#  Dynamic industry uncertainty networks and the business cycle††thanks: We
thank Charlie Cai, Daniel Greenwald, Stephen Terry, Emil Verner and Jacky Qi
Zhang for their valuable comments. We thank the participants at the 3rd
Workshop on Macroeconomic Research (Cracow University), CFE 2020, and
Southwestern Finance Association 2021 Annual Meeting. Mattia Bevilacqua
gratefully acknowledges the support of the Economic and Social Research
Council (ESRC) in funding the Systemic Risk Centre [grant number ES/K002309/1
and ES/R009724/1]. Jozef Barunik gratefully acknowledges support from the
Czech Science Foundation under the EXPRO GX19-28231X project.

Jozef Baruník  Charles University and Czech Academy of Sciences  Mattia
Bevilacqua  London School of Economics, Systemic Risk Centre  Robert Faff
University of Queensland, Australia

###### Abstract

We argue that uncertainty network structures extracted from option prices
contain valuable information for business cycles. Classifying U.S. industries
according to their contribution to system-related uncertainty across business
cycles, we uncover an uncertainty hub role for the communications, industrials
and information technology sectors, while shocks to materials, real estate and
utilities do not create strong linkages in the network. Moreover, we find that
this ex-ante network of uncertainty is a useful predictor of business cycles,
especially when it is based on uncertainty hubs. The industry uncertainty
network behaves counter-cyclically in that a tighter network tends to
associate with future business cycle contractions.

Keywords: Financial Uncertainty, Industry Network, Options Market, Business
Cycle.

##  1 Introduction

Throughout history, industrial structure has witnessed influential economic
and financial cycles in which different sectors seem to take on a leading
role.111The ascendancy of technology and telecommunications is a notable
recent example. The rapidly growing internet sector accounted for $2.1
trillion of the U.S. economy in 2018 or about 10% of the nation’s gross
domestic product (GDP). Tech companies such as Apple, Google, and Amazon are
leading the stock market; any little variation in their quarterly earnings or
stock market prices can move the entire index.  Fluctuations in performance,
valuation and interconnection have been crucially important not only for
swings in financial markets, but also for the real economy and the prediction
of business cycles. Despite the intensifying linkages emerging from economic
activities among industries, we do not fully understand the influence of such
networks on the real economy. Given this motivational backdrop, we develop
forward-looking measures of industry uncertainty network structures, and we
study how industry-specific shocks to option investors’ expectations linked to
investor fear evolve over time as well as how these shocks relate to
macroeconomic activity.

Understanding the differential actions and performance of the largest firms
within an economy’s various industrial sectors offers key insight into the
aggregate economy.222For example, Gabaix (2011) reports that the total sales
of the top 50 firms accounted for 25% of GDP in 2005. As another example, in
December 2004, a $24 billion one-time Microsoft dividend boosted growth in
personal income from 0.6% to 3.7% (Bureau of Economic Analysis, January 31,
2005). Acemoglu et al. (2012) document that significant aggregate fluctuations
can originate from firm-specific microeconomic shocks or disaggregated sectors
due to interconnections between different firms and sectors, functioning as a
potential propagation mechanism of idiosyncratic shocks throughout the
economy. Notably, Atalay (2017) concludes that 83% of the variation in
aggregate output growth is attributable to idiosyncratic industry-level
shocks, while Gabaix (2011) contends that a salient feature of business cycles
is that firms and sectors comove.

To capture uncertainty connected to industrial structures (and especially
driven by large firms), recent literature has exploited information about
volatility or uncertainty shocks highlighting the importance of network
measures to capture the propagation of volatility mechanisms (e.g. Acemoglu et
al., 2012; Carvalho and Gabaix, 2013; Gabaix, 2016; Acemoglu et al., 2017;
Baqaee and Farhi, 2019; Herskovic et al., 2020). Carvalho and Gabaix (2013)
argue that the sector-specific “fundamental” microeconomic volatility has
explanatory power and can serve as an early warning signal of swings in
macroeconomic volatility. Further, financial uncertainty shocks and
fluctuations in risk have been identified as one of the main drivers of the
U.S. business cycle (see Bloom, 2009; Christiano et al., 2014; Leduc and Liu,
2016; Basu and Bundick, 2017; Bloom et al., 2018; Ludvigson et al., 2020).

Despite these enormous efforts, a recent surge of interest relies on
historical or ex-post analysis. In contrast, we propose an ex-ante approach
employing measures extracted from option prices of individual firms. We argue
that information reflected in the implied second moment of firms provides an
enhanced overview of the future risk and performance of economic sectors and
the firms therein, shedding light on their future outlook and evaluation in a
forward-looking manner through the options investors’ expectations directly
related to industry-linked “fears” (e.g. Diebold and Yılmaz, 2014; Baruník et
al., 2020).

To infer such prospective information, we exploit an extensive sample of
option prices to extract option-based uncertainty measures for key major firms
across the full U.S. industrial landscape. We produce aggregate uncertainty
measures for each industry at a daily frequency encompassing three recession
periods (including the most recent Covid-19 crisis). Based on the time-varying
parameter approximating models, we then construct a measure of ex-ante
industry uncertainty network that reflects investor expectations of future
uncertainty relevant to each industry. In doing so, we follow the work of
Diebold and Yılmaz (2014); Barunik and Ellington (2020) who argue that time-
varying variance decompositions characterize well how shocks to uncertainty
create dynamic networks.333Our approach is intimately connected to network
node degrees, mean degrees, and connectedness measures of Diebold and Yilmaz
(2012) and Diebold and Yılmaz (2014). We assess the varying role of industries
contribution to shocks to uncertainty across the business cycles. Our analysis
then quite naturally transitions to the question of how useful are such
uncertainty network measures in predicting economic activity. We argue that,
compared to the currently existing approaches, industry-based uncertainty
network measures represent more informative (leading) propagation channels
highly relevant to predict business cycles given their forward-looking
feature.

The essence of our contribution is linked to two relatively recent strands of
literature. The first strand is concerned with uncertainty measures and their
relationship with business cycle fluctuations (e.g. Bloom, 2009, 2014; Jurado
et al., 2015; Ludvigson et al., 2020).444See also Bachmann et al. (2013),
Bachmann and Bayer (2014), Christiano et al. (2014), Decker et al. (2016),
Bloom et al. (2018), and Arellano et al. (2019). The second strand focuses on
the role of sector-level or firm-to-firm linkages in microeconomic shocks and
their relationship with the aggregate economy and changes in business
conditions (see, e.g. Gabaix, 2011; Acemoglu et al., 2012; Carvalho and
Gabaix, 2013; Barrot and Sauvagnat, 2016; Acemoglu et al., 2017; Baqaee and
Farhi, 2019) or the survey in Carvalho and Tahbaz-Salehi (2019). Notably,
connected to the second strand of literature, are studies on the role of
production networks as a propagation mechanism from individual firms and/or
industries to the real economy (see, e.g. Foerster et al., 2011; Di Giovanni
et al., 2014; Ozdagli and Weber, 2017; Atalay, 2017; Garin et al., 2018; Auer
et al., 2019; Lehn and Winberry, 2020).

However, in contrast to this extant body of work, we adopt purely market-based
networks as a mechanism to dynamically study the propagation of shocks to
uncertainty from industries flowing to the real economy. In addition, a recent
body of work suggests that sector-specific shocks have become more volatile
relative to aggregate shocks. For instance, Garin et al. (2018) develop an
“islands” model with two sectors showing a decline in the contribution of the
aggregate volatility shocks to the business cycle in favour of sector-specific
shocks. Similarly, Lehn and Winberry (2020) show that the empirical network is
dominated by a few “investment hubs” that produce the majority of investment
goods, which are highly volatile, and have large aggregate effects on
fluctuations in the business cycle. Inspired by these latest findings, our
work also builds on a similar framework. Augmenting the definition of “hubs”
from the input-output network literature, we characterize critical
“uncertainty hubs”, as industries that largely transmit and/or receive
uncertainty across the business cycle, versus “non-hubs”, being those
industries that are (largely) neutral across business cycles.

A shock to an uncertainty hub may directly trigger consequences for that
specific industry, for the whole system and the broader economy. While it can
affect production, employment and growth at the hub, it can also generate
larger uncertainty spillovers, changes in prices, growth and production of
other sectors in the system, affecting the broader economy. Our proposed
framework with respect to the role of uncertainty hubs in driving fluctuations
in the aggregate economy is reminiscent of investment-specific technology
shocks (e.g. Greenwood et al., 2000; Justiniano et al., 2010). Given this
framework, we hypothesize that the industry network constructed from
uncertainty hubs contains greater predictability for business cycles compared
to the non-hubs uncertainty network.

The main findings of our paper are as follows. First, we identify industries
showing a stronger (versus weaker) contribution of shocks to uncertainty and
their propagation, thus playing an essential role within the aggregate
industry uncertainty network. Specifically, the communications, IT and
industrials play a key role, being classified as the main uncertainty hubs. In
contrast, materials, utilities and real estate are classified as non-hubs.
Further, our analysis suggests that the hub role of the financial industry is
limited to the global financial crisis (GFC). Second, we have further insights
relevant to the time-series perspective. We find that the ex-ante industry
network rises sharply during the dot com bubble, the GFC, increasing steadily
afterwards. Additionally, we detect a rising importance of hubs-specific
shocks in the last decade implying that shocks to hubs account for the
majority of aggregate fluctuations post-GFC. Third, our empirical exercise
shows that the ex-ante industry uncertainty network generates an important
channel through which sector-specific shocks propagate and it is a useful
leading predictor of business cycles up to one year. Changes in business cycle
indicators occur in line with changes in the comovement of uncertainty across
industries, and they are especially due to changes in the network
connectedness in uncertainty hubs. These results suggest that uncertainty hubs
are a major channel for targeting policy effectiveness. Our results are robust
to several robustness checks and the inclusion of additional control
variables.

The remainder of this paper is organized as follows. Section 2 describes the
data and sampling used in our study. Section 3 sets out the essence of the
TVP-VAR network connectedness method applied to our chosen industry setting.
Section 4 studies the dynamic aggregate uncertainty network connectedness, and
section 5 presents the findings for the dynamic idiosyncratic uncertainty
network connectedness through the business cycle. Section 6 studies the
predictive ability of the networks for the real economy. Section 7 concludes
the paper. Additional results are relegated to the appendix of the paper.

##  2 Industry uncertainty, investor beliefs and option prices

According to Kozeniauskas et al. (2018), conceptually three different types of
uncertainty have been used in existing research: measures of uncertainty about
macroeconomic outcomes (macro uncertainty); measures of the dispersion of firm
outcomes (micro dispersion); and measures of the uncertainty that people have
about what others believe (higher-order uncertainty). A common origin for the
various uncertainty shocks can be found in macro uncertainty, shocks of which
generate positive covariances between all pairs of uncertainty types.

The VIX is a common proxy used in the financial economics literature to
measure the unpredictability of future aggregate outcomes, or macro
uncertainty (see Bloom, 2009; Bekaert et al., 2013; Leduc and Liu, 2016;
Kozeniauskas et al., 2018; Bhattarai et al., 2020). The VIX index (often
referred to as “fear index”), introduced by the Chicago Board Options Exchange
(CBOE), is a model-free forward-looking measure implied by options prices and
reflects investors’ expectations about uncertainty in the stock market. It has
also been linked with future macro outcomes in the aforementioned studies.

To study the dynamic uncertainty network, rather than looking at the whole
U.S. stock market, we develop a forward-looking measure of uncertainty
reflecting investor beliefs derived from option price data at the industry
level. To capture industry uncertainty, we use forward-looking uncertainty
measures that are intimately related to the VIX methodology and derived from
the uncertainty of single firms within the main U.S. industries. More details
on the composition of the 11 U.S. industries are provided in the Appendix,
section A.

According to Kozeniauskas et al. (2018), micro uncertainty describes an
increase in the uncertainty that firms have about their outcomes due to
changes in idiosyncratic variables. To this end, while our proposed network
measure might be influenced by a common structure in volatilites, importantly
our measure contains additional critical information about idiosyncratic
volatilities. Indeed, single firm VIX measures can be a valid proxy for both
macro and micro uncertainty, therefore capturing macro volatility as well as
dispersion of firm-specific outcomes. Moreover, such micro uncertainty is
challenging to measure due to the scarcity of data on firm-specific beliefs.
Our measures overcome this limitation since can be computed at a high (daily)
frequency. Finally, options-based measures of risk are superior to historical
volatility measures with respect to both predictive power and the set of
information they encompass (e.g. Christensen and Prabhala, 1998; Santa-Clara
and Yan, 2010; Baruník et al., 2020).

###  2.1 Extracting model-free industry uncertainty from option prices

For each chosen firm, we compute a model-free implied volatility index as
detailed in the Appendix, section B. This measure reflects expectations about
investor uncertainty regarding the individual firm over the coming 30-day
horizon. We then aggregate the individual firm information to construct a
measure of ex-ante uncertainty at the industry level. Such a measure reflects
the industry expected uncertainty over the next 30 days.

More formally, the ex-ante industry uncertainty measure
$\text{IVIX}^{(\text{Ind})}_{t}$ is constructed by taking the time-varying
weighted average of the main five stocks in each industry and at each point in
time through our sample period as:

$\text{IVIX}^{(\text{Ind})}_{t}=\sum_{s\in
N^{(\text{Ind})}}\mathcal{W}_{t}^{(s)}\text{VIX}_{t}^{(s)}$ (1)

where $\text{Ind}\in\\{1,\ldots,11\\}$ represents the industry we consider,
$s$ is an index for one of the $N^{(\text{Ind})}$ firms included in the given
industry at time $t$, $\text{VIX}_{t}^{(s)}$ is the implied volatility for an
individual stock $s$, and $\mathcal{W}_{t}^{(s)}$ is the time-varying market
capitalization weight of that specific stock $s$ computed as the ratio between
the time-varying market capitalization of the stock and total market
capitalization of all stocks included in the industry.

###  2.2 Data

We use daily data encompassing the sample period January 2000 to May 2020 in
each of the following 11 U.S. industries: consumer discretionary (CD),
communications (CM), consumer staples (CS), energy (E), financials (F), health
care (HC), industrials (IN), information technology (IT), materials (M), real
estate (RE) and utilities (U). More specifically we merge two data sources. We
use options data from OptionMetrics from January 2000 to December 2018 to
compute the individual stock VIXs. We expand the coverage of the VIX time
series from January 2019 until May 2020 aided by the IHS Markit’s Totem
Vanilla Volatility Swap data set from which we collect broker-dealers
consensus prices for the volatility strike of the swaps.555The Totem database
is a service within IHS Markit that gathers a large variety of derivatives
marks from the major broker-dealers and returns consensus prices. In the
volatility swaps service, contributors are requested to price the volatility
strike at which the swap would have an inception price of zero which should be
the traders best estimate of mid-market. Before 2019 January, the majority of
the data in the Totem Vanilla Volatility (or Variance) Swap services were
monthly, therefore would not have served our purpose. This aids the expansion
of the individual stock VIXs since we replace $\text{VIX}_{t}^{(s)}$ with the
volatility strike of the swaps and we then compute the industry uncertainty
measures as in equation 1.666On the equality between VIX and the strike of a
volatility swap see, for instance, Filipović et al. (2016) and Cheng (2019).
For more details on the VIX index, the variance swap market and their
relationship see Carr and Wu (2006) and Carr and Lee (2009). The other
financial information such as market capitalization and trading volume
regarding the selected stocks are collected from Bloomberg.

Overall, our data set includes options prices of 69 U.S. firms. We select the
largest constituent stocks in each U.S. industry at every point in time
according to time-varying market capitalizations, new IPOs, exclusions of the
stocks from the $S\&P500$, or missing data. For instance, for some industries
such as industrials, the same five stocks have been adopted throughout the
sample period. In more dynamic sectors such as IT, we observe several changes
in the stocks ranking within our sample period. In cases where options on a
specific firm have been only issued in recent times, we include the next
ranked firm as a substitute to always ensure at least five stocks for every
sector across our time period in that industry with available data.777The only
exceptions are the materials and real estate industries. For the first, we use
only four stocks, monthly interpolated between 01-2019 and 04-2019 due to data
availability. This is because before 05-2019 for a few stocks in this sector
the submission service was monthly. Between 01- and 04-2019 no single firm
volatility data is available for the real estate industry, hence we replace
$\text{IVIX}^{(\text{RE})}_{t}$ directly with the real estate sector
volatility measure submitted to IHS Markit/Totem. Table B1 in the appendix
shows the included stocks within each industry and their available time
period. Figure B1 in the appendix depicts an example of individual firm
uncertainty $\text{VIX}^{(i)}$.888The CBOE has introduced stock market VIX
series for a few stocks in the U.S. Comparing our calculations, with available
period CBOE counterparts, show a correlation, on average, exceeding 94%. This
minor divergence is likely due to the interpolation among the two closest
expiration dates to 30 days used in the CBOE methodology. For the data
collected by IHS Markit spanning a shorter time frame, the correlation between
the consensus volatility and CBOE VIX series is, on average, above 97%, this
again due to the interpolation used in the CBOE methodology.

The selected stocks account for more than 58% of the U.S. $S\&P500$ market
capitalization, thus being a valid proxy for the 11 U.S. industries, and being
representative of a not trivial fraction of the U.S. GDP (e.g. Gabaix, 2011).
A large representation of the U.S. stock market and its industries is what
matters when studying these as economic and business cycle drivers. We report
the descriptive statistics for the industry uncertainty indexes in Table 1.

Table 1: Industry Uncertainty $\text{IVIX}^{(\text{Ind})}_{t}$ Descriptive
Statistics

Ind | CD | CM | CS | E | F | HC | IN | IT | M | RE | U
---|---|---|---|---|---|---|---|---|---|---|---
Mean | 0.321 | 0.318 | 0.230 | 0.270 | 0.370 | 0.252 | 0.300 | 0.342 | 0.289 | 0.343 | 0.232
Standard Dev. | 0.116 | 0.121 | 0.096 | 0.106 | 0.279 | 0.089 | 0.148 | 0.144 | 0.111 | 0.191 | 0.107
Min | 0.133 | 0.1305 | 0.115 | 0.128 | 0.145 | 0.136 | 0.139 | 0.138 | 0.152 | 0.122 | 0.104
Max | 0.876 | 1.137 | 0.911 | 1.404 | 2.682 | 0.815 | 1.409 | 0.994 | 1.191 | 2.225 | 1.018
Skewness | 1.363 | 1.538 | 1.777 | 3.249 | 3.727 | 1.552 | 2.409 | 1.504 | 2.220 | 2.769 | 2.246
Kurtosis | 4.658 | 5.755 | 6.363 | 21.240 | 20.468 | 5.969 | 11.411 | 5.011 | 11.247 | 17.057 | 9.788

  * •

Notes: This table reports the descriptive statistics for the industry
uncertainty $\text{IVIX}^{(\text{Ind})}_{t}$ index of 11 industries: consumer
discretionary (CD), communications (CM), consumer staples (CS), energy (E),
financials (F), health care (HC), industrials (IN), information technology
(IT), materials (M), real estate (RE) and utilities (U). The time period is
from 03-01-2000 to 29-05-2020, at a daily frequency.

From Table 1 we observe that the financial industry uncertainty shows the
highest mean, followed by the information technology and real estate industry
uncertainty measures with consumer staples and utilities showing the lowest
mean values. The financial sector is also found to be the one with the highest
standard deviation, and skewness of uncertainty measure. On the other hand,
consumer staples, energy, health care and utilities are found to have lower
standard deviations. Consumer discretionary and IT show lower skewness and
kurtosis compared to the other industries uncertainty measures. The minimum
values of industry uncertainty range between 10% and 15%, while the maximum
values present a wider range with financials and real estate leading with the
highest values. As an example, we plot uncertainty measures for the IT,
consumer staples and financial sectors in Figure 1.

Figure 1: Industry Uncertainty Notes: This figure shows
$\text{IVIX}^{(\text{Ind})}_{t}$ series for financials (black), IT (grey) and
consumer staples (light grey) covering the period 03-01-2000 to 29-05-2020 at
a daily frequency.

We observe that the IT sector dominates the other two during the dot com
bubble and technologic boom in the early 2000s. The financial sector shows a
dominant role during the GFC in 2008 and the Eurozone debt crisis in 2010 and
2011. In the most recent times, the IT sector exhibits greater uncertainty in
comparison to the financial sector, showing how uncertainty might come from
technology innovation and growth. Consumer staples show peaks in the early
2000s and during the GFC, however being always below the other two series and
overall low throughout the sample period. All three uncertainty measures
spiked in correspondence to the recent Covid-19 pandemic in March 2020. Among
them, we observe that financials and IT increased during the Covid-19 crisis
in a more pronounced manner than consumer staples impacted less by the
Covid-19 crisis.

##  3 Measurement of dynamic industry uncertainty networks

Industries are connected directly through counterparty risk, contractual
obligations or other general business conditions of the firms. High-frequency
analysis of such networks requires generally unavailable high-frequency
information. In contrast, option prices and uncertainty measured in high
frequencies reflect the decisions of many agents assessing risks from the
existing linkages. Hence the pure market-based approach we use, in contrast to
other network techniques, allows us to monitor the network on a daily
frequency as well as to exploit its forward-looking strength with minimal
assumptions.

Looking at how a shock to the expected uncertainty of a firm $j$ transmits to
future expectations about the uncertainty of a firm $k$, we will define
weighted and directed networks. Aggregating the information about such
networks can provide industry level uncertainty characteristics that will
measure how strongly the investors’ expectations are interconnected.
Importantly, we will focus on the time variation of such networks.

###  3.1 Link to the network literature and causality of proposed measures

The measures that we use are intimately related to modern network theory.
Algebraically, the adjacency matrix capturing information about network
linkages carries all information about the network and any sensible measure
must be related to it. As noted by Diebold and Yılmaz (2014), a variance
decomposition matrix defining network adjacency matrix is then readily used as
a network connectedness that is related to network node degrees and mean
degree. Currently, studies examine, almost exclusively, static networks
mimicking time dynamics with estimation from an approximating window. In
contrast to this approach, we follow Barunik and Ellington (2020) who employ a
locally stationary TVP-VAR that allows us to estimate the adjacency matrix for
a network at each point in time with possibly large dimension. Dynamic
networks defined by such time-varying variance decompositions are then more
sophisticated than classical network structures in several ways.999For
previous literature on the importance of the variation in the transmission of
uncertainty shocks over time (see Caggiano et al., 2014; Mumtaz and
Theodoridis, 2018; Alessandri and Mumtaz, 2019).

In a typical network, the adjacency matrix contains a set of zero and one
entries, depending on the node being linked or not, respectively. In the above
notion, one interprets variance decompositions as weighted links showing the
strength of the connections. In addition, the links are directed, meaning that
the $j$ to $k$ link is not necessarily the same as the $k$ to $j$ link, and
hence, the adjacency matrix is not symmetric. Therefore we can define
weighted, directed versions of network connectedness statistics readily that
include degrees, degree distributions, distances and diameters. Using the
time-varying approximating model, we will define a truly time-varying
adjacency matrix that will describe a dynamic network.

The proposed network connectedness measure is also directly connected to the
vast economic literature about the importance of network effects in
macroeconomics (see Acemoglu et al., 2012; Carvalho and Gabaix, 2013; Gabaix,
2016; Barrot and Sauvagnat, 2016; Acemoglu et al., 2017; Baqaee and Farhi,
2019; Altinoglu, 2020; Acemoglu and Azar, 2020). Herskovic et al. (2020) state
that measuring network effects is crucial to explain the joint evolution of
firm volatility distributions. The network analysis has developed conceptual
frameworks and an extensive set of tools to effectively measure
interconnections among the units of a network, see for instance the survey by
Carvalho and Tahbaz-Salehi (2019).

Moreover, our network connectedness measure improves on shocks to uncertainty
measured ex-post (e.g. Diebold and Yılmaz, 2014). Employing implied measures
of uncertainty gives one access to a different set of information in
uncertainty reflecting market participants’ expectations of future movements
in the underlying asset, a set of information found superior compared to ex-
post measures of uncertainty (see Christensen and Prabhala, 1998). We are
naturally interested in capturing shocks to the ex-ante uncertainty of
industry $j$ that will transmit to future expectations about the uncertainty
of industry $k$.101010Baruník et al. (2020) stated that option based measures
of uncertainty reflect decisions of many agents assessing the risks from the
existing linkages. The options market-based approach allows us to monitor the
network on a daily frequency as well as use its forward-looking strength in
contrast to other network techniques based on balance sheet and other
information which is generally unavailable at high frequency.

Finally, we note that our measures can have a direct causal interpretation.
Rambachan and Shephard (2019) provide an important discussion about the causal
interpretation of impulse response analysis in the time series literature. In
particular, they argue that if an observable time series is shown to be a
potential outcome time series, then generalized impulse response functions
have a direct causal interpretation. Potential outcome series describe at time
$t$ the output for a particular path of treatments.

In the context of our study, paths of treatments are shocks. The assumptions
required for a potential outcome series are natural and intuitive for a
typical economic and/or financial time series: i) they depend only on past and
current shocks; ii) series are outcomes of shocks; and iii) assignment of
shocks depend only on past outcomes and shocks. The dynamic adjacency matrix
we introduce in the next section is a transformation of generalized impulse
response functions. Therefore, the dynamic adjacency matrix and all measures
that stem from manipulations of its elements possess a causal interpretation;
thus establishing the notion of causal dynamic network measures.

###  3.2 Construction of dynamic uncertainty network

To formalize the discussion, we construct a dynamic uncertainty network of
industries from the industry implied volatilities computed for the main U.S.
industries and we interpret the TVP-VAR model approximating its dynamics as a
dynamic network following the work of Barunik and Ellington (2020). In
particular, consider a locally stationary TVP-VAR of lag order $p$ describing
the dynamics of industry uncertainty as

$\mathbf{IVIX}_{t,T}=\boldsymbol{\Phi}_{1}(t/T)\mathbf{IVIX}_{t-1,T}+\ldots+\boldsymbol{\Phi}_{p}(t/T)\mathbf{IVIX}_{t-p,T}+\boldsymbol{\epsilon}_{t,T},$
(2)

where
$\mathbf{IVIX}_{t,T}=\left(\text{IVIX}^{(\text{1})}_{t,T},\ldots,\text{IVIX}^{(\text{N})}_{t,T}\right)^{\top}$
is a doubly indexed $N$-variate time series of industry uncertainties,
$\boldsymbol{\epsilon}_{t,T}=\sum^{-1/2}(t/T)\eta_{t,T}$ with $\eta_{t,T}\sim
NID(0,I_{M})$, and
$\boldsymbol{\Phi}(t/T)=\Big{(}\boldsymbol{\Phi}_{1}(t/T),....,\boldsymbol{\Phi}_{p}(t/T)\Big{)}^{T}$
are the time-varying autoregressive coefficients. Note that $t$ refers to a
discrete time index $1\leq t\leq T$ and $T$ is an additional index indicating
the sharpness of the local approximation of the time series by a stationary
one. Rescaling time such that the continuous parameter $u\approx t/T$ is a
local approximation of the weakly stationary time-series (Dahlhaus, 1996), we
approximate the $\mathbf{IVIX}_{t,T}$ in a neighborhood of a fixed time point
$u_{0}=t_{0}/T$ by a stationary process $\widetilde{\mathbf{IVIX}}_{t}(u_{0})$
as

$\widetilde{\mathbf{IVIX}}_{t}(u_{0})=\boldsymbol{\Phi}_{1}(u_{0})\widetilde{\mathbf{IVIX}}_{t-1}(u_{0})\ldots+\boldsymbol{\Phi}_{p}(u_{0})\widetilde{\mathbf{IVIX}}_{t-p}(u_{0})+\boldsymbol{\epsilon}_{t}.$
(3)

The process has time-varying Vector Moving Average VMA($\infty$)
representation (Dahlhaus et al., 2009; Roueff and Sanchez-Perez, 2016)

$\mathbf{IVIX}_{t,T}=\sum_{h=-\infty}^{\infty}\boldsymbol{\Psi}_{t,T,h}\boldsymbol{\epsilon}_{t-h}$
(4)

where parameter vector
$\boldsymbol{\Psi}_{t,T,h}\approx\boldsymbol{\Psi}_{h}(t/T)$ is a time-varying
impulse response function characterized by a bounded stochastic
process.111111Since $\boldsymbol{\Psi}_{t,T,h}$ contains an infinite number of
lags, we approximate the moving average coefficients at $h=1,\ldots,H$
horizons. The connectedness measures rely on variance decompositions, which
are transformations of the information in $\boldsymbol{\Psi}_{t,T,h}$ that
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, an
identification scheme is crucial in calculating variance decompositions. We
adapt the generalized identification scheme in Pesaran and Shin (1998) to
locally stationary processes.

The following proposition establishes a time-varying representation of the
variance decomposition of shocks from asset $j$ to asset $k$. It is central to
the development of the dynamic network measures since it constitutes a dynamic
adjacency matrix.

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

121212Note to notation: $[\boldsymbol{A}]_{j,k}$ denotes the $j$th row and
$k$th column of matrix $\boldsymbol{A}$ denoted in bold.
$[\boldsymbol{A}]_{j,\cdot}$ 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 $A$.

Suppose $\mathbf{IVIX}_{t,T}$ is a locally stationary process, then the time-
varying generalized variance decomposition of the $j$th variable at a rescaled
time $u=t_{0}/T$ due to shocks in the $k$th variable forming a dynamic
adjacency matrix of a network is

$\Big{[}\boldsymbol{\theta}^{H}(u)\Big{]}_{j,k}=\frac{\sigma_{kk}^{-1}\displaystyle\sum_{h=0}^{H}\Bigg{(}\Big{[}\boldsymbol{\Psi}_{h}(u)\boldsymbol{\Sigma}(u)\Big{]}_{j,k}\Bigg{)}^{2}}{\displaystyle\sum_{h=0}^{H}\Big{[}\boldsymbol{\Psi}_{h}(u)\boldsymbol{\Sigma}(u)\boldsymbol{\Psi}_{h}^{\top}(u)\Big{]}_{j,j}}$
(5)

where $\boldsymbol{\Psi}_{h}(u)$ is a time-varying impulse response function.

###### Proof.

See Appendix C. ∎

It is important to note that proposition 12 defines the dynamic network
completely. Naturally, our adjacency matrix is filled with weighted links
showing strengths of the connections over time. The links are directional,
meaning that the $j$ to $k$ link is not necessarily the same as the $k$ to $j$
link. Therefore the adjacency matrix is asymmetric.

To characterize network uncertainty, we define total dynamic network
connectedness measures in the spirit of Diebold and Yılmaz (2014); Barunik and
Ellington (2020) as the ratio of the off-diagonal elements to the sum of the
entire matrix

$\mathcal{C}^{H}(u)=100\times\displaystyle\sum_{\begin{subarray}{c}j,k=1\\\
j\neq
k\end{subarray}}^{N}\Big{[}\widetilde{}\boldsymbol{\theta}^{H}(u)\Big{]}_{j,k}\Bigg{/}\displaystyle\sum_{j,k=1}^{N}\Big{[}\widetilde{}\boldsymbol{\theta}^{H}(u)\Big{]}_{j,k}$
(6)

where $\Big{[}\widetilde{}\boldsymbol{\theta}^{H}(u)\Big{]}$ is a normalized
$\boldsymbol{\theta}$ by the row sum. This measures the contribution of
forecast error variance attributable to all shocks in the system, minus the
contribution of own shocks. Similar to the aggregate network connectedness
measure that infers the system-wide strengths of connections, we define
measures that will reveal when an individual industry is a transmitter or a
receiver of uncertainty shocks in the system. We use these measures to proxy
dynamic network uncertainty. The dynamic directional connectedness that
measures how much of each industry’s $j$ variance is due to shocks in other
industry $j\neq k$ in the economy is given by

$\mathcal{C}_{j\leftarrow\bullet}^{H}(u)=100\times\displaystyle\sum_{\begin{subarray}{c}k=1\\\
k\neq
j\end{subarray}}^{N}\Big{[}\widetilde{}\boldsymbol{\theta}^{H}(u)\Big{]}_{j,k}\Bigg{/}\displaystyle\sum_{j,k=1}^{N}\Big{[}\widetilde{}\boldsymbol{\theta}^{H}(u)\Big{]}_{j,k},$
(7)

defining the so-called from connectedness. Note one can precisely interpret
this quantity as dynamic from-degrees (or out-degrees in the network
literature) that associates with the nodes of the weighted directed network we
represent by the dynamic variance decomposition matrix. Likewise, the
contribution of asset $j$ to variances in other variables is

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

and is the so-called to connectedness. Again, one precisely interprets this as
dynamic to-degrees (or in-degrees in the network literature) that associates
with the nodes of the weighted directed network that we represent by the
variance decompositions matrix. These two measures show how other industries
contribute to the uncertainty of industry $j$, and how industry $j$
contributes to the uncertainty of others, respectively, in a time-varying
fashion. Further, the net dynamic connectedness showing whether an industry is
inducing more uncertainty than it receives from other industries in the system
can be calculated as the difference between to and from is as
$\mathcal{C}_{j,\textsc{net}}^{H}(u)=\mathcal{C}_{j\rightarrow\bullet}^{H}(u)-\mathcal{C}_{j\leftarrow\bullet}^{H}(u)$
and the agg connectedness measure as
$\mathcal{C}_{j,\textsc{agg}}^{H}(u)=\mathcal{C}_{j\rightarrow\bullet}^{H}(u)+\mathcal{C}_{j\leftarrow\bullet}^{H}(u)$.

Finally, to obtain the time-varying coefficient estimates, and the time-
varying covariance matrices at a fixed time point $u=t_{0}/T$,
$\boldsymbol{\Phi}_{1}(u),...,\boldsymbol{\Phi}_{p}(u)$
$\boldsymbol{\Sigma}(u)$, we estimate the approximating model in (2) using
Quasi-Bayesian Local-Likelihood (QBLL) methods (Petrova, 2019). Specifically,
we use 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 available analytically. 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 for
inference. We detail the estimation algorithm in Appendix D.

##  4 The dynamics of the network connectedness of industry uncertainties

The technological and housing market bubbles, the commodity crash, and the
Covid-19 pandemic are a few major examples that show how a dramatic increase
in uncertainty and different investors’ expectations can rise sharply in many
alternative industries. Being able to temporally and precisely characterize
the industry-based network dynamics is crucial given that industries can
swiftly change their characteristics and macro-economic roles. Working with
dynamic network estimates, we can characterize and assess industry uncertainty
connections in a timely and forward-looking manner according to the precise
events leading to a more or less connected industry uncertainty network. This
measurement provides new insights about the propagation of the ex-ante
uncertainty shocks over different phases of the business cycle, and it
identifies periods in which the U.S. industries’ uncertainty was tightly
connected.

We compute the dynamic aggregate network connectedness through equation 6 and
present its dynamics in Figure 2. We identify several cycles mainly driven by
key events that took place in our sample such as the dot com bubble in the
early 2000s, the housing market bubble, the 2007-2009 GFC, and the most recent
Covid-19 crisis. Some events might be described as bursts that rapidly
subside, others might be characterized by a more continuous pattern and trend.
We also split the time period into inversions, recessions and expansions
following the NBER classification. Inversions are marked between July 2000 and
March 2001 and between September 2006 and December 2007.131313The FOMC raised
the target fed funds rate by 25 basis points on June 29, 2006 and lowered the
target by 50 basis points on September 18, 2007. Adrian and Estrella (2008)
identified September 2006 as the end of the tightening cycle because during
that month the one-month fed futures rate went from higher than the spot rate
to lower. Due to the unprecedented causes of the pandemic recession in 2020,
this has resulted in a downturn with different characteristics and dynamics
than prior recessions. Hence, we are unable to establish an inversion period
for the Covid-19 crisis that we signal only as a recession from February 2020.
See also the NBER website: https://www.nber.org/cycles.html. The recessions
are marked between April 2001 and November 2001, between January 2008 and June
2009, and between February 2020 until the end of the sample, while the other
years are marked as expansions.

Figure 2: Dynamic Network Connectedness of Industry Uncertainties Notes: This
figure shows the dynamic uncertainty network connectedness with respect to the
11 U.S. industries estimated during the 03-01-2000 – 29-05-2020 period at a
daily frequency. Inversions (light grey area) are marked between July 2000 and
March 2001 and between September 2006 and December 2007. The recessions (grey
area) are marked between April 2001 and November 2001, between January 2008
and June 2009, and between February 2020 until the end of the sample, while
the other years are marked as expansions. Note the network connectedness is
plotted with two standard deviation percentiles of the measure.

We document the system to be strongly connected with values fluctuating around
60% in the first half of 2000. The first cycle starts with the burst of the
tech bubble in 2000, and with the network measure climbing from about 60% to
68%, and increasing up to about 80% in the second half of 2001 as a response
to the dot com bubble strengthening U.S. industries’ uncertainty connections
via shock to uncertainty from the technology industry. The index recovers to
the initial level until 2004, hitting minimum values in our sample at the end
of 2004 before spiking again. After that, uncertainty network connectedness
shows a new lower average level, fluctuating around 50% until the second half
of 2007, the only exception being a peak at the end of 2005 which might be due
to the U.S. housing bubble showing a connectedness level up to 67%.

We observe the index recording a significant upward movement from the
beginning of 2007 to 2009 reaching a level close to 80%, in response to the
high uncertainty during the 2007–2009 GFC spreading from the financial
industry to other industries. Several cycles can be detected during the
2007–2009 GFC: the first between the first quarter of 2007 and August 2007
reflecting the U.S. credit crunch; the second in January–March 2008 (panic in
stock and foreign exchange markets, and Bear Stearns’ takeover by JP Morgan),
the collapse of Lehman Brothers in September 2008 showing a spike from 47% to
75% in our network connectedness, and lastly in the first half of 2009 when
the financial crisis started to propagate among all other industries,
increasing the average network connectedness level.

Uncertainty connectedness spikes again in line with the two phases of the
European sovereign debt crisis, in 2010 and the second half of 2011, reaching
one of the highest levels, up to that point, close to 80%. We then observe a
drop in 2012, this being followed by a quite calm period from 2012 to
mid-2013. Connectedness spikes again at the end of 2013 due to trade wars and
energy turmoils reaching levels above 70%, and twice at the end of 2014 and in
mid-2015 reaching levels overcoming 80%. Connectedness peaks in correspondence
of Brexit in 2016 and at the end of 2017, and eventually in 2020 due to the
coronavirus outbreak, reaching its all-time maximum value in March 2020 (a
level of almost 90%), signalling the beginning of the Covid-19 crisis. This
reflects the tight connectedness among all industries in the coronavirus
period since almost all industries have been severely affected.

The fluctuations of the aggregate uncertainty network across crises, market
downturns and expansions open up for a further investigation of the role of
each industry uncertainty network characteristics. The U.S. industries appear
to be more connected after the GFC and even more with the most recent Covid-19
crisis. We still lack an understanding of which industries drive the
tightening or loosening of the network according to different business cycles.
The next sections aim to clarify these points exploiting the precise time-
varying estimation of the uncertainty network first, and its forward-looking
properties in the last section.

##  5 Uncertainty networks across the business cycles

In addition to the aggregate network characteristics, we can classify each
industry based on their expected contribution to shocks to uncertainty in the
system across different phases of the business cycle. Specifically, we are
interested to identify transmitters, receivers as well as industries being
hubs of uncertainty. To this end, we classify industries according to the
$\mathcal{C}_{\textsc{net}}^{H}$ and $\mathcal{C}_{\textsc{agg}}^{H}$
characteristics of dynamic uncertainty network.

A specific shock to uncertainty related to any industry, especially when the
most influential ones are affected, can trigger major consequences for the
other industries generating an aggregate impact on the whole network,
tightening or weakening the uncertainty network, as well as being ultimately
transmitted to the real economy. For instance, a tightening in the industry
network may be connected to drops in real activity, representing a timely
monitoring tool for immediate interventions by the Federal Reserve to sustain
the business cycle.

###  5.1 Hubs, non-hubs and business cycles

In the case of positive or negative net measure, computed as the difference
between to (equation 8) and from (equation 7) values, an industry is deemed to
be an uncertainty transmitter or receiver, respectively. An industry receiving
or transmitting shocks to uncertainty with an intermediate level can be
classified as a moderate transmitter or receiver, respectively, and may
contribute to the uncertainty propagation in the system in a mild manner. An
industry transmitting shocks to the system more (less) then receiving shocks
from the system is labelled as a transmitter (receiver). An industry showing
high values of both directional measures reflected by high agg values is
playing an active role in the transmission of uncertainty shocks and is
denoted as “uncertainty hub”, and is an industry that contributes the most to
uncertainty shocks within the network. Conversely, a neutral industry showing
low agg values is denoted as “uncertainty non-hub”.

Industries might have changed their roles in terms of their contribution to
shocks to uncertainty according to the specific economic cycle. Accordingly,
we average the network characteristics across each of the three business cycle
phases (inversions, recessions and expansions) as well as over the total
period. Table 2 provides the details.

Financials and IT are detected as the main uncertainty hubs in the inversion
and recession periods, reflecting the role of these two industries in the dot
com and GFC, respectively. Also, the consumer discretionary and industrials
are classified as uncertainty hubs during recessions. The IT industry is found
to be the main uncertainty hub during all cycles consistently, showing a time-
invariant role as a key industry in terms of the contribution of shocks to
uncertainty, especially during the dot com and Covid-19 recessions. During
expansion periods, the communication industry also plays a hub role in
addition to the IT industry. In contrast, materials, real estate and utilities
show the smallest values of network statistics and are classified as
uncertainty non-hubs.

The IT, communication and industrial industries are classified as the main
uncertainty hubs within the total period, with positive net characteristics
paired with the highest agg values. This finding highlights the important role
that the information and communication technology (ICT) industries have played
in the last two decades in the system (e.g. Jorgenson, 2001; Bloom et al.,
2012). Consumer discretionary and energy industries can also be classified as
uncertainty hubs given their high agg statistics. Conversely, we find that
financials, materials, real estate and utilities are overall classified as
uncertainty non-hubs within the whole sample. For instance, Table 2 reveals
that the IT industry, the main uncertainty hub, contributes to the network
connectedness characteristics two and three-times more in comparison to the M
and U industries, respectively. Market participants forward-looking
expectations show that shocks to uncertainty in one of the hubs are valued
differently from the corresponding shocks to uncertainty in non-hubs.

Notably, the financial industry uncertainty has been transmitting differently
within different market settings, mainly during inversions and recessions, but
it is overall classified as a non-hub.141414Studies show how the financial and
banking sector may represent a major channel in transmitting the shocks across
markets during crises (e.g. Kaminsky and Reinhart, 1999; Tai, 2004; Baur,
2012). This finding reflects the ability of authorities to influence the U.S.
financial sector in the aftermath of the GFC through accommodative and
unconventional monetary policies. To some, such policy interventions aimed at
restoring the functioning of the financial sector during the Great Recession
might have been key to avoid a second Great Depression (see Bianchi,
2020).151515To note that The IMF October 2017 Global Financial Stability
Report (GFSR) finds that the global financial system continues to strengthen
in response to extraordinary policy support, regulatory enhancements, and the
cyclical upturn in growth. This has also been accompanied by stronger
harmonization of financial regulatory standards (e.g. the Basel capital
framework). The financial sector has seen one of the most-pronounced stock
market booms on record during 2009-2018. It is therefore not surprising that
there has been a low level of uncertainty within the financial industry and,
as a consequence, transmitted to the rest of the system.

Table 2: Aggregate $\mathcal{C}_{\textsc{NET}}$ and $\mathcal{C}_{\textsc{AGG}}$ across business cycles |  | Inversion |  | Recession |  | Expansion |  | Total Period
---|---|---|---|---|---|---|---|---
|  | NET | AGG |  AGG $\%$ |  | NET | AGG |  AGG $\%$ |  | NET | AGG |  AGG $\%$ |  | NET | AGG |  AGG $\%$
CD |  | -1.72 | 16.77 | 9.8 |  | 0.49 | 35.02 | 10.9 |  | -0.98 | 25.61 | 11.0 |  | -0.71 | 29.61 | 11.1
CM |  | -1.55 | 13.07 | 7.6 |  | -0.29 | 28.88 | 9.0 |  | 1.37 | 27.16 | 11.6 |  | 1.55 | 30.49 | 11.5
CS |  | -0.87 | 12.12 | 7.1 |  | -0.15 | 32.70 | 10.2 |  | -1.54 | 20.93 | 9.0 |  | -1.50 | 24.27 | 9.1
E |  | -4.52 | 16.88 | 9.8 |  | -0.43 | 30.70 | 9.5 |  | -0.43 | 25.09 | 10.7 |  | -0.80 | 28.27 | 10.6
F |  | 3.76 | 21.07 | 12.3 |  | 0.01 | 40.55 | 12.6 |  | 0.04 | 18.02 | 7.7 |  | 0.54 | 22.15 | 8.3
HC |  | -0.76 | 14.23 | 8.3 |  | -2.15 | 29.70 | 9.2 |  | -0.29 | 23.63 | 10.1 |  | -0.56 | 26.45 | 9.9
IN |  | 0.92 | 20.63 | 12.1 |  | -0.21 | 40.09 | 12.5 |  | 0.26 | 26.71 | 11.4 |  | 0.17 | 31.47 | 11.9
IT |  | 4.42 | 25.09 | 14.7 |  | 2.77 | 39.74 | 12.4 |  | 1.93 | 28.48 | 12.2 |  | 2.23 | 33.71 | 12.7
M |  | -0.33 | 10.15 | 5.9 |  | -1.77 | 15.99 | 5.0 |  | -1.01 | 14.62 | 6.3 |  | -0.98 | 15.84 | 6.0
RE |  | 1.24 | 12.01 | 7.1 |  | 1.55 | 17.18 | 5.3 |  | 0.97 | 13.89 | 6.0 |  | 0.51 | 14.22 | 5.3
U |  | -0.58 | 9.11 | 5.3 |  | 0.19 | 11.04 | 3.4 |  | -0.24 | 9.41 | 4.0 |  | -0.43 | 9.59 | 3.6
Notes: The table shows the average net and agg values with respect to the 11
U.S. industries’ uncertainty network. When the net measure is positive an
industry can be classified as a net marginal transmitter, while, when
negative, it can be classified as a net marginal receiver. The highest values
of the agg network statistics are associated with uncertainty hubs, while the
lowest with uncertainty non-hubs. The statistics are reported for the business
cycle main phases, namely inversion, recession, expansion, aggregated, and
also for the total period, namely from 03-01-2000 to 29-05-2020, at a daily
frequency.

After having classified the industries based on their contribution to
uncertainty across business cycles, we compute separate forward-looking
networks extracted from uncertainty hubs only and uncertainty non-hubs only.
To this end, we input the $\text{IVIX}^{(\text{Ind})}_{t}$ of hubs only
(consumer discretionary, communications, energy, industrials and IT) and of
non-hubs only (financials, materials, real estate and utilities) in the
network model in equation 6.

Figure 3 plots the dynamics of both network connectedness characteristics. We
observe that the uncertainty network extracted from hubs shows a higher degree
of integration compared to the ones extracted from non-hubs. The difference in
uncertainty produced by hubs versus non-hubs becomes stronger in the post-GFC
and more recent years, implying that shocks in hubs such as ICT industries and
industrials have played a key role increasingly contributing to shocks in
uncertainty within the system and aggregate fluctuations over time. This
finding is consistent with the development of the ICT industries in the last
decades. Such hubs are in fact industries providing services to other
industries, sharing a role in propagating uncertainty shocks to other sectors
(see Bloom et al., 2012). This channel can also be because uncertainty hubs
are sectors likely to have financial importance and a large and fast-growing
market capitalization. Thus, our proposed framework for the role of
uncertainty hubs in driving fluctuations in the aggregate economy is
reminiscent of investment-specific technology shocks (e.g. Greenwood et al.,
2000; Justiniano et al., 2010).

Figure 3: Uncertainty Hubs and non-Hubs Networks Notes: This figure shows the
network connectedness measures extracted from uncertainty hubs (black line)
and non-hubs (grey line) from 03-01-2000 to 29-05-2020, at a daily frequency.
Inversions (light grey area) are marked between July 2000 and March 2001 and
between September 2006 and December 2007. The recessions (grey area) are
marked between April 2001 and November 2001, between January 2008 and June
2009, and between February 2020 until the end of the sample, while the other
years are marked as expansions.

An uncertainty shock can affect production, employment and growth within the
hub, it can also generate larger uncertainty spillovers, changes in prices,
growth and production of other sectors in the system, affecting the broader
economy (e.g. Kozeniauskas et al., 2018). As an example of a few possible
mechanisms, Lehn and Winberry (2020) state that a positive shock to an
investment hub directly increases production and employment in that hub;
because the shock also raises the supply of investment goods, other sectors
increase employment to produce more intermediate inputs for the hub. In
contrast, a shock in a non-hub has a small effect on investment supply
generating smaller spillovers to the rest of the economy. Further, according
to the islands’ framework in Garin et al. (2018), a shock that simultaneously
affects both hubs and non-hubs (both islands) is significantly weaker compared
to shocks in both islands separately, this due to a reallocative shock
mechanism. In our context, this would directly translate into an increase in
uncertainty of both islands (hubs and non-hubs) with the difference that non-
hubs receive shocks to uncertainty, while hubs (mainly providers of services
cross-sectorally) both receive and transmit shocks to uncertainty to a much
larger extent (captured by a higher agg statistic in our model). The
uncertainty transmitted or received by hubs is found to be approximately three
times higher than for non-hubs.

Given the framework and rationale that we put forward, we hypothesize that the
response of business cycles to shocks should be larger for hubs than for non-
hubs. Shocks to uncertainty hubs generate grater effects within the network of
industries and extending their implications towards the real economy.
Investors’ expectations and beliefs generating shocks to uncertainty in hubs
reflect value-added growth into those, future price increases or losses,
productivity shocks and changes in future outcomes which can trigger economic
booms or downturns to a larger extent than in non-hubs. Therefore, the hubs-
based network measure is the natural candidate to be a better leading
indicator of business cycles compared to non-hubs, which is explored in the
next section.

###  5.2 Shocks to uncertainty in specific business cycles

Before moving on, we briefly show a more granular classification of industries
for each business cycle. We report the $\mathcal{C}_{\textsc{net}}$ and
$\mathcal{C}_{\textsc{agg}}$ statistics in Table E1 in the appendix.

The IT industry can be classified as the main uncertainty hub during both
inversion and recession phases related to the dot com bubble (from 2000 to
mid-2002), contributing to a spread in uncertainty within the whole system.
Given the high agg values, CD and F are classified as main uncertainty hubs
during the dot com recession. In contrast, M and U are classified as
uncertainty non-hubs during this first recession. Interestingly, around
2002-2004 we observe a role of uncertainty transmitter for RE since this
period corresponds to the U.S. housing market bubble. The 2007–2008 GFC was
indeed related to the bursting of a real estate bubble, identified precisely
by the dynamic network at the end of 2005.

We then observe a main role as net uncertainty transmitter for F between 2006
and 2007 reflecting the first events related to the GFC and the mortgage
crisis, and especially in correspondence to the collapse of Lehman Brothers in
September 2008. F shows a predominant role during the GFC, both in inversion
and actual recession, contributing to spread uncertainty across the whole
system and being therefore classifiable as uncertainty hub within this period.
After this, the U.S. economy is characterized by a long recovery period in
which the ICT industries are, by far, the main uncertainty hubs. From 2010
until 2015, we observe E taking on a role as an uncertainty hub, and this
could be due to a combination of events throughout this time period,161616For
instance the changes in the U.S. energy policies, U.S. conflicts in the Middle
East, OPEC excessive production, and U.S. oil prices collapse leading to
higher oil price volatility. Moreover, in summer 2011, oil and other
commodities prices have fallen, although from historically high levels, and
this was reflected in the fear that the commodities boom was over, with raw
materials set to drop sharply. emphasized by the spike in uncertainty between
June 2014 and February 2015 associated with the global commodity price crash
and oil price drop. From 2015 until the end of our sample period, industrials,
consumer discretionary, communications and IT are detected as the main
uncertainty hubs within our system, whereas all the other industries either as
mild receivers or as uncertainty non-hubs. Finally, in the most recent
Covid-19 recession, we note that the same industries are found to be the main
uncertainty hubs, while health-care, real estate and utilities as uncertainty
non-hubs.

This analysis further highlights the usefulness of the time-varying parameter
model to more precisely uncover timing in the role of industries in shocks to
uncertainty. Business cycles, economic downturns and crises with different
nature might intensify the role of a specific industry, becoming a key player
in contributing to shocks to uncertainty within the network. It is then
crucial to be able to identify the role of each industry across time.
Moreover, differences in terms of shocks to uncertainty might play a critical
role in predicting business cycles throughout the sample.

##  6 Industry uncertainty networks and business cycle predictability

We study whether the information extracted from options of large firms and
aggregated into ex-ante industry uncertainty networks may contribute to the
predictability of indicators of the business cycle. We draw from previous
literature on the role of aggregated sector-level or firm-level networks of
microeconomic shocks to uncertainty and their relationship with the aggregate
economy and business conditions (e.g. Gabaix, 2011; Acemoglu et al., 2012;
Carvalho and Gabaix, 2013; Barrot and Sauvagnat, 2016; Atalay, 2017; Lehn and
Winberry, 2020). The forward-looking aspect of our network is what we are
interested in exploiting in this section to predict future business cycles in
advance. Further, we also argue that network measures extracted from
uncertainty hubs play a more informative predictive role compared to the ones
extracted from uncertainty non-hubs.

The literature on business cycle indicators, turning points, expansions and
recessions characterizations has a long history. The Business Cycle Dating
Committee (BCDC) of the National Bureau of Economic Research (NBER) provides
probably the largest historical record of the economic activity classification
and business cycles. However, its releases are usually made with about a
year’s delay. It, therefore, represents a very reliable chronology of peaks
and troughs throughout history rather than providing an early warning tool.
The latter is what studies have been aiming to propose for decades. We study
the relationship between uncertainty network connectedness and more timely
indicators of business cycles, both coincident and leading indicators. We
hypothesize that our network connectedness may represent an even more timely
and forward-looking predictor of business cycle indicators given its ex-ante
characteristics from the options market. It may therefore represent both a
good predictor of coincident and leading indicators and it can be also
classified as a leading monitoring tool of the business cycle itself. This
would provide researchers, policy-makers and the public with an even more
timely indicator than the ones already available.

###  6.1 The predictability of business cycle coincident indicators

Berge and Jordà (2011) provides an exhaustive summary of the different
measures available that provide reliable signals about the current state of
the business cycle. Among those, we adopt the Chicago Fed National Activity
Index (CFNAI) and the Aruoba, Diebold, and Scotti (ADS) index of business
conditions (see Aruoba et al., 2009).171717The CFNAI is a monthly index that
tracks the overall economic activity and the inflationary pressure. It is
computed as the first principal component of 85 series drawn from four broad
categories of data all adjusted for inflation. A zero value for the monthly
index has been associated with the national economy expanding at its
historical trend (average) rate of growth; negative values with below-average
growth; positive values with above-average growth. See also Chava et al.
(2020) on the adoption of the CFNAI as business cycle indicator. For more
information see https://www.chicagofed.org/publications/cfnai/index. The
Aruoba-Diebold-Scotti (ADS) Business Condition Index tracks real business
conditions at a high frequency and it is based on economic indicators. The
average value of the ADS index is zero. Progressively positive values indicate
progressively better-than-average conditions, whereas progressively more
negative values indicate progressively worse-than-average conditions. It is
collected from: https://www.philadelphiafed.org/research-and-data/real-time-
center. Specifically, we adopt the 3-month moving average of the Chicago FED
National Activity Index (CFNAI-MA3) and we aggregate the ADS indicator at a
monthly frequency. We adopt these as business cycle coincident indicators and
we study whether the uncertainty network can forecast them.

We also disentangle these business cycles indicators into proxies of
expansions and recessions following the decomposition approach by Berge and
Jordà (2011) who proposed optimal thresholds for CFNAI and ADS equal to -0.72
and -0.80, respectively. Thus, periods of economic expansion are associated
with values of the CFNAI-MA3 (ADS) above –0.72 (-0.80), whereas periods of
economic contraction with values of the CFNAI-MA3 (ADS) below –0.72 (-0.80).
We study whether the predictive ability of the uncertainty network
connectedness varies according to different states of the business cycle. We
aggregate the network connectedness measure at a monthly frequency to match
the frequency of the business cycle indicators and we run the following
predictive regression:

$\mathcal{Y}^{(\ell)}_{t+h}=\beta_{0}+\beta_{\mathcal{C}}\
\mathcal{C}_{t}+\sum_{i=1}^{N}\beta_{X,i}\ X_{t,i}+\epsilon_{t}$ (9)

where $\mathcal{Y}^{(\ell)}_{t+h}$ is one of the business cycle indicators we
select (or their components) with the predictive horizon $h\in{1,3,6,9,12}$
months. The $\mathcal{C}_{t}$ is the industry uncertainty network measure
(note we drop index $H$ here for the ease of notation), $X_{t,i}$ is a set of
control variables including both traditional predictors of business cycles
such as oil price changes (OIL), term-spread as 10-year bond rate minus the
3-month bond rate (TS), unemployment rate (UR), (see also Gabaix, 2011), and
also a potential leading indicator extracted from the financial markets,
namely the changes in the CBOE VIX index being a common proxy for macro
uncertainty in the U.S. (VIX), changes in the $S\&P500$ price index (SPX), the
Bloomberg Commodity price index (COMM) and the S&P Case-Shiller Home price
(CSHP).181818Oil prices, 10-year and 3-month bond rates, unemployment rate and
the S&P Case-Shiller Home price index are collected from the Federal Reserve
Bank of St. Louis economic database at https://fred.stlouisfed.org/; the CBOE
VIX index, $S\&P500$ price index and the Bloomberg Commodity price index are
collected from Bloomberg. Therefore, $X_{t,i}$ is indexed for $i$ up to $N=7$,
the number of control we select, with $i\in(\text{OIL, TS, UR, VIX, SPX, COMM,
CSHP})$.

Table 3 reports the predictive results. We observe that the uncertainty
network is a strong predictor of the aggregate CFNAI-MA3 indicator of the
business cycle up to 12 months in advance also after taking into account the
information of the selected controls. The coefficient associated with our
independent predictor is negative suggesting that a tighter network of
industry uncertainties would lead to a contraction in the business cycle in
the future horizons. The network is therefore found to behave counter-
cyclically, a finding in line with previous studies relating uncertainty
measures with the business cycles (e.g. Bloom et al., 2018). The performance
of the models measured by the adjusted-$R^{2}$ is found to be close to 30% at
the 1-month horizon, then it decreases at the semi-annual horizon, increasing
again at longer horizons such as 9 and 12 months.

Table 3: CFNAI-MA3 Predictive Results

|  | Panel A: CFNAI-MA3
---|---|---
|  | h=1 | h=3 | h=6 | h=9 | h=12
$\mathcal{C}_{t}$ $\mid$ $X_{t}$ |  | -0.013** | -0.024*** | -0.026*** | -0.040*** | -0.028***
|  | (0.006) | (0.007) | (0.007) | (0.007) | (0.007)
Adj. $R^{2}$ |  | 0.294 | 0.156 | 0.067 | 0.176 | 0.168
Obs |  | 243 | 241 | 238 | 235 | 232
|  | Panel B: CFNAI-MA3 Expansion
|  | h=1 | h=3 | h=6 | h=9 | h=12
$\mathcal{C}_{t}$ $\mid$ $X_{t}$ |  | -0.007*** | -0.010*** | -0.012*** | -0.011*** | -0.008***
|  | (0.002) | (0.002) | (0.002) | (0.002) | (0.002)
Adj. $R^{2}$ |  | 0.093 | 0.175 | 0.268 | 0.304 | 0.239
Obs |  | 243 | 241 | 238 | 235 | 232
|  | Panel C: CFNAI-MA3 Recession
|  | h=1 | h=3 | h=6 | h=9 | h=12
$\mathcal{C}_{t}$ $\mid$ $X_{t}$ |  | -0.006 | -0.014** | -0.013** | -0.029*** | -0.020***
|  | (0.006) | (0.007) | (0.007) | (0.007) | (0.007)
Adj. $R^{2}$ |  | 0.287 | 0.135 | 0.059 | 0.144 | 0.148
Obs |  | 243 | 241 | 238 | 235 | 232

  * •

Notes: This table presents the results of the predictive regression in
equation 9 between the industry uncertainty network connectedness and the
3-month moving average of the Chicago FED National Activity Index (CFNAI-MA3),
indicator of business cycle (Panel A). In Panel B and Panel C the results of
the predictive regression with respect to the CFNAI-MA3 expansion and
recession indicators are reported, respectively. We also add a set of
controls, $X$. The five columns of the table represent different
predictability horizons with $h\in(1,3,6,9,12)$. Regressions’ coefficients and
standard errors (in parentheses), and adjusted-$R^{2}$ are reported.
Coefficients are marked with *, **, *** for 10%, 5%, 1% significance levels,
respectively. Intercept and controls results are not reported for the sake of
space. Series are considered at a monthly frequency between 01-2000 and
05-2020.

When we look at the disentangled components of the business cycle indicator,
we observe that uncertainty network can predict well future expansion periods
up to one year in advance. The sign associated with the models’ coefficients
is, again, negative therefore suggesting that expansion periods might contract
when the network is tighter. We notice a greater adjusted-$R^{2}$ performance
of the model for the 6- to 12-month horizons. Regarding U.S. recession
periods, we observe weaker predictability of network across shorter horizons.
However, the uncertainty network is still found to predict well future
recessions from 3-month up to the 12-month horizon, being the higher
adjusted-$R^{2}$ placed again on the longer horizon. Interestingly, we find a
negative sign associated with the coefficients, this implying that increasing
levels of network connectedness will expand the business cycle when in
recession (in this case the dependent variable is below the -0.72
threshold).191919The Chicago Fed suggested -0.7 to be a more accurate
threshold of turning points for the CFNAI indicator. We repeat the empirical
analysis of this section by adopting this threshold. The results are found to
be materially the same.

We validate the predictive ability of uncertainty network connectedness by
showing how it can also similarly predict a different business cycle
coincident indicator. We show the results for the ADS Index in Table F1 in the
appendix. The aggregate uncertainty network measure shows predictability power
for the ADS index from 3 months up to 12 months in advance, with again
negative coefficients and stronger performance at the long horizon. When we
look at the expansion or recession indicators, a stronger predictive ability
is still placed at longer horizons, especially for the expansion indicator.

Another possible business cycle coincident indicator is the industrial
production (IP) growth rate. As a robustness check, we repeat the same
exercise with the IP annualized growth rate, still confirming the predictive
ability of the uncertainty network. We also adopt another business cycle
coincident indicator collected from the Economic Cycle Research Institute
(ECRI), the U.S. coincident indicator (U.S.CI).202020For more information and
data see https://www.businesscycle.com/ecri-reports-indexes/all-indexes. We
take the growth rate of the indicator and show that this leads to similar
results, relegated to the appendix in Table F2. As an additional robustness
check, we also replace the uncertainty network connectedness measure
constructed with time-varying networks with a network measure constructed by
following previous studies (e.g. Diebold and Yilmaz, 2012; Diebold and Yılmaz,
2014) using a moving window. We find that the latter is unable to predict
future business cycles and the expansion and recession components,
highlighting, even more, the importance of precisely characterizing the
network at any point in time without relying on moving windows when it comes
to predicting future levels of the business cycle or the real
economy.212121The all set of results is available from the authors upon
request.

###  6.2 Industry uncertainty networks and leading indicators

Due to its forward-looking nature, we argue that the uncertainty network
connectedness could also potentially serve as a good predictor of business
cycle leading indicators, and be considered a leading indicator itself. We
here check the predictive ability of this index for two business cycle leading
indicators: the U.S. composite leading indicator (CLI) by the OECD.222222The
composite leading indicator is collected from the OECD data base at
https://data.oecd.org/leadind/composite-leading-indicator-cli.htm. CLI
provides early signals of turning points in business cycles showing
fluctuations of the economic activity around its long term potential level.
and the U.S. leading indicator (U.S.LI) computed by the Economic Cycle
Research Institute (ECRI). U.S.LI is available at a weekly frequency and
aggregated here at a monthly frequency, and we take the growth rate of the
indicator. We find similar results, confirming both significance and
coefficients signs, even after adding the set of controls. We repeat the same
predictive exercise of the previous subsection, by running equation 9 where
now the dependent variable is CLI. We report the results in Table 4. We
observe that the predictability of uncertainty network connectedness is even
stronger for the business cycle leading indicators, spanning from 3-month up
to one year and from 1-month up to 9-month horizons, for CLI and U.S.LI,
respectively. The coefficients are still found to be negative confirming our
previous findings.

Table 4: Leading Indicators Predictive Results

|  | Panel A: CLI
---|---|---
|  | h=1 | h=3 | h=6 | h=9 | h=12
$\mathcal{C}_{t}$ $\mid$ $X_{t}$ |  | -0.011 | -0.030*** | -0.051*** | -0.079*** | -0.084***
|  | (0.010) | (0.011) | (0.012) | (0.011) | (0.011)
Adj. $R^{2}$ |  | 0.435 | 0.294 | 0.192 | 0.247 | 0.262
Obs |  | 243 | 241 | 238 | 235 | 232
|  | Panel B: U.S.LI
|  | h=1 | h=3 | h=6 | h=9 | h=12
$\mathcal{C}_{t}$ $\mid$ $X_{t}$ |  | -0.348*** | -0.371*** | -0.417*** | -0.372*** | -0.108
|  | (0.063) | (0.066) | (0.066) | (0.067) | (0.073)
Adj. $R^{2}$ |  | 0.364 | 0.304 | 0.303 | 0.299 | 0.178
Obs |  | 243 | 241 | 238 | 235 | 232

  * •

Notes: This table presents the results of the predictive regression in
equation 9 between the industry uncertainty network connectedness and two
leading indicators of business cycle, namely CLI and U.S.LI, in Panel A and B,
respectively. We also add a set of controls, $X$. The five columns of the
table represent different predictability horizons with $h\in(1,3,6,9,12)$.
Regressions’ coefficients and standard errors (in parentheses), and
adjusted-$R^{2}$ are reported. Coefficients are marked with *, **, *** for
10%, 5%, 1% significance levels, respectively. Intercept and controls results
are not reported for the sake of space. Series are considered at a monthly
frequency between 01-2000 and 05-2020.

Overall, it appears that the uncertainty network can anticipate what is
commonly viewed as a business cycle leading indicator. This opens up some
interesting considerations. Given that the uncertainty network is extracted
from options prices, it is expected that the newly proposed uncertainty
network contains forward-looking information that can be useful as ex-ante
business cycle monitoring indicator. We know that a business cycle leading
indicator should ideally anticipate and predict coincident indicators. We
illustrate in the previous section that the uncertainty network shares such
properties. In this subsection, we also show how our network measure is a good
predictor of leading indicators, such a finding emphasizing even further the
usefulness of its forward-looking information content.

Table 5: Coincident Indicators Predictive Results

|  | Panel A: CFNAI-3M
---|---|---
|  | h=1 | h=3 | h=6 | h=9 | h=12
$\mathcal{C}_{t}$ $\mid$ $X_{t}$ |  | -0.011** | -0.023*** | -0.025*** | -0.040*** | -0.028***
|  | (0.005) | (0.006) | (0.007) | (0.007) | (0.007)
CLI |  | 0.432*** | 0.357*** | 0.265*** | 0.136*** | 0.008
|  | (0.036) | (0.045) | (0.051) | (0.051) | (0.053)
Adj. $R^{2}$ |  | 0.560 | 0.333 | 0.162 | 0.198 | 0.164
Obs |  | 243 | 241 | 238 | 235 | 232
|  | Panel B: ADS
|  | h=1 | h=3 | h=6 | h=9 | h=12
$\mathcal{C}_{t}$ $\mid$ $X_{t}$ |  | -0.018 | -0.026* | -0.034* | -0.053*** | -0.038**
|  | (0.015) | (0.018) | (0.018) | (0.018) | (0.019)
CLI |  | 0.590*** | 0.587*** | 0.425*** | 0.305** | 0.159
|  | (0.109) | (0.130) | (0.134) | (0.135) | (0.138)
Adj. $R^{2}$ |  | 0.351 | 0.105 | 0.066 | 0.080 | 0.074
Obs |  | 243 | 241 | 238 | 235 | 232

  * •

Notes: This table presents the results of the predictive regressions between
the industry uncertainty network connectedness, and the coincident indicators
of business cycle, namely CFNAI and ADS. We present results for regression
equation 9 in which we add a set of controls including also the leading
indicator, CLI. The five columns of the table represent different
predictability horizons with $h\in(1,3,6,9,12)$. Regressions’ coefficients and
standard errors (in parentheses), and adjusted-$R^{2}$ are reported.
Coefficients are marked with *, **, *** for 10%, 5%, 1% significance levels,
respectively. Intercept and controls results are not reported for the sake of
space, the only exception being the CLI control. Series are considered at a
monthly frequency between 01-2000 and 05-2020.

To validate this point, we test whether the existing business cycle leading
indicators might contain a different set of information, mainly at shorter
horizons, compared to our uncertainty network. To this end, we test whether
our measure can predict coincident indicators even after controlling for a
leading indicator (CLI). The results are reported in Table 5. We find that the
uncertainty network predictability holds at every horizon, even after
controlling for CLI. The latter shows a good predictive ability, however up to
the 9-month horizon (in line with the index characteristics description). The
uncertainty network clearly shows characteristics of a complementary (and
rather superior) business cycle leading indicator, spanning predictive power
from 1- to the 12-month horizon in advance, even after controlling for CLI.
For the ADS, as a proxy for a business cycle coincident indicator, we find a
weaker predictive power for the uncertainty network at the short horizon,
however still confirming the anticipatory property of about one quarter. We
repeat the same exercise of Table 5 by adopting the U.S.LI indicator by the
ECRI as control. We obtain similar findings for both CFNAI and ADS and results
are relegated to the paper appendix in Table F3. Overall, the uncertainty
network adds quite a lot in terms of long-horizon predictability compared to
the information content of other leading indicators of business cycles.

###  6.3 Hubs and non-hubs industry connectedness networks

In this subsection, we check whether the predictability power of uncertainty
hubs-based networks may differ from uncertainty non-hubs. We repeat the
empirical analysis of the previous section, now considering only hubs and non-
hubs based networks, $\mathcal{C}_{t}^{\text{hub}}$ and
$\mathcal{C}_{t}^{\text{non-hub}}$, respectively. We hypothesize that the
former leads to greater predictability since reflecting information from the
industries detected to be the main uncertainty contributors within the system.
The uncertainty hubs network is based on the CD, CM, E, IN and IT industries,
while the non-hubs network on F, M, RE and U industries, as illustrated in
section 5 and in Figure 3. Similarly to equation 9, we estimate the following:

$\mathcal{Y}^{(\ell)}_{t+h}=\beta_{0}+\beta_{\text{hub}}\
\mathcal{C}_{t}^{\text{hub}}+\beta_{\text{non-hub}}\
\mathcal{C}_{t}^{\text{non-hub}}+\sum_{i=1}^{N}\beta_{X,i}\
X_{t,i}+\epsilon_{t}$ (10)

where we add the independent variables that characterize uncertainty hubs and
non-hubs based on network connectedness taken jointly and aggregated at a
monthly frequency to match the frequency of the indicators we adopt. We
include the same set of controls $X$.

In Table 6 we observe that the predictability of the hubs network is superior
compared to the non-hubs for the aggregate CFNAI-MA3 and recessions especially
for longer horizons, while for expansions at any horizons. We notice how the
result achieved by looking at the predictive ability of uncertainty network
resemble, or appear even stronger than, the results obtained in the previous
section when looking at the aggregated predictability. In Table G1 in the
appendix we further confirm this finding by showing the results of the
$\mathcal{C}_{t}^{\text{hub}}$ predictor alone, controlling for $X$. The
predictive ability of $\mathcal{C}_{t}^{\text{hub}}$ is found to be stronger
than the one achieved by the aggregate network $\mathcal{C}$. This finding is
reflected by the higher adjusted-$R^{2}$ detected, most of the time, in Table
G1 compared to Table 3. These results suggest that the predictive ability of
the uncertainty network appear to be driven by a few uncertainty hubs.

Table 6: Hubs vs. non-Hubs Network Predictive Results

|  | Panel A: CFNAI-MA3
---|---|---
|  | h=1 | h=3 | h=6 | h=9 | h=12
$\mathcal{C}_{t}^{\text{hub}}$ $\mid$ $X_{t}$ |  | -0.007* | -0.013*** | -0.017*** | -0.021*** | -0.019***
|  | (0.004) | (0.004) | (0.004) | (0.004) | (0.004)
$\mathcal{C}_{t}^{\text{non-hub}}$ $\mid$ $X_{t}$ |  | -0.015*** | -0.021*** | -0.013* | -0.010 | 0.002
|  | (0.006) | (0.006) | (0.007) | (0.007) | (0.007)
Adj. $R^{2}$ |  | 0.319 | 0.214 | 0.115 | 0.183 | 0.183
Obs |  | 243 | 241 | 238 | 235 | 232
|  | Panel B: CFNAI-MA3 Expansion
|  | h=1 | h=3 | h=6 | h=9 | h=12
$\mathcal{C}_{t}^{\text{hub}}$ $\mid$ $X_{t}$ |  | -0.004*** | -0.005*** | -0.005*** | -0.006*** | -0.006***
|  | (0.001) | (0.001) | (0.001) | (0.001) | (0.001)
$\mathcal{C}_{t}^{\text{non-hub}}$ $\mid$ $X_{t}$ |  | 0.001 | -0.001 | -0.005*** | -0.001 | 0.0004
|  | (0.002) | (0.002) | (0.002) | (0.002) | (0.002)
Adj. $R^{2}$ |  | 0.097 | 0.150 | 0.258 | 0.289 | 0.286
Obs |  | 243 | 241 | 238 | 235 | 232
|  | Panel C: CFNAI-MA3 Recession
|  | h=1 | h=3 | h=6 | h=9 | h=12
$\mathcal{C}_{t}^{\text{hub}}$ $\mid$ $X_{t}$ |  | -0.003 | -0.008** | -0.012*** | -0.015*** | -0.012***
|  | (0.004) | (0.004) | (0.004) | (0.004) | (0.004)
$\mathcal{C}_{t}^{\text{non-hub}}$ $\mid$ $X_{t}$ |  | -0.016*** | -0.020*** | -0.009 | -0.009 | 0.002
|  | (0.005) | (0.006) | (0.007) | (0.006) | (0.006)
Adj. $R^{2}$ |  | 0.313 | 0.190 | 0.090 | 0.152 | 0.149
Obs |  | 243 | 241 | 238 | 235 | 232

  * •

Notes: This table presents the results of the predictive regression 10
comparing the predictive ability of the uncertainty hubs vs non-hubs sub-
networks with respect to the 3-month moving average of the Chicago FED
National Activity Index (CFNAI-MA3) in Panel A. In Panel B and Panel C the
results of the predictive regression with respect to the CFNAI expansion and
recession periods are reported, respectively. The five columns of the table
represent different predictability horizons with $h\in(1,3,6,9,12)$.
Regressions’ coefficients and standard errors (in parentheses), and
adjusted-$R^{2}$ are reported. Coefficients are marked with *, **, *** for
10%, 5%, 1% significance levels, respectively. Intercept and controls results
are not reported for the sake of space. Series are considered at a monthly
frequency between 01-2000 and 05-2020.

We then check the relationship between the hubs and non-hubs networks for
leading indicators. The predictive results for CLI are reported in Table 7. We
find that the hub network connectedness strongly predicts CLI up to one year,
whereas the predictive power of the non-hubs network is overall absent. Thus,
the predictive power of the uncertainty hubs network is found to be strong
also for business cycles leading indicators, confirming a clear superior
predictive ability compared to non-hubs.

We further validate the predictive ability of the hubs-based uncertainty
network by including the leading indicator CLI as a control variable in the
multivariate regression when predicting CFNAI-MA3. The hubs network shows
strong predictive ability from 3-month up to one year, complementing the
shorter horizon predictive ability of CLI by expanding it to longer horizons.
The non-hub network shows good predictive power in the short horizon and it,
therefore, appears not to contain additional information compared to other
leading indicators of business cycles e.g. CLI. The hubs-based network shows a
longer horizon predictive power, a useful feature for any business cycle
leading indicators. Our results suggest that the hubs-based network may be
considered as the main driver of the aggregate network, achieving even
stronger predictive power on its own.

Table 7: Hubs vs. non-Hubs Network Predictive Results

|  | Panel A: Leading Indicator CLI
---|---|---
|  | h=1 | h=3 | h=6 | h=9 | h=12
$\mathcal{C}_{t}^{\text{hub}}$ $\mid$ $X_{t}$ |  | -0.012** | -0.024*** | -0.039*** | -0.051*** | -0.056***
|  | (0.005) | (0.006) | (0.006) | (0.006) | (0.006)
$\mathcal{C}_{t}^{\text{non-hub}}$ $\mid$ $X_{t}$ |  | -0.009 | -0.015 | -0.016 | -0.017* | -0.005
|  | (0.008) | (0.009) | (0.010) | (0.010) | (0.010)
Adj. $R^{2}$ |  | 0.447 | 0.335 | 0.269 | 0.325 | 0.339
Obs |  | 243 | 241 | 238 | 235 | 232
|  | Panel B: CFNAI-MA3 controlling for CLI
|  | h=1 | h=3 | h=6 | h=9 | h=12
$\mathcal{C}_{t}^{\text{hub}}$ $\mid$ $X_{t}$ |  | -0.003 | -0.009*** | -0.015*** | -0.020*** | -0.019***
|  | (0.003) | (0.004) | (0.004) | (0.004) | (0.004)
$\mathcal{C}_{t}^{\text{non-hub}}$ $\mid$ $X_{t}$ |  | -0.014*** | -0.021*** | -0.012* | -0.011 | 0.002
|  | (0.004) | (0.005) | (0.007) | (0.007) | (0.007)
CLI |  | 0.424*** | 0.340*** | 0.243*** | 0.109** | -0.017
|  | (0.036) | (0.044) | (0.051) | (0.051) | (0.053)
Adj. $R^{2}$ |  | 0.573 | 0.371 | 0.193 | 0.196 | 0.179
Obs |  | 243 | 241 | 238 | 235 | 232

  * •

Notes: This table presents the results of the predictive regression 10
comparing the predictive ability of the uncertainty hubs vs non-hubs sub-
networks with respect to the leading indicator, CLI (Panel A). In Panel B, the
results with respect to the CFNAI-MA3 coincident indicator controlling also
for the leading indicator, CLI, are reported. The five columns of the table
represent different predictability horizons with $h\in(1,3,6,9,12)$.
Regressions’ coefficients and standard errors (in parentheses), and
adjusted-$R^{2}$ are reported. Coefficients are marked with *, **, *** for
10%, 5%, 1% significance levels, respectively. Intercept and controls results
are not reported for the sake of space, the only exception being the CLI
control. Series are considered at a monthly frequency between 01-2000 and
05-2020.

As a further robustness check, we also repeat the same predictive exercises by
adopting a stricter construction of the hubs and non-hubs networks, including
only CM, IN and IT industries and M, RE and U in the networks respectively. We
report the results in Table G2 in the appendix for all the coincident
indicator CFNAI-MA3, expansions and recessions, and also for the leading
indicator CLI. We corroborate our previous findings and confirm our hypothesis
since the hubs network is found to be more informative in predicting well the
future business cycle indicators.

###  6.4 Predicting the volatility of GDP

Finally, inspired by Carvalho and Gabaix (2013), in this section we check
whether or not our uncertainty network connectedness is also able to predict
future U.S. GDP growth rate and volatility. We calculate the growth rate of
$GDP_{t}$, the U.S. GDP at time $t$ as
$g_{t}=log(GDP_{t+1}/GDP_{t})\quad$where $t$ is expressed in quarterly
frequency, end of the quarter. The volatility of GDP growth is measured as the
annualized GDP standard deviation over 4 quarters. We check whether or not the
aggregate network can predict U.S. GDP indicators in the next $h$ quarters
ahead, with $h\in(1,2,3,4)$ by running the predictive equation 9 at a
quarterly frequency.

We report the empirical results in Table H1 in the appendix. We observe that
the uncertainty network cannot predict the future GDP growth rate 2 and 3
quarters ahead. An intensification of connections leads to a decreasing GDP
growth rate in the following quarters. We then show how the information
embedded in our measures is also useful to predict future GDP volatility. By
repeating the same exercise, we also show that the network predicts future GDP
volatility in the next four quarters, the results being even stronger and
found to be significant up to one year. An increase in connectedness leads to
an increase in GDP volatility, thus confirming the counter-cyclicality of the
uncertainty network.

In Table H2 in the appendix, we show the predictive results of hubs and non-
hubs based uncertainty networks for the GDP growth rate and volatility. Also,
in this case, we find a stronger predictive ability for the hubs network
compared to non-hubs for both the GDP growth rate and GDP volatility. We
confirm the asymmetric predictive ability of the uncertainty network in favour
of the hubs network, while weak and almost absent predictive power for the
non-hubs network. We also confirm stronger results for the hubs network
compared to the aggregate network results of the previous subsection,
emphasizing once more how a network extracted solely from uncertainty hubs
might have even strong predictive power not only for the business cycle
indicators, but also for the volatility of GDP.

##  7 Conclusion

We studied the ex-ante uncertainty network of the U.S. industries constructed
from options-based investors future expectations about the future month
uncertainty. We relied on a novel data set of industry forward-looking
uncertainties and we adopted a time-varying parameter VAR (TVP-VAR) to model
the ex-ante uncertainty network of industries.

We were able to obtain a precise point in time estimation of the uncertainty
network to accurately characterize the specific industry role in shocks to
uncertainty, dynamically over the business cycle. We uncovered a main role for
booming industries such as communications and information technology and we
classified these as uncertainty hubs. Industries such as, financial (having an
important influence mainly limited to the global financial crisis), real
estate, materials and utilities showed a more neutral role and are classified
as uncertainty non-hubs.

We exploited the forward-looking industry connectedness networks
characteristics in predictability. We found the industry uncertainty network
to be a useful tool to predict future business cycles. We identified a greater
predictive ability for the network extracted from uncertainty hubs, being the
main (leading) indicator of business cycles. Such uncertainty networks can
serve as a new tool for regulators and policymakers to monitor the
relationship between industry networks, the business cycle and the real
economy in a precise, timely and forward-looking manner.

Fluctuations and shocks to uncertainty in uncertainty hubs should be more
carefully monitored due to their potential for shaping the industry networks
and impacting the real economy. Our findings suggest a possible direction for
policies and government interventions. Policy actions should target
uncertainty hubs since they are the stronger contributors to uncertainty
shocks and they show a tighter link with the real economy. Policy
interventions aimed at dampening their shocks to uncertainty can potentially
provide a direct channel for boosting business cycles.

## References

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## Appendix

##  Appendix A S&P 500 sectors breakdown

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 (CD): The CD 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, etc. 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 U.S.
came to $4.54 trillion, or about 10.11% of the market. Examples of consumer
discretionary stocks include Amazon and Starbucks.

  * •

Communication Services (CM): 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 U.S. came to $4.42 trillion, or 10.33% of
the market. The CM industry includes stocks such as AT&T and Verizon, but also
the giants Alphabet Inc A and Facebook from 2004 and 2012, respectively. In
fact, the new communication sector of the S&P 500 includes now big companies
such as Facebook and Alphabet Google since these were moved out from the
technology and consumer discretionary sectors, respectively, due to the
changes of the Global Industry Classification Standard (GICS).

  * •

Consumer Staples (CS): The CS 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 U.S.
came to $2.95 trillion, or about 7.18% of the market and includes companies
such as Procter & Gamble.

  * •

Energy (E): The E 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 U.S. came to $3.36 trillion, or about
5.51% of the market. Major energy stocks include Exxon Mobil and Chevron.

  * •

Financial (F): The F 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,
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 U.S. came to $6.89
trillion, or about 13.63% of the market. J.P. MorganChase, GoldmanSachs, and
Bank of America are examples of financial stocks.

  * •

Health Care (HC): The HC 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 HC 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 U.S. came to $5.25 trillion, or about
15.21% of the market. Examples of HC stocks include Johnson & Johnson, and
Pfizer.

  * •

Industrials (IN): The IN 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 IN stocks in the U.S. came to $3.80 trillion, or about
9.33% of the market.

  * •

Information Technology (IT): the 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 IT
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 (M): 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 U.S.
came to $1.77 trillion, or about 2.71% of the market. Major materials stocks
include Dupont.

  * •

Real Estate (RE): The RE 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 RE 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 U.S. came to $1.17 trillion, or 2.96%
of the market. The RE industry includes stocks such as Simon Property Group
and Prologis.

  * •

Utilities (U): The U 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 U.S. came to $1.27
trillion, or about 3.18% of the market. U stocks include many local
electricity and water companies including Dominion Resources.

##  Appendix B Model-free individual implied volatility

Formalizing the implied volatility computation for each stock, we follow
Bakshi et al. (2003) in adopting out-of-money (OTM) call and put option prices
to compute the individual stock $s$th implied variance as

$\sigma^{2}_{\text{VIX}^{(s)}}=\int_{P_{t}}^{\infty}\frac{2(1-\log(K/P_{t}))}{K^{2}}C(t,t+1,K)dK+\int_{0}^{P_{t}}\frac{2(1+\log(P_{t}/K))}{K^{2}}P(t,t+1,K)dK,$
(11)

where $C(.)$ and $P(.)$ denote the time $t$ prices of call and put contracts,
respectively, with time to maturity of one period and a strike price of $K$.

Intuitively, the implied variance measure can be computed in a model-free way
from a range of option prices upon a discretization of formula (11), adopting
call and put option prices with respect to the next 30 days, considering all
available strikes for each individual stock options. We compute the
$\text{VIX}^{(s)}$ for all the stocks in our sample belonging to the 11 U.S.
industries as follows:

$\sigma^{2}_{\text{VIX}^{(s)}}=\frac{2}{T}\sum_{i=1}^{n}\frac{\Delta
K_{i}}{K_{i}^{2}}e^{rT}Q(K_{i})-\frac{1}{T}\left[\frac{F}{K_{0}}-1\right]^{2},\\\
$ (12)

where $T$ is time to expiration, $F$ is the forward index level derived from
the put-call parity as $F=e^{rT}[C(K,T)-P(K,T)]+K$ with the risk-free rate $r$
, $K_{0}$ is the reference price, the first exercise price less or equal to
the forward level $F(K_{0}\leq F)$, and $K_{i}$ is the $i$th out-of-the-money
(OTM) strike price available on a specific date (call if $K_{i}>K_{0}$, put if
$K_{i}<K_{0}$, and both call and put if $K_{i}=K_{0}$). $Q(K_{i})$ is the
average bid-ask of OTM options with exercise price equal to $K_{i}$. If
$K_{i}=K_{0}$, it will be equal to the average between the at-the-money (ATM)
call and put price, relative to the strike price, and $\Delta(K_{i})$ is the
sum divided by two of the two nearest prices to the exercise price $K_{0}$,
namely, $\frac{(K_{i+1}-K_{i-1})}{2}$ for $2\leq i\leq n-1$. The annualized
square roots of the quantities computed for each of the $s$-th individual
firms are then labeled $\text{VIX}^{(s)}$ denoting individual, model-free
implied volatility measures of the expected price fluctuations in the $s$-th
underlying asset’s options over the next month.232323The standard CBOE
methodology considers an interpolation between the two closest to 30-days
expiration dates. We use a simplified formula taking into account only one
expiration date closest to 30-days due to options data availability with
respect to U.S. single stocks.

$\text{VIX}^{(s)}=\sqrt{\frac{365}{30}\sigma^{2}_{\text{VIX}^{(s)}}}$ (13)

Table B1: List of Selected Stocks by Industry

Ticker | Full name | Period |  | Ticker | Full name | Period
---|---|---|---|---|---|---
Consumer discretionary: |  |  |  | Industrial |  |
AMZN | Amazon.com Inc. | 1996-2020 |  | BA | Boeing Company | 1996-2020
HD | Home Depot Inc. | 1996-2020 |  | GE | General Electric Company | 1996-2020
MCD | McDonald’s Corporation | 1996-2020 |  | HON | Honeywell International Inc. | 1996-2020
NKE | NIKE Inc. Class B | 1996-2020 |  | UNP | Union Pacific Corporation | 1996-2020
SBUX | Starbucks Corporation | 1996-2020 |  | UTX | United Technologies Corporation | 1996-2018
|  |  |  | CAT | Caterpillar Inc. | 2019-2020
Communications: |  |  |  | Information Technology |  |
CMCSA | Comcast Corporation Class A | 1996-2004 |  | AAPL | Apple Inc. | 1996-2020
DIS | Walt Disney Company | 1996-2020 |  | ADBE | Adobe Inc. | 1996-2020
EA | Electronic Arts Inc. | 1996-2015 |  | CSCO | Cisco Systems Inc. | 1996-2015
FB | Facebook Inc. Class A | 2012-2020 |  | INTC | Intel Corporation | 1996-2020
GOOG | Alphabet Inc. Class C | 2004-2020 |  | MA | Mastercard Incorporated Class A | 2007-2018
T | AT&T Inc. | 1996-2020 |  | MSFT | Microsoft Corporation | 1996-2020
VZ | Verizon Communications Inc. | 1996-2020 |  | V | Visa Inc. Class A | 2008-2020
Consumer staples: |  |  |  | Materials |  |
COST | Costco Wholesale Corporation | 1996-2018 |  | APD | Air Products and Chemicals Inc | 1996-2018
KO | Coca-Cola Company | 1996-2020 |  | DD | DuPont de Nemours Inc. | 1996-2018
PEP | PepsiCo Inc. | 1996-2020 |  | ECL | Ecolab Inc. | 1996-2018
PG | Procter & Gamble Company | 1996-2020 |  | LIN | Linde plc | 1996-2007
WMT | Walmart Inc. | 1996-2020 |  | PPG | PPG Industries Inc. | 2007-2018
PM | Philip Morris Inc. | 2018-2020 |  | SHW | Sherwin-Williams Company | 1998-2020
|  |  |  | NEM | Newmont Corporation | 2019-2020
|  |  |  | FCX | Freeport-McMoRan Inc. | 2019-2020
|  |  |  | IP | International Paper | 2019-2020
Energy: |  |  |  | Real Estate |  |
COP | ConocoPhillips | 1996-2020 |  | AMT | American Tower Corporation | 1996-2020
CVX | Chevron Corporation | 1996-2020 |  | CCI | Crown Castle International Corp | 1996-2018
SLB | Schlumberger NV | 1996-2020 |  | EQIX | Equinix Inc. | 2006-2020
XOM | Exxon Mobil Corporation | 1996-2020 |  | EQR | Equity Residential | 1996-2020
VLO | Valero Energy Corp | 1996-2015 |  | PLD | Prologis Inc. | 1996-2018
PSX | Phillips 66 | 2012-2020 |  | SPG | Simon Property Group Inc. | 1996-2020
Financial |  |  |  | Utilities |  |
BAC | Bank of America Corp | 1996-2020 |  | AEP | American Electric Power Company Inc. | 1996-2020
BRK | Berkshire Hathaway Inc. Class B | 2010-2020 |  | D | Dominion Energy Inc | 1996-2020
C | Citigroup Inc. | 1996-2020 |  | DUK | Duke Energy Corporation | 2006-2020
GS | Goldman Sachs | 1999-2018 |  | NEE | NextEra Energy Inc. | 1996-2020
JPM | JPMorgan Chase & Co. | 1996-2020 |  | SO | Southern Company | 1996-2020
WFC | Wells Fargo & Company | 1996-2020 |  |  |  |
Health care: |  |  |  |  |  |
ABT | Abbott Laboratories | 1996-2019 |  |  |  |
JNJ | Johnson & Johnson | 1996-2020 |  |  |  |
MRK | Merck & Co. Inc. | 1996-2020 |  |  |  |
PFE | Pfizer Inc. | 1996-2020 |  |  |  |
UNH | UnitedHealth Group Incorporated | 1996-2020 |  |  |  |
ABBV | AbbVie Inc. | 2018-2020 |  |  |  |

  * •

Notes: This table summarizes all the U.S. stocks selected in the paper,
divided by industry. The stock tickers, full names and period of options data
availability are reported.

Figure B1: Individual Firm Uncertainty Notes: This figure illustrates the
individual uncertainty $\text{VIX}^{(i)}$ for Amazon (in black), CocaCola (in
grey) and Disney (in light grey) from 03-01-2000 to 29-05-2020, at a daily
frequency.

##  Appendix C Proofs

###### Proposition 12.

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

$\mathbf{IVIX}_{t,T}=\sum_{h=-\infty}^{\infty}\boldsymbol{\Psi}_{t,T,h}\boldsymbol{\epsilon}_{t-h}$
(14)

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

$\widetilde{\mathbf{IVIX}}_{t}(u)=\sum_{h=-\infty}^{\infty}\boldsymbol{\Psi}_{h}(u)\boldsymbol{\epsilon}_{t-h}$
(15)

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

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

$\boldsymbol{\Omega}^{H}(u)=\sum_{h=0}^{H}\boldsymbol{\Psi}_{h}(u)\boldsymbol{\Sigma}(u)\boldsymbol{\Psi}^{\top}_{h}(u).$
(16)

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,

$\boldsymbol{\xi}^{k,H}(u)=\sum_{h=0}^{H}\boldsymbol{\Psi}_{h}(u)\Big{[}\boldsymbol{\epsilon}_{t+H-h}-\mathbb{E}(\boldsymbol{\epsilon}_{t+H-h}|\boldsymbol{\epsilon}_{k,t+H-h})\Big{]},$
(17)

assuming normal distribution of $\boldsymbol{\epsilon}_{t}\sim
N(0,\boldsymbol{\Sigma})$, we obtain242424Note to notation:
$[\boldsymbol{A}]_{j,k}$ denotes the $j$th row and $k$th column of matrix
$\boldsymbol{A}$ denoted in bold. $[\boldsymbol{A}]_{j,\cdot}$ 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 $A$.

$\mathbb{E}(\boldsymbol{\epsilon}_{t+H-h}|\boldsymbol{\epsilon}_{k,t+H-h})=\sigma_{kk}^{-1}\Big{[}\boldsymbol{\Sigma}(u)\Big{]}_{\cdot
k}\boldsymbol{\epsilon}_{k,t+H-h}$ (18)

and substituting (18) to (17), we obtain

$\boldsymbol{\xi}^{k,H}(u)=\sum_{h=0}^{H}\boldsymbol{\Psi}_{h}(u)\Big{[}\boldsymbol{\epsilon}_{t+H-h}-\sigma_{kk}^{-1}\Big{[}\boldsymbol{\Sigma}(u)\Big{]}_{\cdot
k}\boldsymbol{\epsilon}_{k,t+H-h}\Big{]}.$ (19)

Finally, the local forecast error covariance matrix is

$\boldsymbol{\Omega}^{k,H}(u)=\sum_{h=0}^{H}\boldsymbol{\Psi}_{h}(u)\boldsymbol{\Sigma}(u)\boldsymbol{\Psi}^{\top}_{h}(u)-\sigma_{kk}^{-1}\sum_{h=0}^{H}\boldsymbol{\Psi}_{h}(u)\Big{[}\boldsymbol{\Sigma}(u)\Big{]}_{\cdot
k}\Big{[}\boldsymbol{\Sigma}(u)\Big{]}_{\cdot
k}^{\top}\boldsymbol{\Psi}^{\top}_{h}(u).$ (20)

Then

$\Big{[}\boldsymbol{\Delta}^{H}(u)\Big{]}_{(j)k}=\Big{[}\boldsymbol{\Omega}^{H}(u)-\boldsymbol{\Omega}^{k,H}(u)\Big{]}_{j,j}=\sigma_{kk}^{-1}\sum_{h=0}^{H}\Bigg{(}\Big{[}\boldsymbol{\Psi}_{h}(u)\boldsymbol{\Sigma}(u)\Big{]}_{j,k}\Bigg{)}^{2}$
(21)

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)

$\Big{[}\boldsymbol{\theta}^{H}(u)\Big{]}_{j,k}=\frac{\sigma_{kk}^{-1}\displaystyle\sum_{h=0}^{H}\Bigg{(}\Big{[}\boldsymbol{\Psi}_{h}(u)\boldsymbol{\Sigma}(u)\Big{]}_{j,k}\Bigg{)}^{2}}{\displaystyle\sum_{h=0}^{H}\Big{[}\boldsymbol{\Psi}_{h}(u)\boldsymbol{\Sigma}(u)\boldsymbol{\Psi}^{\top}_{h}(u)\Big{]}_{j,j}}$
(22)

This completes the proof. ∎

##  Appendix D Estimation of the time-varying parameter VAR model

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

$\mathbf{IVIX}_{t,T}=\boldsymbol{\Phi}_{1}(t/T)\mathbf{IVIX}_{t-1,T}+\ldots+\boldsymbol{\Phi}_{p}(t/T)\mathbf{IVIX}_{t-p,T}+\boldsymbol{\epsilon}_{t,T},$
(23)

where
$\boldsymbol{\epsilon}_{t,T}=\boldsymbol{\Sigma}^{-1/2}(t/T)\boldsymbol{\eta}_{t,T}$
with $\boldsymbol{\eta}_{t,T}\sim NID(0,\boldsymbol{I}_{M})$ and
$\boldsymbol{\Phi}(t/T)=(\boldsymbol{\Phi}_{1}(t/T),\ldots,\boldsymbol{\Phi}_{p}(t/T))^{\top}$
are the time-varying autoregressive coefficients. Note that all roots of the
polynomial,
$\chi(z)=\text{det}\left(\textbf{I}_{N}-\sum^{L}_{p=1}z^{p}\mathbf{B}_{p,t}\right)$,
lie outside the unit circle, and $\boldsymbol{\Sigma}^{-1}_{t}$ is a positive
definite time-varying covariance matrix. Stacking the time-varying intercepts
and autoregressive matrices in the vector $\phi_{t,T}$ with
$\overline{\mathbf{IVIX}}^{\top}_{t}=\left(\text{{I}}_{N}\otimes
x_{t}\right),\>x_{t}=\left(1,x^{\top}_{t-1},\dots,x^{\top}_{t-p}\right)$ and
$\otimes$ denotes the Kronecker product, the model can be written as:

$\displaystyle\mathbf{IVIX}_{t,T}=\overline{\mathbf{IVIX}}^{\top}_{t,T}\phi_{t,T}+\boldsymbol{\Sigma}^{-\frac{1}{2}}_{t/T}\boldsymbol{\eta}_{t,T}$
(24)

We obtain the time-varying parameters of the model by employing Quasi-Bayesian
Local Likelihood (QBLL) methods. Estimation of (23) 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 time period $k$ is given by:

$\displaystyle\begin{split}\mathbf{L}_{k}\left(\mathbf{IVIX}|\theta_{k},\boldsymbol{\Sigma}_{k},\overline{\mathbf{IVIX}}\right)&\propto&\\\
|\boldsymbol{\Sigma}_{k}|^{\text{trace}(\mathbf{D}_{k})/2}\exp\left\\{-\frac{1}{2}(\mathbf{IVIX}-\overline{\mathbf{IVIX}}^{\top}\phi_{k})^{\top}\left(\boldsymbol{\Sigma}_{k}\otimes\mathbf{D}_{k}\right)(\mathbf{IVIX}-\overline{\mathbf{IVIX}}^{\top}\phi_{k})\right\\}\end{split}$
(25)

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

$\displaystyle\mathbf{D}_{k}$ $\displaystyle=$
$\displaystyle\text{diag}(\varrho_{k1},\dots,\varrho_{kT})$ (26)
$\displaystyle\varrho_{kt}$ $\displaystyle=$
$\displaystyle\phi_{T,k}w_{kt}/\sum^{T}_{t=1}w_{kt}$ (27) $\displaystyle
w_{kt}$ $\displaystyle=$
$\displaystyle(1/\sqrt{2\pi})\exp((-1/2)((k-t)/H)^{2}),\quad\text{for}\>k,t\in\\{1,\dots,T\\}$
(28) $\displaystyle\zeta_{Tk}$ $\displaystyle=$
$\displaystyle\left(\left(\sum^{T}_{t=1}w_{kt}\right)^{2}\right)^{-1}$ (29)

where $\varrho_{kt}$ is a normalised kernel function. $w_{kt}$ uses a Normal
kernel weighting function. $\zeta_{Tk}$ gives the rate of convergence and
behaves like the bandwidth parameter $H$ in (28), and it is the kernel
function that provides greater weight to observations surrounding the
parameter estimates at time $k$ relative to more distant observations.

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

$\displaystyle\phi_{k}|\boldsymbol{\Sigma}_{k}\backsim\mathcal{N}\left(\phi_{0k},(\boldsymbol{\Sigma}_{k}\otimes\mathbf{\Xi}_{0k})^{-1}\right)$
(30)
$\displaystyle\boldsymbol{\Sigma}_{k}\backsim\mathcal{W}\left(\alpha_{0k},\mathbf{\Gamma}_{0k}\right)$
(31)

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

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

$\displaystyle\phi_{k}|\boldsymbol{\Sigma}_{k},\mathbf{A},\mathbf{Y}$
$\displaystyle\backsim$
$\displaystyle\mathcal{N}\left(\tilde{\theta}_{k},\left(\boldsymbol{\Sigma}_{k}\otimes\mathbf{\tilde{\Xi}}_{k}\right)^{-1}\right)$
(32) $\displaystyle\boldsymbol{\Sigma}_{k}$ $\displaystyle\backsim$
$\displaystyle\mathcal{W}\left(\tilde{\alpha}_{k},\mathbf{\tilde{\Gamma}}^{-1}_{k}\right)$
(33)

with quasi posterior parameters

$\displaystyle\tilde{\phi}_{k}$ $\displaystyle=$
$\displaystyle\left(\mathbf{I}_{N}\otimes\mathbf{\tilde{\Xi}}^{-1}_{k}\right)\left[\left(\mathbf{I}_{N}\otimes\mathbf{A}^{\prime}\mathbf{D}_{k}\mathbf{A}\right)\hat{\phi}_{k}+\left(\mathbf{I}_{N}\otimes\mathbf{\Xi}_{0k}\right)\phi_{0k}\right]$
(34) $\displaystyle\mathbf{\tilde{\Xi}}_{k}$ $\displaystyle=$
$\displaystyle\mathbf{\tilde{\Xi}}_{0k}+\mathbf{A}^{\prime}\mathbf{D}_{k}\mathbf{A}$
(35) $\displaystyle\tilde{\alpha}_{k}$ $\displaystyle=$
$\displaystyle\alpha_{0k}+\sum^{T}_{t=1}\varrho_{kt}$ (36)
$\displaystyle\mathbf{\tilde{\Gamma}}_{k}$ $\displaystyle=$
$\displaystyle\mathbf{\Gamma}_{0k}+\mathbf{Y}^{\prime}\mathbf{D}_{k}\mathbf{Y}+\mathbf{\Phi}_{0k}\mathbf{\Gamma}_{0k}\mathbf{\Phi}^{\prime}_{0k}-\mathbf{\tilde{\Phi}}_{k}\mathbf{\tilde{\Gamma}}_{k}\mathbf{\tilde{\Phi}}^{\top}_{k}$
(37)

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