Testing Stock Returns for Randomness and Dependence

Summary

The document discusses statistical approaches for evaluating whether stock market price changes are consistent with a random walk. It emphasizes testing both randomness and independence, since randomness and dependence are distinct properties and dependence can take forms that autocorrelation tests alone may miss. One proposed approach, called the differential spectrum, compares the distribution of price changes on either side of zero. It suggests assessing symmetry with a Pearson chi-square test, using one side as the observed counts and the other as the expected counts.

The symmetry test is presented as a way to flag possible dependence, not as a complete explanation of its source; further investigation is needed to identify any dependency. The response also cautions against treating random-walk behavior as a permanent market property: patterns may vary over time. It points readers to a financial econometrics text for broader treatment, but supplies no empirical dataset, test results, or guidance on sample size and implementation choices.

Key ideas

• Randomness and independence are separate properties that should both be examined.
• Autocorrelation tests alone may not detect every form of dependence.
• A proposed symmetry check compares positive and negative price-change frequencies using a chi-square test.
• A detected asymmetry is a signal for further investigation, not a full account of dependence.
• Random-walk behavior may shift across different market periods.

Tags

# Proving Random Walk Hypothesis in Stock Market


# Proving Random Walk Hypothesis in Stock Market












Given the time series for a particular stock market, what are the statistical weapons one can bring on to prove, or disprove that random walk hypothesis?

## Answer by Pete (score 12, accepted)

https://quant.stackexchange.com/a/386

Test your historical time series for both randomness and independence. Understand that a time series may be random and independent; non-random and independent; random with dependencies; and non-random with dependencies. A mistake would be to limit dependency tests to autocorrelation. The most general test I know of is called the differential spectrum by Sherry, which works like this:

- histogram the price changes in your time series

- if the price changes are independent, they should be symmetric about 0

- use Pearson's $\chi^2$ test with one sign as "observed" and the other as "expected" for a quantitative measure of symmetry.

However, when you find something with this test, you've still got to hunt for the dependency. But at least the test can tell you if you've got a dependencies or not.

The most important point is that whatever tests you work with, that market can change in the future. So I wouldn't say a market "is" or "isn't" a random walk. Rather, it may phase in and out of random-walkiness for indeterminate amounts of time.

## Answer by Dirk Eddelbuettel (score 6)

https://quant.stackexchange.com/a/380

The first chapter of the book Econometrics of Financial Markets by Campbell, Lo and MacKinlay discusses this very well.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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