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Imagine I have a time series for an animal population and a time series for a climatic variable during the same time period and at the same location. Unfortunately the data are observational (i.e., no experimental manipulation). Now imagine there is a decreasing linear trend in both variables (and no seasonality).

I know before hand that these variables will be correlated because they share a time trend (e.g.,https://stats.stackexchange.com/a/8037/38125). But is there anything more I can infer from this data?

I am looking for a rather low-level conceptual explanation as to whether this is possible and why.

I have 2 thoughts on how this might be accomplished, but haven't figured on my own if either would be valid.... 1) Regress population against both time and the climate variable, or 2) "Detrend" both 'population' and 'climate variable' and regress their residuals against each other

I can provide more detail if necessary but wanted to keep this original post brief. Thanks for your help!

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You got it right that you can't simply regress one on another because it'll probably lead to spurious correlation. That's why in time series analysis the non-stationary series are often transformed to stationary series. An example is so called Box-Jenkins approach, where you difference the non-stationary. Differencing is one of the de-trending methods. There are others.

Once you difference your series, they become stationary so you run the regressions on them.

This is a fairly standard approach. I'd suggest you also pay attention to possible cointegration which tends to complicate the analysis, but has to be taken care of it is present. The idea of a cointegration is very simple as shown here on example of a drunkard and his dog

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  • $\begingroup$ Thanks for the response, this indeed helps. Just out of curiosity I tried both methods I mentioned in my question above...Detrending with residuals from a OLS regression, and simply including 'time' as a covariate in a non-detrended regression, and the coefficent for the climate variable was the same in either case. Does that mean both methods are valid (at least in simple cases?) $\endgroup$
    – Dave M
    Commented Apr 8, 2014 at 21:23
  • $\begingroup$ @DaveM, it's one way to de-trend if you know that the time trend is linear. what if it's not? de-trending is often done in the context of time series decomposition $\endgroup$
    – Aksakal
    Commented Apr 8, 2014 at 21:27
  • $\begingroup$ Is there situations where you lose information by detrending? i.e., let's say in my example above that the data were collected experimentally (so I had good reason to believe there was causation)...Would I still want to detrend? $\endgroup$
    – Dave M
    Commented Apr 9, 2014 at 16:29
  • $\begingroup$ You don't lose information if you detrend properly, because the trend doesn't have to be thrown out. You can analyze trends separately. For instance, you may figure out that trends have different kinds of relationships between each other, or that your detrended series are related to trends etc. One such example is GARCH-like models, where the variance depends on the levels, which are basically trends. $\endgroup$
    – Aksakal
    Commented Apr 9, 2014 at 17:31

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