nag_durbin_watson_stat (g02fcc) (PDF version)
g02 Chapter Contents
g02 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_durbin_watson_stat (g02fcc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_durbin_watson_stat (g02fcc) calculates the Durbin–Watson statistic, for a set of residuals, and the upper and lower bounds for its significance.

2  Specification

#include <nag.h>
#include <nagg02.h>
void  nag_durbin_watson_stat (Integer n, Integer p, const double res[], double *d, double *pdl, double *pdu, NagError *fail)

3  Description

For the general linear regression model
y=Xβ+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-y^=y-Xβ^
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 nag_prob_durbin_watson (g01epc). 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 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:     nIntegerInput
On entry: n, the number of residuals.
Constraint: n>p.
2:     pIntegerInput
On entry: p, the number of independent variables in the regression model, including the mean.
Constraint: p1.
3:     res[n]const doubleInput
On entry: the residuals, r1,r2,,rn.
Constraint: the mean of the residuals ε, where ε=machine precision.
4:     ddouble *Output
On exit: the Durbin–Watson statistic, d.
5:     pdldouble *Output
On exit: lower bound for the significance of the Durbin–Watson statistic, pl.
6:     pdudouble *Output
On exit: upper bound for the significance of the Durbin–Watson statistic, pu.
7:     failNagError *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: p1.
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  Further Comments

If the exact probabilities are required, then the first n-p eigenvalues of HA 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.

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 (g02fcce.c)

9.2  Program Data

Program Data (g02fcce.d)

9.3  Program Results

Program Results (g02fcce.r)


nag_durbin_watson_stat (g02fcc) (PDF version)
g02 Chapter Contents
g02 Chapter Introduction
NAG C Library Manual

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