I have 3 time series: $A$, $B$, and $C$. The three time series measure related quantities but have different units/scales. I have good reason to believe that both $A$ and $B$ are essentially controlled by $C$. $C$ Is quite periodic, and and $A$ and $B$ look pretty periodic but noisier.
Is there a statistical metric for comparing how well the periodicity of two time series match? The posts on cross-correlation and Pearson coefficient suggest that the ARIMA structure of each time series should be discerned independently and the 'within-series' structure should be removed so that the error terms of each time series can tested for correlation while avoiding 'spurious correlation'. However, I think that might be the opposite of what I want to do---I know the cause of periodicity in $C$, and I want to see how well the observed periodicity in $A$ and $B$ match that observed in $C$.
Does it still make sense to remove the 'within-series' structure in this case? Is correlation the right way to be thinking about this problem, or should I be thinking about e.g. Fourier transforms to express periodicity and compare transformed metrics (something I have never done before)?