Firm Networks and Asset Returns

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Firm Networks and Asset Returns Carlos Ramirez 2017-014 Please cite this paper as: Ram´ırez, Carlos (2017). “Firm Networks and Asset Returns,” Finance and Economics DiscussionSeries2017-014. Washington: BoardofGovernorsoftheFederalReserveSystem, https://doi.org/10.17016/FEDS.2017.014r1. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Firm Networks and Asset Returns CARLOS RAMIREZ ∗ February 1, 2018 ∗Board of Governors of the Federal Reserve System. This paper formerly circulated as “Inter-Firm RelationshipsandAssetPrices.” I thankSenayAgca;FernandoAnjos; CelsoBrunetti; ElenaCarletti;FranciscoCisternas;NathanFoley-Fisher;NicolaGennaioli;StefanGissler;BrentGlover;RichardGreen;Anisha Ghosh;BenjaminHolcblat;SteveKarolyi;RobertKieschnick;YongjinKim;MeteKilic;BorghanNarajabad; Artem Neklyudov; Silvio Petriconi; Fulvio Ortu; Emilio Osambela; Ioanid Rosu; Doriana Ruffino; Stefano Sacchetto;AlessioSaretto;JulienSauvagnat;FabianoSchivardi;DuaneSeppi;TatsuroSenga;ChesterSpatt; Claudio Tebaldi; St´ephane Verani; Hannes Wagner; Malcolm Wardlaw; Ariel Zetlin-Jones; an anonymous referee; and seminar participants at the 15th Trans-Atlantic Doctoral Conference at LBS, Carnegie Mellon, INFORMS, Bocconi, IESE, University of Texas at Dallas, Federal Reserve Board, Cornerstone Research, CentralBankofChile,the2016Portsmouth-FordhamConferenceinBankingandFinance,the5thCIRANO- Walton Conference on Networks, Luxembourg School of Finance, ASSET 2016, PUC Chile (FinanceUC), the 2017WarwickFrontiersofFinance Conference,the 2017EuropeanEconomicAssociationMeetings,and the 2017 Northern Finance Association Meetings for their valuable suggestions. I am especially grateful to BurtonHollifield,BryanRoutledgeandR.Ravifortheirhelpfuldiscussions. AliceMooreprovidedexcellent research assistance for some of the empirical sections in this paper. All remaining errors are my own. This article represents the view of the author, and should not be interpreted as reflecting the views of the Board of Governorsof the Federal Reserve System or other members of its staff. E-mail: carlos.ramirez@frb.gov.

Firm Networks and Asset Returns ABSTRACT Thispaperarguesthatchangesinthepropagationofidiosyncraticshocksalongfirmnetworks are important to understanding variations in asset returns. When calibrated to match key features of supplier–customer networks in the United States, an equilibrium model in which investors have recursive preferences and firms are interlinked via enduring relationships generates long-run consumption risks. Additionally, the model matches cross-sectional patterns of portfolio returns sorted by network centrality, a feature unaccounted for by standard asset pricing models.

Firms do not function as isolated entities. Instead, they are interlinked via a variety of material relationships, such as strategic alliances, joint ventures, research and development (R&D) partnerships, and supplier–customer relationships. As shown by recent empirical evidence, these relationships may serve as propagation mechanisms of shocks to individual firms and, in doing so, potentially alter asset returns.1 Despite this evidence, the asset pricing implications of such shock propagation remains, at best, imperfectly understood. In this paper, I develop an equilibrium model to study the asset pricing properties that stem from the propagation of idiosyncratic shocks along firm networks and the extent to which such shock propagation quantitatively explains asset market phenomena. I show that changes in the propagation of idiosyncratic shocks along firm networks are importanttounderstanding variationsinasset returns, bothintheaggregateandinthecross section. In particular, the model generates long-run consumption risk when calibrated to match key characteristics of supplier–customer networks in the United States. Consequently, the model replicates prime characteristics of asset market data, such as a high and volatile risk premium and a low and stable risk-free rate. Additionally, the model matches crosssectional patterns of portfolio returns sorted by network centrality. The model has two main features. First, idiosyncratic shocks propagate via long-lasting relationships. As a consequence, firms’ cash-flow growth rates are related via a firm network. Second, investors have a preference for early resolution of uncertainty and, thus, care about uncertainty regarding firms’ long-term growth prospects. Aside from aggregate shocks, the distribution of aggregate consumption growth is shaped by two characteristics within the model: (a) the topology of the firm network and (b) the propensities of relationships to transmit idiosyncratic shocks, henceforth referred to as propensities. Propensities are assumed to vary over time. Such variation captures temporal changes in relationship-specific characteristics that make firms more susceptible to shocks 1See Hertzel et al. (2008), Jorion and Zhang (2009), Boone and Ivanov (2012), Carvalho et al. (2014), Boyarchenkoand Costello (2015), Todo et al. (2015), Boehm et al. (2015)and Barrotand Sauvagnat(2016) among others. Using French firm-level data from 1990 to 2007, Di Giovanni et al. (2014) provide empirical evidence of the importance of firm-specific shocks in generating aggregate fluctuations. 3

affecting their neighbors. As propensities vary over time, the connectivity of the firm network also varies over time. This variation introduces a time-varying correlation structure among firms’ cash-flow growth rates, which in equilibrium generates stochastic volatility in consumption growth. In the calibrated model, changes in network connectivity are infrequent because firms tend to engage in enduring and stable relationships with their major customers. Then, the nature of these relationships generates long-lasting interdependencies among firms’ cash-flow growth rates. In such an economy, idiosyncratic shocks to one firm have the potential not only to change the current cash flows of every neighboring firm, but also to change the longterm growth prospects of all such firms. Such infrequent changes in network connectivity are what fundamentally drive low-frequency movements in aggregate output growth, which, in equilibrium, generate apersistent component inexpected aggregateconsumption growth. As aresultofinvestorshavingpreferencesforearlyresolutionofuncertainty, themodelgenerates long-run consumption risks. The model accounts for sizable risk premiums because investors fear that extended periods of low economic growth coincide with low asset prices. The model generates a small risk-free rate as a result of investors saving for long periods of low economic growth. Besidesgeneratinglong-runconsumptionrisk, thecalibratedmodelmatchescross-sectional patterns of portfolio returns sorted by network centrality. Central firms command lower risk premiums than peripheral firms because, in the data, relationships of peripheral firms tend to exhibit higher propensities than relationships of central firms, as peripheral firms tend to rely more heavily on their major customers. As a consequence, central firms are less exposed to contagion risk than peripheral firms, commanding lower risk premiums. The model generates a realistic monthly return spread of 0.8% between firms in the lowest and firms in the highest decile of centrality. This economically and statistically significant return spread arises naturally in equilibrium as compensation for contagion risk, a feature unaccounted for by standard asset pricing models. 4

The small and persistent component in expected consumption growth generated by lowfrequency movements in network connectivity provides an equilibrium foundation for longrun risk models in the spirit of Bansal and Yaron (2004). Moreover, the model helps explain the cross section of expected returns, as it provides a mapping between firms’ importance in the network and their contagion risk. Overall, these results suggest that extending standard asset pricing models to take into account how idiosyncratic shocks propagate along firm networks can make significant progress toward generating a unifying framework that simultaneously captures dynamics of the aggregate and the cross section of stock returns. This paper contributes to three strands of the literature. First, the paper develops a new theoretical framework that adds to a growing body of work focused on understanding the effects of economic linkages in asset pricing, for example, Buraschi and Porchia (2012), Ahern (2013), and Herskovic (2017).2 Unlike these papers, however, this model emphasizes relationships at the firm level to explore the asset pricing properties that stem from the propagation of idiosyncratic shocks along firm networks. Second, this paper adds to a body of work that explores how granular shocks may lead to aggregate fluctuations in the presence of linkages among different sectors of the economy, for example, Carvalho (2010), Gabaix (2011), Acemoglu et al. (2012, 2015), Oberfield (2013), Carvalho and Gabaix (2013), Blume et al. (2013), Elliott et al. (2014), Chaney (2014, 2016), and Lim (2016). This paper contributes to this literature by exploring the asset pricing implications of linkages at the firm level and studying how changes in the propagation of idiosyncratic shocks affect not only aggregate variables but also asset returns and aggregate risk premia. Third, this paper adds to recent research that examines the potential sources of long-run risks, for example, Kaltenbrunner and Lochstoer (2010), Kung and Schmid (2015), Bidder 2Buraschi and Porchia (2012) show that more central firms in a market-based network have lower price dividendratiosandhigherexpectedreturns. Usingthenetworkofintersectoraltrade,Ahern(2013)provides evidence that firms in more central industries have greater exposure to systematic risk. Herskovic (2017) focuses on efficiency gains that come from changes in the input-output network and how those changes are priced in equilibrium. My paper, on the other hand, focuses on how changes in the propagation of shocks within a fixed network alter equilibrium asset prices and risk premia. 5

and Dew-Becker (2016), and Collin-Dufresne et al. (2016).3 This paper contributes to this literature by showing that changes in the propagation of idiosyncratic shocks along firm networks can generate long-run risks. I. Baseline Model Though stylized, the baseline model conveys the main intuition for how changes in the propagation of idiosyncratic shocks along firm networks, in combination with recursive preferences, generates long-run consumption risks and implications for the cross section of asset returns. To facilitate exposition, the baseline model abstracts from firms’ production decisions and considers a single-good economy in which firm cash flows are related via a network of long-lasting relationships. Internet Appendix A shows that under some conditions, the main intuition continues to hold within an equilibrium framework where production is explicitly modeled. A. The environment Consider an economy with one perishable good and an infinite time horizon. Time is discrete and indexed by t 0,1,2, . In each period, the single good is produced by n ∈ { ···} infinitely lived firms, with n being potentially large. Firms’ outputs, henceforth cash flows, are related via a network of long-lasting relationships.4 Because I focus on the effect of the firm network on asset returns rather than on strategic network formation, relationships are 3Kaltenbrunner and Lochstoer (2010) shows that long-run risks endogenously arise in a standard production economy model, even when technology growth is i.i.d., because of consumption smoothing. Kung andSchmid(2015)showsthatamodelofendogenousinnovationandR&Disabletogeneratelong-runrisks, while Bidder and Dew-Becker (2016) shows that long-run risks arise in an economy in which investors are pessimistic and not sure about the true model driving the economy. Collin-Dufresne et al. (2016) shows that parameter learning generates long-lasting risks that help explain standard asset pricing puzzles in an economy where investors are uncertain about the structural parameters governing the model economy. 4Long-lasting relationships potentially allow firms to circumvent difficulties in contracting due to unforeseen contingencies, asymmetries of information, and specificity on firms’ investments, for example, Williamson (1979, 1983). 6

assumed to be exogenously determined and fixed before t = 0.5 Besides firms, there is a largenumber ofidentical, infinitely lived individuals whoareaggregatedintoarepresentative investor with Epstein-Zin-Weil preferences who owns all assets in the economy. B. The firm network and firms’ cash flows Firms’ cash flows vary stochastically over time and depend on aggregate and firm-level shocks. Input, labor, and capital decisions are deliberately normalized to 1. Firm i’s cash flow at t, y , follows i,t y i,t log a +z , i 1, ,n , (1) t i,t Y ≡ ∈ { ··· } (cid:18) t−1(cid:19) e e where Y denotes the aggregate output of the economy at t 1, a d i.i.d. (0,2σ2) is a t−1 − t −→ N a shock that affects all firms in the economy at t, and z is a shock that affects firm i at t. it e A key feature of this model is that the firm network determines the dependence structure e among shocks to individual firms. In particular, long-lasting relationships have a dual nature within the model. While relationships may increase firms’ growth opportunities via efficiency gains, relationships may also have additional consequences as they increase a firm’s reliance on its neighbors and, thus, increase a firm’s exposure to negative idiosyncratic shocks affecting a broader set of firms in the economy. To capture such a trade-off, z is assumed to i,t+1 follow e z = α d α ε , i 1, ,n , (2) i,t+1 1 i 2 i,t+1 − ∈ { ··· } e e whereparametersα andα arenon-negativeandequalacrossfirms. Parameter d represents 1 2 i the number of direct relationships of firm i—which may differ across firms. Uncertainty on z isintroducedby aBernoulli randomvariableε . If firmiiseither directly affectedby i,t+1 i,t+1 5See Demange and Wooders (2005), Goyal (2007), and Jackson (2008) for a detailed description of e e network formation models. For models of endogenous formation of production networks, see Oberfield (2013), Chaney (2014, 2016), and Lim (2016), among others. 7

an idiosyncratic shock or affected by an idiosyncratic shock that affects one of its neighbors, then ε = 1. Otherwise, ε = 0. i,t+1 i,t+1 To simplify the modeling, the distribution of ε is determined by the following stochasi,t+1 e e tic process—which abstracts from the temporal propagation of idiosyncratic shocks. At the e beginning of t+1, each firm faces a negative shock independently of other firms with probability 0 < q < 1, which is equal across firms and time invariant. A negative idiosyncratic shock to firm i at t + 1 also affects firm j at t + 1, and, thus, ε = ε = 1 if two i,t+1 j,t+1 things happen: (1) there exists a sequence of relationships that connects i and j in the firm e e network and (2) each relationship in that sequence transmits shocks at t+1.6 The relationship between firms i and j either transmits shocks at t+1 or does not, independently of all other relationships, with probability p . For simplicity, relationships are assumed to be ij,t+1 undirected, and, thus, p = p , (i,j), t. Consequently, p measures the propensity ijt jit ij,t+1 ∀ e ∀ of relation (i,j) to transmit idiosyncratic shocks from firm i (j) to j (i) at t+1.7 e e e At a fundamental level, the value of p captures interdependencies between the cash ij,t+1 flows offirmiandfirmj att+1. Such interdependencies, which cannot bemitigatedthrough e 6Withinthebaselinemodel,onlynegativeidiosyncraticshocksareallowedtopropagateinaprobabilistic manner. However, the baseline model can be easily extended to allow positive and negative shocks to propagate along the network. To do so, define ψ ε 1/2 so that shocks can be positive and i,t+1 i,t+1 ≡ − negative. Then, redefine equation (2) so that e e z = α d α ψ i,t+1 1 i 2 i,t − = α /2+α d α ε , 2 1 i 2 i,t+1 e e − which is similar to equation (2). The cross sectional results in this paper continue to hold as long as the e decrease in firms’ cash flow growth due to negative shocks is larger than the increase in firms’ cash flow growth due to positive shocks. 7Ineachperiod,thisstochasticprocesscanbethoughtofasavariationofeitherareliabilitynetworkora bondpercolationmodel. Inatypicalreliabilitynetworkmodel,theedgesofagivennetworkareindependently removed with some probability. The remaining edges are assumed to transmit a message. A message from node i to j is transmitted as long as there is at least one path from i to j after edge removal—see Colbourn (1987) for more details. Similarly, in a bond percolation model, edges of a given network are removed at randomwith some probability. Edges that are notremovedareassumedto percolate a liquid. The question in percolation is whether the liquid percolates from one node to another in the network—which is similar to the problem of transmitting a message in a reliability context. For more details, see Grimmett (1989), StaufferandAharony(1994),andNewman(2010,Chapter16.1). Blume etal. (2013)analyzeapropagation mechanism similar to the one analyzed here. They focus, however, on strategic network formation issues in a static environment. They provide asymptotic bounds on the welfare of both optimal and stable networks and show that smallamounts of“over-linking”may impose largelosses in welfare to networks’participants. 8

contractual protections, may be driven by the characteristics of the relationship between i and j. Intuitively, the higher the value of p , the higher the likelihood that disruptions ij,t+1 affecting the cash flow of firm i (j) also affect the cash flow of firm j (i) at t+1.8 e Probabilities p are drawn from a Beta distribution with parameters ζ > 0 { ij,t+1 }(i,j) 1,t+1 and ζ > 0 at the beginning of period t+1. Parameters ζ > 0, i = 1,2 , which are 2,t+1 i,t+1 e { } drawn prior to drawing from the Beta distribution, determine the shape of the distribution of propensities across relationships at period t+1. The model timeline at period t is depicted in figure 1. Period t Relationships that ζ and ζ Shocks 1t 2t transmit shocks are drawn propagate are determined p Firm-specific shocks { ijt }(i,j) ∈ Gn are drawn from β(ζ ,ζ ) are realized 1t 2t e Figure 1. Model timeline in period t. n C. Propagation of idiosyncratic shocks and the distribution of ε { i,t }i=1 To fix notation, let denote the network of relationships among n firms—where nodes n e G represent firms and edges represent relationships. Given how idiosyncratic shocks propagate along the network, the joint distribution of ε n is determined by , q, and the process { i,t }i=1 G n driving the stochastic propensity matrix p [p ] . The marginal distribution of ε , t e≡ ijt (i,j) i,t conditional on p , depends on q, the network , and the location of firm i in . In other t n n e G e G e 8In the contexet of supply chains, p may capture restrictions on firm i’s and j’s use of alternative ij,t+1 inputs at t+1. The higher the value of p , the higher the switching costs firms i or j may face at t+1 ij,t+1 and, thus, the higher the likelihood theat a negative shock to firm i (j) also affects firm j (i), provided that firm j (i) may not be able to restructure its production sufficiently fast to overcome firm i (j)’s disruption e in production. 9

words, P ε = 1 p = f (q, ,location of firm i in ), i,t t n n G G (cid:0) (cid:12) (cid:1) e (cid:12)e where P ε = 0 p = 1 P ε = 1 p , and f( ) is a mapping characterized by the stochasi,t t i,t t − · tic proce(cid:0)ss descr(cid:12)ibe(cid:1)d in sect(cid:0)ion I.B,(cid:12) w(cid:1)hich generates a time-varying correlation structure e (cid:12)e e (cid:12)e among firms’ cash-flow growth as p varies over time. t Despite the fact that the mapping f( ) is hard to characterize for large n, its properties e · are easy to describe given the formulation of the stochastic process that generates it. First, in the absence of relationships, P ε = 1 p = P (ε = 1) = q , i and t, so firm-level i,t t i,t ∀ ∀ shocks are independent and ident(cid:0)ically di(cid:12)str(cid:1)ibuted across firms over time. Second, if only e (cid:12)e e one sequence of relationships exists between two firms, the longer the sequence, the smaller the correlation between firm-level shocks.9 D. Temporal changes in shock propagation Tocapturetemporalchangesinrelationship-specific characteristics, theshapeparameters ζ , i = 1,2 , are allowed to vary over time. Variation in the shape parameters may arise it { } from changes in complementarities among firms’ activities or the arrival of new technologies that reshape the economy’s long-term growth prospects. For simplicity, ζ takes two values, it ζ or ζ , with ζ < ζ , and the shape parameter vector ζ [ζ ζ ] follows a four-state iL iH iL iH t 1t 2t ≡ ergodic Markov process with transition matrix Ω and states ζ [ζ ζ ], ζ [ζ ζ ], LL 1L 2L LH 1L 2H ≡ ≡ ζ [ζ ζ ], and ζ [ζ ζ ].10 HL 1H 2L HH 1H 2H ≡ ≡ 9Havingthisfeature—whichissometimescalledcorrelationdecay,asin,forexample,Gamarnik(2013)— greatly helps obtain numerical solutions of the model relatively fast when n is large. 10The main results are robust to variations in the number of values that ζ can take. In unreported it results, I allow ζ to take K values, with K = 3,4,5 , and, hence, the vector ζ follows a 9-, 16-, and 25it t { } ergodic Markov process. In all those cases, the main results continue to hold. 10

II. Aggregate Consumption Growth Aside from aggregate shocks, two features of the model are important to understanding the distribution of aggregate consumption growth: (a) the topology of and (b) how n G idiosyncratic shocks propagate along , captured by the propensity matrix p and its dyn t G namics. In this section, I study how changes in these two features affect the distribution of e aggregate consumption growth and, thus, alter the distribution of the pricing kernel. Let ∆c log Ct+1 andx log Yt+1 denote log consumption and output growth t+1 ≡ Ct t+1 ≡ Yt (cid:16) (cid:17) (cid:16) (cid:17) at t+1, respectively. Rather than assuming that aggregate consumption is the dividend on e e theportfolioofallinvestedwealth, IfollowCampbell(1986),Cecchettietal.(1993),andAbel (1999), and make the slightly more general assumption that the dividend on the aggregate stock market equals aggregate consumption raised to a power. Thus ∆c and x satisfy t+1 t+1 e e 1 x = ∆c , (3) t+1 t+1 τ (cid:18) (cid:19) e e where τ is a constant. Hence, the representative investor is assumed to have access to labor income. As in Abel (1999), (1/τ) represents the leverage ratio on equity. If τ = 1, then the market portfolio is a claim to total wealth. For tractability, consider Y n y 1/n . It then t ≡ i=1 i,t follows from equations (1), (2), and (3) that Q n 1/n y i,t+1 ∆c = τx = τ log t+1 t+1 Y i=1(cid:18) t (cid:19) ! Y e e n n 1 1 = τ a +α d α ε  t+1 1 i 2 i,t+1  n − n ! ! i=1 i=1 X X   e e  = τ a +α | d{¯z } α | W{z } , (4) t+1 1 2 n,t+1 − (cid:16) (cid:17) e f ¯ where d denotes the average number of relationships per firm in the economy, whereas W n,t+1 denotes the average number of firms affected by idiosyncratic shocks at t + 1. It follows f 11

from equation (4) that the distribution of ∆c critically depends on W . Given how t+1 n,t+1 idiosyncratic shocks propagate along the network, the distribution of W is affected by e fn,t+1 p and the topology of . As a result, these two features affect the distribution of ∆c . t+1 G n f t+1 To appreciate the importance of p and the topology of in determining the distribut+1 n e G e tion of ∆c , consider two cases. First, suppose there are no relationships. Then, ε n t+1 e { i,t+1 }i=1 is a sequence ofi.i.d. Bernoulli randomvariables and nW follows a Binomial distribution. n,t+1 e e By the Central Limit Theorem (CLT), √n(W q) is normally distributed as n grows n,t+1 − f large. Provided the absence of relationships, the matrix p is irrelevant to determining the f t+1 distribution of ∆c , as the unconditional mean and variance of √nW are q and q(1−q), t+1 e n,t+1 n respectively. Second, suppose every firm has two relationships and each relationship has f e propensity p, which does not vary over time. Then, ε n is a sequence of dependent { i,t+1 }i=1 Bernoulli random variables and nW approximately follows a Binomial distribution if p n,t+1 e is sufficiently small—see Soon (1996). In this case, p affects the distribution of consumption f growth, as the unconditional mean and variance of W are approximately π and π(1−π), n,t+1 n respectively, where π [0,1] solves the following equation: ∈ f π = q +(1 q)πp(πp+2[p(1 π)+π(1 p)]). − − − Despite the fact that W is the aggregation of shocks to individual firms, there is n,t+1 no guarantee that ∆c is normally distributed, as ε n is a sequence of dependent t+1 f { i,t+1 }i=1 Bernoulli random variables in the presence of relationships. Figure 2 illustrates the previous e e point. Figure2(a)depicts a star network inaneconomy withn = 5firms, whereas figure 2(b) depicts the empirical probability density function of W for the star network depicted n,t+1 in figure 2(a). As figure 2(b) shows, the distribution of W may differ from a normal f n,t+1 distribution if the elements of the matrix p are sufficiently close to 1. In particular, as t+1 f some components in p tend toward one, the distribution of W tends to be bimodal.11 t+1 n,t+1 e 11For a large variety of network topologies, simulation shows that the distribution of ∆c may differ e f t+1 from a normal distribution. In particular, if some elements of the matrix p are sufficiently close to one t+1 and is locally connected—i.e., there is at least one sequence of relationships between any two firms in an n e G e 12

Despite the existence of relationships—and the convoluted dependencies they may generate among firm-level shocks—the topology of and matrix p can be restricted so that (1) n t+1 G the distribution of W can be approximated by well-known distributions, and (2) ∆c n,t+1 t+1 e is normally distributed as the economy grows large. If ∆c is normally distributed, keepf t+1 e ing track of temporal changes in the distribution of ∆c is equivalent to keeping track of t+1 e temporal changes in averages and standard deviations. Then, the dynamics of consumption e growth can be recast as a version of Hamilton (1989) Markov-switching model. Internet Appendix B provides conditions under which W follows a Poisson distribution when n is n,t+1 finite and conditions under which W follows a normal distribution when n grows large. n,t+1 f f III. Asset Pricing To see what and ζ imply for asset returns, I embed the output correlation structure n t G generated by the firm network into a standard asset pricing framework. The representative investor has Epstein-Zin-Weil recursive preferences to account for asset pricing phenomena that are challenging to address with power utility preferences. The asset pricing restrictions on the gross return of firm i, R , are i,t+1 e E M R = 1, (5) t t+1 i,t+1 (cid:16) (cid:17) f e 1−γ 1−γ−1 whereM β e∆ect+1 −ρ 1−ρ R 1−ρ representsthepricingkernelatt+1andR t+1 a,t+1 a,t+1 ≡ h i h i denotes the gross(cid:0)return(cid:1)on aggregate wealth—an asset that delivers aggregate consumption f e e as its dividend each period. To solve the model, I look for equilibrium asset prices so that price–dividend ratios are stationary, as in Mehra and Prescott (1985), Weil (1989), and Kandel and Stambaugh (1991). Because equilibrium values are time-invariant functions of the state of the economy, arbitrarilylargeneighborhoodaroundanygivenfirm—thenanon-negligiblefractionoffirmsintheeconomy are almostsurely affected by negative shocks. Therefore, the distribution of ∆c may exhibit thicker tails t+1 than a normal distribution would. e 13

which is determined by the state of the vector ζ , index t can be eliminated. Hereinafter, t s LL,LH,HL,HH denotes the current state of vector ζ. ∈ S ≡ { } The expected gross return of aggregate wealth in the current state is (see Appendix A for detailed derivations) E (R a | s) = eτ(α1d¯+τσ a 2) ω s,s′ w s a w ′ + a 1 E e−τα2W f n,t+1 s′ , (6) s X ′∈S (cid:18) s (cid:19) (cid:16) (cid:12) (cid:17) (cid:12) where wa is the current price of aggregate wealth and is the solution of the following system s of equations, 1−ρ 1−γ w s a = βeτ(1−ρ)(τ(1−γ)σ a 2+α1d¯) ω s,s′ E e−τ(1−γ)α2W f n,t+1 s′ (w s a ′ +1) 1 1 − − γ ρ . ! s X ′∈S (cid:16) (cid:12) (cid:17) (cid:12) Itfollowsfromtheaboveequationsthattheexpectedreturnandpriceofaggregatewealth are affected by (a) aggregate shocks, parameterized by σ2, (b) the topology of , which a G n ¯ determines d; and (c) the dynamics of ζ , which jointly with , determines the distribution t n G of W . The dynamics of ζ , parameterized by Ω, affect the price and the expected return n,t+1 t of aggregate wealth, as Ω determines the persistence of changes in network connectivity. f Next, I consider the risk-free asset, which pays one unit of the consumption good during the next period with certainty. If R (s) denotes the gross return of the risk-free asset in the f current state, then R (s) solves f ρ−γ R 1 (s) = β 1 1 − − γ ρe−τγ(α1d¯−τγσ a 2) ω s,s′ E eτγα2W f n,t+1 s′ w s a w ′ + a 1 1−ρ . (7) f s X ′∈S (cid:16) (cid:12) (cid:17) (cid:18) s (cid:19) ! (cid:12) Therefore, the equilibrium risk-free rate is also driven by aggregate shocks, the topology of , and the dynamics of ζ , as these three features affect the distribution of W and prices n t n,t+1 G of aggregate wealth. f I now study the cross section of expected asset returns. To do so, it is convenient to express as the union of connected components, which are sets of firms connected via n G 14

at least one sequence of relationships. If i denotes the connected component that firm i Gn belongs to, then can be written as n G i. G n ≡ Gn i [ ∈Gn Define the following averages, 1 1 Wi ε and W−i ε , n,t+1 ≡ n  j,t+1  n,t+1 ≡ n  j,t+1  j X ∈G n i j∈ X Gn\G n i f  e  f  e  where Wi represents the average number of firms in i affected by idiosyncratic shocks n,t+1 Gn at t+1, whereas W−i represents the average number of firms in i (the complement f n,t+1 G n \Gn set of i) affected by idiosyncratic shocks at t+1. If v (s) denotes the current state-price Gn f i of firm i, then the expected gross return of firm i is given by E R i,t+1 s = e(1/τ)((1/τ)σ a 2+α1d¯) ω s,s′v i (s′) E e−(1/τ)α2W f n,t+1 s′ v (s) (cid:16) (cid:12) (cid:17) i s X ′∈S (cid:16) (cid:12) (cid:17) ! e (cid:12) + eσ a 2+α1di ω s,s′ E e−α2εe i,t+1 s′ , (cid:12) (8) v (s) i s′∈S ! X (cid:0) (cid:12) (cid:1) (cid:12) where v (s) solves i ρ−γ v i (s) = β 1 1 − − γ ρe((1/τ)−γ)2σ a 2+α1((1/τ)−γ)d¯ ω s,s′ w s a ′ +1 1−ρ E e−((1/τ)−γ)α2W f n,t+1 s′ v i (s′) wa s X ′∈S (cid:18) s (cid:19) (cid:16) (cid:12) (cid:17) ! + β 1 1 − − γ ρe(1+γ2)σ a 2+α1(di−γd¯) ω s,s′π i (s′)π −i (s′) w s a ′ +1 ρ 1 − − γ ρ , (cid:12) wa s′∈S (cid:18) s (cid:19) ! X with π i (s′) E eα2γ(W f n i ,t+1 −εe i,t+1) s′ and π −i (s′) E eα2γW f n − ,t i +1 s′ . ≡ ≡ (cid:18) (cid:12) (cid:19) (cid:18) (cid:12) (cid:19) To appreciate how firms’ conn(cid:12)ectivity affects expected return(cid:12)s, suppose all firms have (cid:12) (cid:12) (cid:12) (cid:12) the same number of relationships and relationships have the same propensity to transmit idiosyncratic shocks. In this case, the connectivity of any firm is equal to the connectivity 15

of any other firm. Consequently, state-prices are equal across firms, as d = d ¯ , P [ε = i i,t+1 1 s] = p, π = π, and π = π′ i. Therefore, cross-sectional differences in state-prices and i −i | ∀ e expected returns arise solely from differences in firms’ connectivity. As shown in (8), firm i’s state-price and expected return are altered by how frequently other firms are affected by idiosyncratic shocks. To see this effect more clearly, consider two cases. First, consider firms within the same connected component as firm i. If the average number of firms affected by idiosyncratic shocks in i increases, but the likelihood Gn that ε = 1 does not, π increases and, thus, v increases. v increases because firm i is i,t+1 i i i less vulnerable to idiosyncratic shocks affecting firms within the same connected component. e Consequently, firm i’s expected return decreases as a result of the decrease in exposure to contagion risk. Second, consider firms in connected components that are different from the one that firm i belongs to. The higher the average number of firms affected by idiosyncratic shocks in i, the higher π and, thus, the higher v . v increases because firms in i are G n \Gn −i i i Gn not vulnerable to idiosyncratic shock affecting firms in i and, thus, they serve investors G n \Gn to improve their portfolio diversification. Consequently, firm i’s expected return decreases as a result of gains in diversification. Therefore, firms’ expected returns are affected by (a) firms’ vulnerability to idiosyncratic shocks that affect other firms within the same connected component and (b) how frequently firms in other connected components are affected by idiosyncratic shocks. IV. Calibration So far, the model illustrates how changes in the propagation of idiosyncratic shocks along afirmnetworkpotentiallyalterequilibriumassetpricesandexpectedreturns. Inowcalibrate the model to match several features of supplier–customer networks in the United States and explore the extent to which the model quantitatively explains asset market phenomena. Section IV.A describes the data. Section IV.B describes the strategy employed to calibrate 16

and ζ . Section IV.C describes the selection of the rest of the parameters in the model. n t G A. Data A.1. Material relationships among U.S. public firms I use annual data on relationships among U.S. public firms and their major customers to identify material relationships. The Statement of Financial Accounting Standards (SFAS) No.131 requires public firms to report information about customers who represent more than 10% of their annual revenues or sales; firms sometimes report customers below the 10% threshold. Reported customers’ information is available on the COMPUSTAT Segment files. However, sometimes customers’ names are abbreviated inconsistently over time. For these cases, I use a string-matching algorithm, similar to the one used by Atalay et al. (2011), which generates a list of potential customers in COMPUSTAT.12 I then select the best match by inspecting a firm’s name and industry information. The dataset spans from 1976 to 2016 and consists of 8,779 different public firms. Similar to Barrot and Sauvagnat (2016), I consider firms i and j to be connected in all years ranging from the first to the last year that i reports j as one of its major customers. This assumption yields 66,355 unique annual supplier–customer relationships. Table I reports the distribution of firms across major industry groups. More than 65% of companies in the sample are classified as either manufacturing or service firms. Table II reports the evolution of the set ofmost connected firmsover thesampleperiod. Largemanufactures, such asGeneral Motors and Ford, dominate the early eighties. By the end of the sample, the shift in activity from manufacturing to retail and services is widespread, with Walmart and Cardinal Health being the most connected firms. The distribution of firms’ sizes resembles the size distribution of the CRSP universe, but the size distribution of firms’ customers is tilted toward large companies, as firms are only required to report customers that represent more than 10% of 12I thank Enghin Atalay for sharing the soundex code used in Atalay et al. (2011). 17

their annual revenues or sales.13 A.2. Propensity of relationships Pivotal for my analysis is identifying the propensity of relationships to transmit idiosyncratic shocks. Unfortunately, propensities of supplier–customer relationships are unobservable. To deal with this issue, I rely on a composite of two measures to proxy for p . The ijt first measure is the percentage of annual sales that customers represent for their suppliers. e The higher the percentage, the more likely it is that shocks affecting a customer also affect its supplier, other things being equal. The second measure uses information about the specificity of suppliers, as evidence documented by Barrot and Sauvagnat (2016) suggest that input specificity is a key driver in the propagation of idiosyncratic shocks along production networks. Their idea is simple: if supplier i is highly specific, then it is more likely that i is hard to replace in case of distress and, therefore, the likelihood that shocks affecting i also affect j is higher, all other things being equal. With these measures at hand, I proxy for p ijt as e p = % company i’s sales accounted for by j at t Specificity of i at t. (9) ijt × e Percentages of annual sales are obtained from COMPUSTAT. To measure the specificity of suppliers, I construct a composite of three measures of input specificity, which I borrow from Barrot and Sauvagnat (2016).14 Following Barrot and Sauvagnat (2016), I assume that firms are more likely to produce specific goods if they (a) operate in industries producing differentiated goods, (b) have high levels of R&D, or (c) hold a large number of patents. I 13Because firms need to be sufficiently large to represent at least 10% of the annual sales of publicly tradedcompanies, many firms and their relationshipsare overlooked. As a consequence,one may be able to construct, in the most favorablecase, a network that resembles a sparse representationof the U.S. economy. To partially ensure that the topology of the benchmark economy provides a fair representation of the U.S. economy,Icomparethetopologyofthebenchmarkeconomywiththetopologyofnetworksconstructedfrom BEA input–output tables. In unreported results, I show that the network in the benchmark economy does a good job representing some features of the time series of U.S. inter–industry networks and, in doing so, potentially provides a reasonable representation of the aggregate U.S. economy. 14I thank Julien Sauvagnat for sharing this dataset. 18

then compute the specificity of supplier i at t as Rauch(i,t) + R&D(i,t-2)+ Patents(i,t) Specificity of supplier i at t = , 3 (cid:18) (cid:19) where Rauch(i,t) [0,1] denotes the share of differentiated goods produced in the industry ∈ of firm i at t according to Rauch (1999)’s classification of differentiated goods. R&D(i,t) [0,1] denotes the ratio of R&D expenses to sales of firm i at t 2, as innovations may take ∈ − some time to produce changes in the specificity of good i. Patents(i,t) [0,1] denotes the ∈ ratio of the number of patents issued by firm i from t 2 to t to the maximum number of − patents issued by any given firm within firm i’s industry from t 2 to t.15 − A.3. Firm-Level Financial Data MonthlyreturnsandannualfinancialdataonfirmsareobtainedfromtheCRSP/COMPUSTAT Merged Database and COMPUSTAT.16 All continuous variables are winsorized at the 1st and 99th percentiles of their distributions. A.4. Summary Statistics Table III reports summary statistics for the sample. Panel A presents statistics at the annual level. The average and median percentages of sales that customers represent for their suppliers are 19% and 14%, whereas the average and median for suppliers’ specificity scores are 34.2% and 34.9%. The main variable of interest is the propensity of relationships to transmit idiosyncratic shocks. The average and median for this variable are 11.4% and 4.4%, respectively. On average, there are eight years between the first and the last year a firm reports another firm as a major customer. 15 Rauch (1999) classifies inputs into differentiated or homogeneous depending on whether goods are traded on an organized exchange. Each industry is coded as being either sold on an exchange, reference priced, or homogeneous. The ratio of R&D expenses to sales aims to capture the importance of relationship specific investments. The number of patents issued by suppliers aims to capture restrictions on alternative sources of inputs. For more details about the construction of these measures see Barrot and Sauvagnat (2016). 16Accessed via Wharton Research Data Service (WRDS). 19

To examine the persistence of the above variables, Panel B presents statistics regarding autocorrelation coefficients computed at the relationship level. The average first and second autocorrelation coefficients for the percentage of sales that customers represent for their suppliers are 31.5% and 26.8%, and their medians are 27.3% and 25.7%. The average first and second autocorrelation coefficients for suppliers’ specificity scores are 29.5% and 23.8%, with medians of 25.1% and 21.4%. The propensities of relationships are also fairly persistent as their average first and second autocorrelation coefficients are 29.9% and 24.8%, with medians of 25.9% and 23%. B. Uncovering and ζ n t G B.1. Uncovering n G To calibrate , I construct firm networks at an annual frequency over the sample period. n G Nodes represent firms and links represent supplier–customer relationships. Table IVreportsaverages andstandarddeviationsforkey characteristics ofU.S. supplier– customer networks. Onaverage, thereare1,112firms, 1,109relationships, and154connected components per network. For illustration, Internet Appendix C depicts the time series of such networks. As these figures show, U.S. production networks are highly asymmetric in the sense that only a few firms are connected to many others, while most firms have either one or at most two connections. The degree distributions of these networks, which measure the frequency of firms with a given number of customers and suppliers, are highly skewed to the right. Most importantly, this high asymmetry is fairly persistent.17 I use the U.S. supplier–customer network in 2015 (depicted in figure 3) to pin down as its topology matches several of the averages reported in Table IV. Thus, there are n G 17If a power law distribution is fitted to the degree distribution of each network, one obtains SD(exponent of power law distribution fitted to degree distribution) 0.15 = =6%, Mean(exponent of power law distribution fitted to degree distribution) 2.23 which emphasizes the fact that between 1976 and 2016 the level of asymmetry in U.S. production networks has been persistently high. 20

n = 1,110 firms, 1,146 relationships, and 159 connected components in the benchmark economy. The main results continue to hold if supplier–customer networks of other years are fed into the benchmark economy, as the asymmetric structure of U.S. production networks is fairly persistent. B.2. Uncovering ζ t 2016 To calibrate ζ , I use the cross-sectional distributions of propensities p t { ijt }(i,j) t=1976 n o constructed following equation (9). Using these values, I fit a Beta distribution to each cross e sectional distribution. From this procedure, I obtain a time series of estimates, ζ∗ 2016 , { t}t=1976 which are depicted in figure 4. I then fit a vector autoregressive (VAR) process to the time series of estimates. After doing so, I discretize the fitted VAR into a four-state Markov chain using Gospodinov and Lkhagvasuren (2014)’s method and obtain ζ∗ = 0.67, ζ∗ = 0.78, 1L 1H ζ∗ = 3.24, ζ∗ = 4.01, and 2L 2H 0.57 0.27 0.06 0.05   0.25 0.63 0.02 0.12 Ω∗ = ,   0.12 0.02 0.63 0.25       0.05 0.06 0.27 0.57     with a stationary distribution given by P (ζ∗ ) = P (ζ∗ ) = P (ζ∗ ) = P (ζ∗ ) = 0.25. LH HL LL HH C. Selecting the rest of the parameter values The rest of the parameters can be separated mainly into two groups. Parameters in the first group define the preferences of the representative investor, which I select in line with Bansal and Yaron (2004). Thus, β = 0.997, γ = 10 and ρ = 0.65 (IES 1.5). Parameters ≈ in the second group define the dynamics of firms’ cash flows, which I proxy with operating income. I restrict my focus to firms in the supplier–customer database, as relationships are known only for such firms. With these restrictions, I use the following regressions to 21

determine α . First, I run the following OLS regressions, 1 operating income log i,t+1 = Controls+ǫa +ǫind +ǫz (10) operating income t+1 i i,t+1 j j,t! P where ǫa and ǫind capture year and industry fixed effects, respectively. Controls include t+1 i lagged values for firms’ assets, age, and return on assets, to ensure that variation in the error term ǫz is not driven by trends in large, young, or profitable firms. Second, I run the i,t+1 following regressions at the annual frequency, ǫz = β +β d +ǫε , i,t 0 1 i i,t b where ǫz are the residuals obtained from regression (10). I set α = 0.3 so that α equals i,t+1 1 1 the average annual estimate of β over the sample. I set α = 0.3 and q = 0.1 so that the b 1 2 unconditional mean and volatility of consumption growth generated by the model are similar to the ones found in the data. I use annual data on Total Factor Productivity (TFP) growth from the Federal Reserve Bank of San Francisco to determine the volatility of aggregate shocks, σ . I set σ = 1.7 so that σ equals the annual volatility of TFP growth. Finally, I a a a follow Bansal and Yaron (2004) and set τ = 1/3. Table V summarizes the key parameter values in the calibrated model. V. Implications of the Calibrated Model This section quantitatively evaluates the ability of the calibrated model to rationalize features of stock returns. It shows that changes in the propagation of idiosyncratic shocks, within a firm network that captures key characteristics of U.S. supplier–customer networks, are important to understanding variations in stock returns in both the aggregate and the cross section. Section V.A shows that the model generates long-run consumption risks. Section V.B shows that the model also matches cross-sectional patterns of portfolio returns 22

sorted by network centrality. Internet Appendix D describes the methodology used to simulate the model. A. Firm Networks and Long-Run Risks Table VI exhibits moments generated under the benchmark parameterization. By construction, the benchmark economy delivers annual averages and volatilities of consumption and dividend growth similar to those found in the data. It also delivers an average market return of 12.3%, an annual volatility of the market return of 19.5%, an average risk-free rate of 2.16%, an annual volatility of the risk-free rate of 1.8%, an annual equity premium of 10%, and an average Sharpe ratio of 0.51. With the exception of the volatility of the risk-free rate and Sharpe ratio, all values are aligned with those found in the data. Besides matching the above moments, the calibrated model generates a persistent component in expected consumption growth and stochastic consumption volatility similar to those assumed by the long-run risks (LRR) model of Bansal and Yaron (2004). As Bansal and Yaron (2004) and Bansal et al. (2012) show, these two features, together with Epstein- Zin-Weil preferences, help quantitatively explain an array of important asset market phenomena.18 Table VII reports summary statistics of several similarity measures of time series generated with either the calibrated model or the LRR model. To compute averages and standarddeviations, Isamplefromthecalibratedmodel andtheLRRmodel toconstruct two distributions for each similarity measure: one for expected consumption growth, E [∆c ], t t+1 and one for the conditional volatility of consumption growth, Vol [∆c ]. Reported values t t+1 e are based on 300 simulated economies over 620 periods. The first 100 periods are disregarded e to eliminate any bias coming from the initial condition. As table VII suggests, both models generate similar time series for conditional expected consumption growth and conditional 18Since Bansal and Yaron (2004), several authors have used the long-run risk framework to explain an arrayofmarketphenomena. Forinstance,Kiku(2006)providesanexplanationofthevaluepremiumwithin the long-run risks framework. Drechsler and Yaron (2011) show that a calibrated long-run risks model generates a variance premium with time variationand return predictability that is consistent with the data. BansalandShaliastovich(2013)developa long-runrisks model that accounts for bond return predictability and violations of uncovered interest parity in currency markets. 23

consumption volatility. It is important to appreciate that the persistent component in expected consumption growth and stochastic consumption volatility are endogenously generated rather than exogenously imposed, as in many asset pricing models. The calibrated model generates these two features for two reasons: (1) relationships are long-lasting, and (2) the four parameter vectors estimated from the data, ζ∗ , ζ∗ , ζ∗ , and ζ∗ , generate similar cross-sectional LL LH HL HH distributions of p , as figure 5 shows. Consequently, the connectivity of the firm { ijt }(i,j)∈Gn network is fairly stable over time and, thus, the propagation mechanism of idiosyncratic e shocks changes infrequently in the benchmark economy. These infrequent changes generate low-frequency movements in firms’ growth prospects which, in turn, generate a persistent component in aggregate output and expected consumption growth. Changes in the propagation mechanism of idiosyncratic shocks are infrequent because in the data firms tend to engage in enduring and stable relationships with their major customers. For instance, on average, relationships with major customers last more than eightyears. Theinterdependency ofsuchrelationshipsgenerateslong-terminterdependencies among firms’ cash-flow growth rates, fundamentally driving low-frequency movements in aggregate output growth. These low-frequency movements generate persistent changes in aggregate consumption growth in equilibrium. In such an economy, an idiosyncratic shock to a firm has the potential to affect not only the current cash flow growth of all neighboring firms, but also the long-term growth prospects of all such firms, enhancing the temporal effect of idiosyncratic shocks. While themodel endogeneously generates long-runconsumption risks, it doesnot provide a complete micro-foundation of such risks because of the exogenous determination of the relationship structure. Nonetheless, the model provides a novel link between asset returns and firm networks and suggests that changes in the propagation mechanism of idiosyncratic shocks infairly sticky productionnetworks are quantitatively relevant to understanding asset market phenomena. 24

B. Firms’ Centrality and the Cross Section of Risk Premiums Besides endogenizing long-run consumption risks, the model helps in understanding the cross section of expected returns as it provides a mapping between firms’ quantities of priced risk and firms’ importance in the network. To measure the importance of a firm in the network, I define the centrality of firm i at period t as the average number of firms that can be affected by an idiosyncratic shock to firm i at t. This measure captures the relative importance of firm i in propagating idiosyncratic shocks at t. Because the cross-sectional distribution of p changes over time, firms’ centrality scores change over time as { ijt }(i,j)∈Gn well. e To quantitatively assess the effect of a firm’s importance in the network on a firm’s risk– return trade off, I simulate the benchmark economy at a monthly frequency and construct portfolios based on centrality. Firms are assigned into centrality deciles once per year, and the value-weighted portfolios are not rebalanced for the next 12 months. This exercise reveals that a portfolio that is long the lowest centrality decile portfolio andshort the highest centrality decile portfolio generates a statistically significant return of 0.8% per month. Such a return is computed using 200 simulated economies over 1100 monthly observations. I disregard the first 100 observations in each simulation to eliminate any potential bias coming from the initial condition. The above result is explained by the fact that relationships of peripheral firms in the calibrated model (as in data) tend to exhibit higher propensities than relationships of central firms.19 Consequently, peripheral firms tend to have higher exposure to idiosyncratic shocks affecting their neighbors. On average, such contagion risk outweighs the potential benefits peripheral firms receive from their few relationships and, thus, peripheral firms command higher risk premiums than central firms. Central firms, however, seem to benefit from diversification of their neighbors as their relationships exhibit, on average, small propensities. 19Empirical support for this fact can be found if one plots propensity versus centrality of relationships for each annual U.S. supplier–customer network. These plots are depicted in Internet Appendix E. 25

As a result, their contagion risk is outweighed by the benefits generated by their many relationships. Table VIII shows that the calibrated model generates a realistic spread between low and high centrality portfolios as the average monthly return difference between low and high centrality portfolios for firms in the database is 0.82% (with a t-statistic of 4.06). Table VIII reports monthly average raw returns, alphas and loadings from the five–factor model of Fama and French (2015) for two portfolios of stocks sorted by annual centrality as well as the portfolio that is long the lowest centrality decile and short the highest centrality decile. As table VIII suggests, there is a significant negative relation between firms’ centrality andfuturereturns in thedata that cannot becaptured by standardasset pricing modelssuch as the five-factor model. Firms in the lowest centrality decile command an average monthly return of 2.28%, whereas firms in the highest centrality decile command an average monthly return of 1.45%. The 0.82% monthly difference in returns between these two portfolios is economically and statistically significant and appears naturally in an equilibrium context as a compensation for contagion risk.20 VI. Conclusion This paper studies the asset pricing properties that stem from the propagation of idiosyncratic shocks along firm networks. The fundamental insight of this paper is that extending standard asset pricing models to take into account how idiosyncratic shocks propagate along firm networks can make significant progress toward generating a unifying framework that simultaneously captures dynamics of the aggregate and the cross section of stock returns. Acalibratedmodelthatmatcheskeyfeaturesofsupplier–customer networksintheUnited States generates long-run consumption risks, high and volatile risk premiums, and a low 20If one focuses on manufacturing and service firms—as they jointly representmore than 65%of firms in the dataset—the resultstendtobe stronger,whichisconsistentwith empiricalevidencedocumentedbyWu andBirge (2014). See the tables in InternetAppendix E,which reportresults onmanufacturing andservice firms. 26

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Appendix A. Mathematical Derivations This section contains the derivations of formulas in the body of the paper. Let s denote the state t of the parameter vector ζ . Because the firm network is fixed, s determines the equilibrium distribution t t of aggregate consumption growth at t. Because ζ follows a Markov process, the distribution of aggregate t consumption growth varies over time and the dynamics of its moments satisfy the Markov property. Price and Expected Return of Aggregate Wealth: I look for an equilibrium such that pricedividend ratios are stationary. I conjecture that if c is the current aggregateconsumption and s the current state of ζ , then P (c,s) = wac, in which P is the price of aggregate wealth and wa is a number that t a s a s depends onstates. Ifs =sands =s′,the realizedgrossreturnatperiodt+1ofthe assetthatdelivers t t+1 aggregate consumption as its dividend each period, R , equals a,t+1 e P + wa +1 R = a,t+1 C t+1 = s′ C t+1 . a,t+1 P wa a,t s C t e e Setting R =R in equation (5) yields i,t+1 a,t+1 e e 1−γ −ρ 1−ρ 1−γ E β C t+1 R 1−ρ = 1 t a,t+1 " (cid:18) C t (cid:19) # h i  E β C t  +1 −ρ 1 1 − − γ ρ w s a ′ +1 C e t+1 1 1 − − γ ρ s  = 1. ⇒ " (cid:18) C t (cid:19) # (cid:20) w s a C t (cid:21) (cid:12)  (cid:12)  (cid:12)  (cid:12) Because s follows a Markov process, the above equation can be rewritten as t β 1 1 − − γ ρ ω s,s′ E C t+1 1−γ s′ w s a w ′ + a 1 1 1 − − γ ρ = 1. s′∈S (cid:18) C t (cid:19) (cid:12) !(cid:18) s (cid:19) ! X (cid:12) (cid:12) (cid:12) Reordering the above equation yields 1−ρ 1−γ w s a = β ω s,s′ E e(1−γ)∆ect+1 s′ (w s a ′ +1) 1 1 − − γ ρ s X ′∈S (cid:16) (cid:12) (cid:17) ! (cid:12) 1−ρ 1−γ = βeτ(1−ρ)(τ(1−γ)σ a 2+α1d¯) ω s,s′ E e−τ(1−γ)α2W f n,t+1 s′ (w s a ′ +1) 1 1 − − γ ρ (A1) s X ′∈S (cid:16) (cid:12) (cid:17) ! (cid:12) Risk-free Asset: Setting R =R in equation (5) yields i,t+1 f e 1−γ E β C t+1 −ρ 1−ρ R 1 1 − − γ ρ −1 s = 1 . a,t+1 " (cid:18) C t (cid:19) # h i (cid:12) (cid:12)  R f (s)  e (cid:12)  (cid:12) 33

Because s follows a Markov process and P (c,s) = wac, the left-hand side of the above equation can be t a s rewritten as β 1 1 − − γ ρ ω s,s′ E C t+1 −γ s′ w s a w ′ + a 1 ρ 1 − − γ ρ . s′∈S (cid:18) C t (cid:19) (cid:12) !(cid:18) s (cid:19) ! X (cid:12) (cid:12) (cid:12) Therefore, ρ−γ R 1 (s) = β 1 1 − − γ ρ ω s,s′ E e−γ∆ect+1 s′ w s a w ′ + a 1 1−ρ f s X ′∈S (cid:16) (cid:12) (cid:17)(cid:18) s (cid:19) ! (cid:12) ρ−γ = β 1 1 − − γ ρe−τγ(α1d¯−τγσ a 2) ω s,s′ E eτγα2W f n,t+1 s′ w s a w ′ + a 1 1−ρ . (A2) s X ′∈S (cid:16) (cid:12) (cid:17)(cid:18) s (cid:19) ! (cid:12) Firm i’s Expected Return: Consider s =s and s =s′. Equation (5) can be rewritten as t t+1 P = E M P +y i=1, ,n (A3) i,t t t+1 i,t+1 i,t+1 ··· (cid:16) (cid:16) (cid:17)(cid:17) f e where 1−γ t+1 −ρ 1−ρ 1 1 − − γ ρ −1 M β C R t+1 a,t+1 ≡ " (cid:18) C t (cid:19) # h i f e represents the pricing kernel. Dividing equation (A3) by Y yields t P P y i,t = E M X i,t+1 +E M i,t+1 i=1, ,n t t+1 t+1 t t+1 Y t Y t+1 ! (cid:18) Y t (cid:19) ··· e f e f which can be rewritten as y v = E M X v +E M i,t+1 i=1, ,n (A4) i,t t t+1 t+1 i,t+1 t t+1 Y ··· (cid:16) (cid:17) (cid:18) t (cid:19) f e f with v v (s) Pi,t. Because s follows a Markov process and P (c,s) = wac, the first term in the i,t ≡ i ≡ Yt t a s right-hand side of equation (A4) can be rewritten as ρ−γ E t M t+1 X t+1 v i,t+1 = β 1 1 − − γ ρ ω s,s′ w s a w ′ + a 1 1−ρ E e((1/τ)−γ)∆ect+1 s′ v i (s′) (cid:16) (cid:17) s X ′∈ S (cid:18) s (cid:19) (cid:16) (cid:12) (cid:17) ! f e (cid:12) 34

whereas the second term in the right hand side of equation (A4) can be rewritten as E t M t+1 y i,t+1 = eσ a 2+α1diE t M t+1 e−α2εe i,t+1 . Y (cid:18) t (cid:19) (cid:16) (cid:17) f f The expectation term in the right hand side of the above equation can be written as E t M t+1 e−α2εe i,t+1 = β 1 1 − − γ ρ ω s,s′ E C t+1 −γ e−α2εe i,t+1 s′ w s a w ′ + a 1 ρ 1 − − γ ρ (cid:16) (cid:17) s X ′∈ S (cid:18) C t (cid:19) (cid:12) (cid:12) !(cid:18) s (cid:19) ! f (cid:12) ρ−γ = β 1 1 − − γ ρ ω s,s′ E e−γ∆ect+1−α2εe i,t+1 s′ (cid:12) w s a w ′ + a 1 1−ρ . s′∈ S (cid:18) (cid:12) (cid:19)(cid:18) s (cid:19) ! X (cid:12) (cid:12) (cid:12) As a consequence, ρ−γ v i (s) = β 1 1 − − γ ρ ω s,s′ w s a w ′ + a 1 1−ρ E e((1/τ)−γ)∆ect+1 s′ v i (s′) s X ′∈ S (cid:18) s (cid:19) (cid:16) (cid:12) (cid:17) ! (cid:12) ρ−γ + β 1 1 − − γ ρeσ a 2+α1di s′∈ S ω s,s′ E (cid:18) e−γ∆ect+1−α2εe i,t+1 (cid:12) s′ (cid:19)(cid:18) w s a w ′ + s a 1 (cid:19) 1−ρ ! i=1, ··· ,n X (cid:12) (cid:12) (cid:12) To solve for the second expectation in the right-hand side of the above equation, it is convenient to express as a set of connected components. If i denotes the connected component that firm i belongs to, then G n Gn can be written as n G i. G n ≡ Gn i∈ [ Gn Define the following averages, 1 1 Wi ε and W−i ε , n,t+1 ≡ n j,t+1  n,t+1 ≡ n j,t+1  jX∈G n i j∈GXn\G n i f  e  f  e  where Wi represents the average number of firms in i that face firm-level shocks at t+1, whereas n,t+1 Gn W−i representstheaveragenumberoffirmsin i thatfacefirm-levelshocksatt+1. BecauseWi n,t+1f G n \Gn n,t+1 and W−i are independent, f n,t+1 f f E e−γ∆ect+1−α2εe i,t+1 s′ = e−γ(α1d¯−γσ a 2)E eα2γW f n − , i t+1 s′ E eα2γ(W f n i ,t+1 −εe i,t+1) s′ (cid:18) (cid:12) (cid:19) (cid:18) (cid:12) (cid:19) (cid:18) (cid:12) (cid:19) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) = e−γ(α1d¯−γσ a 2) π −i (s (cid:12) (cid:12)′) π i (s′) (cid:12) (cid:12) |× {z }| × {z } 35

Therefore, ρ−γ v i (s) = β 1 1 − − γ ρ ω s,s′ w s a w ′ + a 1 1−ρ E e((1/τ)−γ)∆ect+1 s′ v i (s′) s X ′∈S (cid:18) s (cid:19) (cid:16) (cid:12) (cid:17) ! (cid:12) ρ−γ + β 1 1 − − γ ρeσ a 2(1+γ2)+α1(di−γd¯) s′∈S ω s,s′π i (s′)π −i (s′) (cid:18) w s a w ′ + s a 1 (cid:19) 1−ρ ! i=1, ··· ,n X Using the above computations, the expected one-period gross return of firm i is given by p +y E R s = E i,t+1 i,t+1 s i,t+1 p (cid:16) (cid:12) (cid:17) (cid:18) ve Y i,t e +y (cid:12) (cid:12) (cid:19) e (cid:12) = E i,t+1 t+1 i(cid:12) (cid:12) ,t+1 s ev Y (cid:18) i,t t (cid:12) (cid:19) = 1 ω s,s′v e i (s′)E (cid:12) (cid:12) (cid:12) e(1/τ)∆ect+1 s′ + eσ a 2+α1di ω s,s′ E e−α2εe i,t+1 s′ v (s) v (s) i s X ′∈S (cid:16) (cid:12) (cid:17) ! i s X ′∈S (cid:16) (cid:12) (cid:17) ! = e(1/τ)((1/τ)σ a 2+α1d¯) ω s,s′v i (s′)E (cid:12) e−(1/τ)α2W f n,t+1 s′ + eσ a 2+α1di (cid:12) ω s,s′ E e−α2εe i,t+1 s′ . v (s) v (s) i s X ′∈S (cid:16) (cid:12) (cid:17) ! i s X ′∈S (cid:16) (cid:12) (cid:17) ! (cid:12) (cid:12) Appendix B. Tables and Figures This section contains tables and figures mentioned in the body of the paper. Table I Major Industry Groups The table reports the distribution of firms across major industry groups in the dataset. Major industry groups are defined by the first two digits of firms’ SIC codes. Industry Number of firms Agriculture, forestry, and fishing 31 Construction 94 Finance, insurance, and real estate 569 Manufacturing 4275 Mining 588 Retail 345 Service 1598 Transportation, communications, electric, gas, and sanitary 881 Wholesale 290 Nonclassifiable establishments 108 Total 8,779 36

Table II Most Connected Firms Thetablereportstheaveragenumberofrelationships—consideringcustomersandsuppliers—ofthefivemost connectedfirmsinthefollowingfive-yearintervals: 1976–1980,1981–1985,1986–1990,1991–1995,1996–2000, 2001–2005,2006–2010,and 2011–2015. 1976 to 1995 1976–1980 1981–1985 1986–1990 1991–1995 Name N Name N Name N Name N General Motors 290 General Motors 393 General Motors 395 Walmart 446 Ford 157 IBM 226 AT&T 373 AT&T 437 Sears Roebuck 106 AT&T 206 IBM 303 General Motors 377 JC Penney 90 Ford 191 Ford 237 IBM 344 Sears Holdings 76 Sears Roebuck 184 Chrysler 143 Ford 334 1996 to 2015 1996–2000 2001–2005 2006–2010 2011–2015 Name N Name N Name N Name N Walmart 525 Walmart 585 Walmart 570 Walmart 555 General Motors 305 General Motors 259 Cardinal Health 180 Cardinal Health 187 Ford 275 Ford 200 Mckesson 154 Amerisourcebergen 155 AT&T 270 Daimler 185 Amerisourcebergen 139 Mckesson 144 IBM 253 Home Depot 134 AT&T 135 AT&T 144 37

Table III Descriptive Statistics The table reports descriptive statistics for the sample. The sample contains 8,779 different firms and 17,322 supplier–customer relationships among different pairs of firms from 1976 to 2016. These relationships represent 66,355 unique annual linkages. Panel A reports summary statistics at the annual level for (a) the percentageofsalesthatcustomersrepresentfortheirsuppliers,(b)thespecificityofsuppliers,(c)the Rauch (1999)’s score, (d) the R&D’s score, (e) the patent’s score, (f) p , and (g) the duration of relationships in ijt years. Panel B present summary statistics at the relationship level. AC1 and AC2 report the first and | | | | secondautocorrelationcoefficients of: (a)the percentage ofsalees that customers representfor their supplier, (b) the specificity of suppliers, and (c) the propensity of relationships. In Panels A and B, column Obs denotes the number of non–missing observations used to compute summary statistics. Summary statistics are in percentages; with the exception of duration. All continuous variables are winsorized at the 1st and 99th percentiles of their distributions. Panel A: Annual level Obs Mean 25th Per. Median 75th Per. Min Max % of sales 53,620 19.0 9.8 14.0 22.6 0.6 95.0 Specificity of suppliers 62,447 34.2 0.0 34.9 50.2 0.0 100 Rauch’s score 60,904 56.6 0.0 100 100 0.0 100 R&D score 26,897 10.0 1.0 4.0 14.0 0.0 78.1 Patent score 43,967 13.1 0.0 0.0 6.3 0.0 100 p 65,232 11.4 0.0 4.4 12.1 0.0 100 ijt Duration 66,355 8.37 3 6 12 1 39 e Panel B: Relationship level Obs Mean 25th Per. Median 75th Per. Min Max AC1 % of sales 6,411 31.5 11.9 27.3 49.9 0.0 92.8 | | AC2 % of sales 6,336 26.8 12.0 25.7 41.3 0.0 85.2 | | AC1 specificity 5,032 29.5 13.3 25.1 44.8 0.0 90.1 | | AC2 specificity 5,017 23.8 9.4 21.4 34.3 0.0 77.5 | | AC1 p 6,057 29.9 11.4 25.9 46.3 0.0 94.1 ijt | | AC2 p 6,061 24.8 10.2 23.0 37.7 0.0 84.9 ijt | | e e 38

Table IV Characteristics of Supplier–Customer Networks The table reportscharacteristicsofsupplier–customernetworksgeneratedatanannualfrequencyfrom1976 to 2016. Firms i and j are connected in the network of year t if firm i (j) reports j (i) as a principal customer. Thenumber ofconnectedcomponentspernetworkis computedviaadepth-firstsearchalgorithm as in Tarjan (1972). The benchmark column reports the characteristics of the network in the benchmark economy. Characteristic Mean Standard Deviation Benchmark Number of firms per supplier–customer network 1,112 365 1,110 Number of relationships per supplier–customer network 1,109 393 1,146 Average number of suppliers per firm 0.98 0.06 1.03 Average number of suppliers and customers per firm 1.96 0.13 2.06 Number of connected components per network 154 43 159 Table V Benchmark Parameterization The table reports the list of parameter values in the benchmark parameterization. Parameters in the first groupdefinethepreferencesoftherepresentativeinvestor: β representsthetimediscountfactor,γ represents the coefficient of relative risk aversionfor static gambles, and ρ represents the inverse of the inter-temporal elasticity of substitution. Parameters in the second group describe firms’ cash flows: σ measures the a volatility of aggregateshocks, α measures the marginalbenefits a firm receives from each relationship, and 1 α measures the decrease in a firm’s cash-flow growth if that firm is affected by a negative firm-level shock. 2 Parameters in the third group define the stochastic process that determines the propagation of firm-level shocks. Parameterqmeasureshowfrequentlyfirmsfacenegativeidiosyncraticshocks. Therestofparameters define the crosssectionaldistributionfromwhichpropensities ofrelationshipsaredrawn: ζ , ζ , ζ , and 1L 1H 2L ζ . 2H Preferences Firms’ cash flows Propagation of shocks β γ ρ σ α α ζ ζ q ζ ζ a 1 2 1L 1H 2L 2H 0.997 10 0.65 1.7 0.03 0.3 0.67 0.78 0.1 3.24 4.01 39

Table VI Moments under the Benchmark Parameterization The table reports the first two moments of consumption and dividend growth as well as a set of key asset pricing moments. Column Data reports moments found in the data. Column Model reports moments generatedunder the benchmark parameterizationdescribed in Table V. Column BY2004 reports moments generated under the long-run risks model of Bansal and Yaron (2004). Data on consumption and dividends are obtained from Robert Shiller’s website http://www.econ.yale.edu/ shiller/data.htm. Moments on the return on aggregate wealth, risk-free rate, equity premium, and Sharpe ratio are based on data from 1928 to 2014 and obtained from Aswath Damodaran’s website: http://pages.stern.nyu.edu/ adamodar/. The ∼ annualreturnonaggregatewealthis approximatedby the annualreturnofthe S&P500. The returnonthe risk-free asset is approximated by the yield on three-month T-bills. All values are in percentages with the exception of average Sharpe ratios. Moments Data Model BY2004 Average annual log of consumption growth rate 1.9 1.9 1.8 Annual volatility of log consumption rate 3.5 3.5 2.8 Average annual log dividend growth rate 3.8 3.8 1.8 Annual volatility of the log dividend growth rate 11.63 11.9 12.3 Average annual market return (S&P 500) 11.53 12.3 7.2 Annual volatility of the market return 19 19.5 19.42 Average annual risk-free rate (3-month T-bill) 3.53 2.16 0.86 Annual volatility of risk-free rate 3 1.8 0.97 Average annual equity risk premium 8 10 6.33 Average annual Sharpe ratio 0.4 0.51 0.33 40

Table VII Similarities between the calibrated model and the LRR model The table reports averages and standard deviations of similarity measures between time series generated witheither the calibratedmodelorthe benchmarkparameterizationinthe LRRmodelofBansalandYaron (2004). To compute averages and standard deviations, I sample from the calibrated model and the LRR model to construct two empirical distributions for each similarity measure: one for expected consumption growth, E [∆c ], and one for the conditional volatility of consumption growth, Vol [∆c ]. Reported t t+1 t t+1 values are based on 300 simulated samples over 620 periods. The first 100 periods in each sample are disregarded toeeliminate bias coming from the initial condition. All similarity measures reeport scores computed as 1 , where distance is defined accordingto eachsimilaritymeasure. Let X =(X , ,X ) 1+distance T 1 ··· T and Y = (Y , ,Y ) denote realizations from two time series, X = X and Y = Y . The first T 1 T t t ··· { } { } and second similarity measures focus on the proximity between X and Y at specific points of time. The euclidean distance (ED) is defined as T (X Y )2, whereas the dynamic time warping (DTW) dist=1 t − t tance is defined as min ( m X Yq), where r=((X ,Y ), ,(X ,Y )) is a sequence of m pairs r i=1| ai − bi|P a1 b1 ··· am bm that preserves the order of observations, i.e., a < a and b < b if j > i. DTW seeks to find a map- P i j i j ping such that the distance between X and Y is minimized. This way of computing distance allows two time series that are similar but locally out of phase to align in a nonlinear manner. The third measure focuses on correlation-based distances. It uses the partial autocorrelation function (PACF) to define the distance between time series. In particular, distance is defined as (ρˆ ρˆ )′Ω(ρˆ ρˆ ), where Ω is Xt − Yt Xt − Yt a matrix of weights, whereas ρˆ and ρˆ are the estimated partial autocorrelations of X and Y, respec- Xt Yt p tively. The fourth and fifth measures assume that a specific model generates both time series. The idea is to fit the specific model to each time series and then measure the dissimilarity between the fitted models. The fourth measure computes the distance between two time series as the ED between the truncated AR operators. In this case, distance is defined as k (e e )2, where e = (e , ,e ) and j=1 j,Xt − j,Yt Xt 1,Xt ··· k,Xt e Yt = (e 1,Yt , ··· ,e k,Yt ) denote the vectors of ARq( P k) parameter estimators for X and Y, respectively. The fifth measure computes dissimilarity between two time series in terms of their linear predictive coding in ARIMA processes, as in Kalpakis et al. (2001). The last measure defines distance based on nonparametric spectral estimators. Let f and f denote the spectral densities of X and Y , respectively. The dissim- XT YT T T ilarity measure is given by a nonparametric statistic that checks the equality of the log-spectra of the two time series. It defines distance as n k=1 Z k − µˆ(λ k ) − 2log(1+eZk−µˆ(λk)) − n k=1 Z k − 2log(1+eZk) , where Z = log(I (λ )) log(I (λ )), and µˆ(λ ) is the local maximum log-likelihood estimator of k XT k − PYT k(cid:2) k (cid:3) P (cid:2) (cid:3) µ(λ ) = log(f (λ )) log(f (λ )) computed with local lineal smoothers of the periodograms. All simik XT k − YT k larity measures are computed using the R package TSclust (see Montero and Vilar (2014)). E [∆c ] Vol [∆c ] t t+1 t t+1 Similarity Measure Mean Standard Deviation Mean Standard Deviation ED 0.99 e 0.02 0.96 e 0.01 DTW 0.74 0.10 0.75 0.12 PACF 0.80 0.04 0.78 0.05 ED in AR 0.90 0.12 0.93 0.11 Linear predictive in ARIMA 0.77 0.34 0.75 0.33 Spectral distance 1.00 0.00 1.00 0.00 41

Table VIII Performance of Centrality Portfolios The table reportsmonthly averagerawreturns, alphas andloadings fromthe five-factormodel of Fama and French (2015) for three portfolios constructed by sorting stocks based on centrality: a portfolio that holds stocksonthelowestdecileofcentrality(Low),aportfoliothatholdsstocksonthehighestdecileofcentrality (High), andaportfoliothatislongonstocksonthelowestdecile andshortonstocksonthe highestdecileof centrality(Low-High). Thebottomrowprovidesthet-statisticsforthelowminushighportfolio. Firmsare assigned into deciles at the end of October every year and the value-weighted portfolios are not rebalanced for the next 12 months. The sample is from June 1976 to December 2016. Raw returns and alphas are in percent. Raw 5-Factor Model Decile Return Alpha MKT SMB HML RMW CMA Low 2.28 1.38 0.98 0.41 -0.27 -0.47 0.06 High 1.45 0.55 0.94 -0.23 -0.07 -0.06 0.10 Low - High 0.82 0.43 0.05 0.64 -0.20 -0.40 -0.02 t-statistic [4.06] [2.49] [1.22] [10.49] [-2.58] [-4.84] [-0.21] 42

1.4 q 1.2 p 1 t 0.8 p t q q q p t 0.6 0.4 p t 0.2 q 0 −1 −0.5 0 0.5 1 1.5 2 2.5 3 √nWn (a) Network Topology FDP EmpiricalProbabilityDensityFunctionof√nWn p = 0.10 t p = 0.50 t p = 0.90 t (b) Empirical density function of W n,t Figure 2. The figure illustrates how changes in the propensity of inter-firm relationships to transmit sfhocks at t, p , affect t the distribution of W . Figure 2(a) depicts an economy with n = 5 firms, whereas figure 2(b) depicts estimates of the density n,t function of W for different values of p . These estimates are computed via normal kernel smoothing estimators usingefunction n,t t ksdensity( ) in MATfLAB. · f e 43

Figure 3. The figure shows the customer-supplier network in the benchmark parameterization. 44

Dynamics of z and z 1 2 1980 1990 2000 2010 Years z 1 1 9.0 8.0 7.0 6.0 5.0 5 5.4 4 5.3 3 5.2 2 z 2 Figure 4. The figure shows annual estimates of parameters ζ and ζ . I obtain these estimates by fitting Beta distributions to 1 2 the annual link weight distributions using maximum likelihood. 45

9 8 7 6 5 4 3 2 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Probabilities pe ij noitcnuf ytisneD β(ζ ), β(ζ ), β(ζ ), and β(ζ ) LL LH HL HH LL LH HL HH Figure 5. The figure shows probability density functions for Beta distributions with shape parameter vectors ζ , ζ , ζ , LL LH HL and ζ . HH 46

For Online Publication: Appendix for “Firm Networks and Asset Returns” CARLOS RAMIREZ∗ February 1, 2018 This internet appendix contains supporting results, tables, and figures to supplement the analysis in the paper “Firm Networks and Asset Returns.” Section A presents an equilibrium network model where production is explicitly modeled. Section B provides conditions under which W follows a Poisson or n,+1 Normal distribution. Section C depicts the time series of U.S. supplier–customer networks over the sample f period. Section D provides a description of the algorithm used to simulate the model. Section E presents figures that depict the propensity versus centrality of relationships in U.S. supplier–customer networks as well as tables that support the cross–sectionalresults of the paper. A. A Production-Based Equilibrium Network Model The model embeds a variant of the multisector models of Long and Plosser (1983) and Acemoglu et al. (2012) into a standard asset pricing model with investors with Epstein-Zin-Weil preferences. Section A.A describes the production side of the economy. Section A.B describes investors preferences. Section A.C defines the equilibrium. Section A.D examines the equilibrium distribution of consumption growth. Using approximate analyticalsolutions,section A.E analyzes the assetpricing implications of changes in the propagationofidiosyncraticshocksalongafirmnetwork. SectionA.Fpresentsthe resultsofasimplecalibration exercise to check whether the cross-sectionalresults obtained in the paper can be supported by a calibrated version of the production-based network equilibrium model. A. Production Consider an economy with n different perishable goods and an infinite time horizon, with n being potentially large. Time is discrete and indexed by t 0,1,2, . Goods are produced using both labor and ∈{ ···} intermediateinputs. Eachgoodispotentially usedasaninput inthe productionofeveryothergood. There are n different competitive sectors, each populated by a large number of identical, infinitely lived firms that are aggregated into a representative firm. Within each period, representative firm i produces good i, with i 1,2, ,n . Representative firms, henceforth referred to as firms, buy inputs and produce at the same ∈{ ··· } time. There is one share per firm. FirmsuseCobb-Douglastechnologieswithconstantreturnstoscale. Firmi’soutputatperiodt,denoted ∗BoardofGovernors oftheFederal ReserveSystem. Theinformationinthismanuscriptrepresents theview oftheauthor, andshouldnotbeinterpretedasreflectingtheviewsoftheBoardofGovernorsoftheFederalReserveSystemorothermembers ofitsstaff. E-mail: carlos.ramirez@frb.gov.

by y , is given by it n n y (a z l )χ y (1−χ)wij , with w =1, it ≡ t it it ijt ij j=1 j=1 Y X where log(a ) d i.i.d. (0,σ2) is an aggregate productivity shock at period t, z is a productivity shock t −→ N a it to firm i at period t, l is the amount of labor hired by firm i at period t, and y is the amount of good j it ijt used in the production of good i at period t. Parameter χ (0,1) represents the share of labor. Parameter ∈ w denotes the share of good j used in the production of good i. ij Let p denote the price of good i at period t. Taking input prices as given, firms choose how much it labor and inputs to buy to maximize per-period profits. For simplicity, firms’ choice of labor is deliberately normalized to 1.1 Thus, at period t firm i solves n π max p y p y i,t it it jt ijt ≡{yijt}n j=1 , lit − j=1 X st. l =1. it where π denotes the dividend of firm i at t. i,t A.1. The firm network and firm-level productivity shocks If firm-level productivity shocks are independent across firms, as in Acemoglu et al. (2012), they affect downstream production only via changes in production costs.2 For example, if firm i faces a negative productivity shock at period t, its production decreases and its output price increases, which, in turn, increasesproductioncostsinallfirmsthat(directlyorindirectly)usegoodiasaninputinperiodt. However, if firm-levelproductivity shocksaredependent acrossfirms, they affect downstreamproductionnot only via changes in production costs but also via changes in firms’ productivity. Iassumethatthe dependence structureamongshockstofirm-levelproductivitygrowthisdeterminedby a firm network. In particular, ∆z log zi,t+1 follows i,t+1 ≡ zi,t (cid:16) (cid:17) ∆z = α d α ε , i 1, ,n , i,t+1 1 i 2 i,t+1 − ∈{ ··· } where parameters α and α are non-negative and eqeual across firms. Parameter d represents the number 1 2 i ofrelationshipsoffirmi. Uncertaintyon∆z is introducedby ε whichequalsone iffirmiis affected i,t+1 i,t+1 by a negative firm-level productivity shock at period t+1 and zero otherwise. The distribution of ε is determined as in the baseline modeel. Similarly,propensities p are i,t+1 { ij,t+1 }(i,j) drawnfromaBetadistributionwithparametersζ >0andζ >0,whicharedrawnpriortodrawing 1,t+1 2,t+1 from the Beta distributeion at period t+1. The shape parameter vector ζ [ζ ζ ] followes a four-state t 1t 2t ≡ ergodic Markov process with transition matrix Ω. 1Themainresultscontinue toholdifacompetitivelabormarketisintroduced. 2In principle, firm-level productivity shocks would also affect upstream demand, as they not only change a firm’s output, but also change the amount of input needed to produce any given level of output. Shea (2002) shows that these two effects cancel outwithCobb-Douglastechnologies. 2

B. Representative Investor The economy is populatedby a large number of identical, infinitely lived individuals who are aggregated intoarepresentativeinvestor. Therepresentativeinvestorownsallassetsintheeconomyandisendowedwith n units of labor each period. The representative investor does not benefit from leisure and her preferences are defined over the following consumption bundle: n c1/n, C t ≡ i,t i=1 Y where c denotes her consumption of good i at period t. The representative investor has Epstein-Zin-Weil i,t preferences and, hence, 1−ρ 1− 1 ρ U = (1 β) 1−ρ+βE U1−γ 1−γ t − Ct t t+1 (cid:20) h i (cid:21) represents her utility at period t. Parameter ρ > 0, ρ = 1, represents the inverse of the inter-temporal 6 elasticity of substitution (IES), γ >0 is the coefficient of relative risk aversionfor static gambles, and β >0 measures the subjective discount factor under certainty. The representative investor’s budget constraint is given by n n n p c + (v π )φ = v φ it it it it i,t+1 it i,t − i=1 i=1 i=1 X X X where v denotes the cum-dividend value of firm i at t, and φ denotes the number of shares of firm i i,t i,t owned by the representative investor at the beginning of t (determined at t 1). − C. Competitive Equilibrium DEFINITION 1: A competitive equilibrium of the economy at period t consists of spot prices (p∗ , ,p∗ ), 1t ··· nt a consumption bundle ∗ = c∗ , ,c∗ , and quantities φ∗ ,l∗,y∗, y∗ such that: (a) each Ct 1,t ··· n,t i,t it it ijt j i=1···n firm maximizes per-period pr (cid:0) ofits, (b) the (cid:1) representative inve(cid:16)stor maximi (cid:8) zes u (cid:9) ti(cid:17)lity, and (c) labor and good markets clear; that is, n y∗ = c∗ + y∗ , i, it i,t jit ∀ j=1 X n l∗ = n, it i=1 X φ∗ = 1, i. i,t ∀ D. Equilibrium Consumption Growth The first-order conditions of firm i imply p y∗ = it (1 χ)w y∗. ijt p − ij it (cid:18) jt(cid:19) 3

Substituting these values into the market clearing condition yields n p∗ y∗ = c∗ + jt (1 χ)w y∗ it i,t p∗ − ji jt j=1(cid:18) it(cid:19) X = c∗ (1+q∗) i,t it P where q∗ (1 χ) n j=1 p∗ jt wjiy j ∗ t. Thus, it ≡ − p∗ it c∗ i,t log(y∗) = log c∗ +log(1+q∗) it i,t it 1 1 = log(cid:0)c∗ (cid:1)+q∗ (q∗)2+ (q∗)3+ i,t it− 2 it 3 it ··· (cid:0) (cid:1) Consequently, log y i ∗ t+1 = log c∗ i,t+1 +(q∗ q∗) 1 ( q∗ 2 (q∗)2)+ 1 ( q∗ 3 (q∗)3)+ (cid:18) y i ∗ t (cid:19) c∗ i,t ! it+1− it − 2 it+1 − it 3 it+1 − it ··· (cid:0) (cid:1) (cid:0) (cid:1) c∗ 1 1 = log i,t+1 +(q∗ q∗) 1 (q∗ +q∗)+ ((q∗ )2+q∗ q∗ +(q∗)2)+ . c∗ i,t ! it+1− it (cid:26) − 2 it+1 it 3 it+1 it+1 it it ··· (cid:27) Because φ∗ =1 at equilibrium, the representative investor’s budget constraint implies c∗ =χy∗ when i,t i,t i,t equilibrium prices are different from zero. As a consequence, the second term in the right-hand side of the previous equation is zero. Hence, 1 n y∗ 1 n c∗ log it+1 = log i,t+1 n y∗ n c∗ i=1 (cid:18) it (cid:19) i=1 i,t ! X X at equilibrium. Now, I express 1 n log y i ∗ t+1 as a function of the propagation of idiosyncratic shocks along the n i=1 y∗ it network. Taking the logarithm(cid:16)of y∗(cid:17)and using the first-order conditions of firm i into firm i production P i,t function yields n p∗ log(y∗ ) = χlog(a z )+(1 χ) w log i,t +log(1 χ)+log(w )+log(y∗ ) . i,t t i,t − j=1 ij ( p∗ j,t! − ij i,t ) X Provided that n w =1, the above expression can be reduced to j=1 ij P 1 χ n p∗ 1 χ log(y∗ ) = log(a z )+ − w log i,t + − log(1 χ) i,t t i,t (cid:18) χ (cid:19)  j=1 ij p∗ j,t! (cid:18) χ (cid:19) − X 1 χ   + − w log(w ). ij ij χ (cid:18) (cid:19)j=1 X y∗ Therefore, log i,t+1 can be rewritten as y∗ i,t (cid:16) (cid:17) y∗ a z 1 χ n p∗ p∗ i,t+1 t+1 i,t+1 i,t+1 i,t log = log +log + − w log log . y i ∗ ,t ! (cid:18) a t (cid:19) (cid:18) z i,t (cid:19) (cid:18) χ (cid:19)  j=1 ij ( p∗ j,t+1!− p∗ j,t!) X   4

To simplify the above expression, I use the following normalization on spot prices n n p∗ wij i,t = 1, t. i=1j=1 p∗ j,t! ∀ YY Using the above price normalization and summing over all firms yields 1 n y∗ a 1 n z i,t+1 t+1 i,t+1 log = log + log . n y∗ a n z i=1 i,t ! (cid:18) t (cid:19) i=1 (cid:18) i,t (cid:19) X X C∗ As a result, ∆c log t+1 equals t+1 ≡ C t ∗ (cid:16) (cid:17) e n c∗ 1/n i,t+1 ∆c = log t+1  i=1 c∗ i,t !  Y e 1  n y∗  = log it+1 n y∗ i=1 (cid:18) it (cid:19) X n a 1 z t+1 i,t+1 = log + log a n z (cid:18) t (cid:19) i=1 (cid:18) i,t (cid:19) X n n a 1 1 t+1 = log +α d α ε 1 i 2 i,t+1 (cid:18) a t (cid:19) n i=1 !− n i=1 ! X X e = | {z∆a t+ } 1 +α|1 {zd¯ } |α 2 {zW n,t+1}, − where ∆a denotes innovations to aggregate TFP and d¯denotes the averafge number of relationships per t+1 firm, and W denotes the averagenumber of firms affected by negative shocks to firm-level productivity n,t+1 growth at period t+1. Consequently, innovations to consumption growth are driven by either innovations f in aggregate productivity or innovations in W as in the baseline model; see equation (4). n,t+1 f E. Equilibrium Asset Pricing Although the model is solved numerically, the mechanisms working at equilibrium are shown via approximate analytical solutions. I first derive approximate equalities among variables of interest using log linearizations. I then use those equalities to explore the equilibrium asset pricing implication of changes in the propagationof idiosyncratic shocks along the network. E.1. Approximate Equalities Let Y n y1/n. Define t ≡ i=1 i,t Q y y p p i,t+1 i,t+1 i,t+1 a,t+1 g log , y log , p log , p log , i,t+1 i,t+1 i,t+1 a,t+1 ≡ Y ≡ Y ≡ Y ≡ Y (cid:18) t (cid:19) (cid:18) t+1 (cid:19) (cid:18) t+1 (cid:19) (cid:18) t+1 (cid:19) where p denotes the pribce of aggregatewealth abt t+1. b a,t+1 5

At equilibrium, the following two conditions must hold: n p = p , a,t+1 i,t+1 i=1 X and 1 n y∗ ∆c = log it+1 t+1 n y∗ i=1 (cid:18) it (cid:19) X n n e 1 1 = g + y (IA.1) i,t+1 i,t n n i=1 i=1 X X b =0 (provided Y definition). t | {z } Relationship between p and p : Using a first-order Taylor approximations yields a,t+1 i,t+1 n b b p ϕ + ϕ p , (IA.2) a,t+1 0 i i,t+1 ≈ i=1 X b b whereϕ =E Ppb i,t ,and n ϕ =1. Thetermϕ isselectedtoensurethatfirstorderapproximations i n j=1 pb j,t i=1 i 0 hold in levels(cid:16)as well. (cid:17) P Conditional distribution of g : At equilibrium, i,t+1 g = ∆a +∆z +y . i,t+1 t+1 i,t+1 i,t Then, the conditional distribution of g at t is given by b i,t+1 g d E [∆z ]+y ,2σ2+Var [∆z ] . i,t+1 −→ N t i,t+1 i,t a t i,t+1 (cid:0) (cid:1) Consequently, g can be approximated by b i,t+1 g x +σ η , i,t+1 i,t i,t i,t+1 ≈ d where η (0,1) and i,t+1 −→N x α d α E [ε ]+y , i,t 1 i 2 t i,t+1 i,t ≡ − σ2 2σ2+α2E [ε ](1 E [ε ]). i,t ≡ a 2 t e i,t+1 −b t i,t+1 Giventheinformationattimet,x determinesE [ge ]whileσ edeterminestheconditionalvolatilityof i,t t i,t+1 i,t g . Becausethepropensitymatrixp followsanergodicMarkovprocess,x andσ2 canbeapproximated i,t+1 t i,t i,t by the following autoregressiveprocesses: e x µ +µ x +µ σ ζ , i,t+1 0 1 i,t 2 i,t p,t+1 ≈ σ2 ν +ν σ2 +ν σ ζ , i,t+1 ≈ 0 1 i,t 2 p p,t+1 d where 0 < µ < 1, µ > 0, 0 < ν < 1 and ν > 0. Variable ζ (0,1) represents the uncertainty 1 2 1 2 p,t+1 −→ N 6

coming from unexpected changes in p (network connectivity). Variable η represents the uncertainty t+1 i,t+1 comingfromtheunexpectedchangesinexposureoffirmitoidiosyncraticproductivityshocksaffectingother firms in the economy. e Approximate equalities for equilibrium asset returns: Define the continuous return of firm i at t+1 as p +y i,t+1 i,t+1 r log , i,t+1 ≡ p (cid:18) i,t (cid:19) and the continuous return on the market portfolio at t+1 as p + p +Y a,t+1 t+1 a,t+1 t+1 r log C log . a,t+1 ≡ p ≈ p (cid:18) a,t (cid:19) (cid:18) a,t (cid:19) Using first-order Taylor approximations yields3 r k +ρ p p +ρ ∆c +(1 ρ )g , (IA.3) i,t+1 i i i,t+1 i,t i t+1 i i,t+1 ≈ − − r k p +ρ p +∆c , (IA.4) a,t+1 m a,t m a,t+1 t+1 ≈ − b b e where k n and k ensure that first-ordber approxbimations heold in levels. { i }i=1 m E.2. Equilibrium Asset Returns With the above definitions and approximationsat hand, I study the asset pricing implication of changes in the propagationof idiosyncratic shocks along the network. The pricing kernel is given by θ m θlog(β) ∆c +(θ 1)r . t+1 t+1 a,t+1 ≡ − ψ − with θ 1−γ and ψ = 1. e ≡ 1− ψ 1 ρ The stock price and return of firm i can be determined using the pricing kernel and the representative investor’s first-order condition E [exp(m +r )] = 1. (IA.5) t t+1 i,t+1 I first solve for the return of the market portfolio r substituting r for r in (IA.5). I then a,t+1 i,t+1 a,t+1 solve for the risk-free rate. Finally, I solve for the risk premium of firm i, i 1, ,n . ∀ ∈{ ··· } Return of the Market Portfolio: I conjecture that firm i’s log price-output ratio follows p = a +a x +a σ2 . (IA.6) i,t 0 1 i,t 2 i,t 3Approximation(IA.4)followsdirectlyfromthedividend-ratiomodelofCampbellandShiller(1989). Approximation(IA.3) followsfromCampbellandShiller(1989)oncebnotingthat (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) yi,t yi,t+1 yi,t+1 ri,t+1 ≈ ki+log −ρilog +log , (cid:18) pi,t (cid:19) pi,t+1 (cid:18) yi,t (cid:19) (cid:18) (cid:19) yi,t Yt Yt−1 yi,t+1 Yt+1 Yt yi,t+1 Yt Yt−1 = ki+log −ρilog +log , Yt−1pi,t Yt Yt pi,t+1Yt+1 Yt Yt−1 yi,t = ki+ρipb i,t+1−pb i,t+ρi∆ect+1+(1−ρi)gi,t+1 7

To solvefor constants a , a anda I substitute (IA.1), (IA.2) and(IA.4) into the Euler equation(IA.5). 0 1 2 As η and ζ are conditionally normal, r and m are also normal. Exploiting this normality, i,t+1 p,t+1 a,t+1 t+1 I write down the Euler equation in terms of the state variables x ,σ n . As the Euler equation must { i,t i,t }i=1 hold for all values of the states variables, the terms involving x must satisfy i,t 1 1 1 ϕ 1 a +ρ a µ ϕ 1 = 0 i 1 m 1 1 i − ψ − − − n − ψ (cid:26)(cid:18) (cid:19) (cid:27) (cid:18) (cid:19)(cid:18) (cid:19) ASSUMPTION 1: To a first order approximation, ϕ 1. i ≈ n Assumption 1 is satisfied if is regular (i.e. all firms have the same degree). If exhibits power-law n n G G degree distributions, assumption 1 is also satisfied for a large fraction of firms. If assumption 1 is satisfied, then 1 1 − ψ a . 1 ≈ 1(cid:16) µ ρ(cid:17) 1 m − Assume n is large. Collecting all terms that involve σ2 yields i,t 2 θ 1 1 +ρ2 a2µ2 2 − ψ m 1 2 a 2 ≈ 1 (cid:18)(cid:16) ν ρ + (cid:17) θρ2 a µ ν σ (cid:19). − 1 m 2 m 1 2 2 p Using (IA.6), the innovation to the return of the market portfolio can be written as n n 1 r E [r ] ρ a µ ϕ σ +a ν σ ζ + σ η a,t+1 t a,t+1 m 1 2 i i,t 2 2 p p,t+1 i,t i,t+1 − ≈ ! ! n i=1 i=1 X X n n ρ a µ ϕ σ +a ν σ ζ + ϕ σ η m 1 2 i i,t 2 2 p p,t+1 i i,t i,t+1 ≈ ! ! i=1 i=1 X X n = ρ ∆ ζ + ϕ σ η (IA.7) m p,t p,t+1 i i,t i,t+1 i=1 X where ∆ a µ ( n ϕ σ )+a ν σ . The conditional variance of the market portfolio is given by p,t ≡ 1 2 i=1 i i,t 2 2 p P n n Var [r ] ρ2 ∆2 +Var ϕ σ η +2ρ ∆ ϕ σ Cov (ζ ,η ) t a,t+1 ≈ m p,t t i i,t i,t+1 ! m p,t i i,t t p,t+1 i,t+1 i=1 i=1 X X NOTATION 1: Given two sequences a and b , I write a = o(b ) if an 0 as n , and { n }n { n }n n n bn → → ∞ a =O(b ) if an is bounded from above as n . n n bn →∞ REMARK 1: I(cid:12)f a(cid:12)ssumption 1 is satisfied and (cid:12) (cid:12) n n Var σ η =o(n2) and σ Cov (ζ ,η )=o(n), t i,t i,t+1 i,t t p,t+1 i,t+1 ! i=1 i=1 X X then lim Var [r ] = a2ν2ρ2 σ2. t a,t+1 2 2 m p n→∞ 8

Hence, the volatility of the market portfolio is only driven by changes in network connectivity. Pricing Kernel: Using (IA.1) and (IA.4), I rewrite the pricing kernel in terms of the state variables, θ m θlog(β) ∆c +(θ 1)r t+1 t+1 a,t+1 ≡ − ψ − n θ e θlog(β) ϕ (x +σ η ) i i,t i,t i,t+1 ≈ − ψ ! i=1 X n + (θ 1) k ϕ ϕ a +a x +a σ2 − m − 0 − i 0 1 i,t 2 i,t ! i=1 X (cid:0) (cid:1) n + (θ 1)ρ ϕ + ϕ (a +a µ +a µ x +a µ σ ζ ) m 0 i 0 1 0 1 2 i,t 1 2 i,t p,t+1 − ! i=1 X n + (θ 1)ρ ϕ a ν +a ν σ2 +a ν σ ζ − m i 2 0 2 1 i,t 2 2 p p,t+1 ! i=1 X (cid:0) (cid:1) n + (θ 1) ϕ (x +σ σ ) . i i,t i,t i,t+1 − ! i=1 X Innovations to the pricing kernel are then given by n m E [m ] λ ϕ σ η +λ ∆ ζ , (IA.8) t+1 t t+1 m,q i i,t i,t+1 m,p p,t p,t+1 − ≈ ! i=1 X where λ represents the aggregate market price of risk for each source of risk, namely η n and ζ , { i,t+1 }i=1 p,t+1 which are defined as 1 λ θ 1 1, m,q ≡ − ψ − (cid:18) (cid:19) λ (θ 1)ρ . m,p m ≡ − It follows from (IA.8) that the conditional variance of the pricing kernel is given by n n Var [m ] λ2 Var ϕ σ η +λ2 ∆2 +2λ λ ∆ ϕ σ Cov (η ,ζ )(IA.9) t t+1 ≈ m,q t i i,t i,t+1 ! m,p p,t m,q m,p p,t i i,t t i,t+1 p,t+1 i=1 i=1 X X REMARK 2: If assumption 1 is satisfied and n n Var σ η =o(n2) and σ Cov (ζ ,η )=o(n), t i,t i,t+1 i,t t p,t+1 i,t+1 ! i=1 i=1 X X then lim Var [m ] = λ2 a2ν2σ2. t t+1 m,p 2 2 p n→∞ Hence, the volatility of the pricing kernel is only driven by changes in network connectivity. Equity Premium: The risk premium of the marketreturn is determined by the conditional covariance 9

between the market portfolio and the pricing kernel. It follows that 1 E [r r ] = Cov (m E [m ],r E [r ]) Var (r ) t a,t+1 f,t t t+1 t t+1 a,t+1 t a,t+1 t a,t+1 − − − − − 2 Substituting (IA.7) and (IA.8) in the above equation yields n 1 ρ E [r r ] λ + Var ϕ σ η ρ λ + m ∆2 t a,t+1 − f,t ≈ − m,q 2 t i i,t i,t+1 !− m m,p 2 p,t (cid:18) (cid:19) X i=1 (cid:16) (cid:17) n ∆ (λ ρ +λ +ρ ) ϕ σ Cov (η ,ζ ) (IA.10) p,t m,q m m,p m i i,t t i,t+1 p,t+1 − ! i=1 X REMARK 3: If assumption 1 is satisfied and n n Var σ η =o(n2) and σ Cov (ζ ,η )=o(n), t i,t i,t+1 i,t t p,t+1 i,t+1 ! i=1 i=1 X X then ρ lim E [r r ] = ρ λ + m a2ν2σ2. n→∞ t a,t+1 − f,t − m m,p 2 2 2 p (cid:16) (cid:17) Hence, the equity premium is determined by temporal changes in network connectivity. Risk-free Rate: The risk-free rate satisfies 1 1 θ 1 r = log(β)+ E [∆c ]+ − E [r r ] Var [m ] f,t t t+1 t a,t+1 f,t t t+1 − ψ θ − − 2θ Substituting (IA.9) and (IA.10) in the aboveeequation yields n 1 r log(β)+ ϕ x f,t i i,t ≈ − ψ ! i=1 X n 1 θ 1 ρ − λ + Var ϕ σ η +ρ λ + m ∆2 − θ (cid:18) m,q 2 (cid:19) t X i=1 i i,t i,t+1 ! m (cid:16) m,p 2 (cid:17) p,t ! n 1 θ − ∆ (λ ρ +λ +ρ ) ϕ σ Cov (η ,ζ ) p,t m,q m m,p m i i,t t i,t+1 p,t+1 − θ ! i=1 X n n 1 λ2 Var ϕ σ η +λ2 ∆2 +2λ λ ∆ ϕ σ Cov (η ,ζ ) − 2θ m,q t i i,t i,t+1 ! m,p p,t m,q m,p p,t i i,t t i,t+1 p,t+1 ! i=1 i=1 X X Firms’ Risk Premiums: Aswiththemarketportfolio,theriskpremiumoffirmiisdeterminedbythe conditional covariance between firm i’s return and the pricing kernel. Therefore, 1 E [r r ] = Cov (m E [m ],r E [r ]) Var (r ) (IA.11) t i,t+1 f,t t t+1 t t+1 i,t+1 t i,t+1 t i,t+1 − − − − − 2 10

To simplify the above expression, it becomes convenient to compute the innovations on firm i’s return, r E [r ] ρ (a µ σ +a ν σ )ζ i,t+1 t i,t+1 i 1 2 i,t 2 2 p p,t+1 − ≈ n + ρ ϕ σ η +(1 ρ (1 ϕ ))σ η i j j,t j,t+1 i i i,t i,t+1   − − j6=i X   n = ρ ζ +ρ ϕ σ η +(1 ρ )σ η (IA.12) i i,t p,t+1 i j j,t j,t+1 i i,t i,t+1 ∇   − j=1 X   where a µ σ +a ν σ . Then, it follows from (IA.12) that i,t 1 2 i,t 2 2 p ∇ ≡ n Var (r ) ρ2 2 +ρ2Var ϕ σ η +(1 ρ )2σ2 t i,t+1 ≈ i∇i,t i t  j j,t j,t+1  − i i,t j=1 X   n + ρ (1 ρ )σ Cov (ζ ,η )+ ϕ σ Cov (η ,η ) i i i,t t p,t+1 i,t+1 j j,t t j,t+1 i,t+1 −   j=1 X n   + ρ2 ϕ σ Cov (η ,ζ ) i∇ i,t j j,t t j,t+1 p,t+1 j=1 X REMARK 4: Suppose assumption 1 is satisfied and n n Var σ η =o(n2) and σ Cov (ζ ,η )=o(n), t i,t i,t+1 i,t t p,t+1 i,t+1 ! i=1 i=1 X X then n lim Var (r ) = ρ2 2 +(1 ρ )2σ2 +ρ (1 ρ )σ Cov (ζ ,η )+ ϕ σ Cov (η ,η ) . n→∞ t i,t+1 i∇i,t − i i,t i − i i,t  t p,t+1 i,t+1 j j,t t j,t+1 i,t+1  j=1 X   Hence, firm i’s return volatility depends on σ and the covariance of η with innovations to network i,t i,t+1 connectivity and idiosyncratic productivity shocks to other firms. Additionally, if firm i is such that σ Cov (η ,η ) = o(n), j,t t j,t+1 i,t+1 j=1 X then lim Var (r ) = ρ2 2 +(1 ρ )2σ2 +ρ (1 ρ )σ Cov (ζ ,η ). n→∞ t i,t+1 i∇i,t − i i,t i − i i,t t p,t+1 i,t+1 11

Substituting (IA.8) and (IA.12) in (IA.11) yields n n E [r r ] ρ (λ +λ ∆ ) ϕ σ Cov (η ,ζ ) λ ρ Var ϕ σ η t i,t+1 f,t i m,q i,t m,p p,t j j,t t j,t+1 p,t+1 m,q i t j j,t j,t+1 − ≈ − ∇  −   j=1 j=1 X X n     λ (1 ρ )σ ϕ σ Cov (η ,η ) λ ∆ ρ m,q i i,t j j,t t j,t+1 i,t+1 m,p p,t i i,t − − − ∇ j=1 X 1 λ ∆ (1 ρ )σ Cov (η ,ζ ) Var (r ). (IA.13) m,p p,t i i,t t i,t+1 p,t+1 t i,t+1 − − − 2 REMARK 5: Suppose assumption 1 is satisfied and n n Var σ η =o(n2) and σ Cov (ζ ,η )=o(n). t i,t i,t+1 i,t t p,t+1 i,t+1 ! i=1 i=1 X X Define 1 ϑ λ ∆ ρ ρ2 2 +(1 ρ )2σ2 . i,t ≡ − m,p p,t i ∇ i,t − 2 i∇i,t − i i,t (cid:0) (cid:1) Then, 1 lim E [r r ] = ϑ (1 ρ )σ λ ∆ + ρ Cov (η ,ζ ) t i,t+1 f,t i,t i i,t m,p p,t i t i,t+1 p,t+1 n→∞ − − − 2 (cid:18) (cid:19) n (1 ρ )σ (λ +ρ ) ϕ σ Cov (η ,η ) . i i,t m,q i j j,t t j,t+1 i,t+1 − −   j=1 X   F. Calibration exercise This section analyzes whether the main results obtained in the paper can be supported by a calibrated version of the production-based network equilibrium model. F.1. Values for model parameters I follow Bansal and Yaron (2004)and set the preferences parameters to β =0.997, γ =10, and ρ=0.65. I follow Acemoglu et al. (2012) and set the share of labor to χ = 0.55. As in the paper, I set σ = 1.7 so a that σ equals the annualvolatilityof aggregateTFP growth. To compute the shareof goodj used in good a i, w , I use industry-level data from BEA Input-Output (IO) tables from 1997 to 2015. IO tables { ij }(i,j) are at the annual frequency. I compute the percentage of industry j’s sales purchased by industry i at year t following Ahern and Harford (2014). I then set w equal to the average percentage of industry j’s sales ij purchased by industry i over the sample period. Firms in the same industry are assumed to share the same valuesfor w . Tocalibratethe benchmarktopology,I use the U.S. supplier–customernetworkof2015 { ij }(i,j) as in the paper. Thus, n = 1,100 and the network exhibits a power-law degree distribution. The shape parameter vector ζ and its dynamics are calibrated as in the paper. I set α = 0.1 and α =1 so that the t 1 2 unconditional mean and volatility of consumption growth generated by the calibrated model are similar to the ones found in data. 12

F.2. Implications of the calibrated model Becauseequilibriumconsumptiongrowthcanbedecomposedasinthebaselinemodel,long-runconsumption risks endogenously arise as long as W exhibits a long-run predictable component. This is the case n,t+1 becausetheshapeparametervectorζ followsapersistentprocess,andthus,changesinnetworkconnectivity t f are infrequent. To analyze whether cross-sectionalresults in the paper can be supported by a production-based equilibrium framework, I study how the risk premium of a firm changes in the presence of a small increase in its centrality. Let κ denote the centrality of firm i at t as defined in the paper. If assumption 1 is satisfied it and the following two equalities hold: n n Var σ η =o(n2) and σ Cov (ζ ,η )=o(n). t i,t i,t+1 i,t t p,t+1 i,t+1 ! i=1 i=1 X X then the following approximationis accurate for a large fraction of firms in the economy: ∂E [r r ] ∂ϑ ∂ 1 t i,t+1 f,t i,t − (1 ρ ) σ λ ∆ + ρ Cov (η ,ζ ) i i,t m,p p,t i t i,t+1 p,t+1 ∂κ ≈ ∂κ − − ∂κ 2 i,t i,t i,t (cid:18) (cid:18) (cid:19) (cid:19) n ∂ (1 ρ )(λ +ρ ) σ ϕ σ Cov (η ,η ) , i m,q i i,t j j,t t j,t+1 i,t+1 − − ∂κ    i,t j=1 X    because n is large and the network exhibits a power-law degree distribution. Note that ∂ϑ σ ∂σ i,t = a µ ρ λ (ϕ +∆ )+ρ2 +(1 ρ )2 i,t i,t . ∂κ − 1 2 i m,p i ∇ i,t p,t i∇ i,t − i a µ ∂κ i,t (cid:18) 1 2(cid:19) i,t Additionally, the following first-order approximations hold ∂ 1 ∂σ 1 i,t σ λ ∆ + ρ Cov (η ,ζ ) λ ∆ + ρ +λ a µ ϕ σ Cov (η ,ζ ), i,t m,p p,t i t i,t+1 p,t+1 m,p p,t i m,p 1 2 i i,t t i,t+1 p,t+1 ∂κ 2 ≈ ∂κ 2 i,t (cid:18) (cid:18) (cid:19) (cid:19) (cid:18) i,t(cid:19)(cid:18) (cid:19) n n ∂ ∂σ i,t σ ϕ σ Cov (η ,η ) ϕ σ Cov (η ,η ) . i,t j j,t t j,t+1 i,t+1 j j,t t j,t+1 i,t+1 ∂κ    ≈ ∂κ   i,t i,t j=1 j=1 X X      Under the benchmark parameterization, a > 0, and thus, ∂ϑi,t < 0. In addition, the previous two 1 ∂κi,t derivativesare non-negativein the benchmark parameterization. As a result, ∂E t[ri,t+1−rf,t] 0. This result ∂κi,t ≤ shows that central firms command lower risk premiums than peripheral firms within a calibrated version of the production-based equilibrium framework. A realistic spread between firms in the highest and lowest centrality decile can be generated by fine tuning the values of µ and ν . 0 0 B. Distribution of W and ∆ c n,t+1 t+1 Under specific dependence assumptions, explicit bounds between the distribution of W and welln,t+1 f e knowndistributionscanbefound. Forinstance,effectiveboundsbetweenthedistributionofW andthe n,t+1 Poisson and Binomial distributions can be found if the dependence structure among vari f ables ε n f { i,t+1 }i=1 decreases as the distance between them increases. Chen (1975) finds bounds between the distribution of e 13

the sum of dependent Bernoulli trials and the Poisson distribution. Soon (1996) finds bounds between the distribution of the sum of dependent Bernoulli trials and the Binomial distribution. A. Distribution of W for finite n. n,t+1 Given how idiosyncratic shocks propagate along the network, variables ε n are not independent. f { i,t+1 }i=1 The following definition becomes handy for analyzing the distribution of W as it reconciles the notion n,t+1 of dependence between two random variables ε and ε , and the positieon of firms i and j in . i,t+1 j,t+1 n f G DEFINITION 2 (Dependency Graphs): A graph is said to be a dependency graph for a family of random e G e variables if the following two conditions are satisfied: (i) The set of random variables can be indexed by the nodes of . G (ii) If S and S are two disjoint set of nodes in such that no edge in has one endpoint in S and the 1 2 1 G G other in S , then the corresponding sets of random variables are independent. 2 Note that the above definition does not define a unique dependency graph for every family of random variables. Forinstance,addingoneedgetoadependencygraphyieldsanewdependency graphforthe same family of random variables. It is worth noting that ε n has as its dependency graph. Using the previous definition, the { i,t+1 }i=1 G n following proposition says that if every firm has a sufficiently small probability of being affected by an idiosyncraticshock,thenW e followsapproximatelyaPoissondistributiondespitethefactthat ε n n,t+1 { i,t+1 }i=1 are not independent. f e PROPOSITION 1 (Poisson Approximation of W ): Define n,t+1 π f E ε p i,t+1 i,t+1 t+1 ≡ e λ t+1 ≡ E (cid:2)W e n,t+ (cid:12) (cid:12)1 e p t+(cid:3)1 σ t 2 +1 ≡ V(cid:2)a f r W n,t+ (cid:12) (cid:12)e 1 p t+(cid:3)1 (cid:2) (cid:12) (cid:3) Provided that is the dependency graph of the sequencfe ε (cid:12)e n then G n { i,t+1 }i=1 e n 1 d W ,Po λ min 1, σ2 λ +2 π2 + π π t TV n,t+1 t+1 ≤ λ  t+1− t+1 i,t+1 i,t+1 j,t+1  ∀ (cid:18) (cid:0) (cid:1) (cid:19) (cid:26) t+1(cid:27) X i=1 (i,j)X∈ Rn f  e e e  where d W ,Po λ denotes the total variation distance between the distribution of W and a TV n,t+1 t+1 n,t+1 (cid:18) (cid:19) random variable with P(cid:0)oisson(cid:1) distribution of parameter λ , denoted by Po λ .4 f t+1 t+1 f (cid:0) (cid:1) Proof of Proposition 1. The result follows directly from Janson et al. (2000, Theorem 6.23) 4Thetotal variationdistancebetweenthedistributionoftworandomvariablesX andY isdefinedas (cid:0) (cid:1) (cid:12) (cid:12) dTV X,Y ≡ sup (cid:12)P[X∈A]−P[Y ∈A] (cid:12) A takingthesupremumoverallborelsetsA. IfX andY areintegervalued, then dTV (cid:0) X,Y (cid:1) ≡ 1 X(cid:12) (cid:12)P[X=k]−P[Y =k] (cid:12) (cid:12) 2 k 14

B. Distribution of W as n grows large n,t+1 If variables ε n are independent, the central limit theorem implies that √n(W q) is nor- { i,t+1 }if=1 n,t+1 − mally distributed as n grows large. Unfortunately, variables ε n are not independent. The following propositionsaysethat ifthe number ofrelationshipsofthe mo { st i, c t o + n 1 n }i e = c 1 tedfirmis not too f large,then W n,t+1 follows a normal distribution as n grows large.5 e f PROPOSITION 2 (Asymptotic Normality of W ): Let D denote the highest number of relationships n,t+1 n per firm in the economy. If there are no relationships in define D =1. For each t+1, define n n f G n n µ lim E ε p and σ2 lim Var ε p t+1 ≡n→∞ " i=1 i,t+1 (cid:12) t+1 # t+1 ≡n→∞ " i=1 i,t+1 (cid:12) t+1 # X (cid:12) X (cid:12) e (cid:12)e e (cid:12)e If there exists an integer m 3 such that (cid:12) (cid:12) ≥ 1 n m D n lim =0 (IA.14) n→∞ (cid:18) D n(cid:19) (cid:18) nσ t(cid:19) then, W d µ ,σ2 as n .6 n,t+1 −→N t+1 t+1 →∞ (cid:0) (cid:1) Prooffof Proposition 2. Provided that ε i,t are Bernoulli random variables, ε i,t 1, i. As n is a depen- ≤ ∀ G dency graph for the family ε n , t, it follows from Janson (1988, Theorem 2) that { i,t }i=1 ∀ e (cid:12) (cid:12)e (cid:12) (cid:12) n e ε d µ (t),σ2(t) i,t −→N n n i=1 X (cid:0) (cid:1) e The following corollary provides a more detailed characterization of the types of economies wherein W follows a normal distribution as n gets large. n,t+1 COROLLARY 1: Suppose σ < . If D = o n then log consumption growth is approximately normally f t n ∞ distributed as n gets large. (cid:0) (cid:1) Proof of Corollary 1. If σ < and D = o n then (IA.14) is satisfied. Hence W and ∆c are t n n,t+1 t+1 ∞ normally distributed as n grows large. (cid:0) (cid:1) f e Therefore,aslongasthenumberofrelationshipsofthemostconnectedfirmgrowslessthanlinearlywith n, W and thus, ∆c , follow a normal distribution as n grows large. n,t+1 t+1 f e C. U.S. Supplier–Customer Networks 5SeeBaldiandRinott(1989,Corollary2)forasimilarresulttothatinJanson(1988, Theorem2). 6Ifcondition(IA.14)issatisfied,then{εe i,t}n i=1 canbeinterpretedasam-dependentsequenceofrandomvariables. Namely, ifthedistancebetweenvariablej andkisgreaterthanmthenεe j,t isindependentofεe k,t,forallt. Todoso,however,anotion ofdistancebetwe(cid:8)enεe (cid:9)j,t andεe k,t needstobeproperlydefinedtoreconcilethepositionoffirmsiandjinGnwiththeirposition inthesequence εe l,t n l=1 . 15

(a) 1976 (b) 1977 (c) 1978 (d) 1979 (e) 1980 Figure IA.1. The figure shows supplier–customer networks from 1976 to 1980. 16

(a) 1981 (b) 1982 (c) 1983 (d) 1984 (e) 1985 Figure IA.2. The figure shows supplier–customer networks from 1981 to 1985. 17

(a) 1986 (b) 1987 (c) 1988 (d) 1989 (e) 1990 Figure IA.3. The figure shows supplier–customer networks from 1986 to 1990. 18

(a) 1991 (b) 1992 (c) 1993 (d) 1994 (e) 1995 Figure IA.4. The figure shows supplier–customer networks from 1991 to 1995. 19

(a) 1996 (b) 1997 (c) 1998 (d) 1999 (e) 2000 Figure IA.5. The figure shows supplier–customer networks from 1996 to 2000. 20

(a) 2001 (b) 2002 (c) 2003 (d) 2004 (e) 2005 Figure IA.6. The figure shows supplier–customer networks from 2001 to 2005. 21

(a) 2006 (b) 2007 (c) 2008 (d) 2009 (e) 2010 Figure IA.7. The figure shows supplier–customer networks from 2006 to 2010. 22

(a) 2011 (b) 2012 (c) 2013 (d) 2014 (e) 2015 Figure IA.8. The figure shows supplier–customer networks from 2011 to 2015. 23

D. Simulation of the Model This section describes the algorithm used to simulate the model. Let Z f W , T g Wi ,U h W−i , and e m(ε ) t+1 ≡ n,t+1 i,t+1 ≡ n,t+1 i,t+1 ≡ n,t+1 i,t+1 ≡ i,t+1 (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) denote four random f variableswhich aref f unctions of W , W f i , W−i , and ε , reespectively. Given n,t+1 n,t+1 n,t+1 i,t+1 ζ andq, I use the followingprocedureto compute the conditionalexpectationofZ , T , U , and t+1 t+1 i,t+1 i,t+1 e : f f f e i,t+1 (a) Determine the set of firms that initially face idiosyncratic shocks by drawing a Bernoulli random variable (with success probability q) per each firm. (b) Determine the set of relationships that transmit shocks by drawing a Bernoulli random variable per eachrelationship. Thesuccessprobabilityofrelationship(i,j)att+1isgivenbyp . Probabilities ij,t+1 p are drawn from a Beta distribution of parameter ζ . Higher values of p are { ij,t+1 }(i,j)∈Gn t+1 ij,t+1 assigned to relationships with lower betweenness scores to capture features of theedata. (c) Ceompute W by adding all firms affected by idiosyncratic shock at t+1. Compute We i by n,t+1 n,t+1 adding all firms in i affected by idiosyncratic shock at t+1. Compute W−i by adding all firms in f Gn n,t+1 f the complement set of i affected by idiosyncratic shock at t+1. Firms are considered to be affected Gn f byidiosyncraticshocksaccordingtothepropagationmechanismdescribedinthepaper. UsingW , n,t+1 Wi , and W−i , I compute Z as Z =f W , T as T =g Wi , and U n,t+1 n,t+1 t+1 t+1 n,t+1 i,t+1 i,t+1 n,t+1 i,t+1 f as U =h W−i . (cid:16) (cid:17) (cid:16) (cid:17) f i,t+1 f n,t+1 f f (c) Repeat steps(cid:16)(a), (b),(cid:17)and (c) 10,000 times. I set E[Z t+1 ζ t+1 ], E[T i,t+1 ζ t+1 ], and E[U i,t+1 ζ t+1 ] equal f | | | to their corresponding sample averages. To compute E[e ζ ], I only repeat steps (a) and (b). I i,t+1 t+1 | set E[e ζ ] equals to the sample averageover the 10,000 simulations. i,t+1 t+1 | Whenneeded,oneneedstorepeattheaboveprocedureforeachfirm. Thisturnsouttobecomputationally intensive. To reduce time, one can take advantage of the topology of i. In particular, if i is a tree, the Gn Gn following algorithm can be use to compute the probability that firm i is affected by an idiosyncratic shock. This algorithm exploits the fact that computing such probabilities can be framed as a recursive problem. Algorithm Firm i’s Probability (Gi, s , q) n t ( Description: Algorithm that computes firm i’s probability of facing shocks if Gi is a tree ) ∗ n ∗ Input: Gi (a tree), s (state of the economy), q. n t Output: Firm i’s probability of facing shocks at time t, π∗(s ) i t 1. if firm i has a no connections 2. return π∗(s )=q i t 3. else return Prob(i,Gi,s ,q) n t where Prob(i,Gi,s ,q) corresponds to the following recursive program: n t Algorithm Prob(i,Gi,s ,q) n t ( Description: Recursive algorithm that computes firm i’s probability of facing a shock ) ∗ ∗ Input: Tree Gi wherein node i is the root, s , and q. n t Output: π∗(s ) i t 24

1. Determine the set of children of node i in Gi, say .7 n C i 2. if = i C ∅ 3. return π∗(s )=q i t 4. else if every node in has no children i C 5. return π∗(s )=q+(1 q) 1 E (1 qp ) s i t − − k∈Ci − ikt t 6. else return π∗(s )=q+(1 q) 1 E (1 p Prob(k,T ,s ,q)) s i t − (cid:0) − (cid:2)Qk∈Ci − e ikt (cid:12) (cid:12) (cid:3)(cid:1) i,k t t where T i,k denotes the branch of tree Gi n that star (cid:0) ts at n (cid:2) o Q de k. e (cid:1)(cid:12) (cid:12) (cid:3) E. Figures and Tables 7Inarootedtree,theparentofanodeisthenodeconnectedtoitonthepathtotheroot. Everynodeexcepttheroothas auniqueparent. Achildofanodev isanodeofwhichv istheparent. 25

0.0 0.2 0.4 0.6 0.8 1.0 41 21 01 8 6 4 2 Relationship Betweenness Scores and Specificities in 1980 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 1.0 (a) 1980 51 01 5 0 Relationship Betweenness Scores and Specificities in 1981 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 1.0 (b) 1981 51 01 5 0 Relationship Betweenness Scores and Specificities in 1982 Relationship specificity erocs ssenneewteb pihsnoitaleR (c) 1982 0.0 0.2 0.4 0.6 0.8 51 01 5 0 Relationship Betweenness Scores and Specificities in 1983 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 (d) 1983 51 01 5 0 Relationship Betweenness Scores and Specificities in 1984 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 (e) 1984 51 01 5 0 Relationship Betweenness Scores and Specificities in 1985 Relationship specificity erocs ssenneewteb pihsnoitaleR (f) 1985 Figure IA.9. The figure shows the relation between betweenness and specificity scores of relationships in supplier–customer networks from 1980 to 1985. 26

0.0 0.2 0.4 0.6 0.8 51 01 5 0 Relationship Betweenness Scores and Specificities in 1986 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 (a) 1986 21 01 8 6 4 2 0 Relationship Betweenness Scores and Specificities in 1987 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 1.0 (b) 1987 51 01 5 0 Relationship Betweenness Scores and Specificities in 1988 Relationship specificity erocs ssenneewteb pihsnoitaleR (c) 1988 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 02 51 01 5 0 Relationship Betweenness Scores and Specificities in 1989 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 (d) 1989 02 51 01 5 0 Relationship Betweenness Scores and Specificities in 1990 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 1.0 (e) 1990 51 01 5 0 Relationship Betweenness Scores and Specificities in 1991 Relationship specificity erocs ssenneewteb pihsnoitaleR (f) 1991 Figure IA.10. The figureshows the relationbetweenbetweenness andspecificity scoresofrelationshipsin supplier–customernetworksfrom1986to 1991. 27

0.0 0.2 0.4 0.6 0.8 1.0 01 8 6 4 2 0 Relationship Betweenness Scores and Specificities in 1992 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 1.0 (a) 1992 21 01 8 6 4 2 0 Relationship Betweenness Scores and Specificities in 1993 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 1.0 (b) 1993 51 01 5 0 Relationship Betweenness Scores and Specificities in 1994 Relationship specificity erocs ssenneewteb pihsnoitaleR (c) 1994 0.0 0.2 0.4 0.6 0.8 51 01 5 0 Relationship Betweenness Scores and Specificities in 1995 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 (d) 1995 02 51 01 5 0 Relationship Betweenness Scores and Specificities in 1996 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 1.0 (e) 1996 06 05 04 03 02 01 0 Relationship Betweenness Scores and Specificities in 1997 Relationship specificity erocs ssenneewteb pihsnoitaleR (f) 1997 Figure IA.11. The figureshows the relationbetweenbetweenness andspecificity scoresofrelationshipsin supplier–customernetworksfrom1992to 1997. 28

0.0 0.2 0.4 0.6 0.8 06 05 04 03 02 01 0 Relationship Betweenness Scores and Specificities in 1998 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 (a) 1998 06 05 04 03 02 01 0 Relationship Betweenness Scores and Specificities in 1998 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 (b) 1999 06 05 04 03 02 01 0 Relationship Betweenness Scores and Specificities in 2000 Relationship specificity erocs ssenneewteb pihsnoitaleR (c) 2000 0.0 0.2 0.4 0.6 0.8 1.0 04 03 02 01 0 Relationship Betweenness Scores and Specificities in 2001 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 1.0 (d) 2001 06 05 04 03 02 01 0 Relationship Betweenness Scores and Specificities in 2002 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 1.0 (e) 2002 001 08 06 04 02 0 Relationship Betweenness Scores and Specificities in 2003 Relationship specificity erocs ssenneewteb pihsnoitaleR (f) 2003 Figure IA.12. The figureshows the relationbetweenbetweenness andspecificity scoresofrelationshipsin supplier–customernetworksfrom1998to 2003. 29

0.0 0.2 0.4 0.6 0.8 1.0 001 08 06 04 02 0 Relationship Betweenness Scores and Specificities in 2004 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 1.0 (a) 2004 051 001 05 0 Relationship Betweenness Scores and Specificities in 2005 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 (b) 2005 051 001 05 0 Relationship Betweenness Scores and Specificities in 2006 Relationship specificity erocs ssenneewteb pihsnoitaleR (c) 2006 0.0 0.2 0.4 0.6 0.8 051 001 05 0 Relationship Betweenness Scores and Specificities in 2007 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 (d) 2007 051 001 05 0 Relationship Betweenness Scores and Specificities in 2008 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 (e) 2008 051 001 05 0 Relationship Betweenness Scores and Specificities in 2009 Relationship specificity erocs ssenneewteb pihsnoitaleR (f) 2009 Figure IA.13. The figureshows the relationbetweenbetweenness andspecificity scoresofrelationshipsin supplier–customernetworksfrom2004to 2009. 30

0.0 0.2 0.4 0.6 0.8 1.0 051 001 05 0 Relationship Betweenness Scores and Specificities in 2010 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 1.0 (a) 2010 051 001 05 0 Relationship Betweenness Scores and Specificities in 2011 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 1.0 (b) 2011 052 002 051 001 05 0 Relationship Betweenness Scores and Specificities in 2012 Relationship specificity erocs ssenneewteb pihsnoitaleR (c) 2012 0.0 0.2 0.4 0.6 0.8 1.0 06 05 04 03 02 01 0 Relationship Betweenness Scores and Specificities in 2013 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 1.0 (d) 2013 05 04 03 02 01 0 Relationship Betweenness Scores and Specificities in 2014 Relationship specificity erocs ssenneewteb pihsnoitaleR 0.0 0.2 0.4 0.6 0.8 1.0 (e) 2014 05 04 03 02 01 0 Relationship Betweenness Scores and Specificities in 2015 Relationship specificity erocs ssenneewteb pihsnoitaleR (f) 2015 Figure IA.14. The figureshows the relationbetweenbetweenness andspecificity scoresofrelationshipsin supplier–customernetworksfrom2010to 2015. 31

Table IA.1 Performance of Centrality Portfolios in Manufacturing Thetablereportsmonthlyaveragerawreturns,alphasandloadingsfromthefive-factormodelofFamaandFrench(2015) for three portfolios constructed by sorting manufacturing stocks based on centrality: a portfolio that holds stocks on the lowest decile of centrality (Low), a portfolio that holds stocks on the highest decile of centrality (High), and a portfolio that is long on stocks on the lowest decile and short onstocks on the highest decile of centrality (Low - High). The bottom row provides thet-statisticsforthelowminushighportfolio. ManufacturingfirmsareassignedintodecilesattheendofOctobereveryyear andthevalue-weightedportfoliosarenotrebalancedforthenext12months. ThesampleisfromJune1976toDecember2016. Rawreturnsandalphasareinpercent. Raw 5-Factor Model Decile Return Alpha MKT SMB HML RMW CMA Low 2.21 1.24 1.02 0.53 -0.40 -0.34 0.06 High 1.42 0.46 0.94 -0.19 -0.14 -0.02 0.25 Low-High 0.78 0.38 0.08 0.73 -0.27 -0.32 -0.18 t-statistic [4.37] [3.23] [3.03] [17.36] [-4.91] [-5.62] [-2.19] Table IA.2 Performance of Centrality Portfolios in Service Thetablereportsmonthlyaveragerawreturns,alphasandloadingsfromthefive-factormodelofFamaandFrench(2015) for three portfolios constructed bysorting servicestocks based on centrality: a portfoliothat holds stocks on the lowest decile of centrality(Low),aportfoliothatholdsstocksonthehighestdecileofcentrality(High),andaportfoliothatislongonstockson thelowestdecileandshortonstocksonthehighestdecileofcentrality(Low-High). Thebottomrowprovidesthet-statistics forthelowminushighportfolio. ServicefirmsareassignedintodecilesattheendofOctobereveryyearandthevalue-weighted portfoliosarenotrebalancedforthenext12months. ThesampleisfromJune1976toDecember2016. Rawreturnsandalphas areinpercent. Raw 5-Factor Model Decile Return Alpha MKT SMB HML RMW CMA Low 2.56 1.70 1.11 0.47 -0.56 -0.41 -0.05 High 1.18 0.61 0.84 -0.25 -0.14 -0.24 -0.56 Low-High 1.41 0.67 0.29 0.75 -0.39 -0.17 0.52 t-statistic [4.72] [2.45] [4.37] [7.60] [-3.03] [-1.28] [2.66] Table IA.3 Performance of Centrality Portfolios in Manufacturing and Service Thetablereportsmonthlyaveragerawreturns,alphasandloadingsfromthefive-factormodelofFamaandFrench(2015) for three portfolios constructed by sortingmanufacturing and service stocks based on centrality: a portfoliothat holds stocks on thelowestdecileofcentrality(Low),aportfoliothatholdsstocksonthehighestdecileofcentrality(High),andaportfoliothat is long on stocks on the lowest decile and short on stocks on the highest decile of centrality (Low - High). The bottom row provides thet-statistics forthelow minushighportfolio. Manufacturingandservicefirmsareassignedintodeciles attheend of October every year and the value-weighted portfolios arenot rebalanced for the next 12months. Thesample is fromJune 1976toDecember 2016. Rawreturnsandalphasareinpercent. Raw 5-Factor Model Decile Return Alpha MKT SMB HML RMW CMA Low 2.31 1.36 1.04 0.52 -0.45 -0.36 0.04 High 1.41 0.51 0.93 -0.20 -0.14 -0.04 0.12 Low-High 0.89 0.45 0.11 0.72 -0.31 -0.32 -0.07 t-statistic [5.05] [4.08] [4.39] [18.27] [-6.06] [-5.95] [-0.91] 32

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