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Following case:

We have one time-series A trending upwards. We have one time-series B trending downwards.

Is there any good way to measure their relationship? Making them stationary by first differencing and then potentially measuring negative Pearson correlation?

The assumption is that the increases in A affects the decrease in B. I.e. the rise of temperature lowers the sales of snowboards.

Thanks!

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    $\begingroup$ differencing should work $\endgroup$ – Aksakal Oct 21 '16 at 18:23
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    $\begingroup$ Expanding on @Aksakal, you could first difference then run a vector auto regression. Very likely, lagged terms matter. (i.e. past high temperature affects current snowboard sales). You could then plot impulse response functions to see how how a positive shock to temperature affects snowboard sales over multiple periods. You could also look into controlling for seasonality. Another question is whether you want first differences of snowboards or first differences of log snowboard sales. $\endgroup$ – Matthew Gunn Oct 21 '16 at 18:36
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    $\begingroup$ @MatthewGunn, good point on lags. Run cross correlation (or coherence) plots, that will show correlation of lagged temperatures on sales. Seasonality may be important too, there are sale cycles in winter sport sales, like start of season sales of old stock, end of season sales of rented boards etc. $\endgroup$ – Aksakal Oct 21 '16 at 18:40
  • $\begingroup$ Follwing is some web material on TF modeling autobox.com/pdfs/A.pdf The filtering process to IDENTIFY the structure is discussed here autobox.com/pdfs/WHY-WE-FILTER.ppt and here math.cts.nthu.edu.tw/… $\endgroup$ – IrishStat Mar 22 '19 at 18:18
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Just plot them parametrically, that is, given for time t and for temperatures T; {t0,T0},{t1,T1},...,{tn,Tn} and snowboard sales S; {t0,S0},{t1,S1},{t2,S2},...,{tn,Sn}, plot temperature against snowboard sales {T0,S0},{T1,S1},{T2,S2},...,{Tn,Sn}.

There is more to this question, as well, for example lag plotting. However, the correlation between temperature and sales is unlikely to be spurious, and I would start there.

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