The causal structure between two time series can be represented by a causal graph consisting of nodes for $$x_t, \dots, x_{t - p}, y_t, \dots, y_{t - p}$$, where $$p$$ is the largest lag by which $$x_t$$ or $$y_t$$ can be directly causally influenced. An example of a causal graph is shown in the figure for $$p=2$$.

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Let us consider two scalar stochastic processes $$x_t$$ and $$y_t$$, $$t \in \mathbb{Z}$$, each observed for $$T$$ realizations. We assume that $$x_t$$ and $$y_t$$ are covariance stationary, or that $$\Delta x_t$$ and $$\Delta y_t$$ are covariance stationary. If we exclude the possibility that the future can cause the past, but allow contemporaneous feedback loops due, for example, to temporal aggregation, there are several possibilities regarding the causal structure between $$x_t$$ and $$y_t$$:

- (i) The series $$x_t$$ has a contemporaneous or lagged causal effect on $$y_t$$, i.e., $$x_i \rightarrow y_{i + s}$$ for some $$i, s$$ such that $$i \geq 0, s \geq 0$$.
- (ii) The series $$y_t$$ has a contemporaneous or lagged causal effect on $$x_t$$, i.e., $$y_i \rightarrow x_{i + s}$$ for some $$i, s$$ such that $$i \geq 0, s \geq 0$$.
- (iii) An unmeasured series $$z_t$$ has a contemporaneous or lagged causal effect on both $$x_t$$ and $$y_t$$.
- (iv) The causal structure between $$x_t$$ and $$y_t$$ can be described by any combination of (i)–(iii).
- (v) There is no causal link or path (of any type) linking $$x_t$$ and $$y_{t + s}$$, for any $$s \in \mathbb{N}$$.

Cause-Effect Pairs in Time Series

 Isabelle Guyon, Alexander Statnikov, Berna Bakir Batu (2019)
Cause Effect Pairs in Machine Learning (The Springer Series on Challenges in Machine Learning)
https://www.amazon.com/Effect-Machine-Learning-Springer-Challenges/dp/3030218090

Cause Effect Pairs in Machine Learning

Causal Graph for Time Series

There are two types of approaches to address the cause-effect pair problem in time seires:
 - Vector Autoregressive Analysis
 - Direct Causal Search

Vector autoregressive analysis is the standard approach in econometrics to address this problem, but causal search in this framework is limited.
An alternative valid approach is to apply the causal pairs search directly to time series data, while avoiding the vector autoregressive framework. 

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