It is named after James Durbin and Geoffrey Watson.
The small sample distribution of this ratio was derived by John von Neumann (von Neumann, 1941).
Durbin and Watson (1950, 1951) applied this statistic to the residuals from least squares regressions, and developed bounds tests for the null hypothesis that the errors are serially uncorrelated against the alternative that they follow a first order autoregressive process.
Note that the distribution of this test statistic does not depend on the estimated regression coefficients and the variance of the errors.
is the sample autocorrelation of the residuals at lag 1.
If the Durbin–Watson statistic is substantially less than 2, there is evidence of positive serial correlation.
As a rough rule of thumb, if Durbin–Watson is less than 1.0, there may be cause for alarm.
indicate successive error terms are positively correlated.
, successive error terms are negatively correlated.
In regressions, this can imply an underestimation of the level of statistical significance.
is compared to lower and upper critical values (
To test for negative autocorrelation at significance
is compared to lower and upper critical values (
): Negative serial correlation implies that a positive error for one observation increases the chance of a negative error for another observation and a negative error for one observation increases the chances of a positive error for another.
Their derivation is complex—statisticians typically obtain them from the appendices of statistical texts.
of the regression is known, exact critical values for the distribution of
under the null hypothesis of no serial correlation can be calculated.
are independent standard normal random variables; and the
[3] A number of computational algorithms for finding percentiles of this distribution are available.
[4] Although serial correlation does not affect the consistency of the estimated regression coefficients, it does affect our ability to conduct valid statistical tests.
First, the F-statistic to test for overall significance of the regression may be inflated under positive serial correlation because the mean squared error (MSE) will tend to underestimate the population error variance.
Second, positive serial correlation typically causes the ordinary least squares (OLS) standard errors for the regression coefficients to underestimate the true standard errors.
As a consequence, if positive serial correlation is present in the regression, standard linear regression analysis will typically lead us to compute artificially small standard errors for the regression coefficient.
These small standard errors will cause the estimated t-statistic to be inflated, suggesting significance where perhaps there is none.
The inflated t-statistic, may in turn, lead us to incorrectly reject null hypotheses, about population values of the parameters of the regression model more often than we would if the standard errors were correctly estimated.
If the Durbin–Watson statistic indicates the presence of serial correlation of the residuals, this can be remedied by using the Cochrane–Orcutt procedure.
The Durbin–Watson statistic, while displayed by many regression analysis programs, is not applicable in certain situations.
Durbin's h-test (see below) or likelihood ratio tests, that are valid in large samples, should be used.
The Durbin–Watson statistic is biased for autoregressive moving average models, so that autocorrelation is underestimated.
But for large samples one can easily compute the unbiased normally distributed h-statistic: using the Durbin–Watson statistic d and the estimated variance of the regression coefficient of the lagged dependent variable, provided