# Multiple Regression

Consider the following model, which was estimated on 133 observations
$\begin{tabular}{rllll} y_i=~&~~2.23+~&~~0.63x_{1i}+~~&~~0.17x_{2i}+~&~\hat{\epsilon_i}~\\ &~(1.11)~&~(0.16)~&~(0.06)~&~\\ \end{tabular}$
where $~R^2=0.224$ and standard errors are in parentheses. Test the joint significance of the regression.

Consider the following model, which was estimated on 133 observations
$\begin{tabular}{rllll} y_i=~&~~2.23+~&~~0.63x_{1i}+~~&~~0.17x_{2i}+~&~\hat{\epsilon_i}~\\ &~(1.11)~&~(0.16)~&~(0.06)~&~\\ \end{tabular}$
Standard errors are in parentheses. Assuming $~cov(x_{1i},~x_{2i})=0~$. Test the joint significance of the regression.

Consider the following model, which was estimated on 133 observations
$y_i=1.13+0.44x_{i}-~0.03x^2_{i}+\hat{\epsilon_i}$
Find the value of $~x_i~$ at which the function is maximised.

Consider the following model, which was estimated on 133 observations
$y_i=1.13+0.44x_{i}-~~0.03ln(x_{i})+\hat{\epsilon_i}$
Find the value of $~x_i~$ at which the function is minimised.

Consider the following model:
$y_i=1.13+0.44x_{i}-~0.03x^2_{i}+\hat{\epsilon_i}$
Find $~\delta{y}/\delta{ln(x)}~$.

Consider the following model, which was estimated on 133 observations
$y_i=1.13-0.36x_{1i}+~0.07x^2_{1i}+\hat{\epsilon_i}$
where the variance-covariance matrix is written
$\left~[ \begin{tabular}{lll} 1.01~&~0.11~&~0.04~\\ 0.11~&~0.29~&~-0.03~\\ 0.04~&~-0.03~&~0.10~\\ \end{tabular} \right~]$
Test the hypothesis that the marginal effect of $~y~$ w.r.t $~x_{1i}~$ is zero when $~x_{1i}=2~$.

Consider the following model, which was estimated on 133 observations
$y_i=1.13-0.36x_{1i}-~0.07(1/x_{1i})+\hat{\epsilon_i}$
where the variance-covariance matrix is written $\left~[ \begin{tabular}{lll} 1.01~&~0.11~&~0.04~\\ 0.11~&~0.29~&~-0.03~\\ 0.04~&~-0.03~&~0.10~\\ \end{tabular} \right~]$
Test the hypothesis the marginal effect of $~y~$ w.r.t $~x_{1i}~$ is zero when $~x_{1i}=1~$.

Consider the following model estimated on a sample of 81 observations:
$\begin{tabular}{rlllll} y_i=~&~~1.16+~&~~1.43x_{1i}+~~&~~0.77x_{2i}+~~~&~~0.23x_{3i}+~&~\hat{\epsilon_i}~\\ &~(0.35)~&~(0.26)~&~(0.21)~&~(0.11)~&~\\ \end{tabular}$
where standard errors are in parentheses and RSS=7.36. Calculate the RSS of a model which imposes the hypothesis that the coefficient in $~x_{1i}~$ is unity.

Consider the following model estimated on a sample of 81 observations:
$\begin{tabular}{rlllll} y_i=~&~~1.16+~&~~1.43x_{1i}+~~&~~0.77x_{2i}+~~~&~~0.23x_{3i}+~&~\hat{\epsilon_i}~\\ &~(0.26)~&~(0.44)~&~(0.26)~&~(0.11)~&~\\ \end{tabular}$
where standard errors are in parentheses and RSS=7.36. Imposing the hypothesis that the coefficient on $~x_{1i}~$ is unity and $~x_{2i}~$ is zero yielded a RSS of 7.95. Calculate the value of the test for the hypothesis.

Consider the following a random sample of 2714 individual young workers in the U.S. [National Longitudinal Survey of Youth 1979 (NLSY79)], who were aged between 14 and 21 in 1979. The following output is based on a model to explain earnings:
$\begin{tabular}{lllll} learnings~&~Coef~&~Std~Err~&~t~&~P>|t|~\\ tenure~&~.0338157~&~.0049998~&6.76~&~0.000~\\ tenure2~&~-.0007474~&~.0002304~&~-3.24~&~0.001~~~~\\ age~&~-.0072442~&~.0043806~&~-1.65~&~0.098~~~~\\ female~&~-.2888998~&~.0189968~&~-15.21~&~~0.000~\\ ethnic2~&~.0139359~&~x_1~&~0.34~&~0.735~\\ ethnic3~&~-.0786273~&~~.029267~&~-2.69~&~x_2~\\ school~&~.0818975~~&~.0050734~&~x_3~&~0.000~\\ asvabc~&~.0115505~&~.0013143~&~8.79~&~0.000~~~~~\\ cons~&~1.34236~~&~.1830481~&~7.33~&~0.000~~~~~\\ \end{tabular}$
where: learnings=ln(earnings), female=1 if female and 0 if male, tenure is a measure of work experience, tenure2=tenure*tenure, school is a measure of years of schooling, asvabc=a measure of arithmetic and reading ability, ethnic2=1 if hispanic, 0 otherwise, ethnic3=1 if black, 0 otherwise (ethnic1 taken as the default dummy variable). Construct a 95% CI for the coefficient on school:

Using the result given the earlier question regarding earnings equation for the sample of 2714 individual young workers in the U.S. [National Longitudinal Survey of Youth 1979 (NLSY79)], which of the following is the best interpretation of the coefficient on the variable $~female~$.

Using the result given the earlier question regarding earnings equation for the sample of 2714 individual young workers in the U.S. [National Longitudinal Survey of Youth 1979 (NLSY79)], if the variable $~female~$ is replaced by the variable $~male~$. Which of the following statements would be correct interpretation of the coefficient on $~male~$.

Using the result given the earlier question regarding earnings equation for the sample of 2714 individual young workers in the U.S. [National Longitudinal Survey of Youth 1979 (NLSY79)], which of the following is the best interpretation of the coefficient on the variable $~school~$.

Using the result given the earlier question regarding earnings equation for the sample of 2714 individual young workers in the U.S. [National Longitudinal Survey of Youth 1979 (NLSY79)], suppose you drop the variable ethnic3 and instead include the variable ethnic1. What would be the coefficient estimates on the variables ethnic1 and ethnic2.

• Unrestricted R^2
• DoF unrestricted model
• Number of restrictions
• Unrestricted error variance
• DoF restricted model
• Restricted R^2
• Restricted error variance
• F-test
• Critical Value
• 0.650
• 15.41
• 0.497
• 0.685
• 0.504
• 17.12
• 3
• 1.153
• 34
• 2.911
• 31

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