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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?

Richard Hardy
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Thomas
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1 Answers1

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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:

  1. 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).
  2. Due to the multiple testing problem, the significance level should be adjusted accordingly when doing formal inference.

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

Richard Hardy
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