What's the difference between instantaneous and lagged effect? I am working on causal discovery in time series.
I know the difference between instantaneous and lagged causal effect based their graphical definition. Specifically, if we are studying causal relationship between two time series variables $\textbf{x}=\{x_t,x_{t+1},...\}$ and $\textbf{y}=\{y_t,y_{t+1},...\}$. We say $\textbf{x}$ has a lagged causal effect on $\textbf{y}$ if $x_t \rightarrow y_{t+1}$, $x_{t+1} \rightarrow y_{t+2}$, etc. We say $\textbf{x}$ also has a instantaneous causal effect on $\textbf{y}$ if $x_t \rightarrow y_{t}$, $x_{t+1} \rightarrow y_{t+1}$, etc.
However, I am quite confused about why there is instantaneous causal effect in time series. As the causal precedence principle states, cause must happen before effect, so how it is possible that we have a cause $x_t$ and a effect $y_t$ happen at the same time $t$?
I find an illustration in this paper. The author says instantaneous effect can happen when the measures have a lower time resolution than the causal influences.
But I still do not understand why. What's the 'time resolution of measurement' referred to? Can anyone illustrate my confusion by some intuitive or real world examples?
Thanks a lot.
 A: The time resolution of measurement has to do with the data acquisition or sampling rate: how often are you taking a sample? If you're taking one sample every minute, but the causal influence is on the order of milliseconds, then you can expect the causal influence to occur within the same time chunk. On the other hand, if you're sampling every nanosecond and the causal influence occurs on a time scale of seconds, the causal influence will assuredly not be in the same time chunk.
A: As I understand it, instantaneous causal effects have additional nuances of interpretation not captured by @AdrienKeister's swell answer: An instantaneous effect of $x$ on $y$, being a short run effect, does not affect the equilibrium process of $y$, and does not carry into the future of $y$, and the instantaneous effect is an effect of the change in $x$ (as opposed to the level of $x$) on $y$ which is why both lagged and instantaneous effects can both occur for the same lag length (e.g., in resulting from definitions of $\Delta x_t = x_t - x_{t-1}$, and $x_{t-1}$ in, say, a generalized error correction model).
