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Consider two datasets, say time series $\{X(t),Y(t)|t\geq0\}$, and that the two experience an equal percentage change from one period to the next for all time. That is, if there is an $\alpha \%$ change in $X(t)$ it is accompanied by an $\alpha \%$ change in $Y(t)$ from time $t$ to $t+1$.
Are $X(t)$ and $Y(t)$ perfectly linearly correlated? If so, is there any way to show it explicitly using knowledge of the percentage changes being equal?
Yes, the implication works in one direction:
$ Y(t) = Y(0)\ \rm{[Relative\ Change]} = Y(0) [X(t) / X(0)] = [Y(0) / X(0)] X(t) = Const\ X(t) $
$ \Rightarrow $
time series $X(t)$ and $Y(t)$ are perfectly correlated... This does not work in the opposite direction though. If
$ X(t) = 0 $ and $ Y(t) = X(t) + 1 $
then $X(t)$ and $Y(t)$ are perfectly correlated but any percentage change in $X$ is infinite.
