naginterfaces.library.stat.prob_​durbin_​watson

naginterfaces.library.stat.prob_durbin_watson(n, ip, d)

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

For full information please refer to the NAG Library document for g01ep

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01epf.html

Parameters

nint

$n$, the number of observations used in calculating the Durbin–Watson statistic.

ipint

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

dfloat

$d$, the Durbin–Watson statistic.

Returns

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, $d=⟨value⟩$.

Constraint: $d\ge 0.0$.

Notes

Let $r={(r_1,r_2,\dots ,r_n)}^T$ 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:

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

which can be written as

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

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

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

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

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, $\hat{y}$, can be written as $\hat{y}=Hy$. 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 $u_i$ 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 $4-d$.

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

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