Mathematically, what is going on here?
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$\begingroup$ I think this is a valid question, but you should give us more information before we can comment. $\endgroup$– David AddisonFeb 28 '18 at 1:51
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$\begingroup$ What additional information would be useful? $\endgroup$– Liam DonovanMar 7 '18 at 11:35
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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))
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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.
answered May 9 '20 at 11:43
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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.
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$\begingroup$ 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. $\endgroup$ Mar 7 '18 at 11:34
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$\begingroup$ 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. $\endgroup$– LazyCatMar 7 '18 at 16:55