DDynamic Network Risk
Jozef
Barun´ık ‡ and Michael Ellington † July 14, 2020
Abstract
This paper examines the pricing of short-term and long-term dynamic network riskin the cross-section of stock returns. Stocks with high sensitivities to dynamic net-work risk earn lower returns. We rationalize our finding with economic theory thatallows the stochastic discount factor to load on network risk through the precaution-ary savings channel. A one-standard deviation increase in long-term (short-term)network risk loadings associate with a 7.66% (6.71%) drop in annualized expectedreturns.
JEL Classifications: G10, G12, C58Keywords: Network risk, Firm volatility, Cross section of stock returns ‡ Institute of Economic Studies, Charles University, Opletalova 26, 110 00, and The CzechAcademy of Sciences, IITA, Pod Vod´arenskou Vˇeˇz´ı 4, 182 08, Prague, Czech Republic [email protected]†
University of Liverpool Management School, Chatham Building, Liverpool, L69 7ZH, UK [email protected]
Acknowledgements
We thank Luboˇs P´astor, Lucio Sarno, Oliver Linton, Wolfgang H¨ardle, Luk´aˇs V´acha, RylandThomas, Chris Florackis, Costas Milas, Charlie Cai, and Marcin Michalski for invaluable dis-cussions and comments. We are grateful to Luboˇs Hanus for help in furnishing and convertingestimation codes. We acknowledge insightful comments from many seminar presentations, in-cluding: the Danish National Bank; the 2019 STAT of ML conference; the 13th InternationalConference on Computational and Financial Econometrics; and many more. Jozef Barun´ıkgratefully acknowledges support from the Czech Science Foundation under the 19-28231X (EX-PRO) project. a r X i v : . [ q -f i n . GN ] J u l isclosure Statement: Jozef Barun´ık and Michael Ellington have nothing to disclose.
Idiosyncratic stock return volatilities vary over time and exhibit strong co-movement ( ? ).The synchronous behavior of return volatilities provides the foundation for a commoncomponent explaining cross-sectional variation in stock returns. Economic theory andempirical evidence show that stocks loading positively on volatility risk earn lower re-turns in equilibrium. This is because they act as inter-temporal hedging devices againstfuture uncertainty (Cremers et al., 2015; Herskovic et al., 2016). However, Acemogluet al. (2012) present evidence that network structures forming from idiosyncratic shockpropagation from volatilities can determine systematic fluctuations. The implication hereis that network structures forming around volatility connections create a source of riskfor investors.In this paper, we examine the causal nature of dynamic horizon specific shock propa-gation that determines network structures among idiosyncratic return volatilities; and thepremium investors demand for bearing such exposures. Our analysis provides robust em-pirical evidence that stocks with higher sensitivities to dynamic horizon specific networkrisk earn lower returns. We rationalize our findings with economic theory, that allowsthe stochastic discount factor to load on volatility risk through the precautionary savingschannel (Ang et al., 2006; Campbell et al., 2018). In doing so, we propose a mechanismthat describes this type of investor behaviour by allowing the stochastic discount factorto load on horizon specific network connections among idiosyncratic return volatilities.Viewing stocks as nodes within a network allows us to characterize the nature ofshocks that determine such risk exposures. From a Lucas (1977) perspective, investorsare able to diversify away firm level shocks when the network is completely unconnected orcompletely and equally connected. Therefore, idiosyncratic fluctuations average out andresult in negligible aggregate effects. Hence, an investor holding a large well-diversifiedportfolio has minimal exposure to such shocks, and thus only demands a premium fornon-diversifiable systematic sources of risk. However, relaxing the assumption of symmet-rical connections breaks down the ability of investors to diversify away firm-level shocks,permits the snow-balling of firm-level shocks (Elliott et al., 2014), and induces differencesin the strength of directional connections.To illustrate the above, Figure 1 (a) shows a star network topology where stock 1 iscentral and its shocks propagate to the remaining N assets on the market. In this case,1 x x x x x N (a) Asymmetric network x x x x N (b) Weighted asymmetric network
Figure 1: Emergence of Network Risk:
The sub-figure (a) presents a network containing x , . . . , x N stocks represented as nodes. Here node x influences all other x , . . . , x N nodes.Nodes x , . . . , x N are not connected to each other and also do not influence node x . The sub-figure (b) presents an N -star network where nodes x , . . . , x N are connected to a set of nodesexclusively, and also to one another. Arrows denote the direction of the connection and thedensity of the line denotes the strength of the connections. even if an investor holds a large number of stocks, they still face exposure to stock 1. Thisis a so called directed, or asymmetric network in which the nodes are connected with thesame strength. Figure 1 (b) is an N star network topology where the links among the Ncentral firms have weighted strengths. In this case, an investor faces exposure to shockpropagation and reception of the N central firms resulting in a complex risk structurefrom the network.While the centrality of these nodes is a characteristic for the determination of networkrisk premia, these networks are missing two key ingredients. First, these networks arestatic; and therefore do not capture evolving relationships among firms. Second, whiledistinguishing between short-term and long-term risks is a prominent issue in finance(Bansal and Yaron, 2004) , these networks convey no information on the persistence ofsuch exposures. A network topology hence needs to reflect dynamic linkages due to shocks Note that the assumption that sources of systematic risk are constant across horizons faces rarequestioning within the literature; although studies are emerging. Dew-Becker and Giglio (2016) formalizethe notion of horizon specific risk pricing focusing on investor preferences. This allows the authors toquantify the meaning of long-run in the context of recursive preferences which they go on to show issignificantly priced in equity markets. Bandi and Tamoni (2017) take a different approach and extractthe business cycle component of consumption to obtain factor loadings and show that they have similarpricing ability to those stemming from consumption growth aggregated over a two-year horizon. Bandiet al. (2018) argue that frequency is a source of systematic risk by decomposing betas into horizon specificcomponents. They show a simple horizon specific CAPM may outperform popular multi-factor modelswhilst possessing economic interpretability. t = 1 , . . . , T on a time line. For simplicity we assume that the N central assets arethe same across each layer but will have connections of different strengths over timeand investment horizon. Distinguishing between short-term and long-term layers thatevolve dynamically allows one to relax the assumption that exposures to network risk areconstant over horizons and time. Sh o r t H o r i z o n x x x x N L o n g H o r i z o n x x x x N Sh o r t H o r i z o n x x x x N L o n g H o r i z o n x x x x N t j ↑ t k ↑ T − T Figure 2: A Dynamic Horizon Specific Network:
This figure presents a multi-layerN-star network with snapshot of two time and two horizon specific network layers. Arrowsdenote directions of connections and the line density denotes strength. The curves from the N central nodes allow for connections to spillover across layers. We interpret each layer as anetwork specific to a horizon of interest; for example short-term depicted by light blue colorand long-term depicted by light red color. Therefore, Figure 2 describes an investment opportunity with dynamic horizon spe-cific directional network risk exposures. Investors should require larger compensation, inabsolute value, for risk in the long-run when investing during period t j in comparison toshort-run investments, and vice versa when investing during period t k . Understandingthe dynamics of node centrality and shock propagation in such network is crucial for3n investor because these dynamic connections create a source of systematic risk fromnetwork structures.Our main objective is to investigate the pricing implications of dynamic horizon spe-cific directional network risk for the cross-section of stock returns. In doing so, we de-compose dynamic network connections among the daily realized volatilities of S&P500constituent stock returns into horizon specific components; namely a short-term andlong-term component. We track dynamic horizon specific network connections amongall S&P500 constituents using measures that stem from a large scale time-varying pa-rameter vector autoregressive approximating model. Our approach to tracking networkconnections permits one to examine risk at any horizon of interest and consequently hedgeagainst it. We show that dynamic horizon specific, and aggregate, directional networkrisk constitute sources of risk that price stocks in the cross-section. To the best of ourknowledge, we are the first to propose and explore the asset pricing implications of trulytime-varying directional network risk; whilst also disentangling between short-term andlong-term exposures. An important aspect of our work is the ability to characterize risksin large scale networks composed from hundreds of stocks.To proxy dynamic horizon specific network risk, we construct tradable factors usingdirectional connections among asset return volatilities. Branger et al. (2018) providetheoretical motivations for directional network connections in prices and the channelsthrough which they influence equilibrium returns. We also control for the relative im-portance of nodes to the network, which is similar to concentration defined in Herskovic(2018); a characteristic found to be important for explaining the cross-section of stockreturns. Inherently our proxies for dynamic horizon specific network risk incorporatedynamic network-wide properties.Our main result is that stocks with high sensitivities to dynamic network risk earnlower returns. These sources of risk are statistically significant and economically meaning-ful. Fama-MacBeth regressions indicate a one-standard-deviation increase across stocksin short-term and long-term directional network risk loadings implies a 6.76% and 7.66%drop in expected annual returns respectively. We also conduct portfolio sorts that assignstocks into quintile portfolios according to their short-term (long-term) directional net-work risk betas. The annual Fama and French (2015) five-factor alphas of value weightedhedge portfolios of short-term and long-term directional network risk are -3.43% and-4.05% respectively.Our analysis is robust to using portfolio sorts and Fama-MacBeth regressions thatinclude a battery of additional factors over and above the Fama and French (2015) five-4actor model. Namely, we include: changes in the VIX; momentum; conditional skewness;conditional kurtosis; and market illiquidity. We also control for the variance risk premium,tail risk, idiosyncratic volatility and idiosyncratic skewness. Even with these additionalfactors, dynamic horizon specific network risk maintains statistical and economic signifi-cance; whilst also showing it is a distinct source of systematic risk that is not attributableto volatility risk. We also provide an alternative specification for directional network riskand obtain statistically and economically significant results consistent with our main re-sults. Moreover, we show that one is able to predict future directional network risk, bothhorizon specific and aggregate, by conducting portfolio sorts on past directional networkrisk betas. We obtain statistically significant raw and risk-adjusted returns across allspecifications. This means that investors are able to implement strategies that hedgeagainst these sources of risk in real time.The remainder of this paper proceeds as follows. Section 2 outlines an economywhere the stochastic discount factor loads on dynamic horizon specific directional networkconnections among asset return volatilities and discusses related literature. In Section 3,we describe our methodology in tracking dynamic horizon specific network connections.In Section 4, we discuss data and how we track dynamic horizon specific network risk ofall S&P500 constituents. Our empirical results and extensions are in Sections 5 and 6.Finally, conclusion are given in Section 7. Networks in asset pricing is an emerging literature. Buraschi and Porchia (2012) establisha link between network structure and the cross-section of stock returns using dividends.Meanwhile, Ahern (2013) focuses on a production-based framework using input-outputdata to show that industries in more central positions within the network earn higherreturns. Herskovic (2018) builds on this and deduces an equilibrium asset pricing modelwhere sectors connect through an input-output network. Defining concentration as thedegree to which the network is dominated by a few sectors, and sparsity as the distributionof sectoral connections, the author shows that factors stemming from these network-wideproperties price stocks in the cross-section. Branger et al. (2018) characterize a modelwhere equilibrium expected returns depend on directional network connections of negativeprice jumps. Their model implies that the overall effect of network connections depends5n whether shock propagation and reception dominates a hedging channel . ? assessfirm volatility in a network model where shocks to customers influence their suppliers.Their model allows for asymmetries between strength of customer-supplier linkages andproduces distributions of firm volatility, size and customer concentration consistent withthe data.Our work also relates well with studies regarding volatility risk and time-varyingvolatility (e.g. Ang et al. (2006), and Ang et al. (2006)). These studies show thatinvestors seek to use assets with positive covariance with market volatility as hedgingdevices and will thus accept lower returns. Campbell et al. (2018) allow for stochasticvolatility in an ICAPM framework. They show that asset returns that positively co-vary with a variable forecasting future market volatility have low expected returns inequilibrium. The economic mechanism is that investors reduce current consumption toincrease precautionary savings in light of uncertainty around market returns. Cremerset al. (2015) distinguish between jump and volatility risk showing that they bear differentrisk premia; both of which are negative. Herskovic et al. (2016) extract a common fac-tor from firm level volatility and show that the highest quintile portfolios sorted on thecommon idiosyncratic volatility factor earn 5.4% lower returns per year than the lowestquintile portfolios.We contribute to this literature by focusing on the directional network connectionsamong asset return volatilities. We combine theoretical justifications of negative riskprices for aggregate volatility with an economy where the stochastic discount loads onhorizon specific directional network connections to motivate negative risk prices. Our two-tree endowment economy generates horizon specific connections in the volatil-ity of asset returns and the volatility of consumption growth. Specifically, we extendon Cochrane et al. (2007) and Lucas (1978). Similar to Bansal and Yaron (2004) andBackus et al. (2011), we model asset returns as claims on risk factors in the consumptionprocess. The representative investor has the following general utility over the stream of If the hedging channel dominates the risk premium is negative. This is because less connectedassets during periods of financial turbulence are relatively more favorable to connected assets within thenetwork. U t = E t Z ∞ e − δτ u ( c t + τ ) dτ. (1)Each endowment dividend stream follows a geometric Brownian motion with stochasticvolatility; whose respective drift and diffusion parameters differ. dD ι D ι = µ ι dt + √ v ι,t dZ ι , ι = { S, L } (2) dv ι,t = κ v ι (¯ v ι − v ι,t ) dt + σ v ι √ v ι,t dZ v ι + N X j =1 K ι,j d N ι,j,t (3)where dZ ι are standard Brownian motions that are possibly correlated, C orr (cid:16) dZ S , dZ L (cid:17) = ρ S,L dt . We interpret the endowment trees as short-term ( S ) and long-term ( L ) risk fac-tors within the economy . The diffusion of each tree follows a mean-reverting square rootprocess (Heston, 1993) with N self and mutually exciting jumps N ι,j,t , ι = { S, L } (A¨ıt-Sahalia et al., 2015). ¯ v ι is the unconditional horizon specific component of conditionalvariance and κ v ι captures the speed of mean reversion .The N self and mutually exciting jumps introduce horizon specific network connec-tions in the volatility of consumption growth. The jump intensities are stochastic andfollow Hawkes processes with mean reverting dynamics of the form: d‘ ι,j,t = α ι,j ( ‘ ι,j, ∞ − ‘ ι,j,t ) dt + N X k =1 b ι,j,k d N ι,k,t (4)This means that a horizon specific jumps in asset k causes an increase in the horizonspecific jump intensity of asset j such that ‘ ι,j jumps by b ι,j,k before decaying back towardsthe level ‘ ι,j, ∞ at speed α ι,j . If the increase in ‘ ι,j leads to a jump in asset j , and thereis a non-zero b ι,n,j , the horizon specific shock passes on to asset n . In this manner theshocks can be propagated throughout the entire network, which also permits the initialshock to reach asset k itself.We have N risky assets in the economy that whose dynamics are geometric Brownian One may conjecture that the L tree corresponds to the long-term component of consumption. Thisgenerates a persistent dividend stream and bears long-term risk in the economy. The S tree generates aless persistent dividend stream bearing short-term risks in the economy; µ L > µ S > Again one may conjecture the speed of mean reversion is slower for the volatility process associatedto the long-term part of consumption. k -th asset has the following dynamics: dp k,t p k,t = µ p k dt + √ v p k,S dW S + √ v p k,L dW L (5) dv p k ,ι = κ p k ,ι (¯ v p k ,ι − v p k ,ι ) dt + σ p k ,ι √ v p k ,ι dW ξ + Q ι d N ι,k,t , ι = { S, L } , ξ = { S , L } (6)which is similar to Christoffersen et al. (2009) where the variance of the stock returnis the sum of the two stochastic volatility components. Note that C orr (cid:16) dW S , dW S (cid:17) = ρ W S ,W S dt and C orr (cid:16) dW L , dW L (cid:17) = ρ W L ,W L dt, C orr (cid:16) dW S , dW L (cid:17) = C orr (cid:16) dW L , dW S (cid:17) =0. We add discontinuities to each variance process that also enter horizon specific stochas-tic volatility processes for each consumption dividend stream. Q ι , are the jump sizes ofcompound Poisson processes Q L > Q S > N S,k,t , N L,k,t are mutually independentPoisson processes. Note their intensity parameters are as in (4).The economy also contains a risk-less bond that follows an ordinary differential equa-tion dBB = r f dt with r f being the instantaneous risk-free rate. Applying Itˆo’s lemma toconsumption c t = s t D S + (1 − s t ) D L we obtain consumption dynamics dc t c t = [ s t µ S + (1 − s t ) µ L ] dt + s t √ v S,t dZ S + (1 − s t ) √ v L,t dZ L . (7)Using the results in Appendix A providing first two moments of consumption growthand further details of the economy, the following proposition formalizes how networkconnections influence stochastic discount factor. Proposition 1 (Stochastic Discount Factor Innovations in a Network Economy) . Con-sider an endowment economy with a short run and long run cash flows for consumptionas in Equations 2 – 3 and consumption dynamics follow Equation 7. Then, the time t expected innovation in the stochastic discount factor, E t h d Λ t Λ t i are E t " d Λ t Λ t = − δdt − γ t h s t µ S + (1 − s t ) µ L i dt + 12 η t h s t v S,t + (1 − s t ) v L,t + s t (1 − s t ) √ v S,t √ v L,t ρ S,L i dt (8) with δ > being impatience, γ t ≡ − u ( c t ) c t u ( c t ) > is the coefficient of risk aversion, and η t ≡ u ( c t )( c t ) u ( c t ) > is precautionary saving.Proof. See Appendix B. 8e can see that the stochastic discount factor depends on horizon specific networkconnections through the respective stochastic volatility processes. Note that these expres-sions hold in general for any utility function . We use the above to outline an economythat permits network connections to enter the expression for the stochastic discount fac-tor. In turn, this implies that exposure to horizon specific network risk will be priced inequilibrium.As we outline earlier, the literature concerning volatility risk shows that stocks loadingpositively on volatility act as inter-temporal hedging devices against future uncertainty.In this economy, network connections form among stock return volatilities which meansthat investors will demand compensation for dynamic horizon specific network risk arisingthrough volatility connections of individual assets; over and above the premia for beingexposed to horizon specific consumption growth volatility risk.It is clear from (8) that the stochastic discount factor loads on horizon specific networkconnections through precautionary savings. Therefore, stocks loading positively on hori-zon specific network risk earn lower returns because they act as inter-temporal hedgingdevices against adverse changes in investment opportunities. Importantly in this econ-omy, we show that this is a source of risk that is different to volatility risk. Empirically,Sections 5 and 6 confirm that dynamic horizon specific network risk constitutes a sourceof risk different to measures of volatility risk. Our objective here is to understand how shocks with different persistence propagate acrossa network of assets. Knowing how a shock to stock j transmits to stock k will defineweighted and directed network at a given period of time and at a given horizon. Inturn, one may use this information to build aggregate system-wide measures of networkconnectedness as well as disaggregate measures stemming from individual connectionsthat will characterize various types of risks. In contrast to the network literature inFinance (Elliott et al., 2014; Glasserman and Young, 2016; Herskovic, 2018), we focuson network dynamics over time and across horizons. In doing so, we estimate a large Note also that it is not the purpose of this work to provide a specific, either closed-form or numerical,solution to this model. However, one may do so by specifying, for example, recursive preferences andadopt the methods in Eraker and Shaliastovich (2008) to obtain a numerical solution. Setting oureconomy up in a similar framework to the example in Eraker and Shaliastovich (2008) would yield riskpremia for: horizon specific components of the volatility of consumption growth, and dynamic horizonspecific directional network risk. We leave the solution and calibration of this model for future research. j is due to shocks in variable k . Thus, the variance decompositionmatrix defines the network adjacency matrix and is intimately related to network nodedegrees, mean degrees, and connectedness measures (Diebold and Yilmaz, 2014). Cur-rently studies examine, almost exclusively, static networks mimicking time dynamics withestimation from an approximating window . We employ a locally stationary TVP VARthat allows us to estimate the adjacency matrix for a network, or market, of stocks ateach point in time. We decompose this into horizon specific components that allow us todisentangle short-term and long-term network connections .Dynamic horizon specific networks that we define by time-varying variance decompo-sitions are more sophisticated than classical network structures. In a typical network, theadjacency matrix contains zero and one entries, depending on the node being linked ornot, respectively. In the above notion, one interprets variance decompositions as weightedlinks showing the strength of the connections. In addition, the links are directed, meaningthat the j to k link is not necessarily the same as the k to j link, and hence, the adjacencymatrix is not symmetric. Therefore we can define weighted, directed versions of networkconnectedness statistics readily that include degrees, degree distributions, distances anddiameters.These measures are key to our analysis since risk stems directly from asymmetries Geraci and Gnabo (2018) estimate multiple pairwise time varying parameter models in an attemptto characterize a network of financial stocks using autoregressive coefficients. Dimensionality is a problem using large scale TVP VARs. Chan et al. (2020) propose methods toestimate TVP VARs with 15 variables, and Kapetanios et al. (2019) and Petrova (2019) define largeTVP VARs as 78 and 87 variables respectively. j is due to shocks in variable k . A natural way to describe horizonspecific dynamics (i.e. short-term, and long-term) of the network connections is to con-sider the spectral representation of the approximating model. Stiassny (1996) introducesthe notion of a spectral representation in a relatively restrictive setting, while Barun´ıkand Kˇrehl´ık (2018) define horizon specific connectedness measures for a simple VAR thatwe further generalize to a locally stationary processes.Consider a doubly indexed N -variate time series ( X t,T ) ≤ t ≤ T,T ∈ N with components X t,T = ( X t,T , . . . , X Nt,T ) > that describe all assets in an economy. Here t refers to adiscrete time index and T is an additional index indicating the sharpness of the localapproximation of the time series ( X t,T ) ≤ t ≤ T,T ∈ N by a stationary one. We rescale timeusing the conditions in Dahlhaus (1996) such that the continuous parameter u ≈ t/T isa local approximation of the weakly stationary time-series.We assume assets to follow a locally stationary TVP-VAR of lag order p as X t,T = Φ ( t/T ) X t − ,T + . . . + Φ p ( t/T ) X t − p,T + (cid:15) t,T , (9)where (cid:15) t,T = Σ − / ( t/T ) η t,T with η t,T ∼ N ID (0 , I M ) and Φ ( t/T ) = ( Φ ( t/T ) , . . . , Φ p ( t/T )) > are the time varying autoregressive coefficients. In a neighborhood of a fixed time point u = t /T , we approximate the process X t,T by a stationary process f X t ( u ) as f X t ( u ) = Φ ( u ) f X t − ( u ) + . . . + Φ p ( u ) f X t − p ( u ) + (cid:15) t , (10)with t ∈ Z and under suitable regularity conditions which justifies the notation “locallystationary process”. Importantly, the process has time varying Vector Moving AverageVMA( ∞ ) representation (Dahlhaus et al., 2009; Roueff and Sanchez-Perez, 2016) X t,T = ∞ X h = −∞ Ψ t,T ( h ) (cid:15) t − h (11)11here parameter vector Ψ t,T ( h ) ≈ Ψ ( t/T, h ) is a bounded stochastic process . Theconnectedness measures rely on variance decompositions, which are transformations ofthe information in Ψ t,T ( h ) that permit the measurement of the contribution of shocks tothe 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 adaptthe generalized identification scheme in Pesaran and Shin (1998) to locally stationaryprocesses.We transform the local impulse responses in the system to local impulse transfer func-tions using Fourier transforms Ψ ( u ) e − iω = P h e − iωh Ψ ( u, h ) . The following propositionestablishes a dynamic 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 itconstitutes a dynamic horizon specific adjacency matrix. Proposition 2 (Dynamic Horizon Specific Networks) . Suppose X t , T is a weakly locallystationary process with σ − kk ∞ X h =0 (cid:12)(cid:12)(cid:12)(cid:12)h Ψ ( u, h ) Σ ( u ) i j,k (cid:12)(cid:12)(cid:12)(cid:12) < + ∞ , ∀ j, k. Then the ( j, k ) th elementof the dynamic horizon specific adjacency matrix θ (u,d) at a rescaled time u = t /T andhorizon d = ( a, b ) : a, b ∈ ( − π, π ) , a < b is defined as h θ ( u, d ) i j,k = σ − kk Z ba (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) Ψ ( u ) e − iω Σ ( u ) (cid:21) j,k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dω Z π − π "n Ψ ( u ) e − iω o Σ ( u ) n Ψ ( u ) e + iω o > j,j dω (12) Proof.
See Appendix B.2.It is important to note that h θ ( u, d ) i j,k is a natural dissagregation of traditional vari-ance decompositions to a time-varying and horizon specific adjacency matrix. This isbecause the portion of the local error variance of the j th variable at horizon d due toshocks in the k th variable is scaled by the total variance of the j th variable. Note thatthe quantity in proposition (2) is the squared modulus of weighted complex numbers,thus producing a real quantity. Since Ψ t,T ( h ) contains an infinite number of lags, we approximate the the moving average coefficientsat h = 1 , . . . , H horizons. Note that i = √− Note to notation: [ A ] j,k denotes the j th row and k th column of matrix A denoted in bold. [ A ] j, · denotes the full j th row; this is similar for the columns. A P A , where A is a matrix that denotes thesum of all elements of the matrix A . − π, π ) recovers the time domaincounterpart of the local variance decomposition. The following remark formalizes. Remark 1 (Aggregation of Dynamic Network) . Denote by d s an interval on the realline from the set of intervals D that form a partition of the interval ( − π, π ) , such that ∩ d s ∈ D d s = ∅ , and ∪ d s ∈ D d s = ( − π, π ) . Due to the linearity of integral and the constructionof d s , we have h θ ( u, ∞ ) i j,k = X d s ∈ D h θ ( u, d s ) i j,k . Remark (1) is important as it establishes the aggregation of horizon specific networkconnectedness measures to its time domain counterpart. Thus, short-term ( d = S ) andlong-term ( d = L ) time-varying network characteristics always sum up to an aggregatetime domain counterpart; this makes them directly interpretable. As the rows of thedynamic adjacency matrix do not necessarily sum to one, we normalize the element ineach by the corresponding row sum h e θ ( u, d ) i j,k = h θ ( u, d ) i j,k , P Nk =1 h θ ( u, ∞ ) i j,k .It is important to note that proposition 2 defines the dynamic horizon specific net-work completely. Naturally, our adjacency matrix is filled with weighted links showingstrengths of the connections. The links are directional, meaning that the j to k link isnot necessarily the same as the k to j link. Therefore the adjacency matrix is asymmetricwhich creates undiversifiable risk. Using our notion above, the adjacency matrix evolvesdynamically in time and is horizon specific.To characterize network risk, we define total dynamic network connectedness measuresat a given horizon as the ratio of the off-diagonal elements to the sum of the entire matrix C ( u, d ) = 100 × N X j,k =1 j = k h e θ ( u, d ) i j,k , N X j,k =1 h e θ ( u, ∞ ) i j,k (13)This measures the contribution of forecast error variance attributable to all shocks in thesystem, minus the contribution of own shocks. Similar to the aggregate network con-nectedness measure that infers the system-wide risk, we define measures that will revealwhen an individual asset is a transmitter or a receiver of shocks. We use these measuresto proxy dynamic horizon specific network risk. The dynamic directional connectednessthat measures how much of each asset’s j variance is due to shocks in other assets j = k
13n the economy is given by C j ←• ( u, d ) = 100 × N X k =1 k = j h e θ ( u, d ) i j,k , N X j,k =1 h e θ ( u, ∞ ) i j,k , (14)defining the so-called from connectedness. Note one can precisely interpret this quantityas dynamic from-degrees (or out-degrees in the network literature) that associates withthe nodes of the weighted directed network we represent by the dynamic variance decom-position matrix. Likewise, the contribution of asset j to variances in other variables is C j →• ( u, d ) = 100 × N X k =1 k = j h e θ ( u, d ) i k,j , N X j,j =1 h e θ ( u, ∞ ) i k,j (15)and is the so-called to connectedness. Again, one precisely interprets this as dynamicto-degrees (or in-degrees in the network literature) that associates with the nodes ofthe weighted directed network that we represent by the variance decompositions matrix.These two measures show how other assets contribute to the risk of asset j , and how asset j contributes to the riskiness of others, respectively, in a time-varying fashion at horizon d . Notably one can simply add these measures across all horizons to obtain aggregatetime-varying measures.Finally, to obtain the time-varying coefficient estimates, and the time-varying covari-ance matrices at a fixed time point u = t /T , Φ ( u ) , ..., Φ p ( u ) Σ ( u ), we estimate theapproximating model in (10) using Quasi-Bayesian Local-Likelihood (QBLL) methods(Petrova, 2019).Specifically, we use a kernel weighting function that provides larger weights to obser-vations that surround the period whose coefficient and covariance matrices are of interest.Using conjugate priors, the (quasi) posterior distribution of the parameters of the modelare 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 thatin using (quasi) Bayesian estimation methods, we obtain a distribution of parametersthat we use to construct network measures that provide confidence bands for inference.We detail the estimation algorithm in Appendix C.We provide some details on estimation here. First, the variance decompositions ofthe forecast errors from the VMA( ∞ ) representation require a truncation of the infinitehorizon with a H horizon approximation. As H → ∞ the error disappears (L¨utkepohl,2005). We note here that H serves as an approximating factor and has no interpretation14n the time-domain. We obtain horizon specific measures using Fourier transforms and setour truncation horizon H =100; results are qualitatively similar for H ∈ { , , } .Second in computing our measures, we diagonalize the covariance matrix because ourobjective is to focus on the causal affects of network connections. The Ψ ( u, h ) matrixembeds the causal nature of network linkages, and the covariance matrix Σ ( u ) containscontemporaneous covariances within the off-diagonal elements. In diagonalizing the co-variance matrix we remove the contemporaneous effects and focus solely on causation. Our objective is to explore the pricing implications of dynamic horizon specific direc-tional network risk for stock returns. The economy we outline in Section 2.1 proposes astochastic discount factor that loads on network connections among stock return volatil-ities. Therefore we build a dynamic network among stock return volatilities. In doing so,we use high frequency data for all stocks listed on the S&P500 from July 5, 2005 to August31, 2018 from Tick Data. Specifically, we compute daily returns R t = P D i =1 ( p t,i − p t,i − ),and realized volatility RV t = qP D i =1 ( p t,i − p t,i − ) for each stock on day t , where Ddenotes total number of intraday observations. The i subscripts denote intraday observa-tions which we observe at 5 minutes intervals; and p t,i is the intraday price of the asset .We define short-term as the 1-day to 1-week horizon and long-term as horizons greaterthan 1-week.Figure 3 plots our horizon specific dynamic total connectedness measures from July5, 2005 to August 31, 2018. Overall, there are substantial differences in the levels ofhorizon specific connectedness throughout our estimation sample. In general, long-termconnections are muted during periods of economic/financial tranquillity. However, it isclear that long term connections surge during periods of economic recession or key stockmarket events. For example, long-term total connectedness begins to rise in 2006 andcontinues to do throughout the 2007-2009 recession. Adding to this, we can see long-termconnectedness rising during 2010-2012. This may be attributable to fears of contagion ofthe European sovereign debt crisis, the 2010 flash crash, and when the S&P500 entereda bear market in 2011; albeit short-lived. We can also see during mid to late 2015 short- To obtain our network measures, we estimate the TVP VAR model as in (9) on N =496 stocks with p =2 lags on our T =3278 days of data. We estimate our horizon specific dynamic network measureson a 48-core server. For every t ∈ { , , . . . , T } , we generate 500 simulations of the (quasi) posteriordistribution which results in a total estimation time of 10 days. C ( u , d ) LongShort
Figure 3: Horizon specific dynamic total network connectedness for S&P500 con-stituents
This figure plots the quasi posterior median and 1-standard deviation percentiles of horizonspecific dynamic total network connectedness, C ( u, d ) , d ∈ { S , L } from July 5, 2005 to August31, 2018. S refers to the short-term which we define as 1 day to 1 week; and L refers to long-termwhich we define as horizons > For illustrative purposes, Figure 4, reports net-directional connectedness of: Apple,Netflix, and Google. Net-directional connectedness is the difference between to connect-edness and from connectedness C ( u, d ) j →• − C ( u, d ) j ←• . Net-directional connectednesstells us, on average, how a stock on the market contributes to the network. When C ( u, d ) j →• − C ( u, d ) j ←• > <
0) this tells us that the stock transmits (receives) shocksto (from) the network at horizon band d and time period u after accounting for howreceptive (transmissive) the stock is to the network. Overall, we see differences amonghorizon specific net-directional connectedness measures for these stocks. Apple, is a long-term transmitter during the build up to the 2008 recession before becoming a receiver ofshocks during 2008-2009. Netflix is a short-term transmitter to the network during thebear market of 2011 and fears of contagion of the European sovereign debt crisis. Finally,Google is a strong short-term receiver at the beginning of our sample before becoming along-term receiver during the 2008 recession.Overall, it is clear that our measures provide useful descriptions on the evolutionof connections among financial assets over horizons and time. We can see that totalconnections, particularly over the long-term, rise substantially during key events. Intu-itively, this links well with the observation that financial asset return volatilities exhibit16lustering (Harris, 1991). Our measures show that connections among assets intensifyduring these periods and decompose this into horizon specific bands . This links wellwith Diebold and Yilmaz (2014) and the subsequent literature emerging that uses theirmeasures to track systemic risk. − . . AAPL: C ( u, d ) j →• − C ( u, d ) j ←• LongShort 2005 2010 2016 − . . NFLX: C ( u, d ) j →• − C ( u, d ) j ←• LongShort 2005 2010 2016 − . . GOOGL: C ( u, d ) j →• − C ( u, d ) j ←• LongShort
Figure 4: Horizon specific dynamic net-directional connectedness of Apple, Netflixand Google
The top panel of this figure plots the quasi posterior median of the horizon specific dynamicnet-directional connectedness of: Apple, AAPL; Netflix, NFLX; and Google, GOOGL. Net-directional connectedness is computed as the difference between to connectedness and from connectedness as C ( u, d ) j →• − C ( u, d ) j ←• from July 5, 2005 to August 31, 2018. S refers to theshort-term which we define as 1 day to 1 week; and L refers to long-term which we define ashorizons > Fama and French (1993) and the vast ensuing literature use data on stock characteristicsto create factors that approximate the stochastic discount factor. This search for potentialfactor candidates results in hundreds to choose from (Harvey et al., 2016; McLean andPontiff, 2016). Typically, the approach conducts portfolio sorts and creates long-shortportfolios to approximate risk factors. Despite the success of potential asset pricingfactors, many bear no theoretical grounding . This methodology also permits one to look at disaggregated measures of directional network con-nections. These are also shown to vary across assets, horizons and time. Notably one could go furtherand examine pairwise connections between each asset on the market. There is an emerging literature proposing econometric techniques to select only those factors thatare meaningful (see e.g. Feng et al. (2020)). .With the above in mind, we proceed following the factor literature by conducting port-folio sorts stemming from horizon specific directional network connections. Specifically,we conduct double sorts with daily rebalancing for S&P500 assets based on their rela-tive size and strength of net directional connections using horizon specific, C ( u, d ) j →• −C ( u, d ) j ←• , measures. This allows us to account for both shock propagation and receptioncapacity, and also to extract information from the entire adjacency matrix.Tables 1 reports the average annual returns of value weighted quintile portfolios fromour double sorting procedure. Panels A and B report double sorts on net directional con-nections over the short-term and long-term respectively. We define short-term as 1 day–1 We are not dismissing the possibility that small stocks with strong directional connections can createrisks investors demand compensation for, or that the ‘snowball’ effect in Elliott et al. (2014) for smallstocks cannot lead to systematic shocks; which by definition our proxies for horizon specific directionalnetwork risk can capture. ∞ (i.e. > . Panel C reports averageannual returns of value weighted quintile portfolios sorted on aggregate net directionalconnectedness. from denotes portfolios using assets that are most receptive to shocksfrom other assets in the network. to denotes portfolios using assets that are the strongestshock propagators to other assets in the network. The final column reports quintile port-folios of an average of the from and to portfolios, which summarizes horizon specificdirectional network connections.In general, portfolio returns are monotonically increasing with size. We also see that from portfolios earn higher returns than to portfolios. However, we do not observea monotonic relationship as we move from the from portfolios to the to portfolios;nor should we expect to. As we outline above, shock transmission and reception mayboth create network risks in a market (Branger et al., 2018). Therefore one shouldexpect to see portfolios using assets in the tails of the cross-sectional distribution of netdirectional connections earning relatively lower returns than those using asset in middlequintiles; and overall this is the case. We also report the average annual returns of smallminus big portfolios at each quintile and for the equally weighted average of the from and to portfolios. Note all portfolio returns are negative and statistically significant atconventional levels. Using the results in Table 1, we construct factors in a manner that summarize dynamichorizon specific directional network risk among asset return volatilities whilst controllingfor network concentration (i.e size of nodes). We define short and long horizons as 1 day–1 week, and as horizons > to and from portfolios. These portfolios admitstocks above (below) the 70 th (30 th ) percentile of each respective day’s horizon specificnet-directional connectedness distribution. We characterize these portfolios as to and from portfolios. We then take an average of the to and from portfolios and then along–short position in the respective small and big portfolios. Note that definitions of horizons stems from the frequency with which data is observed in construct-ing the network. While we choose the bands that define horizons in a subjective manner we believe theseassumptions are reasonable. More generally speaking one could use more bands that span the spectrum. able 1: Average annual returns for quintile portfolios on size and horizon specificdirectional network connections Notes: This table reports average annual returns from portfolio double sorts. Specificallyportfolio sorts are separated by size and their horizon specific directional network connectednessmeasures using daily rebalancing from July 5, 2005 to August 31, 2018. Panel A sorts on sizeand d =short-term net-directional connections which we define as 1 day–1 week; Panel B sortson size and d =long-term net-directional connections which we define as 1 week– ∞ ; Panel Csorts on size and uses d =aggregate net-directional connections that sums over short-term andlong-term frequency bands. A: d = Short from /2+1 from to to /21 Small -1.28% -4.87% -7.75% -5.91% -22.32% -11.80% Small-Big -11.07% -15.69% -16.71% -15.72% -27.54% -19.31% t -stat -13.11 -14.58 -13.78 -12.28 -10.08 -12.83 B: d = Long from /2+1 from to to /21 Small -5.05% -5.54% -5.18% -6.25% -20.09% -12.57% -1.29% 4.95% 6.35% 4.85% -1.81% -1.55% Small-Big -11.64% -16.99% -16.41% -18.57% -25.01% -18.32% t -stat -12.61 -13.36 -18.71 -13.06 -8.63 -11.73 C: d = Aggregate from /2+1 from to to /21 Small -4.33% -4.73% -5.18% -5.96% -22.07% -13.20% Small-Big -11.65% -16.46% -14.91% -17.36% -27.56% -19.61% t -stat -14.74 -13.52 -14.97 -14.16 -8.78 -11.2820ormally, the day t directional network risk factor NET( d ) t , at horizon d = { S, L, A } ,where S =short-term which corresponds to horizons of 1-day to 1-week, L =long-termwhich corresponds to horizons 1-week to ∞ , and A is an aggregate that sums over allhorizons (i.e. 1-day to ∞ ), is given byNET( d ) t = (cid:16) from small ( d ) t + to small ( d ) t (cid:17) − (cid:16) from big ( d ) t + to big ( d ) t (cid:17) Table 2: Descriptive statistics of network risk factors
Notes: This table reports descriptive statistics for daily horizon specific and aggregate networkrisk factors from July 5, 2005 to August 31, 2018. Short corresponds to the short-term networkrisk factor which captures directional network risk from 1-day to 1 week. Long is the long-termnetwork risk factor that captures directional network risk from horizons 1-week to ∞ . Aggregate,sums over horizons and thus captures aggregate directional network risk at all horizons. Thetop half of the table reports the annualized expected returns and standard deviations, alongwith sample skewness and kurtosis. The bottom half of this table reports sample correlationson and below the main diagonal, and sample covariances above the main diagonal. Short Long AggregateAnnualized Expected Return -7.34% -7.64% -7.51%
Annualized Standard Deviation
Sample Skewness -0.36 -0.40 -0.22
Sample Kurtosis
Sample Correlations/Covariances Short Long AggregateShort
Long
Aggregate
005 2010 2016 − − % Short 2005 2010 2016 − % Short ⊥ Long2005 2010 2016 − − Long 2005 2010 2016 − − % Aggregate
Figure 5: Horizon specific and aggregate directional network risk factors
The top left and right hand side panels of this figure plot the short-term directional and orthog-onalized short-term directional network (Short ⊥ Long) risk factor daily returns respectively.The bottom left and right hand side panels report the long-term directional network risk fac-tor daily returns and the aggregate directional network risk factor daily returns respectively.The sample period is from July 5, 2005 to August 31, 2018. Short-term is defined as 1 day–1week and long-term is defined as horizons > Dynamic Horizon Specific Network Risk Pricing
This section contains our main results on the pricing of dynamic horizon specific net-work risk in the cross-section of S&P500 returns. We examine the contemporaneous linkbetween factor loadings and returns. Our tests employ individual stocks as our baseassets. We do this for two reasons. First, our focus is on directional network connectionsamong asset return volatilities and the cross-sectional pricing implications of (horizonspecific) directional network risk. Therefore, tracking network connections for a marketrequires this level of granularity. Second, Ang et al. (2020) show that creating portfoliosignores the fact that stocks within portfolios have different betas which can lead to largerstandard errors in cross-sectional risk premia estimates. We first present Fama-MacBethregressions before moving on to consider portfolio sorts.
In our Fama-MacBeth analysis, we run two-stage regressions of daily individual S&P500excess returns on horizon specific and aggregate network risk whilst controlling for anarray of factors existing in the literature. In particular, we control for the Fama andFrench (2015) five-factors; as well as five additional factors. The additional characteristicswe control for are: changes in the VIX; momentum; conditional skewness (Harvey andSiddique, 2000); conditional kurtosis (Dittmar, 2002); and illiquidity (Amihud, 2002).The VIX is from CRSP and the momentum factor is from Ken French’s data library.Conditional skewness is the sample counterpart ofCSKEW = E [( R i,t − µ i,t ) · (MKT t − µ MKT ,t ) ] q var( R i,t ) · var(MKT t )with µ i,t , µ MKT ,t being the average excess return on stock i and the market respectively.We define conditional kurtosis, CKURT, analogously. We construct factors that sortstocks according to their conditional skewness and kurtosis and construct long-shortportfolios using daily rebalancing . Illiquidity is Amihud’s illiquidity measure whichis the ratio of absolute return to trading volume. We compute this for each S&P500constituent and take a cross-sectional average each day as a proxy for market illiquidity.Tables 3 and 4 present cross-sectional pricing results using daily returns of S&P500 We also include firm specific values of CSKEW and CKURT in the time-series regressions to obtainbetas; results are robust to this specification and our dynamic horizon specific directional network riskproxies remain statistically significant and economically meaningful. .Turning to Table 4, a similar story emerges, with the exception of orthogonal short-term directional network risk becoming statistically insignificant in columns 7 and 8. Wealso observe that accounting for additional factors beyond Fama and French (2015) do notdampen the significance or magnitude of the market prices of dynamic horizon specificnetwork risk.We now examine market prices of dynamic aggregate network risk. Table 5 reportsfull sample and rolling regression results in Panels A and B respectively. Again, we finda significantly negative market price of network risk. The estimates change marginallywhen accounting for our additional battery of factors.Overall, models containing only the market risk premium or the Fama and French(2015) five-factor model result in negative prices for the market risk premium. Thissuggests model misspecification. Adding to this, we can see that the intercepts across allspecifications including directional network risk are statistically insignificant. Therebyimplying that these specifications adequately explain cross-sectional return variation.Several implications emerge from this analysis. First, horizon specific network riskis priced in the cross-section of stock returns and the market price of risk is negative.Second, this result is robust across multiple specifications and holds using full-sample androlling window estimates. We note here that the rolling specifications in columns 7 and 8of Table 4 using short-term directional network risk orthogonal to long-term directional By construction, the estimates of market prices of orthogonal short-term directional network riskare smaller in absolute value relative to specifications containing only short-term directional networkrisk. β NET( S ) , β NET( L ) implies anannualized fall in returns of 6.76% and 7.66% respectively . The standard deviation of β NET( S ) =0.81, β NET( L ) =0.79. Taking the market risk premia of -0.034and -0.04 we have: i) -0.034 × β MOM , with the standard deviation results in an annualizedincrease in returns of 9.16% and 8.40% respectively. Note, similar values are implied from alternativespecifications. able 3: Full sample Fama-MacBeth regressions: Dynamic horizon specific networkrisk Notes: This table reports Fama-MacBeth regressions for daily S&P500 stock returns fromJuly 5, 2005 to August 31, 2018. We use Newey West Standard errors with 12-lags, t -ratios insquare brackets below coefficient estimates are adjusted following Shanken (1992). NET( S ) isthe short-term directional network risk factor and NET( L ) is the long-term directional networkrisk factor. For models including both NET( S ) and NET( L ), we use the short-term direc-tional network risk factor that is orthogonal to the long-term directional network risk factor,NET( S ⊥ ). VIX is the daily change in the VIX index; MOM is the momentum factor; CSKEWand CKURT are conditional skewness and conditional kurtosis factors respectively; ILLIQ ismarket illiquidity. Model: S )/NET( S ⊥ ) -0.077 -0.063 -0.022 -0.016[-6.50] [-6.21] [-2.42] [-1.92]NET( L ) -0.079 -0.064 -0.064 -0.056[-6.34] [-5.75] [-4.77] [-4.47]MKT -0.094 -0.083 0.053 0.043 0.065 0.065 0.052 0.061[-2.58] [-2.09] [1.58] [1.30] [1.73] [1.76] [1.56] [1.68]SMB -0.02 -0.006 0.000 0.006 0.009 -0.009 0.004[-1.18] [-0.35] [0.01] [0.42] [0.58] [-0.59] [0.30]HML -0.042 -0.021 -0.01 -0.014 -0.010 -0.027 -0.018[-2.77] [-1.33] [-0.66] [-0.92] [-0.66] [-1.88] [-1.27]RMW 0.047 0.026 0.025 0.011 0.011 0.027 0.012[3.00] [1.67] [1.64] [0.78] [0.80] [1.81] [0.95]CMA -0.036 -0.025 -0.025 -0.011 -0.012 -0.026 -0.011[-3.00] [-2.06] [-2.06] [-1.10] [-1.24] [-2.17] [-1.08]VIX -0.186 -0.182 -0.164[-0.45] [-0.44] [-0.40]MOM 0.146 0.150 0.151[4.06] [4.16] [4.19]CSKEW 0.028 0.038 0.024[1.14] [1.51] [0.97]CKURT 0.005 0.002 0.007[0.21] [0.11] [0.30]ILLIQ 0.008 0.007 0.008[3.05] [2.93] [3.13]Intercept 0.08 0.068 0.01 0.02 -0.005 -0.004 0.007 -0.006[5.05] [3.80] [0.65] [1.29] [-0.31] [-0.26] [0.48] [-0.36] able 4: Rolling Fama-MacBeth regressions: Dynamic horizon specific networkrisk Notes: This table reports daily rolling Fama-MacBeth regressions for daily S&P500 stockreturns from July 5, 2005 to August 31, 2018 using a 3-year window. We use Newey WestStandard errors with 24-lags, t -ratios in square brackets below coefficient estimates are ad-justed following Shanken (1992). NET( S ) is the short-term directional network risk factor andNET( L ) is the long-term directional network risk factor. For models including both NET( S )and NET( L ), we use the short-term directional network risk factor that is orthogonal to thelong-term directional network risk factor, NET( S ⊥ ). VIX is the daily change in the VIX index;MOM is the momentum factor; CSKEW and CKURT are conditional skewness and conditionalkurtosis factors respectively; ILLIQ is market illiquidity. Model: S )/NET( S ⊥ ) -0.043 -0.034 0.007 0.008[-2.40] [-2.24] [0.77] [0.85]NET( L ) -0.048 -0.040 -0.052 -0.044[-2.53] [-2.41] [-2.67] [-2.56]MKT -0.103 -0.015 0.022 0.027 0.015 0.021 0.028 0.020[-1.22] [-0.10] [0.39] [0.44] [0.26] [0.34] [0.47] [0.34]SMB -0.021 0.005 0.008 0.004 0.006 0.009 0.007[-0.85] [0.15] [0.28] [0.13] [0.20] [0.30] [0.24]HML -0.049 -0.025 -0.021 -0.019 -0.017 -0.022 -0.016[-1.55] [-0.77] [-0.70] [-0.70] [-0.62] [-0.73] [-0.62]RMW 0.042 0.023 0.022 0.009 0.008 0.024 0.010[1.39] [0.70] [0.71] [0.34] [0.30] [0.82] [0.41]CMA -0.0007 0.01 0.01 0.001 0.002 0.008 -0.001[0.11] [0.59] [0.60] [0.05] [0.06] [0.48] [-0.07]VIX 0.003 -0.022 -0.017[-0.01] [-0.06] [-0.05]MOM 0.087 0.080 0.080[2.06] [1.96] [1.99]CSKEW 0.031 0.030 0.032[0.47] [0.42] [0.46]CKURT 0.006 0.005 0.007[0.69] [0.66] [0.71]ILLIQ 0.007 0.007 0.007[1.12] [1.09] [1.06]Intercept 0.08 0.027 0.022 0.023 0.030 0.030 0.023 0.032[2.43] [0.94] [0.79] [0.83] [1.25] [1.25] [0.85] [1.35] able 5: Fama-MacBeth regressions: Aggregate directional network risk Notes: Thistable reports Full Sample Fama-MacBeth regressions in Panel A, and Rolling Fama-MacBethregressions in Panel B, for daily S&P500 stock returns. The full sample regressions span fromJuly 5, 2005 to August 31, 2018 and use Newey West Standard errors with 12-lags. Rollingregressions use a 3-year window add 1 day at a time and use Newey West Standard errorswith 24-lags. t -ratios in square brackets below coefficient estimates are adjusted followingShanken (1992). NET( A ) is the aggregate directional network risk factor that sums over short-term and long-term frequency bands. VIX is the daily change in the VIX index; MOM isthe momentum factor; CSKEW and CKURT are conditional skewness and conditional kurtosisfactors respectively; ILLIQ is market illiquidity. A: Full Sample B: Rolling A ) -0.079 -0.065 -0.047 -0.037[-6.36] [-5.94] [-2.44] [-2.32]MKT 0.050 0.066 0.026 0.02[1.48] [1.75] [0.43] [0.33]SMB -0.002 0.008 0.006 0.005[-0.14] [0.49] [0.22] [0.18]HML -0.013 -0.01 -0.023 -0.018[-0.81] [-0.68] [-0.75] [-0.67]RMW 0.025 0.01 0.022 0.008[1.60] [0.73] [0.70] [0.29]CMA -0.026 -0.012 0.009 0.001[-2.16] [-1.23] [0.53] [0.02]VIX -0.173 -0.022[-0.42] [-0.07]MOM 0.145 0.084[3.99] [2.03]CSKEW 0.034 0.03[1.37] [0.45]CKURT 0.004 0.005[0.16] [0.67]ILLIQ 0.007 0.007[2.97] [1.12]Intercept 0.016 -0.003 0.023 0.029[1.10] [-0.20] [0.83] [1.23]29 .2 Portfolio Sorts We now turn to portfolio sorts. Specifically, we estimate betas for S&P500 constituentsand sort directly on factor loading estimates. Our procedure involves estimating rollingregressions using daily data, a 3-year rolling window, and moving forward through thesample on a day-by-day basis. For each of the i S&P500 constituents we estimate thefollowing regressions R i,t = β ,i + β MKT i · MKT t + β NET( d ) i · NET( d ) t + ε i,t , d ∈ { S, L, A } (17)where R i,t is the excess return of the i th S&P500 constituent on day t , MKT˙t is theexcess return on the market portfolio, from Ken French’s data library, on day t . NET( d ) t is the horizon specific network risk factor over the: short-term, S which we define as1-day to 1-week; long-term, L which we define as horizons > A which incorporates network connections over all horizons. We include only the NET( d ) t factor we are sorting on, and only control for the market risk premium to obtain loadingsto reduce the noise in estimation (Ang et al., 2006; Cremers et al., 2015) .From July 10, 2009 we sort stocks into quintiles from the estimates of β NET( d ) i on thatday. We rebalance portfolios monthly during each year and compute annual returnsby summing daily returns throughout each year . Annualized risk-adjusted returns ofquintile portfolios are the alpha of the Fama and French (2015) five-factor model. Wealso report portfolio betas with respect to dynamic horizon specific network risk, Famaand French (2015) five-factors, and the additional battery of factors we use in the Fama-MacBeth regressions; namely the VIX, momentum, conditional skewness, conditionalkurtosis, and market illiquidity. We report results for value-weighted portfolios in Table6, and results for equal-weighted portfolios in Table 7.Panel A of Table 6 shows average annual returns, Fama-French five-factor alphas andportfolio betas for value-weighted quintile portfolios that we sort on β NET( S ) ; as wellas a hedge portfolio that takes a long position in the portfolio containing assets withthe highest 20% loadings, and a short position in the portfolio containing assets withthe lowest 20% loadings. Meanwhile Panel B reports the same characteristics, but for We also conduct the same exercise controlling for the Fama and French (2015) five-factors, as wellas the additional 5 factors we use throughout this paper. Results are qualitatively similar to those wereport and are available on request. Note also that we use Newey West standard errors in constructing t -statistics for all rolling regres-sions with 24 lags. We also start in July 2009 as we use 1-year of data to construct conditional skewnessand conditional kurtosis. β NET( L ) . Table 7 shows comparableresults for equal-weighted portfolios. For completeness, Table 8 shows analogous resultsfor quintile portfolio sorts on aggregate directional network risk betas. Panels A and Bshow value-weighted and equal-weighted characteristics respectively.We are able to draw several conclusions from Tables 6, 7, and 8. First, it is clearthat stocks whose returns load more positively on horizon specific network risk earn lowerreturns. This result is robust across average annual returns, and risk adjusted returns. Weobserve an almost monotonically decreasing relationship in both performance measures.Second, there is a monotonically increasing relationship for portfolio betas with respectto dynamic horizon specific network risk. The dynamic network risk betas associated tothe hedge portfolios all exceed unity and are statistically significant at 1% levels. Thispattern exists after controlling for a battery of factors proposed in the existing literature.Ultimately, these portfolio sensitivities further justify the pricing of dynamic horizonspecific network risk we observe from Fama-MacBeth regressions.Third, we see economically important, and statistically significant raw and risk ad-justed returns stemming from all hedge portfolios in Tables 6, 7, and 8. Specifically,the risk adjusted annual returns for value-weighted hedge portfolios against short-termand long-term dynamic network risk are -3.43% and -4.05% respectively. Furthermore,the equal-weighted portfolios imply annual market prices of short-term and long-termdynamic network risk of -7.14% (-10.27%/1.439) and -7.76% (-10.78%/1.389); of whichare close to the annualized mean of each proxy for network risk we report in Table 2.The same holds true for results in Panel B of Table 8 . A possible reason for the larger returns in value weighted portfolios may stem from the fact that ourproxies for network risk control for concentration, and arguably importance, of assets within the network.From a theoretical standpoint, one could conjecture that those value weighted portfolios containing thelargest stocks on the S&P500 are relatively insulated from shock propagation, or that the feedback loopof own shocks is minimal. able 6: Contemporaneous characteristics of value-weighted portfolios Notes: We create value-weighted portfolios by sorting stocks into quintiles based on daily realized horizon specific network risk betas;short-term in Panel A and long-term in Panel B. Betas are from daily 3-year rolling regressions and the sample spans July 10, 2009 toAugust 31, 2018. Portfolios are rebalanced monthly. The portfolio characteristics are: average annual returns, R p ; annualized Famaand French (2015) five-factor alphas, Ann. α FF ; and average annual betas with respect to dynamic horizon specific network risk, theFama and French (2015) five-factors, the VIX, Momentum, conditional skewness, conditional kurtosis and market illiquidity. Thelatter are obtained from rolling regressions using a 1-year window that add one-month at a time. A: Sorts on NET( S ) Portfolio R p Ann. α FF β NET( S ) β MKT β SMB β HML β RMW β CMA β VIX β MOM β CSKEW β CKURT β ILLIQ β NET( S ) β NET( S ) t -stat -2.06 -4.26 9.69 1.01 3.44 2.73 -1.80 1.46 0.52 -0.09 -1.76 0.83 0.71 B: Sorts on NET( L ) Portfolio R p Ann. α FF β NET( L ) β MKT β SMB β HML β RMW β CMA β VIX β MOM β CSKEW β CKURT β ILLIQ β NET( L ) β NET( L ) t -stat -2.55 -5.30 9.62 1.05 3.27 2.88 -1.32 1.13 0.47 -0.50 -1.50 0.70 0.38 able 7: Contemporaneous characteristics of equal-weighted portfolios Notes: We create equal-weighted portfolios by sorting stocks into quintiles based on daily realized horizon specific network riskbetas; short-term in Panel A and long-term in Panel B. Betas are from daily 3-year rolling regressions and the sample spans July10, 2009 to August 31, 2018. Portfolios are rebalanced monthly. The portfolio characteristics are: average annual returns, R p ;annualized Fama and French (2015) five-factor alphas, Ann. α FF ; and average annual betas with respect to dynamic horizon specificnetwork risk, the Fama and French (2015) five-factors, the VIX, Momentum, conditional skewness, conditional kurtosis and marketilliquidity. The latter are obtained from rolling regressions using a 1-year window that add one-month at a time. A: Sorts on NET( S ) Portfolio R p Ann. α FF β NET( S ) β MKT β SMB β HML β RMW β CMA β VIX β MOM β CSKEW β CKURT β ILLIQ β NET( S ) β NET( S ) -4.21% -11.98% 1.631 0.651 0.255 -0.125 -0.220 0.277 0.007 0.041 -0.303 -0.149 4.1315–1 -10.27% -10.75% 1.439 0.097 0.279 0.160 -0.263 0.134 0.004 -0.017 -0.277 0.005 4.484 t -stat -7.45 -11.80 10.01 1.25 4.18 1.83 -1.91 1.07 0.53 -0.68 -2.04 -0.02 0.65 B: Sorts on NET( L ) Portfolio R p Ann. α FF β NET( L ) β MKT β SMB β HML β RMW β CMA β VIX β MOM β CSKEW β CKURT β ILLIQ β NET( L ) β NET( L ) -4.74% -12.44% 1.594 0.656 0.248 -0.078 -0.162 0.257 0.006 0.011 -0.301 -0.154 3.4095–1 -10.78% -11.25% 1.389 0.100 0.283 0.200 -0.218 0.105 0.004 -0.045 -0.282 -0.003 3.824 t -stat -7.73 -12.59 9.68 1.28 3.91 2.21 -1.58 0.81 0.51 -0.91 -2.00 -0.11 0.41 able 8: Contemporaneous characteristics of value- and equal-weighted portfolios Notes: We create value-weighted portfolios by sorting stocks into quintiles based on daily realized aggregate network risk betas;value-weighted portfolios are in Panel A and equal-weighted portfolios are in Panel B. Betas are from daily 3-year rolling regressionsand the sample spans July 10, 2009 to August 31, 2018. Portfolios are rebalanced monthly. The portfolio characteristics are: averageannual returns, R p ; annualized Fama and French (2015) five-factor alphas, Ann. α FF ; and average annual betas with respect toaggregate directional network risk, the Fama and French (2015) five-factors, the VIX, Momentum, conditional skewness, conditionalkurtosis and market illiquidity. The latter are obtained from rolling regressions using a 1-year window that add one-month at atime. A: Value Weighted Portfolios: Sorts on NET( A ) Portfolio R p Ann. α FF β NET( A ) β MKT β SMB β HML β RMW β CMA β VIX β MOM β CSKEW β CKURT β ILLIQ β NET( A ) β NET( A ) t -stat -2.30 -4.97 9.76 0.92 3.47 2.81 -1.28 1.05 0.53 -0.49 -1.38 0.98 0.53 B: Equal Weighted Portfolios: Sorts on NET( A ) Portfolio R p Ann. α FF β NET( A ) β MKT β SMB β HML β RMW β CMA β VIX β MOM β CSKEW β CKURT β ILLIQ β NET( A ) β NET( A ) -4.40% -12.09% 1.640 0.647 0.256 -0.090 -0.171 0.215 0.007 0.030 -0.264 -0.105 3.5625–1 -10.28% -10.78% 1.410 0.097 0.292 0.186 -0.223 0.070 0.004 -0.032 -0.261 0.050 4.048 t -stat -7.24 -11.88 10.11 1.10 4.17 2.04 -1.56 0.63 0.58 -0.81 -1.76 0.22 0.54 n summary, we provide evidence on the pricing of dynamic horizon specific networkrisk stemming from connections among asset return volatilities. Fama-MacBeth regres-sions and portfolio sorts confirm that stocks with a higher sensitivity to horizon specificnetwork risk earn lower returns. Our results suggest that accounting for only short-termor long-term directional network risk result in similar economic magnitudes and statisticalsignificance. We also demonstrate this result holds for aggregate dynamic network risk.Consistent with economic theory and empirical evidence (Herskovic et al., 2016; Cremerset al., 2015; Campbell et al., 2018), as well as our outline of an economy that gener-ates risk pricing of directional volatility connections, stocks loading positively on horizonspecific network risk earn lower returns as investors seek to hedge against changes ininvestment opportunities. Here we outline and describe a variety of extensions to our main results. We first accountfor additional risk proxies that link with volatility and higher moments of stock returns.We then define an alternative dynamic horizon specific network risk and examine therobustness of our results for Fama-MacBeth regressions and portfolio sorts. Finally, weinvestigate whether we are able to implement an ex-ante strategy to construct hedgeportfolios; thus investigating the ability to predict future network risk.
We investigate whether the contemporaneous relations among dynamic horizon specificnetwork risk and returns remains when controlling for additional risk proxies. In par-ticular, we consider: the variance risk premium (Bollerslev et al., 2009); tail risk (Kellyand Jiang, 2014); idiosyncratic volatility (Ang et al., 2006); and idiosyncratic skewness(Boyer et al., 2010).We define the variance risk premium, VRP, as the difference between the VIX andS&P500 realized variance; the latter we construct from our own realized variance mea-sures where weights relate to each stock’s daily market capitalisation. We proxy tail risk,SKIND, using daily percent change in the CBOE’s skewness index. Idiosyncratic volatil-ity, Idio. Vol, and idiosyncratic skewness, Idio. Skew, are relative to the Fama and French(2015) five-factor model. Specifically, they are the respective sample standard deviationand skewness of the residuals from daily rolling regressions using a 1-year window.35 able 9: Fama-MacBeth regressions: Additional factors
Notes: This table reports FullSample Fama-MacBeth regressions in Panel A, and Rolling Fama-MacBeth regressions in PanelB, for daily S&P500 stock returns. The full sample regressions span from July 5, 2005 to August31, 2018 and use Newey West Standard errors with 12-lags. Rolling regressions use a 3-yearwindow and add 1 day at a time and use Newey West Standard errors with 24-lags. t -ratiosin square brackets below coefficient estimates are adjusted following Shanken (1992). NET( S )(NET( S ⊥ )) is the (orthogonalized) short-term directional network risk factor and NET( L ) isthe long-term directional network risk factor. NET( A ) is the aggregate directional network riskfactor that sums over short-term and long-term frequency bands. VIX is the daily change inthe VIX index; MOM is the momentum factor; CSKEW and CKURT are conditional skewnessand conditional kurtosis factors respectively; VRP is the market variance risk premium; SKINDis the daily change in the CBOE’s skewness index; Idio. Vol is idiosyncratic volatility; and IdioSkew is idiosyncratic skewness. We refrain from reporting coefficients associated to the Famaand French (2015) five-factor model; however each model controls for these. A: Full Sample B: Rolling S )/NET( S ⊥ ) -0.063 -0.016 -0.036 0.007[-6.03] [-2.10] [-2.30] [0.80]NET( L ) -0.062 -0.054 -0.041 -0.046[-5.47] [-4.33] [-2.47] [-2.62]NET( A ) -0.065 -0.04[-5.75] [-2.39]VIX -0.267 -0.267 -0.244 -0.267 0.036 0.007 0.007 0.008[-0.63] [-0.64] [-0.59] [-0.65] [0.06] [-0.01] [-0.00] [-0.01]MOM 0.124 0.128 0.128 0.124 0.092 0.085 0.085 0.088[3.30] [3.38] [3.39] [3.26] [2.15] [2.03] [2.08] [2.09]CSKEW 0.034 0.042 0.029 0.038 0.039 0.038 0.039 0.038[1.25] [1.54] [1.10] [1.42] [0.53] [0.49] [0.51] [0.51]CKURT 0.006 0.005 0.007 0.006 0.001 0.001 0.003 0.001[0.27] [0.23] [0.34] [0.27] [0.58] [0.56] [0.62] [0.56]VRP 0.013 0.012 0.013 0.013 0.008 0.008 0.008 0.008[2.42] [2.31] [2.51] [2.35] [0.93] [0.92] [0.88] [0.94]SKIND 0.462 0.444 0.463 0.451 0.146 0.146 0.14 0.151[2.01] [1.94] [2.01] [1.96] [1.03] [1.04] [1.01] [1.07]Idio. Vol 0.185 0.183 0.184 0.183 0.031 0.031 0.031 0.031[5.20] [5.17] [5.17] [5.17] [1.32] [1.31] [1.31] [1.30]Idio. Skew -0.061 -0.065 -0.064 -0.064 -0.016 -0.015 -0.015 -0.015[-2.76] [-2.86] [-2.84] [-2.81] [-1.32] [-1.27] [-1.26] [-1.28]Intercept -0.009 -0.007 -0.009 -0.007 0.032 0.032 0.033 0.031[-0.52] [-0.45] [-0.52] [-0.42] [1.31] [1.31] [1.37] [1.29]36able 9 presents Fama-MacBeth regressions that account for these additional fourfactors; as well as the Fama and French (2015) five-factors, the daily percent change inthe VIX index, and the MOM, CSKEW and CKURT factors respectively. We removeILLIQ due to the significant positive correlation with VRP. Panel A reports full sampleestimates and Panel B shows results from rolling regressions. We refrain from reportingrisk prices of the Fama and French (2015) five-factors for brevity; however note they aresimilar to our baseline results.Overall, Table 9 suggests that the pricing of dynamic horizon specific network riskis remarkably robust. This is clear from the quantitatively similar risk price estimatesfor both full-sample and rolling specifications. Apart from the risk price estimate oforthogonalized short-term directional network risk in model 7, all Shanken (1992) t -statistics remain significant; which is also consistent with our baseline analysis. The maintakeaway point from Table 9 is that the reward for bearing horizon specific, and indeedaggregate, directional network risk is stable and always negative, as well as statisticallyand economically significant. We explore whether the contemporaneous relation between dynamic horizon specific net-work risk is robust to an alternative definition. This alternative specification takes horizonspecific directional connections and size, but re-formulates our original definition. Specif-cally, we first sort all S&P500 constituents with respect to size as before. However, insteadof taking an equal weighted average of to and from portfolios that are constructed fromassets in the top and bottom 30% percentiles of the daily distribution of net-directionalconnections, we now take the difference between to and from portfolios. Formally, theday t network risk factor NET ( d ) t , at horizon d = { Short , Long , Aggregate } is given byNET ( d ) t = (cid:16) to small ( d ) t − from small ( d ) t (cid:17) − (cid:16) to big ( d ) t − from big ( d ) t (cid:17) (18)We can see from Table 1 that in general to portfolios earn lower returns than from portfolios and that overall portfolios of smaller stocks earn lower returns than those largerstocks. As with our original definition, the dynamic horizon specific network risk factorsummarizes all information regarding directional connections. Instead of imposing equalimportance on shock transmission and reception after controlling for size, these dynamicnetwork risk proxies place a higher weight on shock transmission after accounting foroverall shock reception of the network. 37able 10 shows full sample and rolling regression results for Fama-MacBeth regressionsusing our alternative specifications of horizon specific network risk in Panels A and Brespectively. For the sake of brevity, we report results using short-term and long-termdirectional network risk in the same regression, where short-term directional networkrisk is orthogonal to long-term (i.e. NET ( S ⊥ ) and NET ( L )), and aggregate directionalnetwork risk, NET (T); results using short-term and long-term directional network riskin isolation to one another are available on request. As we can see, estimates for marketrisk prices from full sample and rolling regressions are similar to those in Tables 3 and 5.Note that the statistical significance drops slightly for rolling regressions and is significantat 10% levels when controlling for only Fama and French (2015) five-factors.Moving on to portfolio sorts, we conduct the exact same exercise as in Section 5.2for our alternative factor specification. Table 11 shows contemporaneous characteristicsfor quintile portfolios, and a long-short hedge portfolio that sort on aggregate directionalnetwork risk, β NET ( T ) ; again results for sorts on short-term and long-term directionalnetwork risk betas are available on request. Overall, the same message appears hererelative to Tables 6, 7, and 8. Specifically, we observe a monotonic increase in portfoliosensitivities to aggregate directional network risk after controlling for our battery of otherfactors. Note also that the betas of the hedge portfolios are statistically significant.There are two main differences with our main results that may stem from the definitionof our alternative network risk factors. First, the implied annual market price of riskfrom value-weighted portfolios is consistent with the average annual return of the riskfactor itself; namely the annual market price of risk is -13.35% (-4.34%/0.325) and theaverage annual return of this alternative factor is -15.91%. Second, we do not observea monotonic link between average raw returns and risk adjusted returns. What we dostill see are statistically significant and economically meaningful returns for the hedgeportfolios. This, combined with the monotonic pattern for directional network risk betassuggests it is important to control for these risk factors; which is further justified by theFama-MacBeth regressions in Table 10 . We also conduct our analysis for another alternative specification that ignores relative sizes of assetswithin the network. This simply takes the difference between to and from portfolios. These resultsconform to those we present here, albeit slightly weaker in terms of economic and statistical significance.These are available on request. able 10: Fama-MacBeth regressions: Alternative directional network risk proxies Notes: This table reports Full Sample Fama-MacBeth regressions in Panel A, and Rolling Fama-MacBeth regressions in Panel B, for daily S&P500 stock returns. The full sample regressionsspan from July 5, 2005 to August 31, 2018 and use Newey West Standard errors with 12-lags.Rolling regressions use a 3-year window and add 1 day at a time and use Newey West Standarderrors with 24-lags. t -ratios in square brackets below coefficient estimates are adjusted followingShanken (1992). NET ( S ⊥ ) is the orthogonalized short-term directional network risk factor andNET ( L ) is the long-term directional network risk factor. NET ( A ) is the aggregate directionalnetwork risk factor that sums over short-term and long-term frequency bands. VIX is the dailychange in the VIX index; MOM is the momentum factor; CSKEW and CKURT are conditionalskewness and conditional kurtosis factors respectively; ILLIQ is market illiquidity. A: Full Sample B: Rolling ( S ⊥ ) -0.016 -0.017 0.009 0.005[-1.20] [-1.22] [-0.31] [0.02]NET ( L ) -0.119 -0.101 -0.069 -0.047[-5.37] [-4.62] [-1.88] [-1.45]NET ( A ) -0.124 -0.104 -0.078 -0.048[-5.41] [-4.79] [-2.24] [-1.52]MKT -0.038 0.05 -0.040 0.05 0.002 0.007 -0.007 0.003[-1.03] [1.39] [-1.05] [1.37] [0.03] [0.13] [-0.08] [0.07]SMB -0.008 0.004 -0.009 0.003 -0.010 0.000 -0.013 -0.003[-0.48] [0.27] [-0.55] [0.18] [-0.47] [-0.09] [-0.55] [-0.18]HML -0.009 -0.009 -0.010 -0.01 -0.027 -0.024 -0.026 -0.024[-0.60] [-0.64] [-0.64] [-0.71] [-1.05] [-0.98] [-1.04] [-1.00]RMW 0.035 0.015 0.035 0.015 0.032 0.016 0.033 0.014[2.30] [1.10] [2.27] [1.08] [1.19] [0.68] [1.24] [0.60]CMA -0.031 -0.016 -0.030 -0.015 0.002 -0.005 0.006 -0.002[-2.61] [-1.61] [-2.53] [-1.56] [0.18] [-0.30] [0.40] [-0.14]VIX -0.037 -0.044 0.004 0.05[-0.09] [-0.11] [-0.01] [0.11]MOM 0.153 0.156 0.094 0.094[4.40] [4.46] [2.24] [2.24]CSKEW 0.047 0.049 0.029 0.035[1.86] [1.94] [0.47] [0.57]CKURT -0.005 -0.004 0.012 0.009[-0.21] [-0.17] [0.94] [0.83]ILLIQ 0.007 0.007 0.007 0.006[2.77] [2.74] [1.15] [1.12]Intercept 0.053 -0.002 0.053 -0.003 0.023 0.028 0.024 0.029[3.12] [-0.11] [3.08] [-0.17] [0.87] [1.21] [0.91] [1.24]39 able 11: Contemporaneous characteristics of value- and equal-weighted portfolios Notes: We create value-weighted portfolios by sorting stocks into quintiles based on daily realized aggregate network risk betas; value-weighted portfolios are in Panel A and equal-weighted portfolios are in Panel B. Betas are from daily 3-year rolling regressions and thesample spans July 10, 2009 to August 31, 2018. Portfolios are rebalanced monthly. The portfolio characteristics are: average annualreturns, R p ; annualized Fama and French (2015) five-factor alphas, Ann. α FF ; and average annual betas with respect to dynamichorizon specific network risk, the Fama and French (2015) five-factors, the VIX, Momentum, conditional skewness, conditionalkurtosis and market illiquidity. The latter are obtained from rolling regressions using a 1-year window that add one-month at atime. A: Value Weighted Portfolios: Sorts on NET( A ) Portfolio R p Ann. α FF β NET ( A ) β MKT β SMB β HML β RMW β CMA β VIX β MOM β CSKEW β CKURT β ILLIQ β NET ( A ) β NET ( A ) t -stat -5.54 -3.61 5.34 0.89 0.81 0.79 -1.59 2.21 0.09 -0.50 0.48 -0.41 -1.01 B: Equal Weighted Portfolios: Sorts on NET( A ) Portfolio R p Ann. α FF β NET ( A ) β MKT β SMB β HML β RMW β CMA β VIX β MOM β CSKEW β CKURT β ILLIQ β NET ( A ) β NET( A ) -4.15% -10.42% 0.193 0.674 0.278 -0.063 -0.217 0.360 0.006 -0.144 -0.282 -0.325 -0.1355–1 -10.40% -7.52% 0.389 0.058 0.041 -0.032 -0.175 0.291 0.000 -0.111 -0.035 -0.247 -2.138 t -stat -10.69 -12.28 6.31 1.27 0.98 -0.33 -1.41 2.43 0.11 -2.25 -0.38 -2.15 -0.72 .3 Predicting future network risk Table 12: Average annual portfolio returns sorted by past horizon specific andaggregate directional network risk betas
Notes: This table presents average annual returns of portfolios sorted on past estimates ofhorizon specific and aggregate directional network risk betas. Specifically, on day t we sortstocks according to their network risk betas on day t −
1. We rebalance quintile portfoliosevery month. The table presents average annual returns of value-weighted and equal-weightedportfolios. We also present the annualized Fama and French (2015) five-factor alpha from rollingregressions using a 1-year window and moving forward 1-month at a time, Ann. α FF of the 5–1portfolios. t -statistics associated with the annualized Fama and French (2015) alpha use NeweyWest standard errors with 24 lags and are adjusted according to Shanken (1992). Average Annual Returns β NET( S ) β NET( L ) β NET( A ) VW EW VW EW VW EW β β t -stat -1.93 -7.23 -2.33 -7.44 -2.09 -7.02Ann. α FF t -stat -2.11 -7.62 -3.23 -8.32 -3.05 -7.88Our main focus for this paper is the contemporaneous relationship between dynamichorizon specific network risk and stock returns. Although we find strong evidence infavour of significantly priced directional network risk, contemporaneous analysis tells usnothing with regards the practical use of tracking these network linkages. Therefore,it is necessary to examine whether, in real-time, investors are able to construct hedgeportfolios against such directional network risk.We now conduct portfolio sorts using information on past directional network risk sen-sitivities. Specifically, on day t we sort stocks into quintile portfolios based on β estimateson day t −
1. We rebalance these portfolios monthly as in our baseline analysis through-out each year with the starting day of July 11, 2009 so we can use the beta estimates onJuly 10, 2009. Thus, we follow largely the same procedure as in our contemporaneousportfolio sorts. Table 12 shows average annual returns for portfolios sorted on past β In this paper we investigate the pricing of short-term and long-term dynamic network riskin the cross-section of stock returns. Stocks with a high sensitivity to dynamic networkrisk earn lower returns. We rationalize our finding with economy theory by outlining aneconomy that allows the stochastic discount factor to load on network connections amongthe idiosyncratic volatilities of stock returns through the precautionary savings channel.Our results show that dynamic horizon specific network risk is priced in the cross-section of stock returns and is economically meaningful. In particular, a one-standard-deviation rise across stocks in short-term and long-term directional network risk factorloadings implies a fall in expected annual returns of 6.71% and 7.66% respectively. Wealso show that these patterns hold when considering aggregate dynamic network risk thataccounts for all horizons. Our results are empirically robust to alternative specificationsof dynamic network risk and controlling for a battery of factors over and above the five-factors of Fama and French (2015).There are various important implications that emerge from our study. First, direc-tional network connections in asset return volatility are priced in the cross-section thatare economically significant. Second, these horizon specific connections, and indeed thepricing of this risk, varies substantially throughout time. Nevertheless, we show in realtime that one can predict future dynamic horizon specific network risk and therefore in-vestors can implement strategies to hedge against these exposures. Finally, our methodof decomposing overall network connections among asset return volatilities into horizonspecific components permits investors to examine any horizon of interest. This may beuseful for investment decisions, diversification, and constructing hedge portfolios againstthis source of risk. 42 eferences
Acemoglu, D., V. M. Carvalho, A. Ozdaglar, and A. Tahbaz-Salehi (2012). The networkorigins of aggregate fluctuations.
Econometrica 80 (5), 1977–2016.Ahern, K. R. (2013). Network centrality and the cross section of stock returns.
Availableat SSRN 2197370 .A¨ıt-Sahalia, Y., J. Cacho-Diaz, and R. J. Laeven (2015). Modeling financial contagionusing mutually exciting jump processes.
Journal of Financial Economics 117 (3), 585–606.Amihud, Y. (2002). Illiquidity and stock returns: cross-section and time-series effects.
Journal of Financial Markets 5 (1), 31–56.Ang, A., J. Chen, and Y. Xing (2006). Downside risk.
Review of Financial Studies 19 (4),1191–1239.Ang, A., R. J. Hodrick, Y. Xing, and X. Zhang (2006). The cross-section of volatilityand expected returns.
Journal of Finance 61 (1), 259–299.Ang, A., J. Liu, and K. Schwarz (2020). Using stocks or portfolios in tests of factormodels.
Journal of Financial and Quantitative Analysis 55 (3), 709–750.Backus, D., M. Chernov, and I. Martin (2011). Disasters implied by equity index options.
Journal of Finance 66 (6), 1969–2012.Bandi, F. and A. Tamoni (2017). Business-cycle consumption risk and asset prices.
Available at SSRN 2337973 .Bandi, F. M., S. E. Chaudhuri, A. W. Lo, and A. Tamoni (2018). Measuring horizon-specific systematic risk via spectral betas.Bansal, R. and A. Yaron (2004). Risks for the long run: A potential resolution of assetpricing puzzles.
Journal of Finance 59 (4), 1481–1509.Barun´ık, J. and T. Kˇrehl´ık (2018). Measuring the frequency dynamics of financial con-nectedness and systemic risk.
Journal of Financial Econometrics 16 (2), 271–296.43illio, M., M. Getmansky, A. W. Lo, and L. Pelizzon (2012). Econometric measuresof connectedness and systemic risk in the finance and insurance sectors.
Journal ofFinancial Economics 104 (3), 535–559.Bollerslev, T., G. Tauchen, and H. Zhou (2009). Expected stock returns and variancerisk premia.
Review of Financial Studies 22 (11), 4463–4492.Boyer, B., T. Mitton, and K. Vorkink (2010). Expected idiosyncratic skewness.
Reviewof Financial Studies 23 (1), 169–202.Branger, N., P. Konermann, C. Meinerding, and C. Schlag (2018). Equilibrium assetpricing in directed networks.Buraschi, A. and P. Porchia (2012). Dynamic networks and asset pricing. In
AFA 2013San Diego Meetings Paper .Campbell, J. Y., S. Giglio, C. Polk, and R. Turley (2018). An intertemporal CAPM withstochastic volatility.
Journal of Financial Economics 128 (2), 207–233.Carvalho, V. and X. Gabaix (2013). The great diversification and its undoing.
AmericanEconomic Review 103 (5), 1697–1727.Chan, J. C., E. Eisenstat, and R. W. Strachan (2020). Reducing the state space dimensionin a large tvp-var.
Forthcoming, Journal of Econometrics .Christoffersen, P., S. Heston, and K. Jacobs (2009). The shape and term structure ofthe index option smirk: Why multifactor stochastic volatility models work so well.
Management Science 55 (12), 1914–1932.Cochrane, J. H., F. A. Longstaff, and P. Santa-Clara (2007). Two trees.
Review ofFinancial Studies 21 (1), 347–385.Cremers, M., M. Halling, and D. Weinbaum (2015). Aggregate jump and volatility riskin the cross-section of stock returns.
Journal of Finance 70 (2), 577–614.Dahlhaus, R. (1996). On the kullback-leibler information divergence of locally stationaryprocesses.
Stochastic processes and their applications 62 (1), 139–168.Dahlhaus, R., W. Polonik, et al. (2009). Empirical spectral processes for locally stationarytime series.
Bernoulli 15 (1), 1–39. 44ew-Becker, I. and S. Giglio (2016). Asset pricing in the frequency domain: theory andempirics.
Review of Financial Studies 29 (8), 2029–2068.Diebold, F. X. and K. Yilmaz (2014). On the network topology of variance decomposi-tions: Measuring the connectedness of financial firms.
Journal of Econometrics 182 (1),119–134.Dittmar, R. F. (2002). Nonlinear pricing kernels, kurtosis preference, and evidence fromthe cross section of equity returns.
Journal of Finance 57 (1), 369–403.Elliott, M., B. Golub, and M. O. Jackson (2014). Financial networks and contagion.
American Economic Review 104 (10), 3115–53.Eraker, B. and I. Shaliastovich (2008). An equilibrium guide to designing affine pricingmodels.
Mathematical Finance 18 (4), 519–543.Fama, E. F. and K. R. French (1993). Common risk factors in the returns on stocks andbonds.
Journal of Financial Economics 33 (1), 3–56.Fama, E. F. and K. R. French (2015). A five-factor asset pricing model.
Journal ofFinancial Economics 116 (1), 1–22.Feng, G., S. Giglio, and D. Xiu (2020). Taming the factor zoo: A test of new factors.
Journal of Finance 75 (3), 1327–1370.Geraci, M. V. and J.-Y. Gnabo (2018). Measuring interconnectedness between financialinstitutions with Bayesian time-varying vector autoregressions.
Journal of Financialand Quantitative Analysis 53 (3), 1371–1390.Glasserman, P. and H. P. Young (2016). Contagion in financial networks.
Journal ofEconomic Literature 54 (3), 779–831.Harris, L. (1991). Stock price clustering and discreteness.
Review of Financial Stud-ies 4 (3), 389–415.Harvey, C. R., Y. Liu, and H. Zhu (2016). . . . and the cross-section of expected returns.
Review of Financial Studies 29 (1), 5–68.Harvey, C. R. and A. Siddique (2000). Conditional skewness in asset pricing tests.
Journalof Finance 55 (3), 1263–1295. 45erskovic, B. (2018). Networks in production: Asset pricing implications.
Journal ofFinance 73 (4), 1785–1818.Herskovic, B., B. Kelly, H. Lustig, and S. Van Nieuwerburgh (2016). The common factorin idiosyncratic volatility: Quantitative asset pricing implications.
Journal of FinancialEconomics 119 (2), 249–283.Heston, S. L. (1993). A closed-form solution for options with stochastic volatility withapplications to bond and currency options.
Review of Financial Studies 6 (2), 327–343.Kadiyala, K. R. and S. Karlsson (1997). Numerical methods for estimation and inferencein Bayesian VAR-models.
Journal of Applied Econometrics 12 (2), 99–132.Kapetanios, G., M. Marcellino, and F. Venditti (2019). Large time-varying parametervars: A nonparametric approach.
Journal of Applied Econometrics 34 (7), 1027–1049.Kelly, B. and H. Jiang (2014). Tail risk and asset prices.
The Review of FinancialStudies 27 (10), 2841–2871.Lucas, R. E. (1977). Understanding business cycles. In
Carnegie-Rochester conferenceseries on public policy , Volume 5, pp. 7–29. North-Holland.Lucas, R. E. (1978). Asset prices in an exchange economy.
Econometrica , 1429–1445.L¨utkepohl, H. (2005).
New introduction to multiple time series analysis . Springer Science& Business Media.McLean, R. D. and J. Pontiff (2016). Does academic research destroy stock return pre-dictability?
Journal of Finance 71 (1), 5–32.Pesaran, H. H. and Y. Shin (1998). Generalized impulse response analysis in linearmultivariate models.
Economics letters 58 (1), 17–29.Petrova, K. (2019). A quasi-Bayesian local likelihood approach to time varying parameterVAR models.
Journal of Econometrics .Primiceri, G. E. (2005). Time varying structural vector autoregressions and monetarypolicy.
Review of Economic Studies 72 (3), 821–852.Roueff, F. and A. Sanchez-Perez (2016). Prediction of weakly locally stationary processesby auto-regression. arXiv preprint arXiv:1602.01942 .46hanken, J. (1992). On the estimation of beta-pricing models.
Review of FinancialStudies 5 (1), 1–33.Stiassny, A. (1996). A spectral decomposition for structural VAR models.
EmpiricalEconomics 21 (4), 535–555. 47 nternet Appendix (Not for Publication)A A detailed outline of the asset pricing model
This section outlines an economy that generates horizon specific connections in the volatil-ity of asset returns and consumption growth. We provide this outline as an intuitive gen-eral framework that generates a mechanism for horizon specific network risk to price assetsin equilibrium. Consider an extension of the two tree asset pricing model in Cochraneet al. (2007) that builds on the work of Lucas (1978). Without loss of generality, ourendowment economy has h sources of aggregate risk that we interpret as horizon specificdividend streams from endowment cash flows. We adopt the assumptions in Bansal andYaron (2004) and Backus et al. (2011) and model returns from risky assets (includingdividends) as claims on certain risk factors in the consumption process. To keep thingstractable, we present below a model containing ι = S, L sources of aggregate risk where S corresponds to short-term and L corresponds to long-term risk. Note that accountingfor more horizons in ι results in more tedious algebra.The representative investor has the following general utility over the stream of con-sumption U t = E t Z ∞ e − δτ u ( c t + τ ) dτ. (A.19)Each endowment dividend stream follows a geometric Brownian motions with stochasticvolatility; whose respective drift and diffusion parameters differ. dD ι D ι = µ ι dt + √ v ι,t dZ ι , ι = { S, L } (A.20) dv ι,t = κ v ι (¯ v ι − v ι,t ) dt + σ v ι √ v ι,t dZ v ι + N X j =1 K ι,j d N ι,j,t (A.21)where dZ ι are standard Brownian motions that are possibly correlated, C orr (cid:16) dZ S , dZ L (cid:17) = ρ S,L dt . We interpret the endowment trees as long-term and short-term risk factors withinthe economy. The L tree corresponds to the long-term component of consumption. Thisgenerates a persistent dividend stream and bears long-term risk in the economy. The S tree generates a less persistent dividend stream bearing short-term risks in the economy; µ L > µ S >
0. The diffusion of each tree follows a mean-reverting square root process,similar to the process in Heston (1993). ¯ v ι is long-run conditional variance and κ v ι cap-48ures the speed of mean reversion. We assume the speed of mean reversion is slower forthe volatility process associated to the long-term past of consumption.To introduce horizon specific network connections, we add N self and mutually ex-citing jumps N ι,j,t , ι = { S, L } to the respective square root processes (A¨ıt-Sahalia et al.,2015). These horizon specific self and mutually exciting jumps have constant jump sizes K ι,j > , K ι,j = K ι,k . Their stochastic jump intensities follow Hawkes processes whichby definition have the following F t conditional mean jump rate per unit of time: P [ N ι,j,t +∆ − N ι,j,t = 0 |F t ] = 1 − ‘ ι,j,t ∆ + o (∆) P [ N ι,j,t +∆ − N ι,j,t = 1 |F t ] = ‘ ι,j,t ∆ + o (∆) P [ N ι,j,t +∆ − N ι,j,t > |F t ] = o (∆)With dynamics ‘ ι,j,t = ‘ ι,j, ∞ + N X k =1 Z t −∞ g ι,j,k ( t − s ) d N ι,k,s , k = 1 , . . . , M. here ‘ ι,j, ∞ ≥ j = 1 , . . . , N , the real-valued functions g ι,j,k ( u ) ≥ u ≥ j, k = 1 , . . . , N . This ensures the intensity processes are non-negative withprobability 1. Note here that E [ d N ι,j,s ] = ‘ ι,j,s ds and we have ‘ ι,j = ‘ ι,j, ∞ + N X j =1 ‘ ι,j Z t −∞ g ι,j,k ( t − s ) ds = ‘ ι,j, ∞ + N X j =1 (cid:18)Z ∞ g ι,j,k ( u ) du (cid:19) ‘ ι,j Under this general form, we see that each prior horizon specific jump raises the corre-sponding horizon specific intensities ‘ ι,j . The distribution of the jump processes N j,s are determined by that of the intensities. The compensated processes N ι,j,t − R t −∞ ‘ ι,j,s ds are local martingales. The way in which we impose shock propagation, and therefore thenetwork structure is by imposing exponential decay on the g ι,j,k ( t − s ) formally: g ι,j,k ( t − s ) = b ι,j,k e − α ι,j ( t − s ) , s < t, ι = S, L, j, k = 1 , . . . , N In vector form we have L ι = ( I − G ι ) − L ι, ∞ where G ι contains elements R ∞ g ι,j,k ( u ) du and thevectors L ι , L ι, ∞ contain the corresponding ‘ ι,j , ‘ ι,j, ∞ elements. This ensures stationarity of model. α ι,j > , b ι,j,k > j, k = 1 , . . . , N . Therefore using this functional form of g ι,j,k ( t − s ) we have the following mean reverting dynamics d‘ ι,j,t = α ι,j ( ‘ ι,j, ∞ − ‘ ι,j,t ) dt + N X k =1 b ι,j,k dN ι,k,t (A.22)This means that a horizon specific jumps in asset k causes an increase in the horizonspecific jump intensity of asset j such that ‘ ι,j jumps by b ι,j,k before decaying back towardsthe level ‘ ι,j, ∞ at speed α ι,j . If the increase in ‘ ι,j leads to a jump in asset j , and there is anon-zero b ι,n,j , the horizon specific shock passes on to asset n . In this manner the shockscan be propagated throughout the entire network, and also permits the initial shock toreach asset k itself.The expected value of the jump sizes are E [ K ι,j ] = µ K ι,j in general .We have N risky assets in the economy that whose dynamics are geometric brownianmotions with stochastic volatility. Formally the k -th asset has the following dynamics: dp k,t p k,t = µ p k dt + √ v p k,S dW S + √ v p k,L dW L (A.23) dv p k ,ι = κ p k ,ι (¯ v p k ,ι − v p k ,ι ) dt + σ p k ,ι √ v p k ,ι dW ξ + Q ι d N ι,k,t , ι = { S, L } , ξ = { S , L } (A.24)which is similar to Christoffersen et al. (2009) where the variance of the stock returnis the sum of the two stochastic volatility components. Note that C orr (cid:16) dW S , dW S (cid:17) = ρ W S ,W S dt and C orr (cid:16) dW L , dW L (cid:17) = ρ W L ,W L dt, C orr (cid:16) dW S , dW L (cid:17) = C orr (cid:16) dW L , dW S (cid:17) =0. We add discontinuities to each variance process that also enter horizon specific stochas-tic volatility processes for each consumption dividend stream. Q ι , ι = S, L , are the jumpsizes of compound Poisson processes Q L > Q S > N S,k,t , N L,k,t are mutually inde-pendent Poisson processes. Note their intensity parameters are as in (4).The economy also contains a risk-less bond, B, that follows an ordinary differentialequation: d BB = r f dt. (A.25)with r f being the instantaneous risk-free rate. If the j jump sizes are constant for each horizon ι , then the expected value is E [ K ι,j ] = µ K ι .Note we could further simplify and assume that jump sizes are constant across horizons and assets then E [ K ι,j ] = µ K (Branger et al., 2018). s t = D S D S + D L (A.26)the relative size of the first dividend - i.e. short-term part of the consumption processto aggregate consumption. It is important to note here that since consumption volatil-ity varies throughout time, the states, s t , (1 − s t ) are also time-varying. The functiongenerating consumption is c t = s t D S + (1 − s t ) D L . Applying Itˆo’s lemma, consumptionsdynamics are dc t c t = [ s t µ S + (1 − s t ) µ L ] dt + s t √ v S,t dZ S + (1 − s t ) √ v L,t dZ L (A.27) Proposition 3.
Consider a ι = { S, L } tree endowment economy where cash flows for con-sumption have jump-diffusion dynamics with N self and mutually exciting jumps whoseintensities depend on network connections among N risky assets in the economy. Thenthe first two moments of consumption growth, E t h dc t c t i , E t (cid:20)(cid:16) dc t c t (cid:17) (cid:21) , and the risk-free rate, r f are given by E t " dc t c t = h s t µ S + (1 − s t ) µ L i dt (A.28) E t dc t c t ! = h s t v S,t + (1 − s t ) v L,t + s t (1 − s t ) √ v S,t √ v L,t ρ S,L i dt (A.29) r f = δ + γ t h s t µ S + (1 − s t ) µ L i − η t h s t v S,t + (1 − s t ) v L,t + s t (1 − s t ) √ v S,t √ v L,t ρ S,L i (A.30) with δ > being impatience, γ t ≡ − u ( c t ) c t u ( c t ) > is the coefficient of risk aversion, and η t ≡ u ( c t )( c t ) u ( c t ) > is precautionary saving.Proof. See Appendix B.Naturally, one would assume that jumps affecting the long-term consumption claimwould be larger than those relating to the short-term consumption claim, K L,j > K
S,j with E [ K L,j ] > E [ K S,j ]. Adding to this, one might also assume that the horizon spe-cific intensities would have different mean reversion speeds (i.e. α L,j < α
S,j ) implying51hat intensities exhibit a relatively slower return back to equilibrium in the long-run.This also follows the notion of jump clustering where the network is more receptive toshocks, thereby increasing intensities, and reinforcing the shock propagation mechanism(A¨ıt-Sahalia et al., 2015). It is clear that a jump to either consumption claim reducesconsumption growth. If we wish to examine how the self and mutually exciting jumpsimpact relative consumption shares we can apply Itˆo’s lemma to s t which yields ds t = s t (1 − s t ) (cid:16) µ S − µ L − s t v S,t + (1 − s t ) v L,t + 2 (cid:16) s t − (cid:17) √ v S,t √ v L,t ρ S,L (cid:17) dt + s t (1 − s t ) h √ v S,t dZ S − √ v L,t dZ L i (A.31)The drift of state s t is dependent on: i) current states; ii) current conditional variance ofeach consumption process; and iii) the conditional covariance between the long-term andshort-term parts of consumption. Importantly, horizon specific network connections enterthe conditional variances and therefore influence consumption shares indirectly. If thereare no jumps, then we arrive back at the Cochrane et al. (2007) model with stochasticvolatility. The drift of the long-term consumption claim exhibits what Cochrane et al.(2007) class as “S-shaped mean reversion”. In this case, if s t = (1 − s t ) = dividendshares are most volatile. In our framework the states depend on current levels of variance, v S,t , v
L,t . If s t = , then the terms multiplied by 2 (cid:16) s t − (cid:17) are zero. This means thatcovariance between long-term and short-term consumption claims disappear. B Proofs
B.1 Proofs: Asset pricing with horizon specific volatility con-nections
All proofs for the asset pricing model stem from the first order conditions from therepresentative agent’s utility maximisation problem. This yields an expression for thepricing kernel which we can then combine with the fundamental pricing equation for the k -th asset. After applying Itˆos lemma, we can derive an expression for the risk-free rate,52long with the risk premium of M risky assets in the economy. p k,t u ( c t ) = E t Z ∞ e − δτ u ( c t + τ )d k,t + τ dτ. Λ t ≡ e − δτ u ( c t + τ ) . E t [ d ( p k,t Λ t )] = 0 Proposition 3.
From the fundamental pricing equation we have: r f dt = − E t " d Λ t Λ t r f dt = − E t " − δdt + u ( c t ) c t u ( c t ) dc t c t + 12 u ( c t ) c t u ( c t ) ( dc t ) c t r f dt = δdt − u ( c t ) c t u ( c t ) E t " dc t c t − u ( c t ) c t u ( c t ) E t " ( dc t ) c t r f dt = δdt + γ t E t " dc t c t − η t E t dc t c t ! where δ > γ t ≡ − u ( c t ) c t u ( c t ) > η t ≡ u ( c t )( c t ) u ( c t ) > E t " dc t c t = ( s t µ S + (1 − s t ) µ L ) dt E t dc t c t ! = h s t v S,t + (1 − s t ) v L,t + s t (1 − s t ) √ v S,t √ v L,t ρ S,L i dt Therefore the risk-free rate is given by r f dt = δdt + γ t ( s t µ S + (1 − s t ) µ L ) dt − η t h s t v S,t + (1 − s t ) v L,t + s t (1 − s t ) √ v S,t √ v L,t ρ S,L i dt finally, dividing through by dt delivers the risk-free rate. This completes the proof.53 roposition 1. From the fundamental pricing equation and 3, we can immediately see E t " d Λ t Λ t = − r f dt = − δdt + γ t E t " dc t c t − η t E t dc t c t ! = − δdt − γ t E t " dc t c t + 12 η t E t dc t c t ! inserting the expressions for E t h dc t c t i and E t (cid:20)(cid:16) dc t c t (cid:17) (cid:21) completes the proof. B.2 Proofs: Measurement of dynamic horizon specific networkrisk for large dynamic networks
Proposition 2.
Let us have the VMA( ∞ ) representation of the locally stationary TVPVAR model (Dahlhaus et al., 2009; Roueff and Sanchez-Perez, 2016) X t,T = ∞ X h = −∞ Ψ t,T ( h ) (cid:15) t − h (B.1) Ψ t,T ( h ) ≈ Ψ ( t/T, h ) is a stochastic process satisfying sup ‘ || Ψ t − Ψ ‘ || = O p ( h/t ) for1 ≤ h ≤ t as t → ∞ , hence in a neighborhood of a fixed time point u = t/T the process X t,T can be approximated by a stationary process f X t ( u ) f X t ( u ) = ∞ X h = −∞ Ψ ( u, h ) (cid:15) t − h (B.2)with (cid:15) being iid process with E [ (cid:15) t ] = 0, E [ (cid:15) s (cid:15) t ] = 0 for all s = t , and the local covariancematrix of the errors Σ ( u ). Under suitable regularity conditions | X t,T − f X t ( u ) | = O p ( | t/T − u | +1 /T ).Since the errors are assumed to be serially uncorrelated, the total local covariancematrix of the forecast error conditional on the information at time t − Ω ( u, H ) = H X h =0 Ψ ( u, h ) Σ ( u ) Ψ > ( u, h ) . (B.3)Next, we consider the local covariance matrix of the forecast error conditional on knowl-edge of today’s shock and future expected shocks to k -th variable. Starting from the54onditional forecasting error, ξ k ( u, H ) = H X h =0 Ψ ( u, h ) h (cid:15) t + H − h − E ( (cid:15) t + H − h | (cid:15) k,t + H − h ) i , (B.4)assuming normal distribution of (cid:15) t ∼ N (0 , Σ ), we obtain E ( (cid:15) t + H − h | (cid:15) k,t + H − h ) = σ − kk h Σ ( u ) i · k (cid:15) k,t + H − h (B.5)and substituting (B.5) to (B.4), we obtain ξ k ( u, H ) = H X h =0 Ψ ( u, h ) h (cid:15) t + H − h − σ − kk h Σ ( u ) i · k (cid:15) k,t + H − h i . (B.6)Finally, the local forecast error covariance matrix is Ω k ( u, H ) = H X h =0 Ψ ( u, h ) Σ ( u ) Ψ > ( u, h ) − σ − kk H X h =0 Ψ ( u, h ) h Σ ( u ) i · k h Σ ( u ) i >· k Ψ > ( u, h ) . (B.7)Then h ∆ ( u, H ) i ( j ) k = h Ω ( u, H ) − Ω k ( u, H ) i j,j = σ − kk H X h =0 h Ψ ( u, h ) Σ ( u ) i j,k ! (B.8)is the unscaled local H -step ahead forecast error variance of the j -th component withrespect to the innovation in the k -th component. Scaling the equation with H -step aheadforecast error variance with respect to the j th variable yields the desired time varyinggeneralized forecast error variance decompositions (TVP GFEVD) h θ ( u, H ) i j,k = σ − kk H X h =0 h Ψ ( u, h ) Σ ( u ) i j,k ! H X h =0 h Ψ ( u, h ) Σ ( u ) Ψ > ( u, h ) i j,j (B.9)Next, we derive the frequency representation of the quantity in (B.9) using the factthat unique time varying spectral density of X t,T at frequency ω which is locally thesame as the spectral density of f X t ( u ) at u = t/T can be defined as a Fourier transform Note to notation: [ A ] j,k denotes the j th row and k th column of matrix A denoted in bold. [ A ] j, · denotes the full j th row; this is similar for the columns. A P A , where A is a matrix that denotes thesum of all elements of the matrix A .
55f VMA( ∞ ) filtered series over frequencies ω ∈ ( − π, π ) as S X ( u, ω ) = ∞ X h = −∞ E h X t + h ( u ) X > t ( u ) i e − iωh = n Ψ ( u ) e − iω o Σ ( u ) n Ψ ( u ) e + iω o > , (B.10)where we consider a time varying frequency response function Ψ ( u ) e − iω = P h e − iωh Ψ ( u, h )which can be obtained as a Fourier transform of the coefficients with i = √− H → ∞ , we have time varying generalized forecast error variance decompo-sitions h θ ( u, ∞ ) i j,k = σ − kk ∞ X h =0 h Ψ ( u, h ) Σ ( u ) i j,k ! ∞ X h =0 h Ψ ( u, h ) Σ ( u ) Ψ > ( u, h ) i j,j = AB . (B.11)Starting with frequecy domain counterpart of the nominator A , we will use the stan-dard integral 12 π Z π − π e iω ( r − v ) dω = r = v r = v. (B.12)Using the fact that P ∞ h =0 φ ( h ) ψ ( h ) = π R π − π P ∞ v =0 P ∞ r =0 φ ( r ) ψ ( v ) e iω ( r − v ) dω , we canrewrite (B.11) as 56 − kk ∞ X h =0 h Ψ ( u, h ) Σ ( u ) i j,k ! = σ − kk ∞ X h =0 n X z =1 h Ψ ( u, h ) i j,z h Σ ( u ) i z,k ! = σ − kk π Z π − π ∞ X r =0 ∞ X v =0 n X x =1 h Ψ ( u, r ) i j,x h Σ ( u ) i x,k ! n X y =1 h Ψ ( u, v ) i j,y h Σ ( u ) i y,k e iω ( r − v ) dω = σ − kk π Z π − π ∞ X r =0 ∞ X v =0 n X x =1 h Ψ ( u, r ) e iωr i j,x h Σ ( u ) i x,k ! n X y =1 h Ψ ( u, v ) e − iωv i j,y h Σ ( u ) i y,k dω = σ − kk π Z π − π ∞ X r =0 n X x =1 h Ψ ( u, r ) e iωr i j,x h Σ ( u ) i x,k ! ∞ X v =0 n X y =1 h Ψ ( u, v ) e − iωv i j,y h Σ ( u ) i y,k dω = σ − kk π Z π − π n X x =1 h Ψ ( u ) e iω i j,x h Σ ( u ) i x,k ! n X y =1 h Ψ ( u ) e − iω i j,y h Σ ( u ) i y,k dω = σ − kk π Z π − π (cid:18)h Ψ ( u ) e − iω Σ ( u ) i j,k (cid:19) (cid:18)(cid:16) Ψ ( u ) e iω Σ ( u ) i j,k (cid:19) dω = σ − kk π Z π − π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) Ψ ( u ) e − iω Σ ( u ) (cid:21) j,k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dω (B.13)Hence we have established that A = σ − kk ∞ X h =0 h Ψ ( u, h ) Σ ( u ) i j,k ! = σ − kk π Z π − π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) Ψ ( u ) e − iω Σ ( u ) (cid:21) j,k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dω (B.14)from (B.11), we use the local spectral representation of the VMA coefficients in the secondstep. The rest is a manipulation with the last step invoking the definition of modulussquared of a complex number to be defined as | z | = zz ∗ . Note that we can use thissimplification without loss of generality, because the V M A ( ∞ ) representation that isdescribed by the coefficients Ψ ( u, h ) has a spectrum that is always symmetric.Next, we concentrate on B from (B.11). Using similar steps and the positive semidef-initeness of the matrix Σ ( u ) that ascertains that there exists P ( u ) such that Σ ( u ) = P ( u ) P > ( u ) . X h =0 h Ψ ( u, h ) Σ ( u ) Ψ > ( u, h ) i = ∞ X h =0 h Ψ ( u, h ) P ( u ) ih Ψ ( u, h ) P ( u ) i > = 12 π Z π − π ∞ X r =0 ∞ X v =0 h Ψ ( u, r ) e iωr P ( u ) ih Ψ ( u, v ) e − iωv P ( u ) i > dω = 12 π Z π − π ∞ X r =0 h Ψ ( u, r ) e iωr P ( u ) i ∞ X v =0 h Ψ ( u, v ) e − iωv P ( u ) i > dω = 12 π Z π − π h Ψ ( u ) e iω P ( u ) ih Ψ ( u ) e − iω P ( u ) i > dω = 12 π Z π − π "n Ψ ( u ) e iω o Σ ( u ) n Ψ ( u ) e − iω o > dω (B.15)That establishes the fact that B = ∞ X h =0 h Ψ ( u, h ) Σ ( u ) Ψ > ( u, h ) i j,j = 12 π Z π − π "n Ψ ( u ) e iω o Σ ( u ) n Ψ ( u ) e − iω o > j,j dω (B.16)from (B.11), and we have shown that h θ ( u, ∞ ) i j,k = σ − kk ∞ X h =0 h Ψ ( u, h ) Σ ( u ) i j,k ! ∞ X h =0 h Ψ ( u, h ) Σ ( u ) Ψ > ( u, h ) i j,j = σ − kk Z π − π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) Ψ ( u ) e − iω Σ ( u ) (cid:21) j,k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dω Z π − π "n Ψ ( u ) e iω o Σ ( u ) n Ψ ( u ) e − iω o > j,j dω (B.17)Finally, focusing on a frequency band d = ( a, b ) : a, b ∈ ( − π, π ) , a < b , we have h θ ( u, d ) i j,k = σ − kk Z ba (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) Ψ ( u ) e − iω Σ ( u ) (cid:21) j,k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dω Z π − π "n Ψ ( u ) e iω o Σ ( u ) n Ψ ( u ) e − iω o > j,j dω (B.18)This completes the proof. 58 Estimation of the time-varying parameter VARmodel
To estimate our high dimensional systems, we follow the Quasi-Bayesian Local-Liklihood(QBLL) approach of Petrova (2019). let X t be an N × p lags: X t,T = Φ ( t/T ) X t − ,T + . . . + Φ p ( t/T ) X t − p,T + (cid:15) t,T , (C.1)where (cid:15) t,T = Σ − / ( t/T ) η t,T with η t,T ∼ N ID (0 , I M ) and Φ ( t/T ) = ( Φ ( t/T ) , . . . , Φ p ( t/T )) > are the time varying autoregressive coefficients. Note that all roots of the polynomial, χ ( z ) = det (cid:16) I N − P Lp =1 z p B p,t (cid:17) , lie outside the unit circle, and Σ − t is a positive definitetime-varying covariance matrix. Stacking the time-varying intercepts and autoregressivematrices in the vector φ t,T with ¯ X t = ( I N ⊗ x t ) , x t = (cid:16) , x t − , . . . , x t − p (cid:17) and ⊗ denotesthe Kronecker product, the model can be written as: X t,T = ¯ X t,T φ t,T + Σ − t/T η t,T (C.2)We obtain the time-varying parameters of the model by employing Quasi-Bayesian LocalLikelihood (QBLL) methods. Estimation of (C.1) requires re-weighting the likelihoodfunction. Essentially, the weighting function gives higher proportions to observationssurrounding the time period whose parameter values are of interest. The local likelihoodfunction at time period k is given by:L k (cid:16) X | θ k , Σ k , ¯ X (cid:17) ∝ | Σ k | trace( D k ) / exp (cid:26) −
12 ( X − ¯ X φ k ) ( Σ k ⊗ D k ) ( X − ¯ X φ k ) (cid:27) (C.3)The D k is a diagonal matrix whose elements hold the weights: D k = diag( % k , . . . , % kT ) (C.4) % kt = φ T,k w kt / T X t =1 w kt (C.5) w kt = (1 / √ π ) exp(( − / k − t ) /H ) ) , for k, t ∈ { , . . . , T } (C.6) ζ T k = T X t =1 w kt ! − (C.7)59here % 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 (C.6),and it is the kernel function that provides greater weight to observations surrounding theparameter estimates at time k relative to more distant observations.Using a Normal-Wishart prior distribution for φ k | Σ k for k ∈ { , . . . , T } : φ k | Σ k (cid:118) N (cid:16) φ k , ( Σ k ⊗ Ξ k ) − (cid:17) (C.8) Σ k (cid:118) W ( α k , Γ k ) (C.9)where φ k is a vector of prior means, Ξ k is a positive definite matrix, α k is a scaleparameter of the Wishart distribution ( W ), and Γ k is a positive definite matrix.The prior and weighted likelihood function implies a Normal-Wishart quasi posteriordistribution for φ k | Σ k for k = { , . . . , T } . Formally let A = (¯ x , . . . , ¯ x T ) and Y =( x , . . . , x T ) then: φ k | Σ k , A , Y (cid:118) N (cid:18) ˜ θ k , (cid:16) Σ k ⊗ ˜Ξ k (cid:17) − (cid:19) (C.10) Σ k (cid:118) W (cid:16) ˜ α k , ˜Γ − k (cid:17) (C.11)with quasi posterior parameters˜ φ k = (cid:16) I N ⊗ ˜Ξ − k (cid:17) h ( I N ⊗ A D k A ) ˆ φ k + ( I N ⊗ Ξ k ) φ k i (C.12) ˜Ξ k = ˜Ξ k + A D k A (C.13)˜ α k = α k + T X t =1 % kt (C.14) ˜Γ k = Γ k + Y D k Y + Φ k Γ k Φ k − ˜Φ k ˜Γ k ˜Φ k (C.15)where ˆ φ k = ( I N ⊗ A D k A ) − ( I N ⊗ A D k ) y is the local likelihood estimator for φ k . Thematrices Φ k , ˜Φ k are conformable matrices from the vector of prior means, φ k , and adraw from the quasi posterior distribution, ˜ φ k , respectively.The motivation for employing these methods are threefold. First, we are able toestimate 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 / N ( p + 1 / p =2 and a Minnesota Normal-Wishart prior with ashrinkage value ϕ = 0 .
05 and centre the coefficient on the first lag of each variable to 0.1 ineach respective equation. The prior for the Wishart parameters are set following Kadiyalaand Karlsson (1997). For each point in time, we run 500 simulations of the model togenerate the (quasi) posterior distribution of parameter estimates. Note we experimentwith various lag lengths, p = { , , , } ; shrinkage values, ϕ = { . , . , . } ; andvalues to centre the coefficient on the first lag of each variable, { , . , . , . }}