# Why do I get a different result for cointegration test when I swap the independent and dependent variables?

I'm trying to test whether series x and y are cointegrated using EG method. Both are I(1). When I run the regression y = a + bx + e, then test for stationarity of the residuals (using ADF test) I find the result significant at 10%, so conclude the variables are cointegrated. However when I run the regression x = a + by + c the t statistic is considerably lower and not significant at the 10% level. What does this mean about the series? Are they cointegrated? Is it possible for them to only be cointegrated 'in one direction' if you will?

Any help would be greatly appreciated, thanks!

## 1 Answer

Cointegration is not "directional" because its defining property is intrinsically "nondirectional": a linear combination of the original, integrated series must be a stationary series (here I disregard cointegration of higher orders for simplicity). There is nothing directional in this definition.

However, if the linear combination (more precisely, the coefficients on the different variables) is not known in advance, it has to be estimated. One way of doing that is by regressing one variable on the others by ordinary least squares. This particular estimation technique yields an effect of "direction"; it appears from the formulation of the estimation problem. The errors of $y$ projected on $x$ are not the same as those of $x$ projected on $y$. This is an unfortunate property of the estimation method (unfortunate in the context of cointegration analysis but not necessarily beyond). The problem should go away asymptotically, I think, but it can cause some trouble in finite samples, just as you have found out.

Alternatively, a single cointegrating vector can be estimated as the last principal component (the one with the smallest variance) of the system of variables. Principal component analysis is a "nondirectional" technique so there is no such problem there.