# NAG Library Routine Document

## 1Purpose

g02fcf calculates the Durbin–Watson statistic, for a set of residuals, and the upper and lower bounds for its significance.

## 2Specification

Fortran Interface
 Subroutine g02fcf ( n, ip, res, d, pdl, pdu, work,
 Integer, Intent (In) :: n, ip Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: res(n) Real (Kind=nag_wp), Intent (Out) :: d, pdl, pdu, work(n)
#include nagmk26.h
 void g02fcf_ (const Integer *n, const Integer *ip, const double res[], double *d, double *pdl, double *pdu, double work[], Integer *ifail)

## 3Description

For the general linear regression model
 $y=Xβ+ε,$
 where $y$ is a vector of length $n$ of the dependent variable, $X$ is a $n$ by $p$ matrix of the independent variables, $\beta$ is a vector of length $p$ of unknown arguments, and $\epsilon$ is a vector of length $n$ of unknown random errors.
The residuals are given by
 $r=y-y^=y-Xβ^$
and the fitted values, $\stackrel{^}{y}=X\stackrel{^}{\beta }$, can be written as $Hy$ for a $n$ by $n$ 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).
The Durbin–Watson statistic is
 $d=∑i=1 n-1 ri+1-ri 2 ∑i=1nri2 .$
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
 $d=rTAr rTr$
and the eigenvalues of the matrix $A$ are ${\lambda }_{j}=\left(1-\mathrm{cos}\left(\pi j/n\right)\right)$, for $j=1,2,\dots ,n-1$.
However bounds on the distribution can be obtained, the lower bound being
 $dl=∑i=1 n-pλiui2 ∑i=1 n-pui2$
and the upper bound being
 $du=∑i= 1 n-pλi- 1+pui2 ∑i= 1 n-pui2 ,$
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).
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

## 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 .
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 $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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 $-1$, 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, ${\mathbf{ip}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ip}}\ge 1$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ip}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>{\mathbf{ip}}$.
${\mathbf{ifail}}=2$
On entry, mean of ${\mathbf{res}}=〈\mathit{\text{value}}〉$.
Constraint: the mean of the residuals $\text{}\le \sqrt{\epsilon }$, where
${\mathbf{ifail}}=3$
On entry, all residuals are identical.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation 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.

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.

### 10.1Program Text

Program Text (g02fcfe.f90)

### 10.2Program Data

Program Data (g02fcfe.d)

### 10.3Program Results

Program Results (g02fcfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017