De-trending before regression analysis

I'm currently analysing two sets of time series data (monthly temperature and forest cover over a 10 year period). I first ran a Mann Kendall on each variable (they're non-normal) after removing seasonality, and found that they both show a significant increasing trend - vegetation faster than temperature.

I now plan to run a regression analysis of forest cover (dependent) against temperature (independent) to try to see how much temperature explains cover change. However, I'm confused about whether or not I need to de-trend both of the time series before this because I know that de-trending avoids spurious regressions (which could occur since both factors are increasing), but if forest cover is growing at a higher than temperature, would this difference not be lost if both were de-trended?

Thanks in advance!

Edit: I should have said, I'm interested in the long term temperature trend as a driver rather than inter-annual cycles
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No cake for spunky
If there is a trend over time than you will likely get spurious regression unless cointegration occurs. Generally I think you have to run one of the time series regressions in this case (there are many, ARDL with bound testing is probably best although none are easy).

This is a complex topic to put it mildly. Several Nobel prizes earned this way. :)