The Durbin Watson test looks to check for both positive and negative autocorrelation but for first order only. It should not be used for data that is autocorrelated beyond the 1st order. The following link shows both the hypothesis as well as inference
From this website:
"The Hypotheses for the Durbin Watson test are:
H0 = no first order autocorrelation.
H1 = first order correlation exists.
The Durbin Watson test reports a test statistic, with a value from 0 to 4, where the rule of thumb is:
2 is no autocorrelation.
0 to <2 is positive autocorrelation (common in time series data).
>2 to 4 is negative autocorrelation (less common in time series data).
A rule of thumb is that test statistic values in the range of 1.5 to 2.5 are relatively normal. "
Note that to get a more precise conclusion, we should not just rely on the DW statistic, but rather look at the p-value. Software packages like SAS will give 2 p-values - one for test for positive first order autocorrelation and the second one for the test for negative first order autocorrelation (both p-values add upto 1). If both p-values are more than your selected Alpha (0.05 in most cases), then we can not reject the null hypothesis that "no first order autocorrelation exists.
If any one of the p-values is < 0.05 (or selected Alpha), then we know that the corresponding alternate hypothesis is true (with 1- Alpha certainty).
I hope that helps.