Let us consider a vector $Y_t$ of $k$ time series variables. For example, $Y_t = ( x_t , y_t )'$ , in which case $k = 2$. We assume that $Y_t follows a stochastic process that can be well approximated by a linear VAR process of the form 
$Y_t = \mu +A_1 Y_{t-1} + \cdots + A_p Y_{t-p} + u_t (1)$
where $μ$ is a k × 1 vector of constants, $A_i ( i = 1, …, p )$ is a k × k matrix and $u_t$ is a k × 1 vector of white noise, whose elements are referred to as reduced-form residuals . 

Each element of $u_t$ is in turn assumed to be a linear combination of latent structural shocks, $ε_{1 t} , …, ε_{kt}$ which are the sources of variation of the system.  An usual assumptions is that $ε_{1 t} , …, ε_{kt}$ are mutually independent, although orthogonality is sufficient in many applications. Thus we have: 
$u_t = B ε_t (2)$
where B is a k × k invertible matrix (the impact or mixing matrix) and $ε_t = ( ε_{1 t} , …, ε_{kt} )'$ is a vector of independent shocks. Let W be $B^{−1}$ . Then we get the structural VAR form: 
$WY_t = \mu'+Γ_1 Y_{t-1} + \cdots + Γ_p Y_{t-p} + ε_t (3)$
where $μ' = Wμ$ and $Γ_i = WA_i$ for $i = 1, …, p$. 

The idea of VAR analysis is to follow a two-step procedure: 
 - First Eq. (1) is estimated through standard regression methods to obtain an estimate of the reduced-form residuals $u_t$ . 
 - Second, the parameters of Eq. (3) can be recovered by analyzing the relationships among the elements of $u_t$. Notice that, having estimated (1), knowing B is sufficient for identifying (3).

Structural Vector Autoregressive (VAR) Analysis

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. 

University of Michigan - Ann Arbor

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 $Δx_t$ and $Δy_t$ 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 as regards 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. $x_i → y_{i + s}$ for some $i , s$ such that $i ≥ 0, s ≥ 0$. 
 - (ii) The series $y_t$ has a contemporaneous or lagged causal effect on $x_t$ , i.e. $y_i → x_{i + s}$ for some $i , s$ such that $i ≥ 0, s ≥ 0$.
 - (iii) A not-measured 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

The causal structure between two time series can be represented by a causal graph consisting of nodes for $x_t , …, x_{t − p} , y_t , …, y_{t − p}$ , where $p$ is the largest lag by which $x_t$ or $y_t$ can be directly causally influenced.

An example for causal graphs is shown in the figure, for $p=2$.

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