# How to Interpret Coefficients in Regressions on Filtered Varaibles?

Suppose you have a logged variable $$y_t$$ which is comprised of a trend component $$\tau_t$$ and a cyclical component $$c_t$$. Thus:

$$y_t=\tau_t+c_t.$$

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?