# Risk in Network Economies111I am particularly thankful to Allan Timmermann
for useful comments and discussions throughout the course of this work. I am
also thankful for helpful comments from Alexis Toda, Johannes Wieland, Valerie
Ramey, Ross Valkanov, Joey Engelberg, Munseob Lee, Fabian Trottner, Tjeerd de
Vries, Edoardo Briganti, and Carlos Goes.

Victor Sellemi222PhD Student UC San Diego, email<EMAIL_ADDRESS>
(This Version: July 11, 2022)

###### Abstract

Economic models with input-output networks assume that firm or sector (unit)
growth is driven by a weighted sum of trade partners’ growth and an
independently-drawn idiosyncratic shock. I show that the idiosyncratic risk
assumption in a broad class of network models implicitly generates
restrictions on the network weights which are unrealistic. When allowing for
correlated shocks, units are exposed to an additional risk term which captures
the ability to substitute away from supply and demand shocks propagating
through the network. I provide empirical evidence that changes in
substitutability between trade partners are inversely related to changes in
the panel of realized industry variance. Moreover, I find that supply-side
(demand-side) substitutability is closely related to technological (product)
dispersion of a unit’s suppliers (customers). To synthesize these results, I
propose a production-based asset pricing model in which supply chain
substitutability is a function of dispersion in product/technology space and
correlation in supply and demand shocks is driven by shared customers and
suppliers between firms. The model predicts that assets which are positively
exposed to average propagation of upstream and downstream shocks are useful
hedges and thus earn lower average risk premia. Consistently, I find that
estimated upstream (downstream) propagation factors earn return spreads of
-11.4% (-4.2%) and are negatively associated with aggregate consumption,
output, and dividend growth.

Keywords: Production Networks, Volatility, Systematic Risk, Asset Pricing

##  1 Introduction

The final goods share of consumption in the United States stands at a little
over one-third. Most of what remains is flows of intermediate inputs through
production networks. Recent research finds that production networks play an
important role in shock propagation, business cycles, and systematic risk in
asset markets. However, the relationship between network linkages and
comovement in economic risk is not yet entirely clear, especially at a
granular level. Features of the input-output network are crucial to
understanding the relationship between firm or industry-level risk and
economy-wide aggregate risk.

The benchmark network model assumes that idiosyncratic shocks are drawn
independently across units (firms or industries) in the network, and then
propagate to connected units as a function of network weights. Network weights
capture the relative importance of each connection (edge) between units. As a
result, shocks to any individual unit can have systematic effects. Acemoglu et
al. (2012) argues that fat tails in the distribution of network weights can
inhibit diversification and amplify the systematic effects of idiosyncratic
shocks. Similarly, Gabaix (2011) argues that skewness in the firm size also
inhibits diversification away from shocks.

In this work, I argue that the assumption that shocks are idiosyncratic is not
consistent with realistic input-output network models of the economy. More
specifically, I consider a general reduced-form equation for static
propagation of shocks, which links each unit’s growth to the growth of its
network connections plus a unit-specific shock. This reduced-form is
consistent with a broad class of structural economic models that assume Cobb-
Douglas aggregation of intermediate inputs in a production function. In this
equation, imposing a diagonal structure on the variance-covariance matrix of
shocks generates implicit restrictions on the network weights permitted in the
model.333I provide both mathematical and numerical evidence in support of
this. More specifically, the set of permissible input-output networks has
weights which are overly sparse and/or economically uninteresting. For
example, the idiosyncratic risk assumption might constrain researchers to
studying economies in which pairs of sectors or firms that use each other’s
inputs can only use each other’s inputs.

As a result, I argue that researchers should account for correlation in shocks
when making use of such a network model. Practically speaking, there are
several reasons why shocks to units in the input-output network might be
correlated, especially as the unit definitions become more granular. For
instance, two firms which produce the same goods should experience correlation
in demand shocks at the product level. If the two firms also produce using the
same inputs and technologies, then supply-side shocks associated with that
technology are likely to be correlated as well. Hoberg and Phillips (2016)
show from text data that firms that produce similar products often belong to
different industries, which suggests that industries should also experience
some degree of comovement in demand shocks. Along these lines, Hottman et al.
(2016) show using scanner data that 69% of firms, which account for 99% of
their industries’ output, supply multiple and intersecting product varieties.

Like product similarity, technological and geographic proximity might also
generate comovement in supply and demand shocks. Bloom and Shankerman (2013)
shows that regional shocks to research and development (R&D) incentives have
correlated effects on the growth of firms who operate in closely related
technology spaces. Similarly, firms operating in nearby locations are likely
to be exposed to the same underlying geographic shocks. For instance, Autor et
al. (2013) and Mian and Sufi (2014) provide evidence that local employment
shocks have correlated effects within a region, and Tuzel and Zhang (2017)
studies correlated exposure to regional risk associated with changes in local
prices for factors of production. Even local climate risk could expose
multiple firms to the same regional risks (see e.g., Barrot and Sauvagnat
(2016) and Kruttli et al. (2019)).

When shocks are idiosyncratic, each unit’s growth rate variance is the sum of
unit-specific shock variance and a network-weighted sum of shock variances of
its trade partners. Of course, the former term is unrelated to the presence of
network connections. In the homoskedastic case, the second term simplifies to
a constant times a concentration measure across each unit’s trade partners.
Acemoglu et al. (2012) show that aggregate volatility shocks to this component
decay at a rate slower than
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sqrt{n}$
when weights follow a power law distribution. Herskovic et al. (2020b) focus
on customer concentration, and argue that increases in concentration are
driven by increases in size dispersion. To my knowledge, this is the first
work to investigate the relationship between exposure to correlated shocks in
the production network and realized variance.

In particular, when allowing for non-negligible correlation in shocks, the
expression for growth rate variance gains an additional covariance component,
denoted concentration “between” trade partners. This new term captures the
ability of each unit in the network to substitute away from correlated shocks
to its trade partners. In particular, units are more substitutable (less
concentrated) when they diversify trade between partners that are exposed to
negatively correlated shocks. When units trade with partners that experience
correlated shocks, they are more concentrated when the relative importance of
those trade partners is similar.

Building on this intuition, I estimate concentration between trade partners
using panel data at the disaggregated industry level. Consistent with theory,
I find that this new component explains a significant amount of variation in
the panel of realized industry variance. More concentrated (less
substitutable) industries are more volatile both in terms of market returns
and output growth. This relationship is robust and holds even when controlling
for relevant industry characteristics such as size, centrality, concentration
across trade partners, vertical position in the supply chain, and durability
of output.

This finding alone does not provide any insight on the underlying source of
correlated risks between trade partners. Diving deeper, I consider the results
of Acemoglu et al. (2016), who argue that total factor productivity (TFP)
shocks primarily propagate downstream while government spending shocks
primarily propagate upstream from customers to suppliers. Consistent with this
finding, I show that the elasticity of realized variance to concentration
between trade partners is more precisely estimated on the supply-side when
constructed using pairwise industry correlations in TFP growth. On the other
hand, the elasticity of realized variance to concentration between customers
is more precisely estimated when using correlations in federal procurement
demand shocks.

Additionally, I suppose that correlation between upstream and downstream
propagating shocks is driven by proximity of industries on a latent surface.
For tractability, I assume that correlation between demand shocks is a
function of product similarity, while correlation between supply shocks is a
function of technological similarity. I proxy product similarity using the
text-based scores from Hoberg and Phillips (2016) and technological proximity
following Bloom and Shankerman (2013). Similarly, I find that the elasticity
of realized variance to between concentration is more precisely estimated on
the demand-side using product similarity and on the supply-side using
technological proximity.

These findings suggest a structural foundation for incorporating correlation
in supply and demand shocks in network models of the economy. To fully
investigate the risk implications of this correlation, I propose a production-
based asset pricing model with input-output networks in which firm-level
technology shocks propagate downstream from suppliers to customers and demand
shocks propagate upstream from customers to suppliers. Firms are both
customers and suppliers. Unlike most existing models, I account for both
directions of propagation.444For example, Shea (2002) and Kramarz et al.
(2020), andHerskovic et al. (2020b) focus on upstream propagation of demand
shocks, while Acemoglu et al. (2012) focus on downstream propagation.
Additionally, I introduce a novel mechanism for correlation in shocks in which
the propensity of transmission is a function of customer and supplier
substitutability at the firm level.

In particular, I define substitutability as network weighted sum of latent
distances between a firm’s trade partners. Product distance characterizes
customer substitutability, while technological distance characterizes supplier
substitutability. Shared customers and suppliers between firms induces
comovement in substitutability and thus correlation in propagated shocks. The
proportion of firms that are affected by network propagated shocks is related
to the changes in average propensity and average supply chain
substitutability. Importantly, the model predicts that average propagation in
the upstream and downstream directions represent distinct and negatively
priced sources of systematic risk in the economy.

I test this prediction by calibrating the model and constructing empirical
analogues of upstream and downstream network propagation risk factors.
Consistent with theory, I confirm empirically that upstream and downstream
propagation risk factors are negatively related to aggregate consumption
growth, output growth, and dividend growth. Moreover, a trading strategy which
buys the highest and sells the lowest quintile upstream (downstream)
propagation beta-sorted portfolio generates excess returns of -11.42%
(-4.18%). These factors survive the standard set of robustness checks.

##  2 Idiosyncratic Risk in Input-Output Networks

In an economy where sectors or firms are connected through a network of input-
output linkages, shocks to any individual unit might generate larger
systematic effects. Intuitively, firms or industries with close trade
relationships should also experience some degree of comovement in risk. Recent
research proposes several approaches for modeling the spread of small shocks
from firms or disaggregated sectors.555See e.g., Gabaix (2011), Acemoglu et
al. (2012), Taschereau-Dumouchel (2020), and Baqaee and Farhi (2019) for
discussion on how microeconomic shocks can generate macroeconomic effects.
Since input-output networks are observed in the data, these approaches lend
themselves to empirical studies on the importance of various channels of shock
propagation.666Generally these studies focus on propagation at business-cycle
frequencies.

The benchmark model studied in much of the literature involves static
propagation of shocks through a deterministic network.777Some examples include
Acemoglu et al. (2012), Acemoglu et al. (2016), Ozdagli and Weber (2017),
Herskovic (2018b), Herskovic et al. (2020b). The main idea is that a sector or
firm’s growth rates depend on a network-weighted sum of growth rates of trade
partners and an idiosyncratic shock which is drawn independently from the
other units. Network weights capture the importance of direct trade
relationships between sectors and are generally non-negative. Moreover, if the
entries of the matrix are sales or purchase shares of inputs, these weights
are also bounded above by 1 and in most cases assumed to sum to 1 or less than
1 for every unit. Typically no additional restrictions are imposed on the
input-output network structure (e.g., symmetry or sparsity).

However, in this section I show that in this benchmark model, the assumption
of idiosyncratic shocks across units is not consistent with such a general
class of networks. In particular, stronger restrictions on the input-output
network weights are required when the variance-covariance matrix of
idiosyncratic shocks is diagonal. These additional restrictions are
inconsistent with almost all empirically observed input-output networks, and
cannot be relaxed by adding omitted macroeconomic factors or by accounting for
multiple networks. Additionally, even if these restrictions are satisfied,
there is no definitive empirical evidence that supply and demand shocks have
zero pairwise correlation across all pairs of units.

After formally establishing this result, I explore the implications of
accounting for correlated shocks in this static framework. In this modified
setting, sectors and firms are still exposed to risk from direct and indirect
trade partners, but now can also substitute away from risk by having trade
partners that are differentially exposed to supply and demand shocks. More
specifically, the variance of a sector’s growth rate inherits the standard
network component which is related to the concentration of risk across trade
partners, but also two additional components which capture a trade-off of
concentration and substitutability between trade partners. Intuitively, higher
concentration implies less diversification in supply-chains and should imply
more volatility. On the other hand, high substitutability implies that units
can better average away the effects of shocks across customers or suppliers.
In the following sections, I provide both theoretical and empirical motivation
the researchers should account for correlated shocks when studying risk in
network economies.

###  2.1 Networks and Risk Comovement

In this section, I argue that realistic input-output models of the economy
should account for correlation in supply and demand shocks across units. In
the benchmark static model of sectoral shock propagation, I find that the set
of stable input-output networks that are consistent with the idiosyncratic
risk assumption is unrealistic. Mathematically, in this broad class of
reduced-form linear models, additional restrictions are required on the input-
output network weights to be consistent with an arbitrary covariance matrix of
sector or firm-level growth rates and an arbitrary diagonal covariance matrix
of shocks. Although this result is not immediately intuitive, the assumption
of idiosyncratic shocks implicitly generates a strict relationship on the
interaction between network weights and elements of the variance-covariance
matrix of growth rates.

To illustrate this point, I start from the general reduced-form model of shock
propagation in which a firm’s output growth is driven by a network component
and firm-specific shocks.888Similar models are used in Acemoglu et al. (2012),
Acemoglu et al. (2016), Herskovic (2018a) and Herskovic et al. (2020b). In
Appendix A, I show that this model is consistent with the equilibrium outcome
of a constant returns to scale economy in which Cobb-Douglas producers
experience productivity shocks that propagate downstream from suppliers to
customers and demand shocks that propagate upstream from customers to
suppliers. In particular, for an
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n$-firm
economy, consider the static relationship:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\mathbf{y}=\mathbf{W}\mathbf{y}+\mathbf{u},$
(1)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{y}$
is the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n\times
1$ vector of firm-level output growth,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}$
is the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n\times
n$ network matrix capturing interactions between industries, and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{u}$
is the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n\times
1$ vector of firm-specific supply or demand shocks. This framework is
compatible with either direction of propagation, upstream from customers to
suppliers or downstream from suppliers to customers. The following two
assumptions require that the propagation matrix
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}$
implies is stable, and that firm-specific shocks are idiosyncratic,
respectively.

######  Assumption 2.1 (Stable Weighting Matrix).

The weighting matrix
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}\in
M_{n}$ is non-negative, and has bounded spectral radius
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\rho(\mathbf{W})\leq
1$.

######  Assumption 2.2 (Idiosyncratic Shocks).

Firm-specific shocks
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}u_{i}\sim\mathcal{P}_{i}(0,\sigma_{i}^{2})$
are drawn independently across firms where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{P}_{i}\in
L^{2}$ has finite second moments. In other words, there exists a positive
diagonal matrix
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{D}\in
M_{n}$ such that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{E}[\mathbf{uu}^{\top}]=\mathbf{D}$.

In the following proposition, I characterize a set of additional necessary
restrictions on the matrix
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}\in
M_{n}$ to satisfy Assumption 2.2 and (1).

######  Proposition 2.3 (Necessary Restrictions on
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}$).

Any weighting matrix
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}$
which satisfies 2.1 and 2.2 must have that for any pair of off-diagonal nodes
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{ij}$
with
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i\neq
j$, either
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{ij}w_{ji}\geq
1$ or
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{ij}>0$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{ji}=0$.

###### Proof.

See Appendix C.1. ∎

This proposition highlights a key limitation of equation (1). To apply network
models in a way that is consistent with reality, researchers must either
restrict their focus to a very particular set of networks or allow for
correlation in sectoral or firm-level shocks. Under Assumptions 2.1 and 2.2,
consistent networks have only one-way connections or entries which are overly
sparse and/or economically uninteresting. In the structural model developed in
Appendix A, the entries of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}$
are primitives of the production function and depend on each unit’s sales and
cost shares. For example, To capture the effect of demand shocks propagating
from
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
to
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$,
the implied weight is
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{ij}=\frac{sales_{i\to
j}}{sales_{i}}$. Proposition 2.3 requires that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{ij}w_{ji}=\frac{sales_{i\to
j}}{sales_{i}}\cdot\frac{sales_{j\to i}}{sales_{j}}\geq 1$ for all
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i\neq
j$, which implies that sectors which use each other’s inputs can only use each
other’s inputs.

Intuitively, one might argue that the static network model in (1) is too
parsimonious to capture all the sources of risk comovement in the economy.
Although this is likely true, the restrictions on
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}$
cannot be relaxed by adding omitted macroeconomic factors driving common
variation in risk nor by adding an omitted network component. Moreover,
Proposition 2.3 implies that there is no sufficient statistic that can be
obtained from
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}$
which fully characterizes cross-sectional variation in granular risk, even in
a world where sectoral shocks are identically distributed. See Appendix F for
supporting numerical evidence. In the remainder of this section, I explore the
implications of allowing for correlation in demand and supply shocks across
units.

###  2.2 Granular Volatility with Correlated Shocks

I investigate the volatility predictions of (1) when shocks
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}u_{i}$
are allowed to be correlated across units
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
(i.e.,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\text{var}({\mathbf{u}})$
is not diagonal). Practically speaking, there are several reasons why supply
and demand shocks to units might be correlated, especially at the granular
level. For example, two sectors or firms that produce related goods or
services are likely to experience correlated demand shocks. If the two sectors
produce using the same inputs, then supply-side shocks might be correlated as
well. Hoberg and Phillips (2016) show that firms with similar products might
belong to different industries (according to SIC or NAICS
classifications).999More specifically, Hoberg and Phillips (2016) find that
firms in the newspaper, printing, and publishing industry (SIC3 271) are
similar to firms in the radio broadcasting industry (SIC3 483) and argue that
this is driven by common customers who demand advertising services. They also
find that Disney and Pixar have similar products (movies) although they are in
different industries (business services (SIC3 737) and motion pictures (SIC3
781) industries, respectively). In this case, the differences in industry stem
from the production method and not the product offering. In the even more
simple setting where multiple firms produce the exact same goods and services,
supply and demand shocks at the product level mechanically generate
correlation in supply and demand shocks at the firm level. Hottman et al.
(2016) provide empirical evidence that this is generally the case, with 69% of
firms, which account for 99% of industrial output, supplying multiple (and
intersecting) products.

Like product proximity, both technological and geographic proximity might also
generate correlation in firm and sectoral shocks. For instance, Bloom and
Shankerman (2013) show that shocks to research and development (R&D) have
correlated effects on the productivity and growth of firms with similar
technologies. Similarly, industries or firms operating in nearby geographies
are exposed to the same underlying shocks associated with local labor markets
(see e.g., Autor et al. (2013), Mian and Sufi (2014)), local factor prices
(Tuzel and Zhang (2017), Grigoris (2019)), local technological progress
(Oberfield (2018)), or local weather events (Barrot and Sauvagnat (2016),
Kruttli et al. (2019)).

In the benchmark network model with uncorrelated shocks, the variance of
growth rates depends solely on the concentration of risk across independent
suppliers and/or customers. Herskovic et al. (2020b) provide theoretical and
empirical evidence linking firm volatility and customer concentration in terms
of size dispersion in this setting. However, allowing for correlated shocks
implies two additional variance components. These components capture the
concentration and substitutability of risk between trade partners,
respectively. The distinction between concentration “across” and “between”
trade partners is important. Concentration across refers to the composition of
a unit’s reliance on any particular customer or supplier, while concentration
between refers to how a unit’s the distribution of reliance on a set of
customers or suppliers that are exposed to the same shocks. On the other hand,
substitutability between customers and suppliers captures the distribution of
reliance on a diversified set of customers or suppliers that are exposed to
shocks of the opposite sign.

In other words, concentration between customers and suppliers captures
compounding effects of positively related shocks to similar trade partners,
while substitutability captures mitigating effects of spreading reliance on
trade partners that are exposed to negatively related shocks. Intuitively, a
supplier with major customers that tend to reduce demand at the same time is
more risky than a supplier with some customers that increase demand when the
others reduce it. To see this mathematically, define
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}[h_{ij}]_{ij}$
to be the set of entries in the Leontief inverse matrix
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{H}:=(\mathbf{I}-\mathbf{W})^{-1}$
and recall that equation (1) can equivelently by written as
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{y}=\mathbf{H}\mathbf{u}$.
Note that in this setup the element
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}h_{ij}$
captures the percent change in unit
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$’s
growth after a 1% shock to unit
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$.
Then the variance of unit
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$’s
growth rates can be written:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\text{var}(y_{i})$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\text{var}\bigg{(}\sum_{j=1}^{n}h_{ij}u_{j}\bigg{)}=\sum_{j=1}^{n}h_{ij}^{2}\cdot\text{var}(u_{j})+\sum_{j\neq
k}h_{ij}h_{ik}\cdot\text{cov}(u_{j},u_{k}).$

The first term is the standard expression for variance in this network model
(see e.g., Acemoglu et al. (2012)), while the second term is only non-zero
when inter-industry shocks are correlated. Next, I define the scalar
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s_{jk}$
to be the sign of the pairwise correlation between shocks to
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k$
(i.e.,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s_{jk}:=\text{sgn}(\sigma_{jk})\equiv\text{sgn}(\text{cov}(u_{j},u_{k}))$
where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\text{sgn}(.)$
is the sign function). To build some more intuition on the additional terms, I
can further decompose the covariance term as follows:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\text{var}(y_{i})=\underbrace{h_{ii}^{2}\cdot\sigma_{i}^{2}}_{\text{self}}+\underbrace{\sum_{j\neq
i}h_{ij}^{2}\cdot\sigma_{j}^{2}}_{\text{concentration
``across"}}+\underbrace{\sum_{j\neq
k,s_{jk}=1}h_{ij}h_{ik}\cdot\sigma_{jk}}_{\text{concentration
``between"}}+\underbrace{\sum_{j\neq
k,s_{jk}=-1}h_{ij}h_{ik}\cdot\sigma_{jk}}_{\text{substitutability}}$ (2)

Consider a first order approximation of the Leontief inverse matrix such that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{H}\approx\mathbf{I}+\mathbf{W}$,
where the weights in the propagation matrix
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}$
are related to sales shares when modeling downstream propagation supply-side
shocks, and purchase shares when modeling upstream propagation of demand-side
shocks.101010More specifically, the weight of downstream propagation of
supply-side shocks from supplier
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
to customer
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
is captured by
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{ij}^{d}=sales_{j\to
i}/purchases_{i}$ and the weight of upstream propagation of demand shocks from
customer
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
to supplier
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
is captured by
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{ij}^{u}=sales_{i\to
j}/sales_{i}$. In general, these weights are both asymmetric (i.e.,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{ij}\neq
w_{ji})$ and different depending on the direction of propagation (i.e.,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{ij}^{u}\neq
w_{ij}^{d}$). Suppose also that units are homoskedastic such that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\text{var}(u_{j})=\sigma^{2}$
for all
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\text{cov}(u_{j},u_{k})=\nu\cdot
s_{jk}$ for all
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j\neq
k$, where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sigma$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\nu$
are positive scalars. In this case, the first term
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}h_{ii}\sigma_{i}^{2}=\sigma_{i}^{2}$)
is unrelated to the network and captures the variance of supply or demand
shocks to sector
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$.
On the other hand, the second component (concentration across network
linkages) is non-negative and large when reliance is highly concentrated
across trade partners. Similarly, the third term (concentration between
network linkages) is non-negative and large when reliance is concentrated
between trade partners who experience positively correlated shocks.

Finally, the last term (substitutability of network linkages) is always non-
positive and is large in magnitude when reliance is spread equally between
trade partners who are likely to experience shocks of opposite sign.
Additionally, the sum of the final two terms captures explicitly the trade-off
between concentration and substitutability of correlated supply or demand
shocks. Although this simplification is useful for building intuition, the
more realistic version of the variance decomposition should also take into
account unit heteroskedasticity. That is, two sectors with an equal set of
input-output weights have different network-implied variance only if their
trade partners are exposed to differential volatility in supply or demand
shocks.

Consider for example the Printed Circuit Boards industry (SIC 3672), whose top
3 major manufacturing industry customers include Electronic Components (SIC
3679) and Electronic Computers (SIC 3571), and Communications Equipment (SIC
3669). At first glance, these customers appear very similar, and one might
suspect that a negative demand shock to one customer is likely to be
correlated with a negative shock to the other, which amplifies upstream
propagation to their shared supplier. In other words, the Printed Circuit
Boards industry has high concentration between customers and a harder time
substituting away from upstream effects demand shocks to its major
customers.111111I find that the average product similarity score between these
customer industries is in the top 10% (based on the similarity score developed
in Hoberg and Phillips (2016)). Additionally, I find significant positive
correlation in demand shocks to these industries such as changes of newly
awarded federal defense procurement contracts. On the other hand, the three
most important customers of the Jewelry and Precious Metal industry (SIC 3911)
include Watches, Clocks, and Clockwork Operated Devices (SIC 3873), Perfumes
and Cosmetics (SIC 5048), and Drawing and Insulating of Nonferrous Wire (SIC
3357). In this case, customers produce seemingly unrelated goods (both durable
and non-durable) and there is evidence the demand shocks have zero or negative
pairwise correlation.121212The average pairwise correlation in federal
procurement shocks and Chinese import penetration shocks is -37% and -22%,
respectively for the full set of Jewlery and Precious Metal customers. In
other words, the Jewelry and Precious Metal industry has is able able to
substitute away from demand shocks propagating upstream from any individual
customer.

There are similar examples of high concentration and substitutability on the
supply-side. For instance, the Computer Storage Devices industry (SIC 3572)
has a highly concentrated customer base composed of Electronic Components (SIC
3679), Electronic Coils, Transformers, and other Inductors (SIC 3677),
Semiconductors and Related Devices (SIC 3674), and Electronic Connectors (SIC
3678). This industry is thus more exposed to correlated supply-side risk. On
the contrary, the Meat Packing Plants industry (SIC 2011) can more easily
substitute away from supply-side risk, with a more diversified set of major
suppliers like Poultry Slaughtering and Processing (SIC 2015), Plastics Film
and Sheet (3081), and Paper Mills (2621).

Although these network variance components are intuitive and theoretically
justified if supply and demand shocks are correlated, an important practical
concern is that granular supply and demand shocks are not easily identified
from available data, especially at a high frequency. In the following section,
I address this challenge and propose an empirical methodology for estimating
customer and supplier concentration and substitutability at the industry and
firm levels. I show that both supply and demand channels explain cross-
sectional heterogeneity in risk exposure, beyond what can be explained by
other determinants of variance identified by the literature.

##  3 Empirical Evidence

In this section, I provide empirical estimates of the network-implied variance
components motivated in equation (2). This requires granular data on input-
output relationships and estimates of the variance-covariance matrix of supply
and demand shocks. Consistent with theory, I find that these additional
components explain important variation in realized volatility, controlling for
characteristics such as size, centrality, concentration across trade partners,
vertical position in the supply chain, and durability of output. These results
hold at both industry and firm levels. The main takeaway here is simple. When
accounting for input-output linkages and non-negligible correlation in supply
and demand shocks, heterogeneity in risk exposure is at least in part driven
by differences in the ability of network units to substitute away from
correlated supply and demand shocks.

###  3.1 Setup

Consider the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n$-sector
network model from equation (1) and add a time subscript
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$.
Suppose that in each period I obtain estimates for the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n\times
n$ Leontief Inverse matrices
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\mathbf{H}}_{q,t}$
and the variance-covariance matrices of supply and demand shocks
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\mathbf{\Sigma}}_{q,t}$
where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q=u$
for upstream propagation and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q=d$
for downstream propagation. Then for both supply and demand-side shocks, I can
compute three empirical network-implied variance components, denoted by “self-
originating”, “across”, and “between” risk. The final component sums the final
two covariance terms from (2) and captures the concentration/substitutability
trade-off between trade partners. Low values of concentration between
customers and suppliers implies high substitutability. Then for each industry,
direction, and time triple
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(i,q,t)$,
I compute self-originating risk as:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\hat{\sigma}_{iqt,self}^{2}=[\hat{h}_{qt}]_{ii}^{2}\cdot\hat{\sigma}_{i}^{2},$
(3)

and across and between risk as:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\hat{\sigma}_{iqt,acr}^{2}$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\sum_{j\neq
i}[\hat{h}_{qt}]_{ij}^{2}\cdot\hat{\sigma}_{j}^{2},$ (4)
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\hat{\sigma}_{iqt,bet}^{2}$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\sum_{j\neq
k}[\hat{h}_{qt}]_{ij}\cdot[\hat{h}_{qt}]_{ik}\cdot\hat{\sigma}_{jk}.$ (5)

In the next section, I provide details on the data sources, assumptions, and
methodologies used to estimate
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\mathbf{H}}_{qt}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\mathbf{\Sigma}}_{qt}$.
While the former can be observed directly, I need to make some assumptions to
identify the latter from available data sources. Then I compute all three
components at the industry-level and study their empirical relationship with
realized industry variance. I find that the elasticity of realized variance to
all three components is significant and positive for both directions of
propagation, controlling for a variety of industry characteristics.

###  3.2 Upstream and Downstream Propagation Networks

I begin by constructing the network of input-output linkages at the
disaggregated industry level from the Make and Use tables published by the
Bureau of Economic Analysis (BEA). The goal is to build a directed weighted
network which captures the importance of trade relationships over time and for
the population of industries.131313As far as I know, this is the most
disaggregated database on the entire population of input-output relationships.
Network weights represent the strength of each unit’s reliance on customers
and suppliers, and the network is directed to capture differences in shock
propagation in the upstream (customer to supplier) and downstream (supplier to
customer) directions. In particular, the BEA publishes these tables annually
between 1997-2020 for 66 industry groups.

More specifically, I construct downstream and upstream propagation matrices
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}_{d}=[w_{d}]_{ij}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}_{u}=[w_{u}]_{ij}$
with entries:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle[w_{d}]_{ij}=\frac{sales_{j\to
i}}{costs_{i}},\qquad[w_{u}]_{ij}=\frac{sales_{i\to j}}{sales_{i}},$ (6)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}sales_{i\to
j}$ represents gross trade flows from
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
to
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$,
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}sales_{i}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}costs_{i}$
represent the total sales and costs of industry
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$,
respectively. The downstream (upstream) weights are non-negative and capture
the direct reliance of industry
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
on supplier (customer)
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$.
When weights are large, direct effects of propagated shocks should also be
large. To account for higher order (indirect) network effects as well, I
calculate the strength of network propagation based on the Leontief inverse
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{H}_{q}:=(\mathbf{I}-\mathbf{W}_{q})^{-1}$
of the propagation matrices
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\{\mathbf{W}_{q}:q\in
u,d\\}$.141414See e.g., Baqaee and Farhi (2019) and Herskovic et al. (2020a)
for discussion on the importance of higher order network effects. The entries
of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{H}_{q}=[h_{q}]_{ij}$
capture the total percent effect on
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
of a 1% shock to
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
traveling in the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q$-stream
direction when accounting for all weighted direct and indirect connections.

Appendix D.2 reports summary statistics for observed input-output connections.
At the 66-industry granularity, I find that both propagation and Leontief
inverse weights are highly persistent with an average autocorrelation of more
than 95% for each entry. The cross-sectional correlation is about 8.55%
between upstream and downstream weights and about 11.08% between upstream and
downstream Leontief matrix entries, suggesting that propagation occurs
differently in either direction.

###  3.3 Variance-Covariance Matrix of Supply and Demand Shocks

Unlike the sectoral input-output network, there is no definitive data source
on supply and demand shocks and their variance-covariance matrix. As a
baseline, I implement an empirical analog of the reduced-form equation in (1).
In particular, consider the spatial panel regression:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\tilde{y}_{it}=\delta_{t}+\phi\cdot\tilde{y}_{i,t-1}+\beta_{u}\cdot\sum_{j=1}^{n}w_{u,ij}\tilde{y}_{jt}+\beta_{d}\cdot\sum_{j=1}^{n}w_{d,ij}\tilde{y}_{jt}+\varepsilon_{it},$
(7)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tilde{y}_{it}:=\log(y_{it}/y_{i,t-1})$
is output growth in industry
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
at quarter
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{q,ij}$
is the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(i,j)$
entry of the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q$-stream
propagation matrix
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}_{q}$.
Assuming the variance-covariance matrix of residuals is static, then
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\mathbf{\Sigma}}$
is the empirical variance-covariance matrix of estimated residuals
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\varepsilon}_{it}$.
To ensure that my estimates for network components (4) and (5) are robust to
estimation error in
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\mathbf{\Sigma}}$,
I calculate the average value over samples in which I randomly drop 10% of
pairwise non-zero correlations.151515Note that estimation error from
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\mathbf{\Sigma}}_{q}$
is magnified in estimated network components (4) and (5) at a rate
proportional to the number of nonzero row entries in the Leontief inverse
matrix
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{H}_{q}$.
See Appendix E for more details and alternative specifications.

Pairwise correlation in residuals is centered with a mean value of 0.5% (0.4%)
and a standard deviation of 25% (26%). The largest positive pairwise
correlation is 82% between the Primary Metals (BEA Code 331) and Wholesale
Trade (BEA Code 42) and 81% between Housing (HS) and Educational Services
(61). On the other hand, the largest negative pairwise correlation is -80%
between Primary Metals (331) and Federal Reserve Banks, Credit Intermediation,
and Related Activities (521CI) and -72% between Food and Beverage and Tobacco
Products (311FT) and Wholesale Trade (42).

###  3.4 Network Determinants of Realized Variance

After relaxing the idiosyncratic shock assumption, the benchmark input-output
propagation model predicts that realized variance should depend positively on
three network risk components: risk that is self-originating, risk across
trade partners, and risk between trade partners. In the baseline setup, this
might hold mechanically for self-originating risk since it is estimated from
the variance of residual output growth in equation (7). However, both risk
across and between trade partners contain only variance-covariance information
associated with other industries. I verify these predictions empirically using
panel regressions of the log of realized industry variance on the log of
network components, controlling for a variety of characteristics such as size,
centrality, durability of output, and industry cluster and time fixed
effects.161616I adjust network components by a constant to ensure that the
minimum value is positive so the log is well defined. Industry clusters are
defined by major industry groups (2-digit NAICS code). I measure realized
industry variance using both stock market and output growth data. I define
market variance as the annual return variance of an equal-weighted industry
portfolio and fundamental variance as the variance of quarterly year-on-year
output growth. I obtain similar results when using idiosyncratic variance as
the dependent variable.171717I define idiosyncratic market variance as the
variance of equal weighted residual returns from a Fama and French three-
factor model. Similarly, I define idiosyncratic output growth as the residual
of industry output growth after a regression on aggregate output growth.
Results also replicate for value-weighted industry portfolios, or industry
sales growth, which is constructed as the year-on-year change in the sum of
quarterly sales (reported on Compustat) for all public firms in the industry.
Although the annual variance across quarterly year-on-year output growth and
monthly returns are fairly noisy proxies for true realized cash-flow variance,
the results are robust for several specifications.

I summarize the main results in Table 1. Consistent with theoretical
predictions in equation (2), the elasticity of realized industry variance to
concentration across and between customers are both positive and significant
in all specifications. This holds for both market and output growth measures
of variance. Conditional on both directions of propagation and all controls,
increasing concentration between customers from the median to the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}90^{th}$
percentile increases industry sales growth variance by over 45% (about 0.37
standard deviations) and market variance by 15% (about 0.09 standard
deviations). Similarly, increasing concentration between suppliers from the
median to the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}90^{th}$
percentile increases industry sales growth variance by over 20% (about 0.17
standard deviations) and market variance by 19% (about 0.11 standard
deviations). Without controls, downstream network risk explains 23% of time
series variation in market variance and 22% of time series variation in output
growth variance. Similarly, upstream network risk explains 22% and 31% of
market and output growth variance, respectively. Both directions of
propagation are important for explaining the panel dynamics of industry
variance.

Consistent with the firm-level findings of Herskovic et al. (2020b), I find
that industry variance has a positive elasticity to concentration across
customers and a negative elasticity to average size. A new but related result
is the positive elasticity of variance to concentration across suppliers.
Additionally, Ahern (2013) argues that more central industries have greater
market risk since they are more exposed to aggregate shocks, and thus earn
higher returns on average. On the other hand, my results suggest that more
central industries have less volatile stock returns, but also have less
exposure to aggregate volatility risk and lower idiosyncratic
volatility.181818I find that industries in the highest average upstream
(downstream) centrality decile have 31% (21%) less exposure to systematic
volatility risk than the lowest decile. Average upstream and downstream
centrality are positively correlated (56% cross-sectionally), and industries
who are in the top decile for both average centrality measures have a 52%
lower exposure to aggregate volatility risk than industries in the bottom
decile for both. Moreover, top centrality decile stocks have 25% lower
idiosyncratic volatility, on average. My results are thus consistent with
Ahern (2013), since stocks with lower exposure to aggregate volatility risk or
and lower idiosyncratic volatility earn higher returns on average (see e.g.,
Ang et al. (2006)). Table 7 shows that there is no significant relationship
between centrality and concentration between or across trade partners.

###  3.5 Sources of Correlation in Network Propagating Shocks

So far, I have established both theoretically and empirically the importance
of accounting for correlation in shocks that propagate through the input-
output network. In particular, I show that concentration between trade
partners explains a large amount of variation in the industry panel of
realized variance. However, statistical estimates for the variance-covariance
matrix of shocks do not provide much insight on the underlying sources of
correlation between industries. In this section, I argue that correlation in
supply-side shocks that propagate downstream can be explained by technological
proximity between sectors, while correlation in demand-side shocks that
propagate upstream can be explained by product similarity.

####  3.5.1 Observed Supply and Demand Shocks

Acemoglu et al. (2016) argue that productivity shocks primarily propagate
downstream while government spending and trade shocks primarily propagate
upstream. In this case, these shocks might help to capture differences in
inter-industry correlations which are specific to the direction of
propagation. Along these lines, I construct an annual industry panel of
5-factor total factor productivity (TFP) growth between 1959-2018 from the
NBER-CES Database (Becker et al., 2016). Since this measure of TFP controls
for materials, it does not mechanically encode any information related to
downstream effects such as changes in price and/or quantity. Similarly, I
construct a monthly panel of newly awarded federal procurement contracts
between Jan 2000-Jan 2021 from the universe of contracts published in the
Federal Procurement Data System (FPDS).191919I also consider other observed
shocks in Appendix E.

To focus on inter-industry correlations which are unrelated to common
aggregate factors (e.g., the secular decline in several manufacturing
industries), I estimate the variance-covariance matrix of residuals after an
OLS regression on the cross-sectional average of shocks.202020The cross-
sectional mean approximates the first principal component of shocks when there
are missing values. For shocks
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}dz_{t}$,
I calculate the covariance between sectors
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
as
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\text{cov}(u_{kt},u_{jt})$
where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}u_{kt}$
is the residual in the regression
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}dz_{kt}=\alpha+\beta\cdot\bar{dz}_{.,t}+u_{kt}$.
Endogeneity of shocks is not a major concern assuming any confounding shocks
largely propagate in the same direction in the network. Given such a
confounder, my estimate for the variance-covariance matrix of shocks can be
written as the true estimate plus some measurement error.  I then estimate the
corresponding network components using equations (4) and (5) and study their
relationship with realized variance. Table 8 shows that the elasticity of
realized variance to supplier concentration is positive and more precisely
estimated when calculating supply-side shock covariance as a function of
productivity growth. On the other hand, Table 9 reports more precise estimates
for the elasticity of variance to customer concentration when calculating
demand shock covariance as a function of federal procurement shocks. This
suggests that productivity growth is more informative about upstream network
risk, while changes in government demand are more informative about downstream
network risk.

####  3.5.2 Technological and Product Proximity

On the other hand, suppose that correlation between supply and demand-side
shocks is a function of underlying firm and industry characteristics.
Intuitively, I might assume that correlation in demand shocks propagating
upstream is driven by product similarity and/or geographic proximity of
customers, while correlation in supply shocks propagating downstream is driven
by technological similarity and/or geographic proximity of suppliers.

More generally, I assume that each industry is associated with a vector of
positions in some latent surface
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{z}_{it}\in\Omega_{z}\subset\mathbb{R}^{d}$
and that the correlation between industry shocks can be written as a function
of the distance between these latent vectors.212121Latent surface models are
often used to impute network relationships in microeconomic applications (see
e.g., McCormick and Zheng (2015), Breza et al. (2020)).  Following McCormick
and Zheng (2015), I suppose that industry positions
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{z}_{it}$
lie on the surface of a
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p$-dimensional
latent surface on the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p+1$-dimensional
unit hypersphere
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{S}^{p+1}$.
This implicitly implies that latent positions follow a uniform distribution
across the sphere’s surface. Moreover, since the hypersphere has bounded
surface area, the distance between any two points is bounded. I further assume
that points in the same position have correlation 1 and points on opposite
sides of the sphere have correlation -1.

In practice, I experiment with constructing latent positions of industries
using several combinations of industry variables. For simplicity, my main
results rely on univariate distances in product and technology
space.222222Moreover, contours of the sphere present some calibration
difficulties in higher dimension. I measure product distance using use the
text-based scores developed in Hoberg and Phillips (2016) and technology
distance using patent-based technological proximity scores along the lines of
Bloom and Shankerman (2013). Since both of these scores are available at the
firm-level, I first construct a firm-by-firm product distance network where
distances are inversely related to proximity. To get the distance between
sectors, I use the median length of the shortest weighted path between firms
in the two sectors, rescaled such that the furthest pairwise distance is 1 and
the shortest pairwise distance is zero. I calculate the shortest pairwise path
between any two nodes using Dijkstra’s Algorithm. I calculate these measures
annually.

Transforming distances to correlations, I rescale by the variance estimates
from residuals in equation (7) and recompute network components. When using
product distance to calculate network risk, the elasticity of realized
variance to concentration between customers is significant and positive. On
the other hand, the analogous elasticity to concentration between suppliers is
significantly negative, which suggests that product similarity across
suppliers actually indicates better substitutability away from supply-side
shocks. When approximating correlations based on technological distance,
realized variance has a positive elasticity to concentration between suppliers
and customers, but the elasticity is more precisely estimated on the supply
side. Taken together, these results suggest that technological proximity is a
good proxy for correlated exposure to supply shocks propagating downstream,
while product proximity is a good proxy for correlated exposure to demand
shocks propagating upstream. Along these lines, Table 2 shows that average
technological proximity between sectors is closely related to correlation in
TFP growth shocks, while product similarity is closely related to correlation
in federal defense procurement shocks.

####  3.5.3 Accounting for Dynamics

To account for potential time-variation in industry correlations, I also
compute pairwise inter-industry correlations using the dynamic conditional
correlation (DCC) estimator from Engle (2002). In particular, I estimate
bivariate DCC models for all pairs of industries using Bayesian MCMC following
Fioruci et al. (2013). Since product and technological proximity are computed
at an annual frequency, I am thus able to obtain a one-to-one comparison
between correlation in spatial panel residuals, observed shocks, and distance
based measures. Table 2 reports consistent results when using dynamic
correlations.

Table 1: Network Determinants of Industry Variance Panel A: Market Return
Variance
---
| (1) | (2) | (3) | (4) | (5) | (6)
Self-origin (demand) | 0.055** | 0.056** |  |  | 0.003 | 0.017
| (0.022) | (0.024) |  |  | (0.028) | (0.028)
Across (demand) | 0.094** | 0.083** |  |  | 0.081** | 0.072**
| (0.036) | (0.030) |  |  | (0.024) | (0.021)
Between (demand) | 0.147*** | 0.122** |  |  | 0.197*** | 0.089**
| (0.051) | (0.049) |  |  | (0.053) | (0.038)
Self-origin (supply) |  |  | 0.072*** | 0.073*** | 0.067*** | 0.083**
|  |  | (0.021) | (0.023) | (0.022) | (0.023)
Across (supply) |  |  | 0.216*** | 0.156*** | 0.160*** | 0.154*
|  |  | (0.056) | (0.054) | (0.060) | (0.067)
Between (supply) |  |  | 0.416*** | 0.317*** | 0.210** | 0.196**
|  |  | (0.093) | (0.095) | (0.088) | (0.073)
Size |  | -0.378*** |  | -0.361*** |  | -0.289***
|  | (0.094) |  | (0.091) |  | (0.104)
Upstream centrality |  | -0.182* |  | -0.289*** |  | -0.232**
|  | (0.103) |  | (0.095) |  | (0.101)
Downstream centrality |  | -0.051 |  | -0.086** |  | -0.023
|  | (0.054) |  | (0.043) |  | (0.054)
Durability |  | -0.167 |  | -0.404 |  | -0.637
|  | (0.573) |  | (0.662) |  | (0.615)
Vertical position |  | 1.550** |  | -0.756** |  | 1.970***
|  | (0.701) |  | (0.332) |  | (0.710)
Constant | -6.279 | -3.26 | -4.692 | -1.082 | -5.005 | -3.608
Obs | 1484 | 1484 | 1484 | 1484 | 1484 | 1484
Adj R2 | 0.231 | 0.292 | 0.159 | 0.223 | 0.245 | 0.330
Panel B: Output Growth Variance
| (1) | (2) | (3) | (4) | (5) | (6)
Self-origin (demand) | 0.034 | 0.006 |  |  | 0.026 | 0.006
| (0.026) | (0.026) |  |  | (0.030) | (0.028)
Across (demand) | 0.159** | 0.074 |  |  | 0.157** | 0.095
| (0.069) | (0.024) |  |  | (0.070) | (0.057)
Between (demand) | 0.210*** | 0.281*** |  |  | 0.196*** | 0.280***
| (0.064) | (0.078) |  |  | (0.065) | (0.082)
Self-origin (supply) |  |  | 0.081** | 0.072 | 0.017 | 0.042
|  |  | (0.021) | (0.022) | (0.024) | (0.025)
Across (supply) |  |  | 0.128** | 0.114** | 0.098** | 0.091**
|  |  | (0.053) | (0.033) | (0.021) | (0.031)
Between (supply) |  |  | 0.332** | 0.241*** | 0.221** | 0.198**
|  |  | (0.098) | -0.079 | (0.103) | (0.092)
Size |  | -0.046 |  | -0.185* |  | -0.110
|  | (0.099) |  | (0.098) |  | (0.106)
Upstream centrality |  | -0.127 |  | -0.096 |  | -0.131
|  | (0.108) |  | (0.095) |  | (0.111)
Downstream centrality |  | -0.218*** |  | -0.067 |  | -0.222***
|  | (0.056) |  | (0.042) |  | (0.056)
Durability |  | 0.115 |  | -0.038 |  | 0.024
|  | (0.647) |  | (0.643) |  | (0.703)
Vertical position |  | 1.545** |  | 0.181 |  | 1.375**
|  | (0.654) |  | (0.355) |  | (0.682)
Constant | -5.088 | -6.954 | -6.818 | -5.327 | -6.17 | -6.718
Obs | 1484 | 1484 | 1484 | 1484 | 1484 | 1484
Adj R2 | 0.221 | 0.382 | 0.198 | 0.319 | 0.277 | 0.412
Notes: This table reports panel regressions of realized industry variance on a
variety of characteristics including the log variance of supply and demand
shocks (self-origin), log concentration across trade partners, log
concentration between trade partners, log total output (size), log centrality
of the upstream and downstream propagation networks, durability of output,
vertical position in the supply chain, and industry cluster and year fixed
effects. In Panel A, the dependent variable is the log variance of annualized
monthly returns on an equal-weighted industry portfolio. In Panel B, the
dependent variable is the log variance of total quarterly year-on-year
industry sales growth. I obtain return data from CRSP and GDP data from the
BEA. Concentration between and across trade partners is calculated as the
average value of estimates obtained using equations. 4 and 5. I calculate the
variance-covariance matrix of shocks using residuals from equation 7 and
calculate the average value of components over 1000 random samples each
randomly dropping 10% of pairwise correlations. Following Ahern (2013), I
compute industry centrality as the eigenvector centrality of upstream and
downstream propagation adjacency matrices. I calculate durability as the
proportion of sub-industries classified as durable by Gomes et al. (2009), and
I calculate vertical position of each industry following Antràs et al. (2012)
and Gofman et al. (2020).
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}***$,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}**$,
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}*$
indicate significance at the 1% and 5% and 10% levels, respectively. Standard
errors are clustered at the BEA major 15 major industry group level. Sample is
at an annual frequency from 1997 to 2019 for 66 BEA non-government industries.

Table 2: Sources of Correlation in Supply and Demand Substitutability Panel A:
Upstream Supplier Substitutability (static)
---
Covariance Method | Spatial | TFP | Procurement | Prod Similarity | Tech Proximity
Spatial | 1 | 0.59 | 0.37 | 0.30 | 0.48
TFP |  | 1 | 0.31 | 0.35 | 0.46
Procurement |  |  | 1 | 0.62 | 0.39
Prod Similarity |  |  |  | 1 | 0.27
Tech Proximity |  |  |  |  | 1
Panel B: Downstream Customer Substitutability (static)
Covariance Method | Spatial | TFP | Procurement | Prod Similarity | Tech Proximity
Spatial | 1 | 0.55 | 0.61 | 0.43 | 0.35
TFP |  | 1 | 0.27 | 0.20 | 0.48
Procurement |  |  | 1 | 0.69 | 0.40
Prod Similarity |  |  |  | 1 | 0.24
Tech Proximity |  |  |  |  | 1
Panel C: Upstream Supplier Substitutability (dynamic)
Covariance Method | Spatial | TFP | Procurement | Prod Similarity | Tech Proximity
Spatial | 1 | 0.61 | 0.40 | 0.34 | 0.49
TFP |  | 1 | 0.35 | 0.39 | 0.48
Procurement |  |  | 1 | 0.65 | 0.42
Prod Similarity |  |  |  | 1 | 0.25
Tech Proximity |  |  |  |  | 1
Panel D: Downstream Customer Substitutability (dynamic)
Covariance Method | Spatial | TFP | Procurement | Prod Similarity | Tech Proximity
Spatial | 1 | 0.57 | 0.68 | 0.45 | 0.30
TFP |  | 1 | 0.22 | 0.27 | 0.52
Procurement |  |  | 1 | 0.70 | 0.38
Prod Similarity |  |  |  | 1 | 0.30
Tech Proximity |  |  |  |  | 1
Notes: This table reports the correlation across different measures of
upstream and downstream substitutability measures (negative of concentration
“between”). Substitutability is calculated as the negative value of the log of
(5) (plus a large enough constant) and correlation between two measures
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x_{it}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y_{it}$
is defined by
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\rho}$
from the regression
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y_{it}=\hat{\rho}x_{it}+u_{it}$,
where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x_{it}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y_{it}$
are transformed to have mean zero and standard deviation one. Panels A and B
rely no a static assumption for the variance-covariance matrix across shocks,
while Panels C and D estimate a dynamic conditional variance-covariance matrix
à la Fioruci et al. (2013). Panels A and C report results in the upstream
(supply-side) direction and Panels B and D report results in the downstream
(customer-side) direction. Spatial covariance is based on the panel model in
equation (7). TFP covariance is based on TFP growth measured in Becker et al.
(2016), procurement based on federal government shares interacted with the
procurement proxy in Briganti and Sellemi (2022), product similarity using the
latent distance method and scores from Hoberg and Phillips (2016), and tech
proximity constructed following Bloom and Shankerman (2013).

##  4 Dynamic Network Model of Supply-Chain Substitutability

In this section, I incorporate the supply chain substitutability-concentration
trade-off and my empirical results from the previous section in a structural
dynamic asset pricing model detailed in Appendix A. This model builds on
existing production-based models with input output networks (see e.g., Ramìrez
(2017), Herskovic (2018b), or Gofman et al. (2020)). Unlike existing models, I
introduce a correlation structure in shocks to firm growth rates which
propagate both upstream and downstream in an input-output network.232323To my
knowledge, this is the first network model to feature both correlated shocks
and two directions of propagation. Firms are subject to both productivity
shocks which propagate downstream and demand shocks which propagate upstream.
Shocks are drawn from a joint distribution with finite second moments and in
which correlation across firm-level shocks is induced by shared variation in
firms’ input-output substitutability. More specifically, firms’ ability to
substitute away from productivity (demand) shocks is inversely related to
concentration of trade partners in latent technology (product) space and
correlated between firms who share trade partners.

###  4.1 Setting

Consider a discrete-time economy with
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n$
distinct goods and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n$
firms. Output goods are characterized by vector in product-technology space,
which is fixed exogenously for each good. Firm output (cash flow) depends on
aggregate economic conditions and the cash flows of its customers and
suppliers. There are two kinds of random shocks in this economy, productivity
shocks which propagate downstream from suppliers to customers, and demand
shocks which propagate upstream from customers to suppliers. The input-output
network is captured by two sequences of graphs with
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n$
nodes for each firm and weighted directed edges capturing the importance of
firm-to-firm trade relationships. In the customer (supplier) network, the edge
from
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
to
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
represents the relative reliance of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
on customer (supplier)
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$.

For tractability, I assume that trade relationships are exogenously determined
at the start of each period. Additionally, the model features a representative
investor with constant relative risk aversion (CRRA) preferences who owns all
firms and lives off labor wages and dividends. Next, I describe the process
for firm cash flows, the network structure, and the mechanism of shock
propagation through the input-output network. Then I derive equilibrium
consumption growth and asset prices. For ease of exposition, I provide details
on the production side of the economic since that is the primary source of
risk. Further details are left to Appendix A.

###  4.2 Substitutability and Firms’ Cash Flows

Firms are exposed to undiversifiable aggregate risk factors and risk from
trade partners which can be mitigated with diversification of customers and
suppliers. Every firm is both a customer who purchases inputs from other
firms, and a supplier who produces a single final good. Final goods are
characterized by a latent position in technology-product space, which
fluctuates according to a persistent stationary process discussed in Appendix
B.242424This assumption is justified empirically by the results of Section 3,
which suggest both that distances in product space and technology space change
over time. Latent position dynamics are exogenous to firm and household
decisions and can be interpreted as random changes in product differentiation.
For example, Syverson (2004) argues that the same products might be perceived
differently as a result of intangible factors like delivery speed,
documentation, product support, or branding and advertising.

In reduced-form, firm cash flows are determined by random shocks each period
which propagate stochastically both downstream to customers and upstream to
suppliers. The probability that a shock propagates through the supply chain is
a function of firms’ customer and supplier substitutability. Consistent with
the empirical results from Section 3, I assume that a firm’s customer
substitutability depends on the product diversity of the goods sold by its
customers. Likewise on the supply-side, a firm’s suppliers substitutability
depends on the technological diversity of its suppliers. When supply chains
are highly substitutable, shocks are less likely to propagate.

In particular, firm cash-flow growth has the following reduced-form equation:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\Delta
y_{i,t+1}=\Delta z_{i,t+1}+\Delta g_{i,t+1},$ (8)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Delta
z_{i,t+1}=\log(z_{i,t+1}/z_{it})$ is a shock to productivity and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Delta
g_{i,t+1}=\log(g_{i,t+1}/g_{it})$ is a shock to government demand. I assume
that dependence across shocks is determined by both the firm’s input-output
network and the relative location of its final good in product-technology
space. Productivity growth follows the process:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\Delta
z_{i,t+1}=\gamma_{u}\cdot a_{t+1}-\beta_{u}\cdot\varepsilon_{iu,t+1},$ (9)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{t}\sim_{iid}\mathcal{N}(0,\sigma_{a}^{2})$
is aggregate productivity growth at time
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma_{u}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{u}$
are positive scalars, and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varepsilon_{iut}$
is a Bernoulli shock that negatively affects productivity and originates
upstream. Similarly, government demand growth follows the process:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\Delta
g_{i,t+1}=\gamma_{d}\cdot g_{t+1}-\beta_{d}\cdot\varepsilon_{id,t+1},$ (10)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}g_{t}\sim_{iid}\mathcal{N}(0,\sigma_{g}^{2})$
is aggregate growth in government spending at time
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma_{d}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{d}$
are positive scalars, and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varepsilon_{idt}$
is a Bernoulli shock which negatively affects demand and originates
downstream. In other words,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varepsilon_{idt}$
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varepsilon_{iut}$)
is equal to one when firm
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
experiences a demand (supply) shock which originates at
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
and/or propagates from its downstream customers (upstream suppliers). Shocks
propagating in different directions are independent (i.e.,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varepsilon_{idt}\perp\varepsilon_{iut}$
for all
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$).

###  4.3 Network Structure

The propagation of shocks depends on the sequence of input-output network
connections between firms, defined as follows. The sequence of upstream and
downstream graphs
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\mathcal{G}_{n,u,t})_{n}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\mathcal{G}_{n,d,t})_{n}$
are
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n$-node
graphs with weighted edges given by
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{ijt}^{u}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{ijt}^{d}$,
respectively. Weights capture the importance of the directed relationship
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i\to
j$ from the perspective of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
and are fixed exogenously at the start of period
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$.

To ensure that the input-output network is realistic, I assume that all
weights are between 0 and 1 and introduce some additional restrictions on the
growth rates of input-output connections relative to the number of firms. In
particular, I assume that the number of shared customers and suppliers between
two firms cannot grow at a rate faster than the total number of firms in the
economy
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n$,
and that the maximum number of firm suppliers or customers must grow slower
than the total number of possible edges. First consider the following
definitions.

######  Definition 4.1 (Paths).

A
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k$-path
between nodes
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
in graph
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{G}$
is a length
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k$-sequence
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\{a_{\ell}\\}_{\ell=1}^{k}$
where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{1}=i$,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{k}=j$,
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{a_{\ell}a_{\ell+1}}>0$
for all
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ell=1,...,k-1$.
Denote by
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A_{ij}(\mathcal{G})$
the set of paths between nodes
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
and by
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A_{i}:=\\{k:A_{ki}(\mathcal{G})\neq\emptyset\\}$
the set of nodes for which a path to
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
exists.

######  Definition 4.2 (Maximal Dependency).

The maximal dependency of an
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n$-vertex
graph
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{G}_{n}$
is given by:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\bar{M}_{n}(\mathcal{G}_{n}):=\sup_{i,j}\bigg{[}\text{card}\big{(}A_{i}(\mathcal{G}_{n})\cap
A_{j}(\mathcal{G}_{n})\big{)}\bigg{]}$ (11)

######  Definition 4.3 (Maximal Degree).

The maximal (unweighted) degree in an
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n$-vertex
graph
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{G}_{n}$
is given by:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\bar{D}_{n}(\mathcal{G}_{n})=\sup_{i}\bigg{[}\sum_{j=1}^{n}\text{card}\big{(}A_{ji}(\mathcal{G}_{n})\big{)}\bigg{]}$
(12)

If only direct connections exist, then
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{M}_{n}(\mathcal{G}_{n})=\sup_{i,j}\sum_{k=1}^{n}\mathbbm{1}_{\\{w_{ki}>0\\}}\mathbbm{1}_{\\{w_{kj}>0\\}}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{D}_{n}(\mathcal{G}_{n})=\sup_{i}\sum_{j=1}^{n}\mathbbm{1}_{\\{w_{ji}>0\\}}$.
Given these definitions, the following assumptions formally restricts the
growth rate of input-output connections as the number of firms
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n$
grows. These assumptions are fairly general and relevant for deriving
tractable theoretical properties of the model.

######  Assumption 4.4 (Bounded Growth Rate of Maximal Degree Sequence).

For all
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$,
the maximal degree sequence grows at a rate strictly less than
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n^{2}$:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\bar{D}_{nq}=o(n^{2})$

These assumptions are intuitive and weaker than the restriction that no firms
can serve as a customer or supplier to all other firms. In this case, both the
maximal dependency and the maximal degrees must grow at a rate slower than
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n$.

######  Assumption 4.5 (Bounded Growth Rate of Maximal Dependency).

For all
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$,
the maximal dependency sequence grows at a rate strictly less than
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n$:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\bar{M}_{nq}=o(n)$

###  4.4 Shock Propagation Mechanism

For tractability, productivity and demand shocks propagate in a single
direction within period
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$
and die out in the following period. Network connections induce correlation
across firm-level shocks. At the start of period
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$,
shocks are drawn from distributions
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varepsilon_{idt}\sim\text{Bernoulli}(p_{idt})$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varepsilon_{iut}\sim\text{Bernoulli}(p_{iut})$
where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p_{idt}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p_{iut}$
represent time-varying propensities for firms to experience downstream
(demand-side) or upstream (supply-side) shocks, respectively. Propensities are
a function of the network structure and firm substitutability, both of which
are fixed exogenously at the start of each period.

Intuitively, firms with more substitutability across customers (suppliers)
should have a lower average propensity
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p_{idt}$
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p_{iut}$)
to experience shocks. Mathematically, I assume propensities follow a logistic
(sigmoid) curve:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
p_{iqt}=g(s_{iqt};k_{iq},x_{iq})=\frac{1}{1+\exp\big{\\{}k_{iqt}\cdot(s_{iqt}-x_{iqt})\big{\\}}},\quad
q\in\\{u,d\\},$ (13)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s_{iut}$
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s_{idt}$)
is the supply-side (demand-side) substitutability of firm
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k_{iqt}$
is the sensitivity (steepness) of firm propagation to substitutability, and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x_{iqt}$
is a scalar midpoint. The cross-sectional normalization ensures that firms
with substitutability
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x_{iqt}$
have a 50% chance of being shocked. Substitutability captures network-weighted
dispersion in
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$’s
supplier-technology (customer-product) space, while
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x_{iqt}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k_{iqt}$
jointly characterize the firm-specific risk of firm
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$.
Inverting terms in the “between” concentration measure from (5), I assume
substitutability can be written:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
s_{iqt}=\log\sum_{j\neq k}w_{ijt}^{q}\cdot w_{ikt}^{q}\cdot\delta_{jkt}^{q},$
(14)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{ijt}^{q}$
represents the importance of trade between
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
in the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q$-stream
direction and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\delta_{jkt}^{q}$
is normalized distance between industries
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k$
in latent product
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q=d$)
or technological
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q=u$)
space. See Appendix B for details. Shared customer and supplier connections
induce correlation in substitutability
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s_{iqt}$
across firms. This also implies the shock transmission propensities
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p_{iqt}$
are also correlated. Time-variation in firm product differentiation generates
correlated changes in substitutability across firms who share customers and
suppliers. When there are no network connections, firms are hit by shocks with
probability
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p_{iqt}=1/(1+\text{exp}({-k_{iqt}x_{iqt}}))$.
For remaining sections, I assume that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k_{iqt}=k_{iq}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x_{iqt}=x_{iq}$
are time invariant.

###  4.5 Consumption Growth and the Stochastic Discount Factor

I assume that representative households in this economy own shares in each
firm and have the following preferences:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
u(c_{1t},...,c_{nt},\ell_{t})=\frac{1}{1-\gamma}\cdot\bigg{(}\prod_{i=1}^{n}c_{it}^{\beta_{i}}\bigg{)}^{1-\gamma},$
(15)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}c_{it}$
is the consumption of good
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
with preference weights
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{i}$
such that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sum_{i}\beta_{i}=1$,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma$
is risk aversion, and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}g(.)$
is a decreasing and differentiable function capturing disutility of labor
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ell_{t}$.
In Appendix A, I show that equilibrium consumption growth and output growth
are equal such that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Delta\tilde{c}_{i,t+1}:=\log(c_{i,t+1}/c_{it})=\Delta\tilde{y}_{i,t+1}$
for all
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$.
I also derive an appropriate price normalization such that equilibrium
consumption expenditure is given by
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C_{t}=\prod_{i}c_{it}^{\beta_{i}}=\sum_{i}p_{it}c_{it}$
for a given set of positive prices
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p_{it}$.
Finally, the following proposition derives the expression for growth in
aggregate consumption expenditure under the same assumptions.

######  Proposition 4.6 (Aggregate Consumption and Output Growth).

Assuming
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{i}=1/n$
for all
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
and under the price normalization in Appendix A, aggregate consumption growth
can be written:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\Delta\tilde{c}_{t+1}$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\gamma_{u}\cdot
a_{t+1}+\gamma_{d}\cdot g_{t+1}-\beta_{u}\cdot W_{u,t+1}-\beta_{d}\cdot
W_{d,t+1},$ (16)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W_{ut}=\frac{1}{n}\sum_{i=1}^{n}\varepsilon_{iut}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W_{dt}=\frac{1}{n}\sum_{i=1}^{n}\varepsilon_{idt}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma_{u},\gamma_{d},\beta_{u},\beta_{d}$
are positive scalars.

###### Proof.

See Appendix C.2. ∎

This proposition decomposes aggregate consumption growth into four components.
The first two components capture innovations to aggregate productivity and
demand growth
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{t}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}g_{t}$,
respectively), both of which are positively related to output and consumption
growth. On the other hand, the next two components are negatively related to
output and consumption growth and capture the average impact of bad shocks to
productivity originating upstream
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W_{ut}$),
and the average impact of bad shocks to demand originating downstream
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W_{dt}$).
Combining this result with (35), the log stochastic discount factor (SDF) can
be written:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
m_{t+1}=\log\beta-\gamma\big{(}\gamma_{u}\cdot a_{t+1}+\gamma_{d}\cdot
g_{t+1}-\beta_{u}\cdot W_{u,t+1}-\beta_{d}\cdot W_{d,t+1}\big{)},$ (17)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta$
is the intertemporal discount factor and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma$
is risk aversion. This implies that aggregate productivity and demand growth
have a positive price of risk while average upstream and downstream
propagation have a negative price of risk.

###  4.6 Additional Theoretical Results

This section summarizes some additional relevant theoretical results from the
model. The following proposition states that the conditional distribution of
consumption growth in this model is asymptotically normal as the number of
firms grows.

######  Proposition 4.7 (Distribution of Consumption Growth).

Under Assumption 4.4, the sequence of consumption growth is asymptotically
normal as
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n\to\infty$,
conditional on time
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$
for all
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\Delta\tilde{c}_{t+1}\xrightarrow[]{d}\mathcal{N}(\mu_{c,t+1},\sigma_{c,t+1}^{2}),$
(18)

where:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\mu_{ct}$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle:=\mathbb{E}_{t}[\Delta\tilde{c}_{t}]=\frac{1}{n}\sum_{i=1}^{n}(p_{iut}+p_{idt}),$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\sigma_{ct}^{2}$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle:=\text{var}_{t}[\Delta\tilde{c}_{t}]=\sigma_{g}^{2}+\sigma_{a}^{2}+\text{var}_{t}(W_{ut}+W_{dt}).$

###### Proof.

See Appendix C.4. ∎

Although the conditional mean of consumption growth is known in this model,
there is no closed form expression for the conditional variance term. This
follows from the fact that shock transmission propensities follow a logistic
normal distribution (see Appendix B). After deriving the asymptotic
distribution of consumption growth, the next corollary characterizes the
probability the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W_{nqt}$
deviates from its cross-sectional mean when propensities are known.

######  Corollary 4.8 (Concentration of Network Factors).

Under Assumption 4.5 and if
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{M}_{nqt}>1$,
the propagation factor
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W_{qt}$
can be written:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
W_{nqt}=\mu_{nqt|t}+\epsilon_{nqt|t}$ (19)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mu_{nqt|t}=\frac{1}{n}\sum_{i}p_{iqt|t}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\epsilon_{nqt|t}\sim\mathcal{N}(0,\sigma_{nqt|t}^{2})$
where:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\sigma_{nqt|t}^{2}\leq\frac{\bar{M}_{nqt}}{n}=o(1)$
(20)

Moreover, for any
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k>0$,
the magnitude of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\epsilon_{nqt|t}$
can be upper bounded as follows:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\text{Pr}\big{(}|\varepsilon_{nqt|t}|\geq
2k(\bar{M}_{qt}/n)\big{)}\leq\frac{1}{k^{2}}$

###### Proof.

See Appendix C.3. ∎

##  5 Testable Implications

In this section, I verify the main quantitative predictions of the model using
financial and macroeconomic data. According to equation (17), innovations in
average supply and demand shock propagation have a negative price of risk. In
addition, level changes in these components should be negatively correlated
with aggregate consumption growth.

###  5.1 Data and Calibration

I construct a panel of firms between 1997-2019 whose North American Industry
Classification System (NAICS) are in the set of industries for which BEA
Input-Output accounts are available. I obtain annual and quarterly firm
variables from Compustat and stock return data from CRSP for share codes 10,
11, and 12.252525Firm and return variables are winsorized at the 1% level
unless otherwise specified. I obtain aggregate time series of Total Factor
Productivity growth from Fernald (2012), government demand growth from the
procurement proxy in Briganti and Sellemi (2022), and annual market and risk-
free returns from Kenneth French’s Website.

I begin by computing input-output propagation factors, denoted by
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{W}_{ut}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{W}_{dt}$.
In Section 3, I introduce a latent distance approach to compute the panel of
industry concentration and substitutability between customers and suppliers
from equation (5). Assuming that substitutability is the same for firms in a
given industry, I can then directly compute
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{s}_{iqt}$
for any firm with industry data available. The expression for
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{p}_{iqt}=g(\hat{s}_{iqt};k_{iq},x_{iq})$
follows directly from equation (13) conditional on scalar parameters
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k_{iq}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x_{iq}$.
To calibrate these parameters, I first estimate the following panel
regression:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\Delta\tilde{y}_{i,t+1}$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\gamma_{u}a_{t+1}+\gamma_{d}g_{t+1}+\text{controls}+\epsilon_{i,t+1},$
(21)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Delta\tilde{y}_{i,t+1}$
is year-on-year sales growth,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{t+1}$
is TFP growth, and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}g_{t+1}$
is growth in the federal defense. Controls include year and industry fixed
effects, lagged firm size, age, and return on assets to ensure that changes in
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\epsilon_{i,t+1}$
is unrelated to aggregate economy-wide or industry-level forces or trends in
large, young, or profitable firms.262626Industry fixed effects are at the two-
digit NAICS granularity. Then let
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\epsilon}_{i,t+1}$
denote residual sales growth, and let
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\omega_{iu}$
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\omega_{id}$)
denote the average cost share (sales share) of intermediate inputs in
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$’s
industry, and choose values of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k_{iq}\geq
0$ and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s_{iq}\in\mathbb{R}$
such that:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\widehat{\text{var}}(\hat{\epsilon}_{i,t+1})=\frac{\exp(k_{iq}(\bar{s}_{iq}-x_{iq}))}{\big{(}1+\exp(k_{iq}(\bar{s}_{iq}-x_{iq}))\big{)}^{2}},\quad\text{
and
}\quad\omega_{iq}=\frac{\big{(}1+\exp(k_{iq}(\bar{s}_{iq}-x_{iq}))\big{)}^{2}}{\big{(}1+\exp(-k_{iq}x_{iq}))\big{)}^{2}},$

for
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q\in\\{u,d\\}$
where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{s}_{iq}=\frac{1}{T}\sum_{t}s_{iqt}$
is firm
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$’s
average substitutability over time. The first restriction is based on equation
(8) and ensures that the variance of a typical
Bernoulli($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p_{iqt}$)
shock is equal to residual sales growth variance, while the second restriction
requires
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\omega_{iq}$
proportion of this variance to be attributed to network propagation. Together,
the system of equations uniquely identify
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k_{iq}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x_{iq}$.
Table 16 summarizes the calibrated parameter values. I then approximate each
realized network propagation factor with its cross-sectional empirical mean as
follows:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\hat{W}_{qt}\approx\frac{1}{n}\sum_{i=1}^{n}\hat{p}_{iqt},\quad
q\in\\{u,d\\},$ (22)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{p}_{iqt}$
is the empirical propensity. I use the cross-sectional mean since realized
shocks cannot be identified even when firm propensities
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p_{iqt}$
are known. In practice, this is not a large concern, as Proposition 4.8 shows
that the measurement error can be bounded arbitrarily by increasing the sample
size.272727As a heuristic evaluation of this bound, suppose I restrict our
sample to only firms that show up in the Customer Segments database
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{M}_{nqt}=149$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n=12489$),
then the probability that the measurement error more than 10% is less than 1%.
I plot the estimated series in Figure 1 and report summary statistics in Table
3. See Appendix H for more details.

Table 3: Descriptive Statistics Network Propagation Factors | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W_{ut}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}S_{ut}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W_{dt}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}S_{dt}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}g_{t}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{t}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sigma_{t}^{civ}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sigma_{t}^{mkt}$ | AC(1)
---|---|---|---|---|---|---|---|---|---
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W_{ut}$ | 1 | -0.81 | 0.13 | 0.17 | 0.25 | -0.20 | -0.38 | -0.31 | 0.26
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}S_{ut}$ |  | 1 | -0.36 | -0.03 | -0.21 | -0.16 | 0.62 | 0.52 | 0.20
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W_{dt}$ |  |  | 1 | -0.44 | -0.05 | -0.13 | -0.22 | -0.06 | 0.01
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}S_{dt}$ |  |  |  | 1 | -0.15 | -0.17 | 0.13 | -0.14 | 0.04
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}g_{t}$ |  |  |  |  | 1 | 0.05 | 0.00 | 0.26 | 0.62
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{t}$ |  |  |  |  |  | 1 | -0.19 | -0.43 | 0.30
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sigma_{t}^{civ}$ |  |  |  |  |  |  | 1 | 0.53 | -0.30
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sigma_{t}^{mkt}$ |  |  |  |  |  |  |  | 1 | -0.21
Notes: This table reports the time-series correlation and first-order
autocorrelation of network propagation factors
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W_{ut}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W_{dt}$,
average industry substitutability
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}S_{ut}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}S_{dt}$,
procurement demand growth
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}g_{t}$,
productivity growth
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{t}$,
innovations to common idiosyncratic volatility
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sigma_{t}^{civ}$,
and innovations to market volatility
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sigma_{t}^{mkt}$.
I calculate
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}g_{t}$
as the first log difference in the federal procurement proxy from Briganti and
Sellemi (2022),
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{t}$
as the first difference in the TFP series from Fernald (2012), innovations in
common idiosyncratic volatility as the first log difference in the first
principal component of firm volatility following Herskovic et al. (2016), and
innovations in market volatility as the first log difference in market return
volatility.

(a)

(b)

Figure 1: Network Propagation Risk Factors Notes: This figure plots the time
series of network propagation risk factors (Panel A), the cross-sectional
average industry substitutability (Panel B). Shaded regions indicate NBER-
dated recession periods.

###  5.2 Asset Pricing Tests

To verify the prices of risk predicted in (17), I sort stocks based on their
exposure to factors and form quintile-sorted portfolios. In particular, for
every stock
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
I regress annual excess returns
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}r_{it}-r_{ft}$
on a constant, aggregate demand and productivity growth, and additional
controls.282828I test several specifications including controlling for lag
factor levels. Results are robust to several specifications on the set of
controls, including the baseline without controls. The main regression is
given by:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
r_{it}-r_{ft}=\alpha_{i}+\beta_{ia}a_{t}+\beta_{ig}g_{t}+\beta_{iu}W_{ut}+\beta_{id}W_{dt}+\varepsilon_{it},$
(23)

where equation (17) implies that stocks with high
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{ia}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{ig}$
should have higher expected excess returns and stocks with high
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{iu}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{id}$
should have lower expected excess returns. For each year
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$,
I compute stock exposure to factors on a 15-year rolling window from
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t-14$
to
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$
using (23), and then sort stocks into five portfolios on each beta both
separately (one-way sort) and pairwise (two-way sort). Then I construct value
and equal-weighted portfolios over the subsequent year
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t+1$
and compute average out-of-sample excess returns for each portfolio.

Table 4 provides evidence of a significant return spread in one-way beta
sorted portfolios. In particular, the highest quintile upstream propagation
beta portfolio earns -11.42% lower annual returns than the lowest quintile
portfolio, while the highest quintile downstream propagation beta portfolio
earns -4.18% lower annual returns than the lowest quintile portfolio. Both
return spreads are statistically significant, although more pronounced for
upstream propagation beta sorted portfolios.292929I also test for monotonicity
of returns in upstream and downstream propagation betas, following Patton and
Timmermann (2010). I reject this null hypothesis at the 10% level for upstream
beta sorted portfolios, but fail to reject for downstream beta sorted
portfolios. This is consistent with Herskovic et al. (2020b), who argue that
upstream propagation is the more important channel.

I also observe a return spread in post-sample alphas from the CAPM and Fama
and French (FF3) three factor models, which implies that network propagation
risk is not captured by market returns or FF3 factors. In light of the
variance results of Section 3, I also verify that return spreads are not
explained by market volatility or idiosyncratic volatility factors in Table
17.303030I measure market volatility as the annual volatility of market
returns and idiosyncratic volatility following Herskovic et al. (2016).
Additionally, return spreads cannot be explained by differences in return
volatility, average size, or average book-to-market ratios. Finally, the
average correlation between upstream and downstream propagation betas is 8.6%,
suggesting that the two network factors are distinct sources of risk.

Return spreads are robust to the choice of trailing window length, equal or
value weighting in portfolios, control variables, and show up in double-sorted
portfolios as well. See Appendix H for more details.

Table 4: One-Way Sorted Portfolios on Network Propagation Factors Panel A:
One-way sorts on upstream propagation beta (controlling for
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{t}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}g_{t}$)
---
| 1 (Low) | 2 | 3 | 4 | 5 (High) | H-L | t(H-L) | MR p-val
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{E}[r]-r_{f}$ | 18.10 | 12.81 | 10.59 | 9.42 | 6.69 | -11.42 | -13.22 | 0.07
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{capm}$ | 0.29 | -0.1 | -0.23 | -0.31 | -1.02 | -1.32 | -15.71 | 0.05