Monthly data of oil prices is one of my independent variables. The purpose of doing this analysis is to get the relationship of oil prices with other financial variables (bond spreads, equity prices and others) and not predication. Can the non-stationarity of the independent and in some even the dependent variable still be a problem? and if so, what are the alternatives that I can work with.

At present, I have tried transforming my data by differencing it (which is unfortunately changing the relationship between my variables) or transforming by taking natural log of the data.

Thank you for your assistance in advance!