The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

364
Part Two
Relaxing the Assumptions of the Classical Model
TABLE 10.18
Chemicals in Cheeses 
Obs.
Taste
Acetic
H
2
S
Lactic
1
12.30000
4.543000
3.135000
0.860000
2
20.90000
5.159000
5.043000
1.530000
3
39.00000
5.366000
5.438000
1.570000
4
47.90000
5.759000
7.496000
1.810000
5
5.600000
4.663000
3.807000
0.990000
6
25.90000
5.697000
7.601000
1.090000
7
37.30000
5.892000
8.726000
1.290000
8
21.90000
6.078000
7.966000
1.780000
9
18.10000
4.898000
3.850000
1.290000
10
21.00000
5.242000
4.174000
1.580000
11
34.90000
5.740000
6.142000
1.680000
12
57.20000
6.446000
7.908000
1.900000
13
0.700000
4.477000
2.996000
1.060000
14
25.90000
5.236000
4.942000
1.300000
15
54.90000
6.151000
6.752000
1.520000
16
40.90000
3.365000
9.588000
1.740000
17
15.90000
4.787000
3.912000
1.160000
18
6.400000
5.142000
4.700000
1.490000
19
18.00000
5.247000
6.174000
1.630000
20
38.90000
5.438000
9.064000
1.990000
21
14.00000
4.564000
4.949000
1.150000
22
15.20000
5.298000
5.220000
1.330000
23
32.00000
5.455000
9.242000
1.440000
24
56.70000
5.855000
10.19900
2.010000
25
16.80000
5.366000
3.664000
1.310000
26
11.60000
6.043000
3.219000
1.460000
27
26.50000
6.458000
6.962000
1.720000
28
0.700000
5.328000
3.912000
1.250000
29
13.40000
5.802000
6.685000
1.080000
30
5.500000
6.176000
4.787000
1.250000
Source: http://lib.stat.cmu.edu/
DASL/Datafiles/Cheese.html.
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365
An important assumption of the classical linear regression model (Assumption 4) is that
the disturbances 
u
i
appearing in the population regression function are homoscedastic; that
is, they all have the same variance. In this chapter we examine the validity of this assump-
tion and find out what happens if this assumption is not fulfilled. As in Chapter 10, we seek
answers to the following questions:
1. What is the nature of heteroscedasticity?
2. What are its consequences?
3. How does one detect it?
4. What are the remedial measures?
11.1
The Nature of Heteroscedasticity
As noted in Chapter 3, one of the important assumptions of the classical linear regression
model is that the variance of each disturbance term 
u
i
, conditional on the chosen values
of the explanatory variables, is some constant number equal to 
σ
2
.
This is the assump-
tion of 
homoscedasticity,
or 
equal
(homo) 
spread
(scedasticity), that is, 
equal variance
. 
Symbolically,
E
u
2
i
=
σ
2
i
=
1, 2,
. . .
,
n
(11.1.1)
Diagrammatically, in the two-variable regression model homoscedasticity can be shown
as in Figure 3.4, which, for convenience, is reproduced as Figure 11.1. As Figure 11.1
shows, the conditional variance of 
Y
i
(which is equal to that of 
u
i
), conditional upon the
given 
X
i
, remains the same regardless of the values taken by the variable 
X
.
In contrast, consider Figure 11.2, which shows that the conditional variance of 
Y
i
increases as 
X
increases. Here, the variances of 
Y
i
are not the same. Hence, there is
heteroscedasticity. Symbolically,
E
u
2
i
=
σ
2
i
(11.1.2)
Chapter
11
Heteroscedasticity:
What Happens If the
Error Variance Is
Nonconstant?
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366
Part Two
Relaxing the Assumptions of the Classical Model
Notice the subscript of 
σ
2
, which reminds us that the conditional variances of 
u
i
(
=
conditional variances of 
Y
i
) are no longer constant.
To make the difference between homoscedasticity and heteroscedasticity clear, assume
that in the two-variable model 
Y
i
=
β
1
+
β
2
X
i
+
u
i
,
Y
represents savings and 
X
represents
income. Figures 11.1 and 11.2 show that as income increases, savings on the average also
increase. But in Figure 11.1 the variance of savings remains the same at all levels of
income, whereas in Figure 11.2 it increases with income. It seems that in Figure 11.2 the
higher-income families on the average save more than the lower-income families, but there
is also more variability in their savings.
There are several reasons why the variances of 
u
i
may be variable, some of which are as
follows.
1
1. Following the 
error-learning models,
as people learn, their errors of behavior become
smaller over time or the number of errors becomes more consistent. In this case, 
σ
2
i
is
expected to decrease. As an example, consider Figure 11.3, which relates the number of
typing errors made in a given time period on a test to the hours put in typing practice. As
Figure 11.3 shows, as the number of hours of typing practice increases, the average number
of typing errors as well as their variances decreases.
2. As incomes grow, people have more 
discretionary income
2
and hence more scope
for choice about the disposition of their income. Hence, 
σ
2
i
is likely to increase with
Density
Income
Savings
X
Y
β
1 
+ 
β
2 
X
i
β
β