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# Quantifying randomness: variance ratio

Discussion in 'Strategy Development' started by stephencrowley, Feb 14, 2006.

1. ### stephencrowley

Variance Ratio:
A measure of the randomness of a return series. Variance ratio is computed by dividing the variance of returns estimated from longer intervals by the variance of returns estimated from shorter intervals, (for the same measurement period), and then normalizing this value to one by dividing it by the ratio of the longer interval to the shorter interval. A variance ratio that is greater than one suggests that the returns series is positively serially correlated or that the shorter interval returns trend within the duration of the longer interval. A variance ratio that is less than one suggests that the return series is negatively serially correlated or that the shorter interval returns tend toward mean reversion within the duration of the longer interval.

Anyone use this to screen symbols?

Seems like a good way to find "trendy" stocks.. I highly doubt you'll ever find any that mean-revert.

Here are 2 symbols over 2 days and the corresponding variance ratio profiles:

Blue and green are the same symbol, a few days apart and they are nearly equally "trendy" over both days.. the other symbol is trendy one day and completely random the next.

Problem is, you can't tell in advance when a stock is going to trend or be completely random at the beginning of the day.

However, if the level of randomness is steady over many many days one can make an assumption..

Now here are 4 random walks generated by taking the cumulative sum of a normal distribution with mean 0 variance 1.

As you can see the variance ratio is pretty good at showing that the truly random series are indeed random.

Are there any similiar methods, or better ones for detecting deviations from pure randomness?

Taleb defines it differently. He takes the Hist vol of stock A over 50 days lets say and then takes the hist vol of 60 min bars for an equal amount of time and takes the ratio. MEasures trendiness vs mean revertness of stock . Ex if IBM has a hist vol of 20 daily but 50 intraday, it tells trader that while ibm moves slowly on a daily chart , it around intraday to get to to its desitnation. Interesting angle if u like to scalp gamma.

3. ### gbos

Yes there are other ways to test for non randomness but they are not necessarily better.

One of them is to classify your time series as positive and negative returns and to apply the Runs Test.

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4. ### stephencrowley

Maybe im confused.. but that sounds like the same exact thing. You take the volatility of two overlapping periods..

3. cases

- Ratio rises above 1 as horizon increases.. mean aversion/trending

- Ratio stays around 1 as horizon increases.. random walk

- Ratio approachses 0 as horizon increases.. mean reversion

5. ### stephencrowley

That looks interesting but seems it only applies to coin-flip scenarios and doesnt take into account volatility.

6. ### gbos

It will detect trends. Essentially what you are doing is this&#8230;

Take samples in a short time frame.
If you have a positive return classify the sample as +
If you have a negative return classify the sample as &#8211;

Your time series is now something like this

+++--+--++---+---+++--

If the runs are more than expected the series is mean reverting
If the runs are less than expected the series is trendy

It is not better or worst than the variance method, just different.

7. ### gbos

Taleb mentions one more way similar to the variance method for detecting trends.

What you are doing is taking the high and the low of each time frame and calculating the Parkinson number. This must be equal to 1.67 * volatility. Less means mean reversion, more means trend.

8. ### stephencrowley

Interesting. Which book did he publish this stuff in?

9. ### gbos

Dynamic Hedging page 101 - 107.