Cointegration Testing and Serial Correlation in Pairs Trading
Summary
The document raises practical questions about testing whether two nonstationary stock prices share a long-run equilibrium. Its proposed Engle–Granger sequence first checks that each series is integrated of order one, estimates a long-run regression, and tests its residuals for a unit root. The central concern is that a residual can be stationary while still showing visible serial dependence, and whether that undermines cointegration or a potential pairs-trading use.
It also asks how to choose critical values for tests on raw series, estimated regression residuals, and an augmented regression intended to account for serial correlation. The text is a question rather than a resolved tutorial: it provides no accepted answer, test results, or definitive guidance on critical values, model specification, or trading viability. Readers should therefore treat it as a statement of issues to investigate, not as validation that a pair is cointegrated or ready to trade.
Key ideas
- The Engle–Granger approach checks the integration order of each series before testing a long-run regression residual.
- A stationary residual may still display serial dependence in a plot.
- Critical values for residual-based unit-root tests require care because the residuals are estimated.
- The document asks whether an error-correction model is needed after the residual test but does not resolve the question.
- Statistical evidence of cointegration alone does not establish that a pairs-trading strategy is viable.
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Full text
# Cointegration Test: Residual is stationary but not random?
# Cointegration Test: Residual is stationary but not random?
I am testing cointegration relationship on various pairs of stocks by this following these steps.
- Test for I(1) on a pair of stocks, says X and Y, using Dickey-Fuller test. If both time series are non-stationary, I continue to step 2.
- Run long-run (equilibrium) equation of stock Y on X, estimate parameters and collect residuals.
- Test for unit root of that residual series using same method in step 1. For many pairs I have tested, the tests reject null hypothesis that there is a unit root, implying valid cointegration relationship. (regression on $\Delta \epsilon = \alpha*\epsilon_{t-1} + u_t$ then compare test statistic ($\alpha$/S.E. of estimated $\alpha$) with critical value (from Dickey Fuller's Table, I presume) )
Now, here is my problem. After I finished all that stuffs I try plotting residual series and, clearly, it is not random. Residual looks correlated! It looks like this.
...
My questions:
- Is it safe to conclude that the pair is cointegrated just because the residual is stationary and dont care that the residual is not random?
- Can I continue with the result that the pairs are cointegrated and continue to do pair trading if I want to or I just have to continue to estimate the Error Correction Model in order to complete Engle-Granger 2 step procedure?**
- Which critical value table should I use in step 1 and step 3? I think I should use Dickey Fuller's Table for step 1 as the test is done over the residual term rather than raw data, it is not possible to use standard t-distribution table. For step 3, Im not sure because the test is done over the estimated residual term.
Please bear with me..., there is more
According to this paper, on page 6, http://www2.warwick.ac.uk/fac/soc/economics/staff/gboero/personal/hand2_cointeg.pdf
it says if I test residual for stationary on $$ Y_t = \alpha + \beta_0*X_t + \beta_1*X_{t-1} + C*Y_{t-1} + \epsilon_t, $$ this will prevent the bias due to serial correlation on traditional regression ($y_t = \alpha + \beta_0*x_t + \epsilon_t$). I assume that I will use the same stationary test as in step 3 above.
- Which critical value table should I use for stationary test written in the paper I have shown? Currently, I've got very high test statistic, over 14. I suspect I have done it wrong.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.