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Slide 11.1
Undergraduate Econometrics,2
nd
Edition-Chapter 11
Chapter 11Heteroskedasticity11.1The Nature of Heteroskedasticity
ã
In Chapter 3 we introduced the linear model
12
yx
=β +β
(11.1.1)to explain household expenditure on food (
y
) as a function of household income (
x
).
ã
We begin this section by asking whether a function such as
y
=
β
1
+
β
2
x
is better atexplaining expenditure on food for low-income households than it is for high-incomehouseholds.
ã
Income is less important as an explanatory variable for food expenditure of high-income families. It is harder to guess their food expenditure.
Slide 11.2
Undergraduate Econometrics,2
nd
Edition-Chapter 11
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This type of effect can be captured by a statistical model that exhibitsheteroskedasticity.
12
ttt
yxe
=β +β +
(11.1.2)
ã
We assumed the
e
t
were uncorrelated random error terms with mean zero and constantvariance
σ
2
. That is,()0
t
Ee
=
2
var()
t
e
= σ
cov(,)0
ij
ee
=
(11.1.3)
ã
Including the standard errors for
b
1
and
b
2
, the estimated mean function wasˆ
t
y
= 40.768+0.1283
t
x
(11.1.4) (22.139)(0.0305)
ã
A graph of this estimated function, along with all the observed expenditure-income points (,)
tt
yx
, appears in Figure 11.1.
Slide 11.3
Undergraduate Econometrics,2
nd
Edition-Chapter 11
ã
Notice that, as income (
x
t
) grows, the observed data points (,)
tt
yx
have a tendency todeviate more and more from the estimated mean function.
ã
The least squares residuals, defined by
12
ˆ
ttt
eybbx
= − −
(11.1.5)increase in absolute value as income grows.
[Figure 11.1 here]
ã
The observable least squares residuals ˆ()
t
e
are proxies for the unobservable errors ()
t
e
that are given by
12
ttt
eyx
= −β −β
(11.1.6)
ã
The information in Figure 11.1 suggests that the unobservable errors also increase inabsolute value as income ()
t
x
increases.
ã
Is this type of behavior consistent with the assumptions of our model?
Slide 11.4
Undergraduate Econometrics,2
nd
Edition-Chapter 11
ã
The parameter that controls the spread of
y
t
around the mean function, and measures theuncertainty in the regression model, is the variance
σ
2
.
ã
If the scatter of
t
y
around the mean function increases as
x
t
increases, then theuncertainty about
t
y
increases as
t
x
increases, and we have evidence to suggest thatthe variance is not constant.
ã
Thus, we are questioning the constant variance assumption
2
var()var()
tt
ye
= = σ
(11.1.7)
ã
The most general way to relax this assumption is to add a subscript
t
to
σ
2
, recognizingthat the variance can be different for different observations. We then have
2
var()var()
ttt
ye
= = σ
(11.1.8)
ã
In this case, when the variances for all observations are not the same, we say that
heteroskedasticity
exists. Alternatively, we say the random variable
t
y
and the randomerror
t
e
are
heteroskedastic
.

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