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

TABLE 11.1
Compensation per Employee ($) in Nondurable Manufacturing Industries According to Employment
Size of Establishment, 1958
Employment Size (average number of employees)
Industry
1–4
5–9
10–19
20–49
50–99
100–249 250–499 500–999 1,000–2,499
Food and kindred
products
2,994
3,295
3,565
3,907
4,189
4,486
4,676
4,968
5,342
Tobacco products
1,721
2,057
3,336
3,320
2,980
2,848
3,072
2,969
3,822
Textile mill 
products
3,600
3,657
3,674
3,437
3,340
3,334
3,225
3,163
3,168
Apparel and 
related products
3,494
3,787
3,533
3,215
3,030
2,834
2,750
2,967
3,453
Paper and allied
products
3,498
3,847
3,913
4,135
4,445
4,885
5,132
5,342
5,326
Printing and
publishing
3,611
4,206
4,695
5,083
5,301
5,269
5,182
5,395
5,552
Chemicals and
allied products
3,875
4,660
4,930
5,005
5,114
5,248
5,630
5,870
5,876
Petroleum and
coal products
4,616
5,181
5,317
5,337
5,421
5,710
6,316
6,455
6,347
Rubber and 
plastic products
3,538
3,984
4,014
4,287
4,221
4,539
4,721
4,905
5,481
Leather and 
leather products
3,016
3,196
3,149
3,317
3,414
3,254
3,177
3,346
4,067
Average 
compensation
3,396
3,787
4,013
4,104
4,146
4,241
4,388
4,538
4,843
Standard deviation
742.2
851.4
727.8
805.06
929.9 1,080.6
1,241.2
1,307.7
1,110.7
Average 
productivity
9,355
8,584
7,962
8,275
8,389
9,418
9,795
10,281
11,750
Source:
The Census of Manufacturers,
U.S. Department of Commerce, 1958 (computed by author).
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370
Part Two
Relaxing the Assumptions of the Classical Model
standard deviations of earnings. This can be seen also from Figure 11.6, which plots the
standard deviation of compensation and average compensation in each employment class.
As can be seen clearly, on average, the standard deviation of compensation increases with
the average value of compensation.
11.2
OLS Estimation in the Presence of Heteroscedasticity
What happens to ordinary least squares (OLS) estimators and their variances if we intro-
duce heteroscedasticity by letting 
E
(
u
2
i
)
=
σ
2
i
but retain all other assumptions of the clas-
sical model? To answer this question, let us revert to the two-variable model:
Y
i
=
β
1
+
β
2
X
i
+
u
i
Applying the usual formula, the OLS estimator of 
β
2
is
ˆ
β
2
=
x
i
y
i
x
2
i
=
n
X
i
Y
i
−
X
i
Y
i
n
X
2
i
−
(
X
i
)
2
(11.2.1)
but its variance is now given by the following expression (see Appendix 11A, Section 11A.1):
(11.2.2)
which is obviously different from the usual variance formula obtained under the assump-
tion of homoscedasticity, namely,
(11.2.3)
var (
ˆ
β
2
)
=
σ
2
x
2
i
var (
ˆ
β
2
)
=
x
2
i
σ
2
i
x
2
i
2
3000
600
3500
4000
Mean compensation
4500
5000
800
1000
Standard deviation
1200
1400
FIGURE 11.6
Standard deviation of
compensation and
mean compensation.
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Chapter 11
Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
371
Of course, if
σ
2
i
=
σ
2
for each
i
, the two formulas will be identical. (Why?)
Recall that 
ˆ
β
2
is best linear unbiased estimator (BLUE) if the assumptions of the classi-
cal model, including homoscedasticity, hold. Is it still BLUE when we drop only the
homoscedasticity assumption and replace it with the assumption of heteroscedasticity? It is
easy to prove that 
ˆ
β
2
is still linear and unbiased. As a matter of fact, as shown in Appendix
3A, Section 3A.2, to establish the unbiasedness of 
ˆ
β
2
it is not necessary that the disturbances
(
u
i
) be homoscedastic. In fact, the variance of 
u
i
, homoscedastic or heteroscedastic, plays
no part in the determination of the unbiasedness property. Recall that in Appendix 3A, Sec-
tion 3A.7, we showed that 
ˆ
β
2
is a consistent estimator under the assumptions of the classical
linear regression model. Although we will not prove it, it can be shown that 
ˆ
β
2
is a consistent
estimator despite heteroscedasticity; that is, as the sample size increases indefinitely, 
the estimated 
β
2
converges to its true value. Furthermore, it can also be shown that under
certain conditions (called regularity conditions), 
ˆ
β
2
is 
asymptotically normally distributed
.
Of course, what we have said about 
ˆ
β
2
also holds true of other parameters of a multiple
regression model.
Granted that 
ˆ
β
2
is still linear unbiased and consistent, is it “efficient” or “best”? That is,
does it have minimum variance in the class of unbiased estimators? And is that minimum
variance given by Eq. (11.2.2)? The answer is 
no
to both the questions: 
ˆ
β
2
is no longer best
and the minimum variance is not given by Eq. (11.2.2). Then what is BLUE in the presence
of heteroscedasticity? The answer is given in the following section.