# Persistence in Economic Networks††thanks: We thank Oliver Linton, Wolfgang
Härdle, Melanie Schienle, Catherine Forbes, Lubos Hanus, Lukáš Vácha, Ryland
Thomas, Chris Florackis, Costas Milas, Alex Kostakis and Charlie Cai for
invaluable discussions and comments. We are grateful to Luboš Hanus for help
in furnishing and converting estimation codes. We acknowledge insightful
comments from numerous seminar presentations, such as: the 2019 and 2020
Society for Economic Measurement Conferences; the Danish National Bank; the
2019 STAT of ML conference; the 13th International Conference on Computational
and Financial Econometrics; and many more. Jozef Baruník gratefully
acknowledges support from the Czech Science Foundation under the 19-28231X
(EXPRO) project. For estimation of dynamic horizon specific networks, we
provide packages DynamicNets.jl in JULIA and DynamicNets in MATLAB. The
packages are available at https://github.com/barunik/DynamicNets.jl and
https://github.com/ellington/DynamicNets. Disclosure Statement: Jozef Baruník
and Michael Ellington have nothing to disclose.

Jozef Baruník‡ and Michael Ellington†

This paper studies heterogeneous network structures driven by different
degrees of persistence in economic connections. Using frequency domain
techniques, we introduce measures that characterize network connections
stemming from transitory and persistent components of shocks. Our approach
permits testing for statistical differences in such connections that evolve
over time. We estimate uncertainty networks from the main US sectors and argue
that they track transitory and persistent sources of systemic risk. Hence they
may serve as a useful tool for macro-prudential supervisors and investors
alike.

‡Institute of Economic Studies, Charles University, Opletalova 26, 110 00, and
The Czech Academy of Sciences, IITA, Pod Vodárenskou Věží 4, 182 08, Prague,
Czech Republic.
<EMAIL_ADDRESS>
†University of Liverpool Management School, Chatham Building, Liverpool, L69
7ZH, UK<EMAIL_ADDRESS>
##  1 Introduction

Firms and economic entities create connections through a variety of channels
(see e.g. Richmond, 2019; Garvey et al., 2015; Herskovic et al.,
2020).111Richmond (2019) measure connections through consumption growth.
Garvey et al. (2015) tracks connections describing the supply chain, and
Herskovic et al. (2020) studies network connections among firm volatilities.
Network structures emerging from these relationships are central to risk
measurement and management as well as understanding macroeconomic risks that
form over business cycles. However, those studying networks in economics
almost exclusively model the nature of such connections as constant.

In this paper, we argue that network connections exhibit heterogeneities and
that they are dynamic. Naturally, economic connections show various degrees of
persistence that one can reconcile with preferences of agents that differ
depending on their horizon of interest (Giglio et al., 2015; Dew-Becker and
Giglio, 2016; Bandi et al., 2021). It remains an open question what role such
heterogeneous sources of connections play in networks, and especially how to
infer information about the persistence of networks from data. Our main
contribution provides a novel framework for the measurement of persistence and
time-variation in network structures from a possibly data-rich environment.

Why should an economist be interested in persistence structure of networks?
Understanding the nature of shocks that underlie network connections is
important for a variety of reasons. First, economic theory suggests that the
marginal utility of agents’ preferences depend on cyclical components of
consumption (see e.g. Giglio et al., 2015; Bandi and Tamoni, 2017), and also
relate to investment horizons in their risk attitudes (Dew-Becker and Giglio,
2016). Such behavior is observable, for example, under myopic loss aversion
where the decision of an agent depends upon the horizon of evaluation.
Unanticipated shocks, or news, have the ability to alter such preferences and
therefore may generate transitory and persistent connections of different
strengths.222For example, a shock bearing influence at longer horizons may
reflect permanent changes to expectations around future price movements. Such
a shock may result in a permanent change to a firm’s future dividend payments
(Balke and Wohar, 2002). Conversely a shock that bears influence at shorter
horizons may suggest transient changes to future price movements. Consider now
the shock pertains to a change in an imminent dividend payment only. This
would likely yield a very short-term change to reflect the transient nature of
the news.

Second, firms set different goals over the short- and long-run and investors
view short- and long-run risks differently (see e.g. Drechsler and Yaron,
2011). This behavior provides motivation for the long-run risk asset pricing
literature that Bansal and Yaron (2004) and Bansal et al. (2010) pioneer. The
implication here is that investment horizons can represent a source of
systematic risk investors demand compensation for (contributions along the
lines of this topic include, e.g. Brennan and Zhang, 2018; Chaudhuri and Lo,
2019).

Third, economists seek to understand relationships among variables over
different horizons. For example, labor market economists examine business
cycle linkages between productivity and unemployment within Search and
Matching models (see e.g. Barnichon, 2010; Hagedorn and Manovskii, 2011).
Monetary economists look at the long-run links between money and output and
money and inflation (see e.g. Belongia and Ireland, 2016, 2021). Those
studying the influence of monetary policy shocks for economic activity do so
over the short- medium and long-run (see e.g. Benati and Surico, 2008; Bianchi
et al., 2009).333Meanwhile, policy makers setting interest rates for many
inflation targeting nations operate over the medium-term (i.e. interest rates
set now to influence inflation in 2-year’s time) and also offer prospects for
interest rates in line with longer-term economic objectives (i.e. forward
guidance).

There are a variety of ways to measure and characterise network structures.
Some propose measures stemming from correlations or coefficient estimates (see
e.g. Engle and Kelly, 2012; Geraci and Gnabo, 2018), while others track links
between individual entities and overall market/economic movements (e.g.
Acharya et al., 2012; Adrian and Brunnermeier, 2016; Acharya et al., 2017) .
Notably, Diebold and Yilmaz (2014) provide a unifying framework for measuring
network characteristics using variance decompositions from an approximating
model. Variance decompositions track how shocks influence the future variation
of variables within a system, hence are natural choice for inferring the
network connections from data. The ensuing literature widely adopts the
Diebold and Yilmaz (2014) approach to address an abundance of issues in
finance and economics using network analysis (e.g. Yang and Zhou, 2017;
Baruník et al., 2020).

However, such an approach reveals little on the nature, in terms of
persistence, of shocks underlying network connections. We characterize dynamic
connections that form on the degrees of persistence shocks possess using
spectral decompositions of variance error forecasts and a locally stationary
approximating model.444For an account of the importance of modeling temporal
dependencies in network connections see Geraci and Gnabo (2018). This means
that our dynamic adjacency matrix fully characterizing a network not only
evolves over time, it also differentiates across degrees of persistence shocks
exhibit. By looking at connections in the frequency domain, we are able to
understand the implications that shocks have on network formation over
different horizons. To estimate various network characteristics, we use a
Bayesian time-varying parameter VAR as our approximating model.

Identifying network structures that form due to idiosyncratic shocks is
relevant also because they can determine aggregate fluctuations (Acemoglu et
al., 2012). These connections among individual or firm-level entities create
systemic risks for the sectors and the wider economy (Billio et al., 2012;
Acemoglu et al., 2017). Such risks spur changes in uncertainty which can be
key factors in business cycles and financial distress (Bloom et al., 2018).
Gabaix (2011) shows sectoral co-movements are salient features of business
cycles. Meanwhile, Atalay (2017) finds that the lion’s share of variation in
output growth is due to idiosyncratic industry-level shocks.

This being said, an understanding of the potential longevity of risks emerging
from these connections, along with their importance in driving financial
unrest or business cycles, is incomplete. We demonstrate how our approach
provides measures of transitory and persistent connections using daily firm-
level realized volatilities of S&P500 constituents. We categorize constituents
into their eleven main sectors according to the Global Industry Classification
Standard (GICS) and measure network connections from the transitory and
persistent components of volatility shocks.

Our empirical results document substantial heterogeneities in the persistence
of sectoral uncertainty networks that identify the nature of systemic risks
emerging from the networks. Specifically, we document: i) surges in persistent
network connections when long-lasting financial and economic events occur; and
ii) statistically significant differences between transitory and persistent
network connections across all sectors. Our network measures may serve as an
online monitoring tool for sectoral uncertainty in markets of interest to
macro-prudential supervisors and investors alike.

We make publicly available computationally efficient packages DynamicNets.jl
in JULIA and DynamicNets in MATLAB that allows one to obtain our measures on
data the researcher desires.555The packages are available at
https://github.com/barunik/DynamicNets.jl and
https://github.com/ellington/DynamicNets While our framework is widely
applicable, it is inherently reduced-form in nature. However depending on the
application, one can use existing theoretical models to motivate how network
connections form.666For example, production networks are gaining in popularity
to explain the structural determinants of network formation in economics and
finance (e.g. Herskovic, 2018; Herskovic et al., 2020; Bigio and La’o, 2020;
Gofman et al., 2020). In turn, this can provide guidance on the variables to
include within the empirical model to retrieve our connectedness
measures.777Our approach also resolves the issues regarding inference and
dimensionality from those who mimic time dynamics using an approximating
window (see e.g. Demirer et al., 2018). The Bayesian nature of our framework
allows the user to incorporate prior shrinkage and estimation uncertainty
using the posterior distribution of network measures. This is in sharp
contrast to conventional studies that provide only point estimates which
requires bootstrapping for confidence intervals. However, bootstrapping
suffers from a lack of theoretical understanding and is also infeasible and
computationally inefficient in most cases due to dimensionality issues.

The rest of this paper proceeds as follows. Section 2 derives our network
measures from locally stationary processes, discusses estimation, and proposes
a testing procedure for statistical differences in transitory and persistent
network connectedness. Section 3 provides Monte Carlo evidence that our
measures are able to reliably track connections and correctly identify
statistical differences. In Section 4 we study connections among firm-level
volatilities of S&P500 sectoral constituents and assess the information
content in sectoral connectedness measures over and above leading measures of
uncertainty. Finally Section 5 concludes.

##  2 Measuring Transitory and Persistent Networks

Here we build dynamic network structures using the spectral decomposition of
time-varying variance decomposition matrices. This reveals the impact of
shocks with transitory effects (i.e. a short-term frequency response) and
persistent effects (i.e. a long-term frequency response) thus creating
linkages dependent on the characteristics shocks possess. Our approach allows
the user to retrieve a dynamic adjacency matrix over different frequencies of
interest. This extends on classical network structures where connections are
binary, symmetric, and static (unconditional). The frequency dependent dynamic
adjacency matrices allow for differences in the entire network structures as
well as asymmetries in various measures summarising the properties of the
network.

Algebraically, the adjacency matrix captures all information about the
network, and any sensible measure must relate to it. Network centrality is a
typical metric many studies use (e.g. Richmond, 2019). This provides the user
with information about the relative importance of nodes. Our objective is to
have a unifying framework that allows one to study dynamic adjacency matrices
and summarize dynamic network properties. Specifically we want to measure node
degrees that capture the number of links to other nodes. We also want to track
the distribution of node degrees, which is a network-wide property that
closely relates to network behavior. Regarding the connectedness of the
network, the location of the degree distribution is key, and hence, the mean
of the degree distribution emerges as a benchmark measure of overall network
connectedness.

Our measures rely on locally stationary processes. This assumes the process is
approximately stationary over a short time interval which allows us to
incorporate time-variation into our analysis. In turn we can establish our
measures of frequency dependent time-varying network connections.

Formally, consider a doubly indexed $N$-variate time series
$(\bm{X}_{t,T})_{1\leq t\leq T,T\in\mathbb{N}}$ with components
$\bm{X}_{t,T}=(\bm{X}_{t,T}^{1},\ldots,\bm{X}_{t,T}^{N})^{\top}$ that describe
all variables in an economy. Here $t$ refers to a discrete time index and $T$
is an additional index indicating the sharpness of the local approximation of
the time series $(\bm{X}_{t,T})_{1\leq t\leq T,T\in\mathbb{N}}$ by a
stationary one. Coarsely speaking, we can consider $(\bm{X}_{t,T})_{1\leq
t\leq T,T\in\mathbb{N}}$ to be a weakly locally stationary process if, for a
large $T$, given a set $S_{T}$ of sample indices such that $t/T\approx u$ over
$t\in S_{T}$, the sample $(\bm{X}_{t,T})_{t\in S_{T}}$ approximates the sample
of a weakly stationary time series depending on the rescaled location $u$.
Note that $u$ is a continuous time parameter referred to as the rescaled time
index, and $T$ is interpreted as the number of available observation, hence
$1\leq t\leq T$ and $u\in[0,1]$, see Dahlhaus (1996) for further details.

We assume that the economy follows a locally stationary TVP-VAR of lag order
$p$ as

$\bm{X}_{t,T}=\bm{\Phi}_{1}(t/T)\bm{X}_{t-1,T}+\ldots+\bm{\Phi}_{p}(t/T)\bm{X}_{t-p,T}+\bm{\epsilon}_{t,T},$
(1)

where $\bm{\epsilon}_{t,T}=\bm{\Sigma}^{-1/2}(t/T)\bm{\eta}_{t,T}$ with
$\bm{\eta}_{t,T}\sim NID(0,\bm{I}_{M})$ and
$\bm{\Phi}(t/T)=(\bm{\Phi}_{1}(t/T),\ldots,\bm{\Phi}_{p}(t/T))^{\top}$ are the
time varying autoregressive coefficients. In a neighborhood of a fixed time
point $u_{0}=t_{0}/T$, we approximate the process $\bm{X}_{t,T}$ by a
stationary process $\widetilde{\bm{X}}_{t}(u_{0})$ as

$\widetilde{\bm{X}}_{t}(u_{0})=\bm{\Phi}_{1}(u_{0})\widetilde{\bm{X}}_{t-1}(u_{0})+\ldots+\bm{\Phi}_{p}(u_{0})\widetilde{\bm{X}}_{t-p}(u_{0})+\bm{\epsilon}_{t},$
(2)

with $t\in\mathbb{Z}$ and under suitable regularity conditions
$|\bm{X}_{t,T}-\widetilde{\bm{X}}_{t}(u_{0})|=O_{p}\big{(}|t/T-u_{0}|+1/T\big{)}$
which justifies the notation “locally stationary process.” Crucially, the
process has time varying VMA($\infty$) representation (Dahlhaus et al., 2009;
Roueff and Sanchez-Perez, 2016)

$\bm{X}_{t,T}=\sum_{h=-\infty}^{\infty}\bm{\Psi}_{t,T}(h)\bm{\epsilon}_{t-h}$
(3)

where $\bm{\Psi}_{t,T}(h)\approx\bm{\Psi}(t/T,h)$ is a stochastic process
satisfying $\sup_{\ell}||\bm{\Psi}_{t}-\bm{\Psi}_{\ell}||^{2}=O_{p}(h/t)$ for
$1\leq h\leq t$ as $t\rightarrow\infty$. Since $\bm{\Psi}_{t,T}(h)$ contains
an infinite number of lags, we approximate the moving average coefficients at
$h=1,\ldots,H$ horizons (see the detailed discussion below). The network
characteristics rely on variance decompositions, which are transformations of
the $\bm{\Psi}_{t,T}(h)$ and permit the measurement of the contribution of
shocks to the system.

Since a shock to a variable in the model does not necessarily appear alone,
i.e. orthogonally to shocks to other variables, an identification scheme is
crucial in calculating variance decompositions. We adapt the generalized
identification scheme in Pesaran and Shin (1998) to locally stationary
processes. A natural way to disentangle connections that form over transitory
and persistent components of shocks is to consider a spectral representation
of the approximating model.888Baruník and Křehlík (2018) disentangle long-run
and short-run unconditional network connections using standard VAR models.
Hence instead of impulse responses, we propose to use the (local) frequency
response of a shock. The building block of our measures consider a time-
varying frequency response function
$\bm{\Psi}_{t/T}e^{-i\omega}=\sum_{h}e^{-i\omega h}\bm{\Psi}_{t,T}(h)$ which
we obtain from a Fourier transform of the coefficients with $i=\sqrt{-1}$.

Before introducing our network measures, we define the time varying spectral
density of $\bm{X}_{t,T}$ at frequency $\omega$ which is locally the same as
the spectral density of $\widetilde{\bm{X}}_{t}(u)$ at $u=t/T$ as a Fourier
transform of VMA($\infty$) filtered series as

$\bm{S}_{\bm{X}}(u,\omega)=\sum_{h=-\infty}^{\infty}\mathbb{E}\Big{[}\widetilde{\bm{X}}_{t+h}(u)\widetilde{\bm{X}}_{t}^{\top}(u)\Big{]}e^{-i\omega
h}=\Big{\\{}\bm{\Psi}(u)e^{-i\omega}\Big{\\}}\bm{\Sigma}(u)\Big{\\{}\bm{\Psi}(u)e^{+i\omega}\Big{\\}}^{\top}.$
(4)

The time-varying spectral density is a key quantity for understanding
frequency dynamics. It describes the distribution of the time varying
covariance of $\bm{X}_{t,T}$ over frequency components $\omega$. Using the
spectral representation for the local covariance that is associated with the
local spectral density,

$\mathbb{E}\left[\widetilde{\bm{X}}_{t+h}(u)\widetilde{\bm{X}}_{t}^{\top}(u)\right]=\int_{-\pi}^{\pi}\bm{S}_{\bm{X}}(u,\omega)e^{i\omega
h}d\omega$ (5)

we can naturally introduce time-varying frequency domain counterparts of
variance decompositions. In turn, this allows us to define dynamic adjacency
matrices that contain network structures with different degrees of
persistence.

###  2.1 A Route Towards Heterogeneous Persistence in Network Connections

Characterising network structures using transformations of impulse responses
$\bm{\Psi}_{t,T}(h)$ permits the measurement of the contribution of shocks to
the system. While it may seem natural to choose a forecast horizon
$h=1,\ldots,H$ of interest, the choice is costly in terms of information
aggregation and hence loss. In contrast to cumulative information with
increasing $h$, the spectral representation of impulse responses
$\bm{\Psi}_{t/T}e^{-i\omega}$ contain much richer and precise information.
Switching to the frequency domain allows one to track network structures
stemming from transitory and persistent components of shocks.

To illustrate, consider a simple bivariate system in which connections between
two variables is of interest. Specifically, our focus is on how variable $b$
responds to two different shocks to variable $a$.999Note that we assume the
shock to variable $b$ causes variable $b$ to rise by 1 unit and is persistent;
our analysis below remains the same if own shocks are transitory. The first is
a transitory shock causing variable $b$ to rise by one unit before returning
to zero in the next period. Here, we expect the share in error variation to be
large at short horizons and small at long horizons due to having a purely
transitory impact.

The second is a persistent shock resulting in a unit rise in variable $b$ and
a gradual fall to zero over the impulse horizon. In this case, one expects the
share in error variation to be large at longer-term horizons and smaller at
short horizons. How these quantities in the time- and frequency-domain capture
the responses from these two shocks is the key motivation for our measures.

Figure 1: Impulse Response Functions and Variance Decompositions in time and
frequency domain
Notes: This figure reports impulse response functions (left), variance
decompositions in time domain (middle) and the corresponding spectral
representation of the forecast error variance decompositions (right), of a
variable $b$ within a bivariate system with respect to a shock in the variable
$a$. We consider two types of shocks, a transitory shock (triangles) and a
persistent shock (dots) impacting horizon $h=1,\ldots,H$ and frequency
$\omega\in\\{0,\ldots,\omega\\}$.

Figure 1 shows the impulse response functions of variable $b$ with respect to
shocks to variable $a$ in the left panel, the corresponding share of forecast
error variances in the middle panel, and their corresponding spectral
decompositions of forecast error variance shares in the right panel. Note that
share of forecast error variation (variance decompositions) is almost
indistinguishable in the time domain from the middle panel of Figure 1. A
persistent shock results in a slightly larger value of forecast error
variation relative to the variation due to a transitory shock. At the same
time, if we estimate these quantities from data and consider uncertainty of
the estimates (the Figure plots theoretical values of a simple example), it
would become indistinguishable statistically.

In contrast, the spectral representation of the forecast error variance shares
in the right panel of Figure 1 recovers the heterogeneous impact of both
shocks across frequencies precisely. The transitory shock in variable $a$ has
negligible impact at low frequencies (i.e. near $\omega=0$) indicating that
this shock bears no importance for the long-run variation of variable $b$, and
larger weights at higher frequencies showing the transitory nature of the
connection made through this shock. Vice versa, the persistent shock impacts
low frequencies and correctly identifies a persistent connection between the
variables.

The key implication is that we can build network measures accounting for the
nature of shocks forming network connections. Using spectral decompositions,
we identify transitory and persistent network connections that stay hidden in
the time domain.

###  2.2 Dynamics and Persistence in Adjacency Matrices

The following proposition establishes the time varying spectral representation
of the variance decomposition of shocks from variable $j$ to variable $k$.
This is central to the existence of network measures in the time-frequency
domain.101010Note to notation: $[\bm{A}]_{j,k}$ denotes the $j$th row and
$k$th column of matrix $\bm{A}$ denoted in bold. $[\bm{A}]_{j,\cdot}$ denotes
the full $j$th row; this is similar for the columns. A $\sum A$, where $A$ is
a matrix that denotes the sum of all elements of the matrix $A$.

######  Proposition 1 (Dynamic Adjacency Matrix).

Suppose $\mathbf{\bm{X}_{t,T}}$ is a weakly locally stationary process with
$\sigma_{kk}^{-1}\displaystyle\sum_{h=0}^{\infty}\left|\Big{[}\bm{\Psi}(u,h)\bm{\Sigma}(u)\Big{]}_{j,k}\right|<+\infty,\forall
j,k.$ Then the time-frequency variance decompositions of the $j$th variable at
a rescaled time $u=t_{0}/T$ due to shocks in the $k$th variable on the
frequency band $d=(a,b):a,b\in(-\pi,\pi),a<b$ form a dynamic adjacency matrix
defined as

$\Big{[}\bm{\theta}(u,d)\Big{]}_{j,k}=\frac{\sigma_{kk}^{-1}\displaystyle\int_{a}^{b}\left|\bigg{[}\bm{\Psi}(u)e^{-i\omega}\bm{\Sigma}(u)\bigg{]}_{j,k}\right|^{2}d\omega}{\displaystyle\int_{-\pi}^{\pi}\Bigg{[}\Big{\\{}\bm{\Psi}(u)e^{-i\omega}\Big{\\}}\bm{\Sigma}(u)\Big{\\{}\bm{\Psi}(u)e^{+i\omega}\Big{\\}}^{\top}\Bigg{]}_{j,j}d\omega}$
(6)

where $\bm{\Psi}(u)e^{-i\omega}=\sum_{h}e^{-i\omega h}\bm{\Psi}(u,h)$ is local
impulse transfer function or frequency response function computed as the
Fourier transform of the local impulse response $\bm{\Psi}(u,h)$

###### Proof.

See Appendix A. ∎

It is important to note that $\Big{[}\bm{\theta}(u,d)\Big{]}_{j,k}$ is a
natural disaggregation of traditional variance decompositions to time-varying
frequency bands. This is because a portion of the local error variance of the
$j$th variable at a given frequency band due to shocks in the $k$th variable
is scaled by the variance of the $j$th variable. Note that while the Fourier
transform of the impulse response generally takes on complex values, the
quantity in proposition (1) is the squared modulus of weighted complex
numbers, thus producing a real quantity.

This relationship is an identity which means the integral is a linear
operator, summing over disjoint intervals covering the entire range
$(-\pi,\pi)$ recovers the time domain counterpart of the local variance
decomposition for $h\rightarrow\infty$. The following remark formalizes this
fact.

######  Remark 1 (Aggregation of Adjacency Matrix).

Denote by $d_{s}$ an interval on the real line from the set of intervals $D$
that form a partition of the interval $(-\pi,\pi)$, such that $\cap_{d_{s}\in
D}d_{s}=\emptyset,$ and $\cup_{d_{s}\in D}d_{s}=(-\pi,\pi)$. Due to the
linearity of integral and the construction of $d_{s}$, we have

$\Big{[}\bm{\theta}(u)\Big{]}_{j,k}=\sum_{d_{s}\in
D}\Big{[}\bm{\theta}(u,d_{s})\Big{]}_{j,k}.$

Remark (1) is important as it establishes the aggregation of network
connectedness measures across different frequency bands to its time domain,
total counterpart. Hence one can easily obtain time varying network measures
across any horizon of interest using frequency bands that will always sum up
to an aggregate time domain counterpart.

As the rows of the time-frequency network connectedness do not necessarily sum
to one, we normalize the element in each by the corresponding row sum

$\Big{[}\widetilde{}\bm{\theta}(u,d)\Big{]}_{j,k}=\Big{[}\bm{\theta}(u,d)\Big{]}_{j,k}\Bigg{/}\sum_{k=1}^{N}\Big{[}\bm{\theta}(u)\Big{]}_{j,k}$
(7)

Our notion that we can approximate well the process $\bm{X}_{t,T}$, by a
stationary process $\widetilde{\bm{X}}_{t}(u)$ in a neighborhood of a fixed
time point $u=t/T$, means that all associated local quantities approximate
well their time varying counterparts. Following the arguments in Dahlhaus
(1996), and using mild assumptions, one can easily see that local variance
decompositions at a frequency band $\widetilde{}\bm{\theta}(u,d)$ approximate
well the time-varying variance decompositions of the process $\bm{X}_{t,T}$.

Note that the local generalized variance decompositions form a dynamic
adjacency matrix that defines a time-varying network at a given frequency
band. Thus, we can directly use our measures as time-varying network
characteristics that contain richer information in comparison to typical
network analysis. In our notion, variance decompositions can be viewed as
weighted links showing the strengths of connections. In addition, the links
are directional, meaning that the $j$ to $k$ link is not necessarily the same
as the $k$ to $j$ link, and hence the adjacency matrix is asymmetric. Even
more important, the adjacency matrix is time-varying and frequency specific
that allows the study of time-varying network characteristics at various
frequency bands of the user’s choice. The simplest is to measure transitory
network connections over the short-run and persistent ones over the long-run.

Now we can define network connectedness measures that characterize a time-
varying and frequency specific network. We define local network connectedness
measures at a given frequency band as the ratio of the off-diagonal elements
to the sum of the entire matrix

$\mathcal{C}(u,d)=100\times\displaystyle\sum_{\begin{subarray}{c}j,k=1\\\
j\neq
k\end{subarray}}^{N}\Big{[}\widetilde{}\bm{\theta}(u,d)\Big{]}_{j,k}\Bigg{/}\displaystyle\sum_{j,k=1}^{N}\Big{[}\widetilde{}\bm{\theta}(u)\Big{]}_{j,k}$
(8)

This measures the contribution of forecast error variance attributable to all
shocks in the system, minus the contribution of own shocks over frequency band
$d$ and infers system-wide connectedness over such frequency band. We can also
define measures that reveal when an individual variable in the economy is a
transmitter or receiver of shocks. Local directional connectedness measures
how much of each variables’s $j$ variance is due to shocks in other variables
$k\neq j$ in the economy over frequency band $d$ is given by

$\mathcal{C}_{j\leftarrow\bullet}(u,d)=100\times\displaystyle\sum_{\begin{subarray}{c}k=1\\\
k\neq
j\end{subarray}}^{N}\Big{[}\widetilde{}\bm{\theta}(u,d)\Big{]}_{j,k}\Bigg{/}\displaystyle\sum_{j,k=1}^{N}\Big{[}\widetilde{}\bm{\theta}(u)\Big{]}_{j,k},$
(9)

defining the so-called from connectedness. One can precisely interpret this
quantity as from-degrees (often called out-degrees in the network literature)
associated with the nodes of the weighted directed network represented by the
variance decompositions matrix generalized to a time-varying frequency
specific quantity. Likewise, the contribution of variable $j$ to variances in
other variables is computed as

$\mathcal{C}_{j\rightarrow\bullet}(u,d)=100\times\displaystyle\sum_{\begin{subarray}{c}k=1\\\
k\neq
j\end{subarray}}^{N}\Big{[}\widetilde{}\bm{\theta}(u,d)\Big{]}_{k,j}\Bigg{/}\displaystyle\sum_{k,j=1}^{N}\Big{[}\widetilde{}\bm{\theta}(u)\Big{]}_{k,j}$
(10)

and is the so-called to connectedness. Again, one can precisely interpret this
as to-degrees (often called in-degrees in the network literature) associated
with the nodes of the weighted directed network represented by the variance
decompositions matrix. These two measures show how other variables contribute
to the variation of variable $j$, and how variable $j$ contributes to the
variation of others, respectively, in a time varying fashion at a chosen
frequency band. We note here that taking the difference between to
connectedness and from connectedness summarizes information regarding
directional connections in net-terms. Further, one can track pairwise
connections over frequency bands in an analogous manner to the above as
differences between the $j-k$th element and $k-j$th elements.

Importantly, the following proposition shows one can always reconstruct time
domain network connectedness measures from our frequency-dependent networks.

######  Proposition 2 (Reconstruction of Dynamic Network Connectedness).

Denote by $d_{s}$ an interval on the real line from the set of intervals $D$
that form a partition of the interval $(-\pi,\pi)$, such that $\cap_{d_{s}\in
D}d_{s}=\emptyset,$ and $\cup_{d_{s}\in D}d_{s}=(-\pi,\pi)$. We then have that

$\displaystyle\begin{split}\mathcal{C}(u)&=\sum_{d_{s}\in
D}\mathcal{C}(u,d_{s})\\\ \mathcal{C}_{j\leftarrow\bullet}(u)&=\sum_{d_{s}\in
D}\mathcal{C}_{j\leftarrow\bullet}(u,d_{s})\\\
\mathcal{C}_{j\rightarrow\bullet}(u)&=\sum_{d_{s}\in
D}\mathcal{C}_{j\rightarrow\bullet}(u,d_{s})\end{split}$ (11)

where $\mathcal{C}(u)$ are local network connectedness measures aggregated
over frequencies.

###### Proof.

See Appendix A. ∎

In light of the above, all local frequency connectedness measures
$\mathcal{C}(u,d)$ for $u=t/T$ approximate well the time-varying frequency
connectedness of the process $\bm{X}_{t,T}$.

###  2.3 Obtaining Transitory and Persistent Network Measures

In light of the assumptions that underpin our measures, we conjecture that the
economy follows a stable time-varying parameter heteroskedastic VAR (TVP-VAR)
model as in (1). To obtain the time-varying coefficient estimates at a fixed
time point $u=t_{0}/T$,
$\widehat{\bm{\Phi}}_{1}(u),...,\widehat{\bm{\Phi}}_{p}(u)$, and the time-
varying covariance matrices, $\widehat{\bm{\Sigma}}(u)$, we use the time-
varying parameter VAR in Petrova (2019) that relies on Quasi-Bayesian Local-
Likelihood (QBLL) estimation methods.

This approach uses a kernel weighting function that provides larger weights to
observations that surround the period whose coefficient and covariance
matrices are of interest. Using conjugate priors, the (quasi) posterior
distribution of the parameters of the model are analytical. This alleviates
the need to use a Markov Chain Monte Carlo (MCMC) simulation algorithm and
permits the use of parallel computing. Note also that in using (quasi)
Bayesian estimation methods, we obtain a distribution of parameters that we
use to construct network measures that provide confidence bands. Details of
the model and estimation algorithm are in Appendix B.111111Unlike traditional
TVP VARs time-variation evolves in a non-parametric manner thus making no
assumption on the laws of motion within the model. Typically, the model of
Primiceri (2005), and many extensions, assume parameters evolve as random
walks or autoregressive processes. We provide a computationally efficient
package DynamicNets.jl in JULIA and DynamicNets in MATLAB that allows one to
obtain our measures on data the researcher desires.121212The packages are
available at https://github.com/barunik/DynamicNets.jl and
https://github.com/ellington/DynamicNets

To estimate the elements of dynamic adjacency matrix, we first need to
truncate the infinite VMA($\infty$) representation of the approximating model
with a choice of finite horizon $H$. Here we note that in the frequency domain
quantities, $H$ servers only as an approximation factor, and it has no
interpretation as in the time domain. Hence in the applications we advise
setting the $H$ sufficiently high to obtain a better approximation,
particularly when lower frequencies are of interest. We obtain horizon
specific measures using Fourier transforms and set our truncation horizon
$H=100$. Note, we run all results in this paper for $H\in\\{50,100,200\\}$,
they are qualitatively similar and available upon request.

Next, estimating dynamic network measures requires the user to choose a kernel
and its bandwidth. Typically the larger the bandwidth, the smoother time-
evolution of our frequency specific network measures. Therefore, prior to
tracking dynamic network connections, it is important the user considers the
time-series properties of their data. For example if common peaks (troughs) in
the time-series occur frequently and are transient, then a shorter bandwidth
may be necessary. Conversely, if tracking network connections among data that
evolves gradually over time, like interest rates, a larger bandwidth may be
more appropriate. In the context of our study, we use a Normal kernel and
explore the implication of bandwidth choice for a variety of data generating
processes (DGPs) in Section 3.

It is noteworthy to mention that the choice of a two-sided kernel can come at
a cost; especially if one wishes to use network measures for forecasting
purposes. In these cases one may wish to: i) estimate the dynamic network
recursively throughout time such that the Normal kernel truncates to use only
past values at the time $T$ estimate; or ii) choose a one-sided kernel such as
those in Hahn et al. (2001); Barigozzi et al. (2020).

We estimate the $j,k$ element of our dynamic adjacency matrix at time
$u=t_{0}/T$ and horizon $d=(a,b):a,b\in(-\pi,\pi)$ and $a<b$ such that it
corresponds to the transitory (high frequency band) and persistent (low
frequency band) element of the adjacency matrix respectively as:

$\Big{[}\widehat{}\bm{\theta}(u,d)\Big{]}_{j,k}=\frac{\widehat{\sigma}_{kk}^{-1}\displaystyle\sum_{\omega\in
d}\left(\bigg{[}\widehat{}\bm{\Psi}(u,\omega)\widehat{}\bm{\Sigma}(u)\bigg{]}_{j,k}\right)^{2}}{\displaystyle\sum_{\omega\in(-\pi,\pi)}\Bigg{[}\widehat{}\bm{\Psi}(u,\omega)\widehat{}\bm{\Sigma}(u)\widehat{}\bm{\Psi}^{\top}(u,\omega)\Bigg{]}_{j,j}},$
(12)

where
$\widehat{}\bm{\Psi}(u,\omega)=\sum_{h=0}^{H-1}\sum_{h}\widehat{}\bm{\Psi}(u,h)e^{-i\omega
h}$ is an estimate of the impulse transfer function from Fourier frequencies
$\omega\in\\{aH/2\pi,\ldots,bH/2\pi\\}$ of impulse response functions that
cover a specific horizon.131313Note that $i=\sqrt{-1}$. From this, estimates
of Equations (8)–(10) directly follow. For example if the application uses
daily data, one may define transitory (short-term) as horizons corresponding
to 1–5 days and persistent (long-term) as horizons corresponding to horizons
greater than 5 days. This would require defining the band as
$(a,b)=(2\pi/5,2\pi)$ for the transitory and $(a,b)=(0,2\pi/5)$ for the
persistent networks.

###  2.4 Testing for Heterogeneities in Network Persistence

We now consider how one can determine, from a statistical perspective,
heterogeneities between network connections one computes over different
frequency bands. Since we obtain our dynamic network measures in a Bayesian
setting, there are three alternative approaches for hypothesis testing. The
first is the Bayes factor, the second uses posterior credible intervals, and
the third follows statistical decision theory. We follow the latter and
utilize the work of Li et al. (2014), Li et al. (2015) and Liu et al.
(2020).141414Bayes factors involve comparing the marginal likelihoods of two
competing models and extensively appear in the literature (see e.g. Koop et
al., 2010; Chan, 2020). This is not appropriate in our setting because the
network connectedness measures come from a manipulation of a sequence of
posterior parameters from the same model; we have no alternative model to
specify the marginal likelihood. Using posterior credible intervals is
possible and something we consider in the spirit of Cogley et al. (2010). In
particular we use the joint posterior distribution of network connectedness
measures to compute the probability that connectedness across one frequency
band is larger than an analogous measure across another frequency band. These
results are in Appendix C.

These studies focus on developing test statistics of a point null hypothesis
using the posterior distribution of parameters from a Bayesian model. The
approach requires only the posterior distribution of parameters and has
various advantages. First, they overcome the problem of the
Jeffrey’s-Lindely’s paradox. Second, are not sensitive to the prior and are
pivotal quantities. Third, they are easy to compute. Crucially, these
statistics directly come from quadratic loss functions, as with classical test
statistics, and therefore possess the same distributions as their frequentist
counterparts.

Specifically, Li et al. (2015) develop a Bayesian Lagrange-Multiplier (LM)
type test that is asymptotically equivalent to a classical LM test. Using
similar assumptions, Liu et al. (2020) develop a Bayesian Wald type test that
is asymptotically equivalent to a classical Wald test and requires only the
posterior mean and posterior variance of parameters under the null
hypothesis.151515They also show asymptotic equivalence between their Wald-type
test the LM type test in Li et al. (2015).

Our framework of testing for differences across frequency specific network
measures also utilizes the work of Breitung and Candelon (2006). They develop
a frequency domain Wald test to check for short- and long-run causality. In
particular, they show that one can compute a Wald test on a spectral
decomposition of the VMA($\infty$) representation using a simple set of linear
restrictions. Although the Fourier transforms are nonlinear functions of the
VAR parameters, Breitung and Candelon (2006) show that one can implement a
Wald test based on linear restrictions that possesses the conventional
$\chi^{2}$ distribution.

We combine the assumptions in Li et al. (2015) and Liu et al. (2020) with the
results in Breitung and Candelon (2006), and propose a Wald type test for
heterogeneities between time $u$ network measures across different frequency
bands. This is because our dynamic network measures are manipulations of
impulse transfer functions that directly stem from the VMA($\infty$)
representation.

We test the null hypothesis,
H${}_{0}:\>\mathcal{C}(u,d_{a})=\mathcal{C}(u,d_{b})$ that network
connectedness across frequency band $d_{a}$ and $d_{b}$ are equivalent against
the alternative, H${}_{1}:\>\mathcal{C}(u,d_{a})\neq\mathcal{C}(u,d_{b})$.
This is equivalent to testing
H${}_{0}:\>\mathcal{C}(u,d_{a})-\mathcal{C}(u,d_{b})=0$ against
H${}_{1}:\>\mathcal{C}(u,d_{a})-\mathcal{C}(u,d_{b})\neq 0$. The following
proposition establishes the test statistic and its asymptotic distribution
under the null.

######  Proposition 3 (Testing for Heterogeneities in Network Connectedness).

Let $\bar{\mathcal{D}}(u)$ and
$\mathbf{V}_{\mathcal{D}}\left(\bar{\mathcal{D}}(u)\right)$ denote the time
$u$ posterior mean and variance of the difference between network
connectedness across frequency band $d_{a}$ and $d_{b}$. Then the test
statistic under H0

$\displaystyle\mathbf{W}\left(\bm{X}_{t,T},\mathcal{D}_{0}(u)\right)$
$\displaystyle=$ $\displaystyle
q_{\mathcal{D}}+\left(\bar{\mathcal{D}}(u)-\mathcal{D}_{0}(u)\right)^{\top}\left[\mathbf{V}_{\mathcal{D}}\left(\bar{\mathcal{D}}(u)\right)\right]^{-1}\left(\bar{\mathcal{D}}(u)-\mathcal{D}_{0}(u)\right)$
(13) $\displaystyle=$ $\displaystyle q_{\mathcal{D}}+\mathbf{Wald}$ (14)

where
$\mathbf{Wald}=\left(\bar{\mathcal{D}}(u)-\mathcal{D}_{0}(u)\right)^{\top}\left[\mathbf{V}_{\mathcal{D}}\left(\bar{\mathcal{D}}(u)\right)\right]^{-1}\left(\bar{\mathcal{D}}(u)-\mathcal{D}_{0}(u)\right)$,
$q_{\mathcal{D}}$ is the number of restrictions, and
$\mathcal{D}_{0}(u)=\mathcal{C}(u,d_{a})-\mathcal{C}(u,d_{b})=0$. Under mild
regularity assumptions:

$\displaystyle\mathbf{W}\left(\bm{X}_{t,T},\mathcal{D}_{0}(u)\right)-q_{\mathcal{D}}=\mathbf{Wald}+O_{p}(1)\rightarrow^{d}\chi^{2}\left(q_{\mathcal{D}}\right)$

###### Proof.

See Appendix A. ∎

It is important to note that it is straightforward to generalize Equation (13)
to include multiple restrictions. This may be applicable if one requires
testing equivalence among more than two frequency bands. We also note that one
may also wish to utilize the above to test for differences between directional
network connections over frequency bands. In the context of the above one
would take the difference between net-directional connections, or pairwise
directional connections, over frequency bands and compute the test in an
analogous manner to below.

For estimation purposes, the test statistic only requires the posterior mean
and the posterior variance of $\mathcal{D}(u)$, $\bar{\mathcal{D}}(u)$ and
$\mathbf{V}\left(\bar{\mathcal{D}}\right)$. Let
$\left\\{\mathcal{D}^{[r]}\right\\}^{R}_{r=1}$ denote the posterior draws such
that
$\mathcal{D}^{[r]}=\widehat{\mathcal{C}}^{[r]}(u,d_{a})-\widehat{\mathcal{C}}^{[r]}(u,d_{b})$
is the $r$th posterior draw of the difference between estimates of network
connectedness over frequency band $d_{a}$ and $d_{b}$. Then, the estimate of
our test statistic for heterogeneities between network connectedness is given
by

$\displaystyle\widehat{\mathbf{W}}\left(\bm{X}_{t,T},\mathcal{D}(u)=0\right)=\frac{\frac{1}{R}\sum^{R}_{r=1}\left(\mathcal{D}^{[r]}(u)\right)^{2}}{\frac{1}{R}\sum^{R}_{r=1}\left(\mathcal{D}^{[r]}(u)-\bar{\bar{\mathcal{D}}}(u)\right)^{2}},\>\text{with}\>\bar{\bar{\mathcal{D}}}(u)=\frac{1}{R}\sum^{R}_{r=1}\mathcal{D}^{[r]}(u)$
(15)

Under the null, we have
$\mathbf{W}\left(\bm{X}_{t,T},\mathcal{D}_{0}(u)\right)-q_{\mathcal{D}}\rightarrow^{d}\chi^{2}\left(q_{\mathcal{D}}\right)$
with $q_{\mathcal{D}}=1$ in this particular case.161616 In the classical
setting one uses Wald tests to check a variety of restrictions, such as a
parameter of interest being equal to zero, or equivalence between two
parameters of interest under the null hypothesis. When one tests the latter,
the resulting Wald test is $\backsim\chi^{2}(1)$. Thus, we only need to
compare $\widehat{\mathbf{W}}\left(\bm{X}_{t,T},\mathcal{D}(u)=0\right)$ to
the critical values of the $\chi^{2}\left(1\right)$ distribution. Rejecting
the null implies that the time $u$ network connectedness over frequency band
$d_{a}$ is statistically different to the corresponding time $u$ network
connectedness over frequency band $d_{b}$.

##  3 Monte Carlo Study

In this section, we conduct a Monte Carlo exercise to understand the finite
sample properties of our network connectedness measures. In order to motivate
the need to focus on network connections over different degrees of
persistence, we generate data with different levels of persistence throughout
time and also changes in the covariance structure. This will induce
differences in network connectedness measures we compute over different
frequency bands. Here we concentrate on low and high frequency bands and
consider four different data generating processes (DGP) to highlight their
uses.

For simplicity, we focus on bivariate VAR(2) models with time-varying
parameters and time-varying covariance matrices:

$\displaystyle\bm{X}_{t,T}$ $\displaystyle=$
$\displaystyle\bm{\Phi}_{0}(u)+\bm{\Phi}_{1}(u)\bm{X}_{t-1,T}+\bm{\Phi}_{2}(u)\bm{X}_{t-2,T}+\bm{\epsilon}_{t,T},$
$\displaystyle\bm{\epsilon}_{t,T}$ $\displaystyle=$
$\displaystyle\bm{\Sigma}^{-1/2}(u)\bm{\eta}_{t,T},\>\bm{\eta}_{t,T}\backsim\left(0,\mathbf{I}_{2}\right)$

where $\bm{\Phi}_{0}(u)$ contains the time-varying intercepts and
$\bm{\Phi}_{1}(u)$ and $\bm{\Phi}_{2}(u)$ contain the time-varying
autoregressive parameters. The time-varying covariance matrix
$\bm{\Sigma}(u)=\mathbf{A}^{-1}(u)\mathbf{H}(u)\left(\mathbf{A}^{-1}(u)\right)^{\top}$
with $\mathbf{A}^{-1}(u)$ being a lower triangular matrix with a unit diagonal
and $\mathbf{H}(u)$ is a $2\times 2$ diagonal matrix.

DGPI: Our first DGP has residuals such that,
$\bm{\eta}_{t,T}\backsim\text{NID}\left(0,\mathbf{I}_{2}\right)$. The time-
varying intercepts follow the process:

$\displaystyle\left[\bm{\Phi}_{0}(u)\right]_{j}$ $\displaystyle=$
$\displaystyle 0.0025\sin\left(0.004\pi
t\right)+0.15\sum^{t}_{i=1}\frac{\nu_{i}}{\sqrt{t}},\>\nu_{i}\backsim\text{NID}\left(0,0.001^{2}\right),\>j=1,2$

For the time-varying autoregressive parameters, we have

$\displaystyle\left[\bm{\Phi}_{g}(u)\right]_{j,k}=\begin{cases}0.05\sin\left(0.002\pi
t\right)+0.75\sum^{t}_{i=1}\frac{\kappa_{i}}{\sqrt{t}},\>&t\in\\{1,...,500\\}\>\forall
g,j,k=1,2\\\ 0.45\sin\left(0.002\pi
t\right)+0.75\sum^{t}_{i=1}\frac{\kappa_{i}}{\sqrt{t}},\>&t\in\\{501,...,1000\\}\>j=k=1,\>j=k=2\\\
0.05\sin\left(0.002\pi
t\right)+0.75\sum^{t}_{i=1}\frac{\kappa_{i}}{\sqrt{t}},\>&t\in\\{501,...,1000\\}\>j=1,k=2\>\text{\&}\>j=2,k=1\end{cases}$

with $\kappa_{i}\backsim\text{NID}\left(0,0.0001^{2}\right)$. The (2,1)
element of $\mathbf{A}(u)$ have the following dynamics:

$\displaystyle\left[\mathbf{A}(u)\right]_{2,1}=\begin{cases}0.03\sin\left(0.002\pi
t\right)+0.7\sum^{t}_{i=1}\frac{\upsilon_{i}}{\sqrt{t}},\>&t\in\\{1,...,500\\}\\\
1.5\sin\left(0.002\pi
t\right)+0.7\sum^{t}_{i=1}\frac{\upsilon_{i}}{\sqrt{t}},\>&t\in\\{501,...,1000\\}\end{cases}$

with $\upsilon_{i}\backsim\text{NID}\left(0,0.3^{2}\right)$. The diagonal
elements of $\mathbf{H}(u)$ follow

$\displaystyle\log\left[\mathbf{H}(u)\right]_{j,j}$ $\displaystyle=$
$\displaystyle\mu_{j}+\lambda_{j}\left(\log\left[\mathbf{H}(u-1)\right]_{j,j}-\mu_{j}\right)+\xi_{j,t}$

where
$\xi_{j,t}\backsim\text{NID}\left(\mu_{j},0.1^{2}/(1-\lambda_{j})\right),\>\mu_{j}=0.01,\>\lambda_{j}=0.95$.

This DGP has little to no dependence for the first 500 observations which
means connectedness at both high and low frequency bands will be low and close
to zero. The latter half of the sample sees the AR coefficients in each
equation become persistent as the sin wave becomes negative. Note also that
the contemporaneous relationship intensifies. This induces connections at high
frequency bands while connections at low frequency bands should be low and
close to zero. Note we also allow for non-Gaussian residuals in this DGP such
that we draw $\bm{\eta}_{t,T}$ from a multivariate student-$t$ distribution
with 5 degrees of freedom. These results are in Appendix C.

DGPII: For our second DGP, the time-varying intercepts follow the process:

$\displaystyle\left[\bm{\Phi}_{0}(u)\right]_{j}$ $\displaystyle=$
$\displaystyle 0.25\sin\left(0.004\pi
t\right)+0.15\sum^{t}_{i=1}\frac{\nu_{i}}{\sqrt{t}},\>\nu_{i}\backsim\text{NID}\left(0,0.1^{2}\right),\>j=1,2$

and the time-varying autoregressive parameters follow:

$\displaystyle\left[\bm{\Phi}_{g}(u)\right]_{j,k}=0.25\sin\left(0.004\pi
t\right)+0.75\sum^{t}_{i=1}\frac{\kappa_{i}}{\sqrt{t}},\forall g,j,k=1,2$

with $\kappa_{i}\backsim\text{NID}\left(0,0.3^{2}\right)$. The (2,1) element
of $\mathbf{A}(u)$ and the diagonal elements of $\mathbf{H}(u)$ follow the
processes:

$\displaystyle\left[\mathbf{A}(u)\right]_{2,1}$ $\displaystyle=$
$\displaystyle 0.3\sin\left(0.008\pi
t\right)+0.7\sum^{t}_{i=1}\frac{\upsilon_{i}}{\sqrt{t}}$
$\displaystyle\log\left[\mathbf{H}(u)\right]_{j,j}$ $\displaystyle=$
$\displaystyle\mu_{j}+\lambda_{j}\left(\log\left[\mathbf{H}(u-1)\right]_{j,j}-\mu_{j}\right)+\xi_{j,t},$

where $\upsilon_{i}\backsim\text{NID}\left(0,0.3^{2}\right)$ and
$\xi_{j,t}\backsim\text{NID}\left(\mu_{j},0.1^{2}/(1-\lambda_{j})\right),\>\mu_{j}=0.01,\>\lambda_{j}=0.95$.

This DGP induces two distinct periods of persistence during observations
100-200 and 600-700 which amplifies connectedness at the low frequency band.

DGPIII: Our third DGP is the same as DGPII but relaxes the assumption that
$\bm{\eta}_{t,T}$ are Gaussian. Instead we assume that the residuals follow a
multivariate student-$t$ distribution with $5$ degrees of freedom.

DGPIV: Our fourth DGP is the same as DGPII, but increases the periodicity of
the sin functions in the time-varying autoregressive matrices from
$\sin\left(0.004\pi t\right)$ to $\sin\left(0.006\pi t\right)$. This generates
three distinct periods of persistence during observations 50-150, 300-450, and
700-850.

Figure 2: Dynamic network connectedness measures: True and fitted values
Notes: This figure plots the true network connectedness measures for three
data generating processes following bi-variate TVP VAR(2) models along with
the median and 95% quantiles of estimated network connectedness measures with
bandwidth $W$=8. The left columns report network connectedness on the low-
frequency band, $d\in(0,\>\pi/5)$, the middle columns show network
connectedness on the high-frequency band, $d\in(\pi/5,\>\pi)$, and the right
columns show the aggregate network connectedness such that $d\in(0,\>\pi)$.
DGPI (top row) is a TVP VAR(2) model with Gaussian errors, we introduce a
break in the time-varying autoregressive matrices and contemporaneous
relations from observation 500 that induces large connections across the high
frequency band. DGPII (second row) is a TVP VAR(2) where time-varying
intercepts and autoregressive matrices following sin wave functions with a
stochastic error, time-varying covariance matrix where the off-diagonals
follow sin wave functions with a stochastic error, and the diagonal elements
follow a stationary AR(1) processes. DGPIII (third row) is a TVP VAR(2) model
with student-$t$ errors, time-varying intercepts and autoregressive matrices
following sin wave functions with a stochastic error, time-varying covariance
matrix where the off-diagonals follow sin wave functions with a stochastic
error, and the diagonal elements follow a stationary AR(1) processes. DGPIV
(bottom row) is the same as DGPII, but with an increase in the periodicity of
the respective sin wave functions the time-varying intercepts and
autoregressive matrices follow.

For each of the four DGPs, we generate 100 simulations of length $T=1000$ and
compute the network connectedness measures. We use the median over these
simulations as the true network connectedness. Then, for each of the 100
simulations of DGPI–DGPIV, we fit the TVP-VAR model we outline in Section 2.3.
In fitting this model we take 1000 draws from the posterior distribution,
calculate our network connectedness measures and then save the posterior
median. For this exercise, we compute network connectedness on two frequency
bands that cover the spectrum. The low-frequency band, which empirically
pertains to persistent network connections, is $d\in(0,\>\pi/5)$, and the
high-frequency band, pertaining to transitory network connections, is
$d\in(\pi/5,\>\pi)$. For completeness, we compute the aggregate connectedness
measures that considers the entire spectrum such that $d\in(0,\>\pi)$.

A final noteworthy point is the choice of bandwidth, $W$, for the kernel
weights that induce time-variation into the TVP-VAR model. In this exercise,
we consider $W=\\{8,\>12,\>18\\}$. However, for ease of exposition, we only
report plots of the network connectedness measure estimates using $W$=8 in the
main text, results of fitted values from $W=\\{12,\>18\\}$ are in Appendix C.
The larger the bandwidth, the smoother the network measures become. This is
because larger bandwidths assign weights to a higher number of observations
around the one of interest. In general, we find that larger $W$ results in
poorer fit. This highlights the importance of selecting an appropriate
bandwidth for the kernel weights relative to the data application.

Figure 3: Tests for heterogeneities between network connectedness across
different frequency bands
Notes: This figure plots the test statistics and $+/-$ 2 $\times$ the
numerical standard errors for heterogeneities between network connectedness
measures across different frequency bands from four DGPs. The test statistic
is $\widehat{\mathbf{W}}\left(\bm{X}_{t,T},\mathcal{D}(u)=0\right)-1$ where
$\mathcal{D}(u)=\widehat{\mathcal{C}}(u,d)-\widehat{\mathcal{C}}(u,c)$ with
$d\in(0,\>\pi/5)$ corresponding to the low frequency band and
$c\in(\pi/5,\>\pi)$ corresponding to the high frequency band. The dashed black
line is the 5% critical value from the $\chi^{2}(1)$ distribution =3.84.
Values greater than 3.84 reject the null hypothesis of equivalent network
connections across frequency band $d$ and $c$ in favor of the heterogeneity.
DGPI (top left panel) is a TVP VAR(2) model with Gaussian errors, we introduce
a break in the time-varying autoregressive matrices and contemporaneous
relations from observation 500 that induces large connections across the high
frequency band. DGPII (top right panel) is a TVP VAR(2) where time-varying
intercepts and autoregressive matrices following sin wave functions with a
stochastic error, time-varying covariance matrix where the off-diagonals
follow sin wave functions with a stochastic error, and the diagonal elements
follow a stationary AR(1) processes. DGPIII (bottom left panel) is a TVP
VAR(2) model with student-$t$ errors, time-varying intercepts and
autoregressive matrices following sin wave functions with a stochastic error,
time-varying covariance matrix where the off-diagonals follow sin wave
functions with a stochastic error, and the diagonal elements follow a
stationary AR(1) processes. DGPIV (bottom right panel) is the same as DGPII,
but with an increase in the periodicity of the respective sin wave functions
the time-varying intercepts and autoregressive matrices follow.

Figure 2 reports the true network connectedness measures and the median and
95% quantiles of corresponding estimates from the TVP VAR model using a kernel
bandwidth of $W$=8. We report network connectedness over the low-frequency-
band, the high-frequency-band, and aggregate, in the left, middle, and right
columns respectively. The top row corresponds to DGPI, and the second, third
and fourth rows results from DGPII, DGPIII, and DGPIV respectively. As we can
see, the distribution of estimates for each DGP track the true values
remarkably well. In almost all cases, the true value lies within the 95%
quantiles of the distribution from model estimates. This plot shows that our
method provides an accurate representation of horizon specific network
connectedness, even when the underlying process has complex dynamics and the
true error distribution is non-Gaussian.

Figure 3 further reports the estimates, and numerical standard error bounds of
our test for heterogeneities between high frequency band and low frequency
band network connectedness measures (Equation 15) for each of the four DGPs we
use in the Monte Carlo study. Specifically, for each observation we test the
null hypothesis $\mathcal{D}(u)=0$ where
$\mathcal{D}(u)=\widehat{\mathcal{C}}(u,d)-\widehat{\mathcal{C}}(u,c)$ with
$d\in(0,\>\pi/5)$ corresponding to the low frequency band and
$c\in(\pi/5,\>\pi)$ corresponding to the high frequency band. In each plot, we
also report the 5% critical value from the $\chi^{2}(1)$ distribution of 3.84.
Test statistics exceeding this value reject the null in favor of
heterogeneities between network connectedness measures across frequency
bands.171717We obtain numerical standard errors in a similar manner to Li et
al. (2015). As we can see, our test statistic identifies significant
differences between low and high frequency band network connectedness measures
that correspond with the peaks we observe in Figure 2 for each DGP. We can see
with the non-Gaussian DGP, DGPIII the estimates of the test statistics are
smaller relative to the analogous Gaussian DGP, DGPII. However, there are
still clear rejections.

##  4 Monitoring Persistence in Uncertainty Networks of S&P500 Sectors

Changes in uncertainty can play a key role in driving business cycles and
financial unrest (Bloom et al., 2018). Identifying the sources of such risks
are a focal point for researchers and practitioners. Some quantify systemic
risks emerging from financial markets and sectors (e.g. Billio et al., 2012;
Acemoglu et al., 2015), while others examine how sectoral shocks influence
aggregate fluctuations (e.g. Gabaix, 2011; Acemoglu et al., 2017; Atalay,
2017). In tandem to the above, and in response to financial crises, many
nations implement policies around monitoring systemic risks and financial
stability.

We use our framework to identify new transitory and persistent connectedness
measures within S&P500 sectors, as well as studying such network structures at
a granular level. Production networks provide a feasible economic rationale as
to how firms connect (e.g. Herskovic, 2018). These customer-supplier relations
allow one to understand the structural determinants of such connections, as
well as how they influence risk premia (Gofman et al., 2020) and
characteristics such as firm-size dispersion (Herskovic et al., 2020).
However, building these network structures can often require data that one
observes at a monthly or annual basis, such as the Bureau of Economic Analysis
(BEA) input-output data. This means that tracking network connections on a
daily basis is almost impossible since data relating to these linkages are not
readily available. We adopt a market-based approach which allows us to track
daily network dynamics with minimal assumptions.

We use high frequency data for all stocks listed on the S&P500 from July 5,
2005 to August 31, 2018 from Tick Data. Focusing on sectoral analysis, we
divide the stocks into eleven main sectors according to the Global Industry
Classification Standard (GICS)181818The GICS is an industry taxonomy developed
by MSCI and Standard & Poor’s for use by the global financial community. and
compute realized volatility (RV) for all stocks in the sample. In obtaining
firm-level RVs, we restrict our analysis to five-minute returns during the
business hours of the New York Stock Exchange (NYSE) (i.e. 09:30–16:00). The
data are time-synchronized by the same time stamps, eliminating transactions
executed on Saturdays and Sundays, US Federal holidays, December 24 to 26, and
December 31 to January 2, due to low activity on these days. This leaves us
with 3278 trading days.

After cleaning the data we are left with a cross-section of 496 stocks which
represent the eleven sectors. These are: Consumers Discretionary (COND) with
73 stocks; Consumer Staples (CONS) with 34 stocks; Health Care (HLTH) with 53
stocks; Industrials (INDU) with 73 stocks; Information Technology (INFT) with
67 stocks; Materials (MATR) with 33 stocks; Real Estate (REAS) with 29 stocks;
Financials (SPF) with 66 stocks; Energy (SPN) with 36 stocks; Communication
Services (TELS) with 6 stocks; and Utilities (UTIL) with 26 stocks. Further
details including descriptive statistics (Table 2) for annualized daily RVs
that we compute as $100\times\sqrt{252\times RV_{t}}$ that pools information
across firms within each sector over the sample July 5, 2005 to August 31,
2018 are reported in Appendix D. The energy sector has the highest mean, while
the financials have the highest standard deviation, skewness and kurtosis.
Overall, we can see that these sectoral RVs exhibit substantial differences in
terms of the first four moments, as well as minimum and maximum values.

###  4.1 Transitory and Persistent Network Connectedness of S&P500 Sectors

Here we focus on the overall network connectedness driven by transitory and
persistent shocks in companies at a given sector. For each of the 11 sectors,
we obtain dynamic network measures by estimating a TVP VAR model on all stocks
with two lags on our 3278 days of data. At each day, we take 500 draws from
the (quasi) posterior distribution.191919We estimate our horizon specific
dynamic network measures on a 64-core server, which results in a total
estimation time around 4–5 hours to obtain network estimates from the 11
sectors. We define transitory (short-term) network connections as those
forming over a 1-5 day horizon, and persistent (long-term) network connections
at horizons greater than 5 days (i.e. 5-days to the $\infty$ horizon). Our
choice of these horizons stems from the extant literature on volatility
modeling using high frequency data that shows daily and weekly fluctuations
hold prominent information for future volatility (e.g. Corsi and Renò, 2012).

In Figure 4, we plot the posterior median and 95% confidence bands for
transitory and persistent network connectedness for each sector as in Equation
8. These directly follow from manipulations of estimates of the dynamic
adjacency matrices (see Equation 12). Grey bars on these figures represent
periods where there are statistically significant differences, at 5% levels,
between transitory and persistent network connectedness.202020We plot the
values of these test statistics in Figure 10 in Appendix D. Overall, we
observe substantial differences between these horizon specific networks for
each sector. In general, network connectedness due to the persistent component
of shocks surpasses networks from the transitory component of shocks during
periods of market turbulence. Then, during periods of tranquillity transitory
part of the networks become more prominent.

Figure 4: Dynamics and Persistence in the U.S. Uncertainty Networks
This figure plots the (quasi) posterior median and 95% confidence bands of
network connectedness specific to the transitory, or short-term (in blue) and
persistent, or long-term (in red) shocks to realized volatility of the S&P500
sectors: Consumers Discretionary (COND), Consumer Staples (CONS), Health Care
(HLTH), Industrials (INDU), Information Technology (INFT), Materials (MATR),
Real Estate (REAS), Financials (SPF), Energy (SPN), Comunication Services
(TELS), and Utilities (UTIL) from July 8, 2005 to August 31, 2018. We define
transitory (short-term), as connections made over the 1 day to 1 week horizon;
we characterize connections greater than 1 week as persistent (long-term).
Grey bars indicate periods with significant heterogeneity in persistence
structures.

Comparing these measures across sectors, we can see from the real estate and
financial that surges in network connectedness driven by persistent shocks are
driving uncertainty in the sector for much longer periods of time throughout
the Great Recession relative to other sectors; e.g. CONS and HLTH. In
addition, the magnitude of persistent network connectedness from the real
estate and financial sectors is far greater. Note also that throughout this
period, we observe far more frequent evidence in favor of statistical
differences between networks driven by transitory and persistent shocks for
these sectors; particularly relative to COND, CONS, HLTH, and MATR. Although
the other sectors exhibit spikes in persistent network connectedness, they do
not occur until around February 2008. This highlights how long-term systemic
risks within the real estate and financial sector intensify during this period
and are stronger relative to other sectors.

We also see clear surges in sectoral network connectedness due to persistent
shocks in May to October 2011 and again in 2015-2016. The former is consistent
the S&P500 entering a bear market and the latter with declines in major stock
markets across the globe. Focusing on the 2015–2016, we can see here that
long-term network connections of consumer staples, utilities, real estate and
telecommunications far smaller relative to the other sectors. We expect strong
long-term connections of industrials, materials, energy, information
technology as well as financial sectors during this period because of the fall
in global stock markets interlinking with falling commodity prices and Asian
currencies losing value against the US dollar. With regards to sectoral
network connections due to transitory shocks, the main differences we observe
are in the magnitude of connections. Note however there are subtle differences
in the time-profiles. Common to each sector, we can see that transitory
connections among firm-level uncertainty are substantially larger relative to
the persistent ones during periods of tranquillity.

Delving deeper into our investigation of whether there are significant
differences between transitory and persistent network connectedness for each
sector, two main points emerge. First, we document significant heterogeneity
in persistence for substantial periods of time. Figure 4 shows that during
periods of tranquillity transitory network connectedness among sectors are
stronger relative to connectedness from persistent shocks. However, in periods
of turbulence, our results show that persistent connections intensify in all
sectors, but their magnitudes differ considerably.212121From Figure 10 in
Appendix D we also document that the magnitude of differences evolve
substantially over time. Second, statistical differences between transitory
and persistent connectedness in one sector do not necessarily imply
differences in others. For example over the period 2014–2017, we see clear
differences in rejections that transitory and persistent connectedness are
equivalent among REAS, SPF, SPN, and UTIL.

In general the temporal nature of our findings relate well to Bianchi et al.
(2019) who document a regime dependent impact of systemic risk for financial
markets. They also provide empirical support to Gabaix (2011) and Acemoglu et
al. (2012), and uncover new measures of sectoral uncertainties (or sectoral-
wide risks). Testing for statistical differences between transitory and
persistent sectoral network connectedness adds further substance to our
conjecture that one should consider dynamic network structures that form over
frequency bands.

For researchers, our network connectedness measures may contain useful
information for the real economy or forecasting purposes; there is already
evidence of network connectedness containing predictive content for the real
economy (e.g. Baruník et al., 2020). For practitioners, tracking persistence
in sectoral networks may be useful to inform macro-prudential policies. This
is because one may use these measures as online monitoring tools in order to
investigate the evolution and persistence in sectoral-wide network
connectedness.

Although we provide evidence of substantial heterogeneity in persistence of
network structures, there are commonalities in the time profiles. We attribute
this to high levels of correlation between data used to proxy uncertainty in
our investigation. Herskovic et al. (2016) exploits the correlation structure
of idiosyncratic return volatilities and shows that a common factor among
driving firm-level volatilities has pricing implications. Our network
connectedness measures, by definition, relate to this correlation structure
and provide an aggregate description of the network at each time point.
However, our network measures are richer in information. We are able to obtain
measures that contain information about the entire network structure. These
may refer to directional connections (in- and out- degrees) or concentration
(i.e. a high influence to the overall network of a small number of
firms/nodes); both of which are shown to contain information with economic
implications (see e.g. Herskovic, 2018; Herskovic et al., 2020).

###  4.2 Persistence in Network Connections at a Granular Level

We now move on to examining network structures at a granular level. Tracking
properties of all connections at each point in time, Figures 5–6 plot the
entire network structures driven by transitory and persistent shocks for the
CONS sector at two different dates. The first is October 24, 2008 which
corresponds to the beginning of the global financial crisis, and the second is
one-year later, October 24, 2009. For each plot, arrows show the direction and
strength of the connections, while a transparent (full color) vertex denotes a
stock transmitting (receiving) more shocks than receiving (transmitting). The
size of vertices indicate net-directional connections.

Figure 5: Transitory and Persistent Network of Consumer Staples: October 24,
2008
The left (right) figure depicts network connections among assets constituting
the CONS sector driven by transitory (persistent) shocks during October 24,
2008 corresponding to the day when VIX reached highest value. Arrows denote
the direction of connections and the strength of lines denotes the strength of
connections. Grey (black) vertices denote firms receiving (transmitting) more
shocks than transmitting (receiving) them. The size of the vertices indicate
the net-amount of shocks.

We can see that on October 24, 2008, persistent connections are larger
relative to the transitory ones which indicates that shocks within consumer
staples are creating connections that relate to the long-run. This suggests
that uncertainty within the system has persistence. When comparing this to
connections one-year later, it is clear that transitory connections are
stronger than persistent ones implying that shocks within this sector are
creating connections with a more transient nature. The most important takeaway
from these plots is the stark differences in the entire structure of the
horizon specific networks. For example Archer Daniels Midland (ADM) is a net
recipient of shocks in both the short- and long-term networks on October 24,
2008. Then one-year later, despite weaker connections in the long-term
network, ADM’s role in both networks is larger. Contrastingly Walmart’s (WMT)
role changes in a similar manner. During the Great Recession it was a long-
term net recipient, while a year later it was a relatively weaker net
transmitter (in absolute terms).

Figure 6: Transitory and Persistent network of Consumer Staples: October 24,
2009
The left (right) figure depicts network connections among assets constituting
the CONS sector by transitory (persistent) connectedness. Arrows denote the
direction of connections and the strength of lines denotes the strength of
connections. Grey (black) vertices denote firms receiving (transmitting) more
shocks than transmitting (receiving) them. The size of the vertices indicate
the net-amount of shocks.

Overall, these network structures highlight how a firm’s role can change not
only over time, but also in terms of persistence. Researchers may wish to use
these network structures to assess the pricing implications of such risks. For
instance, users could assess the role of directional connections in an
empirical asset pricing application that builds on the theoretical work of
Branger et al. (2020). Our framework also allows one to track dynamic network
structures that would complement studies such as Herskovic (2018) and Gofman
et al. (2020). The benefit of our approach is that one need not rely on
monthly or annual data to span such networks. Finally, looking at dynamic
adjacency matrices may aid economists in understanding how shocks create
dynamic determine network structures in models of monetary policy (e.g. Pasten
et al., 2020) or the Phillips curve (e.g. Rubbo, 2020).

##  5 Conclusion

This paper proposes a novel framework to measure heterogeneous network
structures driven by different degrees of persistence in economic connections.
Our approach properly accounts for the characteristics of shocks that create
such connections. Using frequency domain techniques, we measure network
structures that form on the transitory and persistent component of shocks. To
ascertain whether transitory and persistent connectedness differ from another
we outline a procedure to test for statistical differences. We conduct Monte
Carlo simulations that indicate our measures are able to reliably track
connectedness and correctly identify statistical differences from various data
generating processes.

Empirically, we show that our transitory and persistent network connectedness
measures enhance our understanding of systemic risks emerging from uncertainty
networks. This is because our approach allows one to track connectedness
across the transitory and persistent component of shocks. This is particularly
useful during periods of elevated uncertainty as our measures indicate whether
systemic risks from network connections are transient or persistent in nature.
Ultimately this could lead to better decision making for macro-prudential
supervisors and market particpants’ investment decisions.

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## For Online Publication: Technical Appendix

##  Appendix A Proofs

###### Proposition 1.

Let us have the VMA($\infty$) representation of the locally stationary TVP VAR
model (Dahlhaus et al., 2009; Roueff and Sanchez-Perez, 2016)

$\bm{X}_{t,T}=\sum_{h=-\infty}^{\infty}\bm{\Psi}_{t,T}(h)\bm{\epsilon}_{t-h}$
(A.1)

$\bm{\Psi}_{t,T}(h)\approx\bm{\Psi}(t/T,h)$ is a stochastic process satisfying
$\sup_{\ell}||\bm{\Psi}_{t}-\bm{\Psi}_{\ell}||^{2}=O_{p}(h/t)$ for $1\leq
h\leq t$ as $t\rightarrow\infty$, hence in a neighborhood of a fixed time
point $u=t/T$ the process $\bm{X}_{t,T}$ can be approximated by a stationary
process $\widetilde{\bm{X}}_{t}(u)$

$\widetilde{\bm{X}}_{t}(u)=\sum_{h=-\infty}^{\infty}\bm{\Psi}(u,h)\bm{\epsilon}_{t-h}$
(A.2)

with $\bm{\epsilon}$ being iid process with $\mathbb{E}[\bm{\epsilon}_{t}]=0$,
$\mathbb{E}[\bm{\epsilon}_{s}\bm{\epsilon}_{t}]=0$ for all $s\neq t$, and the
local covariance matrix of the errors $\bm{\Sigma}(u)$. Under suitable
regularity conditions
$|\bm{X}_{t,T}-\widetilde{\bm{X}}_{t}(u)|=O_{p}\big{(}|t/T-u|+1/T\big{)}$.

Since the errors are assumed to be serially uncorrelated, the total local
covariance matrix of the forecast error conditional on the information at time
$t-1$ is given by

$\bm{\Omega}(u,H)=\sum_{h=0}^{H}\bm{\Psi}(u,h)\bm{\Sigma}(u)\bm{\Psi}^{\top}(u,h).$
(A.3)

Next, we consider the local covariance matrix of the forecast error
conditional on knowledge of today’s shock and future expected shocks to $k$-th
variable. Starting from the conditional forecasting error,

$\bm{\xi}^{k}(u,H)=\sum_{h=0}^{H}\bm{\Psi}(u,h)\Big{[}\bm{\epsilon}_{t+H-h}-\mathbb{E}(\bm{\epsilon}_{t+H-h}|\bm{\epsilon}_{k,t+H-h})\Big{]},$
(A.4)

assuming normal distribution of $\bm{\epsilon}_{t}\sim N(0,\bm{\Sigma})$, we
obtain222222Note to notation: $[\bm{A}]_{j,k}$ denotes the $j$th row and $k$th
column of matrix $\bm{A}$ denoted in bold. $[\bm{A}]_{j,\cdot}$ denotes the
full $j$th row; this is similar for the columns. A $\sum A$, where $A$ is a
matrix that denotes the sum of all elements of the matrix $A$.

$\mathbb{E}(\bm{\epsilon}_{t+H-h}|\bm{\epsilon}_{k,t+H-h})=\sigma_{kk}^{-1}\Big{[}\bm{\Sigma}(u)\Big{]}_{\cdot
k}\bm{\epsilon}_{k,t+H-h}$ (A.5)

and substituting (A.5) to (A.4), we obtain

$\bm{\xi}^{k}(u,H)=\sum_{h=0}^{H}\bm{\Psi}(u,h)\Big{[}\bm{\epsilon}_{t+H-h}-\sigma_{kk}^{-1}\Big{[}\bm{\Sigma}(u)\Big{]}_{\cdot
k}\bm{\epsilon}_{k,t+H-h}\Big{]}.$ (A.6)

Finally, the local forecast error covariance matrix is

$\bm{\Omega}^{k}(u,H)=\sum_{h=0}^{H}\bm{\Psi}(u,h)\bm{\Sigma}(u)\bm{\Psi}^{\top}(u,h)-\sigma_{kk}^{-1}\sum_{h=0}^{H}\bm{\Psi}(u,h)\Big{[}\bm{\Sigma}(u)\Big{]}_{\cdot
k}\Big{[}\bm{\Sigma}(u)\Big{]}_{\cdot k}^{\top}\bm{\Psi}^{\top}(u,h).$ (A.7)

Then

$\Big{[}\bm{\Delta}(u,H)\Big{]}_{(j)k}=\Big{[}\bm{\Omega}(u,H)-\bm{\Omega}^{k}(u,H)\Big{]}_{j,j}=\sigma_{kk}^{-1}\sum_{h=0}^{H}\Bigg{(}\Big{[}\bm{\Psi}(u,h)\bm{\Sigma}(u)\Big{]}_{j,k}\Bigg{)}^{2}$
(A.8)

is the unscaled local $H$-step ahead forecast error variance of the $j$-th
component with respect to the innovation in the $k$-th component. Scaling the
equation with $H$-step ahead forecast error variance with respect to the $j$th
variable yields the desired time varying generalized forecast error variance
decompositions (TVP GFEVD)

$\Big{[}\bm{\theta}(u,H)\Big{]}_{j,k}=\frac{\sigma_{kk}^{-1}\displaystyle\sum_{h=0}^{H}\Bigg{(}\Big{[}\bm{\Psi}(u,h)\bm{\Sigma}(u)\Big{]}_{j,k}\Bigg{)}^{2}}{\displaystyle\sum_{h=0}^{H}\Big{[}\bm{\Psi}(u,h)\bm{\Sigma}(u)\bm{\Psi}^{\top}(u,h)\Big{]}_{j,j}}$
(A.9)

Next, we derive the frequency representation of the quantity in (A.9) using
the fact that unique time varying spectral density of $\bm{X}_{t,T}$ at
frequency $\omega$ which is locally the same as the spectral density of
$\widetilde{\bm{X}}_{t}(u)$ at $u=t/T$ can be defined as a Fourier transform
of VMA($\infty$) filtered series over frequencies $\omega\in(-\pi,\pi)$ as

$\bm{S}_{\bm{X}}(u,\omega)=\sum_{h=-\infty}^{\infty}\mathbb{E}\Big{[}\bm{X}_{t+h}(u)\bm{X}_{t}^{\top}(u)\Big{]}e^{-i\omega
h}=\Big{\\{}\bm{\Psi}(u)e^{-i\omega}\Big{\\}}\bm{\Sigma}(u)\Big{\\{}\bm{\Psi}(u)e^{+i\omega}\Big{\\}}^{\top},$
(A.10)

where we consider a time varying frequency response function
$\bm{\Psi}(u)e^{-i\omega}=\sum_{h}e^{-i\omega h}\bm{\Psi}(u,h)$ which can be
obtained as a Fourier transform of the coefficients with $i=\sqrt{-1}$.

Letting $H\rightarrow\infty$, we have time varying generalized forecast error
variance decompositions

$\Big{[}\bm{\theta}(u)\Big{]}_{j,k}=\frac{\sigma_{kk}^{-1}\displaystyle\sum_{h=0}^{\infty}\Bigg{(}\Big{[}\bm{\Psi}(u,h)\bm{\Sigma}(u)\Big{]}_{j,k}\Bigg{)}^{2}}{\displaystyle\sum_{h=0}^{\infty}\Big{[}\bm{\Psi}(u,h)\bm{\Sigma}(u)\bm{\Psi}^{\top}(u,h)\Big{]}_{j,j}}=\frac{\mathcal{A}}{\mathcal{B}}.$
(A.11)

Starting with frequency domain counterpart of the nominator $\mathcal{A}$, we
will use the standard integral

$\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{i\omega(r-v)}d\omega=\begin{cases}1&\text{
for }r=v\\\ 0&\text{ for }r\neq v.\end{cases}$ (A.12)

Using the fact that
$\sum_{h=0}^{\infty}\phi(h)\psi(h)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\sum_{v=0}^{\infty}\sum_{r=0}^{\infty}\phi(r)\psi(v)e^{i\omega(r-v)}d\omega$,
we can rewrite (A.11) as

$\displaystyle\begin{split}\sigma_{kk}^{-1}\displaystyle\sum_{h=0}^{\infty}&\Bigg{(}\Big{[}\bm{\Psi}(u,h)\bm{\Sigma}(u)\Big{]}_{j,k}\Bigg{)}^{2}=\sigma_{kk}^{-1}\sum_{h=0}^{\infty}\left(\sum_{z=1}^{n}\Big{[}\bm{\Psi}(u,h)\Big{]}_{j,z}\Big{[}\bm{\Sigma}(u)\Big{]}_{z,k}\right)^{2}\\\
&=\sigma_{kk}^{-1}\frac{1}{2\pi}\int_{-\pi}^{\pi}\sum_{r=0}^{\infty}\sum_{v=0}^{\infty}\left(\sum_{x=1}^{n}\Big{[}\bm{\Psi}(u,r)\Big{]}_{j,x}\Big{[}\bm{\Sigma}(u)\Big{]}_{x,k}\right)\left(\sum_{y=1}^{n}\Big{[}\bm{\Psi}(u,v)\Big{]}_{j,y}\Big{[}\bm{\Sigma}(u)\Big{]}_{y,k}\right)e^{i\omega(r-v)}d\omega\\\
&=\sigma_{kk}^{-1}\frac{1}{2\pi}\int_{-\pi}^{\pi}\sum_{r=0}^{\infty}\sum_{v=0}^{\infty}\left(\sum_{x=1}^{n}\Big{[}\bm{\Psi}(u,r)e^{i\omega
r}\Big{]}_{j,x}\Big{[}\bm{\Sigma}(u)\Big{]}_{x,k}\right)\left(\sum_{y=1}^{n}\Big{[}\bm{\Psi}(u,v)e^{-i\omega
v}\Big{]}_{j,y}\Big{[}\bm{\Sigma}(u)\Big{]}_{y,k}\right)d\omega\\\
&=\sigma_{kk}^{-1}\frac{1}{2\pi}\int_{-\pi}^{\pi}\left(\sum_{r=0}^{\infty}\sum_{x=1}^{n}\Big{[}\bm{\Psi}(u,r)e^{i\omega
r}\Big{]}_{j,x}\Big{[}\bm{\Sigma}(u)\Big{]}_{x,k}\right)\left(\sum_{v=0}^{\infty}\sum_{y=1}^{n}\Big{[}\bm{\Psi}(u,v)e^{-i\omega
v}\Big{]}_{j,y}\Big{[}\bm{\Sigma}(u)\Big{]}_{y,k}\right)d\omega\\\
&=\sigma_{kk}^{-1}\frac{1}{2\pi}\int_{-\pi}^{\pi}\left(\sum_{x=1}^{n}\Big{[}\bm{\Psi}(u)e^{i\omega}\Big{]}_{j,x}\Big{[}\bm{\Sigma}(u)\Big{]}_{x,k}\right)\left(\sum_{y=1}^{n}\Big{[}\bm{\Psi}(u)e^{-i\omega}\Big{]}_{j,y}\Big{[}\bm{\Sigma}(u)\Big{]}_{y,k}\right)d\omega\\\
&=\sigma_{kk}^{-1}\frac{1}{2\pi}\int_{-\pi}^{\pi}\left(\Big{[}\bm{\Psi}(u)e^{-i\omega}\bm{\Sigma}(u)\Big{]}_{j,k}\right)\left(\Big{(}\bm{\Psi}(u)e^{i\omega}\bm{\Sigma}(u)\Big{]}_{j,k}\right)d\omega\\\
&=\sigma_{kk}^{-1}\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|\bigg{[}\bm{\Psi}(u)e^{-i\omega}\bm{\Sigma}(u)\bigg{]}_{j,k}\right|^{2}d\omega\end{split}$
(A.13)

Hence we have established that

$\mathcal{A}=\sigma_{kk}^{-1}\displaystyle\sum_{h=0}^{\infty}\Bigg{(}\Big{[}\bm{\Psi}(u,h)\bm{\Sigma}(u)\Big{]}_{j,k}\Bigg{)}^{2}=\sigma_{kk}^{-1}\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|\bigg{[}\bm{\Psi}(u)e^{-i\omega}\bm{\Sigma}(u)\bigg{]}_{j,k}\right|^{2}d\omega$
(A.14)

from (A.11), we use the local spectral representation of the VMA coefficients
in the second step. The rest is a manipulation with the last step invoking the
definition of modulus squared of a complex number to be defined as
$|z|^{2}=zz^{*}$. Note that we can use this simplification without loss of
generality, because the $VMA(\infty)$ representation that is described by the
coefficients $\bm{\Psi}(u,h)$ has a spectrum that is always symmetric.

Next, we concentrate on $\mathcal{B}$ from (A.11). Using similar steps and the
positive semidefiniteness of the matrix $\bm{\Sigma}(u)$ that ascertains that
there exists $\bm{P}(u)$ such that $\bm{\Sigma}(u)=\bm{P}(u)\bm{P}^{\top}(u).$

$\displaystyle\begin{split}\sum_{h=0}^{\infty}\Big{[}\bm{\Psi}(u,h)\bm{\Sigma}(u)\bm{\Psi}^{\top}(u,h)\Big{]}&=\sum_{h=0}^{\infty}\Big{[}\bm{\Psi}(u,h)\bm{P}(u)\Big{]}\Big{[}\bm{\Psi}(u,h)\bm{P}(u)\Big{]}^{\top}\\\
&=\frac{1}{2\pi}\int_{-\pi}^{\pi}\sum_{r=0}^{\infty}\sum_{v=0}^{\infty}\Big{[}\bm{\Psi}(u,r)e^{i\omega
r}\bm{P}(u)\Big{]}\Big{[}\bm{\Psi}(u,v)e^{-i\omega
v}\bm{P}(u)\Big{]}^{\top}d\omega\\\
&=\frac{1}{2\pi}\int_{-\pi}^{\pi}\sum_{r=0}^{\infty}\Big{[}\bm{\Psi}(u,r)e^{i\omega
r}\bm{P}(u)\Big{]}\sum_{v=0}^{\infty}\Big{[}\bm{\Psi}(u,v)e^{-i\omega
v}\bm{P}(u)\Big{]}^{\top}d\omega\\\
&=\frac{1}{2\pi}\int_{-\pi}^{\pi}\Big{[}\bm{\Psi}(u)e^{i\omega}\bm{P}(u)\Big{]}\Big{[}\bm{\Psi}(u)e^{-i\omega}\bm{P}(u)\Big{]}^{\top}d\omega\\\
&=\frac{1}{2\pi}\displaystyle\int_{-\pi}^{\pi}\Bigg{[}\Big{\\{}\bm{\Psi}(u)e^{i\omega}\Big{\\}}\bm{\Sigma}(u)\Big{\\{}\bm{\Psi}(u)e^{-i\omega}\Big{\\}}^{\top}\Bigg{]}d\omega\end{split}$
(A.15)

That establishes the fact that

$\mathcal{B}=\displaystyle\sum_{h=0}^{\infty}\Big{[}\bm{\Psi}(u,h)\bm{\Sigma}(u)\bm{\Psi}^{\top}(u,h)\Big{]}_{j,j}=\frac{1}{2\pi}\displaystyle\int_{-\pi}^{\pi}\Bigg{[}\Big{\\{}\bm{\Psi}(u)e^{i\omega}\Big{\\}}\bm{\Sigma}(u)\Big{\\{}\bm{\Psi}(u)e^{-i\omega}\Big{\\}}^{\top}\Bigg{]}_{j,j}d\omega$
(A.16)

from (A.11), and we have shown that

$\Big{[}\bm{\theta}(u)\Big{]}_{j,k}=\frac{\sigma_{kk}^{-1}\displaystyle\sum_{h=0}^{\infty}\Bigg{(}\Big{[}\bm{\Psi}(u,h)\bm{\Sigma}(u)\Big{]}_{j,k}\Bigg{)}^{2}}{\displaystyle\sum_{h=0}^{\infty}\Big{[}\bm{\Psi}(u,h)\bm{\Sigma}(u)\bm{\Psi}^{\top}(u,h)\Big{]}_{j,j}}=\frac{\sigma_{kk}^{-1}\displaystyle\int_{-\pi}^{\pi}\left|\bigg{[}\bm{\Psi}(u)e^{-i\omega}\bm{\Sigma}(u)\bigg{]}_{j,k}\right|^{2}d\omega}{\displaystyle\int_{-\pi}^{\pi}\Bigg{[}\Big{\\{}\bm{\Psi}(u)e^{i\omega}\Big{\\}}\bm{\Sigma}(u)\Big{\\{}\bm{\Psi}(u)e^{-i\omega}\Big{\\}}^{\top}\Bigg{]}_{j,j}d\omega}$
(A.17)

Finally, focusing on a frequency band $d=(a,b):a,b\in(-\pi,\pi),a<b$, we have

$\Big{[}\bm{\theta}(u,d)\Big{]}_{j,k}=\frac{\sigma_{kk}^{-1}\displaystyle\int_{a}^{b}\left|\bigg{[}\bm{\Psi}(u)e^{-i\omega}\bm{\Sigma}(u)\bigg{]}_{j,k}\right|^{2}d\omega}{\displaystyle\int_{-\pi}^{\pi}\Bigg{[}\Big{\\{}\bm{\Psi}(u)e^{i\omega}\Big{\\}}\bm{\Sigma}(u)\Big{\\{}\bm{\Psi}(u)e^{-i\omega}\Big{\\}}^{\top}\Bigg{]}_{j,j}d\omega}$
(A.18)

This completes the proof. ∎

###### Proposition 2.

Using the Remark 1 and appropriate substitutions, it immediately follows that

$\begin{split}\sum_{d_{s}\in D}\mathcal{C}(u,d_{s})&=\sum_{d_{z}\in
D}\left(\displaystyle\sum_{\begin{subarray}{c}j,k=1\\\ j\neq
k\end{subarray}}^{N}\Big{[}\widetilde{}\bm{\theta}(u,d_{s})\Big{]}_{j,k}\Bigg{/}\displaystyle\sum_{j,k=1}^{N}\Big{[}\widetilde{}\bm{\theta}(u,\infty)\Big{]}_{j,k}\right)\\\
&=\left(\sum_{d_{z}\in D}\displaystyle\sum_{\begin{subarray}{c}j,k=1\\\ j\neq
k\end{subarray}}^{N}\Big{[}\widetilde{}\bm{\theta}(u,d_{s})\Big{]}_{j,k}\right)\Bigg{/}\displaystyle\sum_{j,k=1}^{N}\Big{[}\widetilde{}\bm{\theta}(u,\infty)\Big{]}_{j,k}\\\
&=\displaystyle\sum_{\begin{subarray}{c}j,k=1\\\ j\neq
k\end{subarray}}^{N}\Big{[}\widetilde{}\bm{\theta}(u,\infty)\Big{]}_{j,k}\Bigg{/}\displaystyle\sum_{j,k=1}^{N}\Big{[}\widetilde{}\bm{\theta}(u,\infty)\Big{]}_{j,k}\\\
&=\mathcal{C}(u)\end{split}$ (A.19)

Similarly, quantities $\mathcal{C}_{j\leftarrow\bullet}(u)$ and
$\mathcal{C}_{j\rightarrow\bullet}(u)$ will sum over frequency bands. This
completes the proof. ∎

###### Proposition 3.

The proof immediately follows Theorem 3.1 in Li et al. (2015) and Liu et al.
(2020) by recognizing that network connectedness measures are manipulations of
impulse transfer functions which stem from the VMA($\infty$) coefficients;
which by Breitung and Candelon (2006) one can use Wald tests that follow a
$\chi^{2}(q)$ distribution. ∎

##  Appendix B Estimation Details

To estimate our high dimensional systems, we follow the Quasi-Bayesian Local-
Liklihood (QBLL) approach of Petrova (2019). let $\bm{X}_{t}$ be an $N\times
1$ vector generated by a stable time-varying parameter (TVP) heteroskedastic
VAR model with $p$ lags:

$\bm{X}_{t,T}=\bm{\Phi}_{1}(t/T)\bm{X}_{t-1,T}+\ldots+\bm{\Phi}_{p}(t/T)\bm{X}_{t-p,T}+\bm{\epsilon}_{t,T},$
(B.1)

where $\bm{\epsilon}_{t,T}=\bm{\Sigma}^{-1/2}(t/T)\bm{\eta}_{t,T}$ with
$\bm{\eta}_{t,T}\sim NID(0,\bm{I}_{M})$ and
$\bm{\Phi}(t/T)=(\bm{\Phi}_{1}(t/T),\ldots,\bm{\Phi}_{p}(t/T))^{\top}$ are the
time varying autoregressive coefficients. Note that all roots of the locally
stationary VAR polynomial, lie outside the unit circle, and
$\bm{\Sigma}^{-1}_{t}$ is a positive definite time-varying covariance matrix.
Stacking the time-varying intercepts and autoregressive matrices in the vector
$\phi(t/T)$ with $\bar{\bm{X}}^{\prime}_{t,T}=\left(\text{{I}}_{N}\otimes
x_{t,T}\right),\>x_{t,T}=\left(1,x^{\prime}_{t-1,T},\dots,x^{\prime}_{t-p,T}\right)$
and $\otimes$ denotes the Kronecker product, the model can be written as:

$\displaystyle\bm{X}_{t,T}=\bar{\bm{X}}^{\prime}_{t,T}\phi(t/T)+\bm{\Sigma}^{-\frac{1}{2}}(t/T)\bm{\eta}_{t,T}$
(B.2)

We obtain the time-varying parameters of the model by employing Quasi-Bayesian
Local Likelihood (QBLL) methods. Estimation of (B.1) requires re-weighting the
likelihood function. Essentially, the weighting function gives higher
proportions to observations surrounding the time period whose parameter values
are of interest. The local likelihood function at discrete time period $s$,
where we drop the double time index for notational convenience, is given by:

$\text{L}_{s}\left(\bm{X}_{s}|\phi_{s},\bm{\Sigma}_{s},\bar{\bm{X}}_{s}\right)\propto\\\
|\bm{\Sigma}_{s}|^{\text{trace}(\mathbf{D}_{s})/2}\exp\left\\{-\frac{1}{2}(\bm{X}_{s}-\bar{\bm{X}}^{\prime}_{s}\phi_{s})^{\prime}\left(\bm{\Sigma}_{s}\otimes\mathbf{D}_{s}\right)(\bm{X}_{s}-\bar{\bm{X}}^{\prime}_{s}\phi_{s})\right\\}$
(B.3)

The $\mathbf{D}_{s}$ is a diagonal matrix whose elements hold the weights:

$\displaystyle\mathbf{D}_{s}$ $\displaystyle=$
$\displaystyle\text{diag}(\varrho_{s1},\dots,\varrho_{sT})$ (B.4)
$\displaystyle\varrho_{st}$ $\displaystyle=$
$\displaystyle\zeta_{T,s}w_{st}/\sum^{T}_{t=1}w_{st}$ (B.5) $\displaystyle
w_{st}$ $\displaystyle=$
$\displaystyle(1/\sqrt{2\pi})\exp((-1/2)((k-t)/W)^{2}),\quad\text{for}\>s,t\in\\{1,\dots,T\\}$
(B.6) $\displaystyle\zeta_{Ts}$ $\displaystyle=$
$\displaystyle\left(\left(\sum^{T}_{t=1}w_{st}\right)^{2}\right)^{-1}$ (B.7)

where $\varrho_{st}$ is a normalised kernel function. $w_{st}$ uses a Normal
kernel weighting function. $\zeta_{Ts}$ gives the rate of convergence and
behaves like the bandwidth parameter $W$ in (B.6). it is the kernel function
that provides greater weight to observations surrounding the parameter
estimates at time $s$ relative to more distant observations.

Using a Normal-Wishart prior distribution for $\phi_{s}|\>\bm{\Sigma}_{s}$ for
$s\in\\{1,\dots,T\\}$:

$\displaystyle\phi_{s}|\bm{\Sigma}_{s}\backsim\mathcal{N}\left(\phi_{0s},(\bm{\Sigma}_{s}\otimes\mathbf{\Xi}_{0s})^{-1}\right)$
(B.8)
$\displaystyle\bm{\Sigma}_{s}\backsim\mathcal{W}\left(\alpha_{0s},\mathbf{\Gamma}_{0s}\right)$
(B.9)

where $\phi_{0s}$ is a vector of prior means, $\mathbf{\Xi}_{0s}$ is a
positive definite matrix, $\alpha_{0s}$ is a scale parameter of the Wishart
distribution ($\mathcal{W}$), and $\mathbf{\Gamma}_{0s}$ is a positive
definite matrix.

The prior and weighted likelihood function implies a Normal-Wishart quasi
posterior distribution for $\phi_{s}|\>\bm{\Sigma}_{s}$ for
$s=\\{1,\dots,T\\}$. Formally let
$\mathbf{A}=(\bar{x}^{\prime}_{1},\dots,\bar{x}^{\prime}_{T})^{\prime}$ and
$\mathbf{Y}=(x_{1},\dots,x_{T})^{\prime}$ then:

$\displaystyle\phi_{s}|\bm{\Sigma}_{s},\mathbf{A},\mathbf{Y}$
$\displaystyle\backsim$
$\displaystyle\mathcal{N}\left(\tilde{\phi}_{s},\left(\bm{\Sigma}_{s}\otimes\mathbf{\tilde{\Xi}}_{s}\right)^{-1}\right)$
(B.10) $\displaystyle\bm{\Sigma}_{s}$ $\displaystyle\backsim$
$\displaystyle\mathcal{W}\left(\tilde{\alpha}_{s},\mathbf{\tilde{\Gamma}}^{-1}_{s}\right)$
(B.11)

with quasi posterior parameters

$\displaystyle\tilde{\phi}_{s}$ $\displaystyle=$
$\displaystyle\left(\mathbf{I}_{N}\otimes\mathbf{\tilde{\Xi}}^{-1}_{s}\right)\left[\left(\mathbf{I}_{N}\otimes\mathbf{A}^{\prime}\mathbf{D}_{s}\mathbf{A}\right)\hat{\phi}_{s}+\left(\mathbf{I}_{N}\otimes\mathbf{\Xi}_{0s}\right)\phi_{0s}\right]$
(B.12) $\displaystyle\mathbf{\tilde{\Xi}}_{s}$ $\displaystyle=$
$\displaystyle\mathbf{\tilde{\Xi}}_{0s}+\mathbf{A}^{\prime}\mathbf{D}_{s}\mathbf{A}$
(B.13) $\displaystyle\tilde{\alpha}_{s}$ $\displaystyle=$
$\displaystyle\alpha_{0s}+\sum^{T}_{t=1}\varrho_{st}$ (B.14)
$\displaystyle\mathbf{\tilde{\Gamma}}_{s}$ $\displaystyle=$
$\displaystyle\mathbf{\Gamma}_{0s}+\mathbf{Y}^{\prime}\mathbf{D}_{s}\mathbf{Y}+\mathbf{\Phi}_{0s}\mathbf{\Gamma}_{0s}\mathbf{\Phi}^{\prime}_{0s}-\mathbf{\tilde{\Phi}}_{s}\mathbf{\tilde{\Gamma}}_{s}\mathbf{\tilde{\Phi}}^{\prime}_{s}$
(B.15)

where
$\hat{\phi}_{s}=\left(\mathbf{I}_{N}\otimes\mathbf{A}^{\prime}\mathbf{D}_{s}\mathbf{A}\right)^{-1}\left(\mathbf{I}_{N}\otimes\mathbf{A}^{\prime}\mathbf{D}_{s}\right)y$
is the local likelihood estimator for $\phi_{s}$. The matrices
$\mathbf{\Phi}_{0s},\>\mathbf{\tilde{\Phi}}_{s}$ are conformable matrices from
the vector of prior means, $\phi_{0s}$, and a draw from the quasi posterior
distribution, $\tilde{\phi}_{s}$, respectively.

The motivation for employing these methods are threefold. First, we are able
to estimate large systems that conventional Bayesian estimation methods do not
permit. This is typically because the state-space representation of an
$N$-dimensional TVP VAR ($p$) requires an additional $N(3/2+N(p+1/2))$ state
equations for every additional variable. Conventional Markov Chain Monte Carlo
(MCMC) methods fail to estimate larger models, which in general confine one to
(usually) fewer than 6 variables in the system. Second, the standard approach
is fully parametric and requires a law of motion. This can distort inference
if the true law of motion is misspecified. Third, the methods used here permit
direct estimation of the VAR’s time-varying covariance matrix, which has an
inverse-Wishart density and is symmetric positive definite at every point in
time.

In estimating the model, we use $p$=2 and a Minnesota Normal-Wishart prior
with a shrinkage value $\varphi=0.05$ and centre the coefficient on the first
lag of each variable to 0.1 in each respective equation. The prior for the
Wishart parameters are set following Kadiyala and Karlsson (1997). For each
point in time, we run 500 simulations of the model to generate the (quasi)
posterior distribution of parameter estimates. Note we experiment with various
lag lengths, $p=\\{2,3,4,5\\}$; shrinkage values,
$\varphi=\\{0.01,0.25,0.5\\}$; and values to centre the coefficient on the
first lag of each variable, $\\{0,0.05,0.2,0.5\\}$. Network measures from
these experiments are qualitatively similar. Notably, adding lags to the VAR
and increasing the persistence in the prior value of the first lagged
dependent variable in each equation increases computation time.

##  Appendix C Monte Carlo Study: Additional Results

Figures 7 and 8 report dynamic network connectedness measures under kernel
bandwidths $W$=$\\{8,18\\}$. As we can see, the larger is $W$, the smoother
the measures become.

Figure 7: Dynamic network connectedness measures: True and fitted values
Notes: This figure plots the true network connectedness measures for three
data generating processes following bi-variate TVP VAR(2) models along with
the median and 95% quantiles of estimated network connectedness measures with
bandwidth $W$=12. The left columns report network connectedness on the low-
frequency band, $d\in(0,\>\pi/5)$, the middle columns show network
connectedness on the high-frequency band, $d\in(\pi/5,\>\pi)$, and the right
columns show the aggregate network connectedness such that $d\in(0,\>\pi)$.
DGPI (top row) is a TVP VAR(2) model with Gaussian errors, we introduce a
break in the time-varying autoregressive matrices and contemporaneous
relations from observation 500 that induces large connections across the high
frequency band. DGPII (second row) is a TVP VAR(2) where time-varying
intercepts and autoregressive matrices following sin wave functions with a
stochastic error, time-varying covariance matrix where the off-diagonals
follow sin wave functions with a stochastic error, and the diagonal elements
follow a stationary AR(1) processes. DGPIII (third row) is a TVP VAR(2) model
with student-$t$ errors, time-varying intercepts and autoregressive matrices
following sin wave functions with a stochastic error, time-varying covariance
matrix where the off-diagonals follow sin wave functions with a stochastic
error, and the diagonal elements follow a stationary AR(1) processes. DGPIV
(bottom row) is the same as DGPII, but with an increase in the periodicity of
the respective sin wave functions the time-varying intercepts and
autoregressive matrices follow.

Figure 8: Dynamic network connectedness measures: True and fitted values
Notes: This figure plots the true network connectedness measures for three
data generating processes following bi-variate TVP VAR(2) models along with
the median and 95% quantiles of estimated network connectedness measures with
bandwidth $W$=18. The left columns report network connectedness on the low-
frequency band, $d\in(0,\>\pi/5)$, the middle columns show network
connectedness on the high-frequency band, $d\in(\pi/5,\>\pi)$, and the right
columns show the aggregate network connectedness such that $d\in(0,\>\pi)$.
DGPI (top row) is a TVP VAR(2) model with Gaussian errors, we introduce a
break in the time-varying autoregressive matrices and contemporaneous
relations from observation 500 that induces large connections across the high
frequency band. DGPII (second row) is a TVP VAR(2) where time-varying
intercepts and autoregressive matrices following sin wave functions with a
stochastic error, time-varying covariance matrix where the off-diagonals
follow sin wave functions with a stochastic error, and the diagonal elements
follow a stationary AR(1) processes. DGPIII (third row) is a TVP VAR(2) model
with student-$t$ errors, time-varying intercepts and autoregressive matrices
following sin wave functions with a stochastic error, time-varying covariance
matrix where the off-diagonals follow sin wave functions with a stochastic
error, and the diagonal elements follow a stationary AR(1) processes. DGPIV
(bottom row) is the same as DGPII, but with an increase in the periodicity of
the respective sin wave functions the time-varying intercepts and
autoregressive matrices follow.

###  C.1 DGPI with fat tails

Here we report the connectedness measures for DGPI in the main text with
residuals that follow a multivariate student-$t$ distribution. In Figure 9 we
report the true and estimated connectedness measures for this alternative DGP
at bandwidths $W$=$\\{8,\>12,\>18\\}$. Overall, these plots correspond to the
top row in Figure 2 within the main text.

Figure 9: Dynamic network connectedness measures: True and fitted values DGPI
with fat tails
Notes: This figure plots the true network connectedness measures for DGPI with
residuals following a multivariate student-$t$ distribution. Along with this
the figures report the median and 95% quantiles of estimated network
connectedness measures with bandwidth $W$=8, $W$=12, and $W$=18 in the top,
middle and bottom rows respectively. The left columns report network
connectedness on the low-frequency band, $d\in(0,\>\pi/5)$, the middle columns
show network connectedness on the high-frequency band, $d\in(\pi/5,\>\pi)$,
and the right columns show the aggregate network connectedness such that
$d\in(0,\>\pi)$.

###  C.2 Posterior probabilities for differences in network connectedness
measures

Here we examine whether the network connectedness measures one computes over
frequency bands are different from one another. We use the time $u$ joint
posterior distribution of horizon specific network connectedness to compute
the probability of heterogeneities. Consider the time $u$ network
connectedness measure on frequency band $d$, $\mathcal{C}(u,d)$, and the time
$u$ network connectedness measure on frequency band $c$, $\mathcal{C}(u,c)$
where $d<c$. Then, the probability that the time $u$ network connectedness on
frequency band $d$ is greater than the time $u$ network connectedness measure
on frequency band $c$,
$\text{Pr}\left(\mathcal{C}(u,d)>\mathcal{C}(u,c)\right)$, is given by

$\displaystyle\text{Pr}\left(\mathcal{C}(u,d)>\mathcal{C}(u,c)\right)=\sum\left(\mathcal{C}^{r}(u,d)>\mathcal{C}^{r}(u,c)\right)/\text{R}$

where $\mathcal{C}^{r}(u,d),\>\mathcal{C}^{r}(u,c)$ denote the $r$th draw from
the posterior distribution of each network connectedness measure, and $R$
denotes the total number of draws taken from the posterior distribution.

If $\text{Pr}\left(\mathcal{C}(u,d)>\mathcal{C}(u,c)\right)>$0.95 this tells
us with 95% confidence that the network connectedness measure across frequency
band $d$ is greater than the corresponding measure across frequency band $c$.
Similarly, if $\text{Pr}\left(\mathcal{C}(u,d)>\mathcal{C}(u,c)\right)<$0.05
this tells us with 95% confidence that the network connectedness measure
across frequency band $c$ is greater than the corresponding measure across
frequency band $d$. Note here that
$\text{Pr}\left(\mathcal{C}(u,d)>\mathcal{C}(u,c)\right)=1-\text{Pr}\left(\mathcal{C}(u,c)>\mathcal{C}(u,d)\right)$.
These probabilities provide statistical evidence of whether heterogeneities
exit between network connectedness measures over different frequency bands.

We report the probabilities for differences between network connectedness in
Table 1 at three specific dates, $u$=$\\{400,\>650,\>1000\\}$. First, we
report the true probability that the low-frequency-band network connectedness
measure is greater than the high-frequency-band network connectedness measure,
$\text{Pr}\left(\mathcal{C}(u,d)>\mathcal{C}(u,c)\right)$. Then, we provide
the estimates of sample average probabilities from TVP VAR models with kernel
bandwidths $W$=$\\{8,\>12,\>18\\}$. Again, this indicates that our method
works well at specific observations throughout each DGP. It is noteworthy to
mention here that for bandwidth $W$=$\\{18\\}$ the time profiles relative to
true probabilities become less accurate as can be seen from DGPIII in Panel B.

Table 1: Probability of statistical differences between low-frequency-band and
high-frequency-band network connectedness: Sample average.
Notes: This table reports the probability that the low-frequency-band network connectedness measure is greater than the high-frequency-band network connectedness measure, $\text{Pr}\left(\mathcal{C}(u,d)>\mathcal{C}(u,c)\right)$. Panel A shows results for observation $u$=$400$, and Panels B and C show results for observation $u$=$650$, and $u$=$1000$ respectively. We report the true probabilities for the three DGPs in our Monte Carlo study, followed by the fitted probabilities from TVP VARs with different kernel bandwidths, $W=\\{8,\>12,\>18\\}$. Probabilities greater (less) than 0.95 (0.05) indicate statistical differences at each time period. DGPI (left column) is a TVP VAR(2) model with Gaussian errors, time-varying intercepts and autoregressive matrices following sin wave functions with a stochastic error, time-varying covariance matrix where the off-diagonals follow sin wave functions with a stochastic error, and the diagonal elements follow a stationary AR(1) processes. DGPII (middle column) is a TVP VAR(2) model with student-$t$ errors, time-varying intercepts and autoregressive matrices following sin wave functions with a stochastic error, time-varying covariance matrix where the off-diagonals follow sin wave functions with a stochastic error, and the diagonal elements follow a stationary AR(1) processes. DGPIII (right column) is the same as DGPI, but with an increase in the periodicity of the respective sin wave functions the time-varying intercepts and autoregressive matrices follow.  A: $u$=400 | DGPII | DGPIII | DGPIV
---|---|---|---
True Probability | 0.00 | 0.00 | 0.99
Fitted Probability, $W$=8 | 0.00 | 0.00 | 0.97
Fitted Probability, $W$=12 | 0.00 | 0.00 | 1.00
Fitted Probability, $W$=18 | 0.00 | 0.00 | 1.00
B: $u$=650 | DGPII | DGPIII | DGPIV
True Probability | 1.00 | 1.00 | 0.00
Fitted Probability, $W$=8 | 1.00 | 0.90 | 0.00
Fitted Probability, $W$=12 | 1.00 | 0.98 | 0.00
Fitted Probability, $W$=18 | 1.00 | 1.00 | 0.21
C: $u$=1000 | DGPII | DGPIII | DGPIV
True Probability | 0.00 | 0.00 | 0.00
Fitted Probability, $W$=8 | 0.00 | 0.00 | 0.00
Fitted Probability, $W$=12 | 0.00 | 0.00 | 0.00
Fitted Probability, $W$=18 | 0.00 | 0.00 | 0.00

##  Appendix D S&P 500 Sectors Breakdown and Additional Empirical Results

In this short section we present a U.S. stock market sectors breakdown and
description where the S&P 500 index is used as a proxy for the stock market.
The information in this section are reported as of January 25, 2019. For more
details and updated information, see also
https://us.spindices.com/indices/equity/sp-500.

  * •

Consumer Discretionary: The consumer discretionary sector consists of
businesses that have demand that rises and falls based on general economic
conditions such as washers and dryers, sporting goods, new cars, and diamond
engagement rings. At present, the consumer discretionary sector contains 11
sub-industries: Automobile Components Industry, Automobiles Industry,
Distributors Industry, Diversified Consumer Services Industry, Hotels,
Restaurants & Leisure Industry, Household Durables Industry, Leisure Products
Industry, Multiline Retail Industry, Specialty Retail Industry, Textile,
Apparel & Luxury Goods Industry, Internet & Direct Marketing. The total value
of all consumer discretionary stocks in the United States came to $4.54
trillion, or about 10.11% of the market. Examples of consumer discretionary
stocks include Amazon and Starbucks.

  * •

Communication Services: From telephone access to high-speed internet, the
communication services sector of the economy keeps us all connected. At
present, the communication services sector is made up of five industries:
Diversified Telecommunication Services, Wireless Telecommunication Services,
Entertainment Media, Interactive Media and Services. the total value of all
communication services stocks in the United States came to $4.42 trillion, or
10.33% of the market. The communications industry includes stocks such as AT&T
and Verizon, but also the giants Alphabet Inc A and Facebook from 2004 and
2012, respectively.

  * •

Consumer Staples: The consumer staples sector consists of businesses that sell
the necessities of life, ranging from bleach and laundry detergent to
toothpaste and packaged food. At present, the consumer staples sector contains
six industries: Beverages Industry, Food & Staples Retailing Industry, Food
Products Industry, Household Products Industry, Personal Products Industry,
Tobacco Industry. The total value of all consumer staples stocks in the United
States came to $2.95 trillion, or about 7.18% of the market and includes
companies such as Procter & Gamble.

  * •

Energy: The energy sector consists of businesses that source, drill, extract,
and refine the raw commodities we need to keep the country going, such as oil
and gas. At present, the energy sector contains two industries: Energy
Equipment & Services Industry, and Oil, Gas & Consumable Fuels Industry. The
total value of all energy stocks in the United States came to $3.36 trillion,
or about 5.51% of the market. Major energy stocks include Exxon Mobil and
Chevron.

  * •

Financial: The financial sector consists of banks, insurance companies, real
estate investment trusts, credit card issuers. At present, the financial
sector contains seven industries: Banking Industry, Capital Markets Industry,
Consumer Finance Industry, Diversified Financial Services Industry, Insurance
Industry, Mortgage Real Estate Investment Trusts (REITs) Industry, Thrifts &
Mortgage Finance Industry. The total value of all financial stocks in the
United States came to $6.89 trillion, or about 13.63% of the market.
JPMorganChase, GoldmanSachs, and Bank of America are examples of financial
stocks.

  * •

Health Care: The health care sector consists of drug companies, medical supply
companies, and other scientific-based operations that are concerned with
improving and healing human life. At present, the health care sector contains
six industries: Biotechnology Industry, Health Care Equipment & Supplies
Industry, Health Care Providers & Services Industry, Health Care Technology
Industry, Life Sciences Tools & Services Industry, Pharmaceuticals Industry.
The total value of all health care stocks in the United States came to $5.25
trillion, or about 15.21% of the market. Examples of health care stocks
include Johnson & Johnson, and Pfizer.

  * •

Industrials: The industrial sector comprises railroads and airlines to
military weapons and industrial conglomerates. At present, the industrial
sector contains fourteen industries: Aerospace & Defense Industry, Air Freight
& Logistics Industry, Airlines Industry, Building Products Industry,
Commercial Services & Supplies Industry, Construction & Engineering Industry,
Electrical Equipment Industry, Industrial Conglomerates Industry, Machinery
Industry, Marine Industry, Professional Services Industry, Road & Rail
Industry, Trading Companies & Distributors Industry, Transportation
Infrastructure Industry. The total value of all industrial stocks in the
United States came to $3.80 trillion, or about 9.33% of the market.

  * •

Information Technology: the information technology (IT) sector is home to the
hardware, software, computer equipment, and IT services operations that make
it possible for you to be reading this right now. At present, the information
technology sector contains six industries: Communications Equipment Industry,
Electronic Equipment, Instruments & Components Industry, IT Services Industry,
Semiconductors & Semiconductor Equipment Industry, Software Industry,
Technology Hardware, Storage & Peripherals Industry. The total value of all
information technology stocks in the United States came to $7.10 trillion, or
about 19.85% of the market. It is the largest sector in the S&P 500. Top IT
stocks include Microsoft and Apple.

  * •

Materials: The building blocks that supply the other sectors with the raw
materials it needs to conduct business, the material sector manufacturers,
logs, and mines everything from precious metals, paper, and chemicals to
shipping containers, wood pulp, and industrial ore. At present, the material
sector contains five industries: Chemicals Industry, Construction Materials
Industry, Containers & Packaging Industry, Metals & Mining Industry, Paper &
Forest Products Industry. The total value of all materials stocks in the
United States came to $1.77 trillion, or about 2.71% of the market. Major
materials stocks include Dupont.

  * •

Real Estate: The real estate sector includes all Real Estate Investment Trusts
(REITs) with the exception of Mortgage REITs, which is housed under the
financial sector. The sector also includes companies that manage and develop
properties. At present, the Real Estate sector is made up of two industries:
Equity Real Estate Investment Trusts, Real Estate Management & Development.
The total value of all real estate stocks in the United States came to $1.17
trillion, or 2.96% of the market. The real estate industry includes stocks
such as Simon Property Group and Prologis.

  * •

Utilities: The utilities sector of the economy is home to the firms that make
our lights work when we flip the switch, let our stoves erupt in flame when we
want to cook food, make water come out of the tap when we are thirsty, and
more. At present, the utilities sector is made up of five industries: Electric
Utilities Industry, Gas Utilities Industry, Independent Power and Renewable
Electricity Producers Industry, Multi-Utilities Industry, Water Utilities
Industry. The total value of all utilities stocks in the United States came to
$1.27 trillion, or about 3.18% of the market. Utilities stocks include many
local electricity and water companies including Dominion Resources.

Table 2: Descriptive Statistics
Notes: This table reports descriptive statistics of (annualized) realized volatility estimates for all firms in the Consumers Discretionary (COND), Consumer Staples (CONS), Health Care (HLTH), Industrials (INDU), Information Technology (INFT), Materials (MATR), Real Estate (REAS), Financials (SPF), Energy (SPN), Comunication Services (TELS), and Utilities (UTIL) sectors. All descriptive statistics pool information across all stocks within the sector from July 5, 2005 to August 31, 2018. Mean is the sample average, Std is the sample standard deviation, Min and Max are the minimum and maximum values, Skew and Kurt are the sample skewness and kurtosis respectively.  | COND | CONS | HLTH | INDU | INFT | MATR | REAS | SPF | SPN | TELS | UTIL
---|---|---|---|---|---|---|---|---|---|---|---
Mean | 28.024 | 19.129 | 23.492 | 25.934 | 25.618 | 28.263 | 25.233 | 26.445 | 31.459 | 27.911 | 18.399
Std | 19.487 | 13.016 | 16.822 | 18.611 | 16.242 | 18.921 | 20.579 | 26.289 | 19.71 | 20.261 | 11.371
Min | 4.219 | 2.975 | 0.000 | 3.062 | 1.862 | 4.509 | 3.776 | 1.859 | 4.721 | 5.237 | 4.314
Max | 652.061 | 508.833 | 598.948 | 842.509 | 460.001 | 413.752 | 656.948 | 1526.688 | 752.732 | 294.655 | 302.308
Skew | 4.071 | 4.869 | 5.402 | 4.451 | 3.968 | 3.200 | 4.237 | 6.733 | 3.448 | 2.832 | 4.666
Kurt | 38.155 | 60.767 | 67.314 | 51.261 | 36.928 | 19.088 | 35.824 | 133.412 | 33.112 | 15.423 | 41.019

### Test statistics for differences between sectoral short-term and long-term
network connectedness

Figure 10: Tests for Heterogeneity of Short-term and Long-term of S&P500
Realized Volatility Networks Connectedness
This figure plots the test statistics and numerical standard errors for
heterogeneities between short-term and long-term network connectedness
measures that we compute on the realized volatilities firms within each S&P500
sectors: Consumers Discretionary (COND), Consumer Staples (CONS), Health Care
(HLTH), Industrials (INDU), Information Technology (INFT), Materials (MATR),
Real Estate (REAS), Financials (SPF), Energy (SPN), Comunication Services
(TELS), and Utilities (UTIL) from July 8, 2005 to August 31, 2018.