key: cord-0458465-m5zsmzxl
authors: Barunik, Jozef; Bevilacqua, Mattia; Faff, Robert
title: Dynamic industry uncertainty networks and the business cycle
date: 2021-01-18
journal: nan
DOI: nan
sha: acc58bb231b9a247433c4c8a4135b05181e17c61
doc_id: 458465
cord_uid: m5zsmzxl

This paper introduces new forward-looking uncertainty network measures built from the main US industries. We argue that this network structure extracted from options investors' expectations is meaningfully dynamic and contains valuable information relevant for business cycles. Classifying 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 propagate strongly across the network. We find that a dynamic ex-ante network of uncertainty is a useful predictor of business cycles especially when it is based on uncertainty hubs. The uncertainty network is found to behave counter-cyclically since a tighter network of industry uncertainty tends to associate with future business cycle contractions.

Throughout history, industrial structure in developed economies has witnessed prominent influential economic and financial cycles in which different sectors seem to take on a leading role. 1 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. Given this backdrop, we develop a forward-looking and dynamic measure of industry uncertainty network connectedness, we link this connectedness measure to the US economy and its importance in better understanding how aggregate economic activity is driven by the nuanced interrelations between economic sectors.

Understanding the differential actions and performance of the largest firms within an economy's various industrial sectors offers key insight into the aggregate economy. 2 Many economic fluctuations are attributable to the incompressible "grains" of economic activity, stemming from individual companies (see Gabaix, 2011) . Moreover, Acemoglu et al. (2012) documented that significant aggregate fluctuations can plausibly 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. Further, 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. Notably, Atalay (2017) concludes that 83% of the variation in aggregate output growth is attributable to idiosyncratic industrylevel shocks, while Gabaix (2011) contends that a salient feature of business cycles is that firms and sectors comove.

Industrial network connectedness has been evolving, sometimes strengthening and other 1 The 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.

2 For 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). times weakening, across different time periods and at various points throughout economic cycles. We study the dynamic network concept from a fresh perspective, namely, through the lens of how industry-specific shocks to option buyers' expectations can propagate ex ante uncertainty. More specifically, we ask how does this form of network uncertainty develop over time and what is its relationship with the evolution of business activity, manifesting through business cycles?

To this end, we devise and construct a measure of ex-ante industry uncertainty network connectedness based on information extracted from options market data -data that reflects investor expectations of future uncertainty relevant to each industry. We study the dynamic relations of these uncertainty measures with the business cycle, characterizing each industry based on their expected contribution to shocks to uncertainty to the system across phases of the business cycle. Our analysis then quite naturally transitions to the question of how useful is this measure of network uncertainty in predicting business cycle phases.

Previous studies highlight the importance of network measures to capture the propagation of volatility mechanisms (e.g. Acemoglu et al., 2012; Carvalho and Gabaix, 2013; Gabaix, 2016 ; Barrot and Sauvagnat, 2016; Acemoglu et al., 2017; Baqaee and Farhi, 2019; Herskovic et al., 2020) . For example, measuring network effects is crucial to explain the joint evolution of firm volatility distributions (see Herskovic et al., 2020) . Notably, the survey in Carvalho and Tahbaz-Salehi (2019) entreats researchers to develop models that take such firm-level forces seriously -in the belief that such efforts hold the key to capturing valuable theoretical and empirical richness that is currently missing from the literature. Put simply, our paper is answering this call to action at the industry level.

Fluctuations in risk are the most important shock driving the business cycle (see Christiano et al., 2014) . According to Bloom et al. (2018) , uncertainty is an important factor in business cycles, it is strongly countercyclical, this being true both at the aggregate and the industry level. We augment this existing literature by assessing the propagation of forward-looking industry-specific shocks in uncertainty and the way in which the aggregate network extracted from the idiosyncratic shocks propagate dynamically; and to what extent significant aggregate fluctuations can originate from such shocks.

Previous literature on uncertainty and the related network measures relies on historical or ex post analysis. We contend that such backward-looking approaches are inferior to a forward-looking, ex ante, approach that employs measures extracted from option prices of individual firms. Accordingly, we construct a novel option-based uncertainty measure for each major company, spread across 11 US industries. We create aggregate uncertainty measures for each industry in our sample, covering 20 years at the daily frequency, with three recession periods, including the most recent Covid-19 crisis.

We characterize shocks at the industry uncertainty level and study their propagation mechanism across time. To do this, we adopt the dynamic network measures based on timevarying parameter VAR (TVP VAR) models introduced by Barunik and Ellington (2020) .

This technique estimates the adjacency matrix characterizing a network at each point in time using the variance decomposition matrix. They have a direct causal interpretation that permits the understanding of how shocks to uncertainty create dynamic networks among industry uncertainties, and are thereby useful for various macroeconomic applications e.g. characterizing the propagation of such networks over phases of the business cycle. 3

The essential contribution of our work lies at the intersection of two relatively recent strands of literature. The first strand is on uncertainty measures (e.g. Bloom, 2009; Jurado et al., 2015; Ludvigson et al., 2020) to cite but a few; see also Bloom (2014) for a survey, and especially on the relationship between uncertainty and business cycle fluctuations that began with the work of Bloom (2009) . 4 Some of these studies assess how exogenous changes in volatility are key to generating business cycles and also analyze the possibility of reverse causation between measured uncertainty and business cycles.

The second strand includes studies on the role of sector-level or firm-to-firm linkages in microeconomic shocks and their relationship with the aggregate economy, future economic downturns and changes in business conditions (see, e.g. Acemoglu et al., 2012 Acemoglu et al., , 2017 Baqaee and Farhi, 2019) or the survey in Carvalho and Tahbaz-Salehi (2019) . Connected to this 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 (e.g. Di Giovanni et al., 2014; Ozdagli and Weber, 2017; Carvalho and Tahbaz-Salehi, 2019; Auer et al., 2019; Lehn and Winberry, 2020) . In contrast to these latter works, we adopt pure financial market based networks as a mechanism to study the propagation of shocks to uncertainty from industries to the real economy.

To the best of our knowledge, we are the first to propose an ex ante industry-based uncertainty network measure containing market participant expectations about forwardlooking (next month) industry uncertainty and relate it to business cycles. Being able to temporally and precisely characterize these industry-based network dynamics is important given that industries can swiftly change their characteristics and macro-economic roles. 5

The technological and housing market bubbles, the commodity crash, and the Covid pandemic are a few major examples that show how a dramatic increase in uncertainty and different investor expectations can rise sharply in many alternative industries. Through a TVP-VAR model, industries are more precisely assessed as to their contribution to shocks in uncertainty to the whole system across different phases of the business cycle, shedding crucial light on their mutating interactions and roles.

We identify industries showing a stronger (versus weaker) contribution of shocks to uncertainty, thus playing an essential role within the aggregate industry uncertainty network. Lehn and Winberry (2020) show that the empirical network is dominated by a few "investment hubs" that produce the majority of investment goods, are highly volatile, and are strongly correlated with the cycle. Similarly, and augmenting the definition of "hubs" 5 Pástor and Veronesi (2009) show that stock prices rise with higher uncertainty during times of technological revolutions. The idea that have seen the financial sector always at the top of this pyramid is slightly mutating. On the importance of industry factors and industry risk see especially Griffin and Karolyi (1998) , Griffin and Stulz (2001) , Griffin et al. (2003) and Carrieri et al. (2004). in 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.

When performance varies across industries, especially when the currently most influential ones are affected, this can trigger major consequences for the other industries, tightening or weakening the uncertainty network and being ultimately reflected in the real economy. Therefore, we also investigate whether the ex-ante industry-based uncertainty network might translate uncertainty shocks at the industry-based microeconomic level into fluctuations in macroeconomic aggregates (e.g. Gabaix, 2011; Acemoglu et al., 2012; Carvalho and Gabaix, 2013; Barrot and Sauvagnat, 2016; Atalay, 2017) . To this end, we empirically test whether the aggregate ex ante uncertainty industry network is able to predict business cycles, hypothesizing enhanced predictive ability given that it is built on forward looking option information and through precise time-varying parametrization. We further hypothesize that the industry network constructed from uncertainty hubs achieve greater predictability compared to uncertainty in the non-hubs network.

The main findings of our paper are as follows. We find that the ex ante industry network is countercyclical and rises sharply during the dot com bubble, the global financial crisis (GFC), increasing steadily afterwards. The communications and information technology uncertainty networks play a key role, being classified as the main uncertainty hubs. In contrast, materials, utility and real estate are classified as non-hubs. Notably, the financial industry reveals a key role mainly during the GFC, while other industries are found to be time-varying uncertainty hubs according to specific business cycle phases (e.g. Covid-19 recession). Our empirical exercise shows the usefulness of the ex ante uncertainty network in predicting business cycles up to one year horizons. The results are robust with regard to several checks and the inclusion of additional control variables. Finally, we find that the uncertainty hubs network show greater predictive power with respect to business cycles, compared to their non-hub counterparts.

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 with respect to 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.

To study the dynamic uncertainty network, we develop a forward-looking measure reflecting investor beliefs derived from option price data. To track investor's beliefs and to allow trading on forward-looking volatility, the Chicago Board Options Exchange (CBOE) introduced a volatility index -VIX -extracting expectations from options prices in a model-free manner.

The concept was later formalized by Bakshi and Madan (2000) ; Bakshi et al. (2003) and has quickly gained popularity in the literature as well as among practitioners and policymakers.

To capture industry uncertainty, we use forward-looking uncertainty measures that are intimately related to the VIX methodology. However, rather than looking at the whole U.S. stock market, we measure uncertainty at the industry level, focusing on the main U.S.

industries. Options-based measures of risk are superior to historical volatility measures with respect to both predictive power and set of information they encompass (e.g. Christensen and Prabhala, 1998; Santa-Clara and Yan, 2010; Baruník et al., 2020) .

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

More formally, the ex ante industry uncertainty measure IVIX (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:

where Ind ∈ {1, . . . , 11} represents the industry we consider, s is an index for one of the N (Ind) companies included in the given industry at time t, VIX (s) t is an implied volatility for an individual stock s, and W (s) t 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.

We use daily data encompassing the sample period January 2000 to May 2020 6 in each of the following 11 US industries: consumer discretionary (CD), communications (CM), consumer staples (CS), energy (E), financial (F), health care (HC), industrial (IN), information technology (IT), materials (M), real estate (RE) and utilities (U). 7 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 VIX time series from January 2019 until May 2020 aided by the IHS Markit's Totem Vanilla Volatility Swap data 6 Prior to 2000, there are insufficient available data to compute the individual stock VIX. 7 The new communication service 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). The telecom sector changed its weight from about 2% of the entire S&P 500 to about 11%. The IT sector changes from a roughly 26% to about 20%; the consumer discretionary dropped from 13% to about 11%. The communication sector, still includes existing telecom companies such as Verizon Communications Inc, AT&T Inc. Apple Inc remained in the IT sector, while Amazon remained in the CD sector. set. 8 From the latter, we collect broker-dealers consensus prices with respect to the volatility strike of the swaps. This aids our data collection since they are exactly the measures of individual company uncertainty, namely the individual stock VIXs, we are interested in, therefore allowing us to expand the data frame of this study. 9 Overall, our data set includes options prices with respect to 69 US firms. We select the largest stocks in each US industry based on market capitalization and included in the S&P 500 index. 10 The selected stocks account for more than 58% of the U.S. S&P 500 market capitalization, thus being a valid proxy for the 11 US industries, and being representative of a not trivial fraction of the US GDP (e.g. Gabaix, 2011) . A large representation of the US stock market and its industries is what matters when studying these as economic and business cycle drivers. 11 Table A1 in the appendix shows the included stocks within each industry and their available time period.

The other financial information such as market capitalization and trading volume regarding the selected stocks are collected from Bloomberg.

We roll the constituent stocks of every industry at every point in time according to time-varying market capitalizations, new IPOs, exclusions of the stocks from the S&P 500, or missing data. Figure A1 in the appendix depicts an example of individual company uncertainty indexes. 12 In cases where options on a specific company have been only issued 8 The 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 after having checked for outliers and errors. 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 have not served our purpose.

9 This relationship has also been widely discussed in the literature, see for example Carr and Wu (2006) who provide an excellent history of the VIX index and Carr and Lee (2009) document a history of the development of the variance swap market. Moreover, Filipović et al. (2016) affirm that absent index jumps, the CBOE VIX index is the 30-day variance swap rate on the S&P 500 quoted in volatility units. Cheng (2019) state that the squared VIX at the 30-day horizon, under the assumption of no jumps, equals he 30-day realized variance in the S&P 500 starting from date t. Thus, the squared VIX equals the fair strike on a variance swap. Consequently, the VIX equals the strike of a volatility swap.

10 For example, these do not include Tesla, Royal Dutch Shell or Unilever. 11 Gabaix (2011) states that macroeconomic questions can be clarified by looking at the behavior of large firms. He adopts a sample including annual US Compustat data for the largest 100 firms as of 2007.

12 The CBOE has introduced stock market VIX series for a few stocks in the US. 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 in recent times, we include the next ranked company as a substitute, so as to always ensure at least five stocks for every sector across our time period. 13 in that industry with available data. 14 We report the descriptive statistics for the industry uncertainty indexes in Table 1 . 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 information technology 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 financial and real estate leading with the highest values.

As an example, we plot uncertainty measures for information technology, consumer staples between the consensus volatility and CBOE VIX series is, on average, above 97%, this again due to the interpolation used in the CBOE methodology. 13 The 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. We replace IVIX (RE) t directly with the real estate sector volatility measure submitted to IHS Markit/Totem.

14 With respect to some industries such as industrial, the same five stocks have been adopted throughout the sample period. In more dynamic sectors such as information technology, we observe several changes in the stocks ranking within our sample period ending up with GOOG, MSCF, INTEL, AAPL in the last decade. and financial sector in Figure 1 . 3 Measurement of dynamic industry-level networks of uncertainty Industries are connected directly through counterparty risk, contractual obligations or other general business conditions of the companies. 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 company j transmits to future expectations about the uncertainty of a company 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.

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.

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 network connectedness measure we propose is also directly connected to the vast economic network literature in relation to production networks and granular shocks (see Acemoglu et al., 2012; Carvalho and Gabaix, 2013; Gabaix, 2016; Barrot and Sauvagnat, 2016; Acemoglu et al., 2017; Baqaee and Farhi, 2019; Acemoglu and Azar, 2020) . The network analysis has developed a conceptual framework and an extensive set of tools to effectively measure interconnections among the units of analysis comprising a network, see for instance the survey paper 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. 15

Finally, we note that our measures can have a direct causal interpretation. Rambachan 15 Baruník 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 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. and Shephard (2019) provide an important discussion about 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.

To formalize the discussion, we construct a dynamic uncertainty network of industries from the industry implied volatilities computed for the main US 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

where IVIX t,T = IVIX

are the time varying autoregressive coefficients. Note that t refers to a discrete time index 1 ≤ t ≤ 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 ≈ t/T is a local approximation of the weakly stationary time-series (Dahlhaus, 1996) , we approximate the IVIX t,T in a neighborhood of a fixed time point

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

where parameter vector Ψ t,T,h ≈ Ψ h (t/T ) is a time varying impulse response function characterized by a bounded stochastic process. 16 The connectedness measures rely on variance decompositions, which are transformations of the information in Ψ 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). 17 Suppose IVIX t,T is a locally stationary process, then the time-varying generalized variance decomposition of the jth variable at a rescaled time u = t 0 /T due to shocks in the kth variable forming a dynamic adjacency 16 Since Ψ t,T,h contains an infinite number of lags, we approximate the moving average coefficients at h = 1, . . . , H horizons.

17 Note to notation: [A] j,k denotes the jth row and kth column of matrix A denoted in bold.

[A] j,· denotes the full jth row; this is similar for the columns. A A, where A is a matrix that denotes the sum of all elements of the matrix A.

It is important to note that proposition 1 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

where θ H (u) is a normalized θ 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 in 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 = k in the economy is given by

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

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 C H j,net (u) = C H j→• (u)−C H j←• (u) and the agg connectedness

Finally, to obtain the time-varying coefficient estimates, and the time-varying covariance matrices at a fixed time point

, 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.

Working with a dynamic network estimates, we are able to characterize and time the events leading to more or less connected industry uncertainty providing new insights about the propagation of the ex-ante uncertainty shocks and identifying periods in which the US industries' uncertainty was tightly connected.

In tighter connectedness periods, a specific shock to an uncertainty with respect to any industry may generate an aggregate impact on the whole network of industries as well as on the real economy. We present the dynamic aggregate network connectedness 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. 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 (see NBER website). Hence, we are unable to establish an inversion period with respect to the Covid-19 crisis that we signal only as recession from February 2020. US industries appear to be more connected after the GFC and even more with the most recent Covid-19 crisis. We still lack understanding of which industries are driving such network increase or decrease with respect 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.

In addition to the aggregate network characteristics, we study how specific industries contribute to the ex-ante uncertainty of the system across the US business cycles and we classify them in the network. Specifically, we are interested to identify transmitters, receivers as well as industries being hubs of the uncertainty. We classify industries according to the C H net and C H agg characteristics of dynamic uncertainty network.

In the case of positive or negative net measure 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 labeled as a transmitter (receiver ).

is playing an active role in transmission of uncertainty shocks and is denoted as "uncertainty hub", being 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 changing roles in terms of contributors to shocks to uncertainty in relation to the specific economic cycle. We then average the network characteristics across each of the three business cycle phases (inversions, recessions and expansions) as described in the previous section, as well as over the total period. the important role such industries have played in the last two decades in the system. In addition, CD and E can also be classified as uncertainty hubs due to their high agg statistics.

Conversely, F, M, RE and U industries are uncertainty non-hubs within our sample. This finding shows how financial uncertainty has been transmitting differently across various times, mainly during inversions and recessions, but it is overall a non-hub. After having classified the industries based on their contribution to uncertainty across business cycles, we compute separate networks of uncertainty hubs and non-hubs industries.

Figure 3 plots 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. This is especially evident after the GFC and in more recent years when uncertainty hubs such as CM, IN and IT have indeed played a key role contributing to shock in uncertainty within the system. 

We here briefly show a more granular classification of industries with respect to every business cycle across the whole time period investigating how industry uncertainty contributed to shocks in each of the periods, since they share a very different nature. We report the C net and C agg statistics in Table F1 in 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 leading monitoring tool of business cycle itself. This would provide researchers, policy-makers and the public with a even more timely indicator than the ones already available.

Berge and Jordà (2011) and Scotti (ADS) index of business conditions. We adopt these as coincident indicators

of business cycle and we study whether the uncertainty network is able to forecast them.

Specifically, we adopt the 3-month moving average of the Chicago FED National Activity Index (CFNAI-MA3) as well used proxy of business cycle. 21 The Aruoba-Diebold-Scotti (ADS) Business Condition Index tracks real business conditions at a high frequency and it is based on economic indicators. 22 We aggregate the ADS indicator at monthly frequency.

In addition, we disentangle our business cycles indicators into proxies for expansions and recessions following a more refined 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 above -0.72, whereas periods of economic contraction with values of the CFNAI-MA3 below -0.72.

The same classification applies for ADS with respect to a -0.80 threshold. Aiming to study whether the predictive ability of the uncertainty network connectedness varies according to different states of the business cycle, we also investigate its predictability power with respect to either proxies of expansions or recessions as our dependent variables.

We aggregate network connectedness at a monthly frequency as to match the frequency of the business cycle indicators and we run the following predictive regression:

where Y ( ) t+h is one of the business cycle indicators we select (or their components) with the predictive horizon h ∈ 1, 3, 6, 9, 12 months. The C t is the industry uncertainty network connectedness 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 21 The 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: 1) production and income (23 series), 2) employment, unemployment, and hours (24 series), 3) personal consumption and housing (15 series), and 4) sales, orders, and inventories (23 series). All of the data are 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. For more information see https://www.chicagofed.org/publications/ cfnai/index. 22 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 worsethan-average conditions. It is collected from: https://www.philadelphiafed.org/research-and-data/ real-time-center.

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 potential leading indicators extracted from the financial markets, namely the changes in the CBOE VIX index being a common proxy for uncertainty in the US (VIX), changes in the S&P 500 price index (SPX), the Bloomberg Commodity price index (COMM) and the S&P Case-Shiller Home price (CSHP). 23 Therefore, X t,i is indexed for i up to N = 7, the number of control we select, with i ∈ (OIL, TS, UR, VIX, SPX, COMM, CSHP). Table 3 reports the results. With respect to the aggregate CFNAI-MA3 indicator of business cycle, we observe that uncertainty connectedness is a strong predictor 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.

When we look at the disentangled components of the business cycle indicator, we observe that uncertainty network is able to 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. Now we notice a greater adjusted-R 2 performance of the model with respect to 6-to 12-month horizons.

Regarding US recession periods, we observe a weaker predictability of network across shorter horizons. However, uncertainty network is still found to predict well future recessions from 3-month up to 12-month horizon, being the higher adjusted-R 2 placed again on the longer 23 Oil 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&P 500 price index and the Bloomberg Commodity price index are collected from Bloomberg. 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 ∈ (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 monthly frequency between 01-2000 and 05-2020.

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, thus entering the regression with a negative sign). 24

We validate the predictive ability of uncertainty network connectedness by showing how it can also similarly predict a different coincident indicator of business cycle. We show the results with respect to the ADS Index in Table F1 in appendix. We observe that the aggregate uncertainty network measure shows predictability power with respect to 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, we find weaker predictive power at the short horizons, whereas strong predictive ability is still placed at longer horizons, especially with respect to the expansion indicator.

Another possible coincident indicator of business cycle can be the industrial production growth rate. As a robustness check, we repeat the same exercise with respect to the annualized growth rate of industrial production, still confirming the predictive ability of uncertainty network. We also adopt another business cycle coincident indicator collected from the Economic Cycle Research Institute (ECRI), the US coincident indicator (USCI). 25

We take the growth rate of the indicator and show that this leads to similar results, relegated to 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 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 business cycle or the real economy. 26

Due to its forward-looking nature, we argue that the uncertainty network connectedness may potentially also serve as a good predictor of leading indicators of business cycle, and be considered a leading indicator itself. We here check the predictive ability of this index with respect to two business cycle leading indicators: the US composite leading indicator (CLI) by the OECD. 27 and the US leading indicator (USLI) computed by the Economic Cycle

Research Institute (ECRI). 28 USLI is available at weekly frequency and aggregated here at 25 For more information and data see https://www.businesscycle.com/ecri-reports-indexes/all-indexes. 26 The all set of results is available from the authors upon request. 27 The 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 fluctuation of the economic activity around its long term potential level.

28 For more information and data see https://www.businesscycle.com/ecri-reports-indexes/all-indexes.

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 with respect to leading indicators of business cycle, spanning from 3-month up to one year and from 1-month up to 9-month horizons, with respect to CLI and USLI, respectively. The coefficients are still found to be negative confirming our previous findings. 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 USLI, 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 ∈ (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 monthly frequency between 01-2000 and 05-2020.

Overall, it appears that uncertainty network can anticipate what is commonly viewed as a leading indicator of business cycle. This opens up to some 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 leading indicator of business cycle, as proposed by several papers and institutions, should ideally anticipate and predict coincident indicators. We showed in the previous section that the uncertainty network shares such properties. In this subsection, we show how our network measure is also a good predictor of leading indicators, such a finding emphasizing even further the usefulness of its forward-looking information content.

In order to validate this point we test whether the existing leading indicators of business cycle may contain a different set of information, mainly at shorter horizons, compared to our uncertainty network. We test whether our measure can predict coincident indicators even after controlling for a leading indicator (CLI). The results are reported in Table 5 . 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 ∈ (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 monthly frequency between 01-2000 and 05-2020.

We show how 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 evidence of a complementary and rather superior business cycles leading indicator spanning predictive power from 1-to 12-month horizon in advance, even after controlling for CLI. For the ADS, as a proxy for a coincident business cycle 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.

Finally, we repeat the same exercise of Table 5 by adopting the USLI 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.

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 US GDP growth rate and volatility. We calculate the growth rate of GDP t , the US GDP at time t as g t = log(GDP t+1 /GDP t ) where t is expressed in quarterly frequency. 29 The volatility of GDP growth is measured as the annualized GDP standard deviation over 4 quarters. We check whether or not the aggregate network is able to predict US GDP indicators in the next h quarters ahead, with h ∈ (1, 2, 3, 4) by running the predictive equation 9 at quarterly frequency.

We report the empirical results in Table F1 in appendix. We observe that the uncertainty network is able to predict future GDP growth rate at 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 enclosed in our measures is also useful to predict future GDP volatility. By repeating the same exercise, we also show that network is able to predict 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 uncertainty network.

In this subsection we check whether the predictability power of uncertainty hubs-based networks may differ from the one of uncertainty non-hubs. We repeat the empirical analysis of the previous section, now considering only hubs and non-hubs based networks, C hub t and C non-hub t , respectively. We hypothesize that the former leads to a greater predictability since reflecting information from the industries detected to be the main uncertainty contributors 

where we add the independent variables that characterize uncertainty hubs and non-hubs based on network connectedness taken jointly and aggregated at monthly frequency in order 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 with respect to the aggregate CFNAI-MA3 and recessions especially for longer horizons, while with respect to expansion 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. 31 This shows how the aggregated network connectedness results might be actually driven by few industry uncertainty hubs, these achieving predictability with respect to business cycles which is at times even stronger than the entire aggregated industry network.

30 See also Table 2 and Figure 3 . 31 In Table F1 in the appendix we show the results of the C (H) predictor alone with controlling for X which leads to a similar conclusion and findings similar to the aggregate network C as well. C (H) is found to be able to predict well the business cycle up to one year in advance. The more focused number of uncertainty hubs network C (H) confirms the strong predictive ability also with respect to expansion and recession periods. 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 ∈ (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 monthly frequency between 01-2000 and 05-2020.

We then check the relationship between the hubs and non-hubs networks with respect to leading indicators of the business cycle. The predictive results with respect to CLI are reported in Table 7 . We find that hubs network connectedness is able to strongly predict the leading indicator of business cycle up to one year, whereas the predictive power of non-hubs network is overall absent. Thus, uncertainty hubs show strong predictive power with respect to coincident measures of the business cycle, however this being even stronger with respect to leading indicators of business cycle, confirming a clear superior ability in predicting these indicators compared to non-hubs.

As in the previous section, now we validate the predictive ability of the hubs based network by including the leading indicator CLI to the multivariate regression when predicting CFNAI-MA3 as a control variable. The hubs network shows strong predictive ability from 3-month up to one year being the most timely indicator of the business cycle since it is able to predict coincident indicators one-year in advance. The non-hub network shows good predictive power in the short horizon. On the other hand, the hubs network is found to complement the shorter horizon predictive ability of CLI, expanding it to longer horizons.

Therefore while the predictive ability of the hubs network connectedness spans over longer horizons, up to one year, the predictive ability of non-hubs is found to be limited to shorter horizons up to 6 months, therefore it 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, useful feature for any leading indicators of business cycle.

This further echoes the results obtained with the aggregate uncertainty network, therefore we can conclude that the hubs-based network may be considered as the main driver of the aggregate network, achieving even stronger predictive power on its own.

As a 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 F2 in appendix with respect to all the coincident indicator CFNAI-MA3, expansions and recessions, and also with respect to the leading indicator CLI. We corroborates 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. The new non-hubs however shows better predictability with respect to the aggregate CFNAI-MA3 and recessions.

Finally, we also study the predictive ability of hubs and non-hubs based uncertainty networks with respect to GDP growth rate and volatility as show the results in Table F3 in appendix. Also, in this case, we find a stronger predictive ability for the hubs network compared to non-hubs with respect to both GDP growth rate and GDP volatility. We confirm the asymmetric predictive ability of the uncertainty network in favour of the hubs 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 ∈ (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 monthly frequency between 01-2000 and 05-2020.

network, while weak and almost absent predictive power for 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 with respect to the indicators of the business cycle, but also with respect to the volatility of GDP.

We studied the ex ante uncertainty network of the US industries constructed from optionsbased investors future expectations about one month ahead 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 (important role mainly limited to the global financial crisis), real estate, materials and utilities showed a more neutral role and are classified as uncertainty non-hubs.

Our industry uncertainty network is also forward-looking since constructed from options.

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 also showed that networks extracted from uncertainty hubs are the main drivers of the predictive power of the aggregate network.

We provided new insights about time-varying interactions between the newly constructed ex ante industry uncertainty network for the US and the business cycles. The uncertainty network may serve as new tool for regulators and policy makers in order to monitor the relationship between industry networks, the business cycle, e.g. recessions and expansions, and the real economy, in a precise, timely and forward-looking manner. More focus should be placed on uncertainty hubs since they are the stronger contributors to uncertainty shocks and the main predictors of the real economy. This implies that fluctuations in uncertainty with respect to uncertainty hubs should be more carefully monitored due to its potential for shaping the US industry networks and impacting the real economy.

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.

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

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

• Energy: The energy sector consists of businesses that source, drill, extract, and refine the raw commodities we need to keep the country going, such as oil and gas. At present, the energy sector contains two industries: Energy • Utilities: The utilities sector of the economy is home to the firms that make our lights work when we flip the switch, let our stoves erupt in flame when we want to cook food, make water come out of the tap when we are thirsty, and more. At present, the utilities sector is made up of five industries: Electric Utilities Industry, Gas Utilities Industry, Independent Power and Renewable Electricity Producers Industry, Multi-Utilities Industry, Water Utilities Industry. The total value of all utilities stocks in the United States came to $1.27 trillion, or about 3.18% of the market. Utilities stocks include many local electricity and water companies including Dominion Resources.

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 sth implied variance as

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 VIX (s) for all the stocks in our sample belonging to the 11 US industries as follows:

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 ≤ F ), and K i is the ith out-of-themoney (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 ∆(K i ) is the sum divided by two of the two nearest prices to the exercise price K 0 , namely, (K i+1 −K i−1 ) 2 for 2 ≤ i ≤ n − 1. The annualized square roots of the quantities computed for each of the s-th individual companies are then labeled 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. 32

32 The 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 US single stocks. 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,

assuming normal distribution of t ∼ N (0, Σ), we obtain 33

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

Finally, the local forecast error covariance matrix is

Then

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 jth variable yields the desired time varying generalized forecast error variance decompositions (TVP GFEVD)

This completes the proof.

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

where t,T = Σ −1/2 (t/T )η t,T with η t,T ∼ N ID(0, I M ) and Φ(t/T ) = (Φ 1 (t/T ), . . . , Φ p (t/T )) are the time varying autoregressive coefficients. Note that all roots of the polynomial, χ(z) = det I N − L p=1 z p B p,t , lie outside the unit circle, and Σ −1 t is a positive definite time-varying covariance matrix. Stacking the time-varying intercepts and autoregressive matrices in the vector φ t,T with IVIX t = (I N ⊗ x t ) , x t = 1, x t−1 , . . . , x t−p and ⊗ denotes the Kronecker product, the model can be written as:

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:

The D k is a diagonal matrix whose elements hold the weights:

where kt is a normalised kernel function. w kt uses a Normal kernel weighting function. ζ T k 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 φ k | Σ k for k ∈ {1, . . . , T }:

where φ 0k is a vector of prior means, Ξ 0k is a positive definite matrix, α 0k is a scale parameter of the Wishart distribution (W), and Γ 0k is a positive definite matrix. The prior and weighted likelihood function implies a Normal-Wishart quasi posterior distribution for φ k | Σ k for k = {1, . . . , T }. Formally let A = (x 1 , . . . , x T ) and Y = (x 1 , . . . , x T ) then:

with quasi posterior parameters

whereφ k = (I N ⊗ A D k A) −1 (I N ⊗ A D k ) y is the local likelihood estimator for φ k . The matrices Φ 0k ,Φ k are conformable matrices from the vector of prior means, φ 0k , and a draw from the quasi posterior distribution,φ 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 (MCMC) methods fail to estimate larger models, which in general confine one to (usually) fewer than 6 variables in the system. Second, the standard approach is fully parametric and requires a law of motion. This can distort inference if the true law of motion is misspecified. Third, the methods used here permit direct estimation of the VAR's time-varying covariance matrix, which has an inverse-Wishart density and is symmetric positive definite at every point in time.

In estimating the model, we use p=2 and a Minnesota Normal-Wishart prior with a shrinkage value ϕ = 0.05 and centre the coefficient on the first lag of each variable to 0.1 in each respective equation. The prior for the Wishart parameters are set following Kadiyala and Karlsson (1997 Notes: The table shows the average net and agg network characteristics with respect to the 11 U.S. industries. When the net measure is positive the IV IX (I) can be classified as a net marginal transmitter, while, when negative, it can be classified as a net marginal receiver. The agg statistic is computed as the sum (in absolute values) between to and from. The highest values of the agg statistic are associated with uncertainty hubs, while the lowest with uncertainty non-hubs. The statistics are reported for each business cycle in our sample, namely inversion, recession, expansion separately and also for the total period, namely from 03-01-2000 to 29-05-2020, at a daily frequency. Notes: This table presents the results of the predictive regression in equation 9 between the aggregate network connectedness and the 3month moving average of the ADS indicator of business cycle (Panel A). In Panel B and Panel C the results of the predictive regression with respect to the ADS 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 ∈ (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 monthly frequency between 01-2000 and 05-2020. Notes: This table presents the results of the predictive regressions between the aggregate 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, USLI. The five columns of the table represent different predictability horizons with h ∈ (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, exception for the USLI control. Series are considered at monthly frequency between 01-2000 and 05-2020. Notes: This table presents the results of the predictive regressions between the uncertainty hubs sub-network and the ADS Index, indicator of business cycle (Panel A). In Panel B and Panel C the results of the predictive regression with respect to the ADS expansion and recession indicators are reported, respectively. We present results for both regression equations 10 and ?? in which we add a set of controls. The five columns of the table represent different predictability horizons with h ∈ (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 monthly frequency between 01-2000 and 05-2020. Notes: This table presents the results of the predictive regression 9 between the aggregate network connectedness, and the US GDP growth rate (Panel A) and GDP volatility (Panel B). The four columns of the table represent different predictability horizons with h ∈ (1, 2, 3, 4) quarters. 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 all taken at quarterly frequency, between 01-2000 and 05-2020. Notes: This table presents the results of the predictive regressions ?? between the uncertainty hubs sub-network 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. The five columns of the table represent different predictability horizons with h ∈ (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 monthly frequency between 01-2000 and 05-2020. , 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 monthly frequency between 01-2000 and 05-2020. Notes: This table presents the results of the predictive regression 9 between the hubs and non-hubs networks, and the US GDP growth rate (Panel A) and GDP volatility (Panel B). The four columns of the table represent different predictability horizons with h ∈ (1, 2, 3, 4) quarters. 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 all taken at quarterly frequency, between 01-2000 and 05-2020.

Endogenous production networks

The network origins of aggregate fluctuations

Microeconomic origins of macroeconomic tail risks

Monetary tightening cycles and the predictability of economic activity

Financial frictions and fluctuations in volatility

How important are sectoral shocks?

International inflation spillovers through input linkages

Investment dispersion and the business cycle

Uncertainty and economic activity: Evidence from business survey data

Stock return characteristics, skew laws, and the differential pricing of individual equity options

Spanning and derivative-security valuation

The macroeconomic impact of microeconomic shocks: beyond Hulten's theorem

Input specificity and the propagation of idiosyncratic shocks in production networks

Asymmetric network connectedness of fears. The Review of Economics and Statistics forthcoming

Dynamic networks in large financial and economic systems

Evaluating the classification of economic activity into recessions and expansions

The impact of uncertainty shocks

Fluctuations in uncertainty

Really uncertain business cycles

Volatility derivatives

A tale of two indices

Industry risk and market integration

The great diversification and its undoing

Production networks: A primer

The VIX premium

The relation between implied and realized volatility

Risk shocks

On the Kullback-Leibler information divergence of locally stationary processes

Empirical spectral processes for locally stationary time series

Market exposure and endogenous firm volatility over the business cycle

Firms, destinations, and aggregate fluctuations

Better to give than to receive: Predictive directional measurement of volatility spillovers

On the network topology of variance decompositions: Measuring the connectedness of financial firms

Quadratic variance swap models

The granular origins of aggregate fluctuations

Power laws in economics: An introduction

Momentum investing and business cycle risk: Evidence from pole to pole

Another look at the role of the industrial structure of markets for international diversification strategies

International competition and exchange rate shocks: a cross-country industry analysis of stock returns

Firm volatility in granular networks

Measuring uncertainty

Numerical methods for estimation and inference in Bayesian VAR-models

The investment network, sectoral comovement, and the changing US business cycle

Uncertainty and business cycles: Exogenous impulse or endogenous response?

Monetary policy through production networks: Evidence from the stock market

Technological revolutions and stock prices

Generalized impulse response analysis in linear multivariate models

A quasi-Bayesian local likelihood approach to time varying parameter VAR models

Econometric analysis of potential outcomes time series: instruments, shocks, linearity and the causal response function

Prediction of weakly locally stationary processes by auto-regression

Crashes, volatility, and the equity premium: Lessons from S&P 500 options

Proposition 1. Let us have the VMA(∞) representation of the locally stationary TVP VAR model (Dahlhaus et al., 2009; Roueff and Sanchez-Perez, 2016 )hence in a neighborhood of a fixed time point u = t/T the process IVIX t,T can be approximated by a stationary process IVIX t (u)with being iid process with E[ t ] = 0, E[ s t ] = 0 for all s = t, and the local covariance matrix of the errors Σ(u). Under suitable regularity conditions |IVIX t,T − IVIX t (u)| = O p |t/T − u| + 1/T . 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 byAppendix H Hubs and non-hubs predictive results