Durbin-Watson test

The Durbin-Watson statistic is defined by¹:

\(d = \frac{\sum_{t = 2}^{N}\left( {\widehat{a}}_{t} - {\widehat{a}}_{t - 1} \right)^{2}}{\sum_{t = 1}^{N}{\widehat{a}}_{t}^{2}}\) [1]

where:

${\widehat{a}}_{t}$ – residual from the model.

Since \(\sum_{t = 2}^{N}\left( {\widehat{a}}_{t} - {\widehat{a}}_{t - 1} \right)^{2} \cong \\)2\(\sum_{t = 1}^{N}{\widehat{a}}_{t}^{2} - 2\sum_{t = 2}^{N}{ {\widehat{a}}_{t}{\widehat{a}}_{t - 1}}\), then the approximation $d \cong 2(1 - r_{z,1})$, where \(r_{z,1} = \frac{\sum_{t = 1}^{N}{ {\widehat{a}}_{t}{\widehat{a}}_{t - 1}}}{\sum_{t = 1}^{N}{\widehat{a}}_{t}^{2}}\) is the autocorrelation coefficient of the residuals at lag 1, is true.

The Durbin-Watson statistics is between 0 and 4. When the model provides an adequate description of the data, then $r_{z,1}$ should be close to 0 and therefore the Durbin-Watson statistics is close to 2. When the Durbin–Watson statistic is substantially less than 2, there is evidence of positive serial correlation, while when it is substantially greater than 2 it indicates that the successive error terms are, on average, much different in value from one another, i.e., negatively correlated.

More formally, to test for a positive autocorrelation at significance level $\alpha$, the Durbin-Watson statistics is compared to the lower ($d_{L,\alpha}\ )\ $and upper ($d_{U,\alpha})$ critical values:

• If $d < d_{L,\alpha}$ there is statistical evidence that the error terms are positively autocorrelated.

• If $d > d_{U,\alpha}$ there is no statistical evidence that the error terms are positively autocorrelated.

• If  $d_{L,\alpha}$ $< d < d_{U,\alpha}$ the test is inconclusive.

Positive serial correlation is serial correlation in which a positive error for one observation increases the chances of a positive error for another observation.

To test for negative autocorrelation at significance$\ \alpha$, the test statistic $(4 - d)$ is compared to the lower ($d_{L,\alpha}\ )\ $and upper ($d_{U,\alpha})$ critical values:

• If $\left( 4 - d \right) < d_{L,\alpha}$ there is statistical evidence that the error terms are negatively autocorrelated.

• If $\left( 4 - d \right) > d_{U,\alpha}$ there is no statistical evidence that the error terms are negatively autocorrelated.

• If $d_{U,\alpha} < \left( 4 - d \right) < d_{U,\alpha}$ the test is inconclusive.