8. Heteroskedasticity
We have already seen that homoskedasticity
exists when the error term’s variance, conditional
on all x variables, is constant:
Var (u | X )  
2
Homoskedasticity fails if the variance of the error
term varies in the sample (ie: varies with the x
variables)
-We used Homoskedasticity for t tests, F test, and
confidence intervals, even with large samples
8. Heteroskedasticity
8.1 Consequences of Heteroskedasticity for
OLS
8.2 Heteroskedasticity-Robust Inference
after OLS Estimation
8.3 Testing for Heteroskedasticity
8.4 Weighted Least Squares Estimation
8.5 The Linear Probability Model Revisited
8.1 Consequences of Heteroskedasticity
We have already seen that Heteroskedasticity:
1) Does not cause bias or inconsistency (this
depends on MLR. 1 through MLR. 4)
2) Does not affect R2 or adjusted R2 (since these
estimate the POPULATION variances which
are not conditional on X)
Heteroskedasticity does:
1) Make Var(Bjhat) biased, and therefore
invalidate typical OLS standard errors (and
therefore tests)
2) Make OLS no longer BLUE (a better estimator
may exist)
8.2 Heteroskedasticity-Robust Inference
after OLS Estimation
-Because testing hypothesis is a key element of
econometrics, we need to obtain accurate
standard errors in the presence of
heteroskedasticity
-in the last few decades, econometricians have
learned how to adjust standard errors when
HETEROSKEDASTICITY OF UNKNOWN FORM
exists
-these heteroskedasticity-robust procedures are
valid (in large samples) regardless of eror
variance
8.2 Het Fixing 1
-Given a typical single independent variable
model, heteroskedasticity implies a varying
variance:
yi   0  1 xi  ui
Var (ui | xi )  
2
i
-Rewriting the OLS slope estimator, we can
obtain a formula for its variance:
(
x

x
)
u

i
i
ˆ
1  1 
2
 ( xi  x)
Var ( ˆ1 ) 
2 2
(
x

x
)
 i i
SSTx2
-Recall that
8.2 Het Fixing 1
SST   ( x  x)
2
x
i
-Also notice that given homoskedasticity,
 i2   2
Var ( ˆ1 ) 
 i2
SSTx
-While we don’t know σi2, White (1980) showed
that a valid estimator is:
Vaˆr ( ˆ1 ) 
2 2
(
x

x
)
 i uˆi
SSTx2
8.2 Het Fixing 1
-Given a multiple independent variable model:
y   0  1 x1  ...   k xk  u
-The valid estimator of Var(Bjhat) becomes:
2 2
ˆ
ˆi
r
ij u

Vaˆr ( ˆ j ) 
(8.4)
2
SSR j
-where rijhat2 is the ith residual of a regression of
xj on all other x variables
-where SSRj is the sum of the squared residuals
from that regression
8.2 Het Fixing 1
-The square root of this estimate of variance is
commonly called the HETEROSKEDASTICITYROBUST STANDARD ERROR, but is also called
the White, Huber, or Eickert standard errors due
to its founders
-there are a variety of slight adjustments to this
standard error, but economists generally simply
use the values reported by their program
-this se adjustment gives us
HETEROSKEDASTICITY-ROBUST T STATISTICS:
(estimate

hypothesiz
ed
value)
t
standard error
8.2 Why Bother with Normal Errors?
-One may ask why we bother with normal OLS
errors when heteroskedasticity-robust
standard errors are valid more often:
1) Normal OLS t stats have an exact t
distribution, regardless of sample size
2) Robust t statistics are valid only for large
sample sizes
Note that HETEROSKEDASTICITY-ROBUST F
STATISTICS also exist, often called the
HETEROSKEDASTICITY-ROBUST WALD
STATISTIC and reported by most econ
programs.
8.3 Testing for Heteroskedasticity
-In this chapter we will cover a variety of
modern tests for heteroskedasticity
-It is important to know if heteroskedasticity
exists, as its existence means OLS is no
longer the BEST estimator
-Note that while other tests for
heteroskedasticity exist, the test presented
here are preferred due to their more
DIRECT testing for heteroskedasticity
8.3 Testing for Het
-Consider our typical linear model and a null
hypothesis suggesting homoskedasticity:
y   0  1 x1  ...   k xk  u
H 0 : Var (u | x1 , x2, ..., xk )  
2
Since we know that Var(u|X)=E(u2|X), we can
rewrite the null hypothesis to read:
H 0 : E(u | x1 , x2, ..., xk )  E(u )  
2
2
2
8.3 Testing for Het
-As we are testing whether u2 is related to any
explanatory variables, we can use the linear
model:
u   0  1 x1  ...   k xk  v
2
-where v is an error term with mean zero given
the x’s
-note that the dependent variable is SQUARED
-this changes our null hypothesis to:
H 0 : 1   2  ...   k  0
8.3 Testing for Het
-Since we don’t know the true error of the
regression, but only the residual, our estimation
becomes:
uˆ   0  1 x1  ...   k xk  error
2
-Which is valid for large sample distributions
-The R2 from the above regression is used to
construct an F statistic:
F
2
uˆ 2
R /k
(1  R ) /( n  k  1)
2
uˆ 2
(8.15)
8.3 Testing for Het
-This test F statistic is compared to a critical F*
with k, n-k-1 degrees of freedom
-If the null hypothesis is rejected, there is
evidence to conclude that heteroskedasticity
exists at a given α
-If the null hypothesis is not rejected, there is
insufficient evidence to conclude that
heteroskedasticity exists at a given α
-this is sometimes called the BREUCH-PAGAN
TEST FOR HETEROSKEDASTICITY (BP TEST)
8.3 BP HET TEST
In order to conduct a BP test for het
1) Run a normal OLS regression (y on x’s) and
obtain the square of the residuals, uhat2
2) Run a regression of uhat2 on all independent
variables and save the R2
3) Obtain a test F statistic and compare it to the
critical F*
4) If F>F*, reject the null hypothesis of
homoskedasticity and start correcting for
heteroskedasticity
8.3 BP HET TEST
If we suspect that our model’s heteroskedasticity
depends on only certain x variables,
Only regress uhat2 on those variables
-Keep in mind that the K in the R2 formula and in
the degrees of freedom comes from the
number of independent variables in the uhat2
regression
An alternate test for het is the white test:
8.3 White Test for Het
-Given the statistical modifications covered in
chapter 5, White (1980) proposed another test
for heteroskedasticity
-With 3 independent variables, White proposed a
linear regression with 9 regressors:
uˆ   0  1 x1   2 x2   3 x3   4 x1   5 x2
2
2
2
  6 x3   7 x1 x2   8 x1 x3   9 x2 x3  error
2
-The null hypothesis (homoskedasticity) now sets
all δ (except the intercept) equal to zero
8.3 White Test for Het
-Unfortunately this test involves MANY regressors
(27 regressors for 6 x variables) and as such
may have degrees of freedom issues
-one special case of the White test is to estimate
the regression:
uˆ   0  1 yˆ   2 yˆ  error
2
2
-since this preserves the “squared” concept of
the White test and is particularly useful when
het is suspected to be connected to the level of
the expected value E(y|X)
-this test has a F distribution w/2,n-3 df
8.3 Special White HET TEST
In order to conduct a special White test for het
1) Run a normal OLS regression (y on x’s) and
obtain the square of the residuals, uhat2 and
the predicted values, yhat
2) Run the regression of uhat2 on both yhat and
yhat2 (including an intercept). Record the R2
values
3) Using these R2 values, compute a test F
statistic as in the BP test
4) If F>F*, reject the null hypothesis
(homoskedasticity)
8.3 Heteroskedasticity Note
-Our decision to REJECT the null hypothesis and
suspect heteroskedasticity is only valid if MLR.4
is valid
-if MLR.4 is violated (ie: bad funcitonal form or
omitted variables), one can reject the null
hypothesis even if het doesn’t actually exist
-Therefore always chose functional form and all
variables before testing for heteroskedasticity