When does causation take time? I apologize in advance if this is an elementary question. I am a statistician and would like to use an assumption in a statistical method, but I don't know if the assumption usually holds in nature.
I would like to assume that, if $X$ causes $Y$, then the causal effect from $X$ to $Y$ takes some finite amount of time to complete (i.e. the causal effect is not instantaneous). Is this reasonable? If so, why? I have heard from non-physicists that it is reasonable whenever $X$ and $Y$ are separated in space. Are there any examples in physics where the causal effect is instantaneous?
 A: I think your question is, while useful, slightly adjacent to the point, in a way that I think the following explanation will make clear.
First, for an event to be 'physical' at least classically, it needs to be associated with a location in space and time according to some observer. Let's just take for granted we are in flat spacetime and using Cartesian coordinates; the general case shouldn't matter for you.
Suppose Alice sees event $X$ occur at $t=5$s, $x=10$m (say) and $Y$ with $t=10$s, $x=15$m. Then Bob, who is moving relative to Alice, will generally see a time delay between $X$ and $Y$ which is larger than $5$s and a spatial separation which is shorter than $5$m.
Because of the specific way the transformations work, should the delay between $X$ and $Y$ be shorter than the time it would take light to travel between them, it may happen that Bob sees $Y$ occur before $X$. On the other hand, should the delay be longer than the light travel time, all observers agree on at least the order in which the events occur. 
That temporal ordering depends on the light travel time comes in at a very early level in the physics, and a theory without this property at normal scales would probably be traumatically different from observation.
Now if $X$ causes $Y$ it must do so for all observers (at least, this is generally assumed). Otherwise one would get into weird situations where some observers see people born before their mothers etc. This unambiguous temporal ordering is what physicists are generally referring to when they discuss causality. In particular, two simultaneous events at different locations in space cannot cause one another.
There is nothing (at least nothing so low-level), however, that requires the causal effect of $X$ upon $Y$ to "take some time" having reached $Y$. It is just that "news" that $X$ has occurred takes finite time to propagate. 
A: It depends on what you mean by "it takes some finite amount of time to complete". Physical dynamics is usually given by (systems) of differential equations that specify the first or second time derivative of a quantity as a function of other quantities. That assignment is instantaneous, i.e. there is no time delay between the change of a quantity and the change of a first or second derivative. The time dependence is a consequence of the time scales that are being introduced by the solutions of these equations themselves. This is probably easiest to see with first order linear equations, where the proportionality constant that connects the first derivate of a quantity to the quantity itself becomes the time constant of the system. For second order linear systems we can recover the period and damping time constants etc., so we can extract immediate physical meaning from the equations' coefficients. 
What the structure of these systems guarantees is that there will always be some kind of causality which will make the future depend, in some way, on the past, either with or without memory (i.e. the immediate future may only depend on the immediate past or it may depend on an entire interval or even the entire past). 
One can, of course, discretize these equations to avoid infinitesimal quantities, but that is not as trivial as it sounds, if one wants to keep key aspects of physics (like energy and momentum conservation) intact. Even in the best case scenario such analyses will only approximate the system for finite amounts of time (except in the most trivial cases). 
If you are looking at stochastic processes and their time series, there is a very well worked out theory for those, but it would really depend on your application whether it can be applied successfully. 
A: I can't think of any examples in nature where the causal effect is instantaneous. If X causes Y then the casual effect from X to Y is a finite amount of time. Of course it would depend on what you are measuring. It would still come down to distance and speed. You may know the distance but if you don't know the speed then you can only assume the max is the speed of light.
A: There is a very simple answer to at least a part of your question. The minimum possible time for a causal effect to propagate where event $A$ is a cause of event $B$ is given by the spatial distance $\mathrm{d}_s(A,\,B)$ between the events in spacetime divided by the universal signal speed limit $c$, i.e. $\mathrm{d}_s(A,\,B)/c$. This happens when $B$ lies on the edge of $A$'s future lightcone and the time for the effect to happen at $B$ much smaller than the causation delay. The time for causation can be any time greater than this lower limit. You should take heed that, if the time is less than the lower limit, the time will be observer-dependent, but, even so, the order of two events with a causal relationship cannot be observer-dependent. The time dependence becomes weaker as the time interval approaches the lower limit.
Given you're a statistician, I am probably less rigorous in my usage of the word "causality" than you are, and therefore I suggest you should take a look at a description of what I exactly mean in my answer here (to the extent that I can explain it). You should especially look up the links to the Stanford Dictionary of Philosophy I refer to there,
