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In practice, when you encounter a relationship between historical financial variables that looks good on levels but not on returns, the model you get from it essentially always fails to be predictive. I generally think of this as being due to the historical relationship arising from some confounding third (plus fourth and fifth...) variable effects that ...


The log likelihood function is indeed rather flat in the $\mu$-direction, for small time horizons (you used $T = 1$ it looks like). As you may have noticed, increasing the number of observations but keeping the time horizon the same DOES NOT IMPROVE the accuracy of the estimate of $\mu$ - this is a bit counterintuitive, if you ask me. But, increasing the ...

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