Number of data points required for Cointigration

Some quick comments on this issue. From a statistical point of view, the
phrase "how many data points are required for cointegration" is not well
defined. Technically speaking, if two series are cointegrated then they are
cointegrated for any number of observations. This issue is really about the
size (probability of rejecting the null hypothesis when the null is true)
and power (probability of rejecting the null when the alternative is true)
of tests for cointegration for a given sample size. In intermediate
statistics text books, the chapters on hypothesis testing usually have some
discussion of the relationship between sample size and power. In simple toy
examples you can work out the number of observations required to have power
equal to some specified value (e.g. 0.90). In this case, if the alternative
is true then you can say you can reject the null at the 5% level with
probability 0.90 if the sample size is n=75 (say). Unfortunately, this
exercise is extremely difficult to do with tests for cointegration
(usually,
the null is no cointegration so rejecting the null is evidence of
cointegration). Why? Well there are only general asymptotic results (as
sample size goes to infinity) for tests for no cointegration (e.g.
Engle-granger two step, Johansen rank tests). There are no general finite
sample results (for fixed sample sizes) for power functions. Hence, you
cannot analytically compute a sample size that will give you a certain
power. What to do? Well, you can try to set up some Monte Carlo experiments
with fixed sample sizes to approximate power functions. The problem with
this is that the results are not general. They will depend on the
parameters
used for the Monte Carlo set up (e.g. parameters for serial correlation,
volatility etc). The best you can do is to try to carefully characterize
the
distributions of the series in question and try out some Monte Carlo
experiments for these data. My guess is that you will have the best results
when you use the class of tests that have been found to be optimal
asymptotic tests (where the asymptotic power curve is tangent to the
infeasible power curve of the optimal test at a set power). These tests
have
been developed by Graham Elliot at UCSD and Michael Jansen at UCB.


-----Original Message-----
From: R-SIG-Finance [mailto:r-sig-finance-bounces at r-project.org] On Behalf
Of amol gupta
Sent: Tuesday, January 27, 2015 10:09 AM
To: Paul Teetor
Cc: r-sig-finance at r-project.org
Subject: Re: [R-SIG-Finance] Number of data points required for
Cointigration

Paul

You say that ADF is not really stable . I agree. Other options to explore
are

   - Use other unit root and stationarity  test.
   - Use other cointigration tests like johansen tests.
   - Finding PCA and choosing one of the lower variance portfolio and test
   for stationarity.(I need to understand PCA more.)

I will take some time and test these ideas. Have you tried anyone of these?
If yes, please share you experiences.

Thank you for your insights.

On Tue, Jan 27, 2015 at 5:29 AM, Paul Teetor <paulteetor at yahoo.com> wrote:

Amol,

I don't have a formula or a guideline for determining the number of
data points. But I can share two experiences.

First, when I traded mean-reverting spreads, I used 3 to 5 years of
daily data. That's 750 to 1,250 data points. Less data did not work
well for my spreads.

Second, in my experience, the ADF test was quite unstable. That is, it
might fail to reject for a while, then start rejecting the null
hypothesis when the market showed some trending back to the mean. Then
it would fail to reject again as the market wandered away.

Perhaps smarter people than I have had better luck trading with the
ADF, but for me, it did not provide a complete answer to the question
of mean-reversion.

Paul

Paul Teetor, Elgin, IL USA
http://quantdevel.com/public <http://quanttrader.info/public>

  ------------------------------
 *From:* amol gupta <amolgupta87 at gmail.com>
*To:* "r-sig-finance at r-project.org" <r-sig-finance at r-project.org>
*Sent:* Friday, January 23, 2015 3:24 AM
*Subject:* [R-SIG-Finance] Number of data points required for
Cointigration

Hi

I need help in figuring out the length of historical data that I
should use. I took stock prices(daily close) for two tickers from
yahoo(200 days).I tried finding regression coefficient using PCA and I
use 150 points for PCA. I find a coefficient Beta.

Now to see whether the spread is mean reverting or not I use ADF. If I
use
150 point long spread, It comes out to be nonstationary. If I use 200
points data the outcome is stationary

I again used 200 points to do the PCA and find regression. The spread
comes out to be non stationary. From all these observation *I think*
that this is not a stable relationship.

So following are my questions

   - Is there a way to decide length of historical data to use?
   - Some relationship may be more stable than others. Is there away to
   quantify it?

Any other insight in this regard will be appreciated(time frame, pairs
vs basket). I have attached the plot and the script that was used to
generate the plot.


--
Regards
Amol

If all the seas were ink,
And all the reeds were pens,
And all the skies were parchment,
And all the men could write,
These would not suffice
To write down all the red tape
Of this Government.

--
Regards
Amol

If all the seas were ink,
And all the reeds were pens,
And all the skies were parchment,
And all the men could write,
These would not suffice
To write down all the red tape
Of this Government.

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