g02fc calculates the Durbin–Watson statistic, for a set of residuals, and the upper and lower bounds for its significance.

Syntax

C#
public static void g02fc( int n, int ip, double[] res, out double d, out double pdl, out double pdu, out int ifail )

Visual Basic
Public Shared Sub g02fc ( _ n As Integer, _ ip As Integer, _ res As Double(), _ <OutAttribute> ByRef d As Double, _ <OutAttribute> ByRef pdl As Double, _ <OutAttribute> ByRef pdu As Double, _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub g02fc ( _
	n As Integer, _
	ip As Integer, _
	res As Double(), _
	<OutAttribute> ByRef d As Double, _
	<OutAttribute> ByRef pdl As Double, _
	<OutAttribute> ByRef pdu As Double, _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void g02fc( int n, int ip, array<double>^ res, [OutAttribute] double% d, [OutAttribute] double% pdl, [OutAttribute] double% pdu, [OutAttribute] int% ifail )

F#
static member g02fc : n : int * ip : int * res : float[] * d : float byref * pdl : float byref * pdu : float byref * ifail : int byref -> unit

Parameters

n: Type: System..::..Int32
On entry: $n$ , the number of residuals.

Constraint: $n > ip$ .

ip: Type: System..::..Int32
On entry: $p$ , the number of independent variables in the regression model, including the mean.

Constraint: $ip \geq 1$ .

res: Type: array<System..::..Double>[]()[][]
An array of size [n]
On entry: the residuals, $r_{1}, r_{2}, \dots, r_{n}$ .

Constraint: the mean of the residuals $\leq \sqrt{ε}$ , where $ε = machine precision$ .

d: Type: System..::..Double%
On exit: the Durbin–Watson statistic, $d$ .

pdl: Type: System..::..Double%
On exit: lower bound for the significance of the Durbin–Watson statistic, $p_{l}$ .

pdu: Type: System..::..Double%
On exit: upper bound for the significance of the Durbin–Watson statistic, $p_{u}$ .

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

For the general linear regression model

y = X β + ε,

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 g01ep. 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 g01ep should be made with the parameter d taking the value of

4 - d

; see Newbold (1988).

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

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$

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.

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

The probabilities are computed to an accuracy of at least

4

decimal places.

Parallelism and Performance

None.

Further Comments

If the exact probabilities are required, then the first

n - p

eigenvalues of

H A

can be computed and g01jd 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

Example

A set of

10

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

Example program (C#): g02fce.cs

Example program data: g02fce.d

Example program results: g02fce.r