

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Key Concepts

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

Tax Revenue
Tax revenue is the money collected by a government from taxes imposed on its citizens and businesses. This revenue is critical for funding public services and government operations. Understanding tax revenue in an economic context involves examining different types of taxes, such as property tax, income tax, and sales tax. Each type contributes a share to the total tax revenue.
  • **Property tax**: This is levied on real estate properties and sometimes movable properties like cars. It often funds local services, such as schools and infrastructure.
  • **Income tax**: A tax imposed on individual or business earnings, it's a primary source of revenue for governments.
  • **Sales tax**: Collected at the point of sale, this tax is a percentage of the retail price of goods and services.
These taxes together make up the total tax revenue for a region. Understanding the mix of these tax revenues is crucial, as different combinations can have varied effects on economic variables like employment growth.
Employment Growth
Employment growth refers to the increase in the number of people who are employed within a particular time frame. It's an important economic indicator, reflecting the health of a nation's economy. Job creation can boost consumer spending, enhance economic productivity, and improve overall living standards.
  • **Factors influencing employment growth**: Government fiscal policies, like tax policies, and external economic conditions can significantly influence employment rates.
  • **Impact of taxes on employment**: Different tax policies can encourage or deter businesses from hiring more employees. For example, a high income tax rate might discourage employers from hiring due to increased labor costs, while a favorable property tax rate could encourage businesses to expand and hire more staff.
Understanding these impacts helps policymakers tailor tax mixes that can foster healthy employment growth.
Multicollinearity
Multicollinearity occurs when two or more independent variables in a regression model are highly correlated, meaning they provide redundant information. This can cause issues in statistical models, making it hard to ascertain the individual effect of each variable.
  • **Identifying multicollinearity**: In our exercise, the tax shares must add up to one. Including all in a regression model creates perfect multicollinearity, as each share is perfectly predictable from the others.
  • **Problems caused by multicollinearity**: It can inflate the variance of coefficient estimates and make them highly sensitive to changes in the model, leading to unreliable results.
  • **Solutions**: To avoid this, one share is intentionally omitted. This allows us to examine the impact of the other shares relative to the omitted one, ensuring more stable and interpretable regression results.
Regression Analysis
Regression analysis is a statistical method used to understand the relationship between a dependent variable and one or more independent variables. It helps predict outcomes and determines the strength and direction of the relationships.
  • **Regression equation**: It can be expressed as \( Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + ... + \varepsilon \), where \(Y\) is the dependent variable (such as employment growth), \(X_1, X_2...\) are the independent variables (like tax shares), \(\beta_0, \beta_1, \beta_2...\) are coefficients, and \(\varepsilon\) represents the error term.
  • **Coefficient interpretation**: In our example, \(\beta_1\) indicates the change in the employment growth rate with a unit change in property tax share, assuming all other factors remain constant.
  • **Applications**: This method is widely used in economics to forecast future trends, understand past relationships, and make data-driven decisions.

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Most popular questions from this chapter

The data in WAGE2 on working men was used to estimate the following equation: $$\begin{aligned} \widehat{\text { educ }} &=10.36-.094 \text { sibs }+.131 \text { meduc }+.210 \text { feduc} \\ n &=722, R^{2}=.214 \end{aligned}$$ where \(e d u c\) is years of schooling, sibs is number of siblings, meduc is mother's years of schooling, and feduc is father's years of schooling. i. Does sibs have the expected effect? Explain. Holding meduc and feduc fixed, by how much does sibs have to increase to reduce predicted years of education by one year? (A noninteger answer is acceptable here.) ii. Discuss the interpretation of the coefficient on meduc. iii. Suppose that Man A has no siblings, and his mother and father each have 12 years of education, and Man B has no siblings, and his mother and father each have 16 years of education. What is the predicted difference in years of education between \(B\) and \(A ?\)

Suppose that the population model determining \(y\) is $$ y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+u $$ and this model satisfies Assumptions MLR.1, MLR.2, MLR.3 and MLR.4. However, we estimate the model that omits \(x_{3} .\) Let \(\bar{\beta}_{0}, \bar{\beta}_{1},\) and \(\bar{\beta}_{2}\) be the OLS estimators from the regression of \(y\) on \(x_{1}\) and \(x_{2}\) Show that the expected value of \(\tilde{\beta}_{1}\) (given the values of the independent variables in the sample) is $$\mathbf{E}\left(\tilde{\beta}_{1}\right)=\beta_{1}+\beta_{3} \frac{\sum_{i=1}^{n} \hat{r}_{i 1} x_{i 3}}{\sum_{i=1}^{n} \hat{r}_{i 1}^{2}}$$ where the \(\hat{r}_{i 1}\) are the OLS residuals from the regression of \(x_{1}\) on \(x_{2}\). [Hint: The formula for \(\tilde{\beta}_{1}\) comes from equation \((3.22) .\) Plug \(y_{i}=\beta_{0}+\beta_{1} x_{11}+\beta_{2} x_{12}+\beta_{3} x_{13}+u_{i}\) into this equation. After some algebra, take the expectation treating \(x_{i 3}\) and \(\tilde{r}_{i 1}\) as nonrandom.]

Using the data in GPA2 on 4,137 college students, the following equation was estimated by OLS: $$ \begin{aligned} \widehat{\text {colgpa}} &=1.392-.0135 \text { hsperc }+. .00148 \text { sat } \\\ n &=4.137, R^{2}=.273 \end{aligned} $$ where colgpa is measured on a four-point scale, hsperc is the percentile in the high school graduating class (defined so that, for example, hsperc \(=5\) means the top \(5 \%\) of the class), and sat is the combined math and verbal scores on the student achievement test. i. Why does it make sense for the coefficient on \(h s p e r c\) to be negative? ii. What is the predicted college GPA when hsperc \(=20\) and \(s a t=1,050 ?\) iii. Suppose that two high school graduates, A and B, graduated in the same percentile from high school, but Student A's SAT score was 140 points higher (about one standard deviation in the sample). What is the predicted difference in college GPA for these two students? Is the difference large? iv. Holding hsperc fixed, what difference in SAT scores leads to a predicted colgpa difference of \(.50,\) or one-half of a grade point? Comment on your answer.

Suppose that you are interested in estimating the ceteris paribus relationship between \(y\) and \(x_{1}\). For this purpose, you can collect data on two control variables, \(x_{2}\) and \(x_{3}\). (For concreteness, you might think of \(y\) as final exam score, \(x_{1}\) as class attendance, \(x_{2}\) as GPA up through the previous semester, and \(x_{3}\) as SAT or ACT score. Let \(\tilde{\beta}_{1}\) be the simple regression estimate from \(y\) on \(x_{1}\) and let \(\hat{\beta}_{1}\) be the multiple regression estimate from \(y\) on \(x_{1}, x_{2}, x_{3}\) i. If \(x_{1}\) is highly correlated with \(x_{2}\) and \(x_{3}\) in the sample, and \(x_{2}\) and \(x_{3}\) have large partial effects on \(y,\) would you expect \(\bar{\beta}_{1}\) and \(\hat{\beta}_{1}\) to be similar or very different? Explain. ii. If \(x_{1}\) is almost uncorrelated with \(x_{2}\) and \(x_{3},\) but \(x_{2}\) and \(x_{3}\) are highly correlated, will \(\tilde{\beta}_{1}\) and \(\hat{\beta}_{1}\) tend to be similar or very different? Explain. iii. If \(x_{1}\) is highly correlated with \(x_{2}\) and \(x_{3}\), and \(x_{2}\) and \(x_{3}\) have small partial effects on \(y\), would you expect \(\operatorname{se}\left(\tilde{\beta}_{1}\right)\) or \(\operatorname{se}\left(\hat{\beta}_{1}\right)\) to be smaller? Explain. iv. If \(x_{1}\) is almost uncorrelated with \(x_{2}\) and \(x_{3}, x_{2}\) and \(x_{3}\) have large partial effects on \(y,\) and \(x_{2}\) and \(x_{3}\) are highly correlated, would you expect \(\operatorname{se}\left(\tilde{\beta}_{1}\right)\) or \(\operatorname{se}\left(\hat{\beta}_{1}\right)\) to be smaller? Explain.

i. Consider the simple regression model \(y=\beta_{0}+\beta_{1} x+u\) under the first four Gauss-Markov assumptions. For some function \(g(x),\) for example \(g(x)=x^{2}\) or \(g(x)=\log \left(1+x^{2}\right),\) define \(z_{i}=g\left(x_{i}\right) .\) Define a slope estimator as $$ \tilde{\beta}_{1}=\left(\sum_{i=1}^{n}\left(z_{i}-\bar{z}\right) y_{i}\right) /\left(\sum_{i=1}^{n}\left(z_{i}-\bar{z}\right) x_{i}\right) $$ Show that \(\bar{\beta}_{1}\) is linear and unbiased. Remember, because \(\mathrm{E}(u | x)=0,\) you can treat both \(x_{i}\) and \(z_{i}\) as nonrandom in your derivation. ii. Add the homoskedasticity assumption, MLR.5. Show that $$ \operatorname{Var}\left(\tilde{\beta}_{1}\right)=\sigma^{2}\left(\sum_{i=1}^{n}\left(z_{i}-\bar{z}\right)^{2}\right) /\left(\sum_{i=1}^{n}\left(z_{i}-\bar{z}\right) x_{i}\right)^{2} $$ iii. Show directly that, under the Gauss-Markov assumptions, Var( \(\left.\hat{\beta}_{1}\right) \leq \operatorname{Var}\left(\tilde{\beta}_{1}\right),\) where \(\hat{\beta}_{1}\) is the OLS estimator. [Hint: The Cauchy-Schwartz inequality in Appendix B implies that $$ \left(n^{-1} \sum_{i=1}^{n}\left(z_{i}-\bar{z}\right)\left(x_{i}-\bar{x}\right)\right)^{2} \leq\left(n^{-1} \sum_{i=1}^{n}\left(z_{i}-\bar{z}\right)^{2}\right)\left(n^{-1} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}\right) $$ notice that we can drop \(\bar{x}\) from the sample covariance.
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