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I want to use the Monte Carlo Method described in Aronsons book Evidence based Technical Analysis to test if a given pairs trading strategy is useless. First step there is to randomize the returns of the underlying instrument. Second step is to calculate daily log returns of the strategy as a performence measure.

For the first step: Is it sufficient to randomize the spread that is already calculated or are the underlying instruments to be randomized and hedging coefficient estimation to be repeated?

For the second step: log returns for spreads are not suitable since the spread can have negative values. So may be better use differencies of spread today minus spread yesterday?

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There are two test methods described in Aronson's book; White's Reality Check and a Permutation test. At the heart of both is the idea of a "position vector," e.g. a numerical vector of a series of 1, -1 or 0, which correspond to long, short or neutral positions. For example, a vector of

[ 1 1 1 1 1 0 0 0 -1 -1 -1 -1 ]  

would represent being long for 5 days, out of the market for 3 and finally short for 4 days. This can be directly applied to a pairs trade such as long stock A, short stock B for 5 days, no position for 3 days and finally short A and long B for 3 days. The Monte Carlo aspect of the tests in question is essentially an n number of random permutations of this position vector.

The difference between the two tests is how the null hypothesis sampling distribution is defined and created. For the simpler of the two, the permutation test, the null is that the "rule" has no predictive power and so the randomized position vector is multiplied with the "returns" to give a distribution of "no predictive power returns." In the book log returns are used, but any return can be used; e.g. dollars made per day on a minimum sized pairs position, the tick value of the spread curve etc. This will simply be the test statistic used for comparative purposes, and the test will be comparing apples to apples.

For White's Reality Check, the null is that the "rule's return" is zero, and so the return vector must be detrended such that a continuous long or short position would give a zero return over the test period. The book subtracts the average daily log return from each daily log return to acheive this because log return is the chosen test statistic. However, if another test statistic is chosen, it too must be detrended in an appropriate way, e.g. subtract the average dollars made per day on a minimum sized pairs position from each individual daily dollar return on the same sized position.

It would, therefore, seem to be quite straightforward to apply the standard tests from the book:

i) create your position vector

ii) create your chosen test statistic return vector (detrended if necessary)

iii) apply the test

However, having written all this, I think the more pertinent problem for pairs trading is data mining bias, wherein the search process for stock A and stock B should be subject to testing, rather than testing A with B in the above framework after A and B have been selected.

You may find it interesting to browse my data snooping Github, which has code and various downloaded papers related to this general area.

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I am not familiar with the book you mention, so this may not be helpful or relevant. But the phrasing of your question is very peculiar to me.

A pairs trade's quantitative appeal is determined by analyzing the historical time series of the stocks. Randomizing the stock prices or the spread also randomizes the time order, which invalidates any method of analyzing a pairs trade that I know of. So I am not sure what this is supposed to demonstrate.

Also, the natural logarithm of stock prices is often used in the process of evaluating pairs trades, and it would not make sense to take the logarithm of the spread.

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    $\begingroup$ Pairs trades are generally developed through an extensive search process. Many stock pairs are tried to see if they cointegrate, then all kinds of alternative entry rules and stop rules are tested and the best is chosen. After all this work you have some results but it is difficult to assess the statistical significance. White's Reality Check attempts to do this by (I oversimplify here) simulating a similar search process on random data to see what comes up; if your actual results are not better than the best of the random results you conclude that your actual results are due to chance. $\endgroup$ – Alex C Oct 14 '15 at 3:04
  • $\begingroup$ @AlexC I surmise it is simply a null hypothesis of sorts. I am not sure I would validate my pairs like this, but to each their own. Thank you for the explanation. Regardless, the OP seems to not understand the basics of pairs trading, given his comments about taking the log of the spread. $\endgroup$ – d0rmLife Oct 14 '15 at 3:16
  • $\begingroup$ @Alex C: Your comment describes pretty much what I'm trying to do. My question is just about how to employ this methof for testing the randomness of a pairs strategy. I'm familiar with the methods of selecting pairs and entry/exit logic. @ d0rmLife: The Whites Reality Check uses the mean daily log return of the strategy under test to calculate the test statistic. However, this is not suitable to evaluate a pairs trade because spreads can be negative in opposite to single stock prizes. Simply Adding log returns of all spread legs also doesn't help because the beta is not taken into account. $\endgroup$ – user3276418 Oct 14 '15 at 8:37

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