# Cointegration Test: Residual is stationary but not random?

I am testing cointegration relationship on various pairs of stocks by this following these steps.

1. 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.

2. Run long-run (equilibrium) equation of stock Y on X, estimate parameters and collect residuals.

3. 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:

1. 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?
2. 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?**
3. 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.

1. 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.

Any contribution will be appreciated! Thank you.

• Hi Jack, I applied some tex to the question. But to the content: why do you say that your residual is not random? Is it deterministic? If not then it is random. Or do you mean that it looks a bit periodic? The notion of stationarity is interesting in the context of random processes only. So how can it be stationary but not random? – Ric Jun 9 '15 at 8:45