# Variance ratio test and ADF test for random walk

I am trying to use both ADF test and variance ratio test for random walk. However, the ADF test tells me my financial time series contains unit root, but variance ratio test (lo-mackinlay) rejected that financial time series is random walk (Raito 0.86) at lag 2, but insiganificant at all other meaningful lags. Now my questions are follow:

1, Did I make a mistake by thinking that unit root entails random walk, thus two tests do not conflict each other.

2, If I am correct in thinking that unit root does entails random walk, then the ADF and Variance ratio test tells different story. How can that occur.

3, Also, for robustness check, I did KPSS test and it rejected null hypothesis in favour of alternative, intails the financial time series contain unit roots.

Thanks.

• Hi: I would think that it is possible to have conflicting results for ADF test and Variance Ratio test because there being a unit root does not imply a random walk. The ADF unit root test says that the null is that the first coefficient is 1.0 this doesn't imply a random walk. The variance ratio test OTOH specially assumes that the null is a random walk. But still, it is well known that the various unit root tests can conflict so even just scrapping the VR and using those can be problematic. Some of them assume a unir root for the null and others assume the null is not a unit root. – mark leeds Apr 6 at 15:30
• @mark leeds. Hi thanks for your comment, I was wondering the same thing, but in most material that I looked at, such as lo-mackinlay (1988), they specified their model to be pt = pt-1 +et, and it is a unit root process. Therefore I was wondering what is the difference between Unit root process and random walk, because some materials dont really discuss the difference, do your have any good recommendations of papers or materals that discuss it. Thanks. – Lin Lex Apr 7 at 2:46
• Hi: I'm not an expert in this but the subtlety lies in the fact that the RW does imply a unir root i$p_t$ which is usually the log price at time $t$. The problem is that the ADF uses a model with other lags technically the null is not a random walk. If you want to test for an RW and not use the ADF, then I would check out the DF. That would be testing for an RW but you still have the issue of whether to include an intercept, trend term etc. I would check out Hamilton's text. I remember that being pretty detailed in its unit root coverage. It's all kind of cloudy to me these days. – mark leeds Apr 7 at 19:40