Suppose you have a logged variable $y_t$ which is comprised of a trend component $\tau_t$ and a cyclical component $c_t$. Thus:
Then you apply a filter to that variable to extract the stationary cyclical component $c_t$. Then if you use $c_t$ as a dependent variable in a regression how would one interpret the response of that variable based on some regressor that is:
1.Already stationary and thus not filtered (say some stock market index)
2.Also a filtered variable.
Another side question I have is can you ever apply a filter (for instance the HP or BK filters) on data that isn't logged?