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time sequence plot,
as we have done in Figure 12.8, which shows the residuals
obtained from the log wages–productivity regression (12.5.2). The values of these residu-
als are given in Table 12.5 along with some other data.
Alternatively, we can plot the
standardized residuals
against time, which are also
shown in Figure 12.8 and Table 12.5. The standardized residuals are simply the residuals
(
ˆ
u
t
) divided by the standard error of the regression (
√
ˆ
σ
2
), that is, they are (
ˆ
u
t
/
ˆ
σ
). Notice
that
ˆ
u
t
and
ˆ
σ
are measured in the units in which the regressand
Y
is measured. The values of
the standardized residuals will therefore be pure numbers (devoid of units of measurement)
and can be compared with the standardized residuals of other regressions. Moreover, the
standardized residuals, like
ˆ
u
t
, have zero mean (why?) and
approximately
unit variance.
19
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Chapter 12
Autocorrelation: What Happens If the Error Terms Are Correlated?
431
TABLE 12.5
Residuals: Actual, Standardized, and Lagged
Obs.
S1
SDRES
S1(
1)
Obs.
S1
SDRES
S1(
1)
1960
−
0.036068
−
1.639433
NA
1983
0.014416
0.655291
0.038719
1961
−
0.030780
−
1.399078
−
0.036068
1984
0.001774
0.080626
0.014416
1962
−
0.026724
−
1.214729
−
0.030780
1985
0.001620
0.073640
0.001774
1963
−
0.029160
−
1.325472
−
0.026724
1986
0.013471
0.612317
0.001620
1964
−
0.026246
−
1.193017
−
0.029160
1987
0.013725
0.623875
0.013471
1965
−
0.028348
−
1.288551
−
0.026246
1988
0.017232
0.783269
0.013725
1966
−
0.017504
−
0.795647
−
0.028348
1989
−
0.004818
−
0.219005
0.017232
1967
−
0.006419
−
0.291762
−
0.017504
1990
−
0.006232
−
0.283285
−
0.004818
1968
0.007094
0.322459
−
0.006419
1991
−
0.004118
−
0.187161
−
0.006232
1969
0.018409
0.836791
0.007094
1992
−
0.005078
−
0.230822
−
0.004118
1970
0.024713
1.123311
0.018409
1993
−
0.010686
−
0.485739
−
0.005078
1971
0.016289
0.740413
0.024713
1994
−
0.023553
−
1.070573
−
0.010686
1972
0.025305
1.150208
0.016289
1995
−
0.027874
−
1.266997
−
0.023553
1973
0.025829
1.174049
0.025305
1996
−
0.039805
−
1.809304
−
0.027874
1974
0.023744
1.079278
0.025829
1997
−
0.041164
−
1.871079
−
0.039805
1975
0.011131
0.505948
0.023744
1998
−
0.013576
−
0.617112
−
0.041164
1976
0.018359
0.834515
0.011131
1999
−
0.006674
−
0.303364
−
0.013576
1977
0.020416
0.927990
0.018359
2000
0.010887
0.494846
−
0.006674
1978
0.030781
1.399135
0.020416
2001
0.007551
0.343250
0.010887
1979
0.033023
1.501051
0.030781
2002
0.000453
0.020599
0.007551
1980
0.031604
1.436543
0.033023
2003
−
0.006673
−
0.303298
0.000453
1981
0.020801
0.945516
0.031604
2004
−
0.015650
−
0.711380
−
0.006673
1982
0.038719
1.759960
0.020801
2005
−
0.020198
−
0.918070
−
0.015650
Notes:
S1
=
residuals from the wages–productivity regression (log form).
S1 (
−
1)
=
residuals lagged one period.
SDRES
=
standardized residuals
=
residuals/standard error of estimate.
In large samples (
ˆ
u
t
/
ˆ
σ
) is approximately normally distributed with zero mean and unit vari-
ance. For our example,
ˆ
σ
=
2
.
6755.
Examining the time sequence plot given in Figure 12.8, we observe that both
ˆ
u
t
and the
standardized
ˆ
u
t
exhibit a pattern observed in Figure 12.1
d
, suggesting that perhaps
u
t
are
not random.
To see this differently, we can plot
ˆ
u
t
against
ˆ
u
t
−
1
, that is, plot the residuals at time
t
against their value at time (
t
−
1), a kind of empirical test of the AR(1) scheme. If the
residuals are nonrandom, we should obtain pictures similar to those shown in Figure 12.3.
This plot for our log wages–productivity regression is as shown in Figure 12.9; the under-
lying data are given in Table 12.5. As this figure reveals, most of the residuals are bunched
in the second (northeast) and the fourth (southwest) quadrants, suggesting a strong positive
correlation in the residuals.
The graphical method we have just discussed, although powerful and suggestive, is sub-
jective or qualitative in nature. But there are several quantitative tests that one can use to
supplement the purely qualitative approach. We now consider some of these tests.
II. The Runs Test
If we carefully examine Figure 12.8, we notice a peculiar feature: Initially, we have several
residuals that are negative, then there is a series of positive residuals, and then there are sev-
eral residuals that are negative. If these residuals were purely random, could we observe
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432
Part Two
Relaxing the Assumptions of the Classical Model
–6
–6
–4
–2
Res1(–1)
Res1
0
2
4
–4
–2
0
2
I
II
IV
III
4
FIGURE 12.9
Current residuals
versus lagged
residuals.
such a pattern? Intuitively, it seems unlikely. This intuition can be checked by the so-called
runs test,
sometimes also known as the
Geary test,
a nonparametric test.
20
To explain the runs test, let us simply note down the signs (
or
) of the residuals
obtained from the wages–productivity regression, which are given in the first column of
Table 12.5.
(
−−−−−−−−
)(
+++++++++++++++++++++
)(
−−−−−−−−−−−
)(
+++
)(
−−−
)
(12.6.1)
Thus there are 8 negative residuals, followed by 21 positive residuals, followed by 11 neg-
ative residuals, followed by 3 positive residuals, followed by 3 negative residuals, for a total
of 46 observations.
We now define a
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