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

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

Parameters

Compulsory Input Parameters

1: $n$ – int64int32nag_int scalar: $n$ , the number of observations used in calculating the Durbin–Watson statistic.

Constraint: $n > ip$ .
2: $ip$ – int64int32nag_int scalar: $p$ , the number of independent variables in the regression model, including the mean.

Constraint: $ip \geq 1$ .
3: $d$ – double scalar: $d$ , the Durbin–Watson statistic.

Constraint: $d \geq 0.0$ .

Optional Input Parameters

None.

Output Parameters

1: $pdl$ – double scalar: Lower bound for the significance of the Durbin–Watson statistic, $p_{l}$ .
2: $pdu$ – double scalar: Upper bound for the significance of the Durbin–Watson statistic, $p_{u}$ .
3: $ifail$ – 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,

d < 0.0

$ifail = - 99$: 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

On successful exit at least

4

decimal places of accuracy are achieved.

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 c set to

0.0

and d to the Durbin–Watson statistic.

Example

The values of

n

p

and the Durbin–Watson statistic

d

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

Open in the MATLAB editor: g01ep_example

function g01ep_example


fprintf('g01ep example results\n\n');

% bounds for the significance of a Durbin–Watson statistic.
n  = int64(10);
ip = int64(2);
d  = 0.9238;

[pdl, pdu, ifail] = g01ep(n, ip, d);

fprintf('d_l(n=%4d,p=%4d,d=%7.4f) = %7.4f\n',n,ip,d,pdl);
fprintf('d_u(n=%4d,p=%4d,d=%7.4f) = %7.4f\n',n,ip,d,pdu);

g01ep example results

d_l(n=  10,p=   2,d= 0.9238) =  0.0610
d_u(n=  10,p=   2,d= 0.9238) =  0.0060

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

Chapter Contents

Chapter Introduction

NAG Toolbox