The function may be called by the names: g02fcc, nag_correg_linregm_stat_durbwat or nag_durbin_watson_stat.
For the general linear regression model
is a vector of length of the dependent variable,
is an matrix of the independent variables,
is a vector of length of unknown parameters,
is a vector of length of unknown random errors.
The residuals are given by
and the fitted values, , can be written as for an matrix . 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 can be considered as a time series, the Durbin–Watson test can be used to test for serial correlation in the , see Durbin and Watson (1950), Durbin and Watson (1951) and Durbin and Watson (1971).
The Durbin–Watson statistic is
Positive serial correlation in the will lead to a small value of while for independent errors will be close to . Durbin and Watson show that the exact distribution of depends on the eigenvalues of the matrix where the matrix is such that can be written as
and the eigenvalues of the matrix are , for .
However bounds on the distribution can be obtained, the lower bound being
and the upper bound being
where the are independent standard Normal variables. The lower tail probabilities associated with these bounds, and , are computed by g01epc. The interpretation of the bounds is that, for a test of size (significance) , if the test is significant, if the test is not significant, while if and 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 ; see Newbold (1988).
Durbin J and Watson G S (1950) Testing for serial correlation in least squares regression. I Biometrika37 409–428
Durbin J and Watson G S (1951) Testing for serial correlation in least squares regression. II Biometrika38 159–178
Durbin J and Watson G S (1971) Testing for serial correlation in least squares regression. III Biometrika58 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
1: – IntegerInput
On entry: , the number of residuals.
2: – IntegerInput
On entry: , the number of independent variables in the regression model, including the mean.
3: – const doubleInput
On entry: the residuals, .
the mean of the residuals , where .
4: – double *Output
On exit: the Durbin–Watson statistic, .
5: – double *Output
On exit: lower bound for the significance of the Durbin–Watson statistic, .
6: – double *Output
On exit: upper bound for the significance of the Durbin–Watson statistic, .
7: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, .
On entry, and .
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.
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.
On entry, all residuals are identical.
On entry, mean of .
Constraint: the mean of the residuals , where .
The probabilities are computed to an accuracy of at least decimal places.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
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.
If the exact probabilities are required, then the first eigenvalues of can be computed and g01jdc used to compute the required probabilities with the argument c set to and the argument
d set to the Durbin–Watson statistic .
A set of residuals are read in and the Durbin–Watson statistic along with the probability bounds are computed and printed.