Exam 6: Multiple Linear Regression Analysis

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. $\text { Pooled }$ ANOVA df SS MS F Significance F Regression 4 2.30931+12 5.77328+11 282.1787278 1.2198-215 Residual 4286 8.76901+12 2045965635 Total 4290 1.10783+13 $\text { Male }$ ANOVA df SS MS F Significance F Regression 4 1.54309+12 3.85772+11 131.8489492 9.4649-101 Residual 2136 6.24964+12 2925860190 Total 2140 7.79272+12 $\text { Female }$ ANOVA of SS MS F Significance F Regression 4 6.15055+11 1.53764+11 153.1758618 2.2255-115 Residual 2145 2.15323+12 1003838471 Total 2149 2.76829+12 Figure 6.4 -The tests statistic for the Chow test in Figure 6.4 is

Figure Suppose that in the course of testing whether Age and Family Size are jointly significant,you estimate the results in Figure 6.3. SUMMARY OUTPUT Regrestion Starittict Muftiple R. 0.070494476 R Sequrt 0.004969471 Adigated R. Square 0.001879314 Standard Esror 4.819403326 Observations 970 ANOVA df S5 MS F Signifieance F Regranion 3 112.0566012 37.3522004 1.608161442 0.185858614 Reaidual 966 22436.94237 23.22664841 Total 969 22548.99897 Coeffeientt Srandard Error t Stat P-value Lower 9596 Upper 9596 Intercept 4.049920982 1.042107341 3.886280064 0.000108739 2.004865844 6.094976119 Age 0.015626984 0.010365497 1.507396119 0.131984878 -0.004714504 0.035968471 Family Size -0.093093463 0.084602383 -1.100364552 0.271447442 -0.259119103 0.072932177 Years of Education 0.005642075 0.06474525 0.087142685 0.930576157 -0.121415476 0.132699626 $\text { SUMALARY OUTPUT }$ Regrestion Staritticz Multiple R. 0.005034034 R Square 2.53415-05 Adjuted R. Square -0.00100769 Standard Error 4.826368211 Observations 970 $0.115170636$ ANOVA df SS MS F Signffeance F Regresion 1 0.571425385 0.571425385 0.024531191 0.875573561 Reiidual 968 22548.42754 23.29383011 Total 969 22548.99897 Coefficienty Standard Emor t Star P-value Lower 9596 Upper 9596 Intercept 4.288825645 0.732966216 5.851327866 6.66867E-09 2.850439806 5.727211483 Years of Education 0.009850586 0.062893064 0.156624361 0.875573561 -0.113571873 0.133273045 Figure 6.3 -When testing for the joint significance of Age and Family Size (Figure 6.3),the $\text { USS } S _ { \text {restricted } }$ Is

The test statistic for this Chow test is

Tests of joint significance are based on comparing

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. $\text { Pooled }$ ANOVA df SS MS F Significance F Regression 4 2.30931+12 5.77328+11 282.1787278 1.2198-215 Residual 4286 8.76901+12 2045965635 Total 4290 1.10783+13 $\text { Male }$ ANOVA df SS MS F Significance F Regression 4 1.54309+12 3.85772+11 131.8489492 9.4649-101 Residual 2136 6.24964+12 2925860190 Total 2140 7.79272+12 $\text { Female }$ ANOVA of SS MS F Significance F Regression 4 6.15055+11 1.53764+11 153.1758618 2.2255-115 Residual 2145 2.15323+12 1003838471 Total 2149 2.76829+12 Figure 6.4 -Based on the results of you the Chow test in Figure 6.4 you should

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

The intuition behind a test of joint significance is that if a subset of variables is jointly significant,then

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

Figure Suppose that in the course of testing whether Age and Family Size are jointly significant,you estimate the results in Figure 6.3. SUMMARY OUTPUT Regrestion Starittict Muftiple R. 0.070494476 R Sequrt 0.004969471 Adigated R. Square 0.001879314 Standard Esror 4.819403326 Observations 970 ANOVA df S5 MS F Signifieance F Regranion 3 112.0566012 37.3522004 1.608161442 0.185858614 Reaidual 966 22436.94237 23.22664841 Total 969 22548.99897 Coeffeientt Srandard Error t Stat P-value Lower 9596 Upper 9596 Intercept 4.049920982 1.042107341 3.886280064 0.000108739 2.004865844 6.094976119 Age 0.015626984 0.010365497 1.507396119 0.131984878 -0.004714504 0.035968471 Family Size -0.093093463 0.084602383 -1.100364552 0.271447442 -0.259119103 0.072932177 Years of Education 0.005642075 0.06474525 0.087142685 0.930576157 -0.121415476 0.132699626 $\text { SUMALARY OUTPUT }$ Regrestion Staritticz Multiple R. 0.005034034 R Square 2.53415-05 Adjuted R. Square -0.00100769 Standard Error 4.826368211 Observations 970 $0.115170636$ ANOVA df SS MS F Signffeance F Regresion 1 0.571425385 0.571425385 0.024531191 0.875573561 Reiidual 968 22548.42754 23.29383011 Total 969 22548.99897 Coefficienty Standard Emor t Star P-value Lower 9596 Upper 9596 Intercept 4.288825645 0.732966216 5.851327866 6.66867E-09 2.850439806 5.727211483 Years of Education 0.009850586 0.062893064 0.156624361 0.875573561 -0.113571873 0.133273045 Figure 6.3 -When testing for the joint significance of Age and Family Size (Figure 6.3),the appropriate test statistic is

Figure Suppose that in the course of testing whether Age and Family Size are jointly significant,you estimate the results in Figure 6.3. SUMMARY OUTPUT Regrestion Starittict Muftiple R. 0.070494476 R Sequrt 0.004969471 Adigated R. Square 0.001879314 Standard Esror 4.819403326 Observations 970 ANOVA df S5 MS F Signifieance F Regranion 3 112.0566012 37.3522004 1.608161442 0.185858614 Reaidual 966 22436.94237 23.22664841 Total 969 22548.99897 Coeffeientt Srandard Error t Stat P-value Lower 9596 Upper 9596 Intercept 4.049920982 1.042107341 3.886280064 0.000108739 2.004865844 6.094976119 Age 0.015626984 0.010365497 1.507396119 0.131984878 -0.004714504 0.035968471 Family Size -0.093093463 0.084602383 -1.100364552 0.271447442 -0.259119103 0.072932177 Years of Education 0.005642075 0.06474525 0.087142685 0.930576157 -0.121415476 0.132699626 $\text { SUMALARY OUTPUT }$ Regrestion Staritticz Multiple R. 0.005034034 R Square 2.53415-05 Adjuted R. Square -0.00100769 Standard Error 4.826368211 Observations 970 $0.115170636$ ANOVA df SS MS F Signffeance F Regresion 1 0.571425385 0.571425385 0.024531191 0.875573561 Reiidual 968 22548.42754 23.29383011 Total 969 22548.99897 Coefficienty Standard Emor t Star P-value Lower 9596 Upper 9596 Intercept 4.288825645 0.732966216 5.851327866 6.66867E-09 2.850439806 5.727211483 Years of Education 0.009850586 0.062893064 0.156624361 0.875573561 -0.113571873 0.133273045 Figure 6.3 -When testing for the joint significance of Age and Family Size (Figure 6.3),you should

In multiple linear regression analysis,the number of independent variables should be

How do you perform a Chow test for structural differences between two subsets of the data? What is the rejection rule? What is the intuition for why the test works? Explain.

Figure Suppose that in the course of testing whether Age and Family Size are jointly significant,you estimate the results in Figure 6.3. SUMMARY OUTPUT Regrestion Starittict Muftiple R. 0.070494476 R Sequrt 0.004969471 Adigated R. Square 0.001879314 Standard Esror 4.819403326 Observations 970 ANOVA df S5 MS F Signifieance F Regranion 3 112.0566012 37.3522004 1.608161442 0.185858614 Reaidual 966 22436.94237 23.22664841 Total 969 22548.99897 Coeffeientt Srandard Error t Stat P-value Lower 9596 Upper 9596 Intercept 4.049920982 1.042107341 3.886280064 0.000108739 2.004865844 6.094976119 Age 0.015626984 0.010365497 1.507396119 0.131984878 -0.004714504 0.035968471 Family Size -0.093093463 0.084602383 -1.100364552 0.271447442 -0.259119103 0.072932177 Years of Education 0.005642075 0.06474525 0.087142685 0.930576157 -0.121415476 0.132699626 $\text { SUMALARY OUTPUT }$ Regrestion Staritticz Multiple R. 0.005034034 R Square 2.53415-05 Adjuted R. Square -0.00100769 Standard Error 4.826368211 Observations 970 $0.115170636$ ANOVA df SS MS F Signffeance F Regresion 1 0.571425385 0.571425385 0.024531191 0.875573561 Reiidual 968 22548.42754 23.29383011 Total 969 22548.99897 Coefficienty Standard Emor t Star P-value Lower 9596 Upper 9596 Intercept 4.288825645 0.732966216 5.851327866 6.66867E-09 2.850439806 5.727211483 Years of Education 0.009850586 0.062893064 0.156624361 0.875573561 -0.113571873 0.133273045 Figure 6.3 -When testing for the joint significance of Age and Family Size (Figure 6.3),you should

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

A researcher is interested in estimating how the test scores of elementary schools is related to average class size,parents education level (in years),and percent of English learners at the school.The researcher obtains a sample of 200 California public schools and obtains the following results $\widehat { S c o r e } = 332 - 3$ Class Size $+ 40$ Education Level $- 10$ Percent English Learners =.1437 n=200 a)Interpret the coefficient estimates.Are these results consistent with what you expected? b)Are there any omitted variables in this regression? If so,list them. c)Interpret the R-squared.Is the value of the R-squared high as you expected? d)Calculate the Adjusted R-squared and comment on what it means. e)Predict the test score for a school with average class sizes of 28,an education level of 14 years,and 3% English learners. f)Design an experiment to determine how class size affects test scores.How would the results from that experiment differ from the results presented above?

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.

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

A researcher is interested in estimating how the test scores of elementary schools is related to average class size,parents education level (in years),and percent of English learners at the school.The researcher obtains a sample of 200 California public schools and obtains the following results with standard errors are in parentheses $\hat {Score} = 332 - 3$ Class Size $+ 40$ Education Level $- 10$ Percent English Learners $~~~~~~~~~~~~~~~~~~$ (45) (.78) $~~~~~~~~~~~~~~~~~~$ (14) $~~~~~~~~~~~~~~~~~~$ (2.5) =.1437 =200 a)The F-statistic for this regression is 48.32.Test the overall significance of the regression model at the 5% level.What are the hypothesis,critical value,rejection rule and decision. b)Perform a t-test of for the individual significance for the slopes. c)Comment on the economic significance of the coefficient estimates. d)If you are interested in sending your child to a school with high test scores what would factors would you look for?

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. $\text { Pooled }$ ANOVA df SS MS F Significance F Regression 4 2.30931+12 5.77328+11 282.1787278 1.2198-215 Residual 4286 8.76901+12 2045965635 Total 4290 1.10783+13 $\text { Male }$ ANOVA df SS MS F Significance F Regression 4 1.54309+12 3.85772+11 131.8489492 9.4649-101 Residual 2136 6.24964+12 2925860190 Total 2140 7.79272+12 $\text { Female }$ ANOVA of SS MS F Significance F Regression 4 6.15055+11 1.53764+11 153.1758618 2.2255-115 Residual 2145 2.15323+12 1003838471 Total 2149 2.76829+12 Figure 6.4 -The appropriate critical value for the Chow test in Figure 6.4 is

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