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

(12.6.3)
That is, the probability is 95 percent that the preceding interval will include 
R
. Therefore
we have this rule:
(12.6.2)
Mean:
E
(
R
)
=
2
N
1
N
2
N
+
1
Variance:
σ
2
R
=
2
N
1
N
2
(2
N
1
N
2
−
N
)
(
N
)
2
(
N
−
1)
N
=
total number of observations
=
N
1
+
N
2
N
1
=
number of 
+
symbols (i.e., 
+
residuals)
N
2
=
number of 
−
symbols (i.e., 
− 
residuals)
R
=
number of runs
Decision Rule
Do not reject the null hypothesis of randomness with 95% confidence if 
R
, the number of
runs, lies in the preceding confidence interval; reject the null hypothesis if the estimated 
R
lies outside these limits. (
Note:
You can choose any level of confidence you want.)
Returning to our example, we know that 
N
1
, the number of pluses, is 24 and 
N
2
, the num-
ber of minuses, is 22 and 
R
=
5. Using the formulas given in Eq. (12.6.2), we obtain:
The 95% confidence interval for 
R
in our example is thus:
[24
±
1
.
96(3
.
32)]
=
(17
.
5, 30
.
5)
Obviously, this interval does not include 5. Hence, we can 
reject
the hypothesis that the
residuals in our wages–productivity regression are random with 95% confidence. In other
words, the residuals exhibit autocorrelation. As a general rule, if there is positive autocor-
relation, the number of runs will be few, whereas if there is negative autocorrelation, the
(12.6.4)
E
(
R
)
=
24
σ
2
R
=
11
σ
R
=
3
.
32
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number of runs will be many. Of course, from Eq. (12.6.2) we can find out whether we have
too many runs or too few runs.
Swed and Eisenhart have developed special tables that give critical values of the runs
expected in a random sequence of 
N
observations if 
N
1
or 
N
2
is smaller than 20. These
tables are given in 
Appendix D,
Table D.6. Using these tables, the reader can verify that the
residuals in our wages–productivity regression are indeed nonrandom; actually they are
positively correlated.