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> Are these comments correct? Shouldn't the first sentence be: "An
> n^2 day test, with a perfect random walk, is expected to have a
> standard deviation N times larger than a 1-day test, ASSUMING THAT
> THE 1-DAY TEST IS ALSO A RANDOM WALK."
True.
> Since it's not, I'm confused as to why we are willing to base the
> whole indicator on the 1-day SD.
Excellent point! Why should it be used as the standard?
Alex Saitta's article uses a coin-flip analogy to explain a random
walk -- heads you go north a step, tails you go south. On average, a
fair (50/50) coin will leave you at your starting point, and your
standard deviation will be sqrt(N) after N flips. If you have a
biased (trending) coin, your stdev will be greater than sqrt(N).
However, that assumes a uniform step size after each flip. The
market doesn't move a standard amount every day, though, so we have
to have some way to normalize the results. Saitta used the 1-day
movement -- or, rather, the **standard deviation** of the 1-day
movement -- as the standard unit for the market.
If you chose an arbitrary 1-day (or 1-bar, if you're not using daily
data) move as your standard, I agree that would be very arbitrary and
would make the whole measurement questionable.
But the trend indicator uses the stdev of ALL the 1-day moves in the
sample. That should tend to smooth out the anomalies.
However, your original question still stands: is the 1-day stdev a
valid measure of a random walk, given that the market you're testing
is probably NOT a random walk? VERY good question.
At the time I wrote the indicator, I didn't have access to a copy of
Saitta's article. I wrote it based on a friend's description of the
technique. Now, however, looking at the article, I see that I missed
the final step that Saitta used to determine statistical significance:
He generated 10000 sequences of 1000-bar random data. He then
applied the trending index to those random (i.e. **random walk**)
data series, and observed the results. He defined a "statistically
significant" trend index value to be one that exceeded 85% of the
randomly-generated cases. 85% of the random cases had a 4-day trend
index of 2.04 or less, 9-day index of 3.06 or less, 16-day of 4.09 or
less, and 25-day of 5.11 or less.
So, using that, he decided that a 4-day trend index greater than 2.04
was statistically significant indication of trend.
But I don't think that answers your question. 2.04 or greater was a
statistically significant value **for the true random-walk case**.
Can we assume it's also significant for the case where the 1-day move
is *NOT* a true 1-day random walk? I honestly don't know. I'm going
to have to talk this one over with my more stat-savvy friends.
Anybody out there have an opinion?
Gary
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