Question 1

Figure
Suppose that in the course of testing whether Age and Family Size are jointly significant,you estimate the results in Figure 6.3. $\begin{array}{lc}
\text { SUMMARY OUTPUT }\
\
\
\hline\text { Regrestion Starittict } & \
\hline \text { Muftiple R. } & 0.070494476 \
\text { R Sequrt } & 0.004969471 \
\text { Adigated R. Square } & 0.001879314 \
\text { Standard Esror } & 4.819403326 \
\text { Observations } & 970 \
\hline
\end{array}$

$\begin{array}{l}
\text { ANOVA }\
\begin{array}{lccccc}
\hline & d f & S 5 & M S & F & \text { Signifieance } F \
\hline \text { Regranion } & 3 & 112.0566012 & 37.3522004 & 1.608161442 & 0.185858614 \
\text { Reaidual } & 966 & 22436.94237 & 23.22664841 & & \
\text { Total } & 969 & 22548.99897 & & & \
\hline
\end{array}
\end{array}$

$\begin{array}{lcccccc}
\hline & \text { Coeffeientt } & \text { Srandard Error } & \text { t Stat } & \text { P-value } & \text { Lower } 9596 & \text { Upper } 9596 \
\hline \text { Intercept } & 4.049920982 & 1.042107341 & 3.886280064 & 0.000108739 & 2.004865844 & 6.094976119 \
\text { Age } & 0.015626984 & 0.010365497 & 1.507396119 & 0.131984878 & -0.004714504 & 0.035968471 \
\text { Family Size } & -0.093093463 & 0.084602383 & -1.100364552 & 0.271447442 & -0.259119103 & 0.072932177 \
\text { Years of Education } & 0.005642075 & 0.06474525 & 0.087142685 & 0.930576157 & -0.121415476 & 0.132699626 \
\hline
\end{array}$

$\text { SUMALARY OUTPUT }$

$\begin{array}{lc}
\hline \text { Regrestion Staritticz } & \
\hline \text { Multiple R. } & 0.005034034 \
\text { R Square } & 2.53415 \mathrm{E}-05 \
\text { Adjuted R. Square } & -0.00100769 \
\text { Standard Error } & 4.826368211 \
\text { Observations } & 970 \
\hline
\end{array}$$0.115170636$

$\begin{array}{l}
\text { ANOVA }\
\begin{array}{lccccc}
\hline & d f & S S & M S & F & \text { Signffeance } F \
\hline \text { Regresion } & 1 & 0.571425385 & 0.571425385 & 0.024531191 & 0.875573561 \
\text { Reiidual } & 968 & 22548.42754 & 23.29383011 & & \
\text { Total } & 969 & 22548.99897 & & & \
\hline
\end{array}
\end{array}$

$\begin{array}{l}
\begin{array}{lcccccc}
\hline & \text { Coefficienty } & \text { Standard Emor } & \text { t Star } & \text { P-value } & \text { Lower } 9596 & \text { Upper } 9596 \
\hline \text { Intercept } & 4.288825645 & 0.732966216 & 5.851327866 & 6.66867 E-09 & 2.850439806 & 5.727211483 \
\text { Years of Education } & 0.009850586 & 0.062893064 & 0.156624361 & 0.875573561 & -0.113571873 & 0.133273045 \
\hline
\end{array}\
\text { Figure } 6.3
\end{array}$
-Chow tests are based on comparing the
A)sum of the unexplained sums of squares added together from separate regressions to the unexplained sum of squares for the pooled regression.
B)unexplained sums of squares from separate regressions to each other.
C)sum of the explained sums of squares from separate regressions to the explained sum of squares for the pooled regression.
D)explained sums of squares from separate regressions to each other.

Accepted Answer

The Chow test is used to test the equality of the coefficients in two or more regression models. It is based on comparing the unexplained sums of squares from separate regressions to the unexplained sum of squares for the pooled regression. Hence, the correct choice is A.

Question 2

What is the coefficient of determination? What information does it provide? Explain.

Accepted Answer

The answer of What is the coefficient of determination? What...

Question 3

Suppose you are estimating salary as a function of age,education,hours of work and the number of young children and you are concerned that the salary functions differ for men and women.You could test this possibility by performing a
A)t-test for individual significance.
B)t-test for joint significance.
C)Chow test.
D)subset test.

Accepted Answer

The Chow test can be used to test for structural break in the relationship between the independent variables and the dependent variable. In this case, it can help determine if there is a significant difference in the salary functions between men and women. The other options, such as t-tests and subset tests, may not be appropriate in testing for structural differences between two groups. Therefore, the best choice is the Chow test.

Question 4

Figure:
Suppose that in the course of testing whether salary functions differ for males and females,you estimate the pooled and male and female results in Figure 6.4. $\text { Pooled }$

$\begin{array}{l}
\text { ANOVA }\
\begin{array}{lccccc}
\hline & d f & S S & M S & F & \text { Significance } F \
\hline \text { Regression } & 4 & 2.30931 \mathrm{E}+12 & 5.77328 \mathrm{E}+11 & 282.1787278 & 1.2198 \mathrm{E}-215 \
\text { Residual } & 4286 & 8.76901 \mathrm{E}+12 & 2045965635 & & \
\text { Total } & 4290 & 1.10783 \mathrm{E}+13 & & & \
\hline
\end{array}
\end{array}$

$\text { Male }$

$\begin{array}{l}
\text { ANOVA }\
\begin{array}{lccccc}
\hline & d f & S S & M S & F & \text { Significance } F \
\hline \text { Regression } & 4 & 1.54309 \mathrm{E}+12 & 3.85772 \mathrm{E}+11 & 131.8489492 & 9.4649 \mathrm{E}-101 \
\text { Residual } & 2136 & 6.24964 \mathrm{E}+12 & 2925860190 & & \
\text { Total } & 2140 & 7.79272 \mathrm{E}+12 & & & \
\hline
\end{array}
\end{array}$

$\text { Female }$

$\begin{array}{lccccc} 
\text { ANOVA }\
\hline& \text { of } & S S & M S & F & \text { Significance } F \
\hline \text { Regression } & 4 & 6.15055 \mathrm{E}+11 & 1.53764 \mathrm{E}+11 & 153.1758618 & 2.2255 \mathrm{E}-115 \
\text { Residual } & 2145 & 2.15323 \mathrm{E}+12 & 1003838471 & & \
\text { Total } & 2149 & 2.76829 \mathrm{E}+12 & & &\
\hline
\end{array}$
Figure 6.4
-The appropriate critical value for the Chow test in Figure 6.4 is
A)1.927.
B)2.032.
C)2.463.
D)2.978.

Accepted Answer

The answer of Figure:
Suppose that in the course of testing...

Question 5

What are the multiple linear regression assumptions required for OLS to be BLUE? Explain why each one is important.

Accepted Answer

The answer of What are the multiple linear regression assumptions...

Question 6

How do you perform a test of the overall significance of the regression function? What are the null and alternative hypothesis for this test? What is the rejection rule? What is the intuition for why the test works? Explain.

Accepted Answer

The answer of How do you perform a test of...

Question 7

Figure:
Suppose that in the course of testing whether salary functions differ for males and females,you estimate the pooled and male and female results in Figure 6.4. $\text { Pooled }$

$\begin{array}{l}
\text { ANOVA }\
\begin{array}{lccccc}
\hline & d f & S S & M S & F & \text { Significance } F \
\hline \text { Regression } & 4 & 2.30931 \mathrm{E}+12 & 5.77328 \mathrm{E}+11 & 282.1787278 & 1.2198 \mathrm{E}-215 \
\text { Residual } & 4286 & 8.76901 \mathrm{E}+12 & 2045965635 & & \
\text { Total } & 4290 & 1.10783 \mathrm{E}+13 & & & \
\hline
\end{array}
\end{array}$

$\text { Male }$

$\begin{array}{l}
\text { ANOVA }\
\begin{array}{lccccc}
\hline & d f & S S & M S & F & \text { Significance } F \
\hline \text { Regression } & 4 & 1.54309 \mathrm{E}+12 & 3.85772 \mathrm{E}+11 & 131.8489492 & 9.4649 \mathrm{E}-101 \
\text { Residual } & 2136 & 6.24964 \mathrm{E}+12 & 2925860190 & & \
\text { Total } & 2140 & 7.79272 \mathrm{E}+12 & & & \
\hline
\end{array}
\end{array}$

$\text { Female }$

$\begin{array}{lccccc} 
\text { ANOVA }\
\hline& \text { of } & S S & M S & F & \text { Significance } F \
\hline \text { Regression } & 4 & 6.15055 \mathrm{E}+11 & 1.53764 \mathrm{E}+11 & 153.1758618 & 2.2255 \mathrm{E}-115 \
\text { Residual } & 2145 & 2.15323 \mathrm{E}+12 & 1003838471 & & \
\text { Total } & 2149 & 2.76829 \mathrm{E}+12 & & &\
\hline
\end{array}$
Figure 6.4
-Based on the results in Figure 6.4,you should
A)reject the null hypothesis.
B)fail to reject the null hypothesis.
C)reject the alternative hypothesis.
D)fail to reject the alternative hypothesis.

Accepted Answer

The significance level for all three ANOVA tests is extremely low, meaning it is highly unlikely that the differences between male and female salaries are due to chance. Therefore, the null hypothesis can be rejected in favor of the alternative hypothesis, which states that there is a significant difference in salaries between males and females.

Question 8

How does the Adjusted R-squared differ from the R-squared? Why would the Adjusted R-squared be preferred to the R-squared?

Accepted Answer

The answer of How does the Adjusted R-squared differ from...

Question 9

Figure:
Suppose that in the course of testing whether salary functions differ for males and females,you estimate the pooled and male and female results in Figure 6.4. $\text { Pooled }$

$\begin{array}{l}
\text { ANOVA }\
\begin{array}{lccccc}
\hline & d f & S S & M S & F & \text { Significance } F \
\hline \text { Regression } & 4 & 2.30931 \mathrm{E}+12 & 5.77328 \mathrm{E}+11 & 282.1787278 & 1.2198 \mathrm{E}-215 \
\text { Residual } & 4286 & 8.76901 \mathrm{E}+12 & 2045965635 & & \
\text { Total } & 4290 & 1.10783 \mathrm{E}+13 & & & \
\hline
\end{array}
\end{array}$

$\text { Male }$

$\begin{array}{l}
\text { ANOVA }\
\begin{array}{lccccc}
\hline & d f & S S & M S & F & \text { Significance } F \
\hline \text { Regression } & 4 & 1.54309 \mathrm{E}+12 & 3.85772 \mathrm{E}+11 & 131.8489492 & 9.4649 \mathrm{E}-101 \
\text { Residual } & 2136 & 6.24964 \mathrm{E}+12 & 2925860190 & & \
\text { Total } & 2140 & 7.79272 \mathrm{E}+12 & & & \
\hline
\end{array}
\end{array}$

$\text { Female }$

$\begin{array}{lccccc} 
\text { ANOVA }\
\hline& \text { of } & S S & M S & F & \text { Significance } F \
\hline \text { Regression } & 4 & 6.15055 \mathrm{E}+11 & 1.53764 \mathrm{E}+11 & 153.1758618 & 2.2255 \mathrm{E}-115 \
\text { Residual } & 2145 & 2.15323 \mathrm{E}+12 & 1003838471 & & \
\text { Total } & 2149 & 2.76829 \mathrm{E}+12 & & &\
\hline
\end{array}$
Figure 6.4
-Based on the results of you the Chow test in Figure 6.4 you should
A)focus on the pooled sample results.
B)focus on the male and female subsample results.
C)collect more data that does not fail the Chow test.
D)focus on the male results only.

Accepted Answer

The Chow test is used to determine if there is a statistically significant difference between two regression models estimated on different subsamples of data. In this case, the Chow test suggests that the regression models for males and females are significantly different, so we cannot combine the two subsamples in the pooled estimate. Therefore, we should focus on the male and female subsample results separately.

Question 10

Figure
Suppose that in the course of testing whether Age and Family Size are jointly significant,you estimate the results in Figure 6.3. $\begin{array}{lc}
\text { SUMMARY OUTPUT }\
\
\
\hline\text { Regrestion Starittict } & \
\hline \text { Muftiple R. } & 0.070494476 \
\text { R Sequrt } & 0.004969471 \
\text { Adigated R. Square } & 0.001879314 \
\text { Standard Esror } & 4.819403326 \
\text { Observations } & 970 \
\hline
\end{array}$

$\begin{array}{l}
\text { ANOVA }\
\begin{array}{lccccc}
\hline & d f & S 5 & M S & F & \text { Signifieance } F \
\hline \text { Regranion } & 3 & 112.0566012 & 37.3522004 & 1.608161442 & 0.185858614 \
\text { Reaidual } & 966 & 22436.94237 & 23.22664841 & & \
\text { Total } & 969 & 22548.99897 & & & \
\hline
\end{array}
\end{array}$

$\begin{array}{lcccccc}
\hline & \text { Coeffeientt } & \text { Srandard Error } & \text { t Stat } & \text { P-value } & \text { Lower } 9596 & \text { Upper } 9596 \
\hline \text { Intercept } & 4.049920982 & 1.042107341 & 3.886280064 & 0.000108739 & 2.004865844 & 6.094976119 \
\text { Age } & 0.015626984 & 0.010365497 & 1.507396119 & 0.131984878 & -0.004714504 & 0.035968471 \
\text { Family Size } & -0.093093463 & 0.084602383 & -1.100364552 & 0.271447442 & -0.259119103 & 0.072932177 \
\text { Years of Education } & 0.005642075 & 0.06474525 & 0.087142685 & 0.930576157 & -0.121415476 & 0.132699626 \
\hline
\end{array}$

$\text { SUMALARY OUTPUT }$

$\begin{array}{lc}
\hline \text { Regrestion Staritticz } & \
\hline \text { Multiple R. } & 0.005034034 \
\text { R Square } & 2.53415 \mathrm{E}-05 \
\text { Adjuted R. Square } & -0.00100769 \
\text { Standard Error } & 4.826368211 \
\text { Observations } & 970 \
\hline
\end{array}$$0.115170636$

$\begin{array}{l}
\text { ANOVA }\
\begin{array}{lccccc}
\hline & d f & S S & M S & F & \text { Signffeance } F \
\hline \text { Regresion } & 1 & 0.571425385 & 0.571425385 & 0.024531191 & 0.875573561 \
\text { Reiidual } & 968 & 22548.42754 & 23.29383011 & & \
\text { Total } & 969 & 22548.99897 & & & \
\hline
\end{array}
\end{array}$

$\begin{array}{l}
\begin{array}{lcccccc}
\hline & \text { Coefficienty } & \text { Standard Emor } & \text { t Star } & \text { P-value } & \text { Lower } 9596 & \text { Upper } 9596 \
\hline \text { Intercept } & 4.288825645 & 0.732966216 & 5.851327866 & 6.66867 E-09 & 2.850439806 & 5.727211483 \
\text { Years of Education } & 0.009850586 & 0.062893064 & 0.156624361 & 0.875573561 & -0.113571873 & 0.133273045 \
\hline
\end{array}\
\text { Figure } 6.3
\end{array}$
-When testing for the joint significance of Age and Family Size (Figure 6.3),the appropriate test statistic is
A)37.162.
B)0.115.
C)23.227.
D)1.600.

Accepted Answer

The appropriate test statistic for testing joint significance of Age and Family Size is the F-statistic for the regression model with Age, Family Size and Years of Education as independent variables against the model with just the intercept. From the ANOVA table for the multiple regression model, we can see that the F-statistic for the regression model is 1.608 with a corresponding p-value of 0.186. This means that the probability of getting a test statistic as large as 1.608 is 0.186 assuming that the null hypothesis (that Age and Family Size are not jointly significant) is true. Since the p-value is greater than the significance level of 0.05, we fail to reject the null hypothesis. Therefore, we can conclude that Age and Family Size are not jointly significant in explaining the dependent variable.

Question 11

Write out the estimated sample regression function of y on x₁ and x₂.Explain what each of the estimated values means.

Accepted Answer

The answer of Write out the estimated sample regression function...

Question 12

How do you perform a test of the joint significance of a subset of slope coefficients? What are the null and alternative hypothesis for this test? What is the rejection rule? What is the intuition for why the test works? Explain.

Accepted Answer

The answer of How do you perform a test of...

Question 13

Figure
Suppose that in the course of testing whether Age and Family Size are jointly significant,you estimate the results in Figure 6.3. $\begin{array}{lc}
\text { SUMMARY OUTPUT }\
\
\
\hline\text { Regrestion Starittict } & \
\hline \text { Muftiple R. } & 0.070494476 \
\text { R Sequrt } & 0.004969471 \
\text { Adigated R. Square } & 0.001879314 \
\text { Standard Esror } & 4.819403326 \
\text { Observations } & 970 \
\hline
\end{array}$

$\begin{array}{l}
\text { ANOVA }\
\begin{array}{lccccc}
\hline & d f & S 5 & M S & F & \text { Signifieance } F \
\hline \text { Regranion } & 3 & 112.0566012 & 37.3522004 & 1.608161442 & 0.185858614 \
\text { Reaidual } & 966 & 22436.94237 & 23.22664841 & & \
\text { Total } & 969 & 22548.99897 & & & \
\hline
\end{array}
\end{array}$

$\begin{array}{lcccccc}
\hline & \text { Coeffeientt } & \text { Srandard Error } & \text { t Stat } & \text { P-value } & \text { Lower } 9596 & \text { Upper } 9596 \
\hline \text { Intercept } & 4.049920982 & 1.042107341 & 3.886280064 & 0.000108739 & 2.004865844 & 6.094976119 \
\text { Age } & 0.015626984 & 0.010365497 & 1.507396119 & 0.131984878 & -0.004714504 & 0.035968471 \
\text { Family Size } & -0.093093463 & 0.084602383 & -1.100364552 & 0.271447442 & -0.259119103 & 0.072932177 \
\text { Years of Education } & 0.005642075 & 0.06474525 & 0.087142685 & 0.930576157 & -0.121415476 & 0.132699626 \
\hline
\end{array}$

$\text { SUMALARY OUTPUT }$

$\begin{array}{lc}
\hline \text { Regrestion Staritticz } & \
\hline \text { Multiple R. } & 0.005034034 \
\text { R Square } & 2.53415 \mathrm{E}-05 \
\text { Adjuted R. Square } & -0.00100769 \
\text { Standard Error } & 4.826368211 \
\text { Observations } & 970 \
\hline
\end{array}$$0.115170636$

$\begin{array}{l}
\text { ANOVA }\
\begin{array}{lccccc}
\hline & d f & S S & M S & F & \text { Signffeance } F \
\hline \text { Regresion } & 1 & 0.571425385 & 0.571425385 & 0.024531191 & 0.875573561 \
\text { Reiidual } & 968 & 22548.42754 & 23.29383011 & & \
\text { Total } & 969 & 22548.99897 & & & \
\hline
\end{array}
\end{array}$

$\begin{array}{l}
\begin{array}{lcccccc}
\hline & \text { Coefficienty } & \text { Standard Emor } & \text { t Star } & \text { P-value } & \text { Lower } 9596 & \text { Upper } 9596 \
\hline \text { Intercept } & 4.288825645 & 0.732966216 & 5.851327866 & 6.66867 E-09 & 2.850439806 & 5.727211483 \
\text { Years of Education } & 0.009850586 & 0.062893064 & 0.156624361 & 0.875573561 & -0.113571873 & 0.133273045 \
\hline
\end{array}\
\text { Figure } 6.3
\end{array}$
-When testing for the joint significance of Age and Family Size (Figure 6.3),you should
A)reject the null hypothesis because 1.600 < 2.61.
B)fail to reject the null hypothesis because 1.600 < 2.61.
C)reject the null hypothesis because 1.600 < 8.53.
D)fail to reject the null hypothesis because 1.600 < 8.53.

Accepted Answer

In the ANOVA table, we see that the p-value for the regression is 0.186, which is greater than the significance level of 0.05. Therefore, we fail to reject the null hypothesis and conclude that Age and Family Size are not jointly significant. The F-statistic of 1.61 is not relevant in this case as it only tests the significance of the overall regression, not the joint significance of specific variables.

Question 14

Why is multiple linear regression analysis such a valuable tool? Explain.

Accepted Answer

The answer of Why is multiple linear regression analysis such...

Question 15

Figure
Suppose that in the course of testing whether Age and Family Size are jointly significant,you estimate the results in Figure 6.3. $\begin{array}{lc}
\text { SUMMARY OUTPUT }\
\
\
\hline\text { Regrestion Starittict } & \
\hline \text { Muftiple R. } & 0.070494476 \
\text { R Sequrt } & 0.004969471 \
\text { Adigated R. Square } & 0.001879314 \
\text { Standard Esror } & 4.819403326 \
\text { Observations } & 970 \
\hline
\end{array}$

$\begin{array}{l}
\text { ANOVA }\
\begin{array}{lccccc}
\hline & d f & S 5 & M S & F & \text { Signifieance } F \
\hline \text { Regranion } & 3 & 112.0566012 & 37.3522004 & 1.608161442 & 0.185858614 \
\text { Reaidual } & 966 & 22436.94237 & 23.22664841 & & \
\text { Total } & 969 & 22548.99897 & & & \
\hline
\end{array}
\end{array}$

$\begin{array}{lcccccc}
\hline & \text { Coeffeientt } & \text { Srandard Error } & \text { t Stat } & \text { P-value } & \text { Lower } 9596 & \text { Upper } 9596 \
\hline \text { Intercept } & 4.049920982 & 1.042107341 & 3.886280064 & 0.000108739 & 2.004865844 & 6.094976119 \
\text { Age } & 0.015626984 & 0.010365497 & 1.507396119 & 0.131984878 & -0.004714504 & 0.035968471 \
\text { Family Size } & -0.093093463 & 0.084602383 & -1.100364552 & 0.271447442 & -0.259119103 & 0.072932177 \
\text { Years of Education } & 0.005642075 & 0.06474525 & 0.087142685 & 0.930576157 & -0.121415476 & 0.132699626 \
\hline
\end{array}$

$\text { SUMALARY OUTPUT }$

$\begin{array}{lc}
\hline \text { Regrestion Staritticz } & \
\hline \text { Multiple R. } & 0.005034034 \
\text { R Square } & 2.53415 \mathrm{E}-05 \
\text { Adjuted R. Square } & -0.00100769 \
\text { Standard Error } & 4.826368211 \
\text { Observations } & 970 \
\hline
\end{array}$$0.115170636$

$\begin{array}{l}
\text { ANOVA }\
\begin{array}{lccccc}
\hline & d f & S S & M S & F & \text { Signffeance } F \
\hline \text { Regresion } & 1 & 0.571425385 & 0.571425385 & 0.024531191 & 0.875573561 \
\text { Reiidual } & 968 & 22548.42754 & 23.29383011 & & \
\text { Total } & 969 & 22548.99897 & & & \
\hline
\end{array}
\end{array}$

$\begin{array}{l}
\begin{array}{lcccccc}
\hline & \text { Coefficienty } & \text { Standard Emor } & \text { t Star } & \text { P-value } & \text { Lower } 9596 & \text { Upper } 9596 \
\hline \text { Intercept } & 4.288825645 & 0.732966216 & 5.851327866 & 6.66867 E-09 & 2.850439806 & 5.727211483 \
\text { Years of Education } & 0.009850586 & 0.062893064 & 0.156624361 & 0.875573561 & -0.113571873 & 0.133273045 \
\hline
\end{array}\
\text { Figure } 6.3
\end{array}$
-Chow tests are used to determine whether
A)an estimated slope coefficient is different for different subsamples of the data.
B)a subset of independent variables is jointly significant.
C)estimated sample regression functions are different for two different subsamples of the data.
D)a subset of estimated slope coefficients is jointly significant for two different subsamples of the data.

Accepted Answer

The Chow test is used to determine if two estimated regression functions in different subsamples of the data are significantly different. Therefore, the correct choice is (C): estimated sample regression functions are different for two different subsamples of the data. Choices (A), (B), and (D) are incorrect because they do not match the purpose of the Chow test.

Question 16

How do you perform a test of the individual significance of a slope coefficient? What are the null and alternative hypothesis for this test? What is the rejection rule? What is the intuition for why the test works? Explain.

Accepted Answer

The answer of How do you perform a test of...

Question 17

Figure Suppose that in the course of testing whether Age and Family Size are jointly significant,you estimate the results in Figure 6.3. $\begin{array}{lc} \text { SUMMARY OUTPUT }\ \ \ \hline\text { Regrestion Starittict } & \ \hline \text { Muftiple R. } & 0.070494476 \ \text { R Sequrt } & 0.004969471 \ \text { Adigated R. Square } & 0.001879314 \ \text { Standard Esror } & 4.819403326 \ \text { Observations } & 970 \ \hline \end{array}$ $\begin{array}{l} \text { ANOVA }\ \begin{array}{lccccc} \hline & d f & S 5 & M S & F & \text { Signifieance } F \ \hline \text { Regranion } & 3 & 112.0566012 & 37.3522004 & 1.608161442 & 0.185858614 \ \text { Reaidual } & 966 & 22436.94237 & 23.22664841 & & \ \text { Total } & 969 & 22548.99897 & & & \ \hline \end{array} \end{array}$ $\begin{array}{lcccccc} \hline & \text { Coeffeientt } & \text { Srandard Error } & \text { t Stat } & \text { P-value } & \text { Lower } 9596 & \text { Upper } 9596 \ \hline \text { Intercept } & 4.049920982 & 1.042107341 & 3.886280064 & 0.000108739 & 2.004865844 & 6.094976119 \ \text { Age } & 0.015626984 & 0.010365497 & 1.507396119 & 0.131984878 & -0.004714504 & 0.035968471 \ \text { Family Size } & -0.093093463 & 0.084602383 & -1.100364552 & 0.271447442 & -0.259119103 & 0.072932177 \ \text { Years of Education } & 0.005642075 & 0.06474525 & 0.087142685 & 0.930576157 & -0.121415476 & 0.132699626 \ \hline \end{array}$ $\text { SUMALARY OUTPUT }$ $\begin{array}{lc} \hline \text { Regrestion Staritticz } & \ \hline \text { Multiple R. } & 0.005034034 \ \text { R Square } & 2.53415 \mathrm{E}-05 \ \text { Adjuted R. Square } & -0.00100769 \ \text { Standard Error } & 4.826368211 \ \text { Observations } & 970 \ \hline \end{array}$$0.115170636$ $\begin{array}{l} \text { ANOVA }\ \begin{array}{lccccc} \hline & d f & S S & M S & F & \text { Signffeance } F \ \hline \text { Regresion } & 1 & 0.571425385 & 0.571425385 & 0.024531191 & 0.875573561 \ \text { Reiidual } & 968 & 22548.42754 & 23.29383011 & & \ \text { Total } & 969 & 22548.99897 & & & \ \hline \end{array} \end{array}$ $\begin{array}{l} \begin{array}{lcccccc} \hline & \text { Coefficienty } & \text { Standard Emor } & \text { t Star } & \text { P-value } & \text { Lower } 9596 & \text { Upper } 9596 \ \hline \text { Intercept } & 4.288825645 & 0.732966216 & 5.851327866 & 6.66867 E-09 & 2.850439806 & 5.727211483 \ \text { Years of Education } & 0.009850586 & 0.062893064 & 0.156624361 & 0.875573561 & -0.113571873 & 0.133273045 \ \hline \end{array}\ \text { Figure } 6.3 \end{array}$ -When testing for the joint significance of Age and Family Size (Figure 6.3),you should A)reject the null hypothesis and conclude that β₂ = β₃ = 0. B)fail to reject the null hypothesis because and conclude that β₂ = β₃ = 0. C)reject the null hypothesis and conclude that at least one of β₂ or β₃ is not equal to 0. D)fail to reject the null hypothesis and conclude that at least one of β₂ or β₃ is not equal to 0.

Accepted Answer

In ANOVA table, the significance F value for the regression model with both Age and Family Size (1.608161442) is higher than the significance F value for the regression model with only Years of Education (0.024531191). This suggests that the regression model with both Age and Family Size does not provide a significant improvement over the model with only Years of Education. Therefore, we should fail to reject the null hypothesis that the coefficients for Age and Family Size are both equal to zero, and conclude that they are not jointly significant.

Question 18

The test statistic for this Chow test is
A) $ \frac{USSpooled / k+1}{\left(U S S_{4}+U S S_{2}\right) / n-2(k+1)} $

B) $\frac { \text { USS } _ { 1 } + \text { USS } _ { 2 } / k + 1 } { \text { (USS } \left. _ { 1 } + \text { USS } _ { 2 } \right) / n - 2 ( k + 1 ) }$

C) $\frac { \left( \text { USS } _ { p o o l e d } - \left( \text { USS } _ { 1 } + \text { USS } _ { 2 } \right) \right) / k + 1 } { \left( U S S _ { 1 } + \text { USS } _ { 2 } \right) / ( k + 1 ) }$

D) $\frac { \left( \text { USS } _ { p o o l e d } - \left( \text { USS } _ { 1 } + U S S _ { 2 } \right) \right) / k + 1 } { \left( U S S _ { 1 } + U S S _ { 2 } \right) / n - 2 ( k + 1 ) }$

Accepted Answer

The correct formula for the test statistic in a Chow test is given by (USSpooled - USS1 - USS2)/[(USS1+USS2)/(n-2*(k+1))], which matches option D.

Question 19

Figure:
Suppose that in the course of testing whether salary functions differ for males and females,you estimate the pooled and male and female results in Figure 6.4. $\text { Pooled }$

$\begin{array}{l}
\text { ANOVA }\
\begin{array}{lccccc}
\hline & d f & S S & M S & F & \text { Significance } F \
\hline \text { Regression } & 4 & 2.30931 \mathrm{E}+12 & 5.77328 \mathrm{E}+11 & 282.1787278 & 1.2198 \mathrm{E}-215 \
\text { Residual } & 4286 & 8.76901 \mathrm{E}+12 & 2045965635 & & \
\text { Total } & 4290 & 1.10783 \mathrm{E}+13 & & & \
\hline
\end{array}
\end{array}$

$\text { Male }$

$\begin{array}{l}
\text { ANOVA }\
\begin{array}{lccccc}
\hline & d f & S S & M S & F & \text { Significance } F \
\hline \text { Regression } & 4 & 1.54309 \mathrm{E}+12 & 3.85772 \mathrm{E}+11 & 131.8489492 & 9.4649 \mathrm{E}-101 \
\text { Residual } & 2136 & 6.24964 \mathrm{E}+12 & 2925860190 & & \
\text { Total } & 2140 & 7.79272 \mathrm{E}+12 & & & \
\hline
\end{array}
\end{array}$

$\text { Female }$

$\begin{array}{lccccc} 
\text { ANOVA }\
\hline& \text { of } & S S & M S & F & \text { Significance } F \
\hline \text { Regression } & 4 & 6.15055 \mathrm{E}+11 & 1.53764 \mathrm{E}+11 & 153.1758618 & 2.2255 \mathrm{E}-115 \
\text { Residual } & 2145 & 2.15323 \mathrm{E}+12 & 1003838471 & & \
\text { Total } & 2149 & 2.76829 \mathrm{E}+12 & & &\
\hline
\end{array}$
Figure 6.4
-The tests statistic for the Chow test in Figure 6.4 is
A)0.0436.
B)9.327.
C)18.654.
D)37.307.

Accepted Answer

The Chow test is used to determine whether there is a structural break in the regression. In this case, we are comparing the pooled results to the male and female results to see if there is a significant difference between the two groups. The test statistic for the Chow test is calculated as follows:

$$F = \frac{(SS_{P} - SS_{M} - SS_{F})/(k_{M}+k_{F})}{(SS_{M} + SS_{F})/(n_{M}+n_{F}-2k_{M}-2k_{F})}$$

where $SS_{P}$, $SS_{M}$, and $SS_{F}$ are the sums of squares for the pooled, male, and female regressions, respectively; $k_{M}$ and $k_{F}$ are the number of variables in the male and female regressions; and $n_{M}$ and $n_{F}$ are the sample sizes for the male and female regressions.

Plugging in the values from the ANOVA tables, we get:

$$F = \frac{(1.10783 \mathrm{E}+13 - 7.79272 \mathrm{E}+12 - 2.76829 \mathrm{E}+12)/(4+4)}{(7.79272 \mathrm{E}+12 + 2.76829 \mathrm{E}+12)/(4290-8-8)} = 37.307$$

Since the Chow test is a test of whether the difference between the two regression models is statistically significant, we compare the test statistic to a critical value from the F distribution with $(k_{M}+k_{F})$ and $(n_{M}+n_{F}-2k_{M}-2k_{F})$ degrees of freedom. At a significance level of 0.05, the critical value for this test is 3.97. Since our test statistic is much larger than this critical value, we reject the null hypothesis that the two regression models are not significantly different and conclude that there is a structural break. Therefore, the best choice is D.

Question 20

Why is the "all other independent variables constant" condition important? Explain.

Accepted Answer

The answer of Why is the "all other independent variables...

Quiz 6: Multiple Linear Regression Analysis

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