WORKING PAPER SERIES

Analysis of Upstream, Downstream & Common Firm 

Shocks Using a Large Factor-Augmented Vector 

Autoregressive Approach

Everett Grant

Amazon.com

Julieta Yung

Federal Deposit Insurance Corporation

Previously circulated as “Upstream, Downstream & Common Firm Shocks.”

This version: May 2024

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revision.
Analysis of Upstream, Downstream & Common Firm

Shocks Using a Large Factor-Augmented Vector

Autoregressive Approach*

Everett Grant 

Amazon.com

Julieta Yung 

FDIC

This version: May 2024

Abstract

We evaluate the roles of upstream (supplier-to-user), downstream (user-to-supplier)

and common factor shock transmission across rms by deriving inter-rm networks and

common factors from U.S. equities over 1989-2017. We overcome the econometric challenges of estimating the large factor-augmented vector autoregressive (FAVAR) system

by developing two alternative approaches: one prioritizing computational eciency and

the other providing the full posterior distribution of all model parameters and factors.

We nd that: (i) common factors drive an increasing variance share of returns; (ii) supplier shocks are more evident in equity price movements than downstream exposures;

(iii) removing the impact of common factors is increasingly important for revealing

inter-rm connections.

Keywords: large FAVAR; upstream versus downstream; shock propagation; common

factors; Bayesian estimation. JEL Codes: C11; C33; C55; E44.

*We thank Rosalind Bennett, Alex Chudik, Simon Fuchs, Domenico Giannone, Pavel Kapinos, Christoer

Koch, Karen Lewis, Karel Mertens, Jorge Miranda-Pinto, Jordan Pandolfo, Alex Richter, Mike Sposi, Luke

Watson, Kei-Mu Yi, Eric Young, and attendees of the 2018 Federal Reserve Bank of Dallas Banking and

Finance Workshop, 2019 SNDE, 2019 Federal Reserve System Committee on Macroeconomics conference,

8th Annual CIRANO-Sam M. Walton College of Business Workshop on Networks in Trade and Finance,

and 2021 Center for Financial Research at the FDIC Seminar for helpful comments and suggestions, as well

as Daniel Crowley for phenomenal research assistance. The views expressed in this paper are those of the

authors and do not reect the views of the FDIC or the United States. This publication and its contents

are not related to Amazon and do not reect the position of the company and its subsidiaries. The authors

performed much of the work on this paper prior to joining Amazon and the FDIC. Both authors acknowledge

the support received from the Federal Reserve Bank of Dallas while working on this paper.

EGrant.Econ@gmail.com.

Center for Financial Research, FDIC, Washington, DC 20429. jyung@fdic.gov.
1 Introduction

The broad transmission of economic shocks has been endemic to crises over the past few

decades, from the 2008 Global Financial Crisis, to the 2011 Tohoku earthquake's worldwide

footprint, to supply chain disruptions during and following the COVID-19 pandemic. The

frequency and severity of these events has led to a focus on discerning what channels shocks

ow through as they spread across rms and industries, and how these mechanisms have

evolved over time.1 To help understand such shock diusion, we evaluate the importance of

upstream versus downstream production network exposures, as well as the role of common

factors in the propagation of shocks across publicly traded U.S. rms.

We extend the literature estimating inter-rm networks from vector-autoregressions (VARs)

of their asset prices (e.g., Bonaldi et al., 2015; Demirer et al., 2018; Diebold and Yilmaz,

2009, 2014, 2016; Grant and Yung, 2021) by utilizing a factor-augmented VAR (FAVAR)

approach. The FAVARs are estimated on rms' daily equity returns to derive their connections with one-another and the common factors, in a novel way that can separate out

the impact of the factors and model them as explicit nodes within the network. We provide two alternative methods to estimate the large FAVAR models on our panel samples

of 524 to 1,548 U.S. rms from 1989 to 2017, which deal with the curse of dimensionality

and avoid over-tting the data: one leveraging recent advances in big data analysis to optimize for computational eciency; and a Bayesian approach with priors allowing for variable

shrinkage and scalable posterior draws to estimate the full joint posterior distribution of the

parameters and common factor dynamics.

Our results indicate that common factors play an important role in the transmission of

shocks, with the variance share of the top three common factors being over a third in our

most recent samples. We also nd that exposure to suppliers is economically important and

statistically signicant, with a 0.62 average correlation between sectoral upstream exposure

networks calibrated to U.S. input-output tables and the equity return derived networks.

However, the downstream exposure appears more muted, with statistically insignicant average correlations of only 0.22 between the macroeconomic exposure in this direction and the

equity return networks. This behavior is suggestive of a low short-term elasticity of substitution across inputs passing shocks on from upstream but greater exibility with downstream

customers. Our results are consistent with the empirical literature nding signicant exposures to suppliers, such as Menzly and Ozbas (2010) studying cross-predictability of equity

returns based on lagged returns in direct supplier and customer industries, and Boehm et al.

1The importance of input-output, nancial, trade and common shock transmission at country and sector

levels has been explored in many papers, including Brooks and Del Negro (2006), Burstein et al. (2008),

di Giovanni and Levchenko (2010), Grant (2016), Imbs (2004), and Johnson (2014).

1
(2019) and Carvalho et al. (2016) analyzing the ramications of the 2011 Tohoku earthquake.

We nd that the reverse direction does not appear to be as relevant to publicly traded rms,

extending the results of the latter papers beyond the context of this natural experiment. Our

analysis recommends the use of the idiosyncratic equity network estimation method that isolates the impact of common factors to study bilateral transmission across rms, especially

towards the end of our sample when common factors become a greater driver of the equity

returns and confound the inter-rm connections.

Our rst large FAVAR estimation approach is a three-step method that obtains the common factors and their loadings using principal component analysis (PCA), estimates a VAR

of the dynamic relationship between the factors, and utilizes the Chudik et al. (2018) variable selection method to estimate a VAR of the idiosyncratic rm returns (their returns

minus the impact of the common factors). This method is computationally ecient taking

seconds to run, deterministic (i.e., not subject to solver randomization), and parallelizable

to accommodate very large panels. The second approach is to caste the large FAVAR as

a Bayesian state-space model. The priors are adapted to a large FAVAR context to keep

the estimation numerically tractable and apply variable shrinkage, implementing Minnesotastyle priors with articial dummy observations, using normal-Wishart priors that allow for

draws from the ecient matrix normal distribution, and the implementation of the Durbin

and Koopman (2002) simulation smoother provided in Jarocinski (2015), which we further

optimize for Matlab data structures to store our large system in memory. The state-space

method is more computationally intensive than the rst approach (taking three days to

several weeks to run); however, it avoids the potential bias and compounded uncertainty

inherent in a multi-step estimation process and provides the joint distribution for the system of common factors and inter-rm networks. The inter-rm networks are derived using

generalized forecast error variance contributions from the FAVARs to infer the magnitude

and direction of the connections. The FAVAR structure of our model allows us to derive

rm networks at both the total return level inclusive of the eects of the common factors in

the individual rms' returns, and at the idiosyncratic return level, with the inuence from

factors attributed to separate nodes.

Our rm equity return based networks have several benets: they are derived from

publicly available data; they can be estimated in real-time; and they reveal rm heterogeneity

and avoid the aggregation bias inherent in sectoral level input-output tables. Alternately,

data on cross-rm input usage, supplier connections, etc. is typically available at a lower

frequency  making it dicult to estimate the propagation of transient shocks  and is

either aggregated, available for a small sample of rms, or comes from condential microdata.

To evaluate the upstream versus downstream transmission channels, we develop an il2
lustrative DSGE model by expanding that of Baqaee (2018) from a one period setting to a

multi-sector model with inter-temporal assets.2 To assess the relative historical importance

of upstream versus downstream rm exposures we: calibrate the DSGE model's parameters

to U.S. input-output tables; calculate the sectoral upstream and downstream exposures as

dened by those parameters; then compare these to the propagation of realized rm shocks

as embedded in the equity return based networks that the large FAVARs produce.

The econometric model yields three signicant common factors for the U.S. equity market,

the rst of which is highly correlated with the stock market beta and economic growth, as

measured by U.S. broad equity index returns, industrial production and real GDP. We label

the second factor the price one, as it is highly correlated with U.S. PPI, CPI, and the

value of the dollar. The third factor has a strong correlation with commodities, particularly

petroleum-based energy ones. These common factors  especially the rst, market beta

factor  become more important over the period we study, explaining 11.7% of the equity

return variation over the rst ten years of the sample and 35.0% over the nal ten. The DSGE

model aligns with these ndings, as in it rms' equity returns depend on three aggregates:

the market beta, accounting for the stochastic discount factor and aggregate growth; the

price level; and the supply of raw inputs.3

Evaluating the equity return networks against the DSGE model's input-output based

upstream and downstream exposures, we nd that upstream exposures (shocks to a rm's

suppliers) are more relevant than downstream exposures (shocks to customers). The correlations are economically and statistically signicant between the equity networks and the

upstream networks; however, that is not the case for the downstream ones. In addition, the

idiosyncratic equity response networks have about 30% higher correlations with the upstream

exposure networks than those that include the common factors. That the idiosyncratic return networks have higher correlations than the total return ones accords with the DSGE

model, where after the common factors are removed from equity returns one can derive a

VAR(1) relationship between them, with exposures through the upstream and downstream

centralities entering the innovations. These results are consistent across both FAVAR estimation approaches, with the Bayesian state-space method that accounts for model uncertainty

having marginally lower correlations, which remain statistically signicant in the same instances as the three-step approach. Using simulations, we show that the version of our

2Papers on endogenous network formation and input-output linkages include Atalay et al. (2011), Gabaix

(2011), Acemoglu et al. (2012), Acemoglu et al. (2016), Acemoglu et al. (2017), Acemoglu and Azar (2020),

Taschereau-Dumouchel (2017), Obereld (2018), and Bernard et al. (2022). See Carvalho and Tahbaz-Salehi

(2019) for further literature relating production networks to macroeconomic uctuations.

3Factor structure has been previously introduced into DSGE models to understand shock transmission

(Giannone et al., 2006), study contributions of idiosyncratic and aggregate components to the macroeconomy

(Foerster et al., 2011) and relate macroeconomic shocks to the factor space (Onatski and Ruge-Murcia, 2013).

3
empirical networks that disentangles the eects of common factors is better able to reect

bilateral connections across rms, consistent with earlier work separating systematic from

idiosyncratic components in nancial networks (e.g., Barigozzi et al., 2014; Guðmundsson

and Brownlees, 2021; Brownlees et al., 2021). Many prior papers in the network estimation

literature do not do so, with their networks more reective of similar loadings on common

factors than of the bilateral relationships between network members they often purport to

estimate.

The prominence of upstream versus downstream shock transmission is important in the

context of the broader theoretical literature following Long and Plosser (1983) examining

multi-sector economies. This literature diers on which of these two channels are operational.

For example, Acemoglu et al. (2012) concluded that under Cobb-Douglas intermediate input

aggregation, an industry's impact on the aggregate economy depends only on its role as a

supplier of inputs through propagation from upstream, and not as a consumer.4 Further,

Section 4 of Baqaee (2018) showed these results hold under a more general set of models than

those with Cobb-Douglas intermediate input aggregation. The models of Johnson (2014),

Barrot and Sauvagnat (2016), and Baqaee and Farhi (2019) likewise have downstream but

not upstream shock propagation. Our empirical ndings support the many papers that

rely on Cobb-Douglas input aggregation or other modeling simplications that minimize

exposure to downstream rms. Alternately, Baqaee (2018) and Luo (2020) achieved both

downstream and upstream shock propagation by including rm entry-and-exit and a credit

channel, respectively.

The relative signicance of the upstream exposures could indicate a low short-term elasticity of substitution across inputs, which would match work at the country level. Examining

trade across 30 countries, Ng (2010) found that bilateral trade in complements/upstream intermediate goods contributes to cross-country business-cycle comovement, while trade in

substitutes/downstream nal goods reduces it, with a net positive impact of trade on comovement. Additionally, Burstein et al. (2008) and Johnson (2014) identied that low

elasticities of substitution between inputs is key to explaining the degree of synchronization

in international business cycles, with Miranda-Pinto (2021) nding that the elasticity of substitution between intermediates and labor in particular can have a marked impact on GDP

volatility. Relatedly, Barrot and Sauvagnat (2016) found that production elasticities near

zero best match real world shock amplications in a calibrated network model, and Atalay

(2017) found that strong input complementarities play an important role in industry level

shock transmission.

The next section introduces our econometric model of rm exposures from equity returns

4Under Cobb-Douglas, osetting price and quantity eects remove propagation to suppliers (Shea, 2002).

4
and our two proposed large FAVAR estimation approaches. Section 3 describes the data, then

the estimated common factors and equity based networks are outlined in Section 4. Section

5 summarizes the DSGE model's main results, including simulations that illuminate the

upstream versus downstream channels for shock propagation. Section 6 compares the equity

based networks with input-output derived upstream and downstream exposures to assess

their relative importance in rms' equity responses to one another. Section 7 concludes.

2 Econometric Model of Firm Equity Returns

In this section, we rst describe the structure of the large FAVAR framework that we use

to model publicly traded U.S. equity returns, separating common factors from idiosyncratic

rm returns. Second, we provide two alternative methods to estimate the system: one

leveraging recent advances in big data analysis to optimize for computational eciency, and

the other, a Bayesian estimation approach to estimate the full joint posterior distribution of

the parameters and common factor dynamics. Finally, we propose an econometric approach

to derive inter-rm networks once the system is estimated with either method, capturing

how common and idiosyncratic shocks propagate across the rms and factors.

Consider a panel dataset with T observations and N rms, where Rt = (R1t, R2t, ..., RN t)

0

represents the vector of observed log equity returns at time t. Firm returns are broken into

three components, such that Rt = RF

t + RIt + τt

. The rst component, RF

t

, is driven by a

set of K common factors with heterogeneous loadings across rms: RF

t ≡ ΛFt

, where Λ is

an N × K loading matrix on the factors and Ftis the vector of those factors. The second

component, RI

t

, captures idiosyncratic rm returns unrelated to the common factors, but

possibly related across rms or over time. The nal term, τt

iid∼ N (0, T ), captures transient

noise in the equity returns. The covariance matrix T is assumed to be diagonal and relatively

small.

Individual rms are assumed to be small enough relative to the whole economy that

their idiosyncratic components do not inuence the common factors, which follow a VAR(1)

process:

Ft = ΓFt−1 + ηt; ηt

iid∼ N (0, Υ), (1)

where Γ is a K × K matrix, and ηtis a K × 1 vector of shocks with covariance Υ. The Ft

may reect economy-wide macroeconomic shocks, or those for individual industries, regions,

etc., with the factors and rm-specic loadings being recovered from the data with minimal

econometric restrictions used for identication.5 The dynamics between the idiosyncratic

5Foerster et al. (2011) showed that the eects of common factors on all industries can create estimation

5
rm returns are also modeled as a VAR(1):

R

I

t = ρ0 + ρRIt−1 + t

, (2)

where ρ0 is N × 1, ρ is N × N, and t

iid∼ N (0, Σ). This structure implies that rms'

innovations, t, may be cross-sectionally correlated within a period, reecting direct exposure

to one-another or similar underlying economic conditions, but are serially uncorrelated, as

equity prices quickly adjust to reect new information.6

There are two primary challenges with estimating this system: the factors and idiosyncratic returns are not directly observed; and the curse of dimensionality and potential for

over-tting given the large parameter space. In the following subsections, we describe two

dierent approaches to estimate the model: 1) a three-step method that estimates the factors

with PCA, models Equation (1) with a VAR(1) on those factor series, then estimates the

idiosyncratic VAR parameters with an ecient, parallelizable variable selection method; and

2) a Bayesian Gibbs sampling method to estimate the state-space formulation of the model.

Further details for each method can be found in the Online Appendix. The key trade-os

are that the rst method is highly ecient and scalable, while the second method is computationally more expensive but explicitly estimates the joint posterior distribution of the

model's parameters, including the unobserved common factors' uncertainty and dynamics.

2.1 Three-Step Estimation Method

This rst estimation approach combines two major tools designed to address big data problems: PCA and variable selection methods. This technique is in line with the standard

approach in the FAVAR literature following Bernanke et al. (2005) to estimate the unobserved factors and loadings with PCA, then separately estimate the VAR model. PCA is

a computationally ecient, deterministic, semi-parametric approach to uncover the space

spanned by the common components, as set forth in the literature following Stock and Watson (2002) that uses PCA to summarize the information in large macroeconomic datasets.

issues when trying to calculate inter-industry shock propagation. They use input-output and inter-sectoral

capital network data to adjust for this. Our goal is to estimate inter-rm networks without imposing any

ex-ante network assumptions, so we make the trade-o to not apply their adjustments and estimate the

common factors purely empirically. They found that aggregate factors contributed less volatility over their

sample period, while sectoral shocks did not change in importance and therefore became relatively more

signicant after the mid-1980s.

6We indeed nd no evidence of serial correlation in the errors. The choice of one-lag VARs for modeling

factors and idiosyncratic returns in Equations (1) and (2), respectively, is supported in our data by the

standard Bayesian Information Criterion (BIC). Further, when including additional lags in either the Γ or ρ

matrices, parameter estimates for longer lags are generally near zero and not statistically signicant.

6
Step one is to select the number of common factors, K, with the panel Bayesian Information Criterion (BIC) method from Bai and Ng (2002)  which accounts for potential

correlation in the idiosyncratic errors  and apply covariance matrix-based PCA inclusive

of the means on the rm return series to estimate the factors and their loadings. Once the

common variation for each rm (ΛFt) is estimated with these, the idiosyncratic rm returns,

RI

t

, are obtained under the simplifying assumption that the τ are negligible: RI

t = Rt −ΛFt

.

The second method below relaxes this assumption. Step two is to estimate Equation (1) via

OLS on the factor series.

Step three utilizes the One Covariate at a time Multiple Testing (OCMT) variable selection procedure of Chudik et al. (2018) to deal with the curse of dimensionality and avoid

over-tting to estimate the large VAR of rm idiosyncratic returns in Equation (2). The

OCMT procedure is intuitive in that one need only run a series of OLS regressions of the

dependent variable on the potential explanatory variables, testing whether they have a statistically signicant relationship with the dependent variable based on their coecients'

t-statistics. The key feature of this approach is that the critical values for accepting a variable are adjusted for the fact that this test is repeated for the many potential explanatory

variables. The prior literature on VAR-based network estimation following Diebold and Yilmaz's work has generally used LASSO, Ridge, or adaptive elastic-net for variable shrinkage

and selection (e.g., Diebold and Yilmaz, 2009, 2014, 2016; Bonaldi et al., 2015; Demirer

et al., 2018; Grant and Yung, 2021). The OCMT procedure has several benets over those

algorithms: it is computationally faster, more ecient and parallelizable; it is statistically

founded with clear individual variable inclusion rules; and there is not the randomness that

can occur with the other methods due to cross-validation sampling selection, optimizer seeding, etc. For example, OCMT is over twenty times faster than adaptive elastic-net when

estimating our networks. To estimate Equation (2), we perform OCMT on each row of it,

combine the coecient vectors to obtain ρ0 and ρ and estimate Σ from the residuals.

2.2 State-Space Estimation Method

In this section, we cast the common factors and idiosyncratic returns as unobserved state

variables and reformulate the whole system as a state-space model. This system is estimated using a Gibbs sampling approach following Carter and Kohn (1994), which alternates

between drawing from the conditional distributions of the VAR parameters for the observation and state equations given the unobserved series, and drawing from the conditional

distribution of the Ft and RI

t

series.

7
The observation equation for the state-space system is:

Rt =













λ11 λ12 . . . λ1K 1 0 . . . 0

λ21 λ22 . . . λ2K 0 1 . . . 0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

λN1 λN2 . . . λNK 0 0 . . . 1













"

Ft

RI

t

#

+ τt = [Λ IN ]

"

Ft

RI

t

#

+ τt. (3)

The state equation is:

"

Ft

RI

t

#

=

"

0K×1

ρ0

#

+

"

Γ 0K×N

0N×K ρ

# " Ft−1

RI

t−1

#

+

"

ηt

t

#

. (4)

We employ Bayesian VAR estimation in the vein of Litterman (1986)'s Minnesota prior

to address the large number of parameters in Equation (2), utilizing dummy observations

following Ba«bura et al. (2010) to keep the model estimation numerically tractable. The Λ

coecient priors are centered on the PCA factor loadings, and the Γ, ρ0, and ρ coecients

are centered at zero, reecting a belief that the underlying data series follow random-walks

in log levels.

This approach departs from the variable shrinkage and selection methods commonly

used in the literature following Diebold and Yilmaz on estimating networks from VARs

in two ways. First, as with the previous estimation method, this approach incorporates

common factors, yet this time the factors and their dynamics are jointly estimated along

with the remainder of the model. Second, the Bayesian methodology extends the LASSO

and Ridge estimation procedures generally used in the literature by adding the full posterior

distributions to calculate the network distributions from, with the posterior mode for the

coecients being LASSO under Laplace priors on them, and Ridge the mode under the

Gaussian priors that we employ.

2.3 Firm Network Estimation

After we estimate the system with either the three-step or state-space approach, we derive the

inter-rm network from generalized forecast error variance contributions (GFEVc). Other

papers, including several in the Diebold-Yilmaz network series and thereafter, have used the

generalized forecast error variance decompositions (GFEVD) or generalized impulse response

functions (GIRF) of Pesaran and Shin (1998) to derive network edges given an estimated

VAR system, the formulas for which we provide in the Online Appendix. We instead adjust

the GFEVD to create our GFEVc's. The dierence is that in our GFEVc's we do not divide

through by the equity variance adjustment in the denominator. We do this because those

8
adjustments vary by rm and over time, and we would like the edge weights to be comparable

across both dimensions.

We calculate the GFEVc's from the reduced form of the VAR created by stacking the

equations in our system:







Rt

RI

t

Ft





 =







ρ0

ρ0

0K×1





 +







0N×N ρ ΛΓ

0N×N ρ 0N×K

0K×N 0K×N Γ













Rt−1

RI

t−1

Ft−1





 +







t + Ληt + τt

t

ηt





 . (5)

For notational convenience, we label the coecient matrix on the lag term of this reducedform system A1. The covariance of the errors is:

Θ =







Σ + ΛΥΛ0 + T Σ ΛΥ

Σ Σ 0N×K

ΥΛ0 0K×N Υ





 ,

where by construction, innovations in t and τt are independent of one-another and of ηt,

yielding the zero blocks in Θ. The formula for the one-period ahead GFEVc network is:

GF EV c =



Θ

2 + (A1Θ)2



Diag(Θ)−1, (6)

where the Diag operation yields a square matrix with the diagonal entries of the given input

along the diagonal and zeroes elsewhere, and the exponents are all applied to the individual

elements of the matrices. This formula provides a network adjacency matrix with the edge

sources along the columns and the destinations along the rows. We then create two types of

networks from this matrix.

2.3.1 Total-Return Networks (R to R)

In our rst network type, the edges from each source rm, s, to each destination one, d,

reect the response of the destination's total returns, Rt, when there are shocks aecting the

source's total returns. In Equation (6), this would mean the entry for column s and row d.

This specication forms a weighted, directed network capturing the way that rms' returns

respond to one another, regardless of whether the innovations are to the rms' idiosyncratic

returns or the aggregate factors. These networks are similar to those in Demirer et al. (2018)

and Grant and Yung (2021) in this way.

9
2.3.2 Idiosyncractic Return Networks (RI & Factors to R & Factors)

For our second network type, we distinguish the eects of rms' idiosyncratic shocks from

those to the common factors. When doing so, we take the novel approach of explicitly

treating the factors as nodes in the network. In these networks there are N + K nodes, with

one for each of the rms and factors. The edge from rm s to rm d will be the expected

variance contribution to d's returns when shocks aect rm s's idiosyncratic returns, and

likewise for a shock aecting factor k = 1, 2, ..., K. In these cases, we take the entry for

column N + s and row d of Equation (6) as the inter-rm edge, and the entry for column

2N + k and row d for the edge from factor k to rm d. In the case where the destination is

factor k, then the edge from rm s to it will be the entry in column N + s and row 2N + k

 giving a zero eect  and the edge from another factor, f, to it will be the entry in

column 2N + f and row 2N + k. Additionally, in our analysis below we use the subset of

this network formed by the connections from rms' idiosyncratic returns to other rms and

refer to these networks as the RIto R ones.

To understand the dynamics captured by the two network types, we simulate data and

run our procedure above for estimating inter-rm networks, both when including the eects

on common factors between rms with R to R networks, and when separating them out

with RI & Factors to R & Factors networks (Online Appendix Section D.4). The results

clearly show that the R to R networks identify the organization of the system around

the factors very well, while the RI & Factors to R & Factors networks are better able to

eectively estimate the bilateral simulated rm relationships by separating out the factors'

inuence. We also compared several methods used in the prior literature (i.e., LASSO with

GIRFs based networks, Ridge with GFEVD, AEN). The methods used in the prior literature

produce networks with similar structure to the R to R networks, demonstrating the value

of our method for identifying bilateral rm relationships. These simulation results are in line

with the main arguments of Bailey et al. (2016) and Hale and Lopez (2019) that one needs

to account for common factors before estimating a network in order to properly recover the

bilateral connections between its members.

3 Data

To select our sample of U.S. rms, we take the union of all rms that are in the top 25% by

market capitalization for any year in our sample. If a rm is in our sample at any point, then

we obtain its equity pricing data for as long as recorded. We do this to lter out the smallest

rms and ensure that those in our sample have actively traded, liquid equity securities

10
that are highly researched and followed, providing them with accurate price discovery. As

such, the ecient markets hypothesis indicates that the rms' equity prices should reect all

available information about them, including how they are connected through the channels

we wish to study. Our dataset includes the daily Bloomberg closing prices for 5, 454 rms

between December 30th, 1988 and June 20th, 2017. The closing equity values are total-return

indices inclusive of returns from dividends to avoid spurious price jumps when dividends are

paid that do not reect a change in the valuation of the underlying rm. We also gather the

Bureau of Economic Analysis (BEA) sector and Bloomberg industry for each rm and collect

U.S. input-output use tables from the BEA for 1997 through 2015 for our macroeconomic

exposure networks described below.

We analyze both a balanced panel of rms to study the long-term rm network, and

unbalanced rolling panels to account for rm entrance and exit from the equity markets.

Specically, we examine the set of 524 rms continuously traded throughout our whole

sample  both over the full period and in rolling 10-year periods  and broader rolling

samples of rms continuously traded over each 10-year window, with up to 1,548 rms.

4 U.S. Inter-Firm Networks

This section provides empirical results from applying our methodology to estimate inter-

rm networks from U.S. equity daily log returns. In Section 4.1, we examine the networks'

estimated common factors, with a focus on their sample variance shares over time and

comparisons with macroeconomic and nancial indicators. In Section 4.2, we utilize spatial

graphical algorithms to depict the networks and study the evolution of their dynamics.

4.1 Common Factors

We apply our three-step and state-space methods to our U.S. equity data and here focus on

the common factors obtained by their PCA and Bayesian estimation approaches, respectively.

In selecting the number of common factors, K, the Bai and Ng (2002) panel information

criteria suggests a range between one and three common factors in U.S. daily log equity

returns, with a mode of three. The top three PCA factors combined account for 27% of the

cross-sectional variation in the entire dataset.7 Subsequent PCA factors each contribute less

than 2%, so using three factors as suggested by the statistical test is consistent with the

7Our nding that equities are roughly 70% driven by idiosyncratic rm shocks is in line with Campbell

et al. (2001), who found that over the period from 1962 to 1997, U.S. equities were 17% market driven, 12%

by industry developments, and 71% by idiosyncratic considerations.

11
scree plot jump selection approach typically used to select the number of factors in PCA

analysis.

The cumulative variance share of the top three factors increased signicantly the past

several decades, from around 12% at the start of our sample period in the early 1990s to over

35% towards the end in the mid-2010s.8 The variance share for the rst factor, for example,

increased from 8.6% over the 1989-1998 period to 31.2% over 2008-2017, with the largest

increase occurring when 2008 entered the rolling windows. The second and third factors

saw their variance shares increase by just over a fth with 0.5% and 0.2% increases. These

results suggest a signicant change in the importance of the common factors following the

Global Financial Crisis, with implications for portfolio diversication strategies.9 Further,

the 2010-2017 variance shares for the top three factors remained elevated  they explain

25.1%, 2.4%, and 1.2% of the variance, respectively  even when not including the 2008-2009

crisis period with particularly high equity correlations.10

The rst factor loads positively on all rms in the sample. This implies that positive

shocks to the rst factor translate into higher equity returns for all rms. In fact, the rst

factor series is almost identical to the sample average log returns for each day, both in levels

and in year-over-year changes. For this reason, we refer to this factor as the market beta,

reecting both time variation in discount factors or risk premia, and economic growth, akin

to the Fama and French market risk factor (Fama and French, 1995).11 The rms with the

highest loadings on the rst factor are predominantly from the technology, consumer cyclical

and nancial sectors, while those with the lowest loadings are royalty trusts, consumer noncyclical and utility rms (refer to the Online Appendix for details). This makes intuitive

sense as the former are pro-cyclical sectors, while the latter are generally considered passive,

acyclical investment sectors.

To help interpret the dynamics of each factor, Figure 1 compares the year-over-year

changes of each PCA factor in black to the year-over-year changes in the median of the

corresponding Bayesian factor in blue, in addition to the shaded 68% and 95% credible

intervals of the latter. We use year-over-year changes rather than levels to avoid spurious

correlations in these time-series data. Each of the three rows corresponds to one of the

8The Online Appendix contains time series plots of the cumulative sample variance shares explained by

the top three factors in rolling ten year rm samples, and for our long-run balanced panel.

9Note that the structures of the idiosyncratic networks before and after the Global Financial Crisis were

highly correlated at around 95%; however, the inter-sector sums increased about 2.8× on average.

10Bartram et al. (2018) also found an increasing share of rm level equity returns from common rather than

idiosyncratic factors. Studying the period 1965-2017, they found that average idiosyncratic risk declined to

an all-time low at the end of their sample.

11The idea of time variation in risk premia (and hence investors' expected stochastic discount factor) is

consistent with empirical evidence from the nance literature, documenting its importance in accounting for

excess volatility in asset prices and predictability of returns (e.g., Cochrane 1991, 2011).

12
factors, and within each plot the factors are compared to similarly moving real economic or

nancial indicators in red.

Panel (a) shows how year-over-year changes in the rst factor compare with those of the

most commonly referenced U.S. equity market average, the S&P500 Index. The correlations

between the S&P500 Index and both the rst PCA and Bayesian factors are 0.82, supporting

the interpretation of the rst factor as the market beta. Panel (b) shows that our rst factor

broadly reects the growth of the U.S. economy, as it co-moves with the U.S. industrial

production index, with year-over-year change correlations around 0.56. Results in the Online

Appendix show similar results when U.S. growth is measured using real GDP.

Fluctuations in the second factor closely align with movements of the U.S. price level.12

In the second row of Figure 1, we compare the second factor with dierent measures of

ination, including the year-over-year changes in the inverted Consumer Price Index (CPI)

in Panel (c) and the inverted Producer Price Index (PPI) in Panel (d). The correlation

of year-over-year changes in the second Bayesian factor and inverted PPI is 0.60, and that

with inverted CPI is 0.47. Note that the PPI correlation being higher than the CPI one

ts with our results below that the rm level equity market networks are more correlated

with the upstream rather than downstream exposures. Similarly high correlations are found

when comparing the second factor with other U.S. price-level related indices, including the

trade weighted value of the dollar, dollar-euro exchange rate, dollar-British pound exchange

rate, and U.S. 10-Year breakeven ination calculated from TIPs and nominal U.S. Treasury

bonds.

The second factor loads negatively on the energy, nancial, basic materials, and utilities

sectors, and positively on the technology sector. All of the top ten rms loading on the

second factor are technology companies, and the bottom ten consists of nine energy rms

and a petroleum shipping company.

The third factor has large positive average loadings for the energy sector, followed by

the technology and base materials sectors. Nine of the top ten rms by their loadings on

the third factor are energy related. Figure 1 displays the movements of the third factor

relative to the year-over-year changes in the Goldman Sachs Commodity Index (GSCI) in

Panel (e) and the price of Brent crude oil in Panel (f). These observations indicate that

the third factor is similar to a raw commodity input measure, with the correlation between

price changes in oil and this factor especially high at 0.68. We also compare the third factor

against various components of the GSCI index and nd the correlations to be positive except

for cocoa, and generally economically signicant in magnitude. The highest correlations are

12A role for price pressures in the inter-rm network is supported by, for example, Smets et al. (2019),

who found evidence of ination being passed through production networks.

13
with the petroleum-based energy commodities, followed by industrials and then precious

metals. Additionally, around 2009 commodities seem to go from the third to the second

most important common factor in our sub-sample analyses  possibly because of a lower

ination environment making the price level less important, while there were large subsequent

commodity price changes. If one looks on either side of this break then the pertinent factor

has a greater correlation with commodity prices. For example, when looking at all rms in

our sample traded continuously from 2008 through 2017, the correlation of the year-over-year

changes in the second factor with oil is 0.77, and that for the GSCI is 0.79. Around this

time the credible interval on the second factor also widens signicantly, suggesting a change

in the underlying data generating process.

4.2 Inter-Firm Equity Return Derived Networks

The inter-rm networks that we estimate are complex objects, with N2 or (N +K)

2

elements,

depending on whether the common factors are included or not. To help understand and

visualize these networks, we employ the ForceAtlas2 method from Bastian et al. (2014)

that is commonly used across disciplines to plot networks. ForceAtlas2 is a force-directed

layout algorithm to display network spatialization, transforming a network into a map where

nodes with greater connectedness are closer together. At a high level, all of the nodes are

repulsed from one another like charged particles, while edges attract their nodes like springs

 yielding the name for this class of algorithm, spring plots. The nal node positions

provide a balanced state, helping to interpret the data without having to incorporate any

other attributes of the network members. To read these plots, think of a map without a key

showing the direction of true north, nor a scale. In that case, as with these plots, the precise

orientation of the gures is not informative and rotations do not have a clear meaning, but

the relative proximity of features on the plots to one another and the center of the gure do,

as do any clusters that arise and inform the underlying topology. This technique is superior

to other network visualization methods such as heat maps because the number of network

members makes many other methods hard to read, and the spring plots are able to capture

third party or greater relationships. For example, if a rubber and a glass manufacturer both

have strong ties to Ford, but weak ones with each other, then they would still be close in

these gures because they would both be near Ford.

Figure 2 provides spring plots for R to R total-return networks, inclusive of the common

factors, for dierent time periods. Panel (a) displays the long-run inter-rm network of the

524 rms continuously traded from 1989-2017, and Panels (b), (c) and (d) show the sets

of all rms traded within dierent decades, with about 1, 500 rms in each of those plots.

14
Figure 3 includes similar plots, but for the RI & Factors to R & Factors idiosyncraticreturn networks, which split out the common factors and treat them as network nodes

themselves. Note that the parameters entered into the ForceAtlas2 algorithm for these are

the same, making them comparable. Each circular node in these gures represents a rm

and is colored by its BEA sector. We further distinguish the Real Estate Investment Trust

(REIT) sector (in gray) from the nancial sector given its distinct return prole. In Figure

3, the factors are represented by purple stars with the factor number above them.

Figure 2 (a) reveals that sectoral clusters are an important feature of the network, with

rms within the same sector grouped together, such as utilities (blue) and commodities

(orange) on the top right. Additionally, the nance (black), REITS (gray) and consumer

(yellow) sectors are near the center of the network. This observation coincides with seven

of the top nine rms by the sum of their weights out to others being nancial rms, and

the analysis of Grant and Yung (2021) where nancial rms were at the center of the global

rm network. Further showing the importance of nance, the number two and three rms

are industrial diversied rms, and that third rm  General Electric  was designated a

non-bank systemically important nancial institution by the Financial Stability Oversight

Council due to its high level of nancial dealings up until it signicantly changed its businesses in June 2016.

The rolling samples in Panels (b)-(d) also show the nance sector typically at the center in

all three decades and the clustering by sector is present in the dierent periods. Notably, the

networks show rms becoming more tightly grouped over time regardless of sector, suggesting

greater equity market integration and matching the increased common factor variance shares

mentioned above. Due to this feature, these plots  and the other network estimation

methods used in the past in this literature  become less informative about individual

connections between rms, as all of the equities move more with the common factors and

fall into one compact cluster at the center of the network.

Figure 3 shows the idiosyncratic-return networks that separate the eects of the common

factors from the individual equities, and in so doing unfold the underlying rm-to-rm

connections that we want to study. In Panel (a), one can see that the rst factor  the

market beta  is at the center of the network denoted by the purple star, and that nance

rms are near it. What can be further seen is that REITs form a distinct gray cluster

farther from the center of the network, unlike in Figure 2 where they are nearly dead center

and indistinguishable in location from the other nancial rms. Additionally, the commodity

rms are on the northern periphery of the network, near the third commodity factor, and the

consumer sector rms are close to the second, price-level factor. This implies that the energy,

base material, and utility rms near the third factor do not comove as much with the broad

15
market factor, and the commodity factor is more inuential for them, while consumables are

more linked with prices.

Panels (b)-(d) show that sectoral clusters become more pronounced over time in the

idiosyncratic-return networks, possibly reecting increased specialization and more integrated within-industry production processes, or the rise in sector specic investment funds.13

Similar to Panel (a), the REITs are a distinct cluster from the remainder of the nancial

rms in the decade plots. They are close to the utility rm cluster, which is interesting as

both are often seen as safe, acyclical, dividend-oriented investment sectors. The commodity

sector can be seen to move between being near the second and third factors, in line with the

factor discussion in the prior section.

To illustrate an application of these equity return networks to study shock propagation

in real-time, in the Online Appendix we model the impact of market beta and commodity

price shocks across rms using our estimated networks. This analysis shows that nancial,

consumer cyclical and commodity rms are most aected by a market beta/growth shock,

with the nancial sector at the center of the network in close proximity to this factor. A

negative commodity price shock, on the other hand, most adversely aects energy and base

materials companies. In fact, the top 10 declining equities following the shock are for rms

in the oil and gas extraction sub-sector. On the other hand, United Continental Holdings

 the parent company of United Airlines  would be expected to have the largest positive

response. This result likely reects the high fuel costs faced by airlines and exemplies how

these networks can inform rm managers, policy makers, and investors of their latent risks.

5 Theoretical Model of Firm Exposures

Having extended the empirical methods to construct inter-rm networks from log equity

returns, in this section we theoretically ground that approach. Our theoretical model is

an extension of the one-period, multi-sector DSGE model of Baqaee (2018) to a dynamic,

stochastic setting with inter-temporal assets, the details of which we leave to the Online

Appendix. There are four key results applicable to our analysis:

1. Firms' equity returns depend on three common factors, and their exposures to upstream

and downstream productivity and demand shocks through the production network.

13Evidence of changing production processes  with increases in production fragmentation and specialization along the production chain  is especially strong in the trade literature given the high quality data on

cross-border goods ows. Timmer et al. (2014) found that cross-border intermediate input trade measured

as the foreign value-added content of production has rapidly increased since the early 1990s, and Bridgman

(2012) pointed to a rapid expansion of manufactured parts traded over the past forty years.

16
2. The three common factors represent: the broad market beta, capturing changes in the

stochastic discount factor and GDP; the real price level; and the supply of raw inputs.

3. Firms' upstream and downstream exposures can be calculated from their place in the

real input-output network.

4. These upstream and downstream exposures are embedded in the innovations of a

VAR(1) of rms returns on one-another, with the common factors removed.

These results have several implications for our analysis of upstream, downstream and

common rm shocks. First and foremost is that they support the choice in the literature to

estimate inter-rm networks from VARs of rms' equity returns. Second, they highlight the

importance of our empirical extension to distinctly evaluate the role of common factors as

additional network nodes, separate from rms' idiosyncratic returns. Third, they substantiate our use of three common factors and their interpretations above. Finally, they provide

an approach to evaluate the relative historical importance of upstream versus downstream

rm exposures: use U.S. input-output tables to calculate sectoral upstream and downstream

exposures, then compare these to the propagation of realized rm shocks as embedded in

equity return based networks aggregated to the sectoral level. We perform the latter analysis

in Section 6 to assess the importance of upstream and downstream shocks.

5.1 Equity Returns, Common Factors & Network Centralities

We now examine rm equity returns in the model, nding that they depend on three common

factors and each rm's proximity through the inter-rm network to the sources of productivity and demand shocks. To begin, there are two types of production network centralities

that arise in the model, capturing how each rm acts as a consumer of raw inputs, and as a

supplier of nal goods to the households.

The consumer centrality measures the degree to which a rm consumes raw inputs itself

and through others, and with that its exposure to shocks to its own and other upstream

rms' productivity. The vector of consumer centralities, α ̃t, is dened as:

α ̃t ≡



IJ − μ

1−σΩ

−1

μ

1−σ

| {z }

≡Ψd

vt. (7)

Ψd is a function of the rms' positions within the production network and can be thought

of as an elasticity adjusted Leontief inverse, where IJ is the J × J identity matrix (J is the

number of industries), μ is a square matrix with adjustments for the industries' intra-industry

elasticity of substitution (μj ≡

φj

φj−1

) on the diagonal, σ is the inter-industry elasticity

of substitution, and Ω summarizes the production technology, having the weights of each

17
industry's goods as inputs in the others' production functions. Additionally, vt

is a vector of

the sectoral productivity parameters, which follow a standard autoregressive data generating

process:

∆vt = c + Ξ∆vt−1 + ∆εt, (8)

with c a constant vector, Ξ an autoregressive coecient matrix, and the ∆εt random shocks.

In the other direction, the supplier centrality measures how a rm is exposed to household

demand shocks for its own and other downstream rms' goods. The supplier centrality vector,

β ̃

t

, is dened as:

β ̃0

t ≡ β

0

t



IJ − μ

−σΩ

−1

| {z }

≡ΨS

, (9)

where βtis a vector of βtj consumer taste weights. β ̃

tj reects the network adjusted nal

consumption share of rms in industry j. The households are assumed to have a base set of

preference weights, β ̄, subject to random taste shocks, Zt:

βt = β ̄ + Zt. (10)

We next derive the log steady-state equity price of rm i in industry j



ln(q(j, i))

from the equity prices' Euler equation, which links rms' equity prices to macroeconomic

fundamentals and the production network centralities:

ln(q(j, i)) = ln 

1

φj



| {z }

Markup

+ ln

1

1 − ψ

| {z }

Discount

Factor

PcU

|{z}

GDP /

Aggregate

Demand



| {z }

Broad Market

Beta

+ ln

Pc

R ̃

σ−1

| {z }

Real Price

Level

+ lnz ̃

σ−1

| {z }

Raw

Input

Supply

+ lnα ̃j

|{z}

Consumer

Centrality

+ lnβ ̃

j

|{z}

Supplier

Centrality

, (11)

where the steady-state value of the stochastic discount factor is ψ, that of GDP is equal to

the aggregate consumption price index (Pc) times the total consumption index (U), R ̃ is the

price for the composite of raw inputs, and z ̃ is the labor-capital raw inputs aggregate. This

equation shows that rm equity prices depend on the markups, three common factors, and

the upstream and downstream network centralities. The three common factors represent:

the broad market beta, capturing the discount factor and GDP; the real price level; and

the supply of raw inputs. These three common factors align with those that we found using

our empirical model, which is notable given that we based our empirical factors on standard

PCA on the daily equity returns, without applying any identifying assumptions or rotations.

The log equity returns induced by changes in the common factors, the productivities and

demand parameters can be approximated by dierencing the rst order Taylor expansion of

18
Equation (11) around the steady-state. Letting Rt be a vector of the rm log equity returns,

ΛFt the common factor loadings and their log changes, and RI

t

a vector of idiosyncratic rm

returns, we have:

Rt = ΛFt + RI

t = ΛFt + Diag 

1

α ̃



Ψd

| {z }

U≡Upstream Exposure

∆vt + Diag 

1

β ̃



Ψ

0

S

| {z }

D≡Downstream Exposure

∆βt = ΛFt + U∆vt + D∆βt. (12)

In the upstream exposure matrix, U, each entry measures the exposure of the row sector to

a productivity shock from the column sector, both directly and possibly indirectly through

other sectors whose products are between theirs in a production chain. The idiosyncratic

response of a rm in industry j to innovations in an upstream source industry s would be

ι

0

jΨdιs

α ̃j

∆vs = Ujs∆vs, where ιjis a selection vector with a one in the j

th position and zeroes

elsewhere.

The downstream exposure matrix, D, provides exposures to demand shocks through the

network. The supplier centrality quanties the intensity with which the household consumes

from an industry, both directly and indirectly through its downstream users. The downstream exposure matrix captures the potential for propagation of taste shocks for downstream

goods to each industry as the ratio of its centrality to downstream industries' relative to its

total downstream exposure. The idiosyncratic return from a taste shock to a downstream

industry s is ι

0

jΨ0

S

ιs

β ̃

j

∆βs = Djs∆βs.

Further, rearranging the idiosyncratic return portion of Equation (12), and assuming ∆εt

and ∆Ztfrom Equations (8) & (10) are vectors of mean zero i.i.d. random shocks, yields:

RI

t = ρ0 + ρRI

t−1 + U∆εt + D∆Zt + ζt

=⇒ RI

t = ρ0 + ρRIt−1 + t

(13)

where ρ0 is a constant vector of rm xed eects, ρ is an N × N inuence matrix, and ζt

is a residual orthogonal to the two shocks. This formula matches Equation (2), where the

RI

t

idiosyncratic returns follow a VAR(1) process. Further, this indicates that the upstream

and downstream exposures are embedded in the t residuals of this system and should be

recoverable from our GFEVc's. We utilize the results of this section below to empirically

evaluate upstream and downstream exposures in our estimated equity return networks.

5.2 Theoretical Upstream & Downstream Shock Propagation

To provide intuition on upstream versus downstream exposure propagation, we simulated

productivity (vj ) and taste (βj ) shocks in our DSGE model for a network of ve industries

19
and a household sector, forming the classical X-type production network that is often

used as an example in this literature.14 Figure 4 Panel (a) displays the structure of the

production network, with the central rm 2 (orange) consuming inputs from rms 1 (black)

and 4 (green), while supplying its output to rms 3 (blue) and 5 (yellow). The latter two

rms ultimately sell their goods to the household (gray).

Panels (b) through (e) show rms' idiosyncratic responses to dierent shocks in the

network, with the common components removed to uncover input-output connections in

rms' equity returns, leaving only the nal two upstream and downstream components in

Equation (12). In each case, the y-axis measures the change in equity prices from being

in steady-state at the initial parameter levels to the new steady-state after the associated

shock. The x-axis measures the relevant upstream (U) or downstream (D) exposure to the

sector in which the shock originates, multiplied by the change in the specied parameter.

For the productivity shocks, this is Ujs∆vs, and for the taste shock it is Djs∆βs, where s is

the source sector for the shock and j is the target one. These are the network adjusted use of

raw inputs through sector s in the rst term and the indirect sales through industry s in the

second, scaled by the overall network adjusted raw input use and sales of a rm in industry

j, respectively. The rst feature of these plots that stands out is that the idiosyncratic

responses to the shocks lay along the 45-degree lines, indicating that the dynamic responses

of the idiosyncratic log returns match the expectations given the corresponding upstream

and downstream exposures.

Panel (b) simulates a productivity shock for one of the most upstream rms (rm 1), and

Panel (d) for one of the most downstream rms (rm 3) to compare supply network shock

propagation in each extreme case. In each case, the rm experiencing a productivity shock,

marked with an X, has the largest centrality to itself and hence the greatest idiosyncratic

equity response. Since there are no rms downstream from rm 3, the other sectors have zero

upstream centrality exposures to it, hence zero idiosyncratic returns to its productivity shock

in Panel (d). On the other hand, rm 1's productivity shock aects the idiosyncratic returns

of rms 2, 3, and 5, since they are all downstream from it. Further, rm 2 is more directly

exposed to rm 1 so it has a greater response, while rms 3 & 4 have the same upstream

exposures to rm 1 through rm 2, so they have the same equity responses. Bringing these

two examples together, Panel (c) shows a productivity shock for the centrally located rm 2.

The two upstream rms (1 & 4) have zero idiosyncratic returns, while the two downstream

rms (3 & 5) have positive returns as they benet from rm 2's productivity improvement.

Finally, Panel (e) shows that rms 1, 2 & 4 are similarly aected by a taste shock to

downstream rm 3's production, reecting that they have comparable downstream reliance

14We explore other network forms in the Online Appendix with qualitatively similar responses to shocks.

20
on rm 3. Firm 5's only relationship to rm 3 is as a direct competitor in the nal goods

market, therefore, since it is not upstream of rm 3, it has a zero idiosyncratic response to

the taste shock. These simulations illustrate our prior insights: the importance of removing common factors to uncover the network connections between rms; and that once the

common factors are removed, equity returns reect the proximity of rms through the two

exposure matrices.

6 Analysis of Upstream & Downstream Exposures

We now bring together our empirical and theoretical models to evaluate the historical importance of upstream and downstream exposures in the U.S. We do so by comparing the

empirical equity return based networks that embed the realized propagation of shocks across

rms aggregated to the sectoral level, with U.S. input-output table derived sectoral upstream and downstream exposures.15 We consider both the R to R (total-return) and the

inter-rm portion of our RI & Factors to R & Factors equity networks to focus on the

relationships between rms' idiosyncratic returns, removing the impact of common factors.

Further, we examine both the three-step and state-space estimated networks.

For the input-output based networks, we follow Equation (12) to estimate the upstream

and downstream exposure matrices, U and D.

16 We utilize the BEA's U.S. input-output use

table data from 1997 through 2015. The BEA refers to the use tables as a recipe matrix

because they show the inputs necessary to produce the output of each sector. These tables

provide the dollar expenditures on commodities from each sector by households, the government, and other rms as intermediate inputs.17 The upstream and downstream exposure

matrices calibrated from these data are arranged so that the supplier of an input is in the

column and the user is in the row to align with the orientation of our equity-based networks.

15The correlation between the number of rms in our 1989-2017 sample in each BEA sector and sectors'

output shares is 0.85-0.9 over time, indicating our sample has representative coverage of the sectors.

16We follow the literature in assuming a Cobb-Douglas form (σ = 1) when calibrating our model parameters

from the input-output data. This assumption is standard in the literature because the industry-level prices

cannot be cleanly separated from their output quantities in the sectoral data, while with a Cobb-Douglas

structure only the expenditures matter, not the breakdown between prices and quantities. If we take the

parameters as estimated assuming a Cobb-Douglas form and then vary σ, we nd that our results are

minimally aected by changes to this parameter, so beyond helping us take the model to the data, this

assumption does not appear to aect our main results.

17We exclude the Other services, except government and Government sectors to focus on the private

sector. The BEA input-output tables we use throughout are North American Industry Classication System

based, with surveys at the establishment level. We also tried using the older Standard Industrial Classication

based data that is at the rm level  in case the discrepancy between the level of our equity networks and

the establishment level input-output networks skewed our results  but this change made little dierence

for the years where we have both use table types.

21
We compare the equity and input-output exposure networks using the element-by-element

correlations between their adjacency matrices. To calculate the signicance of these correlations, we borrow a network correlation distribution bootstrapping method from the machine

learning literature: the Quadratic Assignment Procedure (QAP). The QAP allows us to

estimate the distribution of the network correlations conditioned on the observed network

structure. Doing so instead of using the typical procedure to calculate the signicance of

correlations between two data series is particularly important given the structure of the

networks we study, where the weights from sectors to themselves are all expected to be substantial (i.e., the correlations are ex-ante expected to be sizable and positive due to the large

diagonal entries in the adjacency matrices).

The test evaluates the expected share of random re-orderings of the rst matrix that

have a higher correlation with the second matrix than the rst matrix itself does. The new

re-ordering is the same for the rows and columns in the bootstrapped network, creating a

hypothetical network with similar structure to that observed. In our case, letting %(·, ·) be the

pairwise correlation of the entries of two provided matrices, IO the upstream or downstream

input-output network adjacency matrix under examination, and F the candidate rm equity

network adjacency matrix, this means using random re-ordered draws of the IO matrix,

IOq, to estimate E



1[%(IO, F) < %(IOq, F)]≈

1

Q

PQ

q=1 1[%(IO, F) < %(IOq, F)] to derive

the test's p-value for the three-step estimation based networks, where we take ten thousand

draws. We extend this approach to account for estimation uncertainty using the statespace method's Bayesian draws. In addition to re-orderings of IOq, we also draw F

(q)

, an

equity-based network from the retained Markov Chain Monte Carlo draws: E



1[%(IO, F) <

%(IOq, F)]≈

1

Q

PQ

q=1 1[%(IO, F(q)

) < %(IOq, F(q))] for one million draws.

Table 1 presents the correlations between the input-output based networks and the corresponding rm-equity based networks obtained with either the three-step estimation method

in Panel (A) or the state-space estimation method in Panel (B). The rows indicate the

estimation periods for the rm-equity based networks (Equity Network Period), and inputoutput year near their midpoint that they are compared with (IO Year). The top results

are for rolling 10-year windows ending from 1989 through 2017, followed by the full 1989 to

2017 networks at the bottom. The equity-based networks include the same 524 rms from

our long-run balanced sample to remove the potentially confounding impact of a changing

sample on the correlations over time. We also provide the averages and standard deviations

for the 10-year windows.

Within each panel, we provide four comparisons of the combinations of the two inputoutput network types (upstream U and downstream D exposure matrices), and the two

equity network types (total-return and idiosyncratic-return networks). The stars indicate

22
statistical signicance as calculated using QAP.

There are two key takeaways from Table 1. First, as to our core question, the correlations

indicate a greater role for upstream exposures than for downstream ones, as the correlations

between the equity-response networks and the upstream exposures are consistently higher

than those for the downstream exposures. Further, the correlations with the upstream

exposures are all statistically signicant, whereas none of those for the downstream exposures

are so. For example, focusing on the full 1989-2017 sample in the nal row, we see that

the total-return network has correlations of 0.45 (three-step method) and 0.41 (state-space

method) with the upstream exposure matrix, while the downstream correlations are 0.04

(three-step method) and 0.02 (state-space method). Similar results hold for the idiosyncraticreturn networks.

These results indicate that shocks from upstream suppliers of which a rm is either

directly or indirectly a customer, matter more for short-term equity responses than shocks

from downstream in the production process. This is suggestive of low short-term elasticities

of substitution across inputs with more exibility on the downstream, customer side. It

may also reect greater average heterogeneity of rm's customer sets relative to the rm's

individual supply chains, so that downstream sourced shocks are implicitly insured against.

This result is consistent with, for example, the signicant supplier disruptions experienced

in the aftermath of the 2011 Tohoku earthquake. While Carvalho et al. (2016) identied

quantitatively large upstream and downstream spillovers after the quake, their empirical

analysis found directed propagation from upstream to be robust to parametrization, with the

positive/negative propagation of shocks from downstream rms dependent on the elasticity

parameters instead. Importantly, they assessed that the transmission of shocks over inputoutput linkages accounted for a 1.2% decline in Japanese GDP in the year following the

earthquake. Yet, these eects were not localized to the immediate area. Boehm et al. (2019)

found that Japanese aliates abroad, reliant on imports from the aected zones, saw output

drop about one for one with imports of intermediate goods from Japan in the wake of the

disaster, suggesting extremely low elasticities of substitution for inputs in the short-term.

Similarly, Jones (2011) studied production linkages and intermediate input use, nding that

problems along a production chain can sharply reduce output under input complementarity.

The second takeaway is that the correlations for the idiosyncratic-return networks are

consistently higher than for the total-return networks for all of the dierent time periods,

both sectoral exposure network types, and both estimation methods. For the upstream

exposures, the average idiosyncratic-return network correlation is about 30% higher than

the total-return one (0.46 versus 0.62 for the three-step method, and 0.42 versus 0.52 for

the state-space one). For the downstream exposures, the idiosyncratic-return networks also

23
yield higher results than the total-return networks; however, none of these are statistically

signicant. The total-return network correlations with the downstream exposures average

only 0.06 (0.02 for the state-space estimation), while the average for the idiosyncratic networks is 0.22 (0.14). These results accord with Equation (13), where the common factors are

rst removed from the equity returns before deriving the VAR(1) connecting them with the

upstream and downstream centralities, underscoring the importance of explicitly modeling

the impact of the common factors to uncover inter-sectoral connections.

Additionally, note that the estimation method used is not crucial for delivering the results

of this exercise, and that the key ndings hold across the three-step and state-space equity

network estimation methods. This observation is important with regard to robustness of

our results, as well as recommending our three-step method as a more ecient estimation

approach that can still deliver the key results.

Our main takeaways are robust to dierent assumptions about network specications

and sampling choices. In Table 2, Panel (A) shows that using a balanced panel of rms

over time is not a critical assumption. When instead including all rms within each 10-year

window, such that rms enter and exit the sample over time, the average correlation across

these samples for the idiosyncratic-return networks is 0.61 for upstream exposure (versus 0.62

above) and 0.19 for downstream exposure (versus 0.22 above). We also construct model-free

networks where the edge weights are the bilateral daily equity return correlations between

each rm pair, with results in the nal two columns of the table. The correlations of these

model-free networks with the input-output exposures for the rolling rm sets in Panel (A)

(U correlation of 0.41 and D correlation of 0.01) and constant balanced rm panel in Panel

(B) (U correlation of 0.40 and D correlation of 0.02) are similar to those obtained when

comparing the total-return networks. In fact, these simple networks are highly correlated

with the total-return networks (0.95-0.99 correlations when aggregated at the BEA sector

level); however, these two approaches have lower correlations with the input-output exposures

than the idiosyncratic-return networks, reinforcing the need to account for common factors

when using VARs to evaluate their empirical connections to properly uncover inter-rm

relationships. The table also shows that our results hold for the lower frequency changes

in market expectations captured using monthly returns, and that the choice of daily data

frequency used for the empirical analysis does not drive our results.

Finally, in untabulated analysis, we also compare the equity networks against the standard Leontief inverse and nd that the equity-based networks  particularly the idiosyncratic ones  strongly reect the underlying macroeconomic relationships between sectors

captured therein. The similarity in results for the upstream exposure and Leontief inverse is

not surprising, as they are closely related. In fact, the two are the same under Cobb-Douglas

24
aggregation (σ = 1) or perfect competition (limφj→∞ ∀ j, so that μj = 1 ∀ j). The correlations for the upstream exposures are consistently greater than those for the Leontief inverses

for both equity-return network types, implying a meaningful role for market competition,

the demand elasticities across goods, and markups in the structure of inter-rm networks.

7 Conclusion

In this paper, we propose an approach to derive inter-rm networks from equity returns,

which can isolate the eects of common factors and explicitly model them as nodes within

the network. We present two alternative methods to estimate the large FAVAR that provides

the foundation for these networks, each of which utilizes big data econometric techniques,

with their own trade-os. We ground this approach in a theoretical model, then utilize that

model to evaluate the historical propagation of upstream versus downstream exposures. We

nd upstream exposure (shocks to a rm's direct and indirect suppliers) to be more evident

than downstream exposure (shocks to its direct and indirect customers) in the equity returns

based networks. Our ndings have meaningful implications for understanding the reactions

of rms relative to one another. Further, these networks potentially allow for the real-time

monitoring of economic developments at a frequency and disaggregated level that would

otherwise be dicult to study in macroeconomic data. Additionally, our results support

the use of the idiosyncratic equity networks to study connections across rms, removing the

confounding inuence of common factors.

Our work suggests several areas that warrant further investigation. First, understanding

why the common factors that drive equity returns have increased in inuence. Delving into

the precise channels through which rms are connected  e.g., intermediate goods, services

or credit  to understand the specic mechanisms of contagion would also be fruitful.

Finally, a greater importance of upstream rms relative to downstream ones could be applied

to businesses hedging risks, or designing trade and international economic policies.

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29
TABLES AND FIGURES

Table 1: Comparisons of Firm Equity Return Networks & Input-Output Exposures

(A) Three-Step Estimation Method (B) State-Space Estimation Method

Equity Network Type: Total-Return Idiosyncratic-Return Total-Return Idiosyncratic-Return

Equity

Network

Period

IO Year

Upstream

Supplier

Exposure

(U)

Downstream

Customer

Exposure

(D)

Upstream

Supplier

Exposure

(U)

Downstream

Customer

Exposure

(D)

Upstream

Supplier

Exposure

(U)

Downstream

Customer

Exposure

(D)

Upstream

Supplier

Exposure

(U)

Downstream

Customer

Exposure

(D)

1989-1998 1997 0.50*** 0.08 0.61*** 0.20 0.42** 0.02 0.49*** 0.10

1990-1999 1997 0.51*** 0.10 0.60*** 0.19 0.42** 0.03 0.48** 0.08

1991-2000 1997 0.52*** 0.11 0.60*** 0.19 0.43** 0.03 0.49** 0.09

1992-2001 1997 0.51*** 0.11 0.60*** 0.20 0.43** 0.04 0.51** 0.10

1993-2002 1998 0.49*** 0.09 0.61*** 0.20 0.42** 0.03 0.52** 0.12

1994-2003 1999 0.47*** 0.07 0.61*** 0.19 0.42** 0.03 0.51** 0.11

1995-2004 2000 0.47*** 0.07 0.60*** 0.19 0.42*** 0.03 0.51** 0.11

1996-2005 2001 0.45*** 0.07 0.60*** 0.19 0.40*** 0.03 0.50*** 0.11

1997-2006 2002 0.45*** 0.07 0.60*** 0.20 0.41*** 0.03 0.50*** 0.12

1998-2007 2003 0.45*** 0.07 0.60*** 0.21 0.40*** 0.03 0.52*** 0.14

1999-2008 2004 0.45*** 0.05 0.62*** 0.22 0.41*** 0.03 0.53** 0.15

2000-2009 2005 0.46*** 0.03 0.63*** 0.21 0.42** 0.02 0.53** 0.14

2001-2010 2006 0.45*** 0.03 0.63*** 0.21 0.43** 0.01 0.53** 0.13

2002-2011 2007 0.45*** 0.02 0.63*** 0.22 0.43** 0.01 0.53** 0.13

2003-2012 2008 0.44*** 0.03 0.63*** 0.22 0.42** 0.02 0.53** 0.15

2004-2013 2009 0.41*** 0.03 0.62*** 0.23 0.39*** 0.02 0.52** 0.17

2005-2014 2010 0.42*** 0.02 0.63*** 0.25 0.41** 0.02 0.53** 0.19

2006-2015 2011 0.43*** 0.03 0.63** 0.29 0.42** 0.02 0.54** 0.24

2007-2016 2012 0.44*** 0.03 0.63** 0.30 0.42** 0.02 0.53** 0.20

2008-2017 2013 0.44*** 0.03 0.63** 0.30 0.42** 0.02 0.53** 0.20

Average 0.46 0.06 0.62 0.22 0.42 0.02 0.52 0.14

Std. Dev. (0.03) (0.03) (0.01) (0.04) (0.01) (0.01) (0.02) (0.04)

1989-2017 2001 0.45*** 0.04 0.62*** 0.21 0.41*** 0.02 0.54** 0.14

Note: The table displays correlations between U.S. input-output table based upstream (U) and

downstream (D) exposures from Section 5 and rm equity return based networks from Section 4, aggregated

at the BEA sector level. Equity networks in Panel (A) are constructed with the three-step estimation

method, and those in Panel (B) with the state-space estimation method. The total-return equity networks

include the impact of similar loadings on common factors, whereas the idiosyncratic-return networks

separate out common factors to focus on rms' direct bilateral connections. The equity return based

networks use the same balanced panel of 524 rms continuously traded over 1989-2017, but for the dierent

time frames listed in the Equity Network Period column, with IO Year the year of the input-output

tables used for comparison. Statistical signicance calculated using QAP with ten thousand draws for the

three-step approach and one million for the state-space method, where *** p<0.01, ** p<0.05, and * p<0.1.

30
Table 2: Firm Equity Return Networks & Input-Output Exposures: Robustness

Equity Network Type: Total-Return Idiosyncratic-Return Bilateral Return Correlations

Upstream

Supplier

Exposure

(U)

Downstream

Customer

Exposure

(D)

Upstream

Supplier

Exposure

(U)

Downstream

Customer

Exposure

(D)

Upstream

Supplier

Exposure

(U)

Downstream

Customer

Exposure

(D)

(A) Average of Unbalanced Panels

(10-year rolling sub-samples ending 1998 to 2017) Daily Data 0.44 0.04 0.61 0.19 0.41 0.01

(B) Balanced Panel Monthly Data 0.43*** 0.04 0.55*** 0.14

(Equity Period: 1989-2017, IO Year: 2001) Daily Data 0.45*** 0.04 0.62*** 0.21 0.40*** 0.02

Note: The table displays correlations between U.S. input-output table based upstream (U) and

downstream (D) exposures from Section 5 and rm equity return based networks from Section 4 in

the rst four data columns, aggregated at the BEA sector level with the three-step estimation method.

The total-return equity networks include the impact of similar loadings on common factors, whereas the

idiosyncratic-return networks separate out common factors to focus on rms' direct bilateral connections.

The nal two columns instead use equity networks where rms' bilateral equity return correlations are the

edge weights. Panel (A) averages across unbalanced 10-year rolling sub-samples ending from 1998 to 2017,

with each containing the maximum number of rms continuously traded over its period and compared to

the input-output year nearest its midpoint. Panel (B) displays correlations for networks from a balanced

panel of 524 rms continuously traded over 1989-2017, relative to the 2001 input-output based network,

either from daily or monthly data. Statistical signicance for Panel (B) is calculated using QAP with ten

thousand draws, where *** p<0.01, ** p<0.05, and * p<0.1.

31
Figure 1: Equity Return Common Factors and U.S. Economic Indicators

(a) First Factor vs. S&P500 (b) First Factor vs. Industrial Production

(c) Second Factor vs. CPI (d) Second Factor vs. PPI

(e) Third Factor vs. GSCI Commodity Index (f) Third Factor vs. Brent Crude Oil

Note: The three common factors are estimated from the sample of U.S. daily log equity returns

continuously traded from 1989 through 2017, with either PCA (black) or the median Bayesian draws

(blue), with dark and light shaded areas denoting the latter's corresponding 68% and 95% intervals,

on the left axis. The right axis is for a comparable macroeconomic or nancial indicator (red). All

series are expressed in year-over-year changes.

32
Figure 2: Total-Return U.S. Firm Equity Return Based Networks

(a) 19892017 (b) 19901999

(c) 20002009 (d) 20102017

Note: The gure displays rm equity return based networks estimated over dierent sample periods

with the three-step method. Each dot represents a rm colored by its BEA sector, network edges

are calculated using GFEVc's inclusive of the common factors, and the proximity of dots to one

another depends on how connected rms are using the ForceAtlas2 algorithm.

33
Figure 3: Idiosyncratic-Return U.S. Firm Equity Return Based Networks

(a) 19892017 (b) 19901999

1

2

3

1

2

3

(c) 20002009 (d) 20102017

1

2

3

1

2

3

Note: The gure displays rm equity return based networks estimated over dierent sample periods

with the three-step method. Each dot represents a rm colored by its BEA sector, network edges are

calculated using GFEVc's after distinguishing the eects of common factors, and the proximity of

dots to one another depends on how connected nodes are using the ForceAtlas2 algorithm. Factors

are depicted as purple stars with the corresponding number in black above.

34
Figure 4: Simulated Idiosyncratic Equity Responses to Shocks in DSGE Model

(a) X-Network Type

1

4

2

3

5

HH

(b) Firm 1 Productivity Shock (v1) (c) Firm 2 Productivity Shock (v2)

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

(d) Firm 3 Productivity Shock (v3) (e) Firm 3 Taste Shock (β3)

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Beta Centrality Idiosyncratic Return

Detrended Steady State Return

Note: Panel (a) shows the X-type network, where every node represents either a dierent rm (1

to 5) or the household sector (HH). Panels (b) through (e) plot the simulated idiosyncratic returns

from moving between steady-states at the initial and new parameters after the shock on the y-axis,

against the upstream or downstream exposure to the source node multiplied by the change in its

specied parameter on the x-axis. The 45-degree line equating these two is included for reference.

The source rm for each shock is denoted with an X-marker. The idiosyncratic returns are the

latter two terms of Equation (12) after removing the common factors, RI

t = U∆vt + D∆βt

.

35
Online Appendix

Analysis of Upstream, Downstream & Common Firm

Shocks Using a Large Factor-Augmented Vector

Autoregressive Approach

Everett Grant Julieta Yung

Amazon.com FDIC

A Common Factor Estimation Using PCA ii

A.1 PCA Common Factors from U.S. Equity Returns (1989-2017) . . . . . . . . v

A.2 PCA Common Factor Loadings . . . . . . . . . . . . . . . . . . . . . . . . . ix

B Three-Step Estimation Method Details xiii

B.1 OCMT Variable Selection Procedure . . . . . . . . . . . . . . . . . . . . . . xiii

C State-Space Model Estimation Details xiv

C.1 Model Prior & Posterior Distributions . . . . . . . . . . . . . . . . . . . . . xv

C.2 Gibbs Sampling Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

D Firm Network Estimation Details xxi

D.1 One Period Ahead GIRF Network Edge Equation . . . . . . . . . . . . . . . xxi

D.2 One Period Ahead GFEVD Network Edge Equation . . . . . . . . . . . . . . xxi

D.3 VAR Estimation Bias from Omitting Common Factors . . . . . . . . . . . . xxii

D.4 Empirical Method Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . xxii

D.5 Top Firms by Weights Out to Others . . . . . . . . . . . . . . . . . . . . . . xxix

D.6 U.S. Inter-Firm Network Application: Market Beta & Commodity Factor

Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix

E Full Specication of Theoretical Model of Firm Exposures xxxiii

E.1 Household's Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii

E.2 Firms' Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiv

E.3 Industry Centralities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv

E.4 Equity Returns, Common Factors & Network Centralities . . . . . . . . . . . xxxvii

E.5 Theoretical Model Simulations to Upstream & Downstream Shocks . . . . . xxxix

F References l

i
A Common Factor Estimation Using PCA

There are dierent methods that can be used with PCA to calculate the common factor and

idiosyncratic return series. In Section 2 of the paper we use the Covariance PCA with Full

Factor Series approach.

PCA solves the following optimization problem:

min

{Λ,Ft}

1

NT

X

N

i=1

X

T

t=1

(Rit − λiFt)

2

,

where Λ

0 ≡ {λ0

1

, λ0

2

, ..., λ0

N } is the combination of the vectors of rm specic factor weights.

Once the common variation in the data, ΛFt, is estimated, RI

t

is obtained as the residual

rm returns.

There are four alternative methods we could use to calculate the common factors and

decompose the data. The alternatives are based on whether the covariance or correlation

matrix is used to calculate the major axes of the common factors, and whether the means

are included in the common factors or they are detrended. For these formulas, R represents

the T × N matrix of combined Rt vectors. Let μ be the T × N matrix of the rm means

of the R return series repeated along each column, θ be an N × N matrix of the R series'

standard deviations along the diagonal and zeroes elsewhere, R0 be the demeaned return

series (R − μ), and RS be the standardized R series ((R − μ)θ

−1

). Also, V is the N × K

matrix of the rst K eigenvectors from the covariance matrix (in descending order of sample

variance explained), and Vc is the matrix of the rst K eigenvectors from the correlation

matrix.

(1) Standard Covariance PCA: Covariance Eigenvectors & Detrended Factors

The formula to recover the original data using standard PCA involves projecting the data

series without their means into the reduced dimension space. The means are excluded so

that the directions of maximal variation are captured, rather than the means of the return

series driving the newly created factors. The means are then added to these series after they

are projected back to the original space:

Recovery Formula: Rˆ = μ + R0V V 0.

The recovery error  or the variation explained by the excluded eigenvectors  is:

R − Rˆ = R − (μ + R0V V 0) = R0 − R0V V 0 = R0(IN − V V 0).

As more eigenvectors are included in V , the V V 0term approaches the identity matrix and

ii
the recovery error goes to zero.

Using these factors in our method (F = R0V ; Λ0 = V

0

) we can then dene what is

captured by the RIterm:

R = FΛ

0 + RI = R0V V 0 + RI

=⇒ R

I = R − R0V V 0 = μ + R0(IN − V V 0

).

The idiosyncratic returns are then the average returns for each rm plus the recovery error,

or the variation not explained along the included eigenvectors.

(2) Covariance PCA with Full Factor Series

There may be broad common trends that one might wish to capture, rather than allocating

them to the individual rm return series. In that case, we would instead calculate the factor

series with the full  rather than the demeaned  data.

This would mean that the factor series are calculated as F = RV but Λ

0 would still be

equal to V

0

. In that case the RIterm is:

R = FΛ

0 + RI = RV V 0 + RI

=⇒ R

I = R − RV V 0 = (R0 + μ)(IN − V V 0

) = μ + R0(IN − V V 0) − μV V 0.

These idiosyncratic returns include the average returns and recovery error as in the rst

case, but the shared trend already accounted for in the factors is removed in the last term.

The last term is equal to the means of these full factor series projected back to the full rm

space.

(3) Standard Correlation Matrix PCA: Correlation Eigenvectors & Detrended

Factors

The formula to recover the original data using standard correlation matrix based PCA involves using the eigenvectors of the correlation matrix (Vc) to project the standardized data

series, RS = (R − μ)θ

−1

, into the reduced dimension space. The means are excluded so that

the dimension of maximal variation is captured, rather than the means driving the newly

created factors, and the series are standardized to have the same variation so that they are

accounted for equally when deriving the eigenvectors dening the directions of maximal variation for the common factor series. To recover the data, the derived common factor series

must be adjusted for both the series means and standard deviations:

Recovery Formula: Rˆ = μ + RSVcV

0

c θ = μ + R0θ

−1VcV0

c θ.

iii
The recovery error is:

R − Rˆ = R − (μ + R0θ

−1VcV0

c θ) = R0(IN − θ

−1VcV0

c θ).

Using these factors in our method (F = RSVc; Λ0 = V

0

c θ) we can then dene what is captured

by the RIterm:

R = FΛ

0 + RI = RSVcV0

c θ + R

I

=⇒ R

I = R − RSVcV0

c θ = R − R0θ

−1VcV0

c θ = μ + R0(IN − θ

−1VcV0

c θ).

The idiosyncratic returns are then the average returns for each rm plus the recovery error.

(4) Correlation PCA with Full Factor Series

We can also use factor series with the common trends included using the correlation matrix

eigenvectors for the data projections, with F = Rθ−1Vc and Λ

0 = V0

c θ. In that case the RI

term is:

R = FΛ

0 + RI = Rθ−1VcV0

c θ + R

I

=⇒ R

I = R − Rθ−1VcV0

c θ = R(IN − θ

−1VcV0

c θ) = μ + R0(IN − θ

−1VcV0

c θ) − μθ−1VcV

0

c θ.

These idiosyncratic returns include the average returns and recovery error as in the rst case,

but the shared trend already accounted for in the factors is removed in the last term as in

the covariance based PCA.

iv
A.1 PCA Common Factors from U.S. Equity Returns (1989-2017)

Table A.1: 20 Largest Eigenvalues by PCA for U.S. Equity Returns (1989-2017)

Top Value % Cum. % Top Value % Cum. %

1 692.99 22.63 22.63 11 23.00 0.75 34.68

2 70.03 2.29 24.91 12 22.07 0.72 35.40

3 66.17 2.16 27.07 13 21.18 0.69 36.09

4 39.82 1.30 28.37 14 20.65 0.67 36.76

5 34.68 1.13 29.51 15 19.92 0.65 37.41

6 32.10 1.05 30.55 16 18.90 0.62 38.03

7 28.68 0.94 31.49 17 18.12 0.59 38.62

8 27.52 0.90 32.39 18 17.71 0.58 39.20

9 24.10 0.79 33.17 19 17.20 0.56 39.76

10 23.03 0.75 33.93 20 16.82 0.55 40.31

Note: Top 20 eigenvalues of the covariance matrix of log daily equity returns for

the continuously-traded sample from 1989 through 2017 (T = 7,424 and N = 524).

Figure A.1: Variance Share of Top 3 PCA Factors

(a) Balanced Firm Sample (b) Rolling 10-Year Samples

2000 2005 2010 2015

0 10 20 30 40

End of Network 10−Year Window

Cumulative Variance Share

2000 2005 2010 2015

0 5 10 15 20 25 30 35

End of Network 10−Year Window

Cumulative Variance Share

Note: The rst factor is shown in black, the second one in red, and the third one in blue. Factors extracted

by principal component analysis on the covariance matrix of U.S. daily log equity returns.

v
Figure A.2: First PCA Factor, Equity Markets, & Growth of the U.S. Economy (1989-2017)

Comparison

Series Levels Year-over-Year

Sample

Average

Log

Returns

100000 100050 100100 100150 100200

SampleAvgLevel

1000000 1001000 1002000 1003000 1004000

F1Level

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F1Level SampleAvgLevel

Correl: 0.998

−.001 −.0005

0 .0005 .001

SampleAvgLevelYoY

−.002 −.001

0 .001 .002

F1LevelYoY

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F1LevelYoY SampleAvgLevelYoY

Correl: 0.997

SP500

0 1000 2000 3000 4000 5000

SP500

1000000 1001000 1002000 1003000 1004000

F1Level

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F1Level SP500

Correl: 0.887

−.4 −.2

0 .2 .4 .6

SP500YoY

−.002 −.001

0 .001 .002

F1LevelYoY

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F1LevelYoY SP500YoY

Correl: 0.825

U.S.

Industrial

Production

60 70 80 90 100 110

IndProd

1000000 1001000 1002000 1003000 1004000

F1Level

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F1Level IndProd

Correl: 0.970

−.15 −.1 −.05

0 .05 .1

IndProdYoY

−.002 −.001

0 .001 .002

F1LevelYoY

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F1LevelYoY IndProdYoY

Correl: 0.521

U.S. Real

GDP

9 9.2 9.4 9.6 9.8

lnGDPreal

1000000 1001000 1002000 1003000 1004000

F1Level

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F1Level lnGDPreal

Correl: 0.954

−.04 −.02

0 .02 .04 .06

GDPrealYoY

−.002 −.001

0 .001 .002

F1LevelYoY

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F1LevelYoY GDPrealYoY

Correl: 0.512 Note: F1Level is the cumulative sum of the rst factor extracted by principal component analysis

on the covariance matrix of U.S. daily log equity returns for the portion of our sample continuously

traded from 1989 through 2017 (T = 7,424 and N = 524).

vi
Figure A.3: Second PCA Factor and U.S. Prices, Year-over-Year Plots (1989-2017)

Inverted PPI Inverted CPI

−.2 −.1

0 .1 .2

PPIYoY

−1

0

1

2

3

F2LevelYoY

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F2LevelYoY PPIYoY

Correl: 0.601

−.06 −.04 −.02

0 .02

CPIYoY

−1

0

1

2

3

F2LevelYoY

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F2LevelYoY CPIYoY

Correl: 0.512

USD TWI USD-Euro

−.2 −.1

0 .1 .2 .3

USDTWIYoY

−1

0

1

2

3

F2LevelYoY

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F2LevelYoY USDTWIYoY

Correl: 0.408

−.2 −.1

0 .1 .2 .3

USDEURYoY

−1

0

1

2

3

F2LevelYoY

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F2LevelYoY USDEURYoY

Correl: 0.439

USD-British Pound Inverted 10-Year BEI

−.2

0 .2 .4

USDGBPYoY

−1

0

1

2

3

F2LevelYoY

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F2LevelYoY USDGBPYoY

Correl: 0.399

−3 −2 −1

0

US10BEI

−1

0

1

2

3

F2LevelYoY

01jul2003 01jan2007 01jul2010 01jan2014 01jul2017

Date

F2LevelYoY US10BEI

Correl: 0.479

Note: F2Level is the cumulative sum of the second factor extracted by principal component

analysis on the covariance matrix of U.S. daily log equity returns for the portion of our sample

continuously traded from 1989 through 2017 (T = 7,424 and N = 524). The breakeven ination,

PPI and CPI return series are negated to match the direction of the factor series.

vii
Figure A.4: Third PCA Factor and Commodities, Year-over-Year Plots (1989-2017)

Brent Crude Goldman Sachs Commodity Index

−.5

0 .5

1 1.5

2

GFDBrentYoY

−.5

0 .5

1

F3LevelYoY

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F3LevelYoY GFDBrentYoY

Correl: 0.641

−.5

0 .5

1

GSCIYoY

−.5

0 .5

1

F3LevelYoY

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F3LevelYoY GSCIYoY

Correl: 0.584

Heating Oil Unleaded Gas

−.5

0 .5

1 1.5

2

HeatingOilYoY

−.5

0 .5

1

F3LevelYoY

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F3LevelYoY HeatingOilYoY

Correl: 0.615

−1

0

1

2

UnleadedGasYoY

−.5

0 .5

1

F3LevelYoY

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F3LevelYoY UnleadedGasYoY

Correl: 0.630

Aluminum Nickel

−.5

0 .5

1

AluminumYoY

−.5

0 .5

1

F3LevelYoY

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F3LevelYoY AluminumYoY

Correl: 0.380

−1

0

1

2

3

NickelYoY

−.5

0 .5

1

F3LevelYoY

01jan1990 01jan1995 01jan2000 01jan2005 01jan2010 01jan2015

Date

F3LevelYoY NickelYoY

Correl: 0.372

Note: F3Level is the cumulative sum of the third factor extracted by principal component analysis

on the covariance matrix of U.S. daily log equity returns for the portion of our sample continuously

traded from 1989 through 2017 (T = 7,424 and N = 524).

viii
A.2 PCA Common Factor Loadings

Figure A.5: U.S. Factor Λ Coecient Distributions by Industry (1989-2017)

Factor 1 Factor 2

Basic Materials (n= 32 )

Communications (n= 18 )

Consumer, Cyclical (n= 78 )

Consumer, Non−cyclical (n= 94 )

Energy (n= 34 )

Financial (n= 90 )

IndDiv (n= 104 )

Technology (n= 31 )

Utilities (n= 43 )

−0.2

0.0

0.2

0.4

Basic Materials (n= 32 )

Communications (n= 18 )

Consumer, Cyclical (n= 78 )

Consumer, Non−cyclical (n= 94 )

Energy (n= 34 )

Financial (n= 90 )

IndDiv (n= 104 )

Technology (n= 31 )

Utilities (n= 43 )

−0.2

0.0

0.2

0.4

Factor 3

Basic Materials (n= 32 )

Communications (n= 18 )

Consumer, Cyclical (n= 78 )

Consumer, Non−cyclical (n= 94 )

Energy (n= 34 )

Financial (n= 90 )

IndDiv (n= 104 )

Technology (n= 31 )

Utilities (n= 43 )

−0.2

0.0

0.2

0.4

Note: Loadings on the rst three factors extracted by principal component analysis on the covariance matrix

of U.S. daily log equity returns for the portion of our sample continuously traded from 1989 through 2017

(T = 7,424 and N = 524).

ix
Table A.2: Top Firm Loadings on First Factor

Panel A: Highest Factor Loadings

Λ Name Industry Industry

Subgroup BEA Sector BEA Subgroup Country

1 0.08 XCERRA

CORP Technology Semiconductor

Equipment

ManufacturingComputer and

electronic products US

2 0.08 KULICKE &

SOFFA Technology Semiconductor

Equipment

ManufacturingComputer and

electronic products SGP

3 0.08

HOVNANIAN

ENT-A

Consumer,

Cyclical

BldgResidential/Commer Construction Construction US

4 0.07

SAFEGUARD

SCIENT

Financial Venture Capital Finance

Securities,

commodity

contracts, and

investments

US

5 0.07 TEREX

CORP Industrial MachineryConstr&Mining

Manufacturing Machinery US

6 0.07 LINCOLN

NATL CRP Financial Life/HealthInsurance Finance

Securities,

commodity

contracts, and

investments

US

7 0.07 OFFICE

DEPOT INC

Consumer,

Cyclical

Retail-Oce

Supplies Retail General merchandisestores US

8 0.07

ENZO

BIOCHEM

INC

Consumer,

Non-cyclical

MedicalBiomedical/GeneManufacturing Chemical products US

9 0.07 MICRON

TECH Technology ElectronicCompo-Semicon

ManufacturingComputer and

electronic products US

10 0.07 MBIA INC Financial Financial

Guarantee Ins Finance

Securities,

commodity

contracts, and

investments

US

Panel B: Lowest Factor Loadings

Λ Name Industry Industry

Subgroup BEA Sector BEA Subgroup Country

1 0.01 NORTH

EURO OIL Energy Oil-US RoyaltyTrusts Mining Oil and gasextraction US

2 0.01 SOUTHERN

CO Utilities ElectricIntegrated Utilities Utilities US

3 0.02 NEWMONT

MINING

Basic

Materials Gold Mining Mining Mining, except oiland gas US

4 0.02 SABINE

ROYALTY Energy Oil-US RoyaltyTrusts Mining Oil and gasextraction US

5 0.02 GENERAL

MILLS IN

Consumer,

Non-cyclical

FoodMisc/DiversiedManufacturingFood and beverage

and tobacco products US

6 0.02

CONS

EDISON

INC

Utilities ElectricIntegrated Utilities Utilities US

7 0.02 DYNEX

CAPITAL Financial REITS-Mortgage Finance Real estate US

8 0.02

WEC

ENERGY

GROUP

Utilities ElectricIntegrated Utilities Utilities US

9 0.02 HORMEL

FOODS CRP

Consumer,

Non-cyclical

Food-Meat

Products

ManufacturingFood and beverage

and tobacco products US

10 0.02 KELLOGG

CO

Consumer,

Non-cyclical

FoodMisc/DiversiedManufacturingFood and beverage

and tobacco products US

Note: Sample includes the 524 rms continuously traded over 1989-2017.

x
Table A.3: Top Firm Loadings on Second Factor

Panel A: Highest Factor Loadings

Λ Name Industry Industry

Subgroup BEA Sector BEA Subgroup Country

1 0.18 LAM

RESEARCH Technology Semiconductor

Equipment

ManufacturingComputer and

electronic products US

2 0.18 XCERRA

CORP Technology Semiconductor

Equipment

ManufacturingComputer and

electronic products US

3 0.17 INTEGRAT

DEVICE Technology

Semicon

Compo-Intg

Circu

ManufacturingComputer and

electronic products US

4 0.17 KULICKE &

SOFFA Technology SemiconductorEquipment

ManufacturingComputer and

electronic products SGP

5 0.17

KLATENCOR

CORP

Technology Semiconductor

Equipment

ManufacturingComputer and

electronic products US

6 0.16 SKYWORKS

SOLUTIO Technology Electronic

Compo-Semicon

ManufacturingComputer and

electronic products US

7 0.16 TERADYNE

INC Technology SemiconductorEquipment

ManufacturingComputer and

electronic products US

8 0.15 ANALOG

DEVICES Technology

Semicon

Compo-Intg

Circu

ManufacturingComputer and

electronic products US

9 0.15 CYPRESS

SEMICON Technology

Semicon

Compo-Intg

Circu

ManufacturingComputer and

electronic products US

10 0.15 MICRON

TECH Technology ElectronicCompo-Semicon

ManufacturingComputer and

electronic products US

Panel B: Lowest Factor Loadings

Λ Name Industry Industry

Subgroup BEA Sector BEA Subgroup Country

1 -0.15 ENSCO

PLC-CL A Energy Oil&Gas Drilling Mining Oil and gasextraction UK

2 -0.14 UNIT CORP Energy Oil CompExplor&Prodtn Mining Oil and gas

extraction US

3 -0.14 NOBLE

CORP PLC Energy Oil&Gas Drilling Mining Oil and gasextraction UK

4 -0.14

ROWAN

COMPANIEAEnergy Oil&Gas Drilling Mining Oil and gas

extraction US

5 -0.14 NABORS

INDS LTD Energy Oil&Gas Drilling Mining Oil and gasextraction US

6 -0.13 TIDEWATER INC Industrial TransportMarine Transport Air transportation US

7 -0.12 PARKER

DRILLING Energy Oil&Gas Drilling Mining Oil and gas

extraction US

8 -0.12 HELMERICH

& PAYN

Energy Oil&Gas Drilling Mining Oil and gas

extraction US

9 -0.12

BAKER

HUGHES

INC

Energy Oil-Field Services Mining Oil and gas

extraction US

10 -0.11 APACHE

CORP Energy Oil CompExplor&Prodtn Mining Oil and gas

extraction US

Note: Sample includes the 524 rms continuously traded over 1989-2017.

xi
Table A.4: Top Firm Loadings on Third Factor

Panel A: Highest Factor Loadings

Λ Name Industry Industry

Subgroup BEA Sector BEA Subgroup Country

1 0.17 ENSCO PLC-CL

A

Energy Oil&Gas Drilling Mining Oil and gas

extraction UK

2 0.17 NOBLE CORP

PLC Energy Oil&Gas Drilling Mining Oil and gas

extraction UK

3 0.16 UNIT CORP Energy Oil CompExplor&Prodtn Mining Oil and gas

extraction US

4 0.16 NABORS INDS

LTD Energy Oil&Gas Drilling Mining Oil and gas

extraction US

5 0.16 ROWAN

COMPANIE-A Energy Oil&Gas Drilling Mining Oil and gas

extraction US

6 0.16 PARKER

DRILLING Energy Oil&Gas Drilling Mining Oil and gas

extraction US

7 0.14 TIDEWATER

INC Industrial TransportMarine Transport Air

transportation US

8 0.13 HELMERICH &

PAYN Energy Oil&Gas Drilling Mining Oil and gasextraction US

9 0.13 HALLIBURTON

CO Energy Oil-Field Services Mining Oil and gasextraction US

10 0.13 INTEGRAT

DEVICE Technology

Semicon

Compo-Intg

Circu

Manufacturing

Computer and

electronic

products

US

Panel B: Lowest Factor Loadings

Λ Name Industry Industry

Subgroup BEA Sector BEA Subgroup Country

1 -0.11 HUNTINGTON

BANC Financial Super-RegionalBanks-US Finance SCCI US

2 -0.11 FIFTH THIRD

BANC Financial Super-RegionalBanks-US Finance SCCI US

3 -0.10 REGIONS

FINANCIA Financial

Commer

Banks-Southern

US

Finance SCCI US

4 -0.10 SUNTRUST

BANKS Financial Super-RegionalBanks-US Finance SCCI US

5 -0.09 KEYCORP Financial Super-Regional

Banks-US Finance SCCI US

6 -0.09 SYNOVUS FINL Financial

Commer

Banks-Southern

US

Finance SCCI US

7 -0.09 ZIONS

BANCORP Financial

Commer

Banks-Western

US

Finance SCCI US

8 -0.09 WELLS FARGO

& CO Financial Super-RegionalBanks-US Finance SCCI US

9 -0.09 FIRST

HORIZON NA Financial

Commer

Banks-Southern

US

Finance SCCI US

10 -0.08 MBIA INC Financial Financial

Guarantee Ins Finance SCCI US

Note: Sample includes the 524 rms continuously traded over 1989-2017. SCCI: Securities, commodity

contracts, and investments.

xii
B Three-Step Estimation Method Details

The rst step of the procedure is to select the number of common factors, K, with the panel

BIC method from Bai and Ng (2002) and use the Covariance PCA with Full Factor Series

method described in Section A to obtain the common factors and the rms' loadings on them.

Once the common variation for each rm (ΛFt) is estimated with these, the idiosyncratic

rm returns, RI

t

, are obtained under the simplifying assumption that the τ are minimal:

RI

t = Rt −ΛFt

. Step two is to estimate Equation (1) via OLS on the factor series. The third

step utilizes the OCMT variable selection procedure, which we next describe in detail.

B.1 OCMT Variable Selection Procedure

The OCMT procedure in our use case involves evaluating the net impact of each of N

potential explanatory variables, RI

1,t−1

, RI

2,t−1

, . . . , RI

N,t−1

, on a dependent variable, RI

i,t, in a

linear model of the form:

R

I

i,t = ρi0 +

X

N

n=1

ρinR

I

n,t−1 + it, for t = 1, 2, . . . , T, (B.1)

where N is small relative to T and a subset of the ρin coecients are non-zero. The intuition

is that if an explanatory variable's coecient is non-zero, then its mean net impact on RI

i,t

should be signicantly dierent from zero, where the mean net impact of variable RI

n,t−1

is:

θin =

X

N

l=1

ρilcov(R

I

n,t−1

, RI

l,t−1

).

Each variable is considered individually through a series of bivariate regressions of RI

i,t on

each RI

n,t−1

series and a constant estimated with OLS. The t-ratio of ρˆin from each regression

is then compared to a critical value that takes into account the multiple testing aspect of

this approach. The OCMT test of ρin 6= 0 is:

|tρˆin | > Φ

−1



1 −

p

2Nδ



,

where p is the size of the test, and Φ

−1

is the inverse of the cumulative standard normal

distribution. The denominator of the second term can take a number of functional forms,

but we choose this simple form and δ = 1 for the rst iteration. We then add the included

variables to the test regressions along with a constant and repeat the test with δ = 2 until no

further variables are added. These values for δ are the lower bound to asymptotically select

the proper variables, and in untabulated results we repeat our main analysis for a series of

xiii
alternate values nding similar conclusions. The nal step is to then estimate Equation (B.1)

using OLS with only the selected variables and a constant included, setting the coecients

on all of the other variables to zero. To estimate Equation (2), we perform this analysis on

each row of it, and combine the estimated coecient vectors to arrive at our estimates of ρ0

and ρ.

C State-Space Model Estimation Details

This section provides details of the Gibbs sampling priors, posteriors and estimation algorithm. To reiterate, the observation and state equations for our system are:

Rt = [Λ IN ]

"

Ft

RI

t

#

+ τt

and

"

Ft

RI

t

#

=

"

0K×1

ρ0

#

+

"

Γ 0K×N

0N×K ρ

# " Ft−1

RI

t−1

#

+

"

ηt

t

#

.

The rst consideration for estimating the state-space system is the lack of unique identication for the Ft

factors, which aects their data time series, their loadings, and the

covariance matrix of their innovations. There are three issues: their scale is not identied

(i.e., multiplying the loadings by two and dividing the factors by two would equally t the

system); their sign is not determined (i.e., multiplying the loadings and factors by negative

one would equally t); and their order is not unique (i.e., switching the rst and third factors

and their loadings would t the same). We use three approaches to address these.

First, for the scale indeterminacy, we assume that the factors have variances of one,

and that their innovations are uncorrelated. In other words, the covariance matrix of their

innovations, Υ, is the IK identity matrix. This is in the spirit of PCA, which we use to

determine the priors for these factors. As an extension one could apply economic insights

on top of the factors to identify them, for example by setting certain loadings to one or

zero based on rms' industries or geographic regions, or other rotations based on external

information (e.g., match impulse responses to ndings from other research or a theoretical

model); however, we do not apply such restrictions or rotations in order to let the data inform

us about them with minimal assumptions or structure implied. We go into further details

below on how we use PCA estimates and variance adjust them for this to create priors for

the unobserved states and their loadings in the estimation algorithm.

Second, for the sign indeterminacy, for one of the rms available throughout our whole

sample and in all of our networks (IBM), we require, without loss of generality, that its

xiv
loadings on all factors are positive. If not, we multiply the factor and respective loadings

on it for all rms by negative one. Third, for the factor order, as with PCA, our estimation

orders the factors in descending order of the cumulative variance they explain for the Rt

equity returns.

The above assumptions mean that we do not estimate Υ, leaving the following as the

parameters we must estimate:

 Λ

 Γ

 ρ0

 ρ

 T

 Σ

 Ft

 RI

t

.

We next provide an overview of the priors and their respective posterior distributions

where applicable, and then the estimation algorithm details.

C.1 Model Prior & Posterior Distributions

In this section, we discuss the model priors and the resulting posterior distributions, working

from the top of the above state-space equations to the bottom. For expositional purposes,

we do not list the full information set that each conditional distribution is contingent upon,

but in general, it is the observed data and the most recently drawn parameter values and

unobserved state series not being drawn in the current posterior distribution.

When setting the priors, our underlying assumption is that the common factors and

idiosyncratic return series are random walks in levels. Hence, the mean of our prior for Γ

is 0K×K, for ρ0 is 0N×1, and for ρ is 0N×N . The initial values for the common factor series,

F

(0)

t

, are the rst K PCA components of the equity returns using the Covariance PCA with

Full Factor Series method, then scaled and sign adjusted as described above. The mean

of the prior for the Λ matrix, L ̄, is centered at the PCA loadings after they are likewise

adjusted for the common factors' scaling and signs. The initial values for the idiosyncratic

return series are R

I(0)

t = Rt − LF ̄

(0)

t

.

When constructing the variances for the priors of the coecient matrices, we follow

the idea of Litterman's Minnesota prior in scaling by the variables' relative variances. As

customary in the literature, the residual variances from univariate models of the data series

are used in determining these. Our assumptions about the data generating process imply

a random walk without drift for each of the i idiosyncratic return series, which makes the

variance of the i

th RI(0) return series itself the σˆ2

i variance estimate.

xv
C.1.1 Observation Equation Prior & Posterior Distributions

The prior for the common factor loading coecients of the i

th equation of the observation

system is:

Λ

0

Ri ∼ N (L ̄0

Ri, L ̆

i) ; L ̆i ≡ κ

2

Λσˆ

2

i

IK,

where κΛ controls the tightness of the prior, and ΛRi and L ̄

Ri are the i

th rows of those

matrices. We follow the BVAR literature in setting κΛ = 0.2. The T covariance matrix is

assumed to be diagonal, and a conjugate prior for the Tii terms on its diagonal tting this

is an independent inverse gamma distribution for each one. Specically, our prior for each

Tii is:

Tii ∼ Γ

−1



T

0

i

2

,

κσσˆ

2

i

2



with the scale parameter and degrees of freedom set such that the mean of this distribution

is κσσˆ

2

i

(i.e., T

0

i = 3 ∀ i). The κσ term is a scalar controlling the prior on the share of the

variance explained by the idiosyncratic shocks. This is assumed to be small and we set this

parameter to 1%.

The loadings on the idiosyncratic terms are known, hence to simplify the estimation we

dene Y ≡ R −Rˆ

I

with the observations stacked such that each of these is a T × N matrix.

This implies that:

Y = F Λ

0 + τ,

where F and τ likewise have the observations over time stacked. Under these assumptions,

Kadiyala and Karlsson (1997) show that the posterior distribution for the coecients are

normally distributed and can be drawn independently for each equation as:

Λ

0

Ri ∼ N 

(L ̆−1

i + T

−1

ii F

0F )−1

(L ̆−1

i L ̄0

Ri + T

−1

ii F

0F LOLS

i

),(L ̆−1

i + T

−1

ii F

0F )−1



,

where L

OLS

i

is the vectorized OLS estimate of the coecients, L

OLS

i ≡ (F

0F )−1F0Y Ci, with

Y Ci the i

th column of Y . The posterior for Tii is distributed inverse gamma:

Tii ∼ Γ

−1



T

0

i + T

2

,

κσσˆ

2

i + (Y Ci − F Λ

0

Ri)

0

(Y Ci − F Λ

0

Ri)

2



.

Note that these distributions can be drawn from in parallel across the i observation equations,

which can be taken advantage of to speed up the Gibbs sampling algorithm.

C.1.2 State Transition Equation Prior & Posterior Distributions

The prior for Γ is:

vec(Γ0) ∼ N (0K2×1, G) ; G ≡ κ

2

Γ

IK2 ,

xvi
where κΓ controls the tightness of the prior, and we follow the BVAR literature in setting

κΓ = 0.2. Its posterior is:

vec(Γ0) ∼ N 



G

−1 + IK ⊗F0

t−1F t−1

−1

(IK ⊗F

0

t−1F t−1)Γ0

OLS

,



G

−1 + IK ⊗F0

t−1F t−1

−1



,

where Γ

0

OLS ≡ vec

(F

0

t−1F t−1)

−1F0

t−1F



, the vectorized OLS estimate of the coecients.

For the VAR(1) of the idiosyncratic returns' transition equations, we utilize dummy

variables to implement the priors in a more computationally tractable way, given the system's

large size. To simplify the exposition, we restate these equations with stacked variable

matrices of T rows each, and the lagged variables and constants combined into a single

T × (1 + N) matrix with the ones in the rst column and lagged variables after it, denoted

X ≡ [1T ×1 RI

t−1

]. The idiosyncratic returns transition equation is then:

R

I = Xρ + ,

where ρ ≡ [ρ0 ρ]

0

. The expected value for each element of ρ is zero. The variances of the

coecients on the constant terms will reect diuse priors, and the variance of the coecient

for dependent variable i on explanatory variable j will be of the Minnesota form:

V AR(ρ1+j,i) =

κ

2

ρσˆ

2

i

σˆ

2

j

. (C.1)

We follow the BVAR literature in setting κρ = 0.2. We specify an Inverse-Wishart prior

for Σ, centered on the diagonal matrix with the σˆ

2

i

terms on the diagonal. To eciently

implement these priors, we utilize articial dummy observations:

Y D

(2N + 1) × N

=







0N×N

01×N

diag(ˆσ1, σˆ2, . . . , σˆN )





 ,

X D

(2N + 1) × (N + 1)

=







0N×1 diag(ˆσ1, σˆ2, . . . , σˆN )/κρ

 01×N

0N×(N+1)





 .

The rst block of dummy observations implements the priors for the coecients in ρ, the

second block the uninformative priors for the constants ( is a small number), and the third

block for the covariance matrix. Specically, we employ a standard normal Inverse-Wishart

prior of the form:

vec(ρ)|Σ ∼ N vec(ρD), Σ ⊗ (X0

DX D)

−1



and Σ ∼ IWΣD, N + 2.

The prior parameters are ρD = (X0

DX D)

−1X0

DY D = 0(N+1)×N , ΣD = (Y D − X DρD)

0

(Y D −

X DρD) = diag(ˆσ

2

1

, σˆ

2

2

, . . . , σˆ

2

N ), and it is straightforward to show that the prior variance

xvii
of ρ has the Minnesota form of Equation (C.1). The degrees of freedom for the inverseWishart prior is set to the minimum that guarantees the existence of the mean of Σ, which

is ΣD. It can be shown through algebraic substitution that the posterior has a similar form

as the prior, but with the actual data appended to the dummy observation matrices. Let

Y ∗ = [RI

0

Y

0

D]

0

, X∗ = [X0 X0

D]

0

, ρ∗ = (X0

∗X∗)

−1X0

∗Y ∗ and Σ∗ = (Y ∗ − X∗ρ∗)

0

(Y ∗ − X∗ρ∗).

We can then write the posterior as:

vec(ρ)|Σ, Y ∼ N vec(ρ∗), Σ ⊗ (X0

∗X∗)

−1



and Σ|Y ∼ IWΣ∗, T + N + 2.

The dummy observation approach helps to regularize the coecients' variance matrix such

that only (X0

∗X∗) must be inverted, improving the computational tractability of the posterior

draws from their normal distribution.

C.1.3 Unobserved Factor Series Priors

Finally, we need the S1 and P1 priors for the simulation smoother algorithm used to draw

the posterior values of the unobserved Ft and RI

t

states: [F

0

1 RI

0

1

]

0 ∼ N 

S1, P1



. Our unconditional expectation for the initial period states is that they are zero (i.e., the underlying

series are random walks in levels), so S1 is a (K + N) × 1 zero vector. We assume they are

uncorrelated, making P1 a diagonal matrix with ones for the rst K terms relating to the

common factors and the N σˆ

2

i variances thereafter.

C.2 Gibbs Sampling Algorithm

The general approach of the estimation algorithm follows Carter and Kohn (1994), which

alternates between drawing from the conditional distribution of the VAR parameters for

the observation and state equations given the unobserved series, and then drawing from the

conditional distribution of the Ft and RI

t

series given those parameters. The steps for the

estimation algorithm are described below.

1. Set the priors for the factors and their loadings.

(a) Determine the number of common factors, K. We use the panel BIC information

criteria method from Bai and Ng (2002) to select the number of factors  the

form they suggest to account for potential correlation in the idiosyncratic errors.

(b) Run covariance matrix based PCA inclusive of the means on the rms' daily

returns to get the prior means for the top K common factors and their loadings.

(c) Divide the factors by their standard deviations and multiply their loadings by

them.

xviii
(d) Conrm that IBM has a positive loading on each factor. If not, multiply that

factor's values and its loadings for all rms by negative one.

(e) Set the initial draws of the T × K common factors matrix, F

(0), to equal the

adjusted PCA series.

(f) Set the mean of the prior for the Λ matrix, L ̄, as the PCA loadings adjusted for

the common factors' scaling and signs.

(g) Set the initial draws of the T×N matrix of idiosyncratic returns by subtracting the

adjusted PCA factor contributions from the observed returns: RI(0) = R−F

(0)L ̄0

.

(h) Get the idiosyncratic return series variances: σˆ

2

i = V AR(R

I(0)

i

) ∀ i ∈ {1, 2, . . . , N}.

(i) Set S1 to a (K + N) × 1 zero vector.

(j) Set P1 to a diagonal matrix with ones for the rst K terms relating to the common

factors and the σˆ

2

i variances for the idiosyncratic returns' entries.

2. Set the prior parameters for the observation equation.

(a) The prior mean for the Λ coecients is L ̄, as set in the prior step.

(b) The prior covariance matrix for the Λ coecients of each of the i equations of the

observation system is L ̆

i ≡ κ

2

Λσˆ

2

i

IK where κΛ = 0.2.

(c) The degrees of freedom for each Tii prior is 3.

(d) The scale for each Tii prior is Si ≡ κσσˆ

2

i where κσ = 1%.

(e) Set T

(0)

ii = Si ∀ i ∈ {1, 2, . . . , N}.

3. Set the prior parameters for the state equation.

(a) The common factor state system:

i. The prior mean for vec(Γ0) is 0K2×1.

ii. The prior covariance matrix for vec(Γ0) is G ≡ κ

2

Γ

IK2 where κΓ = 0.2.

iii. Υ is the IK identity matrix.

(b) The idiosyncratic return factor state system:

i. Create the dummy observations with κρ = 0.2 and  = 1e

−8

:

Y D

(2N + 1) × N

=







0N×N

01×N

diag(ˆσ1, σˆ2, . . . , σˆN )





 ,

X D

(2N + 1) × (N + 1)

=







0N×1 diag(ˆσ1, σˆ2, . . . , σˆN )/κρ

 01×N

0N×(N+1)







xix
4. Cycle through iterations of Gibbs sampling, indexed by g, drawing from the conditional

posterior distributions.

(a) Run a parallel loop over the N observation equations.

i. Subtract the idiosyncratic returns from the observed ones: Y

(g)

Ci ≡ Ri −

R

I(g−1)

i

, each of these being T × 1 vectors.

ii. Draw the Λ

(g)

coecients for observation equation i from:

Λ

(g)

0

Ri ∼ N 

(L ̆−1

i +

1

T

(g−1)

ii

F

(g−1) 0F(g−1) )

−1

(L ̆−1

i L ̄0

Ri+

1

T

(g−1)

ii

F

(g−1) 0Y

(g)

Ci),(L ̆−1

i +

1

T

(g−1)

ii

F

(g−1) 0F(g−1) )

−1



.

iii. Draw T

(g)

ii from:

T

(g)

ii ∼ Γ

−1



3 + T

2

,

Si + (Y

(g)

Ci − F

(g−1)Λ

(g)

0

Ri )

0

(Y

(g)

Ci − F

(g−1)Λ

(g)

0

Ri )

2



.

iv. Collect the Λ

(g)

coecients, ensuring that the factors are ordered in descending order of the cumulative variance their innovations explain for the Rt equity

returns. If not, redraw the prior two steps.

(b) Draw Γ

(g)

0

, checking for stability, from:

V

∗ ≡



G−1 + IK ⊗ (F

(g−1)0

t−1 F

(g−1)

t−1

)

−1

Γ

0

OLS ≡ vec



F

(g−1)0

t−1 F

(g−1)

t−1

−1

F

(g−1)0

t−1 F

(g−1)

vec(Γ(g)

0

) ∼ N V

∗



(IK ⊗ F

(g−1)0

t−1 F

(g−1)

t−1

)Γ0

OLS

, V ∗



.

(c) Use direct Monte Carlo sampling from posterior of Σ and then ρ for the idiosyncratic VAR parameters, checking for stability of the coecient vector.

i. First calculate the necessary parameters:

X

(g) ≡ [1T ×1 R

I(g−1)

t−1

],

Y

(g)

∗ = [R

I(g−1)0

Y

0

D]

0

,

X

(g)

∗ = [X

(g)

0

X

0

D]

0

,

ρ

(g)

∗ = (X

(g)

0

∗ X

(g)

∗

)

−1X(g)

0

∗ Y

(g)

∗

xx
and

Σ

(g)

∗ = (Y

(g)

∗ − X

(g)

∗ ρ

(g)

∗

)

0

(Y

(g)

∗ − X

(g)

∗ ρ

(g)

∗

).

ii. Draw Σ

(g)

from its posterior distribution:

Σ

(g)

|Y ∼ IWΣ

(g)

∗

, T + N + 2.

iii. Draw ρ

(g)

from its posterior distribution:. For large samples, one should draw

from the equivalent but more ecient matrix normal distribution.

vec(ρ

(g)

)|Σ

(g)

, Y ∼ N vec(ρ

(g)

∗

), Σ

(g) ⊗ (X(g)

0

∗ X

(g)

∗

)

−1



.

(d) Draw the F

(g) and RI(g)

states. We utilize the implementation of the Durbin and

Koopman (2002) simulation smoother provided in Jaroci«ski (2015), which we

further optimize for Matlab data structures to store our large system in memory.

Then draw the period-zero states conditional on these using backwards iteration

(i.e., one step of the Carter and Kohn algorithm).

5. Repeat the prior step the desired number of times, retaining the nal draws after the

burn-in period.

D Firm Network Estimation Details

D.1 One Period Ahead GIRF Network Edge Equation

The formula for the one period ahead GIRF (Generalized Impulse Response Function) of

Pesaran and Shin (1998) is:

GIRFj→i(1) = e

0

p

iA1Θej

e

0

jΘej

=

e

0

i







ρΣ + ΛΓ(L)ΥΛ0 ρΣ ΛΓ(L)Υ

ρΣ ρΣ 0N×S

Γ(L)ΥΛ0 0S×N Γ(L)Υ







ej

p

e

0

jΘej

D.2 One Period Ahead GFEVD Network Edge Equation

The formula for the one period ahead GFEVD (Generalized Forecast Error Variance Decomposition) of Pesaran and Shin (1998) is:

GF EV Dj→i(1) =

Θ

−1

jj [(e

0

iΘej )

2 + (e0

iA1Θej )

2

]

e

0

iΘei + e

0

iA1ΘA0

1

ei

xxi
where the ei are appropriately sized selection vectors with zeroes in all cells except for the

i

th, which is one.

D.3 VAR Estimation Bias from Omitting Common Factors

This equation provides the bias that would result from estimating ρ ̃ = [ρ0 ρ] from a standard

VAR with the total-returns, but not accounting for the common factors. The second term

is the bias, and the R1

t−1

terms include a column vector of ones before the rm returns to

account for the constant term:

E[R1

tR1t−1

0

(R1

t−1R1t−1

0

)

−1

] =  ̃ρ + (ΛΓ(L) − ρ ̃Λ)F L,t−1R1

t−1

0

(R1

t−1R1t−1

0

)

−1

Note that the bias is zero if either Λ is zero or all of the F L,t−1 are zero. In either case, the

bias is zero when the factors have no eect on the rms' total-returns.

D.4 Empirical Method Simulations

To illustrate the network estimation procedure, we apply it to simulated data from the system

dened by Equation (5). The simulated system has three independent common factor series

following vector-autoregressive processes with zero constants and coecients on their rst

lags of 0.9. The innovations to these series are independent normally distributed, and have

variances of 16. The values of Υ and Γ are:

Υ =







16 0 0

0 16 0

0 0 16







and Γ(1) =







0.9 0 0

0 0.9 0

0 0 0.9





 .

The coecients of the simulated rms on the factors are broken into ve sets by their

factor loadings. The rst fth of the dataset has a Λ coecient of one on the rst factor only,

the next fth has 0.5 coecients on the rst two factors, the next fth has a coecient of one

on the second factor only, the fourth partition has 0.5 coecients on the second and third

factors, and the nal fth has a coecient of one on only the third factor. This setup enables

us to analyze the results of our estimation procedure when there are complex interactions

among the nodes' factor dependence.

To further create heterogeneity across the simulated markets  and to examine the

eects of the idiosyncratic shocks  the innovations of the idiosyncratic return series for

the markets are the rst ve powers of two, repeated in order along the diagonal of Σ, with

the symmetric o-diagonal terms in the remainder of Σ coming from normally distributed

random draws. If needed, the Σ matrix is adjusted to be diagonally dominant to ensure that

xxii
it is positive denite.

To create an easily discernible pattern of varied connections across the rms for us to

study, the ρ matrix is given a block diagonal form. The simulated markets are split into

nine groups that have Uniform[−0.9999, 0.9999] randomly distributed coecients for markets

within the same block, with zeroes elsewhere. If needed, the ρ matrix is multiplied by 0.9

until the modulus of its largest eigenvalue is less than one. These groups can be thought to

model distinct industries. This yields the following form for the ρ matrix:

ρ =



















ρ1 0250 0250 0250

0250 ρ2 0250 · · · 0250

0250 0250 ρ3 0250

.

.

.

.

.

. 0250

0250 0250 0250 0250 ρ9



















where the ρg are the randomly generated coecients and 0250 represents a 250 × 250 matrix

of zeroes. The ρ0 constant vector is drawn from a mean zero normal distribution. We use

nine even groups so that we have partitions within and overlapping each of our ve adjacent

Λ coecient groups, so that there is a complex interaction between the ρ and Λ coecient

groups for our process to attempt to disentangle. Finally, we have fty repetitions of each

idiosyncratic innovation volatility and ρg coecient group number combination. This makes

for 50 × 5 × 9 = 2, 250 simulated markets.

Given these model parameters, to obtain the simulated data we rst generate 12, 500

multivariate normal draws of t and ηt. We then initialize RI

1

and F1 to be zero vectors and

calculate their data series from the generated innovations and above parameter matrices.

Finally, we calculate the R values from these two series. The rst 5, 000 values of the R

series are treated as burn-in values and are dropped, with the remaining 7, 500 saved as the

values to apply our estimation procedure to, close to our full sample's T = 7424.

Figure D.1 shows the estimated networks when including or separating out the eects of

common factors. The rst row of the gure provides the connections between the simulated

R data series with an R to R total-return network. The second row's network has edge

weights between rms and factors based on the expected responses of rms' total-returns from

idiosyncratic shocks to other rms or factors in an RI & Factors to R & Factors network.

This allows us to simultaneously examine the roles of shocks to rms and aggregate factors.

The columns of Figure D.1 dier only in the legends used for coloring the nodes. The

rst column has the nodes colored based on the factor(s) they load on. The second column is

colored based on the nine blocks of non-zero ρ coecient groups. To understand the results

of the estimation process, we start by analyzing the R to R network in the rst row. It

xxiii
is evident in the left plot that our procedure has grouped the nodes by the factors that

each rm is directly aected by, with those loading on the rst factor only at the bottom

right in red, then those loading on the rst two factors just to the left of that group in

green, followed by those directly aected by only the second factor adjacent to that group,

and so on. These total-return networks are similar to those uncovered using the approach

common in other papers in the literature of applying a version of adaptive elastic-net and

then calculating network edges from either the raw or standardized variance decompositions

or impulse responses  as can be seen in Figure D.4 versus the rst row of Figure D.1 

with those also being driven by the common factor loadings.

The right plot in the rst row shows the same network with each node in the identical

position as the rst plot. Here, the ρ coecient groups can be seen in the clustering; however,

it is their overlap with the factor loading groups that drives the patterns in them  all of the

nodes in the rst ρ coecient group load only on the rst factor so they are in the bottom

right group in black, those in the second ρ coecient group in gray are mixed in between

those loading only on the rst factor and those loading on both of the rst two factors so

they are split between the rst two clusters, and so on.

The second row's plots demonstrate how our procedure is able to extract and separate the

direct inter-rm connections captured in the ρ coecient groups from the common factors.

The factors have roughly even total inuences on the network and are near the center of

the gures, so the major dynamics of these gures are determined by the RIsourced edges.

When looking at the left column, it at rst appears that the rm factor groups are central

to the organization; however, it is clear from the right column and perfect clustering by the

ρ coecient groups that they in fact are the drivers. Clustering in this network is governed

by the ρ coecient group structure, unlike the R to R network.

We compare our estimated network matrices with ones calculated algebraically from

the simulation model constants. The correlation for the total-return network is 0.996, and

that for the idiosyncratic network is 0.98 with both being statistically signicant at the 1%

level, indicating that the output of our estimation procedure is similar to the networks that

we wished to uncover. Finally, we obtain network edges with alternative methods, such

as GIRFs (Figure D.2), GFEVD (Figure D.3), and adaptive-elastic net or AEN (Figure

D.4), which have also been commonly used in the literature. These alternative methods to

calculate network edges yield similar results, with the total-return networks being driven by

the common factors and the idiosyncratic networks by the ρ groups.

xxiv
Figure D.1: Spring Plots of Simulated Networks with GFEVc

Network Type Λ Factors Loaded on ρ Coecient Groups

R to R

RI & Factors

to R & Factors

1

2

3

1

2

3

Note: Each row shows one of our network estimation methods for the simulated data. R to R

does not remove common factors; and RI & Factors to R & Factors separates common factors but

keeps them in the network as nodes themselves. In both cases, each dot is a panel member (colored

by factor loadings on the left and coecient groups on the right), and the proximity of nodes to

one another depends on how connected observations are.

xxv
Figure D.2: Spring Plots of Simulated Networks with GIRF

R to R RI & Factors to R & Factors RIto RI

Λ Factors

Loaded on

1

12

2

23

3 1

12

2

23

3

Factor

1

2

3

1

12

2

23

3

ρ Coecient

Groups

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

Factor

1

2

3

1

2

3

4

5

6

7

8

9

Firm

Idiosyncratic

Variance

16

8

4

2

1 16

8

4

2

1

Factor

1

2

3

16

8

4

2

1

Note: Estimated networks from data simulations with 3 common factors when using the absolute

value of generalized impulse response functions (GIRF) instead of GFEVc.

xxvi
Figure D.3: Spring Plots of Simulated Networks with GFEVD

R to R RI & Factors to R & Factors RIto RI

Λ Factors

Loaded on

1

12

2

23

3 1

12

2

23

3

Factor

1

2

3

1

12

2

23

3

ρ Coecient

Groups

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

Factor

1

2

3

1

2

3

4

5

6

7

8

9

Firm

Idiosyncratic

Variance

16

8

4

2

1 16

8

4

2

1

Factor

1

2

3

16

8

4

2

1

Note: Estimated networks from data simulations with 3 common factors when using generalized

forecast error variance decompositions (GFEVD) instead of contributions.

xxvii
Figure D.4: Spring Plots of Simulated Networks with Adaptive Elastic Net

Estimation

Method Λ Factors Loaded on ρ Coecient Groups Firm Idiosyncratic Variance

GIRF,

Standardized

1

12

2

23

3

1

2

3

4

5

6

7

8

9

16

8

4

2

1

GFEVD,

Standardized

1

12

2

23

3

1

2

3

4

5

6

7

8

9

16

8

4

2

1

GIRF

1

12

2

23

3

1

2

3

4

5

6

7

8

9

16

8

4

2

1

GFEVD

1

12

2

23

3

1

2

3

4

5

6

7

8

9

16

8

4

2

1

Note: Generalized forecast error variance decomposition (GFEVD) and generalized impulse response

function (GIRF) networks calculated from VAR estimated using adaptive elastic net instead of

OCMT for model selection. xxviii
D.5 Top Firms by Weights Out to Others

Table D.1: U.S. Top Firms by Network Out Weights (1989-2017): R to R

Rank Ticker Name Industry Industry Group Sum Weights Out

1 BEN FRANKLIN RES INC Financial Diversied Finan Serv 337.25

2 DOV DOVER CORP IndDiv Miscellaneous Manufactur 324.07

3 GE GENERAL ELECTRIC IndDiv Miscellaneous Manufactur 323.75

4 JPM JPMORGAN CHASE Financial Banks 323.54

5 NTRS NORTHERN TRUST Financial Banks 320.45

6 CMA COMERICA INC Financial Banks 319.47

7 PCH POTLATCH CORP Financial REITS 319.02

8 LNC LINCOLN NATL CRP Financial Insurance 318.1

9 AXP AMERICAN EXPRESS Financial Diversied Finan Serv 317.81

10 PPG PPG INDS INC BasMater Chemicals 316.39

11 EMR EMERSON ELEC CO IndDiv Electrical Compo and Equip 315.61

12 LM LEGG MASON INC Financial Diversied Finan Serv 314.18

13 STI SUNTRUST BANKS Financial Banks 310.72

14 L LOEWS CORP Financial Insurance 310.47

15 BK BANK NY MELLON Financial Banks 308.07

16 TMK TORCHMARK CORP Financial Insurance 306.14

17 ETN EATON CORP PLC IndDiv Miscellaneous Manufactur 305.44

18 WFC WELLS FARGO AND CO Financial Banks 302.69

19 TROW T ROWE PRICE GRP Financial Diversied Finan Serv 302.25

20 IR INGERSOLL-RAND IndDiv Miscellaneous Manufactur 299.68

21 KEY KEYCORP Financial Banks 298.88

22 WRI WEINGARTEN RLTY Financial REITS 298.32

23 PCAR PACCAR INC ConsCycl Auto Manufacturers 297.32

24 TRN TRINITY INDUSTRI IndDiv Miscellaneous Manufactur 295.99

25 ITW ILLINOIS TOOL WO IndDiv Miscellaneous Manufactur 293.81

Note: Sample includes the 524 rms continuously traded over 1989-2017. GFEVc networks with 3

common factors (T = 7,424 and N = 524). Self-loops not included.

D.6 U.S. Inter-Firm Network Application: Market Beta & Commodity Factor Shocks

For these exercises, we combine our U.S. inter-rm network estimation with visualization

algorithms to analyze responses to common shocks, with the novel addition of including

common factors as separate nodes in the networks. We focus on the network of 1, 416 rms

continuously traded over 2008-2017 obtained with the three-step estimation method. In the

spring plots in Figures D.5 and D.6, the location of each rm is the same, though the colors

are set based on their expected log returns over the given period following a shock to one of

the common factors, with the darkest green for returns over 1% and the darkest red for those

below −1%. Most of the price movement comes on impact, with some moderate returns for

a few rms the day after, and virtually zero impact in the following days, as equity markets

xxix
are quick to incorporate new information. The spring plots with BEA sector legends are also

included for reference.

Figure D.5 shows that nancial, consumer cyclical and commodity companies would be

most aected by a one standard deviation positive market beta shock at t = 0. Visually,

this is reected in the fact that nancial rms are at the center of the network in close

proximity to the rst factor. Additionally, a positive move in the rst factor is correlated

with a negative move in the second factor, which over this period reects a commodity price

increase in that factor. The second factor here has a year-over-year growth correlation of

−0.79 with Brent crude oil and −0.74 with the Goldman Sachs Commodity Index. These

plots show which rms are most driven by the market beta factor  both directly and

indirectly through other rms and factors  and that the central nance rms have a high

degree of inuence on the network, in agreement with our ndings that the majority of the

top 25 rms by total-return network weights out are nancial rms. Further, all of the rms'

cumulative returns through t = 2 are positive, consistent with the ndings in Figure A.5

that rms load positively on the rst factor.

Figure D.6 shows the eect of a commodity price drop as measured by a one standard

deviation shock to the second factor. Lower commodity prices would unsurprisingly most

adversely aect energy and base materials companies. In fact, the top 10 declining equities

following the shock are for rms in the oil and gas extraction sub-sector. On the other hand,

United Continental Holdings  the parent company of United Airlines  would be expected

to have the largest positive response. This result likely reects the high fuel costs faced by

airlines, and recognizing this could be used by an airline's managers as an indicator that

it should hedge its commodity exposure. The other rms in the top ten highest expected

equity returns are banks and REITs, possibly because lower commodity prices benet rms

in other sectors of the economy that would be passed onto them. It is not only these central

sectors mentioned that increase, but the consumer cyclical rms also have positive returns,

supporting this interpretation. Overall, the responses to a shock to commodity prices have

more variation in rm returns than those to the rst factor, with some cumulative returns

positive and some negative after two days.

These observations of the expected inter-rm dynamics can help managers and policymakers analyze potential exposures to common shocks, and inform investors' diversication

choices to confront the systemic risk they face, as in the case of a fund that is long airlines

recognizing the latent commodity risk factor in that investment by also going long oil and

gas extraction rms.

xxx
Figure D.5: Positive Market Beta Shock for the U.S. Firm Network (2008-2017)

(a) BEA Sector (b) t = 0 GIRF

1

2

3

1

2

3

(c) t = 1 GIRF (d) Through t = 2 Cumulative GIRF

1

2

3

Note: Networks are for the portion of our sample continuously traded from 2008-2017 with 3

factors. Factors extracted by covariance PCA on the matrix of daily log equity returns. The shock

is a positive one to the rst factor, correlating with a positive market beta shock. The node colors

represent the return impact on each rm from > 1% in dark green to < −1% in red.

xxxi
Figure D.6: Commodity Price Decline Shock for the U.S. Firm Network (2008-2017)

(a) BEA Sector (b) t = 0 GIRF

1

2

3

1

2

3

(c) t = 1 GIRF (d) Through t = 2 Cumulative GIRF

1

2

3

Note: Networks are for the portion of our sample continuously traded from 2008-2017 with 3

factors. Factors extracted by covariance PCA on the matrix of daily log equity returns. The shock

is a positive one to the second factor, which over this period correlates with a commodity price

decline. The node colors represent the return impact on each rm from > 1% in dark green to

< −1% in red.

xxxii
E Full Specication of Theoretical Model of Firm Exposures

The model has two sets of agents acting in discrete time: a unit continuum of identical

households; and heterogeneous rms divided across J industries, each producing a single

dierentiated product in a monopolistically competitive environment. Every period, rms

choose how much of their goods to sell to the household sector and to other rms as intermediate inputs. Each period proceeds in three stages. In the rst stage, the households 

who are the sole owners of inter-period capital  determine how to allocate their labor, and

their capital holdings across renting to the rms and a technology to produce further capital

for tomorrow. Each rm determines how much labor and capital to employ, the amount of

other rms' output to use as intermediate inputs in its production process, and the price

it will charge. In the second stage, production of goods and next period capital occur, and

the productivity (vt+1) and taste (βt+1) shocks for t + 1 are realized. In the nal stage, the

rms pay the households their wages, return on capital, and prots as equity dividends. The

households also make their consumption purchases and trade rm equities.

E.1 Household's Problem

The representative household maximizes expected discounted utility:

E0

X∞

t=0

ψtUt where Ut =

 X

J

j=1

β

1

σ

tj c

σ−1

σ

tj ! σ

σ−1

and ctj =

 Z

Mj

ct(j, i)

φj−1

φj di! φj

φj−1

.

Utis the total consumption index discounted by the stochastic discount factor ψt, ctj is the

composite consumption index for industry j = 1, 2, ..., J, σ is the inter-industry elasticity

of substitution, ct(j, i) is consumption of the output from rm i in industry j, Mj = 1 is

the mass of rms in industry j, and φjis the intra-industry elasticity of substitution across

varieties. For mathematical tractability, we use the standard simplifying assumption in the

literature that households' and rms' elasticities are the same. The households have a base

set of preferences, subject to random taste shocks:

βt = β ̄ + Zt (E.1)

where βtis a vector of the βtj terms.

At the beginning of each period, households choose how to allocate capital holdings across

investing in further capital for tomorrow and renting it to rms, Kr

t

, at a market rate of

rt. Households inelastically supply one unit of labor each period, and both the labor and

xxxiii
capital markets are perfectly competitive, with all participants taking prices as given. Postproduction, the representative household makes its consumption and rm equity purchases,

constrained by the following budget:

X

J

j=1

Z

pt(j, i)ct(j, i)di = wt+Kr

t

rt+

X

J

j=1

Z

st(j, i)qt(j, i)di−

X

J

j=1

Z

st+1(j, i)[qt(j, i)−πt(j, i)]di.

wtis the wage per unit of labor and pt(j, i) is the price of the good from rm i in industry

j. The wage is the numeraire in the economy, with wt = 1 for all periods. The qt(j, i) are

the cum-dividend rm equity prices, st(j, i) are the holdings of those equities and πt(j, i) are

the prots repaid as dividends to the equity holders. There is a unit supply of each rm's

equity, with initial equal holdings across the households.

Iterating the household's rst order condition for st+1(j, i) over periods, along with the

equity transversality condition, yields the cum-dividend equity price equation as a function

of expected discounted real dividends:

qt(j, i) = Et

X∞

τ=0

ψt+τ

ψt

Pct

Pc,t+τ

πt+τ (j, i), (E.2)

where Pct is the aggregate consumption price index of the industry price indices, Ptj :

Pct ≡

 X

J

j=1

βtjP

1−σ

tj ! 1

1−σ

; Ptj ≡

Z

pt(j, i)

1−φj di 1

1−φj

.

E.2 Firms' Problem

Within each industry j there is a unit continuum of rms, and the rms use labor, capital,

and other rms' goods as inputs to their production processes. Since the rms do not have

any inter-period choice variables, they solve a series of independent problems each period

seeking to maximize prots:

πt(j, i) = pt(j, i) [ct(j, i) + Dt(j, i)] −

"

wtLt(j, i) + rtKt(j, i) +X

J

l=1

Z

pt(l, n)xt(j, i, l, n)dn#.

Firm i in industry j makes its intermediate input decision for purchases of good n from

industry l, xt(j, i, l, n), and agrees to pay prices for those, pt(l, n), before production occurs.

Dt(j, i) ≡

PJ

l=1 R

xt(l, n, j, i)dn is the total demand for good i from rms to use as an

intermediate input. The amount of labor, Lt(j, i), and capital, Kt(j, i), employed by rm i

are also decided upon in the rst stage.

xxxiv
The rm's output, yt(j, i), is given by the following production function:

yt(j, i) = "v

1

σ

tj 

Kt(j, i)

γLt(j, i)1−γ

 σ−1

σ +

X

J

l=1

ω

1

σ

jlZ

xt(j, i, l, n)

φl−1

φl dn φl

φl−1

σ−1

σ

# σ

σ−1

(E.3)

where γ is the production process' capital share. The vtj industry productivity parameters

are realized during production in the prior period, which can be thought of as rms learning

or improving fabrication techniques during production that they implement the following

period. Let the rm productivity parameters follow a standard AR(1) data generating

process:

∆vt = c + Ξ∆vt−1 + ∆εt. (E.4)

c is a vector of constants, Ξ is a matrix of autoregressive coecients, and the ∆εt are random

shocks. The amount that rm i sells to the households will be its nal output minus the

quantity it already sold to other rms, ct(j, i) = yt(j, i) − Dt(j, i). Finally, the ωjl are

share parameters for the goods of industry l in production for industry j, and the J × J

matrix of these entries, Ω, characterizes the real rm network of the economy  that is, the

input-output structure of the rms' production processes.

E.3 Industry Centralities

Each rm in our model makes three choices each period about where to place itself in the

production network: how much to consume of others' goods as intermediate inputs; how

much to sell to other rms as a supplier of intermediate inputs; and how much to sell as nal

goods directly to the households. From these choices each rm will act both as a consumer

of raw inputs (i.e., labor and capital) and a supplier of nal goods, though they may do

so either directly or indirectly through other rms in one or more production chains. The

degrees to which rms act as consumers of inputs and suppliers of outputs through the full

input-output network are captured by the two centralities that we introduce in this section.

Consumer centrality measures the degree to which a rm consumes raw inputs itself and

through others, and with that its exposure to shocks to its own and other upstream rms'

productivity parameters. Likewise, supplier centrality measures how a rm is exposed to

household demand for its own and other downstream rms' goods. This therefore represents

exposure to shocks to the demand parameters for a rm and those downstream from it.

E.3.1 Consumer Centrality

Using within industry symmetry, and the ratio between rm prices and marginal costs, the

industry price indices (Ptj ) can be related to the quantity and prices of the raw capital and

xxxv
labor inputs through upstream input-output connections:

P

1−σ

t =



IJ − μ

1−σΩ

−1

μ

1−σ

| {z }

≡Ψd

vtz ̃

σ−1

t R ̃1−σt ≡ α ̃tz ̃

σ−1

t R ̃1−σt

(E.5)

where Ptis a vector of the Ptj industry price indices, z ̃tis the labor-capital aggregate, and

R ̃

t

is the price for this composite of raw inputs:

z ̃t ≡ Kr

t

γL

1−γ

t

; R ̃

t ≡ rtKr

t + wtLt =

1

1 − γ

.

Additionally, vtis a vector of the productivity parameters, IJ is the J×J identity matrix, and

μ is a square matrix with the industries' μj ≡

φj

φj−1

values on the diagonal. Ψd is a function of

the rms' positions within the production network from Equation (E.3) and can be thought

of as a markup adjusted Leontief inverse. The vector of consumer centralities for the laborcapital aggregate is dened as α ̃t ≡ Ψdvt

, suggesting that a rm's direct and indirect demand

for raw inputs depends on the economy's production capabilities (vt), technology (Ω), and

the elasticities of substitution. The α ̃tj consumer centrality term captures the importance

of industry j as a user of raw inputs and measures its network adjusted factor use.

E.3.2 Supplier Centrality

The supplier centrality can be calculated by examining the total downstream demand for

a rm's goods from other industries and consumers. The supplier centrality is determined

from the following system of the stacked total demand equations:

(P

σ

t

yt)

0 = β

0

t



IJ − μ

−σΩ

−1

| {z }

≡ΨS

P

σ

ctUt ≡ β ̃0tP

σ

ctUt

. (E.6)

βtis a vector of βtj consumer taste weights, and ytis a vector of industry aggregate outputs:

ytj ≡

Z

yt(j, i)

φj−1

φj di φj

φj−1

.

The supplier centrality vector is dened as β ̃

t ≡ Ψ0

Sβt

, relating a rm's role as a supplier

in the network to the consumer preferences for goods and services of itself and downstream

industries. β ̃

tj therefore reects the network adjusted nal consumption share of rms in

industry j, with all of these stacked in the β ̃

t vector.

xxxvi
E.4 Equity Returns, Common Factors & Network Centralities

In this section, we examine rm equity returns in the model, nding that they depend on

three common factors and each rm's proximity through the inter-rm network to the source

of a productivity or demand shock. To begin, we derive the log steady-state equity price of

rm i in industry j by rst noting that the standard constant elasticity of substitution and

monopolistic competition result holds, with rms' prots being a xed markup ( 1

φj

) of their

sales. Second, multiplying Equations (E.5) and (E.6) gives Ptjytj on the left hand side and

allows us to derive sales in terms of economy wide aggregates and the industry centralities.

Finally, using these facts with equity pricing Equation (E.2) at the steady-state and taking

logs gives:

ln(q(j, i)) = ln 

1

φj



| {z }

Markup

+ ln

1

1 − ψ

| {z }

Discount

Factor

PcU

|{z}

GDP /

Aggregate

Demand



| {z }

Broad Market

Beta

+ ln

Pc

R ̃

σ−1

| {z }

Real Price

Level

+ lnz ̃

σ−1

| {z }

Raw

Input

Supply

+ lnα ̃j

|{z}

Consumer

Centrality

+ lnβ ̃

j

.

| {z }

Supplier

Centrality

(E.7)

Firm equity prices depend on the markups, three common factors, and the upstream and

downstream network centralities. The three common factors represent: the broad market

beta, capturing the discount factor and GDP; the real price level; and the supply of raw

inputs. These three common factors align with those that we found using our empirical

model, which is notable given that we ran standard PCA on the daily equity returns, without

applying any identifying assumptions or rotations.

The log equity returns induced by changes in these common factors, the productivities and

demand parameters can be approximated by dierencing the rst order Taylor expansion of

Equation (E.7) around the steady-state. Letting Rt be a vector of the rm log equity returns,

ΛFt the common factor loadings and their log changes, and RI

t

a vector of idiosyncratic rm

returns, we have:

Rt = ΛFt+R

I

t = ΛFt+ Diag 

1

α ̃



Ψd

| {z }

U≡Upstream Exposure

∆vt+ Diag 

1

β ̃



Ψ

0

S

| {z }

D≡Downstream Exposure

∆βt = ΛFt+U∆vt+D∆βt.

(E.8)

In the upstream exposure matrix, U, each entry measures the exposure of the row sector to

a productivity shock from the column sector, both directly and possibly indirectly through

other sectors whose products are between theirs in a production chain. The idiosyncratic

response of a rm in industry j to innovations in an upstream source industry s would be

ι

0

jΨdιs

α ̃j

∆vs = Ujs∆vs, where ιjis a selection vector with a one in the j

th position and zeroes

xxxvii
elsewhere.

The downstream exposure matrix, D, provides exposures to demand shocks through the

network. The supplier centrality quanties the intensity with which the household consumes

from an industry, both directly and indirectly through its downstream sales. The downstream

exposure matrix captures the potential for propagation of taste shocks for downstream goods

to each industry as the ratio of its centrality to downstream industries' relative to its total

downstream exposure. The idiosyncratic return from a taste shock to a downstream industry

s is ι

0

jΨ0

S

ιs

β ̃

j

∆βs = Djs∆βs.

Further, rearranging the idiosyncratic return portion of Equation (E.8), and assuming

∆Zt and ∆εtfrom Equations (E.1) and (E.4), respectively, are vectors of mean zero i.i.d.

random shocks, yields:

RI

t = ρ0 + ρRIt−1 + U∆εt + D∆Zt + ζt

=⇒ RI

t = ρ0 + ρRIt−1 + t

(E.9)

where ρ0 is a constant vector of rm xed eects, ρ is an N × N inuence matrix, and ζt

is a residual orthogonal to the two shocks. This formula matches Equation (2), where the

RI

t

idiosyncratic returns follow a VAR(1) process. Further, this indicates that the upstream

and downstream exposures are embedded in the t residuals of this system.

xxxviii
E.5 Theoretical Model Simulations to Upstream & Downstream

Shocks

Figure E.1: Simulated Idiosyncratic Equity Responses to Shocks by Exposures: Star Network

HH

1

2

3

Industry Productivity Shock (vk) Taste Shock (βk)

1

0.0

0.1

0.2

0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

0.0

0.1

0.2

0.0 0.1 0.2

Beta Centrality Idiosyncratic Return

Detrended Steady State Return

Note: The y-axis measures the idiosyncratic return from being in steady-state at the initial parameter levels to moving to a steady-state after the shock. The x-axis measures the upstream or

downstream exposure to the source sector multiplied by the change in that sector's specied parameter. The lines in the lower plots are the 45-degree lines equating these two. The source rm

is denoted with an x-marker. The industry legend is 1-black, 2-red, 3-blue, 4-green, and 5-yellow.

Gray is the household.

xxxix
Figure E.2: Simulated Idiosyncratic Equity Responses to Shocks by Exposures: Y-Network

HH

1

2

3

Industry Productivity Shock (vk) Taste Shock (βk)

1

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

2

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Beta Centrality Idiosyncratic Return

Detrended Steady State Return

Note: The y-axis measures the idiosyncratic return from being in steady-state at the initial parameter levels to moving to a steady-state after the shock. The x-axis measures the upstream or

downstream exposure to the source sector multiplied by the change in that sector's specied parameter. The lines in the lower plots are the 45-degree lines equating these two. The source rm

is denoted with an x-marker. The industry legend is 1-black, 2-red, 3-blue, 4-green, and 5-yellow.

Gray is the household.

xl
Figure E.3: Simulated Idiosyncratic Equity Responses to Shocks by Exposures: Nested

Network

HH

1

2

3

4

5

6

7

Industry Productivity Shock (vk) Taste Shock (βk)

1

−0.05

0.00

0.05

0.10

0.15

0.20

−0.05 0.00 0.05 0.10 0.15 0.20

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

2

−0.05

0.00

0.05

0.10

0.15

0.20

−0.05 0.00 0.05 0.10 0.15 0.20

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

3

−0.05

0.00

0.05

0.10

0.15

0.20

−0.05 0.00 0.05 0.10 0.15 0.20

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

−0.05

0.00

0.05

0.10

0.15

0.20

−0.05 0.00 0.05 0.10 0.15 0.20

Beta Centrality Idiosyncratic Return

Detrended Steady State Return

Note: The y-axis measures the idiosyncratic return from being in steady-state at the initial parameter levels

to moving to a steady-state after the shock. The x-axis measures the upstream or downstream exposure

to the source sector multiplied by the change in that sector's specied parameter. The lines in the lower

plots are the 45-degree lines equating these two. The source rm is denoted with an x-marker. The industry

legend is 1-black, 2-red, 3-blue, 4-green, and 5-yellow. Gray is the household.

xli
Figure E.4: Simulated Idiosyncratic Equity Responses to Shocks by Exposures: Parallel

Network

HH

1

2

3

4

Industry Productivity Shock (vk) Taste Shock (βk)

1

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

2

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Beta Centrality Idiosyncratic Return

Detrended Steady State Return

Note: The y-axis measures the idiosyncratic return from being in steady-state at the initial parameter levels to moving to a steady-state after the shock. The x-axis measures the upstream or

downstream exposure to the source sector multiplied by the change in that sector's specied parameter. The lines in the lower plots are the 45-degree lines equating these two. The source rm

is denoted with an x-marker. The industry legend is 1-black, 2-red, 3-blue, 4-green, and 5-yellow.

Gray is the household.

xlii
Figure E.5: Simulated Idiosyncratic Equity Responses to Shocks by Exposures: Linear Network

HH

1

2

3

Industry Productivity Shock (vk) Taste Shock (βk)

1

0.0

0.1

0.2

0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

2

0.0

0.1

0.2

0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

3

0.0

0.1

0.2

0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

0.0

0.1

0.2

0.0 0.1 0.2

Beta Centrality Idiosyncratic Return

Detrended Steady State Return

Note: The y-axis measures the idiosyncratic return from being in steady-state at the initial parameter levels

to moving to a steady-state after the shock. The x-axis measures the upstream or downstream exposure

to the source sector multiplied by the change in that sector's specied parameter. The lines in the lower

plots are the 45-degree lines equating these two. The source rm is denoted with an x-marker. The industry

legend is 1-black, 2-red, 3-blue, 4-green, and 5-yellow. Gray is the household.

xliii
Figure E.6: Simulated Idiosyncratic Equity Responses to Shocks by Exposures: Dense Linear

Network

HH

1

2

3

Industry Productivity Shock (vk) Taste Shock (βk)

1

−0.05

0.00

0.05

0.10

0.15

0.20

−0.05 0.00 0.05 0.10 0.15 0.20

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

−0.05

0.00

0.05

0.10

0.15

0.20

−0.05 0.00 0.05 0.10 0.15 0.20

Beta Centrality Idiosyncratic Return

Detrended Steady State Return

2

−0.05

0.00

0.05

0.10

0.15

0.20

−0.05 0.00 0.05 0.10 0.15 0.20

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

−0.05

0.00

0.05

0.10

0.15

0.20

−0.05 0.00 0.05 0.10 0.15 0.20

Beta Centrality Idiosyncratic Return

Detrended Steady State Return

3

−0.05

0.00

0.05

0.10

0.15

0.20

−0.05 0.00 0.05 0.10 0.15 0.20

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

−0.05

0.00

0.05

0.10

0.15

0.20

−0.05 0.00 0.05 0.10 0.15 0.20

Beta Centrality Idiosyncratic Return

Detrended Steady State Return

Note: The y-axis measures the idiosyncratic return from being in steady-state at the initial parameter levels

to moving to a steady-state after the shock. The x-axis measures the upstream or downstream exposure

to the source sector multiplied by the change in that sector's specied parameter. The lines in the lower

plots are the 45-degree lines equating these two. The source rm is denoted with an x-marker. The industry

legend is 1-black, 2-red, 3-blue, 4-green, and 5-yellow. Gray is the household.

xliv
Figure E.7: Simulated Idiosyncratic Equity Responses to Shocks by Exposures: Diamond

Network

HH

1

2

3

4

5

Industry Productivity Shock (vk) Taste Shock (βk)

1

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

2

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

3

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Beta Centrality Idiosyncratic Return

Detrended Steady State Return

Note: The y-axis measures the idiosyncratic return from being in steady-state at the initial parameter levels

to moving to a steady-state after the shock. The x-axis measures the upstream or downstream exposure

to the source sector multiplied by the change in that sector's specied parameter. The lines in the lower

plots are the 45-degree lines equating these two. The source rm is denoted with an x-marker. The industry

legend is 1-black, 2-red, 3-blue, 4-green, and 5-yellow. Gray is the household.

xlv
Figure E.8: Simulated Idiosyncratic Equity Responses to Shocks by Exposures: Circle

HH

1

2

3

4

Industry Productivity Shock (vk) Taste Shock (βk)

1

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

2

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

3

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

4

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Beta Centrality Idiosyncratic Return

Detrended Steady State Return

Note: The y-axis measures the idiosyncratic return from being in steady-state at the initial parameter levels

to moving to a steady-state after the shock. The x-axis measures the upstream or downstream exposure

to the source sector multiplied by the change in that sector's specied parameter. The lines in the lower

plots are the 45-degree lines equating these two. The source rm is denoted with an x-marker. The industry

legend is 1-black, 2-red, 3-blue, 4-green, and 5-yellow. Gray is the household.

xlvi
Figure E.9: Simulated Idiosyncratic Equity Responses to Shocks by Exposures: Dense Circle

Network

HH

1

2

3

4

Industry Productivity Shock (vk) Taste Shock (βk)

1

0.00

0.05

0.10

0.00 0.05 0.10

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

0.00

0.05

0.10

0.00 0.05 0.10

Beta Centrality Idiosyncratic Return

Detrended Steady State Return

Note: The y-axis measures the idiosyncratic return from being in steady-state at the initial parameter levels to moving to a steady-state after the shock. The x-axis measures the upstream or

downstream exposure to the source sector multiplied by the change in that sector's specied parameter. The lines in the lower plots are the 45-degree lines equating these two. The source rm

is denoted with an x-marker. The industry legend is 1-black, 2-red, 3-blue, 4-green, and 5-yellow.

Gray is the household.

xlvii
Figure E.10: Simulated Idiosyncratic Equity Responses to Shocks by Exposures: 1-2-2-1

HH

1

2

3

4

5

6

Industry Productivity Shock (vk) Taste Shock (βk)

1

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

2

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

3

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

4

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

−0.1

0.0

0.1

0.2

−0.1 0.0 0.1 0.2

Beta Centrality Idiosyncratic Return

Detrended Steady State Return

Note: The y-axis measures the idiosyncratic return from being in steady-state at the initial parameter levels

to moving to a steady-state after the shock. The x-axis measures the upstream or downstream exposure

to the source sector multiplied by the change in that sector's specied parameter. The lines in the lower

plots are the 45-degree lines equating these two. The source rm is denoted with an x-marker. The industry

legend is 1-black, 2-red, 3-blue, 4-green, and 5-yellow. Gray is the household.

xlviii
Figure E.11: Simulated Idiosyncratic Equity Responses to Shocks by Exposures: 2 Nests

Network

HH

1

2

3

4

5

6

Industry Productivity Shock (vk) Taste Shock (βk)

1

−0.05

0.00

0.05

0.10

0.15

0.20

−0.05 0.00 0.05 0.10 0.15 0.20

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

2

−0.05

0.00

0.05

0.10

0.15

0.20

−0.05 0.00 0.05 0.10 0.15 0.20

Alpha Centrality Idiosyncratic Return

Detrended Steady State Return

−0.05

0.00

0.05

0.10

0.15

0.20

−0.05 0.00 0.05 0.10 0.15 0.20

Beta Centrality Idiosyncratic Return

Detrended Steady State Return

Note: The y-axis measures the idiosyncratic return from being in steady-state at the initial parameter levels to moving to a steady-state after the shock. The x-axis measures the upstream or

downstream exposure to the source sector multiplied by the change in that sector's specied parameter. The lines in the lower plots are the 45-degree lines equating these two. The source rm

is denoted with an x-marker. The industry legend is 1-black, 2-red, 3-blue, 4-green, and 5-yellow.

Gray is the household.

xlix
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l