where	$y$ is a vector of length $n$ of the dependent variable, $X$ is a $n$ by $p$ matrix of the independent variables, $β$ is a vector of length $p$ of unknown arguments,
and	$ε$ is a vector of length $n$ of unknown random errors.

The residuals are given by

r = y - \hat{y} = y - X \hat{β}

and the fitted values,

\hat{y} = X \hat{β}

, can be written as

H y

for a

n

n

matrix

H

. Note that when a mean term is included in the model the sum of the residuals is zero. If the observations have been taken serially, that is

y_{1}, y_{2}, \dots, y_{n}

can be considered as a time series, the Durbin–Watson test can be used to test for serial correlation in the

ε_{i}

, see Durbin and Watson (1950), Durbin and Watson (1951) and Durbin and Watson (1971).

The Durbin–Watson statistic is

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

Positive serial correlation in the

ε_{i}

will lead to a small value of

d

while for independent errors

d

will be close to

2

. Durbin and Watson show that the exact distribution of

d

depends on the eigenvalues of the matrix

H A

where the matrix

A

is such that

d

can be written as

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

and the eigenvalues of the matrix

A

are

λ_{j} = (1 - \cos (π j / n))

, for

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

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 the

u_{i}

are independent standard Normal variables. The lower tail probabilities associated with these bounds,

p_{l}

and

p_{u}

, are computed by nag_prob_durbin_watson (g01epc). The interpretation of the bounds is that, for a test of size (significance)

α

, if

p_{l} \leq α

the test is significant, if

p_{u} > α

the test is not significant, while if

p_{l} > α

and

p_{u} \leq α

no conclusion can be reached.

The above probabilities are for the usual test of positive auto-correlation. If the alternative of negative auto-correlation is required, then a call to nag_prob_durbin_watson (g01epc) should be made with the argument d taking the value of

4 - d

; see Newbold (1988).

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

Granger C W J and Newbold P (1986) Forecasting Economic Time Series (2nd Edition) Academic Press

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

5 Arguments

1: $n$ – IntegerInput: On entry: $n$ , the number of residuals.

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

Constraint: $p \geq 1$ .
3: $res [n]$ – const doubleInput: On entry: the residuals, $r_{1}, r_{2}, \dots, r_{n}$ .

Constraint: the mean of the residuals $\leq \sqrt{ε}$ , where $ε = machine precision$ .
4: $d$ – double *Output: On exit: the Durbin–Watson statistic, $d$ .
5: $pdl$ – double *Output: On exit: lower bound for the significance of the Durbin–Watson statistic, $p_{l}$ .
6: $pdu$ – double *Output: On exit: upper bound for the significance of the Durbin–Watson statistic, $p_{u}$ .
7: $fail$ – NagError *Input/Output: The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $p = 〈value〉$ .
Constraint: $p \geq 1$ .
NE_INT_2: On entry, $n = 〈value〉$ and $p = 〈value〉$ .
Constraint: $n > p$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_RESID_IDEN: On entry, all residuals are identical.
NE_RESID_MEAN: On entry, the mean of res is not approximately $0.0$ , $mean = 〈value〉$ .

7 Accuracy

The probabilities are computed to an accuracy of at least

4

decimal places.

8 Parallelism and Performance

nag_durbin_watson_stat (g02fcc) 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 nag_prob_lin_chi_sq (g01jdc) used to compute the required probabilities with the argument c set to

0.0

and the argument d set to the Durbin–Watson statistic

d

10 Example

A set of

10

residuals are read in and the Durbin–Watson statistic along with the probability bounds are computed and printed.

NAG Library Function Documentnag_durbin_watson_stat (g02fcc)

▸▿ 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

NAG Library Function Document

nag_durbin_watson_stat (g02fcc)