NAG Library Function Document
nag_prob_durbin_watson (g01epc) calculates upper and lower bounds for the significance of a Durbin–Watson statistic.
||nag_prob_durbin_watson (Integer n,
be the residuals from a linear regression of
independent variables, including the mean, where the
can be considered as a time series. The Durbin–Watson test (see Durbin and Watson (1950)
, Durbin and Watson (1951)
and Durbin and Watson (1971)
) can be used to test for serial correlation in the error term in the regression.
The Durbin–Watson test statistic is:
which can be written as
is given by
with the nonzero eigenvalues of the matrix
Durbin and Watson show that the exact distribution of
depends on the eigenvalues of a matrix
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
are independent standard Normal variables.
Two algorithms are used to compute the lower tail (significance level) probabilities,
, associated with
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 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
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 Biometrika 48 419–426
Newbold P (1988) Statistics for Business and Economics Prentice–Hall
Pan Jie–Jian (1964) Distributions of the noncircular serial correlation coefficients Shuxue Jinzhan 7 328–337
n – IntegerInput
On entry: , the number of observations used in calculating the Durbin–Watson statistic.
ip – IntegerInput
On entry: , the number of independent variables in the regression model, including the mean.
d – doubleInput
On entry: , the Durbin–Watson statistic.
pdl – double *Output
On exit: lower bound for the significance of the Durbin–Watson statistic, .
pdu – double *Output
On exit: upper bound for the significance of the Durbin–Watson statistic, .
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
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
On entry, .
On successful exit at least decimal places of accuracy are achieved.
8 Parallelism and Performance
If the exact probabilities are required, then the first
can be computed and nag_prob_lin_chi_sq (g01jdc)
used to compute the required probabilities with c
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.
10.1 Program Text
Program Text (g01epce.c)
10.2 Program Data
Program Data (g01epce.d)
10.3 Program Results
Program Results (g01epce.r)