Interpreting Johansen Cointegration Vectors in Pairs Trading
Summary
The document asks how to interpret Johansen cointegration vectors when screening pairs of stocks. It describes normalizing the first coefficient to one, then using the resulting coefficients and intercept to construct a spread. The questioner proposes trading opposite positions when that spread moves away from its long-run level, but is unsure what same-sign vector coefficients imply for the stock positions. The excerpt does not resolve that interpretation; the reply instead raises a preliminary issue about testing whether each price series is fractionally integrated.
The reply cautions that conventional unit-root tests may have weak power to distinguish a unit-root process from a fractionally integrated series with long memory. It suggests that this can lead researchers to treat a mean-reverting series as if it had a unit root, and notes that the residual may also be fractionally integrated, affecting its mean-reversion behavior. The answer offers no worked example, trading rule, or resolution of the question about rejecting both rank hypotheses. Its main lesson is to consider integration properties before relying on cointegration results.
Key ideas
- Normalizing a cointegration vector fixes its scale but does not by itself settle how to interpret same-sign coefficients as trading positions.
- The proposed spread combines lagged prices using the cointegration coefficients and an intercept.
- The reply recommends checking for fractional integration before applying conventional cointegration analysis.
- Weak unit-root test power can make long-memory series difficult to distinguish from unit-root processes.
- The excerpt does not explain how to interpret rejection of both rank hypotheses in a two-stock test.
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# Johansen test on two stocks (for pairs trading) yielding annoying results # Johansen test on two stocks (for pairs trading) yielding annoying results I hope you can help me with this one. I am using cointegration to discover potential pairs trading opportunities within stocks and more precisely I am utilizing the Johansen trace test for only two stocks at a time. I have several securities, but for each test I only test two at a time. If two stocks are found to be cointegrated using the Johansen test, the idea is to define the spread as beta' * p(t-1) - c where beta'=[1 beta2] and p(t-1) is the (2x1) vector of the previous stock prices. Notice that I seek a normalized first coefficient of the cointegration vector. c is a constant which is allowed within the cointegration relationship. I am using Matlab to run the tests (jcitest), but have also tried utilizing Eviews for comparison of results. The two programs yields the same. When I run the test and find two stocks to be cointegrated, I usually get output like beta_1 = 12.7290 beta_2 = -35.9655 c = 121.3422 Since I want a normalized first beta coefficient, I set beta1 = 1 and obtain beta_2 = -35.9655/12.7290 = -2.8255 c =121.3422/12.7290 = 9.5327 I can then generate the spread as beta' * p(t-1) - c. When the spread gets sufficiently low, I buy 1 share of stock 1 and short beta_2 shares of stock 2 and vice versa when the spread gets high. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~ The problem ~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Since I am testing an awful lot of stock pairs, I obtain a lot of output. Quite often, however, I receive output where the estimated beta_1 and beta_2 are of the same sign, e.g. beta_1= -1.4 beta_2= -3.9 When I normalize these according to beta_1, I get: beta_1 = 1 beta_2 = 2.728 The current pairs trading literature doesn't mention any cases where the betas are of the same sign - how should it be interpreted? Since this is pairs trading, I am supposed to long one stock and short the other when the spread deviates from its long run mean. However, when the betas are of the same sign, to me it seems that I should always go long/short in both at the same time? Is this the correct interpretation? Or should I modify the way in which I normalize the coefficients? I could really use some help... EXTRA QUESTION: Under some of my tests, I reject both the hypothesis of r=0 cointegration relationships and r<=1 cointegration relationships. I find this very mysterious, as I am only considering two variables at a time, and there can, at maximum, only be r=1 cointegration relationship. Can anyone tell me what this means? All the best, Johan ## Answer by Alex Zaitsev (score 3) https://quant.stackexchange.com/a/18825 Before making regression you have to perform test on fractional integration on each component. The power and size of traditional unit root tests are poor. The tests’ weak power implies that the statistical tests cannot distinguish between a unit root process and a fractionally integrated series with long memory (Baillie, 1996). As a consequence, a mean-reverting time series is incorrectly considered as a unit root process. The residual also could be fractional and therefore bad mean reverting.
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