# A doubt about the cointegration tests

I'm doing Phillips-Ouliaris Cointegration test with po.test function inside tseries library.

I have a simple question about the "cointegration".

When i do PO test I get the p-value result but I don't get the "coefficients" of the cointegration. Is it normal?

I mean, the cointegration is used to know the "relathionship" of two or more series, analyzing p-value I could understand if the series are cointegrated or not, but IF those series are cointegrated how to understand what coefficients establish their relationship?

Do I have to do a linear regression AFTER the cointegration test? (to calculate the coefficients)

Thank you

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## 3 Answers

Are you familiar with Johansen's procedure? Usualy after testing that the vectors are cointegrated, vector error correction model (VECM) should be estimated. Then you get all the coefficients.

In general the coefficients of cointegration are tricky to work with, since they are not unique. For example if two time series $x_t$ and $y_t$ are cointegrated then by definition there exist coefficients $\alpha$ and $\beta$ such that time series $\alpha x_t+\beta y_t$ is stationary. But since stationarity is invariant to linear transformations, $-\alpha$ and $-\beta$ are also cointegration coefficients. How precisely coefficients should be interpreted then depends heavily on the model you are estimating. Coefficients for models based on different economic theory can be interpreted differently.

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 thank you, your answers are very appreciated. So, if the coefficients of cointegration are tricky to work with, do I have to calculate the linear regression after this test? (Maybe after knowing that the series are cointegrated, linear model could give good coefficients) no? – Dail Aug 16 '11 at 7:36 You should specify vector error correction model and estimate it. Estimation is usually done with ordinary least squares, i.e. linear regression, so the coefficients given will be "good". The problem is not with statistical "goodness" of the coefficients, but with their interpretation. – mpiktas Aug 16 '11 at 7:50 What are you trying to estimate? The biggest problem with the vector error correction models lies not in their estimation, but in their specification. – mpiktas Aug 16 '11 at 7:52 I'm tring to estimate the cointegration of two historical series(prices) of stocks, I have those two series with date->price with two years history. I'm tring to understand if they are cointegrated and what coefficients they have to calculate the exposure. What steps can i do? thank you so much! – Dail Aug 16 '11 at 8:01

Check the log of your statistical function. It usually spits out the coefficients before it gives you the p-value. I'm not familiar with the Phillips-Ouliaris test or your 'tseries' library, but for Engle-Granger test or ADF test, it will calculate a regression first in the background, and usually the econometrics software will spit out the results of that regression before testing the residuals for cointegration and printing the p-value. You would HAVE to do the regression first to GET the residuals. It takes the different between the estimated calculated value of the 2nd time series with the actual observed value to get the residual (uhat, û) for that point. Once it has all those residuals, then it runs a test for unit-root or stationarity to see if those two series are cointegrated.

And yes you can do it after also. The coefficients should match the ones discovered during the cointegration test.

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Are you running a statistical arbitrage strategy?

If so, there are other (perhaps more relevant) questions at hand. For example, say your test shows that the two stocks' returns are cointegrated (you are using returns, right?). If the cointegrating vector were such that the two legs of your position don't have opposite market value, then you're potentially holding a whole lot of exposure to generic stock market beta. That's not good for statistical arbitrage purposes, since it's supposed to be a market-neutral strategy.

Instead, I would difference the two returns series yourself, and then run a stationarity test against that difference series. This way, you'll be testing whether going short one and long the other will yield a reasonably safe stat arb. So basically, because of your particular situation, you need to enforce (IMHO) that your two legs be equal magnitude and opposite direction.

There is a small simplification even there, and I suppose you could beta-adjust the two return series to give you a beta-neutral position (slightly better than just MV-neutral) but that could be overkill. The pair trade usually goes down with equal market values in the two directions.

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Why do you need to test for cointegration on the returns? Nearly ALL stocks are cointegrated if you transform them into a series of daily returns first! That's because the returns, from day-to-day, are always going to be about -3% to 3%, and their average is usually 0 (assuming they are not the most volatile). That alone transforms them into a stationary series, and I believe cointegration requires them to be non-stationary, but their residuals are stationary. – Chloe Apr 10 '12 at 7:50