naginterfaces.library.correg.linregm_​stat_​durbwat

naginterfaces.library.correg.linregm_stat_durbwat(ip, res)

linregm_stat_durbwat calculates the Durbin–Watson statistic, for a set of residuals, and the upper and lower bounds for its significance.

For full information please refer to the NAG Library document for g02fc

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g02/g02fcf.html

Parameters

ipint

$p$, the number of independent variables in the regression model, including the mean.

resfloat, array-like, shape $\left(n\right)$

The residuals, $r_1,r_2,\dots ,r_n$.

Returns

dfloat

The Durbin–Watson statistic, $d$.

pdlfloat

Lower bound for the significance of the Durbin–Watson statistic, $p_l$.

pdufloat

Upper bound for the significance of the Durbin–Watson statistic, $p_u$.

Raises

NagValueError

(errno $1$)

On entry, $ip=⟨value⟩$.

Constraint: $ip\ge 1$.

(errno $1$)

On entry, $n=⟨value⟩$ and $ip=⟨value⟩$.

Constraint: $n>ip$.

(errno $2$)

On entry, mean of $res=⟨value⟩$.

Constraint: the mean of the residuals $\le \sqrt{ϵ}$, where $ϵ=\text{machine precision}$.

(errno $3$)

On entry, all residuals are identical.

Notes

For the general linear regression model

$$y=X\beta +ϵ\text{,}$$

where	$y$ is a vector of length $n$ of the dependent variable, $X$ is an $n\times p$ matrix of the independent variables, $\beta$ is a vector of length $p$ of unknown parameters,
and	$ϵ$ is a vector of length $n$ of unknown random errors.

The residuals are given by

$$r=y-\hat{y}=y-X\hat{\beta }$$

and the fitted values, $\hat{y}=X\hat{\beta }$, can be written as $Hy$ for an $n\times n$ 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 ${ϵ}_i$, see Durbin and Watson (1950), Durbin and Watson (1951) and Durbin and Watson (1971).

The Durbin–Watson statistic is

$$d=\frac{{\sum }_{i=1}^{n-1}{(r_{i+1}-r_i)}^2}{{\sum }_{i=1}^n{r}_i^2}\text{.}$$

Positive serial correlation in the ${ϵ}_i$ 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

$$d=\frac{r^{T}Ar}{r^{T}r}$$

and the eigenvalues of the matrix $A$ are ${\lambda }_j=(1-\cos \left(\pi j/n\right))$, for $j=1,2,\dots ,n-1$.

However bounds on the distribution can be obtained, the lower bound being

$$d_l=\frac{{\sum }_{i=1}^{n-p}{\lambda }_i{u}_i^2}{{\sum }_{i=1}^{n-p}{u}_i^2}$$

and the upper bound being

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

where the $u_i$ are independent standard Normal variables. The lower tail probabilities associated with these bounds, $p_l$ and $p_u$, are computed by stat.prob_durbin_watson. 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 $p_l>\alpha$ and $p_u\le \alpha$ 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 stat.prob_durbin_watson should be made with the argument $d$ taking the value of $4-d$; see Newbold (1988).

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