The routine may be called by the names g02fcf or nagf_correg_linregm_stat_durbwat.
3Description
For the general linear regression model
$$y=X\beta +\epsilon \text{,}$$
where
$y$ is a vector of length $n$ of the dependent variable,
$X$ is an $n\times p$ matrix of the independent variables,
$\beta $ is a vector of length $p$ of unknown parameters,
and
$\epsilon $ is a vector of length $n$ of unknown random errors.
The residuals are given by
$$r=y-\hat{y}=y-X\hat{\beta}$$
and the fitted values, $\hat{y}=X\hat{\beta}$, can be written as $Hy$ for an $n\times n$ matrix $H$. 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 ${y}_{1},{y}_{2},\dots ,{y}_{n}$ can be considered as a time series, the Durbin–Watson test can be used to test for serial correlation in the ${\epsilon}_{i}$, see Durbin and Watson (1950), Durbin and Watson (1951) and Durbin and Watson (1971).
Positive serial correlation in the ${\epsilon}_{i}$ will lead to a small value of $d$ while for independent errors $d$ will be close to $2$. Durbin and Watson show that the exact distribution of $d$ depends on the eigenvalues of the matrix $HA$ where the matrix $A$ is such that $d$ can be written as
where the ${u}_{i}$ are independent standard Normal variables. The lower tail probabilities associated with these bounds, ${p}_{\mathrm{l}}$ and ${p}_{\mathrm{u}}$, are computed by g01epf. The interpretation of the bounds is that, for a test of size (significance) $\alpha $, if ${p}_{l}\le \alpha $ the test is significant, if ${p}_{u}>\alpha $ the test is not significant, while if ${p}_{\mathrm{l}}>\alpha $ and ${p}_{\mathrm{u}}\le \alpha $ 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 g01epf should be made with the argument d taking the value of $4-d$; see Newbold (1988).
4References
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
5Arguments
1: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of residuals.
Constraint:
${\mathbf{n}}>{\mathbf{ip}}$.
2: $\mathbf{ip}$ – IntegerInput
On entry: $p$, the number of independent variables in the regression model, including the mean.
Constraint:
${\mathbf{ip}}\ge 1$.
3: $\mathbf{res}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the residuals, ${r}_{1},{r}_{2},\dots ,{r}_{n}$.
Constraint:
the mean of the residuals $\text{}\le \sqrt{\epsilon}$, where $\epsilon =\mathit{machineprecision}$.
4: $\mathbf{d}$ – Real (Kind=nag_wp)Output
On exit: the Durbin–Watson statistic, $d$.
5: $\mathbf{pdl}$ – Real (Kind=nag_wp)Output
On exit: lower bound for the significance of the Durbin–Watson statistic, ${p}_{\mathrm{l}}$.
6: $\mathbf{pdu}$ – Real (Kind=nag_wp)Output
On exit: upper bound for the significance of the Durbin–Watson statistic, ${p}_{\mathrm{u}}$.
7: $\mathbf{work}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
8: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{ip}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ip}}\ge 1$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{ip}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}>{\mathbf{ip}}$.
${\mathbf{ifail}}=2$
On entry, mean of ${\mathbf{res}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: the mean of the residuals $\text{}\le \sqrt{\epsilon}$, where $\epsilon =\mathit{machineprecision}$.
${\mathbf{ifail}}=3$
On entry, all residuals are identical.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
The probabilities are computed to an accuracy of at least $4$ decimal places.
8Parallelism and Performance
g02fcf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
If the exact probabilities are required, then the first $n-p$ eigenvalues of $HA$ can be computed and g01jdf used to compute the required probabilities with the argument c set to $0.0$ and the argument
d set to the Durbin–Watson statistic $d$.
10Example
A set of $10$ residuals are read in and the Durbin–Watson statistic along with the probability bounds are computed and printed.