In statistics, the Durbin–Watson statistic is a test statistic used to detect the presence of autocorrelation at lag 1 in the residuals (prediction errors) from a regression analysis. 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. A similar assessment can be also carried out with the Breusch–Godfrey test and the Ljung–Box test. Computing and interpreting the Durbin–Watson statistic If et is the residual given by the Durbin-Watson test statistic is where T is the number of observations. For large T, d is approximately equal to 2(1 − ), where is the sample autocorrelation of the residuals, d = 2 therefore indicates no autocorrelation. The value of d always lies between 0 and 4. 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. Small values of d indicate successive error terms are positively correlated. If d > 2, successive error terms are negatively correlated. In regressions, this can imply an underestimation of the level of statistical significance. To test for positive autocorrelation at significance α, the test statistic d is compared to lower and upper critical values (dL,α and dU,α): If d < dL,α, there is statistical evidence that the error terms are positively autocorrelated. If d > dU,α, there is no statistical evidence that the error terms are positively autocorrelated. If dL,α < d < dU,α, 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 α, the test statistic (4 − d) is compared to lower and upper critical values (dL,α and dU,α): If (4 − d) < dL,α, there is statistical evidence that the error terms are negatively autocorrelated. If (4 − d) > dU,α, there is no statistical