# NAG Library Function Document

## 1Purpose

nag_prob_durbin_watson (g01epc) calculates upper and lower bounds for the significance of a Durbin–Watson statistic.

## 2Specification

 #include #include
 void nag_prob_durbin_watson (Integer n, Integer ip, double d, double *pdl, double *pdu, NagError *fail)

## 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:    $\mathbf{d}$doubleInput
On entry: $d$, the Durbin–Watson statistic.
Constraint: ${\mathbf{d}}\ge 0.0$.
4:    $\mathbf{pdl}$double *Output
On exit: lower bound for the significance of the Durbin–Watson statistic, ${p}_{l}$.
5:    $\mathbf{pdu}$double *Output
On exit: upper bound for the significance of the Durbin–Watson statistic, ${p}_{u}$.
6:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{ip}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ip}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ip}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>{\mathbf{ip}}$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL
On entry, ${\mathbf{d}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{d}}\ge 0.0$.

## 7Accuracy

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

## 8Parallelism and Performance

nag_prob_durbin_watson (g01epc) is not threaded in any implementation.

## 9Further Comments

If the exact probabilities are required, then the first $n-p$ eigenvalues of $HA$ can be computed and nag_prob_lin_chi_sq (g01jdc) 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 (g01epce.c)

### 10.2Program Data

Program Data (g01epce.d)

### 10.3Program Results

Program Results (g01epce.r)

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