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Durbin-Watson test
A test that the residuals from a linear regression or multiple
regression are independent
...
d
becomes smaller as the serial correlations increase
...

If d < dL

reject H0 : ρ = 0

If d > dU

do not reject H0 : ρ = 0

If dL < d < dU

test is inconclusive
...
Data for Soft Drink Concentrate Sales Example

t
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979

(1)
Annual
Regional
Concentrate
Sales(units)
yt

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

3083
3149
3218
3239
3295
3374
3475
3569
3597
3725
3794
3959
4043
4194
4318
4493
4683
4850
5005
5236

(2)
Annual
Advertising
Expenditures
($×1000)
xt

(3)
LeastSquare
Residuals
et

75
78
80
82
84
88
93
97
99
104
109
115
120
127
135
144
153
161
170
182
20
2 = 7587
...
330 1045
...
603
707
...
7985
2
...
9062
830
...
967
287
...
9491
−1
...
3179
250
...
512
6
...
8605
−1
...
8691
0
...
669
136
...
9405
−0
...
2632
148
...
032
730
...
7270
−4
...
5541
989
...
032 1602
...
1581
23
...
8749
270
...
940 1151
...
3918
−2
...
7674
1348
...
606
74
...
8608
0
...
3306
84
...
848
46
...
3505
−18
...
8988
666
...
063
844
...
8485
20
2 = 8195
...
2065
= 1
...
9154

825000
830445
838750
842940
846315
852240
860760
865925
871640
877745
886520
894500
900400
904005
908525
912160
917630
922220
925910
929610

If we choose α = 0
...
20 and dU = 1
...

∵ d = 1
...
20
∴ We reject H0 and conclude that the errors are positively autocorrelated
...
Critical Values of the Durbin-Watson Statistic
Probability in
Lower Tail
(Significance
Level= α)

dL

15


...
025

...
81

...
08

1
...
23
1
...
70

...
95

1
...
40
1
...
59

...
82

1
...
61
1
...
49

...
69

1
...
84
1
...
39

...
56

1
...
09
2
...
01

...
05


...
08
1
...
15
1
...
41


...
99
1
...
27
1
...
54


...
89
1
...
41
1
...
68


...
79

...
57
1
...
83


...
70

...
74
1
...
99

25


...
025

...
05
1
...
29

1
...
34
1
...
98
1
...
21

1
...
43
1
...
90
1
...
12

1
...
54
1
...
83

...
04

1
...
65
1
...
75

...
95

1
...
77
1
...
01

...
05

1
...
25
1
...
26
1
...
49

1
...
18
1
...
34
1
...
57

1
...
12
1
...
42
1
...
65


...
05
1
...
51
1
...
74


...
98
1
...
61
1
...
83

40


...
025

...
25
1
...
44

1
...
45
1
...
20
1
...
39

1
...
51
1
...
15
1
...
34

1
...
57
1
...
10
1
...
29

1
...
63
1
...
05
1
...
23

1
...
69
1
...
01

...
05

1
...
42
1
...
40
1
...
59

1
...
38
1
...
45
1
...
63

1
...
34
1
...
49
1
...
67

1
...
30
1
...
54
1
...
72

1
...
26
1
...
59
1
...
77

60


...
025

...
38
1
...
55

1
...
54
1
...
35
1
...
51

1
...
57
1
...
32
1
...
48

1
...
61
1
...
28
1
...
44

1
...
65
1
...
25
1
...
41

1
...
69
1
...
01

...
05

1
...
54
1
...
52
1
...
66

1
...
52
1
...
54
1
...
69

1
...
49
1
...
57
1
...
72

1
...
47
1
...
60
1
...
74

1
...
44
1
...
62
1
...
77

100


...
025

...
52
1
...
65

1
...
63
1
...
50
1
...
63

1
...
65
1
...
48
1
...
61

1
...
67
1
...
45
1
...
59

1
...
70
1
...
44
1
...
57

1
...
72
1
...

For purposes of testing hypotheses and constructing confidence intervals we often add the assumption of normality, so
that the εi are NID(0, σ 2)
...
Such data are called time series data
...
The assumption of uncorrelated or independent errors for time series data is often not appropriate
...
Such error terms are said
to be autocorrelated
...
Thus, a simple linear regression model with first-order autoregressive errors
5

would be
yt = β0 + β1xt + εt
εt = ρεt−1 + at
where yt and xt are the observations on the response and
regressor variables at time period t
...
However, if a test for negative autocorrelation is desired, one can use the statistic 4 − d
...
It is
also possible to conduct a two-side test (H0 : ρ = 0 versus
H1 : ρ = 0) by using both one-side tests simultaneously
...

Reference:
1
...
C
...
A
...
G
...

Introduction to Linear Regression Analysis
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6



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