G02FCF (PDF version)
G02 Chapter Contents
G02 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

G02FCF calculates the Durbin–Watson statistic, for a set of residuals, and the upper and lower bounds for its significance.

2  Specification

REAL (KIND=nag_wp)  RES(N), D, PDL, PDU, WORK(N)

3  Description

For the general linear regression model
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
and the fitted values, y^=Xβ^, can be written as Hy for a n by 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 y1,y2,,yn can be considered as a time series, the Durbin–Watson test can be used to test for serial correlation in the εi, see Durbin and Watson (1950), Durbin and Watson (1951) and Durbin and Watson (1971).
The Durbin–Watson statistic is
d=i=1 n-1 ri+1-ri 2 i=1nri2 .
Positive serial correlation in the εi 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 HA where the matrix A is such that d can be written as
d=rTAr rTr
and the eigenvalues of the matrix A are λj=1-cosπj/n, for j=1,2,,n-1.
However bounds on the distribution can be obtained, the lower bound being
dl=i=1 n-pλiui2 i=1 n-pui2
and the upper bound being
du=i= 1 n-pλi- 1+pui2 i= 1 n-pui2 ,
where the ui are independent standard Normal variables. The lower tail probabilities associated with these bounds, pl and pu, are computed by G01EPF. The interpretation of the bounds is that, for a test of size (significance) α, if plα the test is significant, if pu>α the test is not significant, while if pl>α and puα 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 G01EPF should be made with the parameter 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  Parameters

1:     N – INTEGERInput
On entry: n, the number of residuals.
Constraint: N>IP.
2:     IP – INTEGERInput
On entry: p, the number of independent variables in the regression model, including the mean.
Constraint: IP1.
3:     RES(N) – REAL (KIND=nag_wp) arrayInput
On entry: the residuals, r1,r2,,rn.
Constraint: the mean of the residuals ε, where ε=machine precision.
4:     D – REAL (KIND=nag_wp)Output
On exit: the Durbin–Watson statistic, d.
5:     PDL – REAL (KIND=nag_wp)Output
On exit: lower bound for the significance of the Durbin–Watson statistic, pl.
6:     PDU – REAL (KIND=nag_wp)Output
On exit: upper bound for the significance of the Durbin–Watson statistic, pu.
7:     WORK(N) – REAL (KIND=nag_wp) arrayWorkspace
8:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
On entry,NIP,
On entry,the mean of the residuals was >ε, where ε=machine precision.
On entry,all residuals are identical.

7  Accuracy

The probabilities are computed to an accuracy of at least 4 decimal places.

8  Further Comments

If the exact probabilities are required, then the first n-p eigenvalues of HA can be computed and G01JDF 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.

9  Example

A set of 10 residuals are read in and the Durbin–Watson statistic along with the probability bounds are computed and printed.

9.1  Program Text

Program Text (g02fcfe.f90)

9.2  Program Data

Program Data (g02fcfe.d)

9.3  Program Results

Program Results (g02fcfe.r)

G02FCF (PDF version)
G02 Chapter Contents
G02 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012