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

d
test:
Given the level of significance 
α
,
1.
H
0
:
ρ
=
0 versus H
1
:
ρ >
0. Reject 
H
0
at 
α
level if 
d
<
d
U
. That is, there is statistically
significant positive autocorrelation.
2.
H
0
:
ρ
=
0 versus 
H
1
:
ρ <
0. Reject 
H
0
at 
α
level if the estimated (4
−
d
)
<
d
U
, that is,
there is statistically significant evidence of negative autocorrelation.
3.
H
0
:
ρ
=
0 versus 
H
1
:
ρ
=
0. Reject 
H
0
at 2
α
level if 
d
<
d
U
or (4
−
d
)
<
d
U
, that is,
there is statistically significant evidence of autocorrelation, positive or negative.
It may be pointed out that the indecisive zone narrows as the sample size increases,
which can be seen clearly from the Durbin–Watson tables. For example, with 4 regressors
and 20 observations, the 5 percent lower and upper 
d
values are 0.894 and 1.828, respec-
tively, but these values are 1.515 and 1.739 if the sample size is 75.
The computer program SHAZAM performs an 
exact d test,
that is, it gives the 
p 
value,
the exact probability of the computed 
d
value. With modern computing facilities, it is no
longer difficult to find the 
p
value of the computed 
d
statistic. Using SHAZAM (version 9)
for our wages–productivity regression, we find the 
p
value of the computed 
d
of 0.2176 is
practically zero, thereby reconfirming our earlier conclusion based on the Durbin–Watson
tables.
The Durbin–Watson
d
test has become so venerable that practitioners often forget the as-
sumptions underlying the test. In particular, the assumptions that (1) the explanatory vari-
ables, or regressors, are nonstochastic; (2) the error term follows the normal distribution;
(3) the regression models do not include the lagged value(s) of the regressand; and (4) only
the first-order serial correlation is taken into account are very important for the application
of the
d
test. It should also be added that a significant
d
statistic may not necessarily indi-
cate autocorrelation. Rather, it may be an indication of omission of relevant variables from
the model.
If a regression model contains lagged value(s) of the regressand, the 
d
value in such
cases is often around 2, which would suggest that there is no (first-order) autocorrelation in
such models. Thus, there is a built-in bias against discovering (first-order) autocorrelation
in such models. This does not mean that autoregressive models do not suffer from the au-
tocorrelation problem. As a matter of fact, Durbin has developed the so-called