Your question is quite broad I think, but I will try to answer as gretl-specific as I can. You said you're teaching panel regressions, and that's what I think is the natural approach here. Why not just run a fixed-effects regression between the two variables? For example, using the grunfeld sample dataset in gretl to regress 'value' on 'invest' with the standard fixed effects, gretl reports a LSDV-R2 of 0.96 and a within-R2 of 0.37. This is just to calculate a panel-context correlation between the two variables, not as a model per se.

Of course then you have the usual panel-related questions on whether you really want fixed effects and so on, but that's a different (and not directly gretl-related topic).

I don't see this as a time series model in the usual sense, because with 17 regions and 2 panel variables you have 34 time series, but apparently only 17 periods (years) of data. But if you want to have dynamic effects in your panel regression is of course yet another question. Finally, wanting to understand "how regions differ over time" strikes me as an ambitious question in general, because then you would need some interaction effects or maybe even time-varying parameters. Or maybe you just want to calculate some statistics over rolling windows, that's of course feasible (in principle, not saying there's a built-in function for that).

cheers

sven