11institutetext: International Monetary Fund
Washington, DC 20431, USA
11email<EMAIL_ADDRESS>
# Decoding Causality by Fictitious VAR Modeling††thanks: The views expressed
herein are those of the author and should not be attributed to the IMF, its
Executive Board, or its management.

Xingwei Hu

###### Abstract

In modeling multivariate time series for either forecast or policy analysis,
it would be beneficial to have figured out the cause-effect relations within
the data. Regression analysis, however, is generally for correlation relation
and does not accumulate causal effects; also very few researches have focused
on variance analysis for causality discovery. We first set up an equilibrium
for the cause-effect relations using a fictitious vector autoregressive model.
In the equilibrium, long-run relations are identified from noise, and spurious
ones are negligibly close to zero. The solution, called causality
distribution, measures the relative strength causing the movement of all
series or specific affected ones. If a group of exogenous data affects the
others but not vice versa, then, in theory, the causality distribution for
other variables is necessarily zero. The hypothesis test of zero causality is
the rule to decide a variable is endogenous or not. Our new approach has high
accuracy in identifying the true cause-effect relations among the data in the
simulation studies. We also apply the approach to estimating the causal
factors’ contribution to climate change, which accrues direct and indirect
effects over a long time.

###### Keywords:

causality exogeneity endogeneity causal identification causal discovery
climate change.

##  1 Introduction

Causal discovery or identification has been one of the few most fundamental
goals in machine learning and economics (e.g., Eichler, 2013). It is
preliminary for causal inference. In the discovery process, in general, an
automatic algorithm iteratively identifies the latent causations among the
data variables. However, the latency could hide a deep and complicated data
structure over which we merely observe a superficially precedent connection
between any two variables. The pairwise precedence relations indeed
constitutes a directional network with various strength and uncertainty for
each edge. The bilateral relations, however, are not in a logical and
consistent order. This paper applies a network equilibrium (Hu and Shapley,
2003) and defines a new type of causality. The solution integrates direct and
indirect influences among the data, accrues the real causal relations, and
filters out the noisy ones.

Defining causality is a profound convoluted question with many possible
answers which do not satisfy everyone (Granger, 1980). There are dozens of
approaches in the literature. They generally capitalize on the conditional
probability under interventions or the significance of regression coefficients
(e.g., Granger, 1969 and 1980; Pearl, 1995). However, conditional probability
involves a parametric probability distribution and the precision in estimating
them from the data. In a regression, the results depend entirely on the
appropriate selection of variables, and causal factors that are not
incorporated into the regression model cannot be represented in the output.
Adding or dropping a covariate, for example, could make a significant effect
insignificant; it could also make an insignificant one significant. Moreover,
the estimated causal relations are not transitive. For example, if $x$
significantly Granger causes $y$ and $y$ significantly Granger causes $z$;
however, $x$ may not Granger cause $z$. In this causal chain, $y$ could merely
be a pass-through. Direct but artificial ones often marginalize indirect
causal factors. In studying climate change, for example, solar activities have
had little trending vibration in the past hundred years and seem to have no
impact on the up-trending global warming. The aggregate impact over a thousand
years, however, may be no longer negligible. Also, indirect impacts passing
through other determinants to the change have played an essential role in the
last hundred years. A regression model, however, only averages the direct
effect in the estimate sample, and does not automatically add up the small
effects.

In this paper, we propose a new approach that attempts to resolve the above
issues. Our research is based on two presumptions. First, a causal effect
could act on sequential occasions, which could occur over a long time and
happen multiple times. Thus, an aggregate and accrual effect would be suitable
for measurement. Secondly, indirect causal reason passes through one or more
intermediaries and gains causal strength through the spillovers from these
intermediaries. The device we use is a sequence of fictitious vector
autoregressive (VAR) models in which the causal relations are under
investigation. The shocks to the VAR are natural direct interventions;
responses to the shocks are plausible effects. The limit variance
decomposition aggregates both direct and indirect, both short-run and long-run
effects. The decomposition quantifies the bilateral impacts among all
variables. It defines a network with directed edges over which the bilateral
impacts are the flow volumes. Then we define an equilibrium over the network,
which resolves potential conflicts in the bilateral influences or plausible
causations. In the literature, Leamer (1985) suggested that variance
decomposition may contain causal interpretation and stated that Sims (1980)
had already implied that.

The equilibrium has a global and a local version. On a global scale, it
produces a unique solution when the network is highly connected. The vector
solution quantifies the causality distribution for the movement of the time
series variables. When there are both exogenous and endogenous variables, the
values in the solution for any endogenous one would be statistically
insignificant. A hypothesis test can formally conduct this. Similarly, we also
test the disconnection among the variables after element-wisely removing these
endogenous variables from the fictitious VAR model. The local version of
causality distribution attributes each determinant a fair share of
contribution to an effect variable. The precision of these hypotheses tests is
evidenced in our simulation studies. We also apply the distribution to study
climate changes and find which factors are more important than others.

The rest of the paper is organized as follows: Section 2 introduces a
bilateral interaction network among a vector of time series data by a vector
autoregressive model. Section 3 derives the valuation formula, which measures
the amount of causality each time series contains; the solution is the global
causality distribution which applies to the whole vector. Section 4 studies a
causality distribution that measures causal variables’ contributions to any
effect variable. Section LABEL:sect:identification_algorthms provides
algorithms to identify the causal relations. Section
LABEL:sect:simulation_studies uses the simulation studies to test the
effectiveness of the new approach and conducts an empirical study for climate
change. Finally, Section LABEL:sect:conclusion concludes with further
comments. Our exposition is self-contained, and the proofs are in the
Appendix.

##  2 Pairwise Influences Within Multiple Time Series

We study a group of covariance stationary time series, $y_{1}$, $y_{2}$, …,
and $y_{n}$ (i.e., their means, variances, and covariances remain unchanged
over time), which can be adequately described by a linear model with lagged
values of themselves. We index the time series by $N=\\{1,2,\cdots,n\\}$. The
data generating process, however, is a black box; we attempt to identify which
are endogenous, which are exogenous, and which cause which variables. For
convenience, we stack the time series in a vector
$y=\left(\begin{array}[]{c}y_{1}\\\ \vdots\\\ y_{n}\end{array}\right)$ and its
value at time $t$ is $y_{.t}=\left(\begin{array}[]{c}y_{1t}\\\ \vdots\\\
y_{nt}\end{array}\right)$.

###  2.1 Basic Bilateral Relations of Interaction

The bilateral interventions are directional, and all together make a directed
network. Figure 2 is an example in which “$\to$” is for one-way directional
influence, and “$\leftrightarrow$” is for two-way directional influence in
other figures. In the network, nodes are the variables, and the directional
edges are the intervention relations. We study a few basic relations which may
have similarities to the concepts in the theory of discrete-time Markov
chains. For example, an exogeneity class corresponds to an irreducible class
while the endogeneity class corresponds to the transient class. Familiarity
with the Markov chain theory is not required but helpful.

Figure 1: An interactive network.

Figure 2: Exogeneity classes.

####  2.1.1 Exogeneity Classes:

There are a few levels of exogeneity (e.g., Engle et al., 1983). We use the
popular version: a class of variables is exogenous to all others if their
conditional probability is invariant to the changes of the other time series
in the list $y_{.t}$. Otherwise, they are called endogenous to the other
variables. We focus on the disjoint exogeneity classes such that each variable
in a class is endogenous to the other variables in the same class but
exogenous to variables outside of the class. The vector $y_{.t}$ could contain
more than one independent exogeneity class and up to one dependent class of
endogenous variables. The exogeneity classes are exogenous to each other and
also to the endogenous one. The casual relation is a class property in that if
one variable in a class has a causal effect on another variable, then all
variables in the class have the same causal effect. Figure 2 illustrates this
classification of nine time series variables. There are five exogeneity
classes, four of which affect the endogeneity class $\\{y_{5},y_{6}\\}$. The
isolated exogeneity class $\\{y_{7},y_{8}\\}$ evolutes on its own dynamics. If
we model $y_{.t}$ by an unrestricted vector autoregression, however, the
coefficients from other variables could still be significant in the equations
of $y_{7}$ and $y_{8}$ due to the sampling errors. Conversely, $y_{7}$ and
$y_{8}$ may also be significant in other equations in the VAR model. Thus, the
plausible causal linkage may not be reliable from the estimated coefficients
alone.

####  2.1.2 Hierarchy:

In a hierarchy, one causal factor may not directly influence effect ones;
also, all direct causations have a one-way direction. In Figure 4, there are
three levels of hierarchy. $\\{y_{1},y_{2}\\}$ is the exogeneity class but has
no direct causal effect to the third level. After removing
$\\{y_{1},y_{2}\\}$, both $\\{y_{3},y_{4}\\}$ and $\\{y_{5}\\}$ are de facto
exogeneity classes. In a VAR model, restrictions should be placed such that
some coefficients are zeros to ensure one-way directions are in place. Also,
through a sequence of VARs, conditional exogeneity could be finally figured
out.

Figure 3: Hierarchy.

Figure 4: Circular.

####  2.1.3 Circular:

In a circular causal structure, there exist one or more circular causal
chains. In Figure 4, there are two circular causal chains:
$\\{y_{9}\\}\to\\{y_{7},y_{8}\\}\to\\{y_{4}\\}\to\\{y_{9}\\}$ and
$\\{y_{9}\\}\to\\{y_{7},y_{8}\\}\to\\{y_{4}\\}\to\\{y_{5},y_{6}\\}\to\\{y_{9}\\}$.
The series $y_{3}$ is not in any circular chain and has no causal effect on
others. With or without $y_{3}$, the causal relations, among others, should
remain invariant. A circular chain always belongs to the same class;
conversely, a members of a non-singleton exogeneity class always belongs to
some circular chain. The test we demand retains all members in a causal
circular and prevents any outsiders from joining.

####  2.1.4 Periodic:

This is a special circular chain, and periodicity is also a class property.
The circular causal chain has a minimal period $d$ if the chain returns to the
same variable after traveling multiple $d$ periods. Figure 6 has a period $6$.
The number of intermediaries for a circular chain is the same for any member
to recurrent to itself.

Figure 5: Periodic.

Figure 6: Subexogeneity classes.

####  2.1.5 Sub-exogeneity Classes within the endogeneity class :

The endogeneity class may contain one or more subclasses which are exogenous
to the other variables in the endogeneity class. In Figure 6,
$\\{y_{5},y_{6},y_{7},y_{8},y_{9}\\}$ is the endogeneity class. Itself has a
subclass $\\{y_{5},y_{6}\\}$ which is exogenous to $\\{y_{7},y_{8},y_{9}\\}$.
To find the contribution of $y_{5}$ to $y_{7}$, for example, we put
$y_{1},y_{2}$ and $y_{3}$ as exogenous in the VAR of $y_{5},\cdots,y_{9}$, and
ignores $y_{4}$.

###  2.2 The Fictitious VAR Models

The general unrestricted VAR (GUVAR) with $p$ lags is of the form

$y_{.t}=\nu+A_{1}y_{.t-1}+\cdots+A_{p}y_{.t-p}+u_{t}$ (1)

where $y_{.t}$, $\nu$, and $u_{t}$ are $n$ dimensional column vectors, and
$A_{i}$’s are $n\times n$ unknown coefficient matrices. As usual, we assume
that the residual $u_{t}$ is serially uncorrelated and has a constant
variance-covariance $\Sigma$ over time $t$. However, the model (1) is
fictitious and has not differentiated the exogenous and endogenous variables
in the list, as well as the causal relations among them. It is a mechanism to
identify them.

Restrictions could be placed on the short-run coefficient matrices to reflect
the dependence among the component time series. When there are both exogenous
and endogenous variables in the list $y_{.t}$, in theory, the endogenous ones
have no short-run impact on the exogenous ones. Therefore, after some
rearrangements, the true VAR(p) has the pattern for all the coefficient
matrices $A_{1}$, $\cdots$, $A_{p}$:

$\blockarray{*{2}{c}l}\block{*{2}{>{\footnotesize}c<{}}l}Exogenous&Endogenous\\\
\block{[*{2}{c}]>{\footnotesize}l<{}}*&0\bigstrut[t]\ Exogenous\\\ **\
Endogenous\\\ $ (2)

For two or more isolated classes of exogenous variables, the coefficient
matrices of the true VAR(p) have the pattern of block diagonal as

$\blockarray{*{3}{c}l}\block{*{3}{>{\footnotesize}c<{}}l}Class1&Class2\cdots\\\
\block{[*{3}{c}]>{\footnotesize}l<{}}*&0\bigstrut[t]0\ Class1\\\ 0*0\
Class2\\\ 00*\ \hskip 11.38092pt\vdots\\\ $ (3)

In practice, however, we are uncertain if there are any exogenous variables.
Also, an estimated coefficient matrix of (1) would have a nonzero upper right
block, due to the sampling errors.

The VAR(p) in (1) can be rephrased to a VAR(1) structure (4) by stacking
$y_{1},y_{2},\cdots,y_{n}$ into a large vector $Y$,

$Y_{t}=V+AY_{t-1}+U_{t}$ (4)

where

$A=\left[\begin{array}[]{ccccc}A_{1}&A_{2}&\ldots&A_{p-1}&A_{p}\\\
I_{n}&0&\ldots&0&0\\\ 0&I_{n}&&0&0\\\ \vdots&&\ddots&\vdots&\vdots\\\
0&0&\ldots&I_{n}&0\end{array}\right],\ Y=\left[\begin{array}[]{c}y_{1}\\\
\vdots\\\ y_{p}\end{array}\right],\ V=\left[\begin{array}[]{c}\nu\\\ 0\\\
\vdots\\\ 0\end{array}\right],\ \mathrm{and}\
U_{t}=\left[\begin{array}[]{c}u_{t}\\\ 0\\\ \vdots\\\ 0\end{array}\right].$

Besides, $A$ is a $np\times np$ matrix, $I_{n}$ is the $n\times n$ identity
matrix, $Y$, $V$ and $U$ are $np$ dimensional column vectors. Clearly, $U_{t}$
is also serially uncorrelated and has variance-covariance
$\mathrm{cov}(U_{t})=J^{\prime}\Sigma J$ where
$J=\left[\begin{array}[]{llll}I_{n}&0&\ldots&0\end{array}\right]$ is a
$n\times np$ matrix. In general, some elements of a block of an estimated
coefficient matrix may be statistically insignificant; but some may be
significant. Even all elements in the block are insignificant, we can still
not tell if the block is zero statistically.

For the linear engine (1) of the covariance-stationary $y_{.t}$, the driving
forces are the residuals or shocks $u_{t}$, either positive or negative.
Without the forces, the series remain as a constant vector and no cause-effect
relations could be detected. Though human could also engineer shocks (now
called interventions), we ignore the mechanism how they are generated. With a
given $u_{t}$, the whole series vector responds from time $t$ in which a tiny
slice of causal effect would be hardly noticeable. The aggregated effects
through a sequence of $u_{t}$ could be enormous. However, a linear regression
generally averages, not aggregates, the effects from the innovations. We
analyze the variances of the shock series where the regression amplifies or
mitigates these effects.

The shock series is already embedded in the data $y_{.t}$ through the dynamics
(1). Given $Y_{t}$, the h-step ahead vector from (4) is

$Y_{t+h}=\sum\limits_{s=0}^{h-1}A^{s}U_{t+h-s}+A^{h}Y_{t}+\sum\limits_{s=0}^{h-1}A^{s}V,$

and its conditional covariance is

$\mathrm{cov}(Y_{t+h}|Y_{t})=\sum\limits_{s=0}^{h-1}A^{s}\mathrm{cov}(U_{t+h-1})(A^{s})^{\prime}=\sum\limits_{s=0}^{h-1}A^{s}J^{\prime}\Sigma
J(A^{s})^{\prime}.$ (5)

The conditional variance depends on the length of $h$, not on the value of
$Y_{t}$. Let
$X=\lim\limits_{h\to\infty}\mathrm{cov}(Y_{t+h}|Y_{t})=\sum\limits_{s=0}^{\infty}A^{s}J^{\prime}\Sigma
J(A^{s})^{\prime}$, if it exists. Then we have a steady-state variance which
satisfies the Lyapunov equation

$X=AXA^{\prime}+J^{\prime}\Sigma J.$ (6)

The solution to the Lyapunov equation solves the following linear equation
(cf, e.g., Hamilton, 1994, Equations 10.2.13 and 10.2.18)

$\left(I_{n^{2}p^{2}}-A\otimes
A\right)\operatorname{vec}(X)=\operatorname{vec}(J^{\prime}\Sigma J)$

where $\otimes$ is for the Kronecker product of matrices. Therefore,

$\operatorname{vec}(X)=\left(I_{n^{2}p^{2}}-A\otimes
A\right)^{-1}\operatorname{vec}(J^{\prime}\Sigma J).$ (7)

This also implies the covariance of $\operatorname{vec}(X)$ has an even more
complicated form. Let $L$ be a lower triangular matrix obtained by a Cholesky
decomposition of $\Sigma$ such that
$\Sigma=LL^{\prime}=\sum\limits_{j=1}^{n}L_{j}L_{j}^{\prime}$ where $L_{j}$ be
the $j$th column of $L$. Then the contribution of $y_{j}$ to $X$ is

$\left(I_{n^{2}p^{2}}-A\otimes
A\right)^{-1}\operatorname{vec}(J^{\prime}L_{j}L_{j}^{\prime}J).$ (8)

Element-wise ratios between (8) and (7) provide $y_{j}$’s contribution in
percentage to the limit variance of $y_{.t}$.

The covariance matrix $\Sigma$ could also be written as
$\Sigma=(LQ)(LQ)^{\prime}$ for any orthogonal matrix $Q$. However, $LQ$ has an
alignment issue with the order in $y_{.t}$. A common practice is to order the
components of $y_{.t}$ from the most exogenous to the least exogenous (e.g.,
Sims, 1980). This way, the shock from the first component series of $y_{.t}$
affects all others contemporaneously and the $L_{1}$ vector designates the
contemporary effects; the shock from the last component series only affects
the series itself contemporaneously where $L_{n}$ has only one non-zero
element. However, (8) includes not only the direct contemporaneous impact.

For a finite $h$, we take the first $n$ elements from $Y_{t+h}$ in (5) to get

$\mathrm{cov}(y_{t+h}|y_{t})=\sum\limits_{s=0}^{h-1}JA^{s}J^{\prime}\Sigma
J(A^{s})^{\prime}J^{\prime}=\sum\limits_{s=0}^{h-1}\Phi_{s}LL^{\prime}\Phi_{s}^{\prime}=\sum\limits_{s=0}^{h-1}\sum\limits_{j=1}^{n}\Phi_{s}L_{j}L_{j}^{\prime}\Phi_{s}^{\prime}$

where $\Phi_{s}=JA^{s}J^{\prime}$. Thus, the variance of the h-step forecast
of variable $y_{i}$ is

$\mathrm{cov}(y_{i,t+h}|y_{t})=e_{i}^{\prime}\mathrm{cov}(y_{t+h}|y_{t})e_{i}=\sum\limits_{s=0}^{h-1}\sum\limits_{j=1}^{n}\left(e_{i}^{\prime}\Phi_{s}L_{j}\right)^{2}$

where $e_{i}$ is the $i$-th column of $I_{n}$. Therefore, the amount of
forecast variance of variable $y_{i}$, $\mathrm{cov}(y_{i,t+h}|y_{t})$,
accounted for by the shocks to variable $y_{j}$ is given by $\omega_{ij,h}$,

$\omega_{ij,h}=\frac{\sum\limits_{s=0}^{h-1}\left(e_{i}^{\prime}\Phi_{s}L_{j}\right)^{2}}{\sum\limits_{s=0}^{h-1}\sum\limits_{z=1}^{n}\left(e_{i}^{\prime}\Phi_{s}L_{z}\right)^{2}}.$
(9)

The asymptotic distribution of (9) could be derived (e.g.,
L$\mathrm{\ddot{u}}$tkepohl, 2005, page 111); however, it does not account for
the identity $\sum\limits_{j=1}^{n}\omega_{ij,h}=1$ and the non-negativity of
(9). Thus, the distribution cannot be used in the usual way for the hypothesis
test of $\omega_{ij,h}=0$. Theorem 2.1 shows the existence of $\omega_{ij,h}$
in the infinite horizon of $h$. We let the unconditional covariance
$\omega_{ij}=\lim\limits_{h\to\infty}\omega_{ij,h}$ and denote the $n\times n$
matrix $\Omega=[\omega_{ij}]$, which can be exactly calculated from (7) and
(8).

###### Theorem 2.1

If $y_{.t}$ is covariance stationary, then
$\lim\limits_{h\to\infty}\omega_{ij,h}$ exists and does not depend on any
value of $Y_{t}$.

###### Proof

See Appendix A1.

Unlike a causal inference, the discovery of causal relation should be
invariant in some way. It is a YES or NO question. It does not rely on the
unit the variables use or the sign and size of the coefficients in regression.
A causal relation between $y_{i}$ and $y_{j}$ does not merely depend on the
coefficients and their standard errors in the regression of $y_{i}$ and
$y_{j}$. It also relies on the causal structure in $y_{.t}$ and the global
uncertainty in the modeling. The formulas (7), (8), and (9) integrate both the
coefficient matrices and the uncertainty. Meanwhile, (2) and (3) are two
common relational structures in $y_{.t}$. The following two theorems indicate
the invariance of patterns and linearity.

###### Theorem 2.2

If all the coefficient matrices $A_{i}$ have the pattern (2), then $\Omega$
has the same pattern.

###### Proof

See Appendix A2.

###### Corollary 1

If all $A_{i}$ have the diagonal pattern (3), then $\Omega$ is a block lower
triangle matrix.

###### Theorem 2.3

$\Omega$ is invariant under any linear transform of any time series from
$y_{.t}$.

###### Proof

See Appendix A3.

The pattern for the coefficient matrices $A_{i}$ restricts the short-run
direct linkages. If these restrictions carry on in the long run, then the
pattern preserves in $\Omega$ as it integrates both short-run and long-run.
The block diagonal pattern (3) is a special case of (2); thus, Corollary 1
follows. However, $\Omega$ is not necessary a block diagonal; the order of the
exogeneity classes matters. If class $i$ precedes class $j$ in the VAR, then
$i$ has a directional impact on $j$ in $\Omega$; but not vice versa. As an
example of the linearity invariance, the emission of carbon dioxide is a
causal factor to global warming, measured either in the Celsius scale or the
Fahrenheit scale. The invariance also makes it possible to compare the
causality contributions across the variables in $y_{.t}$.

###  2.3 Instantaneous Effects by Structural VAR

When the variables instantaneously affects each other, we could write a
general structural VAR (SVAR) form as

$y_{.t}=\nu+A_{0}y_{.t}+A_{1}y_{.t-1}+\cdots+A_{p}y_{.t-p}+u_{t}.$ (10)

Then

$y_{.t}=(I_{n}-A_{0})^{-1}\left[\nu+A_{1}y_{.t-1}+\cdots+A_{p}y_{.t-p}+u_{t}\right]$
(11)

and

$Y_{t}=(I_{n}-A_{0})^{-1}JV+(I_{n}-A_{0})^{-1}JAY_{t-1}+(I_{n}-A_{0})^{-1}JU_{t}.$
(12)

Taking variance on both sides of (12) and letting $t\to\infty$, we obtain the
Lyapunov equation for $X$,

$X=\left[(I_{n}-A_{0})^{-1}JA\right]X\left[(I_{n}-A_{0})^{-1}JA\right]^{\prime}+J^{\prime}(I_{n}-A_{0})^{-1}\Sigma(I_{n}-A_{0}^{\prime})^{-1}J.$

and its solution is

$\operatorname{vec}(X)=\left(I_{n^{2}p^{2}}-\tilde{A}\otimes\tilde{A}\right)^{-1}\operatorname{vec}(J^{\prime}(I_{n}-A_{0})^{-1}\Sigma(I_{n}-A_{0}^{\prime})^{-1}J).$
(13)

where $\tilde{A}=\left[(I_{n}-A_{0})^{-1}JA\right]$. The proportion
contributed by $y_{j}$ is

$\left(I_{k^{2}p^{2}}-\tilde{A}\otimes\tilde{A}\right)^{-1}\operatorname{vec}(J^{\prime}(I_{n}-A_{0})^{-1}L_{j}L_{j}^{\prime}(I_{n}-A_{0}^{\prime})^{-1}J).$
(14)

The element-wise ratios between (14) and (13) normalizes the contributions.

###### Theorem 2.4

If $y_{.t}$ is covariance stationary and $||(I_{n}-A_{0})^{-1}||\leq 1$, then
both (14) and (13) exist and do not depend on any value of $y_{.t}$.

###### Proof

See Appendix A4.

##  3 A Cause-Effect Equilibrium

The bilateral influence matrix $\Omega$ from the general unrestricted VAR (1)
is a Markov transition matrix, upon which one may define irreducibility,
ergodicity, and periodicity. However, each element of the matrix contains a
heterogeneous uncertainty and, in an empirical analysis, no element in the
matrix would be zero due to the sampling errors. Of course, many elements
would be statistically insignificant when the uncertainty is concerned.
Ignoring the uncertainty, therefore, all variables in $y_{.t}$ are
artificially irreducible, ergodic, and aperiodic.

###  3.1 Global Causality Distribution

There are many inconsistencies in $\Omega$ if the transition has a causal
interpretation. The transitivity does not hold in the variance decomposition
of $\Omega$. Secondly, indirect causal relations are not properly linked.
Also, spurious causal linkages are not removed. Lastly, there exists
heterogeneous uncertainty in $\Omega$.

To overcome the inconsistencies, we consider the $t$-step influence
$\Omega^{t}$ where $\Omega^{t}(i,j)$ is the $y_{j}$’s indirect influence over
$y_{i}$, passing through $t-1$ intermediaries including both $y_{i}$ and
$y_{j}$. The $t$-step matrix make indirect influences direct. In Figure 4, the
first level has a direct influence to the third level in $\Omega^{2}$. The
matrix also integrates any intermediate effects as
$\Omega^{t}=\Omega\Omega^{t-1}=\Omega^{2}\Omega^{t-2}=\cdots.$ The Sun’s
causal effects on global warming, for example, could accumulate for many years
and repeatedly through many intermediates (e.g., oceanic evaporation and
ozone). Finally, for an empirical $\Omega$, the indirect influences eventually
become stable over time,

$\lim\limits_{t\to\infty}\Omega^{t}=1_{n}(\pi_{1},\pi_{2},\cdots,\pi_{n})$

where $1_{n}$ is the $n\times 1$ vector with all $ones$, $\pi_{i}\geq 0$, and
$\sum\limits_{i=1}^{n}\pi_{i}=1$. Therefore, the average direct and indirect
influence also have the same limit,

$\lim\limits_{z\to\infty}\frac{1}{z}\sum\limits_{t=1}^{z}\Omega^{t}=1_{n}(\pi_{1},\pi_{2},\cdots,\pi_{k}).$

In the limits, the $i$th column remains the same across all rows. It is
$y_{i}$’s causality to the movement of the whole vector $y_{.t}$. So we call
$\pi_{i}$ to be $y_{i}$’s global causality.

The row vector $\pi=(\pi_{1},\pi_{2},\cdots,\pi_{k})$ also satisfies the
following counterbalance equilibrium (Hu and Shapley, 2003) over the
directional network $\Omega$:

$\pi=\pi\Omega.$ (15)

In the counterbalance, on the one hand, $y_{i}$’s causality $\pi_{i}$ to the
movements of $y_{.t}$ derives from its effect on all components of $y_{.t}$.
On the other hand, it also distributes its $\pi_{i}$ to others which cause
$y_{i}$’s changes. In the distribution of $\pi_{i}$, the fair division is
through the $i$th row of $\Omega$, which is the percentages of $y_{i}$’s
variability explained by each individual time series. So $y_{j}$ gets
$\pi_{i}\omega_{ij}$ from $\pi_{i}$. Similarly, $\pi_{j}$ also distributes
$\pi_{j}$ to others and $y_{i}$ gets $\pi_{j}\omega_{ji}$. Looping on all $j$
in $N$, we get the following equation

$\pi_{i}=\sum\limits_{j\in N}\pi_{j}\omega_{ji}$ (16)

and (15). So, the $i$th column of $\Omega$ stipulates the weights with which
$y_{i}$ collects its causality distribution.

The causality distribution $\pi$ is a mixed cooperative and non-cooperative
solution. It is noncooperative because $\sum\limits_{i\in N}\pi_{i}=1$; one
series’ gain means another’s loss. It is also cooperative because (16) implies
that a larger $\pi_{j}$ improves $\pi_{i}$ more, as long as $\omega_{ji}>0$
remains the same. Also by (16), increasing $\omega_{ji}$ also enhances
$\pi_{i}$. A long-run of competitions and cooperations finally reaches the
ultimate solution $\pi$.

The distribution $\pi$ could also come from an endogenously weighting.
Ignoring the spillover effects, the $i$th row of $\Omega$ is a proper
assignment of responsibility or reason for $y_{i}$’s movement alone; but it is
not a proper measurement for the whole vector of $y_{.t}$. We assign a number
$\pi_{i}$ to weigh the $i$th row where important variable gets more weight and
less important one gets less weight. Then the weighted assignment of
responsibility is $\pi\Omega$, which also measures each variable’s
contribution to the movement of the vector $y_{.t}$. So $\pi$ is proportional
to $\pi\Omega$. Setting $\sum\limits_{i=1}^{n}\pi_{i}=1$, we get the equation
(15).

Another application of the counterbalance equilibrium can be found in a
sorting algorithm for big data (Hu, 2020). Instead of using $\Omega$ in the
equilibrium, one could also apply the frequency decompositions (Geweke 1982;
Kaminski et al. 2001) or the generalized variance decomposition (Lanne and
Nyberg, 2016). Both these two decompositions also have a unit sum for each
row. An iterative algorithm for (15) is

$\pi^{(t)}=\pi^{(t-1)}\Omega$ (17)

for a given nonnegative vector $\pi^{(0)}$ such that it has a unit sum. The
limit of $\pi^{(t)}$ would be $\pi$ for an empirical $\Omega$.

###  3.2 Noise Filtration by $\pi$

The general unrestricted VAR in (1) is highly artificial as it ignores the
cause-effect relations within the vector $y_{.t}$. Consequently, this may
result in overwhelmingly many insignificant elements in $\Omega$. One solution
is to iteratively identify the causal relations and then re-specify the VAR
model using already identified relations. For example, hypothesis tests could
be used to make the VAR from general to specific and from unrestricted to
restricted. By speculating a possible relation after observing some signals,
one generally sets up a hypothesis on the relation. Finally, he or she
extracts the evidence from the data to approve or reject the hypothesis.

The distribution $\pi$ mitigates the uncertainty in $\Omega$ by its spillover
effects from the third parties. Indeed, the spillover effects take two further
actions: confirmation of real comparative causality and off-setting of noisy
ones. Consequently, $\pi_{i}$ could still be insignificant even if many
elements of $\Omega$’s $i$th column are statistically significant. If $y_{i}$
has a real comparative causality in the directional network, then $\Omega$ has
consistent large values in the $i$’s column, especially for variables with
large values in $\pi$. If the advantage is a noise, on the other hand, then
the sampling errors could largely account for the positive entires in the
$i$th column.

To see the effects of a small variation of $\Omega$ on $\pi$, let us first
shock both $\Omega(j,i)$ and $\Omega(k,i)$ by $\delta$. This can be seen as
$y_{i}$ has some real improvement. Secondly, we shock $\Omega(j,i)$ by
$\delta$ and $\Omega(k,i)$ by $-\delta$. This can be a case of pure white
noise. By Theorem 3.1, the effects add up when two shocks have the same sign
while they offset each other when the shocks are opposite. The small-sized
derivative in Theorem 3.1 also indicates that $\pi_{i}$’s standard error is
small compared to those for the elements in $\Omega$’s $i$th column. In the
theorem, we use three notations for simplicity: the matrix $Z_{i}$ is the
transpose of $\Omega$ with the $i$th row and the $i$th column removed; the
column vector $\alpha_{ji}$ takes the $j$th row from $\Omega$ and then drops
its $i$th element; and $\pi_{-i}$ is the row vector when $\pi_{i}$ is removed
from $\pi$.

###### Theorem 3.1

For any $i,j\in N$,

$\frac{\mathrm{d}\pi_{i}}{\mathrm{d}\omega_{ji}}=\frac{\pi_{j}}{1-\omega_{ji}}\
\frac{1_{n-1}^{\prime}(I_{n-1}-Z_{i})^{-1}\alpha_{ji}}{1+1_{n-1}^{\prime}(I_{n-1}-Z_{i})^{-1}\alpha_{ii}}\geq
0$

and

$\frac{\mathrm{d}\pi_{-i}^{\prime}}{\mathrm{d}\omega_{ji}}=[I_{n-1}-Z_{i}]^{-1}\left[\frac{\mathrm{d}\pi_{i}}{\mathrm{d}\omega_{ji}}\alpha_{ii}-\frac{\pi_{j}}{1-\omega_{ji}}\alpha_{ji}\right].$

###### Proof

See Appendix A5.

The filtration could also be explained by the chains of direct influences. The
unit sum in each row of $\Omega$ places a zero-sum restriction on the noise of
each row. So some noise are positive while others are negative. For a large
$t$,

$\Omega^{t}(i,j)=\sum\limits_{i_{0}=i,i_{1}\in N,\cdots,i_{t-1}\in
N,i_{t}=j}\omega_{i_{0},i_{1}}\omega_{i_{1},i_{2}}\cdots\omega_{i_{t-1},i_{t}}$

which is a sum of $n^{t-1}$ tiny particles. Each particle is the product of
$t$ elements from $\Omega$ in which some contain positive noise and others
contain negative noise. The summation mitigates the noise when the noise
offsets each other. Adding up the small pieces could make a difference;
averaging them, as in a regression, is still small.

##  4 Determinants’ Responsibilities for an Effect

While mutually causal relations exist in a set of data, the one-way cause-
effect linkage is also pervasive. Say, $y_{1}$ has a direct influence on
$y_{2}$. But this does not imply that $y_{2}$ is necessarily essential to
$y_{1}$’s variability. Generally, we may put all direct or indirect
influencing and influenced data into one huge set and analyze the influences
between the classifications inside the huge set. In Figure 8, for example,
classes 1 and 2 influence class 3; however, the latter has no impact on
classes 1 and 2. One question is how much is the relative causal importance of
classes 1 and 2 to class 3. That is, which class is more important to class 3?
And by how much? For example, when applied to climate changes, there are many
causal factors accounting for the changes. How much is attributed to carbon
dioxide emissions? How about the changes in the Sun, the emissions from
volcanoes, and the variations in Earth’s orbit?

|  | Class 1 |  |
---|---|---|---|---
|  | $\mathrel{\rotatebox{90.0}{$\xleftarrow{\rule{34.14322pt}{0.0pt}}$}}$ |  |
Class 2 | $\longrightarrow$ | Class 3 |  | Class 4
Figure 7: Causal-transient class.

$\Omega=\left(\begin{array}[]{c c c c}\Omega_{1}&&&\\\ &\ddots&&\\\
&&\Omega_{k}&\\\ T_{1}&\cdots&T_{k}&T_{e}\end{array}\right)$ Figure 8: General
pattern for $\Omega$.

After all variables $y_{.t}$ are identified, say, there are $k$ endogeneity
classes $C_{1},C_{2},\cdots,C_{k}$, and one endogeneity class $T$. Grouping
variables by their classes, we have the pattern for the transition matrix in
Figure 8. Each $\Omega_{s}$ is a stochastic matrix, estimated from the VAR
with variables from the exogeneity class $C_{s}$. Thus the submatrix
$\Omega_{s}$ itself defines a global causal distribution $\pi^{c_{s}}$ within
the exogeneity class $C_{s}$. The $i$th element of the vector $\pi^{c_{s}}$
quantifies the $i$th variable’s responsibility and causality for the dynamics
of all variables in the class. The matrices $T_{1},\cdots,T_{k}$ and $T_{e}$
are estimated from the VAR with the coefficients matrices pattern of Figure
10. Therefore, $k+1$ VARs are estimated for Figure 8.

$\left(\begin{array}[]{ccccc}*&&&&\\\ &*&&&\\\ &&\ddots&&\\\ &&&*&\\\
&*&\cdots&*&*\end{array}\right)$ Figure 9: Restricted coefficients.

$\Omega=\left(\begin{array}[]{cccccc}\frac{1}{3}&\frac{2}{3}&0&0&0&0\\\
\frac{2}{3}&\frac{1}{3}&0&0&0&0\\\ 0&0&\frac{1}{4}&\frac{3}{4}&0&0\\\
0&0&\frac{1}{5}&\frac{4}{5}&0&0\\\
\frac{1}{4}&0&\frac{1}{4}&0&\frac{1}{4}&\frac{1}{4}\\\
0&\frac{1}{6}&\frac{1}{6}&\frac{1}{3}&\frac{1}{6}&\frac{1}{6}\\\
\end{array}\right).$ Figure 10: An example of $\Omega$.

###  4.1 Causal-transient Members

For any $y_{i}$ in $T$, some $y_{j}$ has causal effect on $y_{i}$ but not vice
versa. So $y_{i}$’s influence on the whole set of times series is transient
and temporary, and it dies out gradually. On the other hand, an exogeneity
class accrues causality from the causal-transient $y_{i}$. Let $T_{e}$ be the
restriction of $\Omega$ to the endogeneity class $T$ (see Figure 8). Without
loss of generality, say, $T$ has $m$ members, namely,
$y_{n-m+1},\cdots,y_{n}$. So $T_{e}$ is the sub-matrix of the last $m$ columns
and the last $m$ rows of $\Omega$. As usual, let $1_{n}$ be the $n\times 1$
vector of ones, and $0_{n}$ the $n\times 1$ zero vector. Also, let $I_{n}$ be
the $n\times n$ identity matrix.

###### Theorem 4.1

$\lim\limits_{t\to\infty}T^{t}_{e}=0_{m}0_{m}^{\prime}$, the $m\times m$ zero
matrix.

###### Proof

See Appendix A6.

###### Corollary 2

$I_{m}-T_{e}$ is invertible; also, if $T\not=\emptyset$, then $y_{.t}$
contains at least one exogeneity class; and finally, $\pi_{i}=0$ for all $i\in
T$.