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

Syntax

[d, pdl, pdu, ifail] = g02fc(ip, res, 'n', n)

[d, pdl, pdu, ifail] = nag_correg_linregm_stat_durbwat(ip, res, 'n', n)

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 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_stat_prob_durbin_watson (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 nag_stat_prob_durbin_watson (g01ep) should be made with the argument 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

Parameters

Compulsory Input Parameters

1: $ip$ – int64int32nag_int scalar: $p$ , the number of independent variables in the regression model, including the mean.

Constraint: $ip \geq 1$ .
2: $res (n)$ – double array: The residuals, $r_{1}, r_{2}, \dots, r_{n}$ .

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

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the array res.
$n$ , the number of residuals.

Constraint: $n > ip$ .

Output Parameters

1: $d$ – double scalar: The Durbin–Watson statistic, $d$ .
2: $pdl$ – double scalar: Lower bound for the significance of the Durbin–Watson statistic, $p_{l}$ .
3: $pdu$ – double scalar: Upper bound for the significance of the Durbin–Watson statistic, $p_{u}$ .
4: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

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.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The probabilities are computed to an accuracy of at least

4

decimal places.

Further Comments

If the exact probabilities are required, then the first

n - p

eigenvalues of

H A

can be computed and nag_stat_prob_chisq_lincomb (g01jd) 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

Example

A set of

10

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

Open in the MATLAB editor: g02fc_example

function g02fc_example


fprintf('g02fc example results\n\n');

res = [ 3.735719   0.912755   0.683626   0.416693   1.990200 ...
       -0.444816  -1.283088  -3.666035  -0.426357  -1.918697];

ip = int64(2);

% Calculate statistic
[d, pdl, pdu, ifail] = g02fc(ip, res);

% Display the results
fprintf(' Durbin-Watson statistic %10.4f\n\n', d);
fprintf(' Lower and upper bound   %10.4f%10.4f\n', pdl, pdu);

g02fc example results

 Durbin-Watson statistic     0.9238

 Lower and upper bound       0.0610    0.0060

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox