g01ep calculates upper and lower bounds for the significance of a Durbin–Watson statistic.
Public Shared Sub g01ep ( _ n As Integer, _ ip As Integer, _ d As Double, _ <OutAttribute> ByRef pdl As Double, _ <OutAttribute> ByRef pdu As Double, _ <OutAttribute> ByRef ifail As Integer _ )
public: static void g01ep( int n, int ip, double d, [OutAttribute] double% pdl, [OutAttribute] double% pdu, [OutAttribute] int% ifail )
- Type: System..::..Int32On entry: , the number of observations used in calculating the Durbin–Watson statistic.Constraint: .
- Type: System..::..Int32On entry: , the number of independent variables in the regression model, including the mean.Constraint: .
- Type: System..::..DoubleOn entry: , the Durbin–Watson statistic.Constraint: .
- Type: System..::..Double%On exit: lower bound for the significance of the Durbin–Watson statistic, .
- Type: System..::..Double%On exit: upper bound for the significance of the Durbin–Watson statistic, .
Let be the residuals from a linear regression of on independent variables, including the mean, where the values 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
where the by matrix is given by
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 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
Errors or warnings detected by the method:
On entry, , or . On entry, .
On successful exit at least decimal places of accuracy are achieved.
The values of , and the Durbin–Watson statistic are input and the bounds for the significance level calculated and printed.