# Find time series that best approximates another time series, question about detrending

The task is to figure out which variable best "approximates" the behavior of the target variable over time, so that it could be potentially used as a substitute later on (e.g. because it's easier to obtain/measure than the target variable).

In my case both the target variable and the candidate variables (the ones to potentially approximate target's variable behavior) exhibit a light, but noticeable, trend:

Question #1: I know that when studying relationships between two time series it is important to get rid of the trend, but given that my task is to simply identify whether one variable approximates the behaviour of the other one, as opposed to "deeply STUDYING" their relationship (e.g. inferring causality, controlling for confounders etc)... do I really need to do that? Because I believe the fact that they share that trend is actually an important manifestation of similarity between the two.

Question #2: If the answer to question #1 is no, but I still decide to do it and obtain correlation between detrended versions of two variables (in my case, differencing worked well) - what would be the interpretation of that correlation as opposed to the one for the original time series? E.g. in my case, I had a correlation of 0.80 before differencing (for the picture above), and 0.60 after differencing - how could I interpret one value as opposed to the other (kind of explaining why the two correlations are different now, what does each one represent)? Would it simply be that in the latter we "control" for the effect of time (as it was a confounder due to shared negative trend) and isolate the actual effects of one variable on the other?

• Could you elaborate what "approximates" means in your sense? Only correlation, i.e. going in the same direction, being useful for forecasts in terms of some loss function? That would change the answer about the detrending. Commented Jul 10, 2020 at 20:18
• Not forecasting, no. Just sheer emulation of behavior, as in similar directionality + similar strength of "jumps", if you will, which appears to be pretty well-captured by classic correlation. Commented Jul 14, 2020 at 20:06