Chapter 09 - Heteroskedasticity

CHAPTER 9

Answers to End of Chapter Problems

9.1 a. The variance of the residuals gets larger as distance increases. Therefore it looks like this model is heteroskedastic.

b. If all of the other assumptions hold (M1-M5) then even if there is heteroskedasticity the model is still unbiased but OLS is no longer BLUE and all of the standard errors, hypothesis tests and confidence intervals are wrong.

c. To detect heteroskedasticity formally you can use the Breusch-Pagan test, the General White’s test, the modified White’s test, or the Goldfield-Quandt test.

d. The most popular method to correct for heteroskedasticity is to use robust standard errors. Unfortunately, Excel does not provide this option. The other method is to use weighted least squares but unfortunately the exact form of heteroskedasticity needs to be known in order to use this method.

9.2 a. Students may do better in larger classes because better students are placed in larger classes because they can handle it or students may do better because the best teachers are placed in the larger classes.

b. This coefficient estimate is likely biased. Parents education is not controlled for in this regression model and as parents education increases we would expect an increase in test scores but we also think that richer communities with higher education rates also have a greater expenditure per student.

c. Unrestricted modelTestScor e i=β0+β1ClassSiz ei+β2Spendin gi+β3Salar y i+εi

Restricted ModelTestScor e i=β0+β1ClassSiz ei+εi∗¿

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

Test statistic:

F−stat=(SSUnexplaine drestricted−SSUnexplaine dunrestricted)/6

SSUnexplainedunrestricted /(n−k−1) Critical Value is Fα ,2 ,n−k−1

Rejection Rule: Reject H0 if F-stat > Fα ,2 ,n−k−1

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Chapter 09 - Heteroskedasticity

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Chapter 09 - Heteroskedasticity

d. This would be a Chow test(1) Estimate the original regression model and obtain SSUnexplainedpooled

TestScor e i=β0+β1ClassSiz ei+β2Spendin gi+β3Salar y i+εi(2) Estimate the above regression model for only the southern schools and obtain SSUnexplained1 (3) Estimate the above regression model for only the northern schools and obtain SSUnexplained2 Formulate the F-stat

F−stat=¿¿Reject H0 if F-stat > Fα ,4 , n−8

e. With heteroskedasticity, OLS is still unbiased but it is not BLUE and the standard errors and all hypothesis testing is wrong. The form of heteroskedasticity is h ( x )=σ 2Salaryi. As salaries increases so does the variance of the error term. Because we know the exact form of heteroskedasticity, we can use weighted least squares. Estimate the model

TestScor ei√Salaryi

=β01

√Salary i+β1

ClassSiz ei√Salary i

+β2

Spendin gi√Salaryi

+β3

Salar y i√Salary i

+εi

√Salaryi

The reason that weighted least squares works is that

Variance( εi√Salary i )= 1

Salary iVariance (ε i)=

1Salary i

σ2Salary i=σ2

The variance of the error term of the weighted least squares model no longer depends on i, or in other words it is now homoskedastic.

9.3 a. With heteroskedasticity, OLS is still unbiased but it is not BLUE and the standard errors and all hypothesis testing is wrong. No, you would not be able to conduct hypothesis tests without either correcting the standard errors or the regression equation.

b. To perform White’s general test:(1) Estimate the original regression y i=β0+β1 x i+εi, obtain the residuals, and then square the residuals.(2) Estimate the regression e i

2=δ 0+δ 1x i+δ 2 x i2+εi and look at the significance of F to test

if δ 1=δ 2=0. If the p-value is large, then you fail to reject the null hypothesis and conclude that the model does not suffer from heteroskedasticity. If the p-value is less than .05 then you reject the null hypothesis and conclude that the model suffers from heteroskedasticity.

c. To perform the Goldfeld-Quandt test you do the following(1) Sort all the data by distance from smallest to largest.(2) Omit the middle n/3 observations(3) Run a regression with only the smallest n/3 observations.(4) Run a regression with only the largest n/3 observations.

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Chapter 09 - Heteroskedasticity

(5) Create the GQ statistic by dividing the larger SSUnexplained from steps (3) or (4) by the smaller SSUnexplained from stets (3) or (4).(6) If the GQ is greater than the tabled value then reject the null hypothesis and conclude the model suffers from heteroskedasticity. If the GQ value is less than the tabled value then fail to reject the null hypothesis and conclude the model is homoskedastic.

d. The form of heteroskedasticity is h ( x )=σ 2distance i. As distance increases so does the variance of the error term. Because we know the exact form of heteroskedasticity, we can use weighted least squares. Estimate the model

Pric e i√Distancei

=β01

√Distancei+ β1

Are ai√Distancei

+β2

Distanc e i√Distancei

+εi

√Salaryi

The reason that weighted least squares works is that

Variance( εi√Distancei )= 1

DistanceiVariance (ε i )=

1Distancei

σ2 Distancei=σ2

The variance of the error term of the weighted least squares model no longer depends on i, or in other words it is now homoskedastic.

9.4 a. To perform the Goldfeld-Quandt test you do the following(1) Sort all the data by population from smallest to largest.(2) Omit the middle n/3 observations(3) Run a regression with only the smallest n/3 observations.(4) Run a regression with only the largest n/3 observations.(5) Create the GQ statistic by dividing the larger SSUnexplained from steps (3) or (4) by the smaller SSUnexplained from stets (3) or (4).(6) If the GQ is greater than the tabled value then reject the null hypothesis and conclude the model suffers from heteroskedasticity. If the GQ value is less than the tabled value then fail to reject the null hypothesis and conclude the model is homoskedastic.

b. If the model is heteroskedastic, OLS is still unbiased (assumptions M1-M5 are still assumed to hold), consistent (again assumptions M1-M5 are still assumed to hold), but not efficient (assumption M6 is needed for efficiency, i.e. for the model to be BLUE).

c. The form of heteroskedasticity is h ( x )=σ 2Pi. As distance increases so does the variance of the error term. Because we know the exact form of heteroskedasticity, we can use weighted least squares. Estimate the model

F i

√Pi

=β01

√Pi

+ β1

Y i

√Pi

+ β2

A i

√Pi

+εi

√Pi

The reason that weighted least squares works is that

Variance( εi√Pi )= 1

PiVariance (εi )=

1Piσ2P i=σ

2

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Chapter 09 - Heteroskedasticity

The variance of the error term of the weighted least squares model no longer depends on i, or in other words it is now homoskedastic.

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Chapter 09 - Heteroskedasticity

9.5 a. It looks like the model suffers from heteroskedasticity. The implications are that OLS is still unbiased, no longer BLUE, and all of the standard errors, hypothesis tests and confidence intervals are wrong.

b. To perform the Goldfeld-Quandt test you do the following(1) Sort all the data by x1 from smallest to largest.(2) Omit the middle 100 observations(3) Run a regression with only the smallest 100 observations.(4) Run a regression with only the largest 100 observations.(5) Create the GQ statistic by dividing the larger SSUnexplained from steps (3) or (4) by the smaller SSUnexplained from stets (3) or (4).(6) If the GQ is greater than the tabled value then reject the null hypothesis and conclude the model suffers from heteroskedasticity. If the GQ value is less than the tabled value then fail to reject the null hypothesis and conclude the model is homoskedastic.

c. To perform White’s general test:(1) Estimate the original regression y i=β0+β1 x1 i+β2 x2i+εi, obtain the residuals, and then square the residuals.(2) Estimate the regression e i

2=δ 0+δ 1 x1 i+δ2 x1 i2 +δ 3 x2 i+δ 4x2 i

2 +δ 5 x1 i x2 i+εi and look at the significance of F to test if δ 1=δ 2=δ 3=δ 4=δ 5=0. If the p-value is large, then you fail to reject the null hypothesis and conclude that the model does not suffer from heteroskedasticity. If the p-value is less than .05 then you reject the null hypothesis and conclude that the model suffers from heteroskedasticity.

d. This is a repeat of part a. The implications are that OLS is still unbiased, no longer BLUE, and all of the standard errors, hypothesis tests and confidence intervals are wrong. The model is still consistent.

e. The easiest way is to just obtain robust standard errors but because we know the form of heteroskedasticity is h ( x )=σ 2Di

2 , weighted least squares is appropriate. Estimate the model

y iDi

=β01Di

+β1

x1i

Di+β2

x2 i

Di+εiDi

The reason that weighted least squares works is that

Variance( εiDi)= 1

Di2 V ariance ( εi )=

1Di

2 σ2 Di

2=σ 2

The variance of the error term of the weighted least squares model no longer depends on i, or in other words it is now homoskedastic.

f. The easiest way is to just obtain robust standard errors but because we know the form of heteroskedasticity is h ( x )=σ 2(Di /w¿¿ i2) , ¿ weighted least squares is appropriate. Estimate the model

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Education.

Chapter 09 - Heteroskedasticity

y iwi

√Di

=β0

wi

√D i

+β1

x1iw i

√D i

+β2

x2iw i

√D i

+εiwi

√Di

The reason that weighted least squares works is that

Variance(w i εi√Di )=

w i2

DiVariance ( εi )=

wi2

D iσ2(Di /w¿¿ i2)=σ2 ¿

The variance of the error term of the weighted least squares model no longer depends on i, or in other words it is now homoskedastic.

Answers to End of Chapter Exercises

E9.1. a.

From this graph it is evident that as GDP per Capita increases then the variance of the error term increases.

b.

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Chapter 09 - Heteroskedasticity

There is a linear relationship between the residuals from the regression and GDP per Capita. This graph also looks like that there is heteroskedasticity.

c. Both of these graphs make it seem like heteroskedasticity is present. The implication of heteroskedasticity is that OLS estimates are still unbiased, OLS is no longer BLUE, and the standard errors, all hypothesis tests, and confidence intervals are wrong.

E9.2. a.

Looking at the significance F or the p-value on GDP Per Capita, they are well below any reasonable significance level (say 1%), we reject H0 and conclude that the model suffers from heteroskedasticity.

b.

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Chapter 09 - Heteroskedasticity

Looking at the significance F it is well below any reasonable significance level (say 1%), we reject H0 and conclude that the model suffers from heteroskedasticity. Note that the predicted medals squared term is not individually statistically significant.

c. A regression with the top 136 observations

Top 136 observations

GQ = (475.9888/138.3969) = 3.44Critical Value < 1.927In this case we reject the null hypothesis and conclude that heteroskedasticity exists.

d. In this case, all three tests can to the exact same decision. It could be the case that one of these tests pick up heteroskedasticity and the others do not but in general they should come to the same decision. The one caveat is the Goldfeld-Quandt test in that if the form of heteroskedasticity is quadratic then

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Chapter 09 - Heteroskedasticity

E9.3. a. The graph is

This looks heteroskedastic.b.

This also looks heteroskedastic.

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Chapter 09 - Heteroskedasticity

c. Both of these graphs look heteroskedastic. The implications are that OLS is still unbiased but no longer BLUE and all standard errors, hypothesis tests, and confidence intervals are wrong.

E9.4 a.

The p-value is 0.8099 and therefore we would conclude that these data are homoskedastic.

b.

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Chapter 09 - Heteroskedasticity

The p-value for this test is 0.2717 and therefore we would conclude that these data are homoskedastic.

c. Performing the Goldfeld-Quandt testThe lowest 40 observations

Upper 40 observations

GQ = (63272672410/1.092E+11) = 1.01Critical Value < 2.091In this case we fail to reject the null hypothesis and conclude these data are homoskedastic.

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Chapter 09 - Heteroskedasticity

d. In this case, all three tests can to the exact same decision. It could be the case that one of these tests pick up heteroskedasticity and the others do not but in general they should come to the same decision. The one caveat is the Goldfeld-Quandt test in that if the form of heteroskedasticity is quadratic then

E9.5 a.Regression results

b. Residuals squared is the dependent variable

The p-value is 0.0222 and therefore this model still suffers from heteroskedasticity

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Chapter 09 - Heteroskedasticity

c. Regression results

d. Dependent variable is residuals squared from the regression in part c

In this case the p-value is greater than 0.05 which means we would conclude at the 5% level that this model is homoskedastic. However, at the 10% level we would conclude that the model still suffers from heteroskedasticity.

The results from this problem highlights why robust standard errors are preferable to weighted least squares if the exact form of heteroskedasticity is not known.

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