I have two time series data: $x_t $ and $y_t$. I have developed an algorithm to detect unusual changes (activities) on my $x_t$ series, so that whenever there is a change detected (based on some criteria), a flag is up.
What I would like to do is to investigate the impact of the detected change at time $t$ in $x_t$ on my other series $y_t$, using historical data. By analyzing the data, I would like to understand whether there is an impact on $y_t$ at all, and if yes, is it effective at the same time step (time $t$) or there is a time lag between the change and the effect.
Can anyone suggest a method for this purpose?