Finally, dollar leads oil?, or oil leads dollar?

From 2001, or so dollar plunged and oil rallied.
From mid 2007 dollar rallied and oil plunged.

Which is the cause, and which the effect?

 

Calculate the returns of both time series and then calculate the lagged-correlation of one vs another and vary the lag parameter and note the peaks. If a peak occurs at a particular lag, your answer is there.

Localize the analysis to the regions of interest and use cross-validation techniques to see whether the situation reversed or not in the two time periods you mentioned.

 

They move pretty much in unison.

I just want to know which is the cause, and which the effect.