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Chapter 4: Problem 8

Consider the multiple regression model with three independent variables, under the classical linear model assumptions MLR.1. MLR.2. MLR.3. MLR.4, MLR.5 and MLR.6: $$y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+u$$ You would like to test the null hypothesis \(\mathrm{H}_{0}: \beta_{1}-3 \beta_{2}=1\) i. Let \(\widehat{\beta}_{1}\) and \(\widehat{\beta}_{2}\) denote the OLS estimators of \(\beta_{1}\) and \(\beta_{2}\). Find \(\operatorname{Var}\left(\widehat{\beta}_{1}-3 \widehat{\beta}_{2}\right)\) in terms of the variances of \(\widehat{\beta}_{1}\) and \(\widehat{\beta}_{2}\) and the covariance between them. What is the standard error of \(\widehat{\beta}_{1}-3 \widehat{\beta}_{2} ?\) ii. Write the \(t\) statistic for testing \(\mathrm{H}_{0}: \beta_{1}-3 \beta_{2}=1\) iii. Define \(\theta_{1}=\beta_{1}-3 \beta_{2}\) and \(\hat{\theta}_{1}=\hat{\beta}_{1}-3 \hat{\beta}_{2} .\) Write a regression equation involving \(\beta_{0}, \theta_{1}, \beta_{2},\) and \(\beta_{3}\) that allows you to directly obtain \(\hat{\theta}_{1}\) and its standard error.

Short Answer

Expert verified
The variance is \(\operatorname{Var}(\widehat{\beta}_{1} - 3\widehat{\beta}_{2}) = \operatorname{Var}(\widehat{\beta}_{1}) + 9\operatorname{Var}(\widehat{\beta}_{2}) - 6\operatorname{Cov}(\widehat{\beta}_{1}, \widehat{\beta}_{2})\). The t-statistic is \(t = \frac{\hat{\theta}_{1} - 1}{SE(\hat{\theta}_{1})}\). The regression equation is \(y = \beta_{0} + \theta_{1}x_{1} + (\beta_{2} + 3\theta_{1})x_{2} + \beta_{3}x_{3} + u\).

Step by step solution

01

Derive the Variance Expression

We need to find \(\operatorname{Var}(\widehat{\beta}_{1} - 3\widehat{\beta}_{2})\). Using the properties of variance, this can be expressed as follows:\[\operatorname{Var}(\widehat{\beta}_{1} - 3\widehat{\beta}_{2}) = \operatorname{Var}(\widehat{\beta}_{1}) + 9\operatorname{Var}(\widehat{\beta}_{2}) - 6\operatorname{Cov}(\widehat{\beta}_{1}, \widehat{\beta}_{2}).\]This formula arises because when finding the variance of a linear combination, \(aX + bY\), the variance is \(a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y) + 2ab \operatorname{Cov}(X, Y)\). Here, \(a = 1\) and \(b = -3\).
02

Calculate the Standard Error

The standard error of \(\widehat{\beta}_{1} - 3\widehat{\beta}_{2}\) is the square root of its variance obtained in Step 1. Thus, it is:\[SE(\widehat{\beta}_{1} - 3\widehat{\beta}_{2}) = \sqrt{\operatorname{Var}(\widehat{\beta}_{1}) + 9\operatorname{Var}(\widehat{\beta}_{2}) - 6 \operatorname{Cov}(\widehat{\beta}_{1}, \widehat{\beta}_{2})}.\]
03

Formulate the t-Statistic

To test the null hypothesis \(\mathrm{H}_{0}: \beta_{1} - 3\beta_{2} = 1\), we use the \(t\)-statistic. The measured parameter is \(\hat{\theta}_{1} = \widehat{\beta}_{1} - 3\widehat{\beta}_{2}\), and the \(t\)-statistic is given by:\[t = \frac{\hat{\theta}_{1} - 1}{SE(\hat{\theta}_{1})}.\]
04

Write the Regression Equation

To obtain \(\hat{\theta}_{1}\) directly, redefine the model in terms of \(\theta_{1}\).Consider the model as:\[y = \beta_{0} + \theta_{1}x_{1} + (\beta_{2} + 3\theta_{1})x_{2} + \beta_{3}x_{3} + u.\]We reparametrize such that: \( \theta_{1} = \beta_{1} - 3\beta_{2} \).In this model, the coefficient for \(x_1\) now directly gives us \(\hat{\theta}_{1}\), facilitating direct estimation.

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Key Concepts

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

OLS estimators
OLS, or Ordinary Least Squares, is a statistical method used to determine the relationship between variables in multiple regression analysis. In simple terms, OLS helps to find the line of best fit through the data points to minimize the differences—called residuals—between observed and predicted values.

When dealing with multiple regression analysis, OLS estimators help us to estimate the coefficients of the independent variables. For each independent variable, OLS provides an estimator, \(\hat{\beta}\). These estimators are crucial because they allow us to predict the dependent variable based on changes in the independent variables.

For example, in the equation \(y=\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+u\), the OLS estimators \(\widehat{\beta}_{1}\) and \(\widehat{\beta}_{2}\) are calculated to find the best approximation of the true coefficients \(\beta_{1}\) and \(\beta_{2}\). This lets us make inferences about how a change in each independent variable \(x_1, x_2\) affects the dependent variable \(y\). Through this process, we achieve a model that correctly describes the relationship highlighted in the dataset.
variance and covariance
In the context of multiple regression, variance and covariance are fundamental concepts to understand the precision and interdependence of the OLS estimators.

**Variance**
Variance measures the dispersion of a set of predictions around their mean value. For OLS estimators, the variance provides insight into the precision of the estimator. A lower variance indicates more reliable estimates.

In our problem, we encountered the task of finding \(\operatorname{Var}(\widehat{\beta}_{1} - 3\widehat{\beta}_{2})\). Using the variance formula for linear combinations, we have:\[\operatorname{Var}(\widehat{\beta}_{1} - 3\widehat{\beta}_{2}) = \operatorname{Var}(\widehat{\beta}_{1}) + 9\operatorname{Var}(\widehat{\beta}_{2}) - 6\operatorname{Cov}(\widehat{\beta}_{1}, \widehat{\beta}_{2}).\]This calculation considers both individual variances and their linear combination.

**Covariance**
Covariance describes how two variables change together. In regression, the covariance between two OLS estimators helps us understand whether changes in one estimator are associated with changes in another.

Thus, when we have the covariance term like \(\operatorname{Cov}(\widehat{\beta}_{1}, \widehat{\beta}_{2})\), it informs us about the joint variability that affects these estimators and should be factored into our analysis of variance in the model.
t-statistic
The t-statistic is an essential concept in hypothesis testing within the context of multiple regression. It tells us how many standard errors our estimated coefficient is away from the hypothesized value under the null hypothesis.

When testing a null hypothesis like \(\mathrm{H}_{0}: \beta_{1}-3\beta_{2}=1\), we use the t-statistic to determine whether the observed coefficients are statistically significantly different from 1.

The formula for the t-statistic in this scenario is:\[t = \frac{\hat{\theta}_{1} - 1}{SE(\hat{\theta}_{1})},\]where \(\hat{\theta}_{1} = \widehat{\beta}_{1} - 3\widehat{\beta}_{2}\) is the estimated difference we are testing and \(SE(\hat{\theta}_{1})\) is the standard error of the estimate.

A larger absolute value of the t-statistic indicates stronger evidence against the null hypothesis. Whether this value is significantly large is often determined by comparing it to critical values from the t-distribution, depending on the desired confidence level and degrees of freedom.
null hypothesis testing
Null hypothesis testing is a statistical method used to determine the validity of a claim, or hypothesis, about a population parameter. It involves a process of making inferences about population parameters based on sample data.

The null hypothesis, denoted as \(\mathrm{H}_{0}\), is a statement of no effect or no difference, which acts as the starting point for statistical testing. In our context, the null hypothesis \(\mathrm{H}_{0}: \beta_{1} - 3\beta_{2} = 1\) suggests that the difference between these coefficients equals one.

During the hypothesis testing process, you:
  • Calculate a test statistic (like the t-statistic) using your sample data.
  • Compare the test statistic to a threshold, often called the critical value, which is derived from a chosen significance level (e.g., \(\alpha = 0.05\)).
  • Decide whether to reject or fail to reject the null hypothesis based on this comparison. If the test statistic is extreme, you might reject \(\mathrm{H}_{0}\), suggesting your sample provides enough evidence to conclude the effect or difference is real.
This testing process helps in determining the statistical significance and reliable interpretations from linear regression models in multiple variable scenarios.

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Most popular questions from this chapter

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 following table was created using the data in CEOSAL2, where standard errors are in parentheses below the coefficients: The variable mktval is market value of the firm, profmarg is profit as a percentage of sales, ceoten is years as CEO with the current company, and comten is total years with the company. i. Comment on the effect of profmarg on CEO salary. ii. Does market value have a significant effect? Explain. iii. Interpret the coefficients on ceoten and comten. Are these explanatory variables statistically significant? iv. What do you make of the fact that longer tenure with the company, holding the other factors fixed, is associated with a lower salary?

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)?

In Example 4.7 . we used data on nonunionized manufacturing firms to estimate the relationship between the scrap rate and other firm characteristics. We now look at this example more closely and use all available firms. i. The population model estimated in Example 4.7 can be written as $$\log (\operatorname{scrap})=\beta_{0}+\beta_{1} \text {hrsemp}+\beta_{2} \log (\text {sales})+\beta_{3} \log (\text {employ})+u$$ Using the 43 observations available for 1987 , the estimated equation is $$\begin{aligned} \widehat{\log (\text {scrap})}=& 11.74-.042 \text { hrsemp}-.951 \log (\text {sales})+.992 \log (\text {employ}) \\ &(4.57)(.019) \\ n=& 43, R^{2}=.310 \end{aligned}$$ Compare this equation to that estimated using only the 29 nonunionized firms in the sample. ii. Show that the population model can also be written as $$\log (s c r a p)=\beta_{0}+\beta_{1} h r s e m p+\beta_{2} \log (s a l e s / e m p l o y)+\theta_{3} \log (e m p l o y)+u$$ where \(\left.\theta_{3}=\beta_{2}+\beta_{3} . \text { [Hint: Recall that } \log \left(x_{2} / x_{3}\right)=\log \left(x_{2}\right)-\log \left(x_{3}\right) .\right]\) Interpret the hypothesis \(\mathrm{H}_{0}: \theta_{3}=0\) iii. When the equation from part (ii) is estimated, we obtain $$\begin{aligned} \widehat{\log (\text {scrap})}=& 11.74-.042 \text { hrsemp}-.951 \log (\text {sales/employ})+.041 \log (\text {employ}) \\ &(4.57)(.019) \\ n=& 43, R^{2}=.310 \end{aligned}$$ Controlling for worker training and for the sales-to-employee ratio, do bigger firms have larger statistically significant scrap rates? iv. Test the hypothesis that a \(1 \%\) increase in sales/employ is associated with a \(1 \%\) drop in the scrap rate.

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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