How to determine which measurements cause which?

Question

Suppose I have two sequences of measurements, $x_1[n]$ and $x_2[n]$ for $0 \le n \le N-1$.

How do I determine if there is a causal relationship between the two?

My first thought was, well... I can just find the correlation between $x_1$ and $x_2$. But almost immediately I was struck by correlation is not causation.

OK, so in Python, we can generate a uniform noise signal x1 and then filter it to get x2. Clearly, in this case, x2 is causally related to x1.

import numpy as np
import scipy.signal as signal

Q = 50
N =1000
x1 = np.random.uniform(-1,1,N)
b,a = signal.iirfilter(1,[2*500*(1-1/(2*Q))/44100,2*500*(1+1/(2*Q))/44100])
x2 = signal.lfilter(b,a,noise1)

x1

x2

Unfortunately, the correlation of the two doesn't tell us much:

Next, I thought about using coherence:

$$ C_{x_1x_2}(\omega) = \frac{|G_{x_1x_2}(\omega)|^2}{G_{x_1x_1}(\omega) G_{x_2x_2}(\omega)} $$

where $G_{ab}$ is the cross-spectral density between $a$ and $b$.

However, this just seems to tell me that the two are causally related... it doesn't tell me the direction of the causation.

The figure above shows what should be obvious: $C_{x_1x_2}(\omega) = C_{x_2x_1}(\omega)$.

So... how do I determine the direction of the causation?

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For example, suppose I want to know whether $x_1$ caused $x_2$, or vice versa.

So either $$ x_1 = h \star x_2 $$ or $$ x_2 = g \star x_1 $$ where $\star$ is convolution, and $h$ and $g$ are the impulse responses of LTI systems. Presumably, $g$ is the inverse of $h$ so that $g \star h = 1$.

Is it possible that both $g$ and $h$ are stable and causal?