nag_durbin_watson_stat (g02fcc) (PDF version)
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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).

## 4  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
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:     nIntegerInput
On entry: $n$, the number of residuals.
Constraint: ${\mathbf{n}}>{\mathbf{p}}$.
2:     pIntegerInput
On entry: $p$, the number of independent variables in the regression model, including the mean.
Constraint: ${\mathbf{p}}\ge 1$.
3:     res[n]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:     ddouble *Output
On exit: the Durbin–Watson statistic, $d$.
5:     pdldouble *Output
On exit: lower bound for the significance of the Durbin–Watson statistic, ${p}_{\mathrm{l}}$.
6:     pdudouble *Output
On exit: upper bound for the significance of the Durbin–Watson statistic, ${p}_{\mathrm{u}}$.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
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.
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.

Not applicable.

## 9  Further 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 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)

nag_durbin_watson_stat (g02fcc) (PDF version)
g02 Chapter Contents
g02 Chapter Introduction
NAG Library Manual