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g02 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_durbin_watson_stat (g02fcc)

## 1  Purpose

nag_durbin_watson_stat (g02fcc) calculates the Durbin–Watson statistic, for a set of residuals, and the upper and lower bounds for its significance.

## 2  Specification

 #include #include
 void nag_durbin_watson_stat (Integer n, Integer p, const double res[], double *d, double *pdl, double *pdu, NagError *fail)

## 3  Description

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 nag_prob_durbin_watson (g01epc). 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 nag_prob_durbin_watson (g01epc) 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

## 5  Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of residuals.
Constraint: ${\mathbf{n}}>{\mathbf{p}}$.
2:    $\mathbf{p}$IntegerInput
On entry: $p$, the number of independent variables in the regression model, including the mean.
Constraint: ${\mathbf{p}}\ge 1$.
3:    $\mathbf{res}\left[{\mathbf{n}}\right]$const doubleInput
On entry: the residuals, ${r}_{1},{r}_{2},\dots ,{r}_{n}$.
Constraint: the mean of the residuals $\text{}\le \sqrt{\epsilon }$, where .
4:    $\mathbf{d}$double *Output
On exit: the Durbin–Watson statistic, $d$.
5:    $\mathbf{pdl}$double *Output
On exit: lower bound for the significance of the Durbin–Watson statistic, ${p}_{\mathrm{l}}$.
6:    $\mathbf{pdu}$double *Output
On exit: upper bound for the significance of the Durbin–Watson statistic, ${p}_{\mathrm{u}}$.
7:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error 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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{p}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$ and ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>{\mathbf{p}}$.
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_RESID_IDEN
On entry, all residuals are identical.
NE_RESID_MEAN
On entry, the mean of res is not approximately $0.0$, $\text{mean}=〈\mathit{\text{value}}〉$.

## 7  Accuracy

The probabilities are computed to an accuracy of at least $4$ decimal places.

## 8  Parallelism and Performance

nag_durbin_watson_stat (g02fcc) is not threaded in any implementation.

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

## 10  Example

A set of $10$ residuals are read in and the Durbin–Watson statistic along with the probability bounds are computed and printed.

### 10.1  Program Text

Program Text (g02fcce.c)

### 10.2  Program Data

Program Data (g02fcce.d)

### 10.3  Program Results

Program Results (g02fcce.r)