NAG Library Routine Document

G02FCF

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 parameters,
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 G01EPF. 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 G01EPF should be made with the parameter 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 Parameters

1: N – INTEGERInput: On entry: $n$ , the number of residuals.
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: RES(N) – REAL (KIND=nag_wp) arrayInput: 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 – REAL (KIND=nag_wp)Output: On exit: the Durbin–Watson statistic, $d$ .
5: PDL – REAL (KIND=nag_wp)Output: On exit: lower bound for the significance of the Durbin–Watson statistic, $p_{l}$ .
6: PDU – REAL (KIND=nag_wp)Output: On exit: upper bound for the significance of the Durbin–Watson statistic, $p_{u}$ .
7: WORK(N) – REAL (KIND=nag_wp) arrayWorkspace
8: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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,	$N \leq IP$ ,
or	$IP < 1$ .

$IFAIL = 2$

On entry,

the mean of the residuals was

> \sqrt{ε}

, where

ε = machine precision

$IFAIL = 3$

On entry,

all residuals are identical.

7 Accuracy

The probabilities are computed to an accuracy of at least

4

decimal places.

8 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 the parameter C set to

0.0

and the parameter D set to the Durbin–Watson statistic

d

9 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 Routine DocumentG02FCF

+− Contents

1 Purpose

2 Specification

3 Description