with the nonzero eigenvalues of the matrix being , for .
Durbin and Watson show that the exact distribution of depends on the eigenvalues of a matrix , where is the hat matrix of independent variables, i.e., the matrix such that the vector of fitted values, , can be written as . However, bounds on the distribution can be obtained, the lower bound being
and the upper bound being
where are independent standard Normal variables.
Two algorithms are used to compute the lower tail (significance level) probabilities, and , associated with and . If the procedure due to Pan (1964) is used, see Farebrother (1980), otherwise Imhof's method (see Imhof (1961)) is used.
The bounds are for the usual test of positive correlation; if a test of negative correlation is required the value of should be replaced by .
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
Farebrother R W (1980) Algorithm AS 153. Pan's procedure for the tail probabilities of the Durbin–Watson statistic Appl. Statist.29 224–227
Imhof J P (1961) Computing the distribution of quadratic forms in Normal variables Biometrika48 419–426
Newbold P (1988) Statistics for Business and Economics Prentice–Hall
Pan Jie–Jian (1964) Distributions of the noncircular serial correlation coefficients Shuxue Jinzhan7 328–337
1: – IntegerInput
On entry: , the number of observations used in calculating the Durbin–Watson statistic.
2: – IntegerInput
On entry: , the number of independent variables in the regression model, including the mean.
3: – doubleInput
On entry: , the Durbin–Watson statistic.
4: – double *Output
On exit: lower bound for the significance of the Durbin–Watson statistic, .
5: – double *Output
On exit: upper bound for the significance of the Durbin–Watson statistic, .
6: – 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, .
On successful exit at least decimal places of accuracy are achieved.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g01epc is not threaded in any implementation.
If the exact probabilities are required, then the first eigenvalues of can be computed and g01jdc used to compute the required probabilities with c set to and d to the Durbin–Watson statistic.
The values of , and the Durbin–Watson statistic are input and the bounds for the significance level calculated and printed.