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Interpreting Unit Root and Variance Ratio Tests for Random Walks

Article Quant Q&A · Author: Lin Lex
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Summary

The document raises a common time-series testing question: whether an ADF test that fails to reject a unit root conflicts with a Lo–MacKinlay variance ratio test that rejects the random-walk hypothesis at one lag. It also reports a KPSS result and asks how these diagnostics should be interpreted together.

The post provides no answer or supporting analysis, so it does not establish whether the tests conflict or explain their different null hypotheses, assumptions, or sensitivities. Its value is as a prompt to distinguish a unit-root model from the broader random-walk property and to examine test results across lags rather than treating one rejection as decisive. The reported outcomes alone cannot identify the data-generating process; the series, test specifications, and broader statistical context are absent.

Key ideas

  • A unit-root test and a variance-ratio test assess different properties of a time series.
  • The post reports rejection of the random-walk null at one lag but not at other meaningful lags.
  • The document asks whether unit roots imply a random walk but does not answer that question.
  • The reported KPSS outcome is not accompanied by test settings or supporting analysis.

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# Variance ratio test and ADF test for random walk


# 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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.