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

12.6
Detecting Autocorrelation
I. Graphical Method
Recall that the assumption of nonautocorrelation of the classical model relates to the pop-
ulation disturbances 
u
t
, which are not directly observable. What we have instead are their
proxies, the residuals 
ˆ
u
t
, which can be obtained by the usual OLS procedure. Although the
40
50
60
80
100
120
140
160
60
70
80
90
100
130
120
110
FIGURE 12.7
Index of compensation
(
Y
) and index of
productivity (
X
),
United States,
1960–2005.
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Part Two
Relaxing the Assumptions of the Classical Model
17
Even if the disturbances 
u
t
are homoscedastic and uncorrelated, their estimators, the residuals, 
ˆ
u
t
,
are heteroscedastic and autocorrelated. On this, see G. S. Maddala, 
Introduction to Econometrics
,
2d ed., Macmillan, New York, 1992, pp. 480–481. However, it can be shown that as the sample
size increases indefinitely, the residuals tend to converge to their true values, the 
u
t
’s. On this see,
E. Malinvaud, 
Statistical Methods of Econometrics,
2d ed., North-Holland Publishers, Amsterdam,
1970, p. 88.
18
Stanford Weisberg, 
Applied Linear Regression,
John Wiley & Sons, New York, 1980, p. 120. 
19
Actually, it is the so-called 
Studentized
residuals that have a unit variance. But in practice the stan-
dardized residuals will give the same picture, and hence we may rely on them. On this, see Norman
Draper and Harry Smith, 
Applied Regression Analysis,
3d ed., John Wiley & Sons, New York, 1998,
pp. 207–208.
1960
1995
1965
1970
1975
1980
1985
1990
2000
2005
–8
–6
–2
–4
0
2
4
SDRES
100*S1
6
Year
FIGURE 12.8
Residuals (magnified
100 times) and
standardized residuals
from the
wages–productivity
regression (log form:
model 12.5.2).
ˆ
u
t
are not the same thing as 
u
t
,
17
very often a visual examination of the 
ˆ
u
’s gives us some
clues about the likely presence of autocorrelation in the 
u
’s. Actually, a visual examination
of 
ˆ
u
t
or (
ˆ
u
2
t
) can provide useful information not only about autocorrelation but also about
heteroscedasticity (as we saw in the preceding chapter), model inadequacy, or specification
bias, as we shall see in the next chapter. As one author notes:
The importance of producing and analyzing plots [of residuals] as a standard part of statistical
analysis cannot be overemphasized. Besides occasionally providing an easy to understand
summary of a complex problem, they allow the simultaneous examination of the data as an ag-
gregate while clearly displaying the behavior of individual cases.
18
There are various ways of examining the residuals. We can simply plot them against
time, the