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

# Syntax

C# |
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public static void g01ep( int n, int ip, double d, out double pdl, out double pdu, out int ifail ) |

Visual Basic |
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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++ |
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public: static void g01ep( int n, int ip, double d, [OutAttribute] double% pdl, [OutAttribute] double% pdu, [OutAttribute] int% ifail ) |

F# |
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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:

which can be written as

where the $n$ by $n$ matrix $A$ is given by

with the nonzero eigenvalues of the matrix $A$ being ${\lambda}_{j}=\left(1-\mathrm{cos}\left(\pi j/n\right)\right)$, for $\mathit{j}=1,2,\dots ,n-1$.

$$d=\frac{\sum _{i=1}^{n-1}{\left({r}_{i+1}-{r}_{i}\right)}^{2}}{\sum _{i=1}^{n}{r}_{i}^{2}}\text{,}$$ |

$$d=\frac{{r}^{\mathrm{T}}Ar}{{r}^{\mathrm{T}}r}\text{,}$$ |

$$A=\left[\begin{array}{rrrrr}1& -1& 0& \dots & :\\ -1& 2& -1& \dots & :\\ 0& -1& 2& \dots & :\\ :& 0& -1& \dots & :\\ :& :& :& \dots & :\\ :& :& :& \dots & -1\\ 0& 0& 0& \dots & 1\end{array}\right]$$ |

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, $\hat{y}$, can be written as $\hat{y}=Hy$. However, bounds on the distribution can be obtained, the lower bound being

and the upper bound being

where ${u}_{i}$ are independent standard Normal variables.

$${d}_{l}=\frac{\sum _{i=1}^{n-p}{\lambda}_{i}{u}_{i}^{2}}{\sum _{i=1}^{n-p}{u}_{i}^{2}}$$ |

$${d}_{u}=\frac{\sum _{i=1}^{n-p}{\lambda}_{i-1+p}{u}_{i}^{2}}{\sum _{i=1}^{n-p}{u}_{i}^{2}}\text{,}$$ |

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**409–428Durbin J and Watson G S (1951) Testing for serial correlation in least squares regression. II

*Biometrika***38**159–178Durbin J and Watson G S (1971) Testing for serial correlation in least squares regression. III

*Biometrika***58**1–19Farebrother R W (1980) Algorithm AS 153. Pan's procedure for the tail probabilities of the Durbin–Watson statistic

*Appl. Statist.***29**224–227Imhof J P (1961) Computing the distribution of quadratic forms in Normal variables

*Biometrika***48**419–426Newbold P (1988)

*Statistics for Business and Economics*Prentice–HallPan 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, ${\mathbf{d}}<0.0$.

# Accuracy

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

# Parallelism and Performance

None.

# Further Comments

# 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