where	$y$ is a vector of length $n$ of the dependent variable, $X$ is an $n \times p$ 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 an

n \times 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 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 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: $n$ – Integer Input: On entry: $n$ , the number of residuals.

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

Constraint: $p \geq 1$ .
3: $res [n]$ – const double Input: 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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_RESID_IDEN: On entry, all residuals are identical.
NE_RESID_MEAN: On entry, mean of $res = ⟨ value ⟩$ .
Constraint: the mean of the residuals $\leq \sqrt{ε}$ , where $ε = machine precision$ .

7 Accuracy

The probabilities are computed to an accuracy of at least

4

decimal places.

8 Parallelism and Performance

g02fcc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

If the exact probabilities are required, then the first

n - p

eigenvalues of

H A

can be computed and 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.

g02fc: FL CL CPP AD

NAG CL Interfaceg02fcc (linregm_​stat_​durbwat)

▸▿ 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 CL Interface
g02fcc (linregm_stat_durbwat)