g02fc calculates the Durbin–Watson statistic, for a set of residuals, and the upper and lower bounds for its significance.
Public Shared Sub g02fc ( _ n As Integer, _ ip As Integer, _ res As Double(), _ <OutAttribute> ByRef d As Double, _ <OutAttribute> ByRef pdl As Double, _ <OutAttribute> ByRef pdu As Double, _ <OutAttribute> ByRef ifail As Integer _ )
public: static void g02fc( int n, int ip, array<double>^ res, [OutAttribute] double% d, [OutAttribute] double% pdl, [OutAttribute] double% pdu, [OutAttribute] int% ifail )
- Type: System..::..Int32On entry: , the number of residuals.Constraint: .
- Type: System..::..Int32On entry: , the number of independent variables in the regression model, including the mean.Constraint: .
- Type: array<System..::..Double>()An array of size [n]On entry: the residuals, .Constraint: the mean of the residuals , where .
- Type: System..::..Double%On exit: the Durbin–Watson statistic, .
- 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, .
For the general linear regression model
|where|| is a vector of length of the dependent variable,
is a by matrix of the independent variables,
is a vector of length of unknown parameters,
|and||is a vector of length of unknown random errors.|
The residuals are given by
and the fitted values, , can be written as for a by 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 g01ep. 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.
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
Errors or warnings detected by the method:
On entry, , or . On entry, the mean of the residuals was , where . On entry, all residuals are identical.
The probabilities are computed to an accuracy of at least decimal places.
A set of residuals are read in and the Durbin–Watson statistic along with the probability bounds are computed and printed.