﻿ g01ep Method
g01ep calculates upper and lower bounds for the significance of a Durbin–Watson statistic.

# Syntax

C#
```public static void g01ep(
int n,
int ip,
double d,
out double pdl,
out double pdu,
out int ifail
)```
Visual Basic
```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 _
)```
Visual C++
```public:
static void g01ep(
int n,
int ip,
double d,
[OutAttribute] double% pdl,
[OutAttribute] double% pdu,
[OutAttribute] int% ifail
)```
F#
```static member g01ep :
n : int *
ip : int *
d : float *
pdl : float byref *
pdu : float byref *
ifail : int byref -> unit
```

#### Parameters

n
Type: System..::..Int32
On entry: $n$, the number of observations used in calculating the Durbin–Watson statistic.
Constraint: ${\mathbf{n}}>{\mathbf{ip}}$.
ip
Type: System..::..Int32
On entry: $p$, the number of independent variables in the regression model, including the mean.
Constraint: ${\mathbf{ip}}\ge 1$.
d
Type: System..::..Double
On entry: $d$, the Durbin–Watson statistic.
Constraint: ${\mathbf{d}}\ge 0.0$.
pdl
Type: System..::..Double%
On exit: lower bound for the significance of the Durbin–Watson statistic, ${p}_{l}$.
pdu
Type: System..::..Double%
On exit: upper bound for the significance of the Durbin–Watson statistic, ${p}_{u}$.
ifail
Type: System..::..Int32%
On exit: ${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

# Description

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:
 $d=∑i=1n-1ri+1-ri2∑i=1nri2,$
which can be written as
 $d=rTArrTr,$
where the $n$ by $n$ matrix $A$ is given by
 $A=1-10…:-12-1…:0-12…::0-1…::::…::::…-1000…1$
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
 $dl=∑i=1n-pλiui2∑i=1n-pui2$
and the upper bound being
 $du=∑i=1n-pλi-1+pui2∑i=1n-pui2,$
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 $4-d$.

# References

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

# Error Indicators and Warnings

Errors or warnings detected by the method:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{n}}\le {\mathbf{ip}}$, or ${\mathbf{ip}}<1$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{d}}<0.0$.
${\mathbf{ifail}}=-9000$
An error occured, see message report.

# Accuracy

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

# Parallelism and Performance

None.

If the exact probabilities are required, then the first $n-p$ eigenvalues of $HA$ can be computed and g01jd used to compute the required probabilities with c set to $0.0$ and d to the Durbin–Watson statistic.

# Example

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

Example program (C#): g01epe.cs

Example program data: g01epe.d

Example program results: g01epe.r