I know this might be an "amateur" question, but I am pretty surprised to see the following fact:

I have a dependent variable, let's call it Y.

Then I have an independent variable, let's call it X.

X is the log return of another quantity, while Y can be either used as a raw (level -- X1) or a log return ( X2 ).

So, my results are significant for the level of the independent variable (X1), but not the log return (X2).

What is going on ? It is fine to use the level (X1) for my results, right ?



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 have not made it into the model.

Consider a relationship that works in return space

$$ \Delta y = \beta \Delta x + \epsilon $$

then in integrating the processes we get

$$ y(T)-y(0) = \beta(x(T)-x(0)) + \int_0^T \epsilon \, dt $$

and it will be a fairly clean relationship since integration is a smoothing process.

On the other hand, if we have $$ y = \beta x + \epsilon $$ then we differentiate to get increments

$$ \frac{dy}{dt} = \beta \frac{dx}{dt} + \frac{d\epsilon}{dt} $$ which on the face of it looks promising. However, taking the derivative is a "roughening" operation, so any kind of noise in $\epsilon$ gets "blown up" by the differentiation process, essentially swamping the model with noise terms.

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  • $\begingroup$ and why is that not the case with returns as well ? $\endgroup$ – adrCoder Sep 30 '15 at 13:25
  • $\begingroup$ Good question...updated. $\endgroup$ – Brian B Sep 30 '15 at 13:34
  • $\begingroup$ In my case it is $$ y = \beta \frac{dx}{dt} + \frac{d\epsilon}{dt} $$ in other words the independent variable is defined as the return, while the dependent is in levels. Does that still create an issue ? $\endgroup$ – adrCoder Sep 30 '15 at 13:37

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