Runs test and Durbin-Watson test yield different outcomes I have analysed the market return using the runs test and the Durbin-Watson test to determine whether the return series follow the random walk or not.
The problem I have found is that some return series is rejected by the runs test's null hypothesis, however, it is accepted by Durbin-Watson test's null hypothesis, and vice versa in some other cases.
How should I interpret this result?
 A: First of all, these tests are not specifically intended for determining whether a process is a random walk. But once you take first-differences of a random walk, you should obtain an i.i.d. sequence. Then departures from i.i.d.-ness can be established using the two tests.
The Durbin-Watson test (DW) and the runs test (R) examine different aspects of dependence. DW assesses autocorrelation at lag 1, while R assesses the distribution of the length of runs. (I wonder how you have transformed your data to produce the signs for running R.) Thus DW and R are not exactly the same. (Deriving the relationship between them is perhaps a little involved, so I will not attempt it here.)
The two important things are: 


*

*Under independence, both DW and R should show independence. But under dependence, these tests may or may not capture the dependence; this depends on the type of dependence* (DW and R have power against their specific alternatives but not necessarily against other types of alternatives).

*Due to the multiple testing problem, the significance level should be adjusted accordingly when doing formal inference.



 *Wow, so many dependencies in one sentence... 
