Lecture 18: Heteroskedasticity
BUEC 333
Professor David Jacks
1
Three assumptions necessary for unbiasedness:
1.) correct specification;
2.) zero mean error term;
3.) exogeneity of independent variables.
Three assumptions necessary for efficiency:
4.) no perfect collinearity;
5.) no serial correlation;
6.) homoskedasticity.
Violatin’ the classical assumptions
2
Since heteroskedasticity (SC) violates 6.) and this
implies that OLS is not BLUE, we want to know:
1.) What is the nature of the problem?
2.) What are the consequence of the problem?
3.) How is the problem diagnosed?
4.) What remedies for the problem are available?
We now consider these in turn…
Violatin’ the classical assumptions
3
Recall what homoskedasticity actually refers to:
all error terms have the same variance; that is,
Var(εi) = σ2 for all i = 1, 2, 3,…, n.
Heteroskedasticity occurs when different
observations’ error terms (εi) have different values
for their variances; that is, Var(εi) = σi2.
Heteroskedasticity
4
Heteroskedasticity most often occurs in crosssectional (CS) data, but is also present in TS data.
CS data: data where observations are for the same
time periods (e.g., a particular day, month, or year)
but are for different observational units (e.g.,
countries, firms, individuals, or provinces).
Heteroskedasticity
5
This also suggest that when modeling salaries, a
$1 million error for Jaromir Jagr (who earned
$10.3 million that year) is “small.”
But a $1 million error is huge for someone like
Brent Sopel (who earned $226,050).
We might then expect players who played many
games and who scored many points to have larger
salary
Heteroskedasticity
6
Just as in the case of SC, there are two basic types
of heteroskedasticity: pure and impure.
Pure heteroskedasticity arises if the model is
correctly specified, but the errors themselves are
heteroskedastic.
Example: the true DGP is Yi = β0 + β1X1i + εi
and we estimate a univariate regression model
accordingly
Pure heteroskedasticity
7
Not surprisingly, there are many ways to
characterize the heteroskedastic variance term, σi2.
Simplest specification: discrete heteroskedasiticity
Pure heteroskedasticity
8
A more common specification is to assume that
the error variance is proportional to variable Z
which may or may not be one of the included
independent variables.
In this case, Var(εi) = σi2 = σ2 Zi2 with each
observation’s error being drawn from its own
distribution with zero mean and variance σ2 Zi2.
Pure heteroskedasticity
9
Pure heteroskedasticity
10
Pure heteroskedasticity
11
Just like impure SC, impure heteroskedasticity can
arise if the model is mis-specified, and the
specification error induces heteroskedasticity.
For example, suppose the DGP is
Yi   0  1 X 1i   2 X 2i  i
Instead, we estimate Yi   0  1 X 1i   where:
1.)  i*   2 X 2i   i
2.)  i is a classical error term, but X 2i has a
*
i
Impure heteroskedasticity via omitted variables
12
Both forms of heteroskedasticity violate
Assumption 6 of the CLRM, and hence OLS is not
the BLUE.
What more can we say?
1.) OLS estimates remain unbiased…but only if
the problem is with pure heteroskedasticity.
Consequences of heteroskedasticity
13
OLS estimates, however, will be biased if the
problem is with impure heteroskedasticity brought
about by correlated omitted variables.
Impure heteroskedasticity represents violation of
Assumption 1.
In this case, the heteroskedasticity problem is of
secondary importance
Consequences of heteroskedasticity
14
2.) Even if unbiased, the sampling variance of the
OLS estimator is inflated.
OLS may attribute systematic variation in the
dependent variable to the independent variables,
even though this is really due to the error term.
Because positive “mistakes” of this kind are as
likely as negative ones, the estimator remains
Consequences of heteroskedasticity
15
3.) Because of #2 above, the estimated value of
the sampling variance—and consequently, the
calculated standard error—is wrong.
By not accounting for this extra sampling
variation, we get a skewed sense of how spread
out our OLS estimates really are.
Generally speaking, this works in the direction of
OLS providing us with a sense of “too much”
Consequences of heteroskedasticity
16
Many formal tests available which make
assumptions about the particular form of the
heteroskedasticity.
But even before proceeding along these lines, we
should think hard about possible omitted variables
as possible source of apparent heteroskedasticity.
We should also make a habit of always looking at
Testing for heteroskedasticity
17
Of the tests available, by far the most useful is the
White test.
Its usefulness is derived from its generality:
it is designed to test for heteroskedasticity of an
unknown form
Its generality also explains its popularity:
this is has been the standard for statistical
packages (including Eviews) for the past 30 years.
The White test for heteroskedasticity
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We can think of the White test as being composed
of three steps:
1.) Estimate a regression model of interest, call
this Equation 1, and collect the residuals.
Suppose our population regression model is:
Equation 1: Yi   0  1 X 1i   2 X 2i   i
We will then estimate the following:
The White test for heteroskedasticity
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2.) Now, calculate the squared residuals.
Run an auxiliary regression (Equation 2 ) of the
squared residuals on the original independent
variables, their squared terms, and their cross
products (implies hetero. can depend in very
general ways on all of the variables in our model).
In our example, Equation 2 is:
ei2   0  1 X 1i   2 X 2i   3 X 12i   4 X 22i   5 X 1i X 2i  ui
The White test for heteroskedasticity
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3.) Finally, test overall significance of Equation 2.
Our test statistic: n * R  the sample size * R
2
2
2
2
Under the null of no heteroskedasticity, this test
statistic has a χ2 distribution with k* degrees of
freedom
Critical values of χ2 found in standard sources, and
standard reasoning applies: if calculated value of
test statistic exceeds critical value, reject the null.
The White test for heteroskedasticity
21
Let’s return to our NHL data and suppose that
salaries are a function of age and points.
. reg salary age points
Source
SS
df
MS
Model
Residual
5.5423e+14
5.2403e+14
2
643
2.7711e+14
8.1498e+11
Total
1.0783e+15
645
1.6717e+12
salary
Coef.
age
points
_cons
56254.93
39941.76
-1148843
Std. Err.
8934.183
1711.216
237466.9
The White test in action
t
6.30
23.34
-4.84
Number of obs
F( 2,
643)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
646
340.03
0.0000
0.5140
0.5125
9.0e+05
P>|t|
[95% Conf. Interval]
0.000
0.000
0.000
38711.23
36581.51
-1615148
73798.63
43302
-682538.9
22
6000000
4000000
0
2000000
Residuals
-2000000
0
50
100
150
Points
The White test in action
23
4.00e+13
ei2
3.00e+13
2.00e+13
0
1.00e+13
0
50
100
150
Points
The White test in action
24
Source
SS
df
MS
Model
Residual
1.8395e+27
2.4183e+27
5
640
3.6789e+26
3.7786e+24
Total
4.2578e+27
645
6.6012e+24
ei2
Coef.
age
points
square_of_~e
square_of_~s
age_times_~s
_cons
-4.28e+11
-1.55e+08
9.16e+09
1.62e+09
-1.93e+09
5.23e+12
Std. Err.
2.30e+11
2.77e+10
4.22e+09
1.18e+08
9.98e+08
3.11e+12
t
-1.86
-0.01
2.17
13.75
-1.93
1.68
Number of obs
F( 5,
640)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.063
0.996
0.030
0.000
0.054
0.093
=
=
=
=
=
=
646
97.36
0.0000
0.4320
0.4276
1.9e+12
[95% Conf. Interval]
-8.80e+11
-5.45e+10
8.76e+08
1.39e+09
-3.89e+09
-8.81e+11
2.40e+10
5.42e+10
1.75e+10
1.85e+09
3.22e+07
1.13e+13
Here:   0.05
Critical value of   11.07
2
5
The White test in action
25
If evidence of pure heteroskedasticity—whether
through a formal test or just by looking at residual
plots—you have several options available to you:
1.) Use OLS and “fix” the standard errors.
We know OLS is unbiased in this case but the
usual formulas for the standard errors is wrong
Remedies for heteroskedastic errors
26
Just as in the case of serial correlation, we can get
consistent estimates of the standard errors using
Newey-West standard errors.
What consistency means: estimators get arbitrarily
close to their true value (in a probabilistic sense)
when the sample size goes to infinity.
If you know the source of the heteroskedasticity:
Remedies for heteroskedastic errors
27
Heteroskedasticity as a very common problem in
econometrics.
At best, heteroskedasticity presents problems
related to the efficiency of OLS estimators.
At worst, heteroskedasticity presents problems
related to both the bias and efficiency of OLS
estimators.
Conclusion
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