A doubt about the cointegration tests

Question

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

Answer 1

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.

Answer 2

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.

Answer 3

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.