# If two price series are cointegrated but not correlated, how do I find the hedge ratio?

Mathematically, what is going on here?

• I think this is a valid question, but you should give us more information before we can comment. – David Addison Feb 28 '18 at 1:51
• What additional information would be useful? – Liam Donovan Mar 7 '18 at 11:35

If two price series are cointegrated, you can run the linear regression to calculate the hedge ratio.

The linear regression is in the form: $$y = \beta X + \epsilon$$

where, $$y$$ is the dependent variable, $$X$$ is the independent variables, and $$\beta$$ is the slope that we want to estimate and $$\epsilon$$ is the error term.

In Python, the OLS function from the statsmodels package is used to calculate the hedge Ratio:

statsmodels.api.OLS (dependent_variable(y), independent_variable(X))


To find the hedge ratio you can run the Johansen Cointegration test and the eigen-vector corresponding to the largest eigenvalue would give you the hedge ratio, you can see with a bit mathematical intuition that it means “The best linear combination of the two series to maintain a stationary series”. The eigenvector would ideally be something like (1, -ve) for a pair since hedge ratio has to be negative. Also before this you’d need to check whether the original pair was individually non-stationary.

Think of the case, when one time series is a constant equal to zero, and the second time series is stationary. You still can run regression to find the hedge ratio is zero, and you don't need to hedge a stationary price with a constant.

• So if I have two non-stationary cointegrated series, they must be correlated? That makes sense intuitively, but I don't know if there's a counter-example I'm not thinking of. – Liam Donovan Mar 7 '18 at 11:34
• Depends on what you mean by correlated. Mine was just an example, I'm sure it's possible to contrive another example, where both series are non-stationary, but the correlation is pretty low. – LazyCat Mar 7 '18 at 16:55