can be considered as a time series. The Durbin–Watson test (see Durbin and Watson (1950), Durbin and Watson (1951) and Durbin and Watson (1971)) can be used to test for serial correlation in the error term in the regression.

The Durbin–Watson test statistic is:

d = \frac{\sum_{i = 1}^{n - 1} {(r_{i + 1} - r_{i})}^{2}}{\sum_{i = 1}^{n} r_{i}^{2}},

which can be written as

d = \frac{r^{T} A r}{r^{T} r},

where the

n

n

matrix

A

is given by

A = [\begin{array}{r} 1 & - 1 & 0 & \dots & : \\ - 1 & 2 & - 1 & \dots & : \\ 0 & - 1 & 2 & \dots & : \\ : & 0 & - 1 & \dots & : \\ : & : & : & \dots & : \\ : & : & : & \dots & - 1 \\ 0 & 0 & 0 & \dots & 1 \end{array}]

with the nonzero eigenvalues of the matrix

A

being

λ_{j} = (1 - \cos (π j / n))

, for

j = 1, 2, \dots, n - 1

Durbin and Watson show that the exact distribution of

d

depends on the eigenvalues of a matrix

H A

, where

H

is the hat matrix of independent variables, i.e., the matrix such that the vector of fitted values,

\hat{y}

, can be written as

\hat{y} = H y

. However, bounds on the distribution can be obtained, the lower bound being

d_{l} = \frac{\sum_{i = 1}^{n - p} λ_{i} u_{i}^{2}}{\sum_{i = 1}^{n - p} u_{i}^{2}}

and the upper bound being

d_{u} = \frac{\sum_{i = 1}^{n - p} λ_{i - 1 + p} u_{i}^{2}}{\sum_{i = 1}^{n - p} u_{i}^{2}},

where

u_{i}

are independent standard Normal variables.

Two algorithms are used to compute the lower tail (significance level) probabilities,

p_{l}

and

p_{u}

, associated with

d_{l}

and

d_{u}

. If

n \leq 60

the procedure due to Pan (1964) is used, see Farebrother (1980), otherwise Imhof's method (see Imhof (1961)) is used.

The bounds are for the usual test of positive correlation; if a test of negative correlation is required the value of

d

should be replaced by

4 - d

4

References

Durbin J and Watson G S (1950) Testing for serial correlation in least squares regression. I Biometrika 37 409–428

Durbin J and Watson G S (1951) Testing for serial correlation in least squares regression. II Biometrika 38 159–178

Durbin J and Watson G S (1971) Testing for serial correlation in least squares regression. III Biometrika 58 1–19

Farebrother R W (1980) Algorithm AS 153. Pan's procedure for the tail probabilities of the Durbin–Watson statistic Appl. Statist. 29 224–227

Imhof J P (1961) Computing the distribution of quadratic forms in Normal variables Biometrika 48 419–426

Newbold P (1988) Statistics for Business and Economics Prentice–Hall

Pan Jie–Jian (1964) Distributions of the noncircular serial correlation coefficients Shuxue Jinzhan 7 328–337

5

Arguments

1: $n$ – IntegerInput: On entry: $n$ , the number of observations used in calculating the Durbin–Watson statistic.

Constraint: $n > ip$ .
2: $ip$ – IntegerInput: On entry: $p$ , the number of independent variables in the regression model, including the mean.

Constraint: $ip \geq 1$ .
3: $d$ – Real (Kind=nag_wp)Input: On entry: $d$ , the Durbin–Watson statistic.

Constraint: $d \geq 0.0$ .
4: $pdl$ – Real (Kind=nag_wp)Output: On exit: lower bound for the significance of the Durbin–Watson statistic, $p_{l}$ .
5: $pdu$ – Real (Kind=nag_wp)Output: On exit: upper bound for the significance of the Durbin–Watson statistic, $p_{u}$ .
6: $work (n)$ – Real (Kind=nag_wp) arrayWorkspace
7: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $ip = 〈value〉$ .
Constraint: $ip \geq 1$ .

On entry, $n = 〈value〉$ and $ip = 〈value〉$ .
Constraint: $n > ip$ .

$ifail = 2$: On entry, $d = 〈value〉$ .
Constraint: $d \geq 0.0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

On successful exit at least

4

decimal places of accuracy are achieved.

8

Parallelism and Performance

g01epf is not threaded in any implementation.

9

Further Comments

If the exact probabilities are required, then the first

n - p

eigenvalues of

H A

can be computed and g01jdf used to compute the required probabilities with c set to

0.0

and d to the Durbin–Watson statistic.

10

Example

The values of

n

p

and the Durbin–Watson statistic

d

are input and the bounds for the significance level calculated and printed.

NAG Library Routine Document

g01epf (prob_durbin_watson)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description