NAG Library Function Document

nag_durbin_watson_stat (g02fcc)

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_prob_durbin_watson (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 nag_prob_durbin_watson (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 – IntegerInput: On entry: $n$ , the number of residuals.
Constraint: $n > p$ .
2: p – IntegerInput: On entry: $p$ , the number of independent variables in the regression model, including the mean.
Constraint: $p \geq 1$ .
3: res[n] – const doubleInput: 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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.
NE_RESID_IDEN: On entry, all residuals are identical.
NE_RESID_MEAN: On entry, the mean of res is not approximately $0.0$ , $mean = ⟨value⟩$ .

7 Accuracy

The probabilities are computed to an accuracy of at least

4

decimal places.

8 Parallelism and Performance

Not applicable.

9 Further Comments

If the exact probabilities are required, then the first

n - p

eigenvalues of

H A

can be computed and nag_prob_lin_chi_sq (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.

NAG Library Function Documentnag_durbin_watson_stat (g02fcc)

+− Contents

1 Purpose

2 Specification

3 Description