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Chapter 07 - Qualitative Variables and Non-Linearities in Multiple Linear Regression Analysis

CHAPTER 7

Answers to End of Chapter Problems

7.1 a. First I would advise the Polk University not to focus too much on the R-squared and standard error. Regression models should be built using economic theory and by looking at other studies, not by trying to maximize R-squared or by trying to minimize the standard error.

b. This would add two new variables.utilityusage=β0+β1x1+β2 x2+ β3 x3+ β4gasheater+β5before1974+ε

The researcher will make two additional variables. The first is gasheater that will have a 1 if the house has a gas heater and a 0 otherwise. The second is before 1974 that will have a 1 if the house was built before 1974 and a 0 otherwise.

c. The two new variables assume that only the intercept changes and not the slopes whentrying to explain utility usage.

β4: On average, holding x1, x2, x3, and before 1974 constant, a house with a gaswater heater has utility bills $β4 more expensive (less expensive depending on the sign of the coefficient) than a house with an electric water heater.

β5: On average, holding x1, x2, x3, and gas heater constant, a house built before1974 has utility bills $β5 more expensive (less expensive depending on the sign of the coefficient) than a house built after 1974.

7.2 a. The journal obtained these estimates using regression analysis. They likely took a sample of managerial accountants, hopefully from a variety of companies and ran a regression. The estimated model likely looked like

Salaryi=31,865+20,811 topmanagement i+3,604 seniormanagemen ti−11,419entrymanagemen ti+1,105 experienc ei+7,600master si−12,467nocollege i+11,527 professionalcer t i+8,667male i

b. A managerial accountant would put their own attributes into the regression equationabove and predict their salary. They would also want to obtain a confidence interval foran individual around this prediction and see if their current salary was below theprediction interval. If it is, they would conclude that they were significantly underpaid.

7.3 a. On average, holding number of competitors and population constant, a restaurant with a drive up window has $15,300 more in sales than a restaurant without a drive up window.

b. The predicted sales is $56,100.

c. The predicted sales is $41,600.

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7.4 a. Hypothesis:

H 0 : β1=0 no discriminationH 1: β1≠0 discrimination

Test Statistic t-statistic = -2.2/1= -2.2. Rejection RuleReject H0 if the |t-stat| > 1.96.Rejection RuleBecause 2.2 > 1.96 we reject H0 and conclude that males are paid statistically less than females. Females, on average, are paid $2,200 more than males.

b. Linear in the parameters, simple random sampling, no perfectmulticollinearity, E(ε)=0, and the zero conditional mean assumption holds.We are also assuming that the model is homoskedastic because otherwisethe standard errors are wrong.

c. On average, male full professors are paid $18,400 more than maleassistant professors. (notice that -2.2 is subtracted for both so the effect ofthat difference drops away).

d. Model (1) suffers from perfect multicollinearity because male andfemale are both in the model with an intercept and all three options for professors are in the model as well.Model (1) cannot be estimated.Model (2) is odd because there is no intercept but this also means that both male and female can be in the model together. They should also add DA into the model but with an intercept, the assumption E(ε)=0 is violated.Neither of the initial models are superior than the initial model.

e. This model does not control for years of experience, number of publications, different departments, if the faculty member holds an administrative position, etc. Because these factors are omitted and also related to the variables in the model, the estimates are biased. Therefore the finding the discrimination exists should not be trusted because these results are on average, wrong.

7.5 a. On average, holding the other factors in the model constant, if a person ages by one year, their salary increases by .40762*($1000) or $407.62.

b. Hypothesis:H 0 : β1=0 H 1: β1≠0

Test Statistic t-statistic = 0.40787/0.1027= 3.9695.


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Rejection RuleReject H0 if the |t-stat| > 2.Rejection RuleBecause 3.9695 > 2 we reject H0 and conclude that age is statistically significant at the 5% level.

c. The results from regression (1) and (2) are different because once the other variables are controlled for in regression (2) the effect of age is no longer statistically significant.

d. Hypothesis:H 0 : β2¿ β3=β4=β5 ¿ β6¿ β7=0H 1: at least one β i isnot equal¿0

Test statistic:

F−stat=(5207.7−2549.9)/6(2549.9) /(80−8)

=12.50778

Critical Value is Fα ,q , n−k−1=F0.05 , 6,72=2.227Rejection Rule: Reject H0 if F-stat > 2.227.Decision: Because 12.51 < 2.227 we reject H0 and conclude that model (2) is preferable at a 5% significance level.

e. Marginal effect of −57.737+2.9376 Agei−0.0317 AgeSquared i−0.0317 experienc ei.

7.6 a. Set male=0 in the data set and female = 0.salary=β0+ β1male+β2 education+ε

β1 interpretation: On average, holding education constant, if a person is male then their salary on average is β1 higher (or lower depending on the sign) than females.β2 interpretation: On average, holding male constant, if a person gets one more year of education then salary increases by β2 dollars. Note that this effect is the same between males and females.

b. Now interact education and malesalary=β0+ β1male+β2 education+β3 (male )(education)+ε

β1 interpretation: On average, holding education constant, if a person is male then their salary on average is β1 higher (or lower depending on the sign) than females.β2 interpretation: On average, for men, if a man gets one more year of education then salary increases by β2 dollars. β2+ β3 interpretation: On average, for women, if a women gets one more year of education then salary increases by β2+ β3 dollars. β3 is the difference in slopes between men and women.

c. 7-4

Copyright © 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill

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7.7 a. On average, holding other independent variables constant, if a faculty member has moved their salary is 11.34% higher relative to those who have not moved. This variable is statistically significant.

On average, holding other independent variables constant, if a faculty member gets one more year of experience then their salary is (1.13% - 0.03%experience) higher relative to those who have not moved. This variable is statistically significant.

On average, holding other independent variables constant, if a faculty member is a male their salary is 0.66% higher relative to a female. This variable is statistically insignificant.

On average, holding other independent variables constant, if a faculty member has one more top 5 article their salary increases by 3.57% higher. This variable is statistically significant.

b. After controlling for moved and the other independent variables, on average, if a faculty member has moved to California their salary is 12.71% higher relative to those who haven’t moved to California. This variable is statistically significant.After controlling for moved and the other independent variables, on average, if a faculty member has moved from California their salary is 12.65% higher relative to those who haven’t moved from California. This variable is statistically significant.

c. Hypothesis:H 0 : β3=β4 ¿ β5 ¿ β6=0H 1: at least one β i aboveis not equal ¿0

Test Statistic

F−stat= (.5025−.4794 )/4(1−.5025)/ (1024−12)

= 0.2620.00049

=532.95

Rejection RuleReject H0 if the F-stat > 2.463.Rejection Rule


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Because 532.95> 2.463 we reject H0 and conclude that model (2) is statistically preferred to model (1).

d. 0.0115 -0.0003experience = 0 or where experience is equal to 38.33. Given that this is a very long career, most faculty members will never hit the point where salaries hit diminishing returns.

e. Using model (2), I would advise a faculty member to move as often as possible, move both to and from California, get more experience, and publish as many top 5 articles as possible.

Answers to End of Chapter Exercises

E7.1. a.

GPA i= β0+ β1HoursStudied i+ β2Hours Studied Square di

GPA i=2.6576+0.0455Hours Studiedi−0.0003Hours Studied Square d i

Not surprisingly, GPA goes up initially as hours studied increases but there are diminishing returns to hours studied that can be see through the negative sign on the hours studied squared term. Because there is a quadratic effect, the effect that hours studied has on GPA is dependent on the value of hours studied. If hours studied is 4, then the marginal effect on GPA is


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0.0455−(2 )0.0003 (3 )= 0.0437 or as hours studied goes from 3 to 4 the impact on GPA is 0.0437 points. If hours studied is 10, then the marginal effect on GPA is 0.0455−(2 )0.0003 (9 )= 0.0401 or as hours studied goes from 9 to 10 the impact on GPA is 0.0401 points.The hours studied squared term is statistically significant at the 1% level because the p-value of 0.0019 is less than 0.01.

b. To find where hours studied reaches a maximum (or where diminishing marginal returns sets in) set 0.0455−0.0003 (2 )Hours StudiedSquare d i=0 or when hours studied is 75.83.

c.

Looking through the p-values, work is only statistically significant at the 10% level (not the 5% level) and male is statistically insignificant with a p-value of 0.391. The rest of the independent variables are statistically significant at the 5% level. To increase GPA, a student should study more but play fewer video games and text less.The marginal effect of hours studied is now


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0.0254−(2 ) 0.0002Hours Studied Square di. Using the same values as before, if hours studied is 4, then the marginal effect on GPA is 0.0254−(2 ) 0.0002 (3 )= 0.0242 or as hours studied goes from 3 to 4 the impact on GPA is 0.0242 points. If hours studied is 10, then the marginal effect on GPA is 0.0254−(2 ) 0.0002 ( 9 )= 0.0218 or as hours studied goes from 9 to 10 the impact on GPA is 0.0218 points. This function now reaches a maximum at 63.5 hours studied.

E7.2 Correlations of Annual Salary with each of the independent variables.

Female Age Prior Exper Beta Exper Education

Salary -0.177040.908980

00.66881784

3 0.8179845480.64982165

6

Age has the strongest positive linear relationship with salary. This is a typical in labor economics.Female has the weakest relationship with annual salary. It is interesting that the linear relationship between female and salary is negative suggesting that females may suffer from discrimination at Beta.Except for possibly Female, all the explanatory variables have a strong linear relationship with salary.

E7.3 a.

PCT i= β0+ β1FG %i

PCT i=−1.22+3.9576 FG %i


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b. Intercept: On average, if field goal percentage is equal to 0 then the estimated value of proportion of games won is -1.22.FG%: On average, if the proportion of field goals by 1% then the proportion of games won increases by 3.95%.c.

PCT i= β0+ β1FG %i+ β2Opp 3 pt %i+ β3OppTOi

PCT i=−1.235+4.8166 FG%i−2.5895Opp3 pt%i+0.0344OppTOi

d. These results imply that for a team to increase the proportion of games won, they should increase their field goal percentage, decrease the opponents three point percentage and increase the number of turnovers made by the opponent.

e. PCT i=−1.235+4.8166 (.45 )−2.5895(.34 )+0.0344(17)PCT i=0.6377 or the team is predicted to win 63.77% of their games.

f. The estimated regression equation does seem to provide a fairly good fit. The R-squared is 56.38% and the standard error is 0.097. There are likely other factors that explain winning percentage but this model looks to do relatively well.

g. Hypothesis: H 0 : β1=β2¿ β3=0H 1: at least one β i isnot equal¿0

Test Statistic F-statistic = 10.77 and the p-value of the F-test is 9.9443E-05.


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Rejection RuleReject H0 if the p-value < 0.05.Rejection RuleBecause 9.9443E-05< 0.05 we reject H0 and conclude that at least one of number of FG%, Opp 3 pt%, or Opp TO explains PCT at the 5% level.h. Using the p-values to perform the t-test, all of p-values are less than 5% and therefore in each case the null hypothesis of no relationship is reject and all three independent variables are statistically significant at the 5% level.

E7.4 a.


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b.

c. Hypothesis:H 0 : β1=β2¿ β3=0H 1: at least one β i isnot equal¿0

Test Statistic F-statistic = 6.34 and the p-value of the F-test is 0.0031.Rejection RuleReject H0 if the p-value < 0.05.Rejection RuleBecause 0.0031< 0.05 we reject H0 and conclude that at least one of number of weight, speed and position explains Rating at the 5% level.

d. The R-squared is 47.55%, which is high for cross sectional data and all of the coefficient estimates are statistically significant. There are likely other factors that also affect rating that are omitted.

e. Yes. The p-value on guard is 0.019, which is certainly less than 5%.

f. Ratingi=11.9556+0.022Weight i−2.278 Speed i−.7324Guard i

Ratingi=11.9556+0.022 (300 )−2.278(5.1)−0.7324 (0)Ratingi=6.99

This player is just between starting and making the team as backup.


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E7.5 a.

HousingPricei=−141,534.7+573.124SqFee t i−118,287.55 Bedroomsi+32,007.71Bathroomsi+9.56 LotSizei−195,986.59Pool i

Square Feet: On average, holding bedrooms, bathrooms, lot size, and pool constant, if a house increases by 1 square foot then the price increases by $573.12. This is statistically significant at the 5% level because the p-value of 1.93E-13 < 0.05.Bedrooms: On average, holding square feet, bathrooms, lot size, and pool constant, if the number of bedrooms in a house increases by 1 then the price drops by $118,287.55. This is statistically significant at the 5% level because the p-value of 0.007 < 0.05. This is a counter intuitive finding as you would expect if the number of bedrooms increases then the price of a house would increase as well (instead of decrease).Bathrooms: On average, holding square feet, bedrooms, lot size, and pool constant, if the number of bedrooms in a house increases by 1 then the price drops by $118,287.55. This is statistically insignificant at the 5% level because the p-value of 0.5635 > 0.05. This is a counter intuitive finding as you would expect the number of bathrooms to be statistically significant.Lot Size: On average, holding square feet, bedrooms, bathrooms, and pool constant, if a the lot size increases by 1 square foot then the price increases by $9.56. This is statistically insignificant at the 5% level because the p-value of 0.2462 > 0.05.


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Pool: On average, holding square feet, bedrooms, bathrooms, and lot size constant, a house with a pool costs $195,986.59 less than a house without a pool. This is statistically insignificant at the 5% level because the p-value of 0.064 > 0.05 (but it is significant at the 10% level). This is also a counter intuitive result because we would expect that a house with a pool would be more valuable.

b.

Now the statistically significant variables at the 5% level are square feet, bedrooms, bedrooms*lotsize, and pool. Lot size is statistically significant at the 10% level.

Marginal effect of LotSize:−63.5+18 Bedroomsi

Notice this is marginal effect changes as the number of bedrooms change.

Marginal effect of Bedrooms:−232,382+18 Lotsiz e i

Notice this is marginal effect changes as the lot size changes.

If instead you interact bedrooms and square footage then the results are


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Now the statistically significant variables at the 5% level are bedrooms, square feet*bedrooms, and pool.

Marginal effect of Square Feet:95.41+124.92Bedrooms i

Notice this is marginal effect changes as the number of bedrooms change.

Marginal effect of Bedrooms:−340,050.44+124.92SquareFee t i

Notice this is marginal effect changes as square feet changes.


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E7.6 a.

ln (HousingPrice)i=12.15+0.0007 SqFee ti−0.063Bedrooms i −0.000004 LotSize i−0.175 Pooli

Square Feet: On average, holding bedrooms, bathrooms, lot size, and pool constant, if the square feet of a house increases by 1 square foot then the price increases by .07%. This is statistically significant at the 5% level because the p-value of 8.45E-13 < 0.05.Bedrooms: On average, holding square feet, bathrooms, lot size, and pool constant, if the number of bedrooms in a house increases by 1 then the price drops by 6.3%. This is statistically insignificant at the 5% level because the p-value of 0.18 > 0.05. Lot Size: On average, holding square feet, bedrooms, bathrooms, and pool constant, if the lot size increases by 1 square foot then the price decreases by 0.0004%. This is statistically insignificant at the 5% level because the p-value of 0.674 > 0.05.Pool: On average, holding square feet, bedrooms, bathrooms, and lot size constant, a house with a pool costs 17.4% less than a house without a pool. This is statistically insignificant at the 5% level because the p-value of 0.13 > 0.05.


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b.

ln (HousingPrice)i=3.58+1.33 ln ¿ +0.000008 LotSizei−0.1245Pooli

Square Feet: On average, holding bedrooms, bathrooms, lot size, and pool constant, if the square feet of a house increases by 1% then the price increases by 1.33%. This is statistically significant at the 5% level because the p-value of 2.22E-21 < 0.05.Bedrooms: On average, holding square feet, bathrooms, lot size, and pool constant, if the number of bedrooms in a house increases by 1 then the price drops by 10.34%. This is statistically significant at the 5% level because the p-value of 0.047 < 0.05. Lot Size: On average, holding square feet, bedrooms, bathrooms, and pool constant, if the lot size increases by 1 square foot then the price decreases by 0.0008%. This is statistically insignificant at the 5% level because the p-value of 0.377 > 0.05.Pool: On average, holding square feet, bedrooms, bathrooms, and lot size constant, a house with a pool costs 12.45% less than a house without a pool. This is statistically insignificant at the 5% level because the p-value of 0.30 > 0.05.


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E7.7

Looking over the results, the p-value for the F-test (0.3322) is greater than 0.05, which implies that all independent variables are jointly insignificant. Looking over the t-tests for individual significant, only sequel is statistically significant at the 10% level and none of the variables are statistically significant at the 5% level. Perform an F-test will all variables except for sequel dropped out of the model yields a restricted regression of


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Now sequel is statistically significant at the 5% level.

Hypothesis:H 0 : β1=β2¿ β3=β4=β5 ¿ β7¿ β8=0H 1: at least one β i isnot equal¿0

Test statistic:

F−stat=(3.6731E+12−3.31375E+12) /7(3.31375E+12)/44

=0.6817

Critical Value is Fα ,q , n−k−1=F0.05 , 7,44=2.226Rejection Rule: Reject H0 if F-stat > 2.226.Decision: Because 0.6817 < 2.226 we fail to reject H0 and conclude that the coefficients are jointly equal to 0. Therefore, the preferred regression model includes only sequel.


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wordpress.viu.cawordpress.viu.ca/.../files/2016/09/SMChap007.docx · Web viewBecause 3.9695 > 2 we reject H 0 and conclude that age is statistically significant at the 5% level