Transmission Channel Analysis in Dynamic Models

May 29, 2024

Enrico Wegner2,∗, Lenard Lieb1, Stephan Smeekes2, and Ines Wilms2

1Department of Macro, International & Labour Economics, School of Business and
Economics Maastricht

2Department of Quantitative Economics, School of Business and Economics
Maastricht

∗Corresponding author<EMAIL_ADDRESS>
###### Abstract

We propose a framework for the analysis of transmission channels in a large
class of dynamic models. To this end, we formulate our approach both using
graph theory and potential outcomes, which we show to be equivalent. Our
method, labelled Transmission Channel Analysis (TCA), allows for the
decomposition of total effects captured by impulse response functions into the
effects flowing along transmission channels, thereby providing a quantitative
assessment of the strength of various transmission channels. We establish that
this requires no additional identification assumptions beyond the
identification of the structural shock whose effects the researcher wants to
decompose. Additionally, we prove that impulse response functions are
sufficient statistics for the computation of transmission effects. We also
demonstrate the empirical relevance of TCA for policy evaluation by
decomposing the effects of various monetary policy shock measures into
instantaneous implementation effects and effects that likely relate to forward
guidance.

JEL Codes: C32, C54, E52, E60

Keywords: transmission channel, policy evaluation, impulse response function,
structural vector autoregression, DSGE, macroeconomic shocks

##  1 Introduction

In this paper we propose a formal framework for analysing transmission
channels of causal effects that is applicable in a large family of dynamic
models. Our framework allows to dynamically decompose the effects of
interventions into a set of disjoint dynamic partial effects, where the choice
of decomposition – the intervention’s transmission channel – is determined by
the research question.

Our first contribution is the development of a formal framework that allows
the study of precisely defined transmission channels. While there is an
extensive literature in macroeconomics and time series econometrics on
methodology for studying the dynamic effects of interventions of shocks [63,
52, 56, see e.g.], the literature on formal analysis of transmission channels
is relatively sparse. In macroeconomics, transmission channels are typically
loosely defined as a collection of economic mechanisms that indirectly affect
key macroeconomic outcomes through intermediate variables. However, a coherent
quantitative framework for analysing transmission channels is missing.
Instead, many studies analyse transmission channels qualitatively. From a
policy perspective however, careful study of transmission channels is crucial.
Given a particular total dynamic effect of the shock of interest, the relative
importance of different transmission channels through which these total
effects are realised can have major implications for designing effective
policies. It is therefore crucial to have an unequivocal definition of
transmission channels and accompanying methodology to decompose total effects
accordingly.

Our second contribution is to formulate transmission channel analysis (TCA
henceforth) in terms of impulse response analysis. Not only do we formulate
TCA as a decomposition of total effects obtained through impulse response
analysis, we rigorously establish that the calculation of impulse responses
between different variables in the system is sufficient for the analysis of
all transmission channels. Importantly, we also prove that only a single shock
driving the impulse responses – the initial shock of interest – has to be
structurally identified, and reduced-form impulse responses otherwise suffice.
Consequently, TCA can be performed under the same conditions as traditional
impulse response analysis, only requiring a structural identification scheme
for a single shock plus a specified dynamic model. We prove that this
equivalence holds for a large class of dynamic models, including structural
vector autoregressive (SVAR) and linearised dynamic stochastic general
equilibrium (DSGE) models.

Our third contribution is of a more technical nature. In order to develop the
results discussed above, we develop a graphical approach to study transmission
channels. Specifically, we show how to connect the impulse-response-space
representation of the (structural) model’s equilibrium dynamics to a Directed
Acyclic Graph (DAG). This involves a re-parameterisation of the dynamic
equilibrium representation, which allows us to uniquely define a transmission
channel as a collection of paths along the graph, connecting causal effects
from one variable (resp. shock) to another variable and over time. To
complement the graphical framework, we also develop an alternative
representation of transmission channels in terms of potential outcomes. We
prove that both representations are equivalent implying that transmission
channels can be defined either as paths through a graph or as a specific
potential outcome. Apart from this specific result that is used throughout our
theoretical analysis, the graphical and potential outcomes frameworks are of
interest in themselves for the analysis of dynamic causal effects beyond our
specific transmission questions.

Our final contribution is to demonstrate the empirical usefulness of the
framework for the purpose of policy evaluation. To this end, we examine two
prominent monetary policy shock series that likely capture two specific
channels through which monetary policy takes effect. While the narrative shock
series proposed by [66] is likely capturing instantaneously implemented
changes in the policy instrument, the external instrument suggested in [47]
is, due to its construction, rather expected to correlate with longer-term
interest rates, and hence, is likely to pick up the forward guidance component
of monetary policy. In our application we find that both shock series indeed
work predominantly through different policy channels. We confirm that
immediate implementation effects drive the total responses when using the
monetary policy shock series proposed by [66], while other dimensions of
monetary policy, such as forward guidance, drive the total responses triggered
by the shock series constructed by [47].

Our paper touches upon various strands of academic literature. First, as
already alluded to above, it is most closely related to the macroeconometric
literature on the analysis of dynamic causal effects and (informal)
transmission channels. For example, [47] infer qualitatively to what extend
the effects of monetary policy are driven by credit costs by studying the
impulse responses of term premia and credit spreads to monetary policy shocks.
Similarly, impulse responses of various variables to government spending or
tax (news) shocks are investigated to understand the transmission mechanisms
of fiscal policies. For example, [62] infers from the response of components
of aggregate private consumption the importance of the wealth effect of
spending shocks. [55], on the other hand, study the transmission mechanism for
different types of tax changes, finding that cuts in personal and corporate
income taxes both boost output via the investment channel, whereas only
personal income cuts drive aggregate demand via an expansion in private
consumption. Contrary to those qualitative approaches, we provide a logically
coherent methodology to define and quantitatively assess the strength of
transmission channels.

A related stream of literature uses impulse response analysis for studying
counterfactual questions [54, 67, 51, see e.g.]. While transmission channel
and counterfactual analysis superficially appear to be related, it is
important to highlight their fundamental difference. Investigating
counterfactual questions requires the researcher to specify a different model,
describing a different equilibrium, where specific behavioural changes result
in the counterfactual scenario. TCA, however, focuses on decomposing impulse
responses within a given equilibrium rather then understanding the difference
between responses across different equilibria. Therefore, TCA evaluates the
relative importance of specific variables in transmitting the total effect of
a particular identified economic shock, whereas counterfactual analysis
investigates total causal effects that such shocks would have, had the policy
response been different. As such, TCA complements counterfactual analysis;
indeed, TCA can be used within a counterfactual study to investigate
transmission channels across the different equilibria induced by the
counterfactual behavioural rules.

TCA also shares similarities with mediation analysis [49, 38, 43, 58, 48, see,
for example,] as it investigates causal effects going through intermediate
variables. In contrast to TCA, mediation analysis focuses on causal models
that do not exhibit any feedback mechanisms, limiting its usefulness for
studying transmission channels in dynamic general equilibrium macroeconomics.

The use of DAGs for transmission channel analysis is related to the graphical
causal analysis literature within computer science [57, see]. Contrary to that
literature, we do not require that the causal system - the set of behavioural
equations - can be cast in a DAG. Instead, we show that for a large class of
linear macroeconomic models, there always exists a DAG representation of the
equilibrium equations rather than of the behavioural equations. This DAG
representation, obtained using a QL-decomposition of the contemporaneous
matrix, is related to the orthogonal reduced-form parameterisation of [36].
Importantly, this allows us to cover non-recursive models which exhibit
contemporaneous feedback effects.

Finally, our work relates to the recent literature which uses potential
outcomes in macroeconomics to gain more insights into total causal effects.
[61] use potential outcomes to investigate the non-parametric meaning of
common macroeconomic causal estimands. [41] use potential outcomes to
investigate state dependent causal effects. Lastly, [35] use potential
outcomes to estimate the effect of monetary policy semi-parametrically.
Contrary to the aforementioned, our focus is on the decomposition of total
effects rather than on total effects themselves.

The paper is organised as follows. Section 2 contains an example of a simple
three-variable system illustrating the main ideas behind TCA. The general TCA
framework is then developed in Section 3. Section 4 contains our main
theoretical results, while Section 5 applies the framework empirically to
monetary policy evaluation. Section 6 concludes, while supplementary results
are contained in the appendix.

##  2 TCA: An Illustrative Example

We present the main idea and intuition behind TCA using a textbook version of
the three-equation New Keynesian model [46] including the output gap
$\mathrm{x}_{t}$, inflation $\mathrm{\pi}_{t}$ and nominal interest rates
$\mathrm{i}_{t}$. The model is given by

$\begin{split}\text{IS:
}\quad&\mathrm{x}_{t}=\mathbb{E}[x_{t+1}]+\alpha_{1}(\bm{\theta})[\mathrm{i}_{t}-\mathbb{E}[\pi_{t+1}]]+\varepsilon^{d}_{t}\\\
\text{PC:
}\quad&\mathrm{\pi}_{t}=\mathbb{E}[\pi_{t+1}]+\alpha_{2}(\bm{\theta})\mathrm{x}_{t}+\varepsilon^{s}_{t}\\\
\text{MR:
}\quad&\mathrm{i}_{t}=\alpha_{3}\mathrm{x}_{t}+\alpha_{4}\mathrm{\pi}_{t}+\varepsilon^{i}_{t},\end{split}$
(1)

and consists of an IS equation and a Phillips curve (PC) equation, together
summarising the equilibrium in the goods market, as well as a (Taylor-type)
rule specifying the central bank’s interest rate policy (MR). The coefficients
of the IS and PC equations, $\alpha_{1}$ and $\alpha_{2}$, depend on deep
structural parameters $\bm{\theta}$ that specify the behaviour of firms and
consumers in the economy. The coefficients of the interest rate rule,
$\alpha_{3}$ and $\alpha_{4}$, have a behavioural interpretation since they
explicitly specify the policy of the central bank. The set of deep structural
parameters in the model is thus
$\bm{\vartheta}=\\{\bm{\theta},\alpha_{3},\alpha_{4}\\}$. The demand, supply
and interest rate shocks are respectively given by $\varepsilon^{d}_{t}$,
$\varepsilon^{s}_{t}$ and $\varepsilon^{i}_{t}$.

For ease of exposition, we assume that the structural shocks
$\varepsilon^{d}_{t}$, $\varepsilon^{s}_{t}$ and $\varepsilon^{i}_{t}$ are
white noise and mutually uncorrelated. The equilibrium in (1) then exhibits
the static representation

$\underbrace{\begin{bmatrix}1&0&-\alpha_{1}(\bm{\theta})\\\
-\alpha_{2}(\bm{\theta})&1&0\\\
-\alpha_{3}&-\alpha_{4}&1\end{bmatrix}}_{\bm{A}}\underbrace{\begin{bmatrix}\mathrm{x}_{t}\\\
\mathrm{\pi}_{t}\\\
\mathrm{i}_{t}\end{bmatrix}}_{\bm{y}_{t}}=\underbrace{\begin{bmatrix}\varepsilon^{d}_{t}\\\
\varepsilon^{s}_{t}\\\
\varepsilon^{i}_{t}\end{bmatrix}}_{\bm{\varepsilon}_{t}}.$ (2)

A core task of macroeconomists is to assess dynamic causal effects of
structural shocks on relevant, endogenously determined economic variables.
Generally the focus is on total effects, i.e. total changes in equilibrium
quantities triggered by an economically interpretable exogenous event. Since
model (2) is static, total effects of structural shocks die out after a single
period, and impact effects of the three structural shocks are summarised by
the columns of the impulse response matrix

$\bm{\varPhi}(\bm{\vartheta})=\begin{bmatrix}\frac{1}{\eta(\bm{\vartheta})}&\frac{\alpha_{1}(\bm{\theta})\alpha_{4}}{\eta(\bm{\vartheta})}&\frac{\alpha_{1}(\bm{\theta})}{\eta(\bm{\vartheta})}\\\
\frac{\alpha_{2}(\bm{\theta})}{\eta(\bm{\vartheta})}&\frac{1-\alpha_{1}(\bm{\theta})\alpha_{3}}{\eta(\bm{\vartheta})}&\frac{\alpha_{1}(\bm{\theta})\alpha_{2}(\bm{\theta})}{\eta(\bm{\vartheta})}\\\
\frac{\alpha_{2}(\bm{\theta})\alpha_{4}+\alpha_{3}}{\eta(\bm{\vartheta})}&\frac{\alpha_{4}}{\eta(\bm{\vartheta})}&\frac{1}{\eta(\bm{\vartheta})}\\\
\end{bmatrix},$

where
$\eta(\bm{\vartheta})=-\alpha_{1}(\bm{\theta})\alpha_{2}(\bm{\theta})\alpha_{4}-\alpha_{1}(\bm{\theta})\alpha_{3}+1$.

The total effect (TE) of a demand shock on interest rates, quantitatively
given by

$\begin{split}\text{{(TE)}}&\quad\bm{\varPhi}(\bm{\vartheta})_{3,1}=(\alpha_{2}(\bm{\theta})\alpha_{4}+\alpha_{3})/\eta(\bm{\vartheta}),\end{split}$
(3)

can be decomposed into two transmission channels. The first channel, labelled
the indirect transmission channel, measures how much of the interest rate
response to a demand shock can be explained by the response of inflation to a
demand shock ceteris paribus, i.e. holding all other endogenous and exogenous
variables constant. The second channel, labelled the direct transmission
channel, is the remainder of the total effect that is not explained by the
indirect channel. It measures how much of the interest rate response to a
demand shock cannot be explained ceteris paribus by the response in inflation
to the demand shock. 111In the following, for brevity we suppress explicitly
mentioning the ceteris paribus condition, but it should be kept in mind that
this is intended when we discuss effects.

Throughout, we take the deep structural parameterisation $\bm{\vartheta}$ as
given. This is contrary to counterfactual analysis where changes in total
effects $\bm{\varPhi}(\bm{\vartheta})$ are analysed under changes in deep
structural parameters $\bm{\vartheta}$, hence under different equilibria. By
taking $\bm{\vartheta}$ and thus the dynamic equilibrium representation as
given, TCA focuses on explaining the effects within a specific equilibrium
rather then understanding effects across different equilibria. TCA is
therefore, by definition, not subject to the Lucas Critique. To keep notation
simple, we subsequently suppress the dependence of $\alpha_{1}$ and
$\alpha_{2}$ on $\bm{\theta}$. Furthermore, we first discuss TCA under
simplified equilibrium dynamics ($\alpha_{1}(\bm{\theta})=0$) in Section 2.1
before turning to the more general equilibrium dynamics
($\alpha_{1}(\bm{\theta})\neq 0$) in Section 2.2.

###  2.1 A Recursive Model

Consider model (2) with $\alpha_{1}=0$ which then implies that $\bm{A}$ is
lower-triangular; we label this the recursive (R) model. Equilibrium dynamics
of the output gap are fully determined by the demand shock and the demand
shock’s total effect on interest rates is given by
$\alpha_{2}\alpha_{4}+\alpha_{3}$.

In this model there exist two transmission channels that jointly explain the
total effect. First, the direct transmission channel consists of the effect of
the demand shock on the output gap which is directly carried forward to
interest rates. Second, the indirect transmission channel consists of the
effect of the demand shock on the output gap, which is carried forward to
inflation and in turn to interest rates.

TCA quantifies the effect that goes through each transmission channel, the
_transmission effect_. To simplify this task, we cast the analysis of
transmission channels in a graphical framework by connecting the model to an
associated Directed Acyclic Graph (DAG), as visualised in Figure 1 panel (R).
The three key ingredients of the graph are the nodes, edges, and path
coefficients. The nodes represent either the variables in the model (solid
black circles) or the shocks (shaded blue squares). The edges are directed and
represent the effects. Edges originating from shocks represent causal effects;
their path coefficient represents the direct causal effect size of the
structural shock on the destination variable. Edges originating from variables
we label as carrying effects since they carry the causal effect forward; their
path coefficients quantify the direct effect of a unit increase in the origin
variable (irrespective of what drove this increase) on the destination
variable.

$\mathrm{x}$$\mathrm{\pi}$$\mathrm{i}$$\alpha_{2}$$\alpha_{4}$$\alpha_{3}$$\varepsilon^{d}$$\varepsilon^{s}$$\varepsilon^{i}$$1$$1$$1$
(R)

$\mathrm{x}$$\mathrm{\pi}$$\mathrm{i}$$\alpha_{2}$$\alpha_{4}$$\alpha_{3}$$\varepsilon^{d}$$\varepsilon^{s}$$\varepsilon^{i}$$1$$1$$1$
(TE)

$\mathrm{x}$$\mathrm{\pi}$$\mathrm{i}$$\alpha_{2}$$\alpha_{4}$$\alpha_{3}$$\varepsilon^{d}$$\varepsilon^{d}$$\varepsilon^{s}$$\varepsilon^{i}$$1$$1$$1$
(DE)

$\mathrm{x}$$\mathrm{\pi}$$\mathrm{i}$$\alpha_{2}$$\alpha_{4}$$\alpha_{3}$$\varepsilon^{d}$$\varepsilon^{s}$$\varepsilon^{i}$$1$$1$$1$
(IE)

Figure 1: Illustrative recursive example (R). We analyse the total effect (TE)
of a demand shock on interest rates, which is decomposed into a direct (DE)
and indirect effect (IE) through inflation. Gray nodes and edges are
irrelevant for the effect shown in each panel.

We aim to decompose the total effect of the demand shock $\varepsilon_{t}^{d}$
on interest rates (TE, $\alpha_{2}\alpha_{4}+\alpha_{3}$) into _direct
effects_ (DE) and _indirect effects_ (IE) that go through the direct and
indirect channels respectively. The total effect
($\alpha_{2}\alpha_{4}+\alpha_{3}$) can be read of Figure 1 panel (TE) by
multiplying the path coefficients of the edges along each path from
$\varepsilon_{t}^{d}$ to $\mathrm{i}_{t}$ and then adding up the results
across all paths. The direct channel is visualised in Figure 1 panel (DE), the
indirect channel in panel (IE). Transmission effects can be read from the
graph in a similar way as the total effect. The transmission effects of the
direct and indirect channel are respectively given by

$\text{{(DE)}}\quad\alpha_{3},\qquad\text{{(IE)}}\quad\alpha_{2}\alpha_{4}.$
(4)

The transmission effects of the disjoint direct and indirect channels thus add
up to the total effect; thereby decomposing the total effect.

###  2.2 The Non-Recursive Model

Consider the general three-equation New Keynesian model (2) with
$\alpha_{1}\neq 0$. The contemporaneous matrix $\bm{A}$ is no-longer lower-
triangular; the model is now non-recursive (NR). Equilibrium dynamics of the
output gap are no-longer fully determined by the demand shock, as visualised
in Figure 2 panel (NR) by two incoming edges into $\mathrm{x}_{t}$. This
complicates the equilibrium dynamics and TCA cannot be performed directly on
this graph since the feedback loop (or cycle) between the output gap and
inflation in Figure 2 panel (NR) implies that no logically consistent
definition of a transmission channel exists.

$\mathrm{x}$$\mathrm{\pi}$$\mathrm{i}$$\varepsilon^{d}$$\varepsilon^{s}$$\varepsilon^{i}$
(NR)

$\mathrm{x}$$\mathrm{\pi}$$\mathrm{i}$$\varepsilon^{d}$$\varepsilon^{s}$$\varepsilon^{i}$
(NR’)

$\mathrm{x}$$\mathrm{\pi}$$\mathrm{i}$$\varepsilon^{d}$$\varepsilon^{s}$$\varepsilon^{i}$
(TE)

$\mathrm{x}$$\mathrm{\pi}$$\mathrm{i}$$\varepsilon^{d}$$\varepsilon^{s}$$\varepsilon^{i}$
(DE)

$\mathrm{x}$$\mathrm{\pi}$$\mathrm{i}$$\varepsilon^{d}$$\varepsilon^{s}$$\varepsilon^{i}$
(IE)

Figure 2:  Illustrative non-recursive example (NR) together with the re-
parameterised acyclic graph suitable for TCA (NR’). We analyse the total
effect (TE) of a demand shock on interest rates, which is decomposed into a
direct (DE) and indirect effect (IE) through inflation. Gray nodes and edges
are irrelevant for the effect shown in each panel.

While simply restricting $\alpha_{1}$ to zero would break the cycle in the
graph, it would also change the equilibrium since it would require an explicit
change in the deep structural parameters $\bm{\vartheta}$. Restricting
$\alpha_{1}=0$ would thus correspond to a counterfactual analysis, comparing
the total dynamic effect across two equilibria; one equilibrium under
$\alpha_{1}\neq 0$ and one under $\alpha_{1}=0$. TCA, in contrast, takes
$\bm{\vartheta}$ and thus the dynamic equilibrium representation as given. To
decompose the equilibrium dynamics leading to the total effects, TCA requires
model (2) to be re-parameterised into an equivalent equilibrium representation
that lends itself to logically consistent definitions of transmission channels
- a parameterisation that does not involve a feedback loop (cycle) and that
can then be represented by a DAG.

Since our focus is on the transmission of shocks in an equilibrium
representation, any equivalent equilibrium representation can be used that
permits the quantification of the transmission channels of interest. The re-
parameterisation step is therefore crucially determined by the research
question. To quantify the effect corresponding to the direct and indirect
transmission channel, we seek a representation of the form

$\bm{L}\begin{bmatrix}\mathrm{x}_{t}\\\ \mathrm{\pi}_{t}\\\
\mathrm{i}_{t}\end{bmatrix}=\bm{Q}^{\prime}\begin{bmatrix}\varepsilon^{d}_{t}\\\
\varepsilon^{s}_{t}\\\ \varepsilon^{i}_{t}\end{bmatrix},$ (5)

where equilibrium interest rates depend on inflation and the output gap; here
$\bm{L}$ is a lower-triangular matrix and $\bm{Q}$ is a rotation matrix. Model
(5) describes the same dynamic equilibrium as model (2) and is suitable to
perform TCA for the research question at hand, but the interpretation of
individual equations may be different and the re-parameterised relationships
among variables may no longer have a behavioural interpretation. For example,
the re-parameterised formulation of the “inflation equation” and the “interest
rate equation” may now include the demand shock; and the re-parameterised
relationship between the interest rate and inflation does not characterise the
central bank’s policy reaction anymore.

Representation (5) can be obtained using a QL-decomposition of the matrix
${\bm{A}}$ which specifies the contemporaneous relationships among the
endogenous variables. Multiplying both sides of (5) by $\bm{Q}$ and noting
that $\bm{Q}\bm{Q}^{\prime}=\bm{I}$, we obtain

$\bm{Q}\bm{L}\begin{bmatrix}\mathrm{x}_{t}\\\ \mathrm{\pi}_{t}\\\
\mathrm{i}_{t}\end{bmatrix}=\begin{bmatrix}\varepsilon^{d}_{t}\\\
\varepsilon^{s}_{t}\\\ \varepsilon^{i}_{t}\end{bmatrix}$

where $\bm{A}=\bm{Q}\bm{L}$ is unique given a variable ordering and assuming
that $\bm{A}$ is non-singular.

Model (5) can now be represented as a DAG, see Figure 2 panel (NR’) with path
coefficients omitted for clarity. Unlike to the DAG of Section 2.1, edges now
exist from all shocks to all variables. Focusing on the total effect of the
demand shock on interest rates in Figure 2 (TE), edges exist from the demand
shock to the output gap, as before, but also to inflation and interest rates;
their interpretation remains the same as in Section 2.1. There are two paths
from the demand shock to the output gap not going through inflation that form
the direct transmission channel in Figure 2 (DE) and two paths going through
inflation that together form the indirect channel in Figure 2 (IE). The
corresponding transmission effects can be computed as before and are
respectively given by

$\begin{split}\text{{(DE)}}&\quad(\alpha_{3}+\alpha_{1}(1-\eta))/((1+\alpha_{1}^{2})\eta)\\\
\text{{(IE)}}&\quad(\alpha_{2}\alpha_{4})/((1+\alpha_{1}^{2})\eta).\end{split}$
(6)

Finally, note that the transmission channels and effects obtained through
model (5) depend on the ordering of the variables in the QL-decomposition.
Importantly, this ordering is not needed to identify the structural shock of
interest and its meaning is therefore fundamentally different from the
ordering for the Cholesky decomposition in recursively identified structural
models. Instead, it is required to quantify the transmission channels of
interest, as determined entirely by the research question. Given its crucial
role for TCA, we encode this variable ordering in a permutation matrix
$\bm{T}$ which we label the transmission matrix.

Figure 3 gives all possible permutations of the transmission matrix in the
example of this section. With $\bm{A}^{*}=\bm{A}\bm{T}^{\prime}$ and
$\bm{y}_{t}^{*}=\bm{T}\bm{y}_{t}$, the QL-decomposition of $\bm{A}^{*}$
results in six equivalent equilibrium representations, whose DAG
representations are visualised in panels (a) to (f). The choice of
transmission matrix determines the possible paths through the DAG and
therefore transmission channels up for analysis. To answer our research
question we need an equilibrium representation in which the interest rate
depends on inflation and the output gap. This holds for the transmission
matrices $\bm{T}$ in panels (a) and (c), thereby directly ruling out the
choice of $\bm{T}$ in the other panels. By ordering the output gap before
inflation as in panel (a), we also enforce that inflation is dependent on the
output gap, making it the only natural and logical choice to quantify our two
transmission effects of interest.

$\mathrm{x}$$\mathrm{\pi}$$\mathrm{i}$$\varepsilon^{d}$$\varepsilon^{s}$$\varepsilon^{i}$$\bm{T}=\begin{bmatrix}1&0&0\\\
0&1&0\\\ 0&0&1\end{bmatrix}$ (a)

$\mathrm{x}$$\mathrm{i}$$\mathrm{\pi}$$\varepsilon^{d}$$\varepsilon^{s}$$\varepsilon^{i}$$\bm{T}=\begin{bmatrix}1&0&0\\\
0&0&1\\\ 0&1&0\end{bmatrix}$ (b)

$\mathrm{\pi}$$\mathrm{x}$$\mathrm{i}$$\varepsilon^{d}$$\varepsilon^{s}$$\varepsilon^{i}$$\bm{T}=\begin{bmatrix}0&1&0\\\
1&0&0\\\ 0&0&1\end{bmatrix}$ (c)

$\mathrm{\pi}$$\mathrm{i}$$\mathrm{x}$$\varepsilon^{d}$$\varepsilon^{s}$$\varepsilon^{i}$$\bm{T}=\begin{bmatrix}0&1&0\\\
0&0&1\\\ 1&0&0\end{bmatrix}$ (d)

$\mathrm{i}$$\mathrm{\pi}$$\mathrm{x}$$\varepsilon^{d}$$\varepsilon^{s}$$\varepsilon^{i}$$\bm{T}=\begin{bmatrix}0&0&1\\\
0&1&0\\\ 1&0&0\end{bmatrix}$ (e)

$\mathrm{i}$$\mathrm{x}$$\mathrm{\pi}$$\varepsilon^{d}$$\varepsilon^{s}$$\varepsilon^{i}$$\bm{T}=\begin{bmatrix}0&0&1\\\
1&0&0\\\ 0&1&0\end{bmatrix}$ (f)

Figure 3: Panels (a) through (f) show all possible permutations of the three
variables $\mathrm{x}_{t}$, $\mathrm{\pi}_{t}$ and $\mathrm{i}_{t}$ as encoded
in the transmission matrix $\bm{T}$. The associated DAG is shown above the
matrix where gray nodes and edges are irrelevant for the transmission of a
demand shock.

##  3 The General Framework

In this section we formalise the intuition developed in the previous section.
Section 3.1 introduces the general dynamic model we consider together with its
impulse-response representation we use to compute total effects. Section 3.2
then introduces the associated DAG which we use to decompose the total effect
into transmission channels, and formally defines transmission channels and
transmission effects.

###  3.1 Dynamic Models and Total Effects

For our general setup, we assume $\bm{y}_{t}$ to be a $K$-dimensional
stationary stochastic process with a structural Vector Autoregressive Moving
Average (VARMA) representation given by

$\bm{A}_{0}\bm{y}_{t}=\sum_{i=1}^{\ell}\bm{A}_{i}\bm{y}_{t-i}+\sum_{j=1}^{q}\bm{\varPsi}_{j}\bm{\varepsilon}_{t-j}+\bm{\varepsilon}_{t},$
(7)

where $\\{\bm{A}_{i}\\}_{i=1}^{\ell}$ and $\\{\bm{\varPsi}_{j}\\}_{j=1}^{q}$
are $K\times K$ coefficient matrices which are statistically identified using
any common scheme such as the echelon form (e.g., [60]), $\bm{A}_{0}$ is a
contemporaneous coefficient matrix assumed to be (partially) identified using
some economic identification scheme, and $\bm{\varepsilon}_{t}$ is a $K\times
1$ vector of white noise.

The structural VARMA model (7) encompasses a wide range of popular dynamic
models used in macroeconomic research. The SVAR is recovered by setting
$\bm{\varPsi}_{j}=\bf 0$ (for $j=1,\ldots,q$), while linearised DSGE models
can be represented as structural VARMAs with cross-equation restrictions [65,
45].

We make two standard assumptions on $\bm{\varepsilon}_{t}$ and $\bm{A}_{0}$.

######  Assumption 1.

The white noise vector $\bm{\varepsilon}_{t}$ consists of $K$ structural
shocks satisfying
$\operatorname{\mathbb{E}}[\bm{\varepsilon}_{t}]=\bm{0}_{K}$,
$\operatorname{\mathbb{E}}[\bm{\varepsilon}_{t}\bm{\varepsilon}_{t}^{\prime}]=\bm{I}_{K}$
and
$\operatorname{\mathbb{E}}[\bm{\varepsilon}_{t}\bm{\varepsilon}_{t-r}^{\prime}]=\bm{O}_{K}$
for all $r\geq 1$.

######  Assumption 2.

$\bm{A}_{0}$ is non-singular.

Assumption 1 assures that impulse responses with respect to the white noise
disturbances $\bm{\varepsilon}_{t}$ have a causal interpretation. Assumption 2
then assures that these causally interpretable impulse responses can be
obtained from reduced-form impulse response functions. Section 4.2 uses this
last implication to show how transmission effects can be obtained from a
single identified shock. Hence, $\bm{A}_{0}$ may be only partially identified.

While all equivalent equilibrium representations can be used to analyse total
effects, their decomposition into transmission channels requires a precise
definition of which variables depend on which other variables in equilibrium.
TCA thus involves choosing the ordering of variables using the transmission
matrix $\bm{T}$.

######  Definition 1.

The _transmission matrix_ $\bm{T}$ denotes the $K\times K$ permutation matrix
defining the ceteris paribus dependencies of the $K$ variables in the chosen
equilibrium representation; it is fixed and specified by the researcher.

Having a fixed and completely specified transmission matrix assures a fixed
equilibrium representation. However, Section 4.2 shows that, under certain
conditions when two distinct transmission matrices lead to the same
transmission effect, the transmission matrix only needs to be partially
specified, so only a partial variable ordering is required in such case.

Under the conditions formulated above, the QL-decomposition of
$\bm{A}_{0}^{*}=\bm{A}_{0}\bm{T}^{\prime}$ is unique, implying that the
structural VARMA (7) can be written in unique lower-triangular form

$\bm{L}\bm{y}^{*}_{t}=\sum_{i=1}^{\ell}\bm{Q}^{\prime}\bm{A}_{i}^{*}\bm{y}^{*}_{t-i}+\sum_{j=1}^{q}\bm{Q}^{\prime}\bm{\varPsi}_{j}\bm{\varepsilon}_{t-j}+\bm{Q}^{\prime}\bm{\varepsilon}_{t},$
(8)

where $\bm{L}$ is a $K\times K$ lower-triangular matrix,
$\bm{y}^{*}_{t}=\bm{T}\bm{y}_{t}$, $\bm{A}^{*}_{i}=\bm{A}_{i}\bm{T}^{\prime}$
for $i=1,\dots,\ell$ and $\bm{Q}$ is a $K\times K$ rotation matrix.

Throughout the paper, we focus on transmission channels of the total effect of
shock $\varepsilon_{i,t}$ to $y^{*}_{j,t+h}$ over the fixed finite impulse-
response horizon $h$. Then the lower-triangular form (8) can be re-written in
systems form, capturing all dynamics of the shock $\varepsilon_{i,t}$ up to
horizon $h$. Defining
$\bm{x}=(\bm{y}_{t}^{*^{\prime}},\dots,\bm{y}_{t+h}^{*^{\prime}})^{\prime}$
and
$\bm{\varepsilon}=(\bm{\varepsilon}_{t}^{\prime},\dots,\bm{\varepsilon}_{t+h}^{\prime})^{\prime}$,
the systems form is given by

$\bm{x}=\bm{B}\bm{x}+\bm{\varOmega}\bm{\varepsilon},$ (9)

where $\bm{B}$ is lower-triangular with zeros on the diagonal,
$\bm{\varOmega}$ is lower-block-triangular, and both are respectively given
by222Note that $\bm{I}=\bm{I}_{K}$ and $\bm{O}=\bm{O}_{K}$. For expositional
simplicity, we henceforth omit the dimension subscript.

$\begin{array}[]{ccc}\bm{B}=\begin{bmatrix}\bm{I}-\bm{D}\bm{L}&\bm{O}&\dots&\bm{O}\\\
\bm{D}\bm{Q}^{\prime}\bm{A}^{*}_{1}&\bm{I}-\bm{D}\bm{L}&\dots&\bm{O}\\\
\vdots&\ddots&\ddots&\vdots\\\
\bm{D}\bm{Q}^{\prime}\bm{A}^{*}_{h}&\dots&\bm{D}\bm{Q}^{\prime}\bm{A}^{*}_{1}&\bm{I}-\bm{D}\bm{L}\end{bmatrix},&&\bm{\varOmega}=\begin{bmatrix}\bm{D}\bm{Q}^{\prime}&\bm{O}&\dots&\bm{O}\\\
\bm{D}\bm{Q}^{\prime}\bm{\varPsi}_{1}&\bm{D}\bm{Q}^{\prime}&\dots&\bm{O}\\\
\vdots&\ddots&\ddots&\vdots\\\
\bm{D}\bm{Q}^{\prime}\bm{\varPsi}_{h}&\dots&\bm{D}\bm{Q}^{\prime}\bm{\varPsi}_{1}&\bm{D}\bm{Q}^{\prime}\end{bmatrix},\end{array}$

(10)

with $\bm{D}=\text{diag}(\bm{L})^{-1}$, where $\text{diag}(\bm{X})$ is a
diagonal matrix of the diagonal of $\bm{X}$, $\bm{A}^{*}_{i}=\bm{O}$ for
$i>\ell$ and $\bm{\varPsi}_{j}=\bm{O}$ for $j>q$; see Appendix A for further
details.

The total effects can then be obtained from the impulse-response
representation

$\bm{x}=\bm{\varPhi}\bm{\varepsilon},\qquad\bm{\varPhi}=(\bm{I}-\bm{B})^{-1}\bm{\varOmega}.$
(11)

Under our assumptions the impulse response matrix $\bm{\varPhi}$ always
exists. The aim of TCA is to decompose these total effects into transmission
channels. However, before turning to the graphical representation used for
this decomposition, we first revisit the examples of Section 2 in our general
framework.

######  Example 1.

For the recursive model of Section 2.1, $\bm{T}=\bm{I}$,
$\bm{B}=\bm{I}-\bm{D}\bm{A}$ and $\bm{\varOmega}=\bm{D}$ with
$\bm{D}=\text{diag}(\bm{A})^{-1}=\bm{I}$. The total effect of
$\varepsilon^{d}_{t}$ on $\mathrm{i}_{t}$ is therefore
$\alpha_{2}\alpha_{4}+\alpha_{3}$.

######  Example 2.

For the non-recursive model of Section 2.1with transmission matrix
$\bm{T}=\bm{I}$,

$\begin{array}[]{ccc}\bm{B}=\begin{bmatrix}0&0&0\\\
\frac{\left(\alpha_{1}^{2}+1\right)\alpha_{2}+\alpha_{1}\alpha_{4}(1-\alpha_{1}\alpha_{3})}{\alpha_{1}^{2}\left(\alpha_{4}^{2}+1\right)+1}&0&0\\\
\frac{\alpha_{1}+\alpha_{3}}{\alpha_{1}^{2}+1}&\frac{\alpha_{4}}{\alpha_{1}^{2}+1}&0\\\
\end{bmatrix}&\text{and}&\bm{\varOmega}=\begin{bmatrix}\frac{1}{1-\alpha_{1}(\alpha_{2}\alpha_{4}+\alpha_{3})}&\frac{\alpha_{1}\alpha_{4}}{1-\alpha_{1}(\alpha_{2}\alpha_{4}+\alpha_{3})}&\frac{\alpha_{1}}{1-\alpha_{1}(\alpha_{2}\alpha_{4}+\alpha_{3})}\\\
-\frac{\alpha_{1}\alpha_{4}}{\alpha_{1}^{2}\left(\alpha_{4}^{2}+1\right)+1}&\frac{\alpha_{1}^{2}+1}{\alpha_{1}^{2}\left(\alpha_{4}^{2}+1\right)+1}&-\frac{\alpha_{1}^{2}\alpha_{4}}{\alpha_{1}^{2}\left(\alpha_{4}^{2}+1\right)+1}\\\
-\frac{\alpha_{1}}{\alpha_{1}^{2}+1}&0&\frac{1}{\alpha_{1}^{2}+1}\\\
\end{bmatrix}.\end{array}$

The total effect of $\varepsilon^{d}_{t}$ on $\mathrm{i}_{t}$ is therefore
$(\alpha_{2}\alpha_{4}+\alpha_{3})/\eta$, where
$\eta=-\alpha_{1}\alpha_{2}\alpha_{4}-\alpha_{1}\alpha_{3}+1$.

###  3.2 Dynamic Graphs and Transmission Channels

To decompose the total effects given by $\bm{\varPhi}$ in equation (11) into
transmission channels, we introduce the graph
$\mathcal{G}(\bm{B},\bm{\varOmega})$ associated with system (9). The graph
$\mathcal{G}(\bm{B},\bm{\varOmega})$ is a DAG and has three key ingredients:
nodes, directed edges and path coefficients. A node exists for each shock
$\varepsilon_{i,r}$ at time $r=t$ and for each variable $y_{i,r}$ at time
points $r=t,\dots,t+h$, where $h$ is the fixed horizon. Let
$\bm{x}=(\bm{y}_{t}^{*^{\prime}},\dots,\bm{y}_{t+h}^{*^{\prime}})^{\prime}$
and
$\bm{\varepsilon}=(\bm{\varepsilon}_{t}^{\prime},\dots,\bm{\varepsilon}_{t+h}^{\prime})^{\prime}$
as in equation (9). A directed edge $x_{i}\to x_{j}$ then exists between
variable $x_{i}$ and variable $x_{j}$ whenever $\bm{B}_{j,i}\neq 0$, where
$\bm{B}_{j,i}$ denotes the $(j,i)$ element of $\bm{B}$. The path coefficient
of this edge is given by $\omega_{x_{i},x_{j}}=\bm{B}_{j,i}$ and is
interpreted as the direct effect of a unit increase in $x_{i}$ (irrespective
of the cause) on $x_{j}$. Similarly, a directed edge $\varepsilon_{i}\to
x_{j}$ exists from the structural shock $\varepsilon_{i}$ to the variable
$x_{j}$ whenever $\bm{\varOmega}_{j,i}\neq 0$. The path coefficient of this
edge is given by $\omega_{\varepsilon_{i},x_{j}}=\bm{\varOmega}_{j,i}$ and is
similarly interpreted as the direct causal effect of a unit increase in
$\varepsilon_{i}$ on $x_{j}$.

######  Example 3.

Consider the trivariate SVAR(1) given by

$\bm{A}_{0}\bm{y}_{t}=\bm{A}_{1}\bm{y}_{t-1}+\bm{\varepsilon}_{t},$

with $\bm{A}_{0}$ and $\bm{A}_{1}$ being $3\times 3$ coefficient matrices,
$\bm{y}_{t}=(y_{1,t},y_{2,t},y_{3,t})^{\prime}$ and
$\bm{\varepsilon}_{t}=(\varepsilon_{1,t},\varepsilon_{2,t},\varepsilon_{3,t})^{\prime}$.
Fixing the horizon $h=1$ and the transmission matrix $\bm{T}=\bm{I}$,

$\begin{split}\bm{x}&=(y_{1,t},y_{2,t},y_{3,t},y_{1,t+1},y_{2,t+1},y_{3,t+1})^{\prime}\\\
\bm{\varepsilon}&=(\varepsilon_{1,t},\varepsilon_{2,t},\varepsilon_{3,t},\varepsilon_{1,t+1},\varepsilon_{2,t+1},\varepsilon_{3,t+1})^{\prime},\end{split}$

and the corresponding graph $\mathcal{G}(\bm{B},\bm{\varOmega})$ is shown in
Figure 4(a). Due to the dynamic nature of the SVAR(1), an edge exists from
each component of $\bm{y}_{t}$ (i.e. $x_{1}$ to $x_{3}$ in the graph) to each
component of $\bm{y}_{t+1}$ (i.e. $x_{4}$ to $x_{6}$ in the graph).

######  Example 4.

Assume, instead of a trivariate SVAR(1), a trivariate SVARMA(1, 1)

$\bm{A}_{0}\bm{y}_{t}=\bm{A}_{1}\bm{y}_{t-1}+\bm{\varepsilon}_{t}+\bm{\varPsi}_{1}\bm{\varepsilon}_{t-1},$

with $\bm{A}_{0}$, $\bm{A}_{1}$ and $\bm{\varPsi}_{1}$ being $3\times 3$
coefficient matrices, $\bm{y}_{t}=(y_{1,t},y_{2,t},y_{3,t})^{\prime}$ and
$\bm{\varepsilon}_{t}=(\varepsilon_{1,t},\varepsilon_{2,t},\varepsilon_{3,t})^{\prime}$.
Fixing the horizon $h=1$ and the transmission matrix $\bm{T}=\bm{I}$, $\bm{x}$
and $\bm{\varepsilon}$ are the same as in Example 3, but the graph is given in
Figure 4(b). Similar to the graph corresponding to the SVAR(1), edges exist
from each component of $\bm{y}_{t}$ to each component of $\bm{y}_{t+1}$.
Unlike the SVAR case, edges from structural shocks can now also skip time
periods, thereby going from each component of $\bm{\varepsilon}_{t}$ to each
component of $\bm{y}_{t+1}$. One such edge is given by $\varepsilon_{1}\to
x_{4}$, which is still interpreted as a direct causal effect of a unit
increase in $\varepsilon_{1}$ on $x_{4}$.

$x_{1}$$x_{2}$$x_{3}$$x_{4}$$x_{5}$$x_{6}$$\varepsilon_{1}$$\varepsilon_{2}$$\varepsilon_{3}$
(a)

$x_{1}$$x_{2}$$x_{3}$$x_{4}$$x_{5}$$x_{6}$$\varepsilon_{1}$$\varepsilon_{2}$$\varepsilon_{3}$
(b)

Figure 4: Panels (a) and (b) show the DAG corresponding to a SVAR(1) and a
SVARMA(1, 1) respectively, where
$\bm{x}=(y_{1,t},y_{2,t},y_{3,t},y_{1,t+1},y_{2,t+1},y_{3,t+1})^{\prime}$.
Compared to panel (a), panel (b) has edges from all shocks
$\varepsilon_{\cdot,t}$ to all variables in period $t+1$, $y_{\cdot,t+1}$,
where $\cdot$ is used as a placeholder.

Since each edge describes a direct effect, paths – consisting of chained edges
– can be interpreted as descriptions of causal flows. The collection
$\mathcal{P}_{\varepsilon_{i},x_{j}}$ of all paths connecting the structural
shock $\varepsilon_{i}$ to the variable $x_{j}$ is thus a set of causal
descriptions; so is any subset $P_{\varepsilon_{i},x_{j}}$ of this collection.
We are now ready to define a transmission channel.

######  Definition 2.

Let $\mathcal{G}(\bm{B},\bm{\varOmega})$ be the graph induced by model (9). A
subset of paths from $\varepsilon_{i}$ to $x_{j}$,
$P_{\varepsilon_{i},x_{j}}\subset\mathcal{P}_{\varepsilon_{i},x_{j}}$, is
called a _transmission channel_.

Through TCA we quantify transmission effects as the effects flowing through
transmission channels. To this end, we define the effects along a specific
path $p$ and the effects along a collection of paths
$P_{\varepsilon_{i},x_{j}}$, the latter of which leads to the definition of
the transmission effect.

######  Definition 3.

Let $\mathcal{G}(\bm{B},\bm{\varOmega})$ be a graph induced by model (9),
$\mathcal{P}_{\varepsilon_{i},x_{j}}$ be the collection of all paths in
$\mathcal{G}(\bm{B},\bm{\varOmega})$ from $\varepsilon_{i}$ to $x_{j}$, and
$\xi$ be the shock size.

  1. 1.

The _path-specific effect_ of a path $p\in\mathcal{P}_{\varepsilon_{i},x_{j}}$
is

$\mathcal{Q}_{\xi}(p)=\xi\sum_{u\to v\in p}\omega_{u,v},$

where $\omega_{u,v}$ is the path coefficient of the edge going from node $u$
to node $v$ on path $p$.

  2. 2.

The _total path-specific effect_ of a sub-collection of paths
$P_{\varepsilon_{i},x_{j}}\subseteq\mathcal{P}_{\varepsilon_{i},x_{j}}$ is

$\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}})=\sum_{p\in
P_{\varepsilon_{i},x_{j}}}\mathcal{Q}_{\xi}(p).$

  3. 3.

The _transmission effect_ of the transmission channel
$P_{\varepsilon_{i},x_{j}}\subset\mathcal{P}_{\varepsilon_{i},x_{j}}$ is the
total path-specific effect of $P_{\varepsilon_{i},x_{j}}$.

######  Example 1 (continued).

The graph $\mathcal{G}(\bm{B},\bm{\varOmega})$ corresponding to the recursive
model of Section 2.1 is given in Figure 1 panel (R). Due to its recursive
nature, all shocks have only a single directed edge to one variable. The
indirect effect of $\varepsilon^{d}_{t}$ through $\mathrm{\pi}_{t}$ onto
$\mathrm{i}_{t}$, in Figure 1 panel (IE), is given by the set of paths
$P_{\varepsilon^{d}_{t},\mathrm{i}_{t}}=\\{\varepsilon^{d}_{t}\to\mathrm{x}_{t}\to\mathrm{\pi}_{t}\to\mathrm{i}_{t}\\}$
with total path-specific effect
$\mathcal{Q}_{1}(P_{\varepsilon^{d}_{t},\mathrm{i}_{t}})=\alpha_{2}\alpha_{4}$.
This is the transmission effect of the indirect transmission channel.

######  Example 2 (continued).

The graph $\mathcal{G}(\bm{B},\bm{\varOmega})$ corresponding to the non-
recursive model of Section 2.2 is given by Figure 2 panel (NR’). Due to its
non-recursive nature, edges now exist from structural shocks to several
variables. These direct edges are interpreted as direct causal effects of a
unit increase in $\varepsilon^{d}_{t}$. The indirect effect of
$\varepsilon^{d}_{t}$ through $\mathrm{\pi}_{t}$ on $\mathrm{i}_{t}$, in
Figure 2 panel (IE), is now given by
$P_{\varepsilon^{d}_{t},\mathrm{i}_{t}}=\\{\varepsilon^{d}_{t}\to\mathrm{x}_{t}\to\mathrm{\pi}_{t}\to\mathrm{i}_{t},\varepsilon^{d}_{t}\to\mathrm{\pi}_{t}\to\mathrm{i}_{t}\\}$.
The total path-specific effect of $P_{\varepsilon^{d}_{t},\mathrm{i}_{t}}$,
and thus the transmission effect of the indirect channel, is given by
$\mathcal{Q}_{1}(P_{\varepsilon^{d}_{t},\mathrm{i}_{t}})=(\alpha_{2}\alpha_{4})/((1+\alpha_{1}^{2})\eta)$.

Given the formal definition of TCA in dynamic models (Definitions 2 and 3) for
system (9) and its associated graph $\mathcal{G}(\bm{B},\bm{\varOmega})$, we
now provide some intuition for our three main results that we formally prove
in Section 4.2. For ease of exposition, consider a transmission channel
$P_{\varepsilon_{i},x_{j}}$ of $\varepsilon_{i}$ on $x_{j}$ through $x_{k}$
consisting of all paths starting at $\varepsilon_{i}$, going to $x_{k}$ and
ending in $x_{j}$. First, note that the total effect of $\varepsilon_{i}$ on
$x_{j}$ equals the total path-specific effect of
$\mathcal{P}_{\varepsilon_{i},x_{j}}$. This ensures the coherence of our
framework.

Second, transmission channels of the total effect of $\varepsilon_{i}$ only
require $\varepsilon_{i}$ itself to be structurally identified. Edges leaving
the structural shock $\varepsilon_{i}$ are causal effects of
$\varepsilon_{i}$, thereby requiring structural identification. Edges leaving
any intermediate variable $x_{m}$ on a path $p\in P_{\varepsilon_{i},x_{j}}$,
however, are effects of a unit increase in $x_{m}$, irrespective of the cause.
Therefore, they do not require structural identification. Thus, only the
single edge on any path $p\in P_{\varepsilon_{i},x_{j}}$ originating from
$\varepsilon_{i}$ requires structural identification.

Third, all transmission effects of transmission channels decomposing the total
effect of $\varepsilon_{i}$ on $x_{j}$ can be calculated by combining the
structural impulse responses of $\varepsilon_{i}$ together with non-structural
impulse responses; thus, impulse responses are sufficient for the calculation
of any transmission effect. All paths $p$ corresponding to the transmission
channel $P_{\varepsilon_{i},x_{j}}$ go through the intermediate variable
$x_{k}$, through which the transmission channel $P_{\varepsilon_{i},x_{j}}$
can be split. The first part consists of all paths connecting
$\varepsilon_{i}$ with $x_{k}$, while the second part consists of all paths
connecting $x_{k}$ to $x_{j}$. The first part is the total effect of
$\varepsilon_{i}$ on $x_{k}$ obtained through the structural impulse response
$\bm{\varPhi}_{k,i}$. The second part corresponds to the total effect of a
unit increase in variable $x_{k}$ on $x_{j}$; in other words, the reduced-form
impulse response. In the next section, we prove these three main findings
making use of the potential outcomes framework.

##  4 Main Results

In this section, we establish four main results on TCA. To this end, we use
the potential outcomes framework as a convenient mathematical tool. In Section
4.1, we introduce potential outcomes for TCA and show its equivalence with the
graphical definitions and representation of TCA. In Section 4.2, we then show
that (i) the total effect of $\varepsilon_{i}$ on $x_{j}$ is the total path-
specific effect of $\mathcal{P}_{\varepsilon_{i},x_{j}}$, (ii) IRFs are
sufficient statistics to compute transmission effects since structural IRFs
can be combined with IRFs obtained using a Cholesky decomposition to obtain
any transmission effect, and (iii) only the shock of interest
$\varepsilon_{i}$ needs to be structurally identified for transmission effects
to be identified. The proofs of all results are provided in Appendix B.

###  4.1 Transmission Effects as Potential Outcomes

We introduce potential outcomes for TCA by adapting notation used in the
mediation analysis literature [43]. We show that the potential outcomes
framework and the more intuitive, graphical framework for TCA used in Section
3 are equivalent. Throughout this section, we consider the effects from the
structural shock $\varepsilon_{i}$ to the variable $x_{j}$.

####  4.1.1 Notation

Recall system (9), $\bm{x}=\bm{B}\bm{x}+\bm{\varOmega}\bm{\varepsilon}$. Since
$\bm{B}$ is lower-triangular with zeros on the diagonal, $x_{j}$ directly
depends on the realisations of $x_{i}$ for $i<j$, which, in turn, directly
depends on the realisations of $x_{k}$ for $k<i$. All variables may,
additionally, depend on realisations of the shock of interest
$\varepsilon_{i}$. However, shocks themselves are independent of all other
variables and shocks. Thus, chains can be traced back from the variable of
interest $x_{j}$ to the structural shock $\varepsilon_{i}$. Potential outcomes
are statements about these chains, with the potential outcome for $x_{j}$
taking the form

$x_{j}(x_{1}(\epsilon_{1}),x_{2}(x_{1}(\epsilon_{2}),\epsilon_{3}),\dots)=x_{j}^{*}(\epsilon_{1},\epsilon_{2},\epsilon_{3},\dots,\epsilon_{\eta(j)})=x_{j}(\bm{\epsilon}^{(j)}),$

where $\bm{\epsilon}^{(j)}$ is the potential outcomes assignment vector. The
elements of this vector, $\epsilon_{l}$ for $l=1\dots\eta(j)$, are the
realisations of the structural shock experienced by the $l$th nested chain,
where $\eta(j)=2^{j-1}$ is the number of chains moving back from $x_{j}$ to
$\varepsilon_{i}$.

######  Example 2 (continued).

Consider the non-recursive model in equation (2) with associated graph in
Figure 5 panel (a), where all except the demand shock have been omitted for
clarity. The first step to introducing potential outcomes is to bring the
model into general notation. Thus, $\varepsilon_{1}=\varepsilon_{t}^{d}$,
$x_{1}=\mathrm{x}_{t}$ , $x_{2}=\mathrm{\pi}_{t}$ and $x_{3}=\mathrm{i}_{t}$,
see Figure 5 panel (a’). The potential outcome for the interest rate $x_{3}$
is then given by

$x_{3}(x_{1}(\epsilon_{1}),x_{2}(x_{1}(\epsilon_{2}),\epsilon_{3}),\epsilon_{4})=x_{3}^{*}(\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4})=x_{3}^{*}(\bm{\epsilon}^{(3)}),$

where $\epsilon_{l}$ ($l=1,\ldots,4$) are the realisations of the structural
shock $\varepsilon_{1}$. Note that the different realisations of the
structural shock do not have to be the same.

Figure 5 panels (b) to (e) depict the nested chains corresponding to the shock
realisations $\epsilon_{1}$ to $\epsilon_{4}$ respectively. Panel (b) shows
that $x_{3}$ depends on the realisation of $x_{1}$, which, in turn, depends on
a realisation of the structural shock $\varepsilon_{1}$, denoted by
$\epsilon_{1}$. Panels (c) and (d) both start from the dependence of $x_{3}$
on the realisation of $x_{2}$. Panel (c) then considers the chain where
$x_{2}$ depends on the realisation of $x_{1}$, which in turn depends on the
realisation of the structural shock $\epsilon_{2}$. Panel (d) considers the
chain where $x_{2}$ depends on the realisation of the structural shock
$\epsilon_{3}$. Lastly, panel (e) shows that $x_{3}$ directly depends on the
realisation of the structural shock $\epsilon_{4}$.

$\mathrm{x}$$\mathrm{\pi}$$\mathrm{i}$$\varepsilon^{d}$ (a)

$x_{1}$$x_{2}$$x_{3}$$\varepsilon_{1}$ (a’)

$x_{1}$$x_{2}$$x_{3}$$\varepsilon_{1}$$x_{3}(x_{1}(\epsilon_{1}),{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}\pgfsys@color@gray@stroke{0.7}\pgfsys@color@gray@fill{0.7}x_{2}(x_{1}(\epsilon_{2}),\epsilon_{3}),\epsilon_{4}})$$=x_{3}^{*}(\epsilon_{1},{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}\pgfsys@color@gray@stroke{0.7}\pgfsys@color@gray@fill{0.7}\epsilon_{2},\epsilon_{3},\epsilon_{4}})$
(b)

$x_{1}$$x_{2}$$x_{3}$$\varepsilon_{1}$$x_{3}({\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}\pgfsys@color@gray@stroke{0.7}\pgfsys@color@gray@fill{0.7}x_{1}(\epsilon_{1})},x_{2}(x_{1}(\epsilon_{2}){\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}\pgfsys@color@gray@stroke{0.7}\pgfsys@color@gray@fill{0.7},\epsilon_{3}}){\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}\pgfsys@color@gray@stroke{0.7}\pgfsys@color@gray@fill{0.7},\epsilon_{4}})$$=x_{3}^{*}({\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}\pgfsys@color@gray@stroke{0.7}\pgfsys@color@gray@fill{0.7}\epsilon_{1},}\epsilon_{2}{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}\pgfsys@color@gray@stroke{0.7}\pgfsys@color@gray@fill{0.7},\epsilon_{3},\epsilon_{4}})$
(c)

$x_{1}$$x_{2}$$x_{3}$$\varepsilon_{1}$$x_{3}({\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}\pgfsys@color@gray@stroke{0.7}\pgfsys@color@gray@fill{0.7}x_{1}(\epsilon_{1})},x_{2}({\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}\pgfsys@color@gray@stroke{0.7}\pgfsys@color@gray@fill{0.7}x_{1}(\epsilon_{2}),}\epsilon_{3}){\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}\pgfsys@color@gray@stroke{0.7}\pgfsys@color@gray@fill{0.7},\epsilon_{4}})$$=x_{3}^{*}({\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}\pgfsys@color@gray@stroke{0.7}\pgfsys@color@gray@fill{0.7}\epsilon_{1},\epsilon_{2}},\epsilon_{3}{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}\pgfsys@color@gray@stroke{0.7}\pgfsys@color@gray@fill{0.7},\epsilon_{4}})$
(d)

$x_{1}$$x_{2}$$x_{3}$$\varepsilon_{1}$$x_{3}({\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}\pgfsys@color@gray@stroke{0.7}\pgfsys@color@gray@fill{0.7}x_{1}(\epsilon_{1}),x_{2}(x_{1}(\epsilon_{2}),\epsilon_{3}),}\epsilon_{4})$$=x_{3}^{*}({\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}\pgfsys@color@gray@stroke{0.7}\pgfsys@color@gray@fill{0.7}\epsilon_{1},\epsilon_{2},\epsilon_{3},}\epsilon_{4})$
(e)

Figure 5: Panel (a) shows the DAG of the non-recursive model in Example 2 with
all except the demand shock omitted, while panel (a’) shows the same DAG but
in general notation. Panels (b) through (e) show, on top, the paths (in black)
corresponding to each element in the potential outcome and its assignment
vector (highlighted in black below the graph).

As evident from the example, keeping track of the various dependencies and the
corresponding shock realisations in the potential outcomes assignment vector
$\bm{\epsilon}^{(j)}$ quickly leads to notational clutter. For notational
convenience, we therefore introduce additional indexing notation to keep track
of elements in the assignment vector that correspond to dependencies involving
certain intermediate variables. Defining $H(j)=2^{j}-1$, the entries
$H(k_{1}-1)+1$ to $H(k_{1})$ in the assignment vector $\bm{\epsilon}^{(j)}$
correspond to the dependencies of the variable of interest $x_{j}$ on variable
$x_{k_{1}}$, which we jointly collect in $\bm{\epsilon}^{(j\cdot
k_{1})}=(\epsilon^{(j)}_{H(k_{1}-1)+1},...,\epsilon^{(j)}_{H(k_{1})})$. In the
subset of dependencies of $x_{j}$ on $x_{k_{1}}$, we can further single out
the dependencies on another variable $x_{k_{2}}$, $k_{2}<k_{1}$, that is
ranked higher up in the system. Then $\bm{\epsilon}^{(j\cdot k_{1}\cdot
k_{2})}$ collects the entries in the assignment vector corresponding to the
dependencies of $x_{j}$ on $x_{k_{1}}$ and $x_{k_{1}}$ in turn on $x_{k_{2}}$.
Continuing in this way, we thus define

$\begin{split}\bm{\epsilon}^{(j)}&=(\epsilon_{1},...,\epsilon_{\eta(j)})\\\
\bm{\epsilon}^{(j\cdot
k_{1})}&=(\epsilon^{(j)}_{H(k_{1}-1)+1},...,\epsilon^{(j)}_{H(k_{1})})\\\
\bm{\epsilon}^{(j\cdot k_{1}\cdot k_{2})}&=(\epsilon^{(j\cdot
k_{1})}_{H(k_{2}-1)+1},...,\epsilon^{(j\cdot k_{1})}_{H(k_{2})})\\\
\vdots\end{split}$

######  Example 2 (continued).

Figure 5 panels (c) and (d) show the two nested dependencies of $x_{3}$ on the
intermediate variable $x_{2}$. The two elements in the potential outcomes
assignment vector $\bm{\epsilon}^{(3)}$ corresponding to the shock
realisations experienced by these two nested dependencies are
$\bm{\epsilon}^{(3\cdot 2)}=(\epsilon_{2},\epsilon_{3})$. We can then further
single out the single element in the potential outcome assignment vector that
corresponds to the dependency of $x_{3}$ on $x_{2}$ and $x_{2}$ on $x_{1}$, as
given by $\epsilon^{(3\cdot 2\cdot 1)}=\epsilon_{2}$. Note that we do not
introduce additional notation for the final dependency on the structural shock
since this is by definition always the last element.

####  4.1.2 Potential Transmission Channels

Each potential outcomes assignment vector $\bm{\epsilon}^{(j)}$ defines a
thought experiment in which some chains experience a shock while others may
not. The most basic thought experiment is the one in which all
$\epsilon_{i}=\xi$, implying that all chains experience the same shock
realisation as commonly investigated in impulse response analysis in
macroeconomics. A conceptually different thought experiment is one in which
$\epsilon_{k}=0$, for some $1\leq k\leq\eta(j)$, implying that some of the
nested dependencies may experience a non-zero shock of size $\xi$, while
others do not. Causality is then carried along a subset of all nested chains –
equivalently, along a subset of paths through the graph – in other words,
along a transmission channel. Throughout the paper we assume that a nested
chain is either shut off ($\epsilon_{k}=0$) or is active and together with all
other active ones experiences the same realised shock size
($\epsilon_{l}=\xi)$, as formalised in Assumption 3.

######  Assumption 3.

The potential outcomes assignment vector $\bm{\epsilon}^{(j)}$ consists of
components $\epsilon_{i}\in\\{0,\xi\\}$ for all $1\leq i\leq\eta(j)$.

We are now ready to define transmission channels in potential outcome
notation.

######  Definition 4.

Let $\bm{\epsilon}^{(j)}$ be a potential outcomes assignment vector with
$\epsilon_{i}$ satisfying Assumption 3. If $\epsilon_{k}=0$ and
$\epsilon_{l}=\xi$ for some $k,l$, then $\bm{\epsilon}^{(j)}$ is a
transmission channel.

######  Example 2 (continued).

Consider the potential outcomes assignment vector $\bm{\epsilon}^{(3)}$ for
the non-recursive case displayed in Figure 5. If
$\bm{\epsilon}^{(3)}=(0,1,0,0)$ implying $\epsilon^{(3\cdot 2\cdot 1)}=1$,
then only the second nested chain experiences a shock. This chain is shown in
Figure 5 panel (c) and corresponds to a transmission channel in which the
shock changes $x_{1}$ which in turn changes $x_{2}$ which finally changes
$x_{3}$. If $\bm{\epsilon}^{(3)}=(0,1,1,0)$, then we obtain the indirect
transmission channel which corresponds to the combination of the second and
third transmission channel shown in Figure 5 panels (c) and (d) respectively.

####  4.1.3 Causal Effects

The causal effect of a potential outcomes assignment vector measures the
effect of a thought experiment by comparing it to a baseline. Under Assumption
3, a natural baseline is $\epsilon_{i}=0$ for all $i=1,\dots,\eta(j)$ such
that no chain experiences a shock. If $\bm{\epsilon}^{(j)}$ is a transmission
channel according to Definition 4, then the transmission effect compares the
reaction of $x_{j}$ under the thought experiment,
$x_{j}^{*}(\bm{\epsilon}^{(j)})$, to the reaction of $x_{j}$ under the
baseline, $x_{j}^{*}(\bm{0})$. This is the causal effect of
$\bm{\epsilon}^{(j)}$ on $x_{j}$.

######  Definition 5.

The causal effect of the structural shock $\varepsilon_{i}$ on the variable of
interest $x_{j}$ is given by

$\mathcal{C}_{\varepsilon_{i},x_{j}}(\bm{\epsilon}^{(j)})=\operatorname{\mathbb{E}}[x^{*}_{j}(\bm{\epsilon}^{(j)})-x^{*}_{j}(\bm{0})],$

where $\bm{\epsilon}^{(j)}$ is a potential outcomes assignment vector and
$\bm{0}$ is a conformable vector of zeros. If $\bm{\epsilon}^{(j)}$ is a
transmission channel according to Definition 4, then
$\mathcal{C}_{\varepsilon_{i},x_{j}}(\bm{\epsilon}^{(j)})$ is the transmission
effect of that transmission channel.

######  Example 2 (continued).

Let $\bm{\epsilon}^{(3)}=(e_{1},e_{2},e_{3},e_{4})$. Then

$\begin{split}\operatorname{\mathbb{E}}[x_{3}^{*}(\bm{\epsilon}^{(3)})]&=B_{3,1}\operatorname{\mathbb{E}}[x_{1}^{*}(\epsilon^{(3\cdot
1)})]+B_{3,2}\operatorname{\mathbb{E}}[x_{2}^{*}(\bm{\epsilon}^{(3\cdot
2)})]+\Omega_{3,1}e_{4}\\\
&=B_{3,1}\Omega_{1,1}e_{1}+B_{3,2}B_{2,1}\operatorname{\mathbb{E}}[x_{1}^{*}(\epsilon^{(3\cdot
2\cdot 1)})]+B_{3,2}\Omega_{2,1}e_{3}+\Omega_{3,1}e_{4}\\\
&=B_{3,1}\Omega_{1,1}e_{1}+B_{3,2}B_{2,1}\Omega_{1,1}e_{2}+B_{3,2}\Omega_{2,1}e_{3}+\Omega_{3,1}e_{4}\end{split}$

The transmission effect of the indirect transmission channel,
$\bm{\epsilon}^{(3)}=(0,1,1,0)$ is therefore
$\mathcal{C}_{\varepsilon_{i},x_{j}}(\bm{\epsilon}^{(3)})=B_{3,2}(B_{2,1}\Omega_{1,1}+\Omega_{2,1})=(\alpha_{2}\alpha_{4})/((1+\alpha_{1}^{2})\eta)$,
while the transmission effect of the direct transmission channel,
$\bm{\epsilon}^{(3)}=(1,0,0,1)$, is given by
$\mathcal{C}_{\varepsilon_{i},x_{j}}(\bm{\epsilon}^{(3)})=B_{3,1}\Omega_{1,1}+\Omega_{3,1}=(\alpha_{3}+\alpha_{1}(1-\eta))/((1+\alpha_{1}^{2})\eta)$.

####  4.1.4 Equivalence to the Graphical Framework

Finally, in Theorem 1, we show that transmission channels defined in
Definitions 2 and 4, and transmission effects defined in Definitions 3 and 5
are equivalent.

######  Theorem 1.

Assume $\bm{y}_{t}$ is generated by model (7) and Assumptions 1 to 3 hold,
such that $\bm{y}_{t}$ can be equivalently represented in static form (9). Let
$\mathcal{G}(\bm{B},\bm{\varOmega})$ be the graph induced by the model.
Consider the effect from the structural shock $\varepsilon_{i}$ to the
variable of interest $x_{j}$, then we have the following equivalence:

  1. (i)

Given a collection of paths
$P_{\varepsilon_{i},x_{j}}\subseteq\mathcal{P}_{\varepsilon_{i},x_{j}}$, there
exists a potential outcomes assignment vector $\bm{\epsilon}^{(j)}$ such that
$\mathcal{Q}_{\xi}(P_{\varepsilon_{j},x_{j}})=\mathcal{C}_{\varepsilon_{i},x_{j}}(\bm{\epsilon}^{(j)})$.

  2. (ii)

Given a potential outcomes assignment vector $\bm{\epsilon}^{(j)}$, there
exists a collection of paths
$P_{\varepsilon_{i},x_{j}}\subseteq\mathcal{P}_{\varepsilon_{i},x_{j}}$ such
that
$\mathcal{C}_{\varepsilon_{i},x_{j}}(\bm{\epsilon}^{(j)})=\mathcal{Q}_{\xi}(P_{\varepsilon_{j},x_{j}})$.

######  Example 2 (continued).

Figure 5 makes use of the equivalence result between the graphical and the
potential outcomes frameworks. Panels (c) through (e) show on top the path
that is either shut-off ($\epsilon_{i}=0$; gray edges) or active
($\epsilon_{i}=\xi$; black edges) with the potential outcomes underneath
highlighting which element in the potential outcomes assignment vector
$\bm{\epsilon}^{(3)}$ corresponds to the path under consideration.

###  4.2 Properties of TCA

The previous section introduced potential outcomes and showed in Theorem 1
that the potential outcomes and graphical framework for TCA are equivalent. In
this section, we continue with the potential outcomes framework and use its
equivalence with the graphical framework to derive three main results. We
focus throughout on the (transmission) effect of the structural shock
$\varepsilon_{i}$ on $x_{j}$.

####  4.2.1 Total Effects

First, since causal effects flow along paths in the associated graph
$\mathcal{G}(\bm{B},\bm{\varOmega})$, intuitively, the total causal effect
should be the effect along all paths connecting $\varepsilon_{i}$ and $x_{j}$,
$\mathcal{P}_{\varepsilon_{i},x_{j}}$. Theorem 2 shows that this is the case.

######  Theorem 2.

Let $\bm{1}_{\eta(j)}$ be a $\eta(j)$ dimensional vector of ones,
$P^{(1)}_{\varepsilon_{i},x_{j}},\dots,P^{(k)}_{\varepsilon_{i},x_{j}}$ be a
partition of $\mathcal{P}_{\varepsilon_{i},x_{j}}$ and
$\bm{e}_{1}\dots\bm{e}_{k}$ be $\eta(j)$ dimensional vectors in
$\\{0,\xi\\}^{\eta(j)}$ such that
$\sum_{m=1}^{k}\bm{e}_{m}=\xi\bm{1}_{\eta(j)}$. Then, under the same
conditions as in Theorem 1,

  1. (i)

$\mathcal{Q}_{\xi}(\mathcal{P}_{\varepsilon_{i},x_{j}})=\mathcal{C}_{\varepsilon_{i},x_{j}}(\xi\bm{1}_{\eta(j)})=\xi\bm{\varPhi}_{j,i}$

  2. (ii)

$\sum_{m=1}^{k}\mathcal{Q}_{\xi}(P^{(m)}_{\varepsilon_{i},x_{j}})=\xi\bm{\varPhi}_{j,i}$

  3. (iii)

$\sum_{m=1}^{k}\mathcal{C}_{\varepsilon_{i},x_{j}}(\bm{e}_{m})=\xi\bm{\varPhi}_{j,i}$.

Part (i) states that the total effect given by entry $\bm{\varPhi}_{j,i}$ in
the impulse response matrix is equal to the total path-specific effect of the
collection of all paths
$\mathcal{Q}_{1}(\mathcal{P}_{\varepsilon_{i},x_{j}})$, which is in turn equal
to the causal effect of the potential outcomes assignment vector
$\bm{\epsilon}^{(j)}=\bm{1}_{\eta(j)}$ by Theorem 1. Multiplication by $\xi$
simply accounts for the shock size. Part (ii) states that the total path-
specific effect of each set in the partition
$P^{(1)}_{\varepsilon_{i},x_{j}},\dots,P^{(k)}_{\varepsilon_{i},x_{j}}$ of
$\mathcal{P}_{\varepsilon_{i},x_{j}}$ decomposes the total effect. Thus, since
each set $P_{\varepsilon_{i},x_{j}}^{(i)}$ in the partition is a transmission
channel by Definition 2, disjoint transmission channels decompose the total
effect. Part (iii) implies that the same statement can be made for a set of
potential outcomes assignment vectors $\bm{e}_{1}\dots\bm{e}_{k}$ such that
$\sum_{m=1}^{k}\bm{e}_{m}=\xi\bm{1}_{\eta(j)}$, which is the potential
outcomes equivalent to the partition
$P^{(1)}_{\varepsilon_{i},x_{j}},\dots,P^{(k)}_{\varepsilon_{i},x_{j}}$.

####  4.2.2 IRF Sufficiency

While Theorem 2 shows that total effects are decomposed by non-overlapping
transmission channels, it says nothing about how the transmission effects and
can be obtained. This is the subject of Theorem 3.

######  Theorem 3.

Let $\tilde{\bm{\varPhi}}$ be the impulse responses obtained using a Cholesky
identification scheme following the ordering in $\bm{T}$. Then, under the same
conditions as in Theorem 1, for some function $f:\mathbb{R}^{hK\times
hK}\times\mathbb{R}^{hK\times hK}\to\mathbb{R}$, and for all $x_{r}$, $x_{s}$,
$x_{j}$ and $\varepsilon_{i}$ with $r<s$,

  1. (i)

$\mathcal{Q}_{1}(\mathcal{P}_{x_{r},x_{s}})=\tilde{\bm{\varPhi}}_{s,r}\tilde{\bm{\varPhi}}_{r,r}^{-1}$

  2. (ii)

$\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}})=f(\bm{\varPhi},\tilde{\bm{\varPhi}})$.

Theorem 3 shows that the transmission effect of any transmission channel is a
combination of the two impulse response matrices $\bm{\varPhi}$ and
$\tilde{\bm{\varPhi}}$; thus, IRFs are sufficient statistics for the
calculation of transmission effects. This extends known sufficiency results of
IRFs for FEVDs [52], policy counterfactual analysis [54], and the analysis of
policy optimality [37]. Note that although the sufficiency result makes use of
a Cholesky identification scheme, the Cholesky scheme is not used to identify
a structural shock; it is simply a computational tool. Only the impulse
response matrix $\bm{\varPhi}$ is structural, while the matrix
$\tilde{\bm{\varPhi}}$ contains orthogonalised impulse responses – they are
orthogonalised with respect to the transmission matrix $\bm{T}$ rather than a
structural ordering.

The Cholesky impulse response matrix $\tilde{\bm{\varPhi}}$ used to compute
transmission effects is sensitive to the ordering in $\bm{T}$; however, the
transformation $f$ in Theorem 3 does not always make use of all columns in
$\tilde{\bm{\varPhi}}$. In cases in which $f$ only makes use of some columns
of $\tilde{\bm{\varPhi}}$, results by [40] imply that only a partial variable
ordering is needed. More precisely, if $f$ makes use of the columns
$k_{1},\dots k_{m}$ then variables ordered between $x_{k_{i}}$ ($1\leq i<m$)
and $x_{k_{j}}$ ($i<j\leq m)$ can be reordered, so can variables ordered
before $x_{k_{1}}$ or after $x_{k_{m}}$.

######  Example 2 (continued).

Consider the transmission effect of the indirect channel for the non-recursive
case displayed in Figure 5. This effect can be computed in two stages. In the
first stage the effect follows all paths from the demand shock to inflation
and is given by the total effect $\alpha_{2}/\eta$ of the demand shock on
inflation; hence a structural impulse response from $\bm{\varPhi}$. In the
second stage, the effect is carried further from inflation to interest rates
and is given by $\alpha_{4}/(1+\alpha_{1}^{2})$; hence a Cholesky impulse
response from $\tilde{\bm{\varPhi}}$ obtained following the ordering in
$\bm{T}=\bm{I}$. Multiplying both IRFs results in the indirect effect in
equation (6).

####  4.2.3 Identification Requirements

As the final step in our theoretical development, Theorem 4 shows that TCA
only requires the shock of interest to be identified.

######  Theorem 4.

Under the same conditions as in Theorem 1, transmission effects can be
obtained if the $i$th column of the impulse response matrix,
$\bm{\varPhi}_{\cdot,i}$, is known.

TCA provides additional insights into what drives equilibrium dynamics due to
a structural shock $\varepsilon_{i}$. This implies that TCA only requires the
identification of $\varepsilon_{i}$, the structural intervention under
investigation. This stands in contrast to counterfactual policy analysis (e.g.
[54] or [67]), where in addition to $\varepsilon_{i}$ further identified
shocks are needed to mimic the counterfactual policy rule in a different
dynamic equilibrium.

Theorem 3 and 4 together imply that TCA is feasible whenever standard
macroeconomic analysis of total dynamic causal effects is feasible. In
addition, the theoretical properties established provide guidance for the
practical computation of transmission effects either by aggregating path
effects, or by calculating the relevant impulse responses. While both
approaches are valid, their computational efficiency will depend on the
specific application. Appendix C goes into more details about the
computational aspects and proposes an algorithm that exploits the established
properties to yield a computationally efficient procedure.

##  5 Forward Guidance of Monetary Policy

In this section we illustrate how transmission channel analysis can be used to
quantify the forward guidance content of monetary policy shock measures. Our
analysis is inspired by [54] who analyse the propagation of an investment-
specific technology shock under various monetary policy rules. To simulate the
monetary policy rule counterfactuals, they require a set of monetary policy
shocks constituting various forward guidance horizons. To this end, they use
the [66] and [47] monetary policy shocks and argue that RR measures more of
the contemporaneous implementation effects of monetary policy, while GK has a
larger forward guidance content. However, they are not able to provide
quantitative support for this hypothesis. Using TCA, we split the total
effects of the RR and GK shocks into contemporaneous and non-contemporaneous
effects; therefore providing quantitative support for the hypothesis of [54].
To conduct TCA we estimate an SVAR(4) in the federal funds rate
$\mathrm{i}_{t}$, the output gap $\mathrm{x}_{t}$, inflation
$\mathrm{\pi}_{t}$ and commodity prices $\mathrm{p}_{t}$, and identify the
monetary policy shock using either RR or GK as an internal instrument [59].

The contemporaneous effect is the part of the total effect of a monetary
policy shock that can be explained by the reaction of the federal funds rate
to the monetary policy shock within the same period. Accordingly, we define
this transmission channel to be all affects flowing through the federal funds
rate contemporaneously. The non-contemporaneous channel is its compliment,
that is, everything that cannot be explained by the contemporaneous
adjustments in the federal funds rate. Since the federal funds rate is
implicitly held constant contemporaneously, the non-contemporaneous effect
also captures forward guidance. Defining the non-contemporaneous effect as the
remainder of the total effect implies that the contemporaneous and non-
contemporaneous effect add to the total effect; hence, they decompose the
total effect.

Since the contemporaneous effect measures the effect of the monetary policy
shock going through the federal funds rate contemporaneously, the federal
funds rate should be ordered first in the transmission matrix $\bm{T}$.
Ordering the federal funds rate first implies that all other equilibrium
quantities may be affected by both implementation and forward guidance
policies, i.e. there exists at least one path to all other variables that does
not go through the federal funds rate and at least one path that does go
through the federal funds rate. This is visualised in Figure 6 panel (a); all
except the first two periods and the monetary policy shock have been omitted
for clarity. Figure 6 panel (a’) shows the same model but in the general
notation of Section 3. The contemporaneous effect consists of all paths going
through $x_{1}$, visualised in panel (b). The non-contemporaneous effect
consists of the remaining paths, visualised in panel (c). Paths of the non-
contemporaneous effect all start at the monetary policy shock
$\varepsilon_{1}$ and follow a direct edge to $x_{j}$ ($j=2,3,4$); the paths
skip the federal funds rate contemporaneously. Thus, if the path coefficients
of the edges $\varepsilon_{1}\to x_{j}$ ($j=2,3,4$) were all zero, then no
non-contemporaneous effect would exist and all of the effect of a monetary
policy shock would be due to a contemporaneous adjustment in the federal funds
rate.

$\mathrm{i}_{0}$$\mathrm{x}_{0}$$\mathrm{\pi}_{0}$$\mathrm{p}_{0}$$\varepsilon^{mp}_{0}$$\mathrm{i}_{1}$$\mathrm{x}_{1}$$\mathrm{\pi}_{1}$$\mathrm{p}_{1}$
(a)

$x_{1}$$x_{2}$$x_{3}$$x_{4}$$\varepsilon_{1}$$x_{5}$$x_{6}$$x_{7}$$x_{8}$ (a’)

$x_{1}$$x_{2}$$x_{3}$$x_{4}$$\varepsilon_{1}$$x_{5}$$x_{6}$$x_{7}$$x_{8}$ (b)

$x_{1}$$x_{2}$$x_{3}$$x_{4}$$\varepsilon_{1}$$x_{5}$$x_{6}$$x_{7}$$x_{8}$ (c)

Figure 6: Each panel shows the DAG of the SVAR(4) used to investigate the
forward guidance content of a RR and GK identified monetary policy shock. The
federal funds rate $\mathrm{i}_{t}$ is ordered before the output gap
$\mathrm{x}_{t}$, inflation $\mathrm{\pi}_{t}$ and commodity prices
$\mathrm{p}_{t}$. Panel (a’) through (c) represent the same graph in the same
variables as panel (a) but use the general notation of Section 3. Panel (b)
highlights the contemporaneous transmission channel going through a
contemporaneous adjustment in the federal funds rate. Panel (c) represents the
non-contemporaneous transmission channel. For clarity, all panels only
visualise the first two periods and omit other shocks besides the monetary
policy shock in the first period.

While the federal funds rate should be ordered first in the transmission
matrix $\bm{T}$, the relative ordering of the remaining variables does not
matter. As all paths go through $x_{1}$, the contemporaneous federal funds
rate, the collection of all paths going from the shock $\varepsilon_{i}$ to
$x_{1}$, $\mathcal{P}_{\varepsilon_{i},x_{1}}$ equals the total
contemporaneous effect of the monetary policy shock on the federal funds rate,
$\Phi_{1,1}$, by Theorem 2. By Theorem 3, the collection of all paths going
from $x_{1}$ to $x_{j}$ ($j\geq 2)$, $\mathcal{P}_{x_{1},x_{j}}$ can be
obtained from the Cholesky impulse responses as
$\tilde{\Phi}_{j,1}\tilde{\Phi}_{1,1}^{-1}$.

By Lemma 1, the contemporaneous effect can then be obtained by multiplying the
two effects as $\Phi_{1,1}\tilde{\Phi}_{j,1}\tilde{\Phi}_{1,1}^{-1}$ and non-
contemporaneous effect as the remainder
$\Phi_{j,1}-\Phi_{1,1}\tilde{\Phi}_{j,1}\tilde{\Phi}_{1,1}^{-1}$. The total
effect $\bm{\varPhi}_{\cdot,1}$ is independent of the ordering chosen in the
transmission matrix $\bm{T}$, and the Cholesky impulse response
$\tilde{\bm{\varPhi}}_{\cdot,1}$ is invariant to re-orderings of the variables
$\mathrm{x}_{t},\mathrm{\pi}_{t},\mathrm{p}_{t}$ [40]. Therefore, all
transmission matrices that order the federal funds rate first will arrive at
the same transmission effects. Figure 6 corresponds to an arbitrary choice of
one of those transmission matrices.

RR |
---|---
GK |
Figure 7: Total effect of a RR (top) or GK (bottom) identified monetary policy
shock, shown as a black scatter-line and normalised to increase the federal
funds rate (FFR) by 25bp on impact. The blue and yellow stacked bars decompose
this total effect into a contemporaneous and non-contemporaneous effect, where
the contemporaneous effect is the effect going through a contemporaneous
adjustment in the federal funds rate.

Figure 7 shows the total, contemporaneous, and non-contemporaneous effects of
the RR and GK shock, all normalised to a 25bp contemporaneous increase in the
federal funds rate. The total effect is depicted as a scatter-line in black,
while the contemporaneous and non-contemporaneous effects are depicted as
stacked bars in blue and yellow respectively. Comparing the effects reveals
that the federal funds rate response to a GK identified monetary policy shock
is humped shaped, which is not the case for an RR identified shock. While the
humped shaped response of the GK shock is almost entirely due to non-
contemporaneous effects, the response of the federal funds rate to an RR
identified shock shows almost no non-contemporaneous effects.

The comparison also reveals a difference in the inflation response. The effect
on inflation is largely driven by the contemporaneously implemented monetary
policy component for the RR shock series, whereas for GK the non-
contemporaneous component dominates the total response of inflation. For the
latter case, it is interesting to observe that prices rise for some time
before somewhat delayed implementation effects kick in and lead to decreasing
inflation. Positive effects on output and inflation due to Odyssean forward
guidance have been discussed in the literature, for instance in [53] or [44].
It is important to note however that the non-contemporaneous component of
monetary policy shocks does not only capture forward guidance and the
decomposed response of inflation may also be driven by other (non-
contemporaneous) components of monetary policy captured by the GK instrument.
To better identify the forward guidance component of monetary policy shocks
the specification of a separate channel is necessary, characterising
intermediate variables that are likely to describe the mechanism through which
forward guidance is affecting output or prices.

Concluding, we find clear evidence that the RR shock series is mostly driven
by contemporaneous implementation effects, while the GK shock series has a
much larger forward guidance component. Our results therefore quantitatively
confirm the hypothesis of [54].

##  6 Concluding Remarks

We develop a new coherent framework for transmission channel analysis for a
large class of dynamic models by connecting them to a graphical
representation. Within the graphical representation, transmission channels are
defined as paths connecting the shock of interest to the outcome variable;
transmission effects are the effects flowing along these paths. We show that
this graphical representation is equivalent to a potential outcomes
representation which we use to prove that impulse responses are sufficient for
the computation of all transmission effects.

Current methods to analyse transmission channels are purely based on
qualitative inspections of various impulse responses. These qualitative
assessments cannot be used to quantitatively compare the importance of various
transmission channels. Our framework opens up the way for a formal
quantitative analysis of transmission channels, therefore extending the
toolbox of the empirical macroeconomic researcher. Of particular relevance for
applied work is our result that TCA does not require any additional
identification requirements beyond the identification of the structural shock
of interest.

A crucial component of TCA is a precise definition of the transmission channel
of interest. We showed how this can be formalised through the transmission
matrix which defines the ceteris paribus ordering of the variables in
equilibrium; this ordering is entirely defined by the research question. We
also show that under some cases, a partial ordering is sufficient. This is the
case in our empirical application in which we show that the [66] shock series
is mostly driven by contemporaneous implementation effects, while the [47]
shock series is largely driven by other aspects of monetary policy such as
forward guidance effects.

While our proposed framework is applicable to a large set of commonly used
dynamic macroeconomic models, including SVARs, DSGEs, and local projections,
it is currently not applicable to general state space models, which are used
to model dynamics via a system of latent variables. While some state space
representations, such as those used for DSGE models, can be rewritten into
SVARMA form, this rewriting cannot accommodate all such models; in particular,
one cannot apply our framework directly to dynamic factor models. In state
space models, transmission channels and effects can still be defined in an
equivalent way using either the graphical or the potential outcomes
representation. However, this definition comes with statistical identification
problems that are beyond the scope of this paper. We therefore leave this
extension for future research.

Our proposed framework relies on linearity of the dynamic model and is thus
inherently subject to potential shortcomings of linear models. This might be
considered a limitation as some evidence points to nonlinear effects of
policies. For example, effects of monetary expansions and contractions are
clearly distinct [42, 35, 68]. This difference may well be driven by the
difference in the transmission of an expansionary and a contractionary shock.
Under the current setup, TCA cannot make a difference in expansionary and
contractionary shocks – it simply analyses the average transmission effects
unconditional of the sign. However, most of macroeconomic analysis still
relies on linear dynamic models, and TCA is applicable in a very wide class of
these. TCA in its current form can therefore already complement the vast
majority of empirical macroeconomic analyses. In addition, extending TCA to
nonlinear models, particularly when allowing for interaction effects, requires
a complete rethink of how to define transmission channels. This is another
exciting avenue for future research.

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##  Appendix A From structural VARMA to Systems Form

In this appendix, we provide details how the lower-triangular from of the
structural VARMA given by

$\bm{L}\bm{y}^{*}_{t}=\sum_{i=1}^{\ell}\bm{Q}^{\prime}\bm{A}_{i}^{*}\bm{y}^{*}_{t-i}+\sum_{j=1}^{q}\bm{Q}^{\prime}\bm{\varPsi}_{j}\bm{\varepsilon}_{t-j}+\bm{Q}^{\prime}\bm{\varepsilon}_{t},$
(A.1)

as presented in equation (8) can be re-written into the general systems form

$\bm{x}=\bm{B}\bm{x}+\bm{\varOmega}\bm{\varepsilon},$ (A.2)

as presented in equation (9).

Let $h$ be the fixed finite horizon of the transmission channels under
consideration. Equation (A.1) for time points $t,\dots,t+h$ can then be
represented as the system of equations

$\begin{cases}\bm{L}\bm{y}^{*}_{t}=\sum_{i=1}^{\ell}\bm{Q}^{\prime}\bm{A}^{*}_{i}\bm{y}^{*}_{t-i}+\sum_{j=1}^{q}\bm{Q}^{\prime}\bm{\varPsi}_{j}\bm{\varepsilon}_{t-j}+\bm{Q}^{\prime}\bm{\varepsilon}_{t}\\\
\vdots\\\
\bm{L}\bm{y}^{*}_{t+h}=\sum_{i=1}^{\ell}\bm{Q}^{\prime}\bm{A}^{*}_{i}\bm{y}^{*}_{t+h-i}+\sum_{j=1}^{q}\bm{Q}^{\prime}\bm{\varPsi}_{j}\bm{\varepsilon}_{t+h-j}+\bm{Q}^{\prime}\bm{\varepsilon}_{t+h},\end{cases}$
(A.3)

which contains all information of the dynamic system up to horizon $h$.
Multiplying all equations on the left-hand-side and right-hand-side by
$\bm{D}=\text{diag}(\bm{L})^{-1}$, where $\text{diag}(\bm{X})$ is a diagonal
matrix containing the diagonal of $\bm{X}$, and moving
$(\bm{I}-\bm{D}\bm{L})\bm{y}^{*}_{t+i}$ to the right-hand-side in equations
$i=0,\dots,h$, we obtain

$\begin{cases}\bm{y}^{*}_{t}=(\bm{I}-\bm{D}\bm{L})\bm{y}^{*}_{t}+\sum_{i=1}^{\ell}\bm{D}\bm{Q}^{\prime}\bm{A}^{*}_{i}\bm{y}^{*}_{t-i}+\sum_{j=1}^{q}\bm{D}\bm{Q}^{\prime}\bm{\varPsi}_{j}\bm{\varepsilon}_{t-j}+\bm{D}\bm{Q}^{\prime}\bm{\varepsilon}_{t}\\\
\vdots\\\
\bm{y}^{*}_{t+h}=(\bm{I}-\bm{D}\bm{L})\bm{y}^{*}_{t+h}+\sum_{i=1}^{\ell}\bm{D}\bm{Q}^{\prime}\bm{A}^{*}_{i}\bm{y}^{*}_{t+h-i}+\sum_{j=1}^{q}\bm{D}\bm{Q}^{\prime}\bm{\varPsi}_{j}\bm{\varepsilon}_{t+h-j}+\bm{D}\bm{Q}^{\prime}\bm{\varepsilon}_{t+h}.\end{cases}$
(A.4)

Given a shock in period $t$, then variables $\bm{y}^{*}_{t-i}$ for $i>0$ are
unaffected by the shock, and thus do not play a role in the propagation of the
shock. We may therefore ignore them by setting their corresponding
coefficients equal to zero. System (A.4) then becomes

$\begin{cases}\bm{y}^{*}_{t}=(\bm{I}-\bm{D}\bm{L})\bm{y}^{*}_{t}+\bm{D}\bm{Q}^{\prime}\bm{\varepsilon}_{t}\\\
\vdots\\\
\bm{y}^{*}_{t+h}=(\bm{I}-\bm{D}\bm{L})\bm{y}^{*}_{t+h}+\sum_{i=1}^{\min(h,p)}\bm{D}\bm{Q}^{\prime}\bm{A}^{*}_{i}\bm{y}^{*}_{t+h-i}+\sum_{j=1}^{\min(h,q)}\bm{D}\bm{Q}^{\prime}\bm{\varPsi}_{j}\bm{\varepsilon}_{t+h-j}+\bm{D}\bm{Q}^{\prime}\bm{\varepsilon}_{t+h}.\end{cases}$

(A.5)

System (A.5) is then equivalent to the general form (A.2), with
$\bm{x}=(\bm{y}_{t}^{*^{\prime}},\dots,\bm{y}_{t+h}^{*^{\prime}})^{\prime}$,
$\bm{\varepsilon}=(\bm{\varepsilon}_{t}^{\prime},\dots,\bm{\varepsilon}_{t+h}^{\prime})^{\prime}$,
and

$\begin{array}[]{ccc}\bm{B}=\begin{bmatrix}\bm{I}-\bm{D}\bm{L}&\bm{O}&\dots&\bm{O}\\\
\bm{D}\bm{Q}^{\prime}\bm{A}^{*}_{1}&\bm{I}-\bm{D}\bm{L}&\dots&\bm{O}\\\
\vdots&\ddots&\ddots&\vdots\\\
\bm{D}\bm{Q}^{\prime}\bm{A}^{*}_{h}&\dots&\bm{D}\bm{Q}^{\prime}\bm{A}^{*}_{1}&\bm{I}-\bm{D}\bm{L}\end{bmatrix},&&\bm{\varOmega}=\begin{bmatrix}\bm{D}\bm{Q}^{\prime}&\bm{O}&\dots&\bm{O}\\\
\bm{D}\bm{Q}^{\prime}\bm{\varPsi}_{1}&\bm{D}\bm{Q}^{\prime}&\dots&\bm{O}\\\
\vdots&\ddots&\ddots&\vdots\\\
\bm{D}\bm{Q}^{\prime}\bm{\varPsi}_{h}&\dots&\bm{D}\bm{Q}^{\prime}\bm{\varPsi}_{1}&\bm{D}\bm{Q}^{\prime}\end{bmatrix},\end{array}$

(A.6)

as given in equation (10) of Section 3.

##  Appendix B Proofs of Main Theorems

Section B.1 first introduces some notation and preliminary results based on
Boolean algebra needed in the proofs of the main theorems. The proofs of
Theorems 1 to 4 are then presented in Section B.2.

### B.1 Boolean Algebra and Transmission Channels

Definition 2 defines transmission channels as a sub-collection of paths. In
practice, it is more convenient to define transmission channels by verbal
statements about which variables lie on the paths and which do not. Boolean
algebra hereby plays an important role in translating verbal statements of
transmission channels into sub-collections of paths. In this section, we
introduce the basics of Boolean algebra333For a more in depth overview of
Boolean algebra and mathematical logic see [39, 64]. needed in the context of
TCA. The main result of this section is Lemma 1 which provides an algebra for
transmission channels and is key to proving Theorems 2, 3, and 4.

Let $b$ be a Boolean formula that is true if and only if a path satisfies it,
and false otherwise. We denote $\mathcal{P}_{u,v}$ the collection of all paths
in $\mathcal{G}(\bm{B},\bm{\varOmega})$ from variable or shock $u$ to variable
$v$. Then $P_{u,v}(b)\subseteq\mathcal{P}_{u,v}$ represents the sub-collection
of paths of $\mathcal{P}_{u,v}$ that satisfy the Boolean formula $b$. By
Definition 2, $P_{u,v}(b)$ is a transmission channel as long as $u$ is a shock
and as long as at least one path is not satisfying the Boolean formula $b$.

A Boolean formula can consist of the following elements:

  1. 1.

$u$: denotes that variable $u$ must be on the paths.

  2. 2.

$b^{\prime}\land u$: denotes that the paths must satisfy the Boolean formula
$b^{\prime}$, and variable $u$ must be on the paths.

  3. 3.

$b^{\prime}\land b^{\prime\prime}$: denotes that both Boolean formulas
$b^{\prime}$ and $b^{\prime\prime}$ must be satisfied by the paths.

  4. 4.

$b^{\prime}\lor u$: denotes that the paths must either satisfy the boolean
formula $b^{\prime}$ or $u$ must be on the paths, or both.

  5. 5.

$b^{\prime}\lor b^{\prime\prime}$: denotes that the paths must either satisfy
the boolean formula $b^{\prime}$ or $b^{\prime\prime}$ or both.

  6. 6.

$\neg u$: denotes that variable $u$ cannot be on the paths.

  7. 7.

$\neg b^{\prime}$: denotes that the Boolean formula $b$ cannot be satisfied by
the paths.

In line with Definition 3, denote the total path-specific effect of the
collection $P_{u,v}(b)$ by

$\mathcal{Q}_{\xi}[P_{u,v}(b)]=\xi\sum_{p\in P_{u,v}(b)}\prod_{(k\to l)\in
p}\omega_{kl},$

where $\xi$ is the shock size, and $\omega_{k,l}$ is the path coefficient of
the edge going from node $k$ to node $l$ on path $p$.

Employing standard set theory arguments and Boolean algebra, several
properties of the total path-specific effect can be demonstrated. These
properties are collected in the next lemma.

######  Lemma 1.

Suppose $\bm{y}_{t}$ is generated by model (7) and Assumptions 1 through 3 are
satisfied. Let $\mathcal{G}(\bm{B},\bm{\varOmega})$ be the graph induced by
model (9). Denote with $\mathcal{P}_{u,v}$ the set of all paths connecting
variable or shock $u$ to variable $v$ in $\mathcal{G}(\bm{B},\bm{\varOmega})$
and by $P_{u,v}(b)\subseteq\mathcal{P}_{u,v}$ the subset of paths satisfying a
Boolean formula $b$. Further, denote $\mathcal{Q}_{\xi}[P_{u,v}(b)]$ the total
path-specific effect of the paths in $P_{u,v}(b)$. Then, the following
properties hold:

  1. (i)

Let $b=x_{k}$. Then
$\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}(b)]=\mathcal{Q}_{\xi}[\mathcal{P}_{\varepsilon_{i},x_{k}}]\mathcal{Q}_{1}[\mathcal{P}_{x_{k},x_{j}}]$
with $k\neq j$.

  2. (ii)

Let $i_{1}<i_{2}<...<i_{k}$ and $i_{k}\neq j$. Let
$b=\wedge_{m=1}^{k}x_{i_{m}}$. Then

$\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}(b)]=\mathcal{Q}_{\xi}[\mathcal{P}_{\varepsilon_{i},x_{i_{1}}}]\mathcal{Q}_{1}[\mathcal{P}_{x_{i_{1}},x_{i_{2}}}]\dots\mathcal{Q}_{1}[\mathcal{P}_{x_{i_{k}},x_{j}}].$

  3. (iii)

Let $P_{\varepsilon_{i},x_{j}}\subset\mathcal{P}_{\varepsilon_{i},x_{j}}$ and
$P_{\varepsilon_{i},x_{j}}^{\prime}\subset\mathcal{P}_{\varepsilon_{i},x_{j}}$
be two disjoint sets of paths. Then
$\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}\cup
P_{\varepsilon_{i},x_{j}}^{\prime}]=\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}]+\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}^{\prime}]$.

  4. (iv)

Let $P_{\varepsilon_{i},x_{j}}\subset\mathcal{P}_{\varepsilon_{i},x_{j}}$ and
$P_{\varepsilon_{i},x_{j}}^{\prime}\subset\mathcal{P}_{\varepsilon_{i},x_{j}}$.
Then $\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}\backslash
P_{\varepsilon_{i},x_{j}}^{\prime}]=\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}]-\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}\cap
P_{\varepsilon_{i},x_{j}}^{\prime}]$.

  5. (v)

Let $b,b^{\prime}$ be two Boolean formulas. Then
$P_{\varepsilon_{i},x_{j}}(b)\cap
P_{\varepsilon_{i},x_{j}}(b^{\prime})=P_{\varepsilon_{i},x_{j}}(b\wedge
b^{\prime})$.

  6. (vi)

Let $b,b^{\prime}$ be two Boolean formulas. Then
$P_{\varepsilon_{i},x_{j}}(b\vee
b^{\prime})=P_{\varepsilon_{i},x_{j}}(b)\cup(P_{\varepsilon_{i},x_{j}}(b^{\prime})\backslash
P_{\varepsilon_{i},x_{j}}(b\wedge b^{\prime}))$.

  7. (vii)

Let $b,b^{\prime}$ be two Boolean formulas. Then
$P_{\varepsilon_{i},x_{j}}(b\wedge\neg
b^{\prime})=P_{\varepsilon_{i},x_{j}}(b)\backslash
P_{\varepsilon_{i},x_{j}}(b\wedge b^{\prime})$.

  8. (viii)

Let $b=\neg x_{k}$. Then
$\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}(b)]=\mathcal{Q}_{\xi}[\mathcal{P}_{\varepsilon_{i},x_{j}}]-\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}(x_{k})]$
with $i\neq j$.

###### Proof of Lemma 1(i).

Each path $p\in P_{\varepsilon_{i},x_{j}}(b)$ can be split into two parts:
$p_{1}$ which goes from $\varepsilon_{i}$ to $x_{k}$, and $p_{2}$ which goes
from $x_{k}$ to $x_{j}$. So we have
$P_{\varepsilon_{i},x_{j}}(b)=\mathcal{P}_{\varepsilon_{i},x_{k}}\otimes\mathcal{P}_{x_{k},x_{j}}$
where $\otimes$ denotes the cross product of paths: For
$p_{1}\in\mathcal{P}_{\varepsilon_{i},x_{k}}$ and
$p_{2}\in\mathcal{P}_{x_{k},x_{j}}$, $p_{1}\to
p_{2}\in\mathcal{P}_{\varepsilon_{i},x_{k}}\otimes\mathcal{P}_{x_{k},x_{j}}$.
The total path specific effect can then be written as

$\begin{split}\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}(b)]&=\xi\sum_{p\in
P_{\varepsilon_{i},x_{j}}(b)}\prod_{(u\to v)\in p}\omega_{uv}\\\
&=\xi\sum_{p_{1}\in\mathcal{P}_{\varepsilon_{i},x_{k}}}\sum_{p_{2}\in\mathcal{P}_{x_{k},x_{j}}}\left(\prod_{(u\to
v)\in p_{1}}\omega_{uv}\right)\left(\prod_{(u\to v)\in
p_{2}}\omega_{uv}\right)\\\
&=\xi\sum_{p_{1}\in\mathcal{P}_{\varepsilon_{i},x_{k}}}\left\\{\prod_{(u\to
v)\in
p_{1}}\omega_{uv}\right\\}\sum_{p_{2}\in\mathcal{P}_{x_{k},x_{j}}}\left\\{\prod_{(u\to
v)\in p_{2}}\omega_{uv}\right\\}\\\
&=\mathcal{Q}_{\xi}[\mathcal{P}_{\varepsilon_{i},x_{k}}]\mathcal{Q}_{1}[\mathcal{P}_{x_{k},x_{j}}].\qed\end{split}$

###### Proof of Lemma 1(ii).

We prove this by induction. We know from Lemma 1(i) that it holds for $k=1$.
Now suppose it holds for some $k$. We then need to show that it holds for
$k+1$. To do so, first define $b_{k}=\wedge_{m=1}^{k}x_{i_{m}}$ and
$b_{k+1}=\wedge_{m=1}^{k+1}x_{i_{m}}$. Any path $p\in
P_{\varepsilon_{i},x_{j}}(b_{k+1})$ can then be split into three parts:
$p_{1}\in P_{\varepsilon_{i},x_{i_{k}}}(b_{k})$,
$p_{2}\in\mathcal{P}_{x_{i_{k}},x_{i_{k+1}}}$, and
$p_{3}\in\mathcal{P}_{x_{i_{k+1}},x_{j}}$. The following is then an extension
of the proof to 1(i):

$\begin{split}&\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}(b_{k+1})]=\xi\sum_{p\in
P_{\varepsilon_{i},x_{j}}(b_{k+1})}\prod_{(u\to v)\in p}\omega_{u,v}\\\
&=\xi\sum_{p_{1}\in
P_{\varepsilon_{i},x_{i_{k}}}(b_{k})}\sum_{p_{2}\in\mathcal{P}_{x_{i_{k}},x_{i_{k+1}}}}\sum_{p_{3}\in\mathcal{P}_{x_{i_{k+1}},x_{j}}}\left(\prod_{(u\to
v)\in p_{1}}\omega_{u,v}\right)\left(\prod_{(u\to v)\in
p_{2}}\omega_{u,v}\right)\left(\prod_{(u\to v)\in p_{3}}\omega_{u,v}\right)\\\
&=\xi\sum_{p_{1}\in P_{\varepsilon_{i},x_{i_{k}}}(b_{k})}\left\\{\prod_{(u\to
v)\in
p_{1}}\omega_{u,v}\right\\}\sum_{p_{2}\in\mathcal{P}_{x_{i_{k}},x_{i_{k+1}}}}\left\\{\prod_{(u\to
v)\in
p_{2}}\omega_{u,v}\right\\}\sum_{p_{3}\in\mathcal{P}_{x_{i_{k+1}},x_{j}}}\left\\{\prod_{(u\to
v)\in p_{3}}\omega_{u,v}\right\\}\\\
&=\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{i_{k}}}(b_{k})]\mathcal{Q}_{1}[\mathcal{P}_{x_{i_{k}},x_{i_{k+1}}}]\mathcal{Q}_{1}[\mathcal{P}_{x_{i_{k+1}},x_{j}}].\end{split}$

By the induction hypothesis,

$\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{i_{k}}}(b_{k})]=\mathcal{Q}_{\xi}[\mathcal{P}_{\varepsilon_{i},x_{i_{1}}}]\mathcal{Q}_{1}[\mathcal{P}_{x_{i_{1}},x_{i_{2}}}]\dots\mathcal{Q}_{1}[\mathcal{P}_{x_{i_{k-1}},x_{i_{k}}}].$

Thus, the statement holds for $k+1$. This implies that it holds for all
$k\in\mathbb{N}$, concluding the proof. ∎

###### Proof of Lemma 1(iii).

Since for any $p\in P_{\varepsilon_{i},x_{j}}\cup
P_{\varepsilon_{i},x_{j}}^{\prime}$, we have $p\in P_{\varepsilon_{i},x_{j}}$
or $p\in P_{\varepsilon_{i},x_{j}}^{\prime}$ but never in both, we can write

$\displaystyle\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}\cup
P_{\varepsilon_{i},x_{j}}^{\prime}]$ $\displaystyle=\xi\sum_{p\in
P_{\varepsilon_{i},x_{j}}\cup P_{\varepsilon_{i},x_{j}}^{\prime}}\prod_{(u\to
v)\in p}\omega_{u,v}$ $\displaystyle=\xi\sum_{p\in
P_{\varepsilon_{i},x_{j}}}\prod_{(u\to v)\in p}\omega_{u,v}+\xi\sum_{p\in
P_{\varepsilon_{i},x_{j}}^{\prime}}\prod_{(u\to v)\in p}\omega_{u,v}$
$\displaystyle=\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}]+\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}^{\prime}].\qed$

###### Proof of Lemma 1(iv).

Make the following two observations:

  1. 1.

$P_{\varepsilon_{i},x_{j}}\backslash P_{\varepsilon_{i},x_{j}}^{\prime}$ and
$P_{\varepsilon_{i},x_{j}}\cap P_{\varepsilon_{i},x_{j}}^{\prime}$ are
disjoint.

  2. 2.

$P_{\varepsilon_{i},x_{j}}\backslash
P_{\varepsilon_{i},x_{j}}^{\prime}\cup(P_{\varepsilon_{i},x_{j}}\cap
P_{\varepsilon_{i},x_{j}}^{\prime})=P_{\varepsilon_{i},x_{j}}$.

Using these observations and 1(iii), we have

$\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}]=\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}\backslash
P_{\varepsilon_{i},x_{j}}^{\prime}\cup(P_{\varepsilon_{i},x_{j}}\cap
P_{\varepsilon_{i},x_{j}}^{\prime})]=\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}\backslash
P_{\varepsilon_{i},x_{j}}^{\prime}]+\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}\cap
P_{\varepsilon_{i},x_{j}}^{\prime}].$

The result follows immediately. ∎

###### Proof of Lemma 1(v).

First, let $p\in P_{\varepsilon_{i},x_{j}}(b)\cap
P_{\varepsilon_{i},x_{j}}(b^{\prime})$. Then $p$ is in both
$P_{\varepsilon_{i},x_{j}}(b)$ and $P_{\varepsilon_{i},x_{j}}(b^{\prime})$.
This can only be if $p$ satisfies both $b$ and $b^{\prime}$. But then $p$ is
also in $P_{\varepsilon_{i},x_{j}}(b\wedge b^{\prime})$. Second, let $p\in
P_{\varepsilon_{i},x_{j}}(b\wedge b^{\prime})$. Then $p$ satisfies both $b$
and $b^{\prime}$. Thus $p\in P_{\varepsilon_{i},x_{j}}(b)$ and $p\in
P_{\varepsilon_{i},x_{j}}(b^{\prime})$. But then $p\in
P_{\varepsilon_{i},x_{j}}(b)\cap P_{\varepsilon_{i},x_{j}}(b^{\prime})$. The
result now follows from the above two observations. ∎

###### Proof of Lemma 1(vi).

First, let $p\in P_{\varepsilon_{i},x_{j}}(b\vee b^{\prime})$. Then $p$
satisfies either $b$, or $b^{\prime}$ or both.

  * •

If $p$ satisfies $b$, then $p\in P_{\varepsilon_{i},x_{j}}(b)$ and thus $p\in
P_{\varepsilon_{i},x_{j}}(b)\cup(P_{\varepsilon_{i},x_{j}}(b^{\prime})\backslash
P_{\varepsilon_{i},x_{j}}(b\wedge b^{\prime}))$.

  * •

If $p$ satisfies $b$ and $b^{\prime}$, then $p$ satisfies $b$ and thus $p\in
P_{\varepsilon_{i},x_{j}}(b)$ and therefore $p\in
P_{\varepsilon_{i},x_{j}}(b)\cup(P_{\varepsilon_{i},x_{j}}(b^{\prime})\backslash
P_{\varepsilon_{i},x_{j}}(b\wedge b^{\prime}))$.

  * •

If $p$ satisfies $b^{\prime}$ but not $b$, then $p\in
P_{\varepsilon_{i},x_{j}}(b^{\prime})\backslash
P_{\varepsilon_{i},x_{j}}(b\wedge b^{\prime})$ and thus $p\in
P_{\varepsilon_{i},x_{j}}(b)\cup(P_{\varepsilon_{i},x_{j}}(b^{\prime})\backslash
P_{\varepsilon_{i},x_{j}}(b\wedge b^{\prime}))$.

Therefore, $p\in P_{\varepsilon_{i},x_{j}}(b\vee b^{\prime})\Rightarrow p\in
P_{\varepsilon_{i},x_{j}}(b)\cup(P_{\varepsilon_{i},x_{j}}(b^{\prime})\backslash
P_{\varepsilon_{i},x_{j}}(b\wedge b^{\prime}))$.

Second, let $p\in
P_{\varepsilon_{i},x_{j}}(b)\cup(P_{\varepsilon_{i},x_{j}}(b^{\prime})\backslash
P_{\varepsilon_{i},x_{j}}(b\wedge b^{\prime}))$.

  * •

If $p\in P_{\varepsilon_{i},x_{j}}(b)$, then $p$ satisfies $b$ or $p$
satisfies $b$ and $b^{\prime}$.

  * •

If $p\in P_{\varepsilon_{i},x_{j}}(b^{\prime})\backslash
P_{\varepsilon_{i},x_{j}}(b\wedge b^{\prime})$, then $p$ satisfies
$b^{\prime}$ but not $b$.

  * •

$P_{\varepsilon_{i},x_{j}}(b)$ and
$P_{\varepsilon_{i},x_{j}}(b^{\prime})\backslash
P_{\varepsilon_{i},x_{j}}(b\wedge b^{\prime})$ are disjoint and thus $p$ never
falls into both.

Thus, $p$ always satisfies $b$ or $b^{\prime}$ or both, and thus $p\in
P_{\varepsilon_{i},x_{j}}(b)\cup(P_{\varepsilon_{i},x_{j}}(b^{\prime})\backslash
P_{\varepsilon_{i},x_{j}}(b\wedge b^{\prime}))\Rightarrow p\in
P_{\varepsilon_{i},x_{j}}(b\vee b^{\prime})$. Both points together prove the
statement. ∎

###### Proof of Lemma 1(vii).

First, let $p\in P_{\varepsilon_{i},x_{j}}(b\wedge\neg b^{\prime})$. Then $p$
satisfies $b$ but not $b^{\prime}$. Thus $p\in P_{\varepsilon_{i},x_{j}}(b)$
but $p\notin P_{\varepsilon_{i},x_{j}}(b\wedge b^{\prime})$. Therefore, $p\in
P_{\varepsilon_{i},x_{j}}(b)\backslash P_{\varepsilon_{i},x_{j}}(b\wedge
b^{\prime})$. Second, let $p\in P_{\varepsilon_{i},x_{j}}(b)\backslash
P_{\varepsilon_{i},x_{j}}(b\wedge b^{\prime})$. Then $p$ satisfies
$b\wedge\neg(b\wedge b^{\prime})$. A truth table then shows that $p$ satisfies
$b\wedge\neg b^{\prime}$ and therefore $p\in
P_{\varepsilon_{i},x_{j}}(b\wedge\neg b^{\prime})$. Both points together prove
the statement. ∎

###### Proof of Lemma 1(viii).

Note that $P_{\varepsilon_{i},x_{j}}(\neg
x_{k})=\mathcal{P}_{\varepsilon_{i},x_{j}}\backslash
P_{\varepsilon_{i},x_{j}}(x_{k})$. Then by Property 1(iv) we have

$\begin{split}\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}(\neg
x_{k})]&=\mathcal{Q}_{\xi}[\mathcal{P}_{\varepsilon_{i},x_{j}}\backslash
P_{\varepsilon_{i},x_{j}}(x_{k})]=\mathcal{Q}_{\xi}[\mathcal{P}_{\varepsilon_{i},x_{j}}]-\mathcal{Q}_{\xi}[\mathcal{P}_{\varepsilon_{i},x_{j}}\cap
P_{\varepsilon_{i},x_{j}}(x_{k})]\\\
&=\mathcal{Q}_{\xi}[\mathcal{P}_{\varepsilon_{i},x_{j}}]-\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}(x_{k})].\qed\end{split}$

### B.2 Proofs of Main Results

###### Proof of Theorem 1.

Under the conditions of Theorem 1, we can define the causal effect of
$\varepsilon_{i}$ on $x_{j}$ as

$\mathcal{C}_{\varepsilon_{i},x_{j}}(\bm{\epsilon}^{(j)})=\operatorname{\mathbb{E}}[x_{j}^{*}(\bm{\epsilon}^{(j)})-x_{j}^{*}(\bm{0})],$

which can then be written as

$\begin{split}\mathcal{C}_{\varepsilon_{i},x_{j}}(\bm{\epsilon}^{(j)})&=\sum_{k_{1}=1}^{j-1}\left\\{\bm{B}_{j,k_{1}}\operatorname{\mathbb{E}}[x^{*}_{k_{1}}(\bm{\epsilon}^{(j\cdot
k_{1})})-x^{*}_{k_{1}}(\bm{0}_{\eta(k_{1})})]\right\\}+\bm{\varOmega}_{j,i}\bm{\epsilon}^{(j)}_{\eta(j)}\\\
&=\sum_{k_{1}=1}^{j-1}\left\\{\bm{B}_{j,k_{1}}\mathcal{C}_{\varepsilon_{i},x_{k_{1}}}(\bm{\epsilon}^{(j\cdot
k_{1})})\right\\}+\bm{\varOmega}_{j,i}\bm{\epsilon}^{(j)}_{\eta(j)}\\\
&=\bm{\varOmega}_{j,i}\bm{\epsilon}^{(j)}_{\eta(j)}+\sum_{k_{1}=1}^{j-1}\bm{B}_{j,k_{1}}\bm{\varOmega}_{{k_{1}},i}\bm{\epsilon}^{(j\cdot
k_{1})}_{\eta(k_{1})}+\sum_{k_{1}=1}^{j-1}\sum_{k_{2}=1}^{k_{1}-1}\left\\{\bm{B}_{j,k_{1}}\bm{B}_{k_{1},k_{2}}\mathcal{C}_{\varepsilon_{i},x_{k_{2}}}(\bm{\epsilon}^{(j\cdot
k_{1}\cdot k_{2})})\right\\}.\end{split}$

Denote for $l=0,...,j-1$,

$S(l)=\begin{cases}\bm{\varOmega}_{j,i}\bm{\epsilon}^{(j)}_{\eta(j)}&l=0\\\
\sum_{k_{1}=1}^{j-1}\sum_{k_{2}=1}^{k_{1}-1}\ldots\sum_{k_{l}=1}^{k_{l-1}-1}\bm{B}_{j,k_{1}}\bm{B}_{k_{1},k_{2}}\ldots\bm{B}_{k_{l-1},k_{l}}\bm{\varOmega}_{{k_{l}},i}\bm{\epsilon}^{(j\cdot
k_{1}\cdot\ldots\cdot k_{l-1}\cdot k_{l})}_{\eta(k_{l})}&l\geq 1.\end{cases}$

It then follows that

$\mathcal{C}_{\varepsilon_{i},x_{j}}(\bm{\epsilon}^{(j)})=\sum_{l=1}^{j-1}S(l).$
(B.1)

The following observations now lead to the equivalence proof:

  1. 1.

$S(l)$ enumerates all paths that include $l$ intermediate variables.

  2. 2.

Due to the structure of $\bm{B}$, there never exists an edge from node
$x_{r}\to x_{s}$ if $r>s$. Thus, there exists no path from $\epsilon_{i}$ to
$x_{j}$ that has more than $j-1$ intermediate variables.

  3. 3.

This last observation implies that the right-hand-side of equation (B.1)
enumerates all paths that exist in $\mathcal{G}(\bm{B},\bm{\varOmega})$ from
$\epsilon_{i}$ to $x_{j}$.

  4. 4.

The product
$\bm{B}_{jk_{1}}\bm{B}_{k_{1}k_{2}}...\bm{B}_{k_{l-1}k_{l}}\bm{\varOmega}_{{k_{l}},i}$
in $S(l)$ is the product of path coefficients along the path $\epsilon_{i}\to
x_{k_{l}}\to\ldots\to x_{k_{1}}\to x_{j}$ and thus equals the path-specific
effect of this path.

  5. 5.

$\epsilon^{(j\cdot k_{1}\cdot\ldots\cdot k_{l-1}\cdot k_{l})}_{\eta(k_{l})}$
acts as a path selector. If $\epsilon^{(j\cdot k_{1}\cdot\ldots\cdot
k_{l-1}\cdot k_{l})}_{\eta(k_{l})}=0$, such that the path is not selected,
then the term
$\bm{B}_{jk_{1}}\bm{B}_{k_{1}k_{2}}...\bm{B}_{k_{l-1}k_{l}}\bm{\varOmega}_{{k_{l}},i}$
drops out of the right-hand-side of equation (B.1).

According to observations 1 to 3, the potential outcomes assignment vector has
as many entries as there exist paths in the graph
$\mathcal{G}(\bm{B},\bm{\varOmega})$. Next, take a set of paths
$P_{\varepsilon_{i},x_{j}}$ as given. By observations 4 and 5, we can set the
elements in the potential outcomes assignment vector $\bm{\epsilon}^{(j)}$
selecting the paths in $P_{\varepsilon_{i},x_{j}}$ equal to $\xi$ and all
other elements to zero. This then implies
$\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}})=\mathcal{C}_{\varepsilon_{i},x_{j}}(\bm{\epsilon}^{(j)})$.
Finally, take a potential outcomes assignment vector $\bm{\epsilon}^{(j)}$
with $\epsilon^{(j)}_{k}\in\\{\xi,0\\}$ as given. Then, by observations 3 and
5, this potential outcomes assignment vector selects a set of paths
$P_{\varepsilon_{i},x_{j}}$ such that
$\mathcal{C}_{\varepsilon_{i},x_{j}}(\bm{\epsilon}^{(j)})=\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}})$.
∎

###### Proof of Theorem 2(i).

Recall from Theorem 1 that
$\mathcal{C}_{\varepsilon_{i},x_{j}}(\bm{\epsilon}^{(j)})=\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}]$.
Taking $\bm{\epsilon}^{(j)}=\xi\bm{1}_{\eta(j)}$ immediately results in
$P_{\varepsilon_{i},x_{j}}=\mathcal{P}_{\varepsilon_{i},x_{j}}$. This, in
turn, implies by Theorem 1 that
$\mathcal{C}_{\varepsilon_{i},x_{j}}(\xi\bm{1}_{\eta(j)})=\mathcal{Q}_{\xi}(\mathcal{P}_{\varepsilon_{i},x_{j}})$.
Next, note that
$\mathcal{C}_{\varepsilon_{i},x_{j}}(\xi\bm{1}_{\eta(j)})=\operatorname{\mathbb{E}}[x_{j}|\varepsilon_{i}=\xi]-\operatorname{\mathbb{E}}[x_{j}|\varepsilon_{i}=0]=\xi\bm{\varPhi}_{j,i}$,
which completes the proof. ∎

###### Proof of Theorem 2(ii).

As $P_{\varepsilon_{i},x_{j}}^{(1)},\dots,P_{\varepsilon_{i},x_{j}}^{(k)}$ is
a partition,
$P_{\varepsilon_{i},x_{j}}^{(1)},\cup_{r=2}^{k}P_{\varepsilon_{i},x_{j}}^{(r)}$
is a partition as well. By the first part we therefore have

$\xi\bm{\varPhi}_{j,i}=\mathcal{Q}_{\xi}(\mathcal{P}_{\varepsilon_{i},x_{j}})=\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}^{(1)}\cup(\cup_{r=2}^{k}P_{\varepsilon_{i},x_{j}}^{(r)})].$

Since $P_{\varepsilon_{i},x_{j}}^{(1)}$ and
$\cup_{r=2}^{k}P_{\varepsilon_{i},x_{j}}^{(r)}$ are disjoint, Lemma 1(iii)
implies

$\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}^{(1)}\cup(\cup_{r=2}^{k}P_{\varepsilon_{i},x_{j}}^{(r)})]=\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}^{(1)}]+\mathcal{Q}_{\xi}[\cup_{r=2}^{k}P_{\varepsilon_{i},x_{j}}^{(r)}].$

Applying the same logic recursively to
$\mathcal{Q}_{\xi}[\cup_{r=2}^{k}P_{\varepsilon_{i},x_{j}}^{(r)}]$ results in

$\xi\bm{\varPhi}_{j,i}=\sum_{r=1}^{k}\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}}^{(r)}),$

completing the proof of the second part. ∎

###### Proof of Theorem 2(iii).

Note that each element in the potential outcomes assignment vector
$\bm{\epsilon}^{(j)}$ selects whether a specific path is active or not. Thus,
for each collection $P_{\varepsilon_{i},x_{j}}^{(r)}$ there exists a potential
outcomes assignment vector $\bm{e}_{r}$ that has a $\xi$ at those elements
selecting that paths present in $P_{\varepsilon_{i},x_{j}}^{(r)}$ and $0$ at
all other elements. Since no path can be present in two collections of the
partition, $\sum_{r=1}^{k}\bm{e}_{r}=\xi\bm{1}_{\eta(j)}$. Applying Theorems 1
and 2(ii) yields
$\sum_{r=1}^{k}\mathcal{C}_{\varepsilon_{i},x_{j}}(\bm{e}_{m})=\xi\bm{\varPhi}_{j,i}$.
∎

###### Proof of Theorem 3(i).

After substituting $\bm{Q}^{\prime}\bm{A}_{i}^{*}=\bar{\bm{A}}$,
$\bm{Q}^{\prime}\bm{\varPsi}_{i}\bm{Q}=\bar{\bm{\varPsi}}_{i}$ and
$\bm{\gamma}_{t}=\bm{Q}^{\prime}\bm{\varepsilon}_{t}$, equation (8) can be
written as

$\bm{L}\bm{y}_{t}^{*}=\sum_{i=1}^{\ell}\bar{\bm{A}}_{i}\bm{y}_{t-i}^{*}+\bm{\gamma}_{t}+\sum_{j=1}^{q}\bar{\bm{\varPsi}}_{j}\bm{\gamma}_{t}.$
(B.2)

Since $\bm{L}$ is lower-triangular, this system can be obtained from the
reduced form system using a Cholesky identification scheme following the
ordering in the transmission matrix $\bm{T}$.

Following similar steps to Appendix A, the system (B.2) can be written as

$\bm{x}=\tilde{\bm{B}}\bm{x}+\tilde{\bm{\varOmega}}\bm{\gamma},$ (B.3)

where
$\bm{\gamma}=(\bm{\gamma}_{t}^{\prime},\dots,\bm{\gamma}_{t+h}^{\prime})^{\prime}$,
$\bm{x}=(\bm{y}_{t}^{\prime},\dots,\bm{y}_{t+h}^{\prime})^{\prime}$ and
$\tilde{\bm{B}}=\bm{B}$, and

$\tilde{\bm{\varOmega}}=\begin{bmatrix}\bm{D}&0&\dots&0\\\
\bm{D}\bar{\bm{\varPsi}}_{1}&\bm{D}&\dots&0\\\ \vdots&\ddots&\ddots&\vdots\\\
\bm{D}\bar{\bm{\varPsi}}_{h}&\dots&\bm{D}\bar{\bm{\varPsi}}_{1}&\bm{D}\end{bmatrix},$
(B.4)

with $\bar{\bm{A}}_{i}=\bm{O}$ for $i>\ell$, and
$\bar{\bm{\varPsi}}_{j}=\bm{O}$ for $j>q$, and
$\bm{D}=\text{diag}(\bm{L})^{-1}$, where $\text{diag}(\bm{X})$ is a diagonal
matrix of the diagonal of $\bm{X}$.

Since $\bm{B}=\tilde{\bm{B}}$, the system (B.3) induces a graph
$\mathcal{G}(\tilde{\bm{B}},\tilde{\bm{\varOmega}})$ that has the same path-
coefficients for any edge $x_{j}\to x_{i}$ as the graph
$\mathcal{G}(\bm{B},\bm{\varOmega})$.

Because the shocks in $\bm{\gamma}$ are uncorrelated, we can use the graph
$\mathcal{G}(\tilde{\bm{B}},\tilde{\bm{\varOmega}})$ to investigate the effect
of the shock $\gamma_{i}$ on $x_{j}$ along all paths in the graph,
$\mathcal{P}_{\gamma_{i},x_{j}}$. This effect is given by the total path-
specific effect

$\sum_{p\in\mathcal{P}_{\gamma_{i},x_{j}}}\prod_{u\to v\in
p}\tilde{\omega}_{u,v},$ (B.5)

where $\tilde{\omega}_{u,v}$ is the path coefficient of the edge connecting
$u$ to $v$.

Since $\tilde{\bm{\varOmega}}$ is diagonal, any path in
$\mathcal{P}_{\gamma_{i},x_{j}}$ must always first go into $x_{i}$ before it
can go through any other variables. Thus equation (B.5) can also be written as

$\tilde{\omega}_{\gamma_{i},x_{i}}\sum_{p\in\mathcal{P}_{x_{i},x_{j}}}\prod_{u\to
v\in p}\tilde{\omega}_{u,v}.$ (B.6)

Because, $\tilde{\bm{B}}=\bm{B}$, all path coefficients along paths connecting
$x_{k}$ to $x_{l}$ for all $k,l$, are the same as in the graph
$\mathcal{G}(\bm{B},\bm{\varOmega})$, $\tilde{\omega}_{u,v}=\omega_{u,v}$ for
all $u\neq\gamma_{i}$. Thus, (B.6) can also be written as

$\tilde{\omega}_{\gamma_{i},x_{i}}\sum_{p\in\mathcal{P}_{x_{i},x_{j}}}\prod_{u\to
v\in p}\omega_{u,v}.$ (B.7)

Letting $\gamma_{i}$ take the role of $\varepsilon_{i}$ in Theorem 2, one can
show that the total path-specific effect of $\mathcal{P}_{\gamma_{i},x_{j}}$
in the graph $\mathcal{G}(\tilde{\bm{B}},\tilde{\bm{\varOmega}})$ equals the
total effect of $\gamma_{i}$ on $x_{j}$, which is given by the impulse
response matrix

$\tilde{\bm{\varPhi}}_{j,i}=[(\bm{I}-\tilde{\bm{B}})^{-1}\tilde{\bm{\varOmega}}]_{j,i}=[(\bm{I}-\bm{B})^{-1}\tilde{\bm{\varOmega}}]_{j,i}.$
(B.8)

Similarly, the effect of $\gamma_{i}$ on $x_{i}$, which consists of only the
direct edge $\gamma_{i}\to x_{i}$ and thus equals
$\tilde{\omega}_{\gamma_{i},x_{i}}$, is a total effect, and equal to the
impulse response of $\gamma_{i}$ on $x_{i}$,

$\tilde{\bm{\varPhi}}_{i,i}=[(\bm{I}-\bm{B})^{-1}\tilde{\Omega}]_{i,i}.$ (B.9)

By combining (B.7), (B.8) and (B.9),

$\tilde{\bm{\varPhi}}_{i,i}\sum_{p\in\mathcal{P}_{x_{i},x_{j}}}\prod_{u\to
vinp}\omega_{u,v}=\tilde{\bm{\varPhi}}_{j,i}.$ (B.10)

Rearranging, (B.10) gives the final result,

$\mathcal{Q}_{1}(P_{x_{i},x_{j}})=\sum_{p\in\mathcal{P}_{x_{i},x_{j}}}\prod_{u\to
vinp}\omega_{u,v}=\tilde{\bm{\varPhi}}_{j,i}\tilde{\bm{\varPhi}}_{i,i}^{-1}.$
(B.11)

To complete the proof, observe that since (B.2) is the system obtained from a
Cholesky identification scheme following the ordering given in $\bm{T}$, the
impulse response matrix
$\tilde{\bm{\varPhi}}=(\bm{I}-\bm{B})^{-1}\tilde{\bm{\varOmega}}$ is given by
a Cholesky identification following the ordering given in $\bm{T}$. ∎

###### Proof of Theorem 3(ii).

Let $b$ be a Boolean formula. We say $b$ is in disjunctive normal form (DNF)
if $b$ can be written as

$b=b_{1}\vee b_{2}\vee...\vee b_{n},$

where $b_{i}$ includes only conjunctions, $\wedge$, and negations, $\neg$.
Further, we say $b$ can be written in $\text{DNF}_{n}$ if $b$ can be written
in disjunctive normal form involving only $n-1$ disjunctions, $\vee$.

According to Theorem 3.8.8 in [39] and Theorem 2.1 in [64], every Boolean
formula $b$ can be written in DNF. Thus, for every Boolean formula there
exists a $k\in\mathbb{N}$ such that $b$ can be written in $DNF_{k}$.
Additionally, for any collection of paths
$P_{\varepsilon_{i},x_{j}}\subseteq\mathcal{P}_{\varepsilon_{i},x_{j}}$ there
exists a boolean formula $b$ such that
$P_{\varepsilon_{i},x_{j}}(b)=P_{\varepsilon_{i},x_{j}}$. Thus, to prove part
two of Theorem 3 it suffices to shows that for any $DNF_{k}$ with
$k\in\mathbb{N}$, there exists a function $f:\mathbb{R}^{hK\times
hK}\times\mathbb{R}^{hk\times hk}\to\mathbb{R}$ sucht that
$\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}}(b))=f(\bm{\varPhi},\tilde{\bm{\varPhi}})$.
We will prove this by induction.

Let $k=1$ such that the Boolean formula $b$ consists of only conjunctions and
negations; the Boolean formula $b$ is $\text{DNF}_{1}$. If $b$ has no
negations, then Lemma 1(ii) with Theorems 2 and 3(i) imply that there exists a
function $f$ such that
$\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}}(b))=f(\bm{\varPhi},\tilde{\bm{\varPhi}})$.
If $b$ has a single negation then we may write $b=\tilde{b}\land\neg x_{r}$
for some variable $x_{r}$. Then

$\begin{split}\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}(b)]&=\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}(\tilde{b}\land\neg
x_{r})]=\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}(\tilde{b})\setminus
P_{\varepsilon_{i},x_{j}}(\tilde{b}\land x_{r})]\\\
&=\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}(\tilde{b})]-\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}(\tilde{b}\land
x_{r})].\end{split}$

The second line above follows by Lemma 1(vii) while the last line follows by
Lemma (1(iv)) and (1(v)). Since $\tilde{b}$ and $\tilde{b}\land x_{r}$ are
Boolean formulas consisting of only conjunctions, there exist functions
$f_{1}$ and $f_{2}$ such that
$\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}}(\tilde{b}))=f_{1}(\bm{\varPhi},\tilde{\bm{\varPhi}})$
and $\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}}(\tilde{b}\land
x_{r}))=f_{2}(\bm{\varPhi},\tilde{\bm{\varPhi}})$. Thus, let
$f(\bm{\varPhi},\tilde{\bm{\varPhi}})=f_{1}(\bm{\varPhi},\tilde{\bm{\varPhi}})-f_{2}(\bm{\varPhi},\tilde{\bm{\varPhi}})$.
Then
$\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}}(b))=f(\bm{\varPhi},\tilde{\bm{\varPhi}})$.

Finally, if $b$ consists of $m$ negations, then we may write
$b=\bar{b}\land\neg x_{r}$ for some variable $x_{r}$, where $\bar{b}$ consists
of $m-1$ negations. We can thus again write

$\begin{split}\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}(b)]&=\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}(\bar{b}\land\neg
x_{r})]\\\ &=\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}(\bar{b})\setminus
P_{\varepsilon_{i},x_{j}}(\bar{b}\land x_{r})]\\\
&=\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}(\bar{b})]-\mathcal{Q}_{\xi}[P_{\varepsilon_{i},x_{j}}(\bar{b}\land
x_{r})].\end{split}$

By the induction hypothesis, there exists a function $f_{1}$ sucht that
$\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}}(\bar{b}))=f_{1}(\bm{\varPhi},\tilde{\bm{\varPhi}})$.
Additionally, since $\bar{b}\land x_{r}$ consists of a only $m-1$ negations,
the induction hypothesis also implies that there exists a function $f_{2}$
such that $\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}}(\bar{b}\land
x_{r}))=f_{2}(\bm{\varPhi},\tilde{\bm{\varPhi}})$. Thus, let
$f(\bm{\varPhi},\tilde{\bm{\varPhi}})=f_{1}(\bm{\varPhi},\tilde{\bm{\varPhi}})-f_{2}(\bm{\varPhi},\tilde{\bm{\varPhi}})$.
Then
$\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}}(b))=f(\bm{\varPhi},\tilde{\bm{\varPhi}})$.
Thus, if $b$ is $\text{DNF}_{1}$ with $m\in\mathbb{N}$ negations, then there
exists a function $f$ sucht that
$\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}}(b))=f(\bm{\varPhi},\tilde{\bm{\varPhi}})$.

Next, suppose there exists a function $f$ such that
$\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}}(\bar{b}))=f(\bm{\varPhi},\tilde{\bm{\varPhi}})$
if $\bar{b}$ is $\text{DNF}_{k}$ and let the Boolean formula $b$ be
$\text{DNF}_{k+1}$. We can then write $b=b_{1}\lor\bar{b}$ where $b_{1}$ is
$\text{DNF}_{1}$. By Lemma 1(vi), it follows that

$P_{\varepsilon_{i},x_{j}}(b)=P_{\varepsilon_{i},x_{j}}(b_{1})\cup(P_{\varepsilon_{i},x_{j}}(\bar{b})\backslash
P_{\varepsilon_{i},x_{j}}(b_{1}\land\bar{b})).$

Note that $P_{\varepsilon_{i},x_{j}}(b_{1})$ and
$P_{\varepsilon_{i},x_{j}}(\bar{b})\backslash
P_{\varepsilon_{i},x_{j}}(b_{1}\land\bar{b})$ are disjoint. Thus, by Lemma
1(iii) and 1(iv),

$\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}}(b))=\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}}(\underbrace{b_{1}}_{\text{DNF}_{1}}))+\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}}(\underbrace{\bar{b}}_{\text{DNF}_{k}}))-\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}}(\underbrace{b_{1}\land\bar{b}}_{\text{DNF}_{k}})).$

Since there exists a function $f_{1}$ such that
$\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}}(b_{1}))=f_{1}(\bm{\varPhi},\tilde{\bm{\varPhi}})$,
and since the induction hypothesis implies that there exist functions $f_{2}$
and $f_{3}$ such that
$\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}}(\bar{b}))=f_{2}(\bm{\varPhi},\tilde{\bm{\varPhi}})$
and
$\mathcal{Q}_{\xi}(P_{\varepsilon_{i},x_{j}}(b_{1}\land\bar{b}))=f_{3}(\bm{\varPhi},\tilde{\bm{\varPhi}})$,
there exists a function