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1. Ch. 7: Heteroskedasticity
Justin Heflin
University of Kentucky
Eco 392
Nov 4, 2024

2. Class Announcements/Today’s Plan
Class Announcements
•In-Class Lab 9 Wednesday 11/6
Today’s Plan
•What is Heteroskedasticity?
•Pure vs. Impure
Heteroskedasticity
•The Consequences of
Heteroskedasticity
•Detecting Heteroskedasticity

3. What is Heteroskedasticity?
Classical Assumption 5: observations of the error term are drawn from a
distribution with a constant variance
Heteroskedasticityrefers to situations where the
observation is from the mean) Yi−ˆYi)
Heteroskedasticity can occur in both cross-sectional and time-series data
structures

4. Heteroskedasticity vs. Homoskedasticity
✪Homoskedasticity: residuals do not deviate far from the fitted line
✪Heteroskedasticity: a fan out or cone shape indicates the presence of
heteroskedasticity

5. Pure vs. Impure Heteroskedasticity
HeteroskedasticityPure HeteroskedasticityImpure Heteroskedasticity
Pure heteroskedasticityis caused by the error term of the correctly specified
equation
Impure heteroskedasticityis caused by a specification error such as an omitted
variable, too many variables included, or incorrect functional form, and that
causes the non-constant variance

6. Pure vs. Impure Heteroskedasticity
Classical Assumption 5 assumes that:
VAR(ϵi) =σ
2
=a constant(i=1, 2, ...,N) (7.1)
With heteroskedasticity, the variance of the error term depends on exactly which
observation is being discussed:
VAR(ϵi) =σ
2
i(i=1, 2, ...,N) (7.2)

7. Pure vs. Impure HeteroskedasticityHeteroskedasticity often occurs in data sets in which there is a wide range of
values of the dependent variable (large difference between min. and max.)
•The larger the disparity, the more likely the error term observations
associated will have different variances
Time Series Example:
•Retail online sales for the past 30 years
•The number of sales over the past 10 years would be significantly larger due to
an increase in online shopping−→potentially skew the residuals−→hetero.
Cross-Sectional Example:
•Wages
•Wages of all fast-food employees in Lexington probably would not deviate too
much, range of wages would be relatively small
•Wages ofall employeesin Lexington, there would be a wide range of values
due to all the differences in salaries
➞It would result in an unequal distribution of values and increase the chances of
heteroskedasticity

8. Consequences of Heteroskedasticity
1
Pure heteroskedasticity does not cause bias in the coefficient estimates
•Lack of bias does not guarantee “accurate” coefficient estimates,
heteroskedasticity increases the variance of the estimates (wider distribution)
2
Heteroskedasticity typically cause OLS to no longer be the minimum variance
estimator
3
Heteroskedasticity causes OLS estimates of the SE(ˆβ)s to be biased (i.e., too
small), leading to unreliable hypothesis testing and confidence intervals
•A Ψoo small” SE(ˆβ) will cause a Ψoo high” t-score for a particular coefficient
•Increases the chance of rejectingH0=⇒more likely to make aType Ierror

9. Testing for Heteroskedasticity
Two tests to detect heteroskedasticity
1
The Breusch-Pagan Test
2
The White Test
❂Both belong to a general group of tests based on the LM test
The best place to start in correcting a heteroskedasticity problem is to carefully
examine the specification of your equation for potential errors that could be
causing impure heteroskedasticity

10. Testing for Heteroskedasticity
TheBreusch-Pagan testinvestigates whether the squared residuals can be
explained by possible proportionality factor (exogenous variable)
Using the Bruesch-Pagan test to investigate the possibility of heteroskedasticity
involves three steps:
1
Obtain the residuals from the estimated equation:
ei=Yi−ˆYi (8.3)
2
Use the squared residuals as the dependent variable in an auxiliary equation
with the original independent variables on the right-hand side:
e
2
i=α0+α1X1i+α2X2i+ui (8.4)

11. Testing for Heteroskedasticity
e
2
i=α0+α1X1i+α2X2i+ui (8.4)
3
Test the overall significance of Equation 8.4 with a chi-square test:
H0:α1=α2=0
HA:H0is false
•The null hypothesis is homoskedasticity, because ifα1=α2=0, then the
variance equalsα0, which is a constant
The test statistic here isNR
2
•Sample size (N) multiplied by theR
2
from Equation 8.4
Similar to the LM test,ifNR
2
is greater than or equal to the critical chi-square
value, then we reject the null hypothesis of homoskedasticity