1. Lecture 16 Preview: Heteroskedasticity
Regression Model
Standard Ordinary Least Squares (OLS) Premises
Estimation Procedures Embedded within the Ordinary Least Squares (OLS) Estimation Procedure
What Is Heteroskedasticity?
Heteroskedasticity and the Ordinary Least Squares (OLS) Estimation Procedure: The Consequences
The Mathematics
Our Suspicions
Confirming Our Suspicions: A Simulation
Accounting for Heteroskedasticity: An Example
Justifying the Generalized Least Squares (GLS) Estimation Procedure
Robust Standard Errors: An Alternative Approach

2. Regression Model
yt = Dependent variable
xt = Explanatory variable
et = Error term
yt = Const + xxt + et
Const and x are the parameters
t = 1, 2, …, T
The error term is a random variable representing random influences: Mean[et] = 0
Standard Ordinary Least Squares (OLS) Premises
Error Term Equal Variance Premise: The variance of the error term’s probability distribution for each observation is the same.
Error Term/Error Term Independence Premise: The error terms are independent.
Explanatory Variable/Error Term Independence Premise: The explanatory variables, the xt’s, and the error terms, the et’s, are not correlated.
OLS Estimation Procedure Includes Three Estimation Procedures
Value of the parameters, Const and x:
bx =
bConst =
Question: What happens when the error term equal variance premise is violated?
Variance of the error term’s probability distribution, Var[e]:
SSR
EstVar[e] =
Degrees of Freedom
Variance of the coefficient estimate’s probability distribution, Var[bx]:
EstVar[bx] =
Good News: When the standard premises are satisfied each of these procedures is unbiased.
Good News: When the standard premises are satisfied the OLS estimation procedure for the coefficient value is the best linear unbiased estimation procedure (BLUE).
Crucial Point: When the ordinary least squares (OLS) estimation procedure performs its calculations, it implicitly assumes that the three standard (OLS) premises are satisfied.

3. Error Term Equal Variance Premise: The variance of the error term’s probability distribution for each observation is the same; all the variances equal Var[e]:
Var[e1] = Var[e2] = … = Var[eT] = Var[e]
Let us review precisely what this means.
What Is Heteroskedasticity?
Consider the error terms of Professor Lord’s three students:
 Lab 16.1
Heter = 0: No Heteroskedasticity
For each student, the mean equals 0.
This indicates that each student’s error term indeed reflects random influences.
The variances are equal.
No heteroskedasticity is present; the error term equal variance premise is satisfied.

4. Error Term Equal Variance Premise: The variance of the error term’s probability distribution for each observation is the same; all the variances equal Var[e]:
Var[e1] = Var[e2] = … = Var[eT] = Var[e]
 Lab 16.1
Heter = 1
For each student, the mean equals 0.
This indicates that each student’s error term indeed reflects random influences.
The variances are not equal.
Heteroskedasticity is present; the error term equal variance premise is violated.
Question: Why might heteroskedasticity exist in this case?

5. Consequences of Heteroskedasticity
How does the presence of heteroskedasticity affect the estimation procedure for the
bx =
value of the coefficient?
SSR
variance of the error term’s probability distribution?
EstVar[e] =
Degrees of Freedom
variance of the coefficient estimate’s probability distribution?
EstVar[bx] =
More specifically, are the three estimation procedures embedded in the ordinary least squares (OLS) estimation procedure still unbiased in the presence of heteroskedasticity?
Estimation Procedure for the Value of the Coefficient
Question: In the presence of heteroskedasticity, is the OLS estimation procedure for the value of the coefficient unbiased?
That is, does Mean[bx] still equal x?
Review: Arithmetic of Means
Mean of a constant plus a variable: Mean[c + x] = c + Mean[x] Mean of a constant times a variable: Mean[cx] = c Mean[x] Mean of the sum of two variables: Mean[x + y] = Mean[x] + Mean[y]

6. Mean of the Coefficient Estimate’s Probability Distributionbx
Mean[c + x] = c + Mean[x]
Rewrite fraction as a product
Mean[cx] = cMean[x]
Mean[x+y] = Mean[x] + Mean[y]
Mean[cx] = cMean[x]
Mean[e1] = Mean[e2] = Mean[e3] = 0
Question: Have we relied on the error term equal variance premise to show that the OLS estimation procedure for the coefficient value is unbiased? No
Question: In the presence of heteroskedasticity, should we expect the OLS estimation procedure for the coefficient value still to be unbiased?
Yes

7. This equation is estimating a “single” Var[e].
OLS Estimation Procedure: Variance of the Coefficient Estimate’s Probability Distribution
Question: In the presence of heteroskedasticity, is the OLS estimation procedure for the variance of the coefficient estimate’s probability distribution unbiased?
Recall the two step strategy we used to estimate the variance of the coefficient estimate’s probability distribution:
Step 1: Estimate the variance of the error term’s probability distribution from the available information.
Step 2: Apply the relationship between the variances of the coefficient estimate’s and error term’s probability distributions:
SSR
EstVar[e] =
Var[bx] =
Degrees of Freedom
EstVar[e]
This equation is estimating a “single” Var[e].
What does Var[e] equal?
EstVar[bx] =
Var[e1] = … = Var[eT] = Var[e]
Strategy: The strategy the ordinary least squares (OLS) estimation procedure uses is based on the premise that there is a “single” Var[e].
Question: Has the OLS estimation procedure relied on the error term equal variance premise to estimate the variance of the coefficient estimate’s probability distribution?
Yes
Question: In the presence of heteroskedasticity, might the OLS estimation procedure for the coefficient estimate’s variance be flawed?
Yes

8. Our Suspicions: OLS estimation procedure for estimating the
coefficient value should be unbiased.
The variance calculation is based on a false premise.
variance of the coefficient estimate’s probability distribution may be flawed.
Act Coef
Is the estimation procedure for the coefficient value unbiased?
Unbiased estimation procedure: After many, many repetitions of the experiment the average (mean) of the estimates equals the actual value.
2 0 2
Mean (average) of the value estimates from all repetitions.
Repetition
Coefficient estimate for this repetition:
Coef Value Est
Variance of the estimated coefficient values estimates from all repetitions.
Mean
bx =
Var
Sum Sqr XDev
Is the estimation procedure for the variance of the coefficient estimate’s probability distribution unbiased?
SSR
EstVar[e] =
Degrees of Freedom
SSR
EstVar[bx] =
Coef Var Est
Estimate of the variance for the coefficient estimate’s probability distribution calculated from this repetition
Mean
Average of the variance estimates from all repetitions
“Single” Var[e] premise

9. Is the OLS estimation procedure for the coefficient’s value unbiased?
Simulation Results
 Lab 16.2
Is OLS estimation procedure for the variance of the coefficient estimate’s probability distribution unbiased?
Is the OLS estimation procedure for the coefficient’s value unbiased?
Mean (Average) Variance of the Average of Actual of the Estimated Estimated Coef Estimated Variances, Heter Value Values, bx, from Values, bx, from EstVar[bx], from Each Factor of x All Repetitions All Repetitions All Repetitions
2.0
2.0
2.5
2.5 1
2.0
2.0
3.6
2.9
When heteroskedasticity is absent
Nothing but good news
When heteroskedasticity is present
Good news:
OLS estimation procedure for the coefficient value is unbiased.
Bad news:
OLS procedure for estimating the variance of the coefficient estimate’s probability distribution is flawed because it is based on a false premise.
Consequently, all calculations based on the variance of the coefficient estimate’s probability distribution will be flawed:
standard errors
t-statistics
tail probabilities

10. Accounting for Heteroskedasticity
Step 1: Apply the Ordinary Least Squares (OLS) Estimation Procedure.
Estimate the model’s parameters with the ordinary least squares (OLS) estimation procedure.
Step 2: Consider the Possibility of Heteroskedasticity.
Ask whether there is reason to suspect that heteroskedasticity may be present.
Use the ordinary least squares (OLS) regression results to “get a sense” of whether hetereoskedasticity is a problem by examining the residuals.
If the presence of hetereoskedasticity is suspected, formulate a model to explain it.
Use the Breusch-Pagan-Godfrey approach by estimating an artificial regression to test for the presence of heteroskedasticity.
Step 3: Apply the Generalized Least Squares (GLS) Estimation Procedure.
Apply the model of heteroskedasticity and algebraically manipulate the original model to derive a new, tweaked model in which the error terms do not suffer from heteroskedasticity.
Use the ordinary least squares (OLS) estimation procedure to estimate the parameters of the tweaked model.
An Example: GDP and Internet Use
Theory: Higher per capita GDP increases Internet use.
Model: LogUsersInternett = Const + GDPGdpPCt + et
Theory: GDP > 0.
1992 Internet Data: Cross section data of Internet use and gross domestic product for 29 countries in 1992.
LogUsersInternett Log of Internet users per 1,000 people in nation t
GdpPCt Per capita GDP (1,000’s of real “international” dollars) in nation t

11. Ordinary Least Squares (OLS)
Step 1: Apply the Ordinary Least Squares (OLS) Estimation Procedure.
Theory: Higher per capita GDP increases Internet use.
 EViews
Model: LogUsersInternett = Const + GDPGdpPCt + et
Ordinary Least Squares (OLS)
Dependent Variable: LogUsersInternet
Explanatory Variable(s):
Estimate SE
t-Statistic
Prob
GdpPC
0.0046
Const 
0.4475
Number of Observations 29
Estimated Equation: EstLogUsersInternet =  GdpPC
Interpretation: We estimate that a $1,000 increase in real per capita GDP results in a 10.1 percent increase in Internet users.
Critical Result: The GdpPC coefficient estimate equals The positive sign of the coefficient estimate suggests that higher per capita GDP increases Internet use. This evidence supports the theory.
H0: GDP = 0 Per capita GDP does not affect Internet use
H1: GDP > 0 Higher per capita GDP increases Internet use
Prob[Results IF H0 True]: What is the probability that the GdpPC estimate from one repetition of the experiment will be .101 or more, if H0 is true (that is, if the per capita GDP has no effect on the Internet use, if GDP actually equals 0)?

12. Ordinary Least Squares (OLS)
Prob[Results IF H0 True]: What is the probability that the GdpPC estimate from one repetition of the experiment will be .101 or more, if H0 is true (that is, if the per capita GDP has no effect on the Internet use, if GDP actually equals 0)?
H0: GDP = 0
H1: GDP > 0
t-distribution
Mean = 0
SE = .0326
DF = 27
.0046/2
.0046/2
Using the tails probability:
bGDP
.0046
= .0023
Prob[Results IF H0 True] =
.101
.101 2
.101
Would we reject H0 at the traditional levels?
Yes
Question: Could there be a potential problem?
Question: What do we know about EstVar[bGDP] when heteroskedasticity is present?
Answer: It is based on a false premise; EstVar[bGDP] could be inaccurate.
Question: What is the SE[bGDP]?
Answer: The square root of EstVar[bGDP]
The tails probability is based on the SE[bGDP].
Consequently, the tails probability could be inaccurate also.
Our calculation of Prob[Results IF H0 True] could be misleading us.
Ordinary Least Squares (OLS)
Dependent Variable: LogUsersInternet
Explanatory Variable(s):
Estimate SE
t-Statistic
Prob
GdpPC
0.0046
Const 
0.4475
Number of Observations 29
Degrees of Freedom 27

13.  EViews Step 2: Consider the Possibility of Heteroskedasticity.
Is reason to suspect that heteroskedasticity may be present?
Yes.
When the per capita GDP is low, individuals have little to spend on any goods other than the basic necessities.
Individuals have little to spend on Internet use and consequently Internet use will be low.
When the per capita GDP is high, individuals can afford to purchase more goods.
Naturally, consumer tastes vary from nation to nation.
In some high per capita GDP nations, individuals will opt to spend much on Internet use.
In other high per capita GDP nations, individuals will opt to spend little on Internet use.
Model: LogUsersInternett = Const + GDPGdpPCt + et
 EViews
Two nations with virtually the same level of per capita GDP have quite different rates of Internet use.
The error term in the model would capture these differences.
As per capita GDP increases we would expect the variance of the error term’s probability distribution to increase.

14. Use the ordinary least squares (OLS) regression results to “get a sense” of whether hetereoskedasticity is a problem by examining the residuals.
We can think of the residuals as the estimated errors.
The error terms, the et’s, are unobservable
The residuals, the Rest’s, are observatible
 yt = Const + xxt + et
 Rest = yt  Estyt
Estyt = bConst + bxxt
 et = yt  (Const + xxt)
 Rest = yt  (bConst + bxxt)
Our suspicions appear to be borne out.
 EViews

15. Ordinary Least Squares (OLS)
If the presence of hetereoskedasticity is suspected, formulate a model to explain it.
Heteroskedasticity Model: (et  Mean[et])2 = Const + GDPGdpPCt + vt
Theory: GDP > 0
 Since Mean[et] = 0.
Use the Breusch-Pagan-Godfrey approach by estimating an artificial regression to test for the presence of heteroskedasticity.
We can think of the  residuals as the estimated errors.
ResSqrt = Const + GDPGdpPCt + vt
Aside: Statistical software makes it easy to do this.
 EViews
Ordinary Least Squares (OLS)
Dependent Variable: ResSqr
Explanatory Variable(s):
Estimate SE
t-Statistic
Prob
GdpPC
0.0118
Const  
0.2651
Number of Observations 29
Critical Result: The GdpPC coefficient estimate equals The positive sign of the coefficient estimate suggests that higher per capita GDP increases the squared deviation of the error term from its mean. This evidence supports the view that heteroskedasticity is present.
H0: GDP = 0 Per capita GDP does not affect the squared deviation of the residual
H1: GDP > 0 Higher per capita GDP increases the squared deviation of the residual
Based on these results we assume that the variance of the error terms probability distribution is proportion to GdpPC:
.0118
Prob[Results IF H0 True] = = 2
Heteroskedasticity Model: Var[et] = VGdpPCt where V equals a constant

16. Step 3: Apply the Generalized Least Squares (GLS) Estimation Procedure.
Apply the model of heteroskedasticity and algebraically manipulate the original model to derive a new, tweaked model in which the error terms do not suffer from heteroskedasticity.
Original Model: LogUsersInternett = Const GDPGdpPCt et
For now, do not worry about why we divide by ;
it will become clear shortly.
Divide by
Arithmetic of variances: Var[cx] = c2Var[x]
Heteroskedasticity Model: Var[et] = VGdpPCt where V equals a constant
Crucial Point: The tweaked model does not suffer from heteroskedasticity. That is why we divided by
= V

17. Ordinary Least Squares (OLS)
Use the ordinary least squares (OLS) estimation procedure to estimate the parameters of the tweaked model.
NB: The tweaked model does not include a constant term.
 EViews
Ordinary Least Squares (OLS)
Dependent Variable: AdjLogUsersInternet
Explanatory Variable(s):
Estimate SE
t-Statistic
Prob
AdjGdpPC
0.0002
AdjConst  
0.1183
Number of Observations 29
H0: GDP = 0
H1: GDP > 0
.0002 =
Prob[Results IF H0 True] = 2
The Ordinary Least Squares (OLS) and Generalized Least Squares (GLS) Estimates
GDP Estimate SE t-Statistic Tails Prob
Ordinary Least Squares (OLS)
Generalized Least Squares (GLS)

18. Is the estimation procedure for the coefficient’s value unbiased?
Justifying the Generalized Least Squares (GLS) Estimation Procedure
Is the estimation procedure for the variance of the coefficient estimate’s probability distribution unbiased?
Is the estimation procedure for the coefficient’s value unbiased?
Recall our simulation:
Mean (Average) Variance of the Average of Actual of the Estimated Estimated Coef Estimated Variances, Heter Estim Value Values, bx, from Values, bx, from EstVar[bx], from Each Factor Proc of x All Repetitions All Repetitions All Repetitions
OLS
2.0
2.0
2.5
2.5 1
OLS
2.0
2.0
3.6
2.9 1
GLS
2.0
2.0
2.3
2.3
 Lab 16.4
Questions: Is the estimation procedure:
Std Premises
Hetero
an unbiased estimation procedure for the
OLS
OLS
GLS
coefficient value?
Yes
Yes
Yes
variance of the coefficient estimate’s probability distribution?
Yes No
Yes
for the coefficient value the best linear unbiased estimation procedure (BLUE)?
Yes No
Yes

19. Justifying the Generalized Least Squares (GLS) Estimation Procedure
Two issues emerge with the ordinary least squares (OLS) estimation procedure when heteroskedasticity is present:
The standard error calculations made by the ordinary least squares (OLS) estimation procedure are flawed.
While the ordinary least squares (OLS) for the coefficient value is unbiased, it is not the best linear unbiased estimation procedure (BLUE).
Recall our simulation:
Mean (Average) Variance of the Average of Actual of the Estimated Estimated Coef Estimated Variances, Heter Estim Value Values, bx, from Values, bx, from EstVar[bx], from Each Factor Proc of x All Repetitions All Repetitions All Repetitions
OLS
2.0
2.0
2.5
2.5 1
OLS
2.0
2.0
3.6
2.9 1
GLS
2.0
2.0
2.3
2.3
 Lab 16.4
Questions: Is the estimation procedure:
Std Premises
Hetero
an unbiased estimation procedure for the
OLS
OLS
GLS
coefficient value?
Yes
Yes
Yes
variance of the coefficient estimate’s probability distribution?
Yes No
Yes
for the coefficient value the best linear unbiased estimation procedure (BLUE)?
Yes No
Yes

20. Ordinary Least Squares (OLS)
Robust Standard Errors: An Alternative Approach
Two issues emerge with the ordinary least squares (OLS) estimation procedure when heteroskedasticity is present:
The standard error calculations made by the ordinary least squares (OLS) estimation procedure are flawed.
While the ordinary least squares (OLS) for the coefficient value is unbiased, it is not the best linear unbiased estimation procedure (BLUE).
 EViews
Huber-White robust standard errors:
Dependent Variable: LogUsersInternet
Explanatory Variable(s):
Estimate SE
t-Statistic
Prob
GdpPC
0.0044
Const  
0.3627
White heteroskedasticity-consistent SEs
Number of Observations 29
Standard errors based on the equal error term variance premise:
Ordinary Least Squares (OLS)
Dependent Variable: LogUsersInternet
Explanatory Variable(s):
Estimate SE
t-Statistic
Prob
GdpPC
0.0046
Const 
0.4475
Number of Observations 29