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892
Appendix D
Statistical Tables
TABLE D.6A
Critical Values of Runs in the Runs Test
N
2
N
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
2
2
2
2
2
2
2
2
2
2
3
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
4
2
2
2
3
3
3
3
3
3
3
3
4
4
4
4
4
5
2
2
3
3
3
3
3
4
4
4
4
4
4
4
5
5
5
6
2
2
3
3
3
3
4
4
4
4
5
5
5
5
5
5
6
6
7
2
2
3
3
3
4
4
5
5
5
5
5
6
6
6
6
6
6
8
2
3
3
3
4
4
5
5
5
6
6
6
6
6
7
7
7
7
9
2
3
3
4
4
5
5
5
6
6
6
7
7
7
7
8
8
8
10
2
3
3
4
5
5
5
6
6
7
7
7
7
8
8
8
8
9
11
2
3
4
4
5
5
6
6
7
7
7
8
8
8
9
9
9
9
12
2
2
3
4
4
5
6
6
7
7
7
8
8
8
9
9
9
10
10
13
2
2
3
4
5
5
6
6
7
7
8
8
9
9
9
10
10
10
10
14
2
2
3
4
5
5
6
7
7
8
8
9
9
9
10
10
10
11
11
15
2
3
3
4
5
6
6
7
7
8
8
9
9
10
10
11
11
11
12
16
2
3
4
4
5
6
6
7
8
8
9
9
10
10
11
11
11
12
12
17
2
3
4
4
5
6
7
7
8
9
9
10
10
11
11
11
12
12
13
18
2
3
4
5
5
6
7
8
8
9
9
10
10
11
11
12
12
13
13
19
2
3
4
5
6
6
7
8
8
9
10
10
11
11
12
12
13
13
13
20
2
3
4
5
6
6
7
8
9
9
10
10
11
12
12
13
13
13
14
Note:
Tables D.6A and D.6B give the critical values of runs
n
for various values of
N
1
(
+
symbol) and
N
2
(
−
symbol). For the one-sample runs test, any value
of
n
that is equal to or smaller than that shown in Table D.6A or equal to or larger than that shown in Table D.6B is significant at the 0.05 level.
Source: Sidney Siegel,
Nonparametric Statistics for the Behavioral Sciences,
McGraw-Hill Book Company, New York, 1956, table F, pp. 252–253. The tables have been
adapted by Siegel from the original source: Frieda S. Swed and C. Eisenhart, “Tables for Testing Randomness of Grouping in a Sequence of Alternatives,”
Annals of
Mathematical Statistics,
vol. 14, 1943. Used by permission of McGraw-Hill Book Company and
Annals of Mathematical Statistics
.
TABLE D.6B
Critical Values of Runs in the Runs Test
N
2
N
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
2
3
4
9
9
5
9
10
10
11
11
6
9
10
11
12
12
13
13
13
13
7
11
12
13
13
14
14
14
14
15
15
15
8
11
12
13
14
14
15
15
16
16
16
16
17
17
17
17
17
9
13
14
14
15
16
16
16
17
17
18
18
18
18
18
18
10
13
14
15
16
16
17
17
18
18
18
19
19
19
20
20
11
13
14
15
16
17
17
18
19
19
19
20
20
20
21
21
12
13
14
16
16
17
18
19
19
20
20
21
21
21
22
22
13
15
16
17
18
19
19
20
20
21
21
22
22
23
23
14
15
16
17
18
19
20
20
21
22
22
23
23
23
24
15
15
16
18
18
19
20
21
22
22
23
23
24
24
25
16
17
18
19
20
21
21
22
23
23
24
25
25
25
17
17
18
19
20
21
22
23
23
24
25
25
26
26
18
17
18
19
20
21
22
23
24
25
25
26
26
27
19
17
18
20
21
22
23
23
24
25
26
26
27
27
20
17
18
20
21
22
23
24
25
25
26
27
27
28
guj75772_appD.qxd 05/09/2008 11:09 AM Page 892
Appendix D
Statistical Tables
893
EXAMPLE 2
In a sequence of 30 observations consisting of 20
+
signs (
=
N
1
) and 10
−
signs (
=
N
2
),
the critical values of runs at the 0.05 level of significance are 9 and 20, as shown by
Tables D.6A and D.6B, respectively. Therefore, if in an application it is found that the
number of runs is equal to or less than 9 or equal to or greater than 20, one can reject
(at the 0.05 level of significance) the hypothesis that the observed sequence is random.
TABLE D.7
1% and 5% Critical Dickey–Fuller
t
(
=
τ
) and
F
Values for Unit Root Tests
Sample
t
nc
*
t
c
*
t
ct
*
F
†
F
‡
Size
1%
5%
1%
5%
1%
5%
1%
5%
1%
5%
25
−
2.66
−
1.95
−
3.75
−
3.00
−
4.38
−
3.60
10.61
7.24
8.21
5.68
50
−
2.62
−
1.95
−
3.58
−
2.93
−
4.15
−
3.50
9.31
6.73
7.02
5.13
100
−
2.60
−
1.95
−
3.51
−
2.89
−
4.04
−
3.45
8.73
6.49
6.50
4.88
250
−
2.58
−
1.95
−
3.46
−
2.88
−
3.99
−
3.43
8.43
6.34
6.22
4.75
500
−
2.58
−
1.95
−
3.44
−
2.87
−
3.98
−
3.42
8.34
6.30
6.15
4.71
∞
−
2.58
−
1.95
−
3.43
−
2.86
−
3.96
−
3.41
8.27
6.25
6.09
4.68
*Subscripts nc,
c
, and ct denote, respectively, that there is no constant, a constant, and a constant and trend term in the regression Eq. (21.9.5).
†
The critical
F
values are for the joint hypothesis that the constant and
δ
terms in Eq. (21.9.5) are simultaneously equal to zero.
‡
The critical
F
values are for the joint hypothesis that the constant, trend, and
δ
terms in Eq. (21.9.5) are simultaneously equal to zero.
Source: Adapted from W. A. Fuller,
Introduction to Statistical Time Series,
John Wiley & Sons, New York, 1976, p. 373 (for the
τ
test), and D. A. Dickey and W. A. Fuller,
“Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root,”
Econometrica,
vol. 49, 1981, p. 1063.
guj75772_appD.qxd 05/09/2008 11:09 AM Page 893
In this appendix we show the computer output of
EViews
, MINITAB, Excel, and STATA,
which are some of the popularly used statistical packages for regression and related statis-
tical routines. We use the data given in Table E.1 from the textbook website to illustrate the
output of these packages. Table E.1 gives data on the civilian labor force participation rate
(CLFPR), the civilian unemployment rate (CUNR), and real average hourly earnings in
1982 dollars (AHE82) for the U.S. economy for the period 1980 to 2002.
Although in many respects the basic regression output is similar in all these packages,
there are differences in how they present their results. Some packages give results to sev-
eral digits, whereas some others approximate them to four or five digits. Some packages
give analysis of variance (ANOVA) tables directly, whereas for some other packages they
need to be derived. There are also differences in some of the summary statistics presented
by the various packages. It is beyond the scope of this appendix to enumerate all the dif-
ferences in these statistical packages. You can consult the websites of these packages for
further information.
E.1
EViews
Using Version 6 of
EViews,
we regressed CLFPR on CUNR and AHE82 and obtained the
results shown in Figure E.1.
This is the standard format in which
EViews
results are presented. The first part of this
figure gives the regression coefficients, their estimated standard errors, the
t
values under
the null hypothesis that the corresponding population values of these coefficients are
zero, and the
p
values of these
t
values. This is followed by
R
2
and adjusted
R
2
. The other
summary output in the first part relates to the standard error of the regression, residual
sum of squares (RSS), and the
F
value to test the hypothesis that the (true) values of all
the slope coefficients are simultaneously equal to zero. Akaike information and Schwartz
criteria are often used to choose between competing models. The lower the value of these
criteria, the better the model is. The method of maximum likelihood (ML) is an alterna-
tive to the method of least squares. Just as in OLS we find those estimators that minimize
Appendix
E
Computer Output
of EViews, MINITAB,
Excel, and STATA
894
guj75772_appE.qxd 05/09/2008 11:10 AM Page 894
Appendix E
Computer Output of EViews, MINITAB, Excel, and STATA
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