No, I'm afraid it doesn't. The straight line cannot lag or anticipate, since it's straight. Also, since there is no correlation, there cannot be any autocorrelation of the regression. Finally, if if it did make sense, the residual errors will be so large that the tiny variations between two tracks measured by DW will be totally drowned.
In econometrics, you tent to look for autocorrelations over the base period (the month in your data) or some meaningful time frame (yearly variations in crop and weather related values). There are also several techniques to test for randomness of absolute returns (are too many positive or negative returns following each other).
DW gives the impression that it doesn't require a time frame in the hypothesis. You want a measure of autocorrelation without choosing a time frame, so you try to bend DW to your needs.
I don't know what I can add, really. You can make any computation you like, but it will not be a test, it will be exploratory statistics, or, if you are really lucky, descriptive statistics.
(°v°)
