Interpreting the Automatic Variance Ratio Test Statistic

Summary

The document clarifies how to interpret the statistic returned by an automatic variance ratio test applied to financial returns. The null hypothesis is that returns are serially uncorrelated, corresponding to a variance ratio of one; the alternative is serial correlation. Under the stated asymptotic result, the standardized test statistic converges in distribution to a standard normal variable, so the reported statistic can be assessed against that reference distribution.

The formula depends on the estimated variance ratio, sample size, and a lag truncation point. The convergence statement is asymptotic and relies on the sample size, lag, and their ratio growing; it should not be read as an exact finite-sample distribution. The test is two-sided, so departures above or below the null value may count as evidence against it. The answer explains the statistical interpretation but does not give a complete implementation, lag-selection procedure, or advice about whether the square-root-of-time variance rule is appropriate for any particular series.

Key ideas

• The test’s null is that returns are serially uncorrelated and the variance ratio equals one.
• The automatic test statistic has an asymptotic standard normal distribution under the null.
• Its construction uses sample size, an estimated variance ratio, and a lag truncation point.
• The test is two-sided, and its normal reference is an asymptotic approximation.

Tags

# Variance Ratio Test in R


# Variance Ratio Test in R












I would like to conduct a variance ratio test for a financial time series in order to examine whether I can apply the square root rule for the variance with the software R. I used the Automatic Variance Ratio Test `vrtest::Auto.Vr` and got a statistic of `-0.01`. Now I am wondering, is that the z-score, which is distributed standard normal under the Null hypothesis, that this ratio is `1` (or equivalent that there is no autocorrelation)? It is not specified in the describtion, I just found this:

```
Usage:
Auto.VR(y)
Arguments:
y financial return time series
Value:
stat Automatic variance ratio test statistic
```

## Answer by rbm (score 3)

https://quant.stackexchange.com/a/32816

TL;DR: the test statistic's distribution is $N(0,1)$

A bit more information about the Automatic Variance Ratio Test:

$H_0$: ${\Delta}r_t$ is serially uncorrelated (where ${\Delta}r_t=r_t-r_{t-1}$)

$H_1$: ${\Delta}r_t$ is serially correlated

The test statistic is $VR=\sqrt{T/l}[\hat{VR}(l)-1]/\sqrt{2} \quad {\xrightarrow{d}} \quad N(0,1)$

The $d$ over the arrow is important, i.e. the $VR$ converges in distribution to standard normal (hence $d$ does not imply the convergence in mean square or convergence in probability) as $T$, $l$ and $T/l$ approach infinity. The $l$ is the lag truncation point. The paper has detail on formulae for both $VR$ and $l$.

Final note: the test is two sided.

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.