# NAG Library Routine Document

## 1Purpose

g01epf calculates upper and lower bounds for the significance of a Durbin–Watson statistic.

## 2Specification

Fortran Interface
 Subroutine g01epf ( n, ip, d, pdl, pdu, work,
 Integer, Intent (In) :: n, ip Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: d Real (Kind=nag_wp), Intent (Out) :: pdl, pdu, work(n)
#include <nagmk26.h>
 void g01epf_ (const Integer *n, const Integer *ip, const double *d, double *pdl, double *pdu, double work[], Integer *ifail)

## 3Description

Let $r={\left({r}_{1},{r}_{2},\dots ,{r}_{n}\right)}^{\mathrm{T}}$ be the residuals from a linear regression of $y$ on $p$ independent variables, including the mean, where the $y$ values ${y}_{1},{y}_{2},\dots ,{y}_{n}$ 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:
 $d=∑i=1 n-1 ri+1-ri 2 ∑i=1nri2 ,$
which can be written as
 $d=rTAr rTr ,$
where the $n$ by $n$ matrix $A$ is given by
 $A= 1 -1 0 … : -1 2 -1 … : 0 -1 2 … : : 0 -1 … : : : : … : : : : … -1 0 0 0 … 1$
with the nonzero eigenvalues of the matrix $A$ being ${\lambda }_{j}=\left(1-\mathrm{cos}\left(\pi j/n\right)\right)$, for $\mathit{j}=1,2,\dots ,n-1$.
Durbin and Watson show that the exact distribution of $d$ depends on the eigenvalues of a matrix $HA$, where $H$ is the hat matrix of independent variables, i.e., the matrix such that the vector of fitted values, $\stackrel{^}{y}$, can be written as $\stackrel{^}{y}=Hy$. 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 ${u}_{i}$ are independent standard Normal variables.
Two algorithms are used to compute the lower tail (significance level) probabilities, ${p}_{l}$ and ${p}_{u}$, associated with ${d}_{l}$ and ${d}_{u}$. If $n\le 60$ 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 $d$ should be replaced by $4-d$.

## 4References

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

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of observations used in calculating the Durbin–Watson statistic.
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:     $\mathbf{d}$ – Real (Kind=nag_wp)Input
On entry: $d$, the Durbin–Watson statistic.
Constraint: ${\mathbf{d}}\ge 0.0$.
4:     $\mathbf{pdl}$ – Real (Kind=nag_wp)Output
On exit: lower bound for the significance of the Durbin–Watson statistic, ${p}_{l}$.
5:     $\mathbf{pdu}$ – Real (Kind=nag_wp)Output
On exit: upper bound for the significance of the Durbin–Watson statistic, ${p}_{u}$.
6:     $\mathbf{work}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
7:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . 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  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  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, ${\mathbf{d}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{d}}\ge 0.0$.
${\mathbf{ifail}}=-99$
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

On successful exit at least $4$ decimal places of accuracy are achieved.

## 8Parallelism and Performance

g01epf is not threaded in any implementation.

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 c set to $0.0$ and d to the Durbin–Watson statistic.

## 10Example

The values of $n$, $p$ and the Durbin–Watson statistic $d$ are input and the bounds for the significance level calculated and printed.

### 10.1Program Text

Program Text (g01epfe.f90)

### 10.2Program Data

Program Data (g01epfe.d)

### 10.3Program Results

Program Results (g01epfe.r)