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!