$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{ff3}$ | 0.08 | -0.09 | -0.22 | -0.29 | -0.54 | -0.63 | -8.61 | 0.09
Volatility (%) | 15.54 | 13.89 | 13.59 | 13.03 | 19.66 | - | - | -
Book-to-market | 0.52 | 0.56 | 0.53 | 0.55 | 0.50 | - | - | -
Market value ($bn) | 6.46 | 16.99 | 10.62 | 15.15 | 9.11 | - | - | -
Panel B: One-way sorts on downstream 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
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{E}[r]-r_{f}$ | 13.54 | 13.23 | 11.02 | 9.77 | 9.36 | -4.18 | -7.56 | 0.25
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{capm}$ | -0.04 | -0.18 | -0.28 | -0.38 | -0.60 | -0.56 | -4.78 | 0.00
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{ff3}$ | -0.11 | -0.14 | -0.23 | -0.28 | -0.36 | -0.25 | -3.62 | 0.03
Volatility (%) | 15.44 | 13.95 | 18.58 | 12.99 | 13.88 | - | - | -
Book-to-market | 0.52 | 0.56 | 0.55 | 0.53 | 0.51 | - | - | -
Market value ($bn) | 15.84 | 7.45 | 4.54 | 17.6 | 12.72 | - | - | -
Panel C: One-way sorts on upstream propagation beta (no controls)
| 1 (Low) | 2 | 3 | 4 | 5 (High) | H-L | t(H-L) | MR p-val
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{E}[r]-r_{f}$ | 15.15 | 12.61 | 11.46 | 9.44 | 7.23 | -7.91 | -11.57 | 0.07
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{capm}$ | 0.09 | -0.15 | -0.17 | -0.28 | -1.09 | -1.18 | -17.53 | 0.26
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{ff3}$ | -0.12 | -0.14 | -0.18 | -0.22 | -0.58 | -0.46 | -9.96 | 0.31
Volatility (%) | 15.26 | 14.23 | 13.57 | 12.61 | 20.97 | - | - | -
Book-to-market | 0.54 | 0.58 | 0.52 | 0.52 | 0.50 | - | - | -
Market value ($bn) | 6.87 | 17.48 | 10.92 | 16.38 | 6.56 | - | - | -
Panel D: One-way sorts on downstream propagation beta (no controls)
| 1 (Low) | 2 | 3 | 4 | 5 (High) | H-L | t(H-L) | MR p-val
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{E}[r]-r_{f}$ | 12.66 | 11.94 | 11.8 | 8.34 | 5.13 | -7.53 | -8.65 | 0.42
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{capm}$ | -0.15 | -0.18 | -0.19 | -0.4 | -0.64 | -0.49 | -11.37 | 0.41
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{ff3}$ | -0.10 | -0.21 | -0.21 | -0.29 | -0.32 | -0.22 | -4.76 | 0.44
Volatility (%) | 14.09 | 13.9 | 14.44 | 13.22 | 31.95 | - | - | -
Book-to-market | 0.54 | 0.49 | 0.58 | 0.51 | 0.54 | - | - | -
Market value ($bn) | 15.97 | 12.79 | 6.33 | 16.88 | 6.34 | - | - | -
Notes: This table reports average excess returns and post-sample alphas in
annual percentages for value-weighted portfolios sorted into quintiles on
annual upstream and downstream propagation factors. Sample is between
1997-2021 for more than 10,000 stocks belonging to the BEA 66 non-government
industry classifications. Panels A and B control for productivity growth and
federal procurement demand growth, while Panels C and D have no controls. I
also report average return volatility, book-to-market ratio and market value
for each portfolio. To test for significant return spreads, I report
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$-statistics
for the null hypothesis
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H_{0}:xr_{5}=xr_{1}$,
where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}xr_{q}$
is the average return of the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q^{th}$
quintile single sorted portfolio. Moreover, I report p-values for the test
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H_{0}:xr_{q+1}<xr_{q}\forall
q\leq 4$, calculated via bootstrap following Patton and Timmermann (2010).

###  5.3 Verifying Macroeconomic Predictions

Equation (16) predicts that upstream and downstream propagation factors should
be negatively correlated with consumption, output growth, and aggregate
dividend growth. To test this, I construct aggregate series between 1997-2021
for consumption and output growth from the National Income and Product
Accounts (NIPA) and corporate dividend growth from BEA data. Then I regress
each outcome on network propagation factors, controlling for aggregate
productivity and federal procurement demand growth. I standardize each
variable to have zero mean and unit standard deviation. Consistent with the
predictions of the model, Table 5 reports negative and statistically
significant coefficients on both upstream and downstream propagation risk
factors. The factors explain a large portion of time variation in consumption,
output, and dividend growth with
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R^{2}$
values of 56%, 68%, and 26%, respectively.

The coefficients on downstream propagation are -0.17
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t=-1.89$),
-0.60
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t=-2.57$),
and -0.01
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>for aggregate consumption, output, and dividend growth regressions,
respectively. On the other hand, the coefficients on upstream propagation are
not significant for aggregate consumption and output growth regressions,
although the coefficient in the dividend growth regression is
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t=-1.80$).
Additionally, the coefficients on upstream propagation are significant when
the dependent variable is limited to only durable consumption or output
growth, -0.111
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>and -1.38
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t=-2.39$),
respectively. This suggests that durable consumption is more sensitive to
upstream (supply-side) risk.

Table 5: Network Propagation and Macroeconomic Factors Variable | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Delta c_{t}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Delta c_{t}^{dur}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Delta c_{t}^{nondur}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Delta y_{t}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Delta y_{t}^{dur}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Delta y_{t}^{nondur}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Delta D_{t}$
---|---|---|---|---|---|---|---
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W_{ut}$ | 0.003 | -0.111** | -0.158 | -0.112 | -1.383** | -0.416 | -0.033*
| (0.103) | (0.048) | (0.087) | (0.275) | (0.607) | (0.229) | (0.021)
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W_{dt}$ | -0.174* | -0.031 | -0.022 | -0.598** | -0.387 | -0.058 | -0.010
| (0.074) | (0.054) | (0.101) | (0.222) | (0.684) | (0.267) | (0.021)
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{t}$ | 0.321** | 0.128** | 0.197** | 1.018** | 1.607** | 0.520** | 0.018
| (0.114) | (0.055) | (0.108) | (0.271) | (0.689) | (0.285) | (0.010)
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}g_{t}$ | 0.271 | -0.056 | 0.604 | -0.011 | -0.704 | 1.592 | 0.029
| (0.332) | (0.276) | (0.638) | (0.956) | (3.450) | (1.682) | (0.109)
Intercept | -0.485 | -0.111 | -0.205 | 1.218 | 4.09 | 2.023 | 0.042
Obs | 24 | 24 | 24 | 24 | 24 | 24 | 24
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R^{2}$ | 0.56 | 0.537 | 0.376 | 0.679 | 0.537 | 0.376 | 0.257
Notes: This table reports results of OLS regressions of aggregate consumption
growth, output growth, and dividend growth on input-output network propagation
risk factors, controlling for productivity and federal procurement demand
growth. The columns represent different dependent variables corresponding to
aggregate PCE growth, durable consumption growth, non-durable consumption
growth, output growth, durable output growth, non-durable output growth, and
dividend growth. All series are standardized to have zero mean and unit
variance. Sample is at an annual frequency between 1997-2021.

##  6 Conclusion

In this work, I propose a production-based asset pricing model with input-
output networks in which productivity shocks propagate downstream from
suppliers to customers and demand shocks propagate upstream from customers to
suppliers. This model is consistent with a benchmark reduced-form equation
that links output growth of each node (industry or firm) in the network to
output growth of other nodes and a node-specific shock. Almost all research in
this area assumes that node-specific shocks are idiosyncratic- that is, drawn
independently across units. In this work, I prove that the idiosyncratic shock
assumption in this reduced-form equation is not consistent with realistic
input-output networks. I argue that researchers who make use of network models
should account for potential inter-node correlation in propagated shocks.

When accounting for non-negligible correlation in shocks, the variance
expression for output growth gains an additional component which depends on a
weighted sum of covariances of shocks between each node’s trade partners. As a
result, units in the network are exposed to risk associated with the
homogeneity of their customer and supplier connections. This generates cross-
sectional differences in nodes’ ability to substitute away from correlated
shocks propagating through the network. I define substitutability as the
negative value of the new covariance term, termed concentration “between”
suppliers. Consistent with theory, I provide empirical evidence that customer
and supplier substitutability can explain differences in realized variance
across industries. In particular, higher substitutability (lower between-
concentration) explains lower realized variance, conditional on other input-
output and variance related characteristics.

Although the covariance between shocks is defined in a purely statistical
sense, I also investigate industry characteristics which can provide a
structural explanation for high correlation in shocks. To this end, I argue
that technological proximity explains correlation in supply-side shocks that
propagate downstream, while product similarity explains correlation in demand-
side shocks that propagate upstream. As a result, supply chain
substitutability can be calculated as a function of distances between units in
latent technology and product space. Although there is evidence of other
sources of correlation in shocks, this simplification is simple, tractable,
and can be incorporated in theoretical models to provide a structural origin
for correlation in shocks.

I incorporate this mechanism in an extension of the network model which
directly models the propagation of shocks between firms in the input-output
network. The propensity that shocks propagate is a decreasing function in
substitutability, such that firms with more substitutable supply chains are
less likely to experience shocks. Moreover, demand shock substitutability is
captured by network-weighted dispersion of a firm’s customers in latent
product space, while supply shock substitutability is captured by network-
weighted dispersion of a firm’s suppliers in latent technology space. In the
extended model, systematic risk is driven by the average propagation of
upstream and downstream transmitted shocks.

Under some reasonable assumptions, I calibrate the model to data on publicly
traded firms and construct empirical analogues of upstream and downstream
propagation risk factors. I provide evidence that these risk factors are
negatively priced in the cross-section of returns. Additionally, consistent
with the model’s theoretical predictions, both factors are associated with
lower aggregate consumption growth, lower output growth, and lower dividend
growth. Empirical results suggest the importance of both directions of
propagation as significant sources of systematic risk in the economy.

Future work might investigate in more depth the sources of correlation in
supply and demand shocks and generate more granular estimates of the agents’
ability to substitute away from them.

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##  Appendix A General Equilibrium Model of Input-Output Linkages

In this section, I show that (1) can be cast as an outcome of a production-
based asset pricing model. This model provides a structural foundation for the
theoretical contributions of this work and is closely related to Acemoglu et
al. (2012), Acemoglu et al. (2016), Ramìrez (2017), and Herskovic (2018b).
Consider a competitive economy with
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n$
production units (firms or industries) with Cobb-Douglas technology,
representative households with constant relative risk aversion (CRRA)
preferences over a basket of goods and who work and own shares in all firms
and live off wages and dividends, and a government that finances purchases
with a lump-sum tax. In this economy, Hicks-neutral productivity shocks
propagate downstream from suppliers to customers, while government demand
shocks propagate upstream from customers to suppliers.

###  A.1 Production

Production 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
output is a constant returns to scale function of labor and intermediate
inputs:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
y_{it}=\exp(z_{it})\ell_{it}^{\alpha_{i\ell}}\prod_{j=1}^{n}x_{ijt}^{w_{ijt}},$
(24)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x_{ijt}$
is the amount of product
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
used as input by 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 time
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ell_{it}$
is labor input, and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}z_{it}$
is a Hicks-neural productivity shock, respectively. I assume that 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
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$,
the labor share of production is positive (i.e.,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{i\ell}>0$)
and intermediate input shares are non-negative
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{ijt}\geq
0$) and sum to the capital share of production (i.e.,
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sum_{j=1}^{n}w_{ijt}=1-\alpha_{i\ell}$).

Taking both spot market prices and input shares as given, production units
optimize dividends (denoted
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}D_{it}$)
as a function of input and labor purchases:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
D_{it}=\max_{\\{x_{ijt}\\}_{j=1}^{n},\ell_{it}}p_{it}y_{it}-\sum_{j=1}^{n}p_{jt}x_{ijt}-p_{wt}\ell_{it}$
(25)

subject to (24) and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ell_{it}\in(0,1)$.
Suppose further that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}M_{t+1}$
is the stochastic discount factor (SDF) that prices all assets in the economy.
Then the cum dividend value of firm
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
(denoted
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}V_{it}$)
satisfies the following Bellman equation:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
V_{it}=D_{it}+\mathbb{E}_{t}[M_{t+1}V_{i,t+1}].$ (26)

###  A.2 Government

The government purchases goods
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}G_{it}$
from each unit
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
at time
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$
and finances them via a lump-sum tax
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}T_{t}$.
Taking prices as given, the government’s financing constraint implies that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}T_{t}=\sum_{i=1}^{n}p_{it}G_{it}$.

###  A.3 Households

Assume that the representative household owns shares in each unit and has the
following preferences:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
u(c_{1t},...,c_{nt},\ell)=\frac{1}{1-\gamma}\cdot\bigg{(}\prod_{i=1}^{n}c_{it}^{\beta_{i}}\bigg{)}^{1-\gamma}\cdot
g(\ell_{t}),$ (27)

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 and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}g(.)=(1-\ell_{t})^{\nu}$
is a decreasing and differentiable function capturing disutility of labor.
Households also have a time-discount factor of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta$
and cannot store goods from one period to another. In equilibrium, households
hold a zero net position in a risk-free asset and choose to own
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\vartheta_{it}$
in each unit according to the following budget constraint:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
T_{t}+p_{wt}\ell_{t}+\sum_{i=1}^{n}p_{it}c_{it}+\sum_{i=1}^{n}\vartheta_{i,t+1}(V_{it}-D_{it})=\sum_{i=1}^{n}\vartheta_{i,t}V_{it},$
(28)

where the right hand side is total value of investments and the left hand side
is the sum of taxes paid, wages earned, cost of consumption, and unrealized
capital gains, respectively. The household’s optimization problem satisfies
the Bellman equation:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
U_{t}=\max_{\\{c_{it},\vartheta_{i,t+1},\ell\\}_{i=1}^{n}}u(.)+\beta\mathbb{E}_{t}[U_{t+1}],$
(29)

subject to (28).

###  A.4 Equilibrium

The competitive equilibrium of the economy consists of spot market prices
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\{p_{it}\\}_{i=1}^{n}$,
consumption bundles
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\{c_{it}\\}_{i=1}^{n}$,
share holdings
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\{\vartheta_{it}\\}_{i=1}^{n}$,
labor supply
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ell_{t}$,
wages
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p_{wt}$,
and input bundles
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\{x_{ijt}\\}_{i,j=1}^{n}$
such that both production units and households exhibit optimal behavior and
good/asset markets clear.

####  A.4.1 Market Clearing

In equilibrium, all good markets clear such that:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
y_{it}=\underbrace{c_{it}}_{\text{final consumption
demand}}+\underbrace{\sum_{j=1}^{n}x_{jit}}_{\text{intermediate
demand}}+\underbrace{G_{it}}_{\text{government consumption}},$

and all asset markets clear
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\vartheta_{it}=1$
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
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$.

####  A.4.2 Producer Optimality

Taking prices as given, 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
first order dividend maximizing conditions satisfy

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
w_{ijt}=\frac{p_{jt}x_{ijt}}{p_{it}y_{it}}\equiv\frac{sales_{j\to
i}}{sales_{i}}$ (30)

and

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\alpha_{i\ell}=\frac{p_{wt}\ell_{it}}{p_{it}y_{it}}$
(31)

####  A.4.3 Consumer Optimality

Given the Cobb-Douglas aggregation in preferences over goods (i.e.,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C_{t}:=\prod_{i}c_{it}^{\beta_{i}}$),
utility maximizing households consume
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{i}$
of income on good
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
and hold shares fixed at
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\vartheta_{it}=1$.
More specifically, letting
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda_{t}$
be the Lagrange multiplier for the period
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$
household budget constraint, the first-order condition for consumption is
written:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\lambda_{t}=\frac{C_{t}^{-\gamma}}{p_{it}}\cdot\frac{\partial
C_{t}}{\partial c_{it}}.$ (32)

This implies that equilibrium consumption satisfies:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
p_{it}c_{it}=\beta_{i}\bigg{(}p_{wt}\ell_{t}^{*}+\sum_{j=1}^{n}D_{jt}-T_{t}\bigg{)}.$
(33)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ell_{t}^{*}$
solves:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\frac{p_{wt}\ell_{t}^{*}}{p_{wt}\ell_{t}^{*}+\sum_{j=1}^{n}D_{jt}-T_{t}}=-\frac{\ell_{t}^{*}g^{\prime}(\ell_{t}^{*})}{g(\ell_{t}^{*})}$
(34)

####  A.4.4 Asset Prices

From (32), the stochastic discount factor can be written:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
M_{t+1}=\beta\frac{\lambda_{t+1}}{\lambda_{t}}=\beta\bigg{(}\frac{C_{t+1}}{C_{t}}\bigg{)}^{-\gamma}\frac{p_{it}\cdot\partial
C_{t+1}/\partial c_{i,t+1}}{p_{i,t+1}\cdot\partial C_{t}/\partial c_{it}}.$
(35)

Following Herskovic (2018a), I assume that prices are normalized such that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p_{it}=\partial
C_{t}/\partial c_{it}$, or equivalently that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\prod_{j}p_{jt}^{\beta_{j}}=\prod_{j}\beta_{j}^{\beta_{j}}$
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
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$.
This implies the the utility aggregator is equal to the household’s
consumption expenditure
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>further assume that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>Then (35) simplifies to:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
M_{t+1}=\beta\bigg{(}\frac{\sum_{i=1}^{n}p_{i,t+1}c_{i,t+1}}{\sum_{i=1}^{n}p_{it}c_{it}}\bigg{)}^{-\gamma}.$
(36)

####  A.4.5 Shock Propagation

I now derive closed form expressions for the effects of productivity and
government demand shocks on output growth in this model. The main takeaway is
that output growth is captured by the following reduced form expression:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\mathbf{d}\log\mathbf{y}_{t}=\mathbf{H}_{down,t}\mathbf{dz}_{t}+\mathbf{H}_{up,t}\mathbf{dG}_{t},$

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{H}_{.,t}$
are
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n\times
n$ Leontief inverse propagation matrices. I provide a derivation for each
component separately.

##### Productivity Shocks

Totally differentiate the expression in (24) to obtain:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
d\log y_{it}=dz_{it}+\alpha_{i\ell}d\log\ell_{it}+\sum_{j=1}^{n}w_{ijt}d\log
x_{ijt}$ (37)

Totally differentiating (30), (31), and (33) and plugging in to this
expression yields:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
d\log y_{it}$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=dz_{it}+\alpha_{i\ell}d\log\ell_{it}+\sum_{j=1}^{n}w_{ijt}(d\log
y_{it}+d\log p_{it}-d\log p_{jt})$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=dz_{it}+\alpha_{i\ell}(d\log
y_{it}-d\log c_{it})+\sum_{j=1}^{n}w_{ijt}(d\log y_{it}-d\log c_{it}+d\log
c_{jt}).$

Given constant returns to scale
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{i\ell}+\sum_{j}w_{ij}=1$),
this expression can be further simplified as follows:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
d\log c_{i}=dz_{it}+\sum_{j=1}^{n}w_{ijt}d\log c_{j}$

or in vector notation:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\mathbf{d}\log\mathbf{c}_{t}=\mathbf{dz}_{t}+\mathbf{W}_{t}\mathbf{d}\log\mathbf{c}_{t},$

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}_{t}$
has entries
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{ijt}$.
Note that market clearing and profit maximization conditions together imply
that:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\frac{y_{jt}}{c_{jt}}=1+\sum_{i=1}^{n}w_{ijt}\frac{\beta_{i}y_{it}}{\beta_{j}c_{it}}$

which implies that equilibrium consumption growth is equal to equilibrium
output growth:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\mathbf{d}\log\mathbf{c}_{t}=\mathbf{d}\log\mathbf{y}_{t}$
(38)

and thus that:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\mathbf{d}\log\mathbf{y}_{t}=(\mathbf{I}-\mathbf{W}_{t})^{-1}\mathbf{dz}_{t},$
(39)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{H}_{down,t}:=(\mathbf{I}-\mathbf{W}_{t})^{-1}$
is the Leontief inverse of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}_{t}$.
Here, the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}_{t}$
matrix determines the strength of downstream propagation of productivity
shocks.

##### Demand Shocks

To study the effects of government spending shocks in the model, normalize
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{z}_{t}=0$
and consider the unit cost function for
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
C_{it}(\mathbf{p}_{t},p_{wt})=A_{it}p_{wt}^{\alpha_{i\ell}}\prod_{j=1}^{n}p_{jt}^{w_{ijt}},\quad\text{where
}A_{it}=\alpha_{i\ell}^{-\alpha_{i\ell}}\prod_{j=1}^{n}w_{ijt}^{-w_{ijt}}.$

The zero productivity normalization implies zero dividends for production
units, and combined with the price normalization for wages, this implies that:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\log
p_{it}=\log A_{it}+\sum_{j=1}^{n}w_{ijt}\log p_{jt}$

Conditional on productivity vector
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{z}_{t}$
and defining the vector
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{a}_{t}$
with entries
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\log
A_{it}$, prices are a function of the network and cost but not government
purchases:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\log\mathbf{p}_{t}=(\mathbf{I}-\mathbf{W}_{t})^{-1}\mathbf{a}_{t}.$

Setting
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\nu=1$
Then (34) and the fact that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}T_{t}=\sum_{i}p_{it}G_{it}$
implies that:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\ell_{t}=\frac{1}{2}+\frac{1}{2}\sum_{i=1}^{n}p_{it}G_{it}$
(40)

and thus:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
p_{it}c_{it}=\beta_{i}[p_{wt}\ell_{t}-T_{t}]=\frac{\beta_{i}}{2}\bigg{(}1-\sum_{j=1}^{n}p_{jt}G_{jt}\bigg{)}.$

Differentiating and combining with the resource constraint and profit
maximization conditions yields:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\frac{d(p_{it}y_{it})}{p_{it}y_{it}}=\sum_{j=1}^{n}w_{jit}\frac{d(p_{jt}y_{jt}}{p_{it}y_{it}}+\frac{dG_{it}}{y_{it}}-\frac{\beta_{i}}{2}\sum_{j=1}^{n}\frac{d(p_{jt}G_{jt})}{p_{it}y_{it}}.$

Since prices are constant (i.e.,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}d(p_{it}y_{it})/p_{it}y_{it}=d\log
y_{it}$), I can write in vector notation:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\mathbf{d}\log\mathbf{y}_{t}=\mathbf{H}_{up,t}\mathbf{dG_{t}},$
(41)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{H}_{up,t}=(\mathbf{I}-\mathbf{W}_{t}^{\top})^{-1}\mathbf{\Lambda}_{t}$
is the upstream propagation Leontief inverse and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{\Lambda}_{t}$
is a scaling matrix with diagonal entries
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(1-\beta_{i}/2)/p_{it}y_{it}$
and off-diagonal entries
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-(\beta_{i}/2)/p_{it}y_{it}$
for row indices
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$.

##  Appendix B Product Varieties in Latent Space

Suppose that each industry (or firm) is associated with a random position
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}z_{it}:=(\cos\theta_{it},\sin\theta_{it})$
on a circular surface on the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}3$-dimensional
hypersphere
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{S}^{p+1}$.
Suppose that the surface represents the space of varieties in production
technology space. I also assume each unit corresponds to a position in latent
product variety space, and that product positions are independent of
technological positions. The stochastic process for positions is the same in
both spaces and depends on changes in the angle
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\theta_{it}$
as follows:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\theta_{it}=\rho\cdot\theta_{i,t-1}+\epsilon_{it},\qquad\epsilon_{it}\sim_{iid}\mathcal{N}(0,\sigma_{\theta}^{2}),$
(42)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\theta_{it}$
is measured in radians. The distance between two points
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
can then be written:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(43)

This implies a correlation structure in the distances between units as
follows:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\text{corr}(\delta_{ijt},\delta_{kmt})=\begin{cases}0&\text{if
}i\notin\\{k,m\\}\cap j\notin\\{k,m\\},\\\ 1&\text{if }i\in\\{k,m\\}\cup
j\in\\{k,m\\},\end{cases},$ (44)

or equivalently that:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\text{cov}(\delta_{ijt},\delta_{kmt})=\begin{cases}0&\text{if
}i\notin\\{k,m\\}\cap j\notin\\{k,m\\},\\\
\sqrt{\text{var}(\delta_{ijt})\text{var}(\delta_{kmt})}&\text{if
}i\in\\{k,m\\}\cup j\in\\{k,m\\},\end{cases},$ (45)

where:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\sigma_{d}^{2}:=\text{var}(\delta_{ijt})$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\text{var}(\theta_{it})+\text{var}(\theta_{jt})-\mathbb{E}[|\theta_{it}-\theta_{jt}|]^{2}=\frac{2\sigma_{\theta}^{2}}{1-\varphi^{2}}-\frac{(4/\pi)\sigma_{\theta}^{2}}{1-\varphi^{2}},$

Define the set of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$’s
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q$-stream
located trade partners by
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A_{iqt}:=\\{k:A_{ki}(\mathcal{G}_{qt})\neq\emptyset\\}$.
Note that in technology space,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q$
refers to upstream propagation. Combining with equation (14) yields:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\text{cov}(s_{iqt},s_{jqt})$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\sum_{k\neq\ell}\sum_{m\neq
p}w_{ikt}w_{i\ell t}w_{jmt}w_{jpt}\cdot\text{cov}(\delta_{k\ell
t},\delta_{mpt})$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\sum_{k,\ell\in
A_{iqt};m,p\in A_{jqt}}w_{ikt}w_{i\ell
t}w_{jmt}w_{jpt}\cdot\sigma_{d}^{2}\cdot\mathbbm{1}\\{(m\in A_{iqt})\cup(p\in
A_{iqt})\\}$

Additionally,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s_{iqt}$
are jointly distributed as a folded truncated normal with variance-covariance
matrix
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{\Sigma}_{qt}=[\text{cov}(s_{iqt},s_{jqt})]_{ij}$
and mean vector
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\boldsymbol{\mu}_{qt}=[\mu_{iqt}]_{i}$
with entries:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\mu_{iqt}:=\mathbb{E}[s_{iqt}]$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=-\sigma_{d}\sqrt{8/\pi}\cdot\sum_{j<k;j,k\in
A_{iqt}}w_{ijt}w_{ikt}.$

When
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i=j$,
I can further simplify as follows:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\text{var}(s_{iqt})$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=4\sigma_{d}^{2}\cdot\sum_{j<k;j,k\in
A_{iqt}}(w_{ijt}w_{ikt})^{2}.$

Next, using equation (13), I can write the log-odds function as follows:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
l_{iqt}:=\log\bigg{(}\frac{p_{iqt}}{1-p_{iqt}}\bigg{)}$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=k\bigg{(}s_{iqt}-\frac{1}{n}\sum_{i=1}^{n}s_{iqt}\bigg{)}=k\bigg{(}\hat{e}_{in}^{\top}-\frac{1}{n}\iota_{n}^{\top}\bigg{)}\mathbf{s}_{qt},$

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{e}_{in}$
is the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$th
column of an
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n\times
n$ identity matrix and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\iota_{n}$
is an
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n\times
1$ vector of ones. Equivalently, the vector
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{l}_{qt}:=(l_{1qt},...,l_{nqt})^{\top}$
can be written:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\mathbf{l}_{qt}$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=k\bigg{(}\mathbf{I}_{n}-\frac{\iota_{n}\iota_{n}^{\top}}{n}\bigg{)}\mathbf{s}_{qt}\sim\mathcal{N}\big{(}\mathbf{B}_{k,n}\boldsymbol{\mu}_{qt},\mathbf{B}_{k,n}^{\top}\mathbf{\Sigma}_{qt}\mathbf{B}_{k,n}\big{)},$

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{B}_{k,n}:=k(\mathbf{I}_{n}-\iota_{n}\iota_{n}^{\top}/n)$.
Notice that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p_{iqt}=F(l_{iqt})$
where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}F(x)=(1+\exp(-x))^{-1}$
and thus the vector
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{p}_{qt}$
has a logistic normal distribution and thus no closed form representation for
its mean vector and variance-covariance matrix. However, I can write
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\text{median}(\mathbf{p}_{qt})=F(\mathbf{B}_{k,n}\boldsymbol{\mu}_{qt})$.

##  Appendix C Proofs

###  C.1 Proof of Proposition 2.3

I begin by characterizing the family of matrices
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}\in
M_{n}$ that are consistent with Assumptions 2.1 and 2.2 with a sequence of if
and only if relationships. First define
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{\Sigma}_{y}:=\mathbb{V}[\mathbf{y}]$
to be the variance-covariance matrix of sectoral production, and notice that
Assumption 2.2 implies that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{\Sigma}_{y}=(\mathbf{I}-\mathbf{W})^{-1}\mathbf{D}(\mathbf{I}-\mathbf{W}^{\top})^{-1}$
is non-negative. This is because
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{\Sigma}_{y}$
is the product of non-negative matrices by definition since
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\mathbf{I}-\mathbf{W})^{-1}=\mathbf{I}+\mathbf{W}+\mathbf{W}^{2}+\mathbf{W}^{3}+...\geq
0$ and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{D}\geq
0$. The former decomposition is only possible when
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\rho(\mathbf{W})\leq
1$. Rearranging terms, I obtain:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\mathbf{D}$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\mathbb{V}[(\mathbf{I}-\mathbf{W})\mathbf{y}]=(\mathbf{I}-\mathbf{W})\mathbf{\Sigma}_{y}(\mathbf{I}-\mathbf{W}^{\top}).$

For ease of notation, let
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}h_{ij}:=\mathbbm{1}_{\\{i=j\\}}-w_{ij}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\rho_{ij}\sigma_{i}\sigma_{j}$
to be the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(i,j)$
entries of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{H}:=\mathbf{I}-\mathbf{W}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{\Sigma}_{y}$,
respectively. Then the matrix
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{Q}_{n}:=\mathbf{H\Sigma
H}^{\top}=[q_{ij}]$ is diagonal with entries:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
q_{ij}=\sum_{k=1}^{n}\sum_{m=1}^{n}h_{ik}h_{jm}\rho_{km}\sigma_{k}\sigma_{m}$

Notice that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{Q}_{n}$
is symmetric and since it is also diagonal, then the following must hold:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\text{det}\big{(}\mathbf{Q}_{n}\big{)}=\prod_{i=1}^{n}\bigg{(}\sum_{k=1}^{n}\sum_{m=1}^{n}h_{ik}h_{im}\rho_{km}\sigma_{k}\sigma_{m}\bigg{)}.$

Additionally,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{P}\mathbf{Q}_{n}\mathbf{P}^{\top}$
is also diagonal for any permutation matrix
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{P}\in
M_{n}$ so I can exchange the ordering of units without loss of generality.
Consider the Laplace expansion of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{Q}_{n}$
by minors along row
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$.
Set
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i=1$
arbitrarily and notice that:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\text{det}\big{(}\mathbf{Q}_{n}\big{)}=q_{11}\cdot\text{det}\big{(}\mathbf{Q}_{n}[1,1]\big{)}+\sum_{k=2}^{n}(-1)^{1+k}q_{1k}\cdot\text{det}\big{(}\mathbf{Q}_{n}[1,k]\big{)},$

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{Q}_{n}[i,j]$
is a size
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n-1$
sub-matrix of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{Q}_{n}$
obtained by deleting row
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
and column
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$.
Since the principal sub-matrix
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{Q}_{n}[1,1]$
is also diagonal, its determinant is
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\prod_{i=2}^{n}q_{ii}$
and thus the second term in the above expression must be zero. This implies
that the determinant for any size
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}2$
principal sub-matrix of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{Q}_{n}$
with size
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n-2$
index set
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha$
and entries
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{ij}$
must satisfy:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\text{det}\big{(}\mathbf{Q}_{n}[\alpha]\big{)}=a_{11}a_{22}-a_{12}a_{21}=a_{11}a_{22}$

Note that the indices (1,2) refer without loss of generality to any arbitrary
pair of sectors in the set
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\{1,...,n\\}$,
and all of the following results must hold for all
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n\text{C}2$
pairwise combinations of units. Substitute terms and notice that all principal
sub-matrices of a symmetric matrix must also be symmetric (i.e.,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{12}=a_{21}$)
to obtain the equivalent restriction:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
0=a_{12}$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=w_{12}\sigma_{2}^{2}-\rho_{12}\sigma_{1}\sigma_{2}(1+w_{21}w_{12})+w_{21}\sigma_{1}^{2}$

Or identically:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
w_{21}\mathbb{V}[y_{1}]+w_{12}\mathbb{V}[y_{2}]-\text{cov}(y_{1},y_{2})(1+w_{12}w_{21})=0,$
(46)

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{V}[y_{i}]$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\text{cov}(y_{i},y_{j})$
are the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(i,i)$th
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(i,j)$th
entries of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{\Sigma_{y}}$,
respectively. I consider two different cases.

##### Case 1.

At least one of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{21}$
or
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{12}$
is zero. If both are zero, then (46) holds trivially. However, if
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{ij}=0$
for all
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i\neq
j$, then
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{W}$
no longer has full rank. If only one is zero (e.g.,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{12}=0$),
this implies that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\text{cov}(y_{1},y_{2})=w_{21}\mathbb{V}[y_{1}]$
which is consistent with (1).

##### Case 2.

Both
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{21}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{12}$
are positive. Redefine
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tilde{y}_{1}:=y_{1}/\sqrt{w_{21}}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tilde{y}_{2}:=y_{2}/\sqrt{w_{12}}$
and apply the Cauchy-Schwarz Inequality:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\frac{(\mathbb{V}[\tilde{y}_{1}]+\mathbb{V}[\tilde{y}_{2}])^{2}}{(w_{21}w_{12})\cdot(1+w_{21}w_{12})^{2}}=\text{cov}(\tilde{y}_{1},\tilde{y}_{2})^{2}\leq\mathbb{V}(\tilde{y}_{1})\mathbb{V}(\tilde{y}_{2})$
(47)

For now write
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}c:=w_{21}w_{12}(1+w_{21}w_{12})^{2}$:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\mathbb{V}[\tilde{y}_{1}]^{2}+\mathbb{V}[\tilde{y}_{2}]^{2}+2\mathbb{V}[\tilde{y}_{1}]\mathbb{V}[\tilde{y}_{2}]\leq
c\mathbb{V}(\tilde{y}_{1})\mathbb{V}(\tilde{y}_{2})$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\mathbb{V}[\tilde{y}_{1}]^{2}+\mathbb{V}[\tilde{y}_{2}]^{2}+(2-c)\mathbb{V}[\tilde{y}_{1}]\mathbb{V}[\tilde{y}_{2}]\leq
0$

The left hand side is a quadratic equation in
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{R}_{+}^{2}$
of the variables
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{V}[\tilde{y}_{1}]$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{V}[\tilde{y}_{2}]$.
Suppose without loss of generality that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{V}[\tilde{y}_{1}]=\mathbb{V}[\tilde{y}_{2}]>0$,
then
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}c$
must satisfy:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle(4-c)\mathbb{V}[\tilde{y}]^{2}\leq
0,$

which implies that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}c\geq
4$, or equivalently:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
w_{21}w_{12}(1+w_{12}w_{21})^{2}\geq 4$

Recall that since
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}$
is a non-negative matrix, any pair of weights which satisfy this inequality
must have
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{12}w_{21}\geq
1$. Note that due to (47), this restriction is necessary but not sufficient.
Moreover, since the spectral radius
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\rho(\mathbf{W})\leq
1$, there must be at least one row or column bounded above by one (see e.g.,
Theorem 8.1.22 in Horn and Johnson (2013)).

###  C.2 Proof of Proposition 4.6

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
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$,
equilibrium consumption expenditure is given by
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C_{t}=\sum_{i=1}^{n}p_{it}c_{it}=\prod_{i=1}^{n}c_{it}^{1/n}$.
Given the price normalization in Section A.4.4. from Appendix A, I can write:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\Delta
c_{t+1}:=\log(C_{t+1}/C_{t})$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\log\bigg{(}\prod_{i=1}^{n}\bigg{(}\frac{c_{i,t+1}}{c_{it}}\bigg{)}^{1/n}\bigg{)}$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\log(c_{i,t+1}/c_{it})$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\log(y_{i,t+1}/y_{it})$

Plugging in the reduced form from (8) yields:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\Delta
c_{t+1}$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\big{(}\Delta
z_{i,t+1}+\Delta g_{i,t+1}\big{)}$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\gamma_{u}\cdot\frac{1}{n}\sum_{i=1}^{n}a_{t+1}+\gamma_{d}\cdot\frac{1}{n}\sum_{i=1}^{n}g_{t+1}-\beta_{u}\cdot\frac{1}{n}\sum_{i=1}^{n}\varepsilon_{iu,t+1}-\beta_{d}\cdot\frac{1}{n}\sum_{i=1}^{n}\varepsilon_{id,t+1}$

###  C.3 Proof of Proposition 4.8

The proof of Proposition 4.7 established the asymptotic normality of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W_{nqt}$
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
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$.
I omit subscripts and write:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\sigma_{n}^{2}=\text{var}(W_{n})$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\frac{1}{n^{2}}\sum_{i=1}^{n}\text{var}[\varepsilon_{i}]+\frac{1}{n^{2}}\sum_{i\neq
j}\text{cov}(\varepsilon_{i},\varepsilon_{j})$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\frac{1}{n^{2}}\sum_{i=1}^{n}p_{i}(1-p_{i})+\frac{1}{n^{2}}\sum_{i\neq
j}\text{cov}(\varepsilon_{i},\varepsilon_{j})\cdot\mathbbm{1}\\{\mathbbm{1}\\{i,j\in
A_{i}(\mathcal{G}_{nq})\cap A_{j}(\mathcal{G}_{nq})\\}\\}$

When
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n>1$.
This quantity can be bounded above by:

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

where the final equality follows from Assumption 4.5. Moreover, for
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k>0$
Chebyshev’s Inequality yields:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\text{Pr}\big{(}|W_{n}-\mathbb{E}[W_{n}]|\geq
2k\bar{M_{n}}/n\big{)}$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\leq\text{Pr}\big{(}|W_{n}-\mathbb{E}[W_{n}]|\geq
k\sigma_{n}\big{)}\leq\frac{1}{k^{2}}$

###  C.4 Proof of Proposition 4.7

This proof is an application of Theorem 2 from Janson (1988). Fix
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q$
and omit the time-subscript without loss of generality. Since
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varepsilon_{iq}\sim\text{Bernoulli}(p_{iq})$,
the mean and variance of firm cash flow shocks conditional on
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p_{iq}$
can be written:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\mathbb{E}[\varepsilon_{iq}]=p_{iq},\quad\text{var}(\varepsilon_{iq})=p_{iq}(1-p_{iq}),$

and the covariance between firm shocks can be written:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\text{cov}(\varepsilon_{iq},\varepsilon_{jq})$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\mathbb{E}[\varepsilon_{iq}\varepsilon_{jq}]-\mathbb{E}[\varepsilon_{iq}]\mathbb{E}[\varepsilon_{jq}]=\underbrace{\text{Pr}(\varepsilon_{iq}=1,\varepsilon_{jq}=1)}_{=:p_{ijq}}-p_{iq}p_{jq}>0.$

Therefore, the mean and variance of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W_{nq}=\frac{1}{n}\sum_{i=1}^{n}\varepsilon_{iq}$
can be written:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\mu_{nq}:=\mathbb{E}[W_{nq}]$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}[\varepsilon_{iq}]=\frac{1}{n}\sum_{i=1}^{n}p_{iq}$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\sigma_{nq}^{2}:=\text{var}\big{[}W_{nq}\big{]}$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\frac{1}{n^{2}}\sum_{i=1}^{n}\text{var}[\varepsilon_{iq}]+\frac{1}{n^{2}}\sum_{i\neq
j}\text{cov}(\varepsilon_{iq},\varepsilon_{jq})$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\geq\frac{1}{n^{2}}\sum_{i=1}^{n}\text{var}[\varepsilon_{iq}],$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\frac{1}{n^{2}}\sum_{i=1}^{n}p_{iq}(1-p_{iq})$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\geq\frac{1}{n}\cdot\min_{i}p_{iq}$

Let
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{D}_{nq}$
denote the maximal number of edges incident to a single vertex in graph
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{G}_{nq}$.
Theorem 2 from Janson (1988) requires that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{X}_{n,m}=o(1)$
for some integer
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m$,
where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{X}_{n,m}$
is given by:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\mathcal{X}_{n,m}$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle:=\bigg{(}\frac{n}{\bar{D}_{nq}}\bigg{)}^{1/m}\cdot\frac{\bar{D}_{nq}}{n\sigma_{nq}}=\frac{\bar{D}_{nq}^{1-1/m}}{n}\cdot\frac{1}{n\sigma_{nq}}\leq\frac{\bar{D}_{nq}^{1-1/m}}{n}.$

Assumption 4.4 implies:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\mathcal{X}_{n,2}=o(1),$

and therefore:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle(W_{nq}-\mu_{nq})/\sigma_{nq}\xrightarrow{d}\mathcal{N}(0,1)$

Combine this result with the assumptions that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{t}\sim\mathcal{N}(0,\sigma_{a}^{2})$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}g_{t}\sim\mathcal{N}(0,\sigma_{g}^{2})$
to get the final distribution of consumption growth.

##  Appendix D Upstream and Downstream Propagation Matrices

###  D.1 Construction from BEA Data

In this section, I discuss a simple method for constructing from the data the
upstream and downstream propagation matrices developed in the previous
section.323232Similar procedures are discussed in Acemoglu et al. (2016),
Ozdagli and Weber (2017), and Gofman et al. (2020). These matrices capture the
strength of a connection between an industry and its customer or supplier
industries. Shocks transmitted from customer
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
to supplier
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
(supplier
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
to customer
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$)
should depend on the strength of the connection in the upstream (downstream)
direction. I construct these matrices directly from the BEA make and use
tables described in Horowitz, Planting, et al. (2006). Consider again an
economy with
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n$
industries.

##### The Make Table

I extract from the BEA make table an
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n\times
n$ industry-by-commodity matrix 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(MAKE)_{ij}=OUT_{i\to
j}\equiv\text{dollar value of commodity
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
produced by industry
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$}$

Note that the BEA makes a slight distinction between commodities and
industries, since in principle an industry might produce another industry’s
commodity as a by-product of its own output. Next, I denote the total
production of commodity
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
by
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}OUT_{j}:=\sum_{i=1}^{n}OUT_{i\to
j}$. Using the notation in Horowitz, Planting, et al. (2006), I define the
_market share matrix_ with the following entries:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle(MKTSHARE)_{ij}=\frac{(MAKE)_{ij}}{OUT_{j}}=\frac{OUT_{i\to
j}}{OUT_{j}}$

Here, 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 describes the share of industry
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
in the total production of commodity
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$.
Equivalently, I can write
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}MKTSHARE=MAKE\odot(\boldsymbol{\iota}_{n}\cdot\mathbf{S})$
where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\odot$
denotes the Hadamard (elementwise) product and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}S=(OUT_{1}^{-1},...,OUT_{n}^{-1})$
is the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}1\times
n$ scaling vector.

##### The Use Table

Similarly, I extract from the BEA use table an
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n\times
n$ commodity-by-industry matrix 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(USE)_{ij}=IN_{i\to
j}\equiv\text{dollar value of commodity
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
used as input by industry
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$}$

Define the total output of industry
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
by
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>calculate total industry output from the BEA use table as the sum of total
intermediates, scrap, and value added. An industry’s value added is defined by
the BEA as the “market value it adds in production, or the difference between
the price at which it sells its products and the cost of the inputs it
purchases from other industries”.  Then I construct the _input requirement
matrix_ by rescaling the value of an industry’s inputs by the industry’s total
value as measured by output. The entries of this matrix are given by:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle(INPUTREQ)_{ij}=\frac{(USE)_{ij}}{y_{j}}=\frac{IN_{i\to
j}}{y_{j}}.$ (48)

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 above matrix describes the importance of industry
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$’s
inputs from industry
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
relative to
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$’s
total size.

##### Scrap Adjustment

The BEA input-output tables include scrap as a commodity which includes any
by-products of production with zero market demand. I therefore redefine the
total output of an industry as the non-scrap output. Mathematically, this
adjustment is implemented using the non-scrap ratio, calculated as follows:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\theta_{i}=\frac{y_{i}-scrap_{i}}{y_{i}},$

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}scrap_{i}$
denotes the total scrap produced by industry
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$.
I then write the entries of the scrap-adjusted market share matrix as follows:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\tilde{(MKTSHARE)}_{ij}=\frac{OUT_{i\to
j}}{OUT_{j}}\cdot\frac{1}{\theta_{j}}$

##### Direct Requirements

From the market share and input requirements matrices, I then construct the
industry-by-industry direct requirement table, denoted by
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}$:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\underbrace{\mathbf{W}}_{\text{industry}\times\text{industry}}=\underbrace{\tilde{MKTSHARE}}_{\text{industry}\times\text{commodity}}\cdot\underbrace{(INPUTREQ)}_{\text{commodity}\times\text{industry}}.$

To understand this construction, consider the entries of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}$:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle(\mathbf{W})_{ij}=\frac{1}{y_{j}}\sum_{k=1}^{n}\underbrace{\frac{OUT_{i\to
k}}{OUT_{k}\cdot\theta_{k}}}_{\text{share of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
in producing
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k$}}\cdot\underbrace{IN_{k\to
j}}_{\text{use of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k$
by
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>
Here, 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 captures industry
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$’s
total dependence on inputs from industry
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
relative to it’s total output
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y_{j}$.
The key assumption here is as follows. If industry
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
purchases
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}X$
of commodity
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}K$,
then the proportion of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}K$
coming from industry
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
is equal to
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$’s
adjusted market share of production of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}K$.
This is a reasonable assumption on average. Given this assumption, the
following identity holds:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle(\mathbf{W})_{ij}\equiv\frac{SALES_{i\to
j}}{SALES_{j}}\iff(\mathbf{W}^{\top})_{ij}=\frac{SALES_{j\to i}}{SALES_{i}}$

The downstream weighting matrix is thus defined:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\mathbf{A}\equiv\mathbf{W}_{down}:=\mathbf{W}^{\top},$

whose
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(i,j)$
entry in the above matrix represents the dependence 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 input from industry
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
(i.e., shocks to supplier
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
propagate downstream to customer
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
according to the corresponding downstream weight). The sum of row
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
in this matrix is equal to
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x_{i}/y_{i}$
where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x_{i}$
is industry
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$’s
total input purchases relative to its size (normalized by
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$’s
total output). For the upstream weighting matrix, I require the intermediate
rescaling matrix
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{R}$
with entries
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\mathbf{R})_{ij}=y_{j}/y_{i}$.
The upstream weighting matrix is thus defined:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\hat{\mathbf{A}}^{\top}\equiv\mathbf{W}_{up}:=\mathbf{W}\odot\mathbf{R},$

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\odot$
is the Hadamard (elementwise) product. 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 in the above matrix represents the dependence 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 sales to industry
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
(i.e., shocks to customer
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}j$
propagate upstream to supplier
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$
according to the corresponding upstream weight)

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle(\mathbf{W}\odot\mathbf{R})_{ij}=\frac{SALES_{i\to
j}}{SALES_{i}}.$

###  D.2 Descriptive Statistics

(a)

(b)

Figure 2: Weighted In-Degree Distribution This figure visualizes the empirical
distribution of weighted in and out-degrees across 66 non-government
industries as defined in the Bureau of Economic Analysis (BEA) Make and Use
Tables. The left panel shows the Gaussian kernel density estimate of the
distribution, and the right panel shows the empirical counter-cumulative
distribution function.

(a)

(b)

Figure 3: Weighted Out-Degree Distribution This figure visualizes the
empirical distribution of weighted in and out-degrees across 66 non-government
industries as defined in the Bureau of Economic Analysis (BEA) Make and Use
Tables. The left panel shows the Gaussian kernel density estimate of the
distribution, and the right panel shows the empirical counter-cumulative
distribution function.

(a)

(b)

Figure 4: Distribution of Downstream Centrality This figure visualizes the
empirical distribution of weighted log eigenvector centrality of the upstream
propagation matrix across 66 non-government industries as defined in the
Bureau of Economic Analysis (BEA) Make and Use Tables. The left panel shows
the Gaussian kernel density estimate of the distribution, and the right panel
shows the empirical counter-cumulative distribution function.

(a)

(b)

Figure 5: Distribution of Upstream Centrality This figure visualizes the empirical distribution of weighted log eigenvector centrality of the upstream propagation matrix across 66 non-government industries as defined in the Bureau of Economic Analysis (BEA) Make and Use Tables. The left panel shows the Gaussian kernel density estimate of the distribution, and the right panel shows the empirical counter-cumulative distribution function. Table 6: Correlation of Input-Output Shares Entry | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{d,ij,t-1}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{u,ij,t-1}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}h_{d,ij,t-1}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}h_{u,ij,t-1}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{d,ij,t}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{u,ij,t}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}h_{d,ij,t}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}h_{u,ij,t}$
---|---|---|---|---|---|---|---|---
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{d,ij,t-1}$ | 1 | 0.086 | 0.970 | 0.096 | 0.991 | 0.086 | 0.961 | 0.096
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{u,ij,t-1}$ |  | 1 | 0.094 | 0.967 | 0.084 | 0.994 | 0.093 | 0.961
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}h_{d,ij,t-1}$ |  |  | 1 | 0.110 | 0.962 | 0.095 | 0.990 | 0.111
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}h_{u,ij,t-1}$ |  |  |  | 1 | 0.095 | 0.962 | 0.109 | 0.994
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{d,ij,t}$ |  |  |  |  | 1 | 0.086 | 0.970 | 0.096
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{u,ij,t}$ |  |  |  |  |  | 1 | 0.095 | 0.967
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}h_{d,ij,t}$ |  |  |  |  |  |  | 1 | 0.111
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}h_{u,ij,t}$ |  |  |  |  |  |  |  | 1
Note: This table report correlations between entries in the upstream and
downstream propagation matrices
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W_{qt}$
and their Leontief inverses
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H_{qt}$.
I construct annual matrices from the BEA Input-Output Accounts for 66 non-
government industries between 1997-2020.

(a)

(b)

Figure 6: Visualizing Weighted Degree by Industry The left panel shows the
average weighted in-degree by sector as constructed from the downstream
propagation network
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\textbf{W}_{down}$.
The right panel shows the average weighted in-degree by sector as constructed
from the upstream propagation network
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\textbf{W}_{up}$.
The sectors are numbered identically in both panels, according to the BEA 66
non-government industry classification.

##  Appendix E Determinants of Network Variance

Table 7: Correlation in Predictors of Realized Variance Predictor | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{vol}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C_{u}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C_{d}$ | Between (d) | Between (s) | Size | Across (s) | Across (d) | VP | mkt ivol | mkt vol
---|---|---|---|---|---|---|---|---|---|---|---
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{vol}$ | 1 | -0.329 | -0.130 | -0.308 | 0.072 | 0.285 | 0.061 | -0.128 | -0.225 | 0.287 | 0.406
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C_{u}$ |  | 1 | 0.284 | 0.356 | -0.251 | 0.131 | -0.441 | 0.319 | 0.423 | -0.383 | -0.396
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C_{d}$ |  |  | 1 | 0.588 | -0.018 | -0.128 | -0.116 | 0.288 | 0.289 | -0.003 | -0.022
Between (d) |  |  |  | 1 | -0.205 | -0.113 | -0.156 | 0.704 | 0.708 | -0.270 | -0.311
Between (s) |  |  |  |  | 1 | -0.133 | 0.056 | -0.163 | -0.313 | 0.073 | 0.054
Size |  |  |  |  |  | 1 | -0.230 | 0.018 | -0.111 | -0.386 | -0.275
Across (s) |  |  |  |  |  |  | 1 | -0.088 | -0.159 | 0.205 | 0.255
Across (d) |  |  |  |  |  |  |  | 1 | 0.627 | -0.224 | -0.249
VP |  |  |  |  |  |  |  |  | 1 | -0.279 | -0.322
mkt ivol |  |  |  |  |  |  |  |  |  | 1 | 0.965
mkt vol |  |  |  |  |  |  |  |  |  |  | 1
Notes: This table reports the average correlation between predictors in the
panel regression from Table 1.
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C_{u}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C_{d}$
refer to upstream and downstream centrality, the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(d)$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(s)$
labels denote demand and supply-side concentration, size is average output,
and VP is vertical position. Idiosyncratic volatility is calculated from FF3
residual returns.

Table 8: Network Determinants of Industry Variance (TFP growth shocks) Panel
A: Market Return Variance
---
| (1) | (2) | (3) | (4) | (5) | (6)
Self-origin (demand) | 0.365*** | 0.246*** |  |  | 0.217*** | 0.119***
| (0.038) | (0.039) |  |  | (0.021) | (0.022)
Across (demand) | 0.100*** | 0.084*** |  |  | 0.085*** | 0.119***
| (0.020) | (0.018) |  |  | (0.020) | (0.019)
Between (demand) | 0.043** | 0.010 |  |  | 0.091*** | -0.076***
| (0.018) | (0.022) |  |  | (0.019) | (0.020)
Self-origin (supply) |  |  | 0.039* | 0.149*** | 0.217*** | 0.119***
|  |  | (0.029) | (0.038) | (0.021) | (0.022)
Across (supply) |  |  | 0.119*** | 0.147*** | 0.270*** | 0.170***
|  |  | (0.021) | (0.026) | (0.027) | (0.037)
Between (supply) |  |  | 0.085*** | 0.042** | 0.086*** | 0.179***
|  |  | (0.019) | (0.019) | (0.025) | (0.034)
Size |  | -0.202*** |  | -0.325*** |  | -0.183***
|  | (0.030) |  | (0.023) |  | (0.034)
Upstream centrality |  | -0.145** |  | 0.129** |  | -0.238***
|  | (0.062) |  | (0.051) |  | (0.066)
Downstream centrality |  | 1.956*** |  | 0.296** |  | 2.137***
|  | (0.145) |  | (0.125) |  | (0.165)
Durability |  | 1.088*** |  | 0.389*** |  | 1.164***
|  | (0.153) |  | (0.108) |  | (0.156)
Vertical position |  | 3.890*** |  | 0.243*** |  | 4.063***
|  | (0.116) |  | (0.083) |  | (0.134)
Constant | -2.123 | -0.634 | -2.821 | -0.096 | -1.473 | -0.693
Obs | 2359 | 1626 | 2861 | 2138 | 2114 | 1471
Adj $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R^{2}$ | 0.225 | 0.607 | 0.083 | 0.191 | 0.288 | 0.633
Panel B: Cash Flow Variance
| (1) | (2) | (3) | (4) | (5) | (6)
Self-origin (demand) | 0.363*** | 0.009 |  |  | 0.190*** | 0.017
| (0.072) | (0.093) |  |  | (0.037) | (0.049)
Across (demand) | 0.038 | -0.050* |  |  | 0.011 | 0.099**
| (0.039) | (0.039) |  |  | (0.040) | (0.042)
Between (demand) | 0.136*** | 0.128*** |  |  | 0.156*** | 0.184***
| (0.037) | (0.042) |  |  | (0.039) | (0.047)
Self-origin (supply) |  |  | 0.026 | 0.286*** | 0.190*** | 0.017
|  |  | (0.065) | (0.076) | (0.037) | (0.049)
Across (supply) |  |  | 0.250*** | 0.384*** | 0.309*** | 0.107*
|  |  | (0.051) | (0.062) | (0.061) | (0.091)
Between (supply) |  |  | 0.093** | 0.169*** | 0.202*** | 0.162**
|  |  | (0.044) | (0.055) | (0.057) | (0.086)
Size |  | -0.161** |  | -0.333*** |  | -0.155**
|  | (0.064) |  | (0.052) |  | (0.075)
Upstream centrality |  | -0.069 |  | 0.562*** |  | -0.100
|  | (0.150) |  | (0.132) |  | (0.162)
Downstream centrality |  | 2.193*** |  | -0.070 |  | 2.315***
|  | (0.343) |  | (0.305) |  | (0.379)
Durability |  | 2.204*** |  | 1.172*** |  | 2.307***
|  | (0.272) |  | (0.182) |  | (0.285)
Vertical position |  | 3.502*** |  | 1.142* |  | 3.590***
|  | (0.268) |  | (0.181) |  | (0.320)
Constant | -4.201 | -0.711 | -5.246 | 0.023 | -4.123 | -0.75
Obs | 2359 | 1626 | 2861 | 2138 | 2114 | 1471
Adj $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R^{2}$ | 0.074 | 0.248 | 0.017 | 0.08 | 0.087 | 0.271
Notes: This table reports panel regressions of realized industry variance on a
variety of characteristics including the average log variance of supply and
demand shocks, log concentration across trade partners, log concentration
between trade partners, log total employment (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. I
calculate network components as the average value of 1000 bootstrap samples
that randomly drop 10% of estimated pairwise non-zero correlations. 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 sales data from Compustat. Concentration
between and across trade partners are calculated as in 4 and 5, where I
calculate the variance-covariance matrix of supply and demand shocks directly
from four-factor TFP growth in the NBER-CES Database (Becker et al., 2016).
Following Ahern (2013), I compute industry centrality as the eigenvector
centrality of upstream and downstream propagation adjacency matrices. I obtain
durability classifications from Gomes et al. (2009) and calculate vertical
position of each industry as in 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%, 5%, and 10% levels, respectively. Standard
error are clustered at the BEA 15 major industry group level. Sample is at an
annual frequency from 1988 to 2017 for 479 BEA manufacturing industries.

Table 9: Network Determinants of Industry Variance (federal procurement
shocks) Panel A: Market Return Variance
---
| (1) | (2) | (3) | (4) | (5) | (6)
Self-origin (demand) | 1.058*** | 0.903*** |  |  | 0.166* | 0.558***
| (0.157) | (0.200) |  |  | (0.111) | (0.121)
Across (demand) | 0.088** | 0.314*** |  |  | 0.062* | 0.356***
| (0.041) | (0.059) |  |  | (0.044) | (0.070)
Between (demand) | 0.149*** | 0.153*** |  |  | 0.437*** | 0.261***
| (0.040) | (0.039) |  |  | (0.055) | (0.077)
Self-origin (supply) |  |  | 0.219*** | 0.363*** | 0.166* | 0.558***
|  |  | (0.044) | (0.056) | (0.111) | (0.121)
Across (supply) |  |  | 0.142*** | 0.094* | 0.072 | 0.088
|  |  | (0.036) | (0.052) | (0.072) | (0.079)
Between (supply) |  |  | 0.150*** | 0.062 | -0.063 | -0.13
|  |  | (0.034) | (0.039) | (0.075) | (0.077)
Size |  | 0.059 |  | -0.242*** |  | -0.033
|  | (0.053) |  | (0.047) |  | (0.053)
Upstream centrality |  | -2.434** |  | -4.885*** |  | -2.140*
|  | (1.053) |  | (1.116) |  | (1.269)
Downstream centrality |  | 10.76*** |  | 13.06*** |  | 10.55***
|  | (2.686) |  | (2.501) |  | (3.043)
Durability |  | 4.128*** |  | 1.991*** |  | 3.884***
|  | (0.511) |  | (0.407) |  | (0.460)
Vertical position |  | 11.74*** |  | 4.334*** |  | 10.66***
|  | (0.915) |  | (0.595) |  | (0.930)
Constant | -6.278 | -3.489 | -5.461 | -4.234 | -6.781 | -3.419
Obs | 839 | 666 | 1003 | 828 | 839 | 666
Adj R2 | 0.595 | 0.76 | 0.071 | 0.331 | 0.666 | 0.762
Panel B: Cash Flow Variance
| (1) | (2) | (3) | (4) | (5) | (6)
Self-origin (demand) | 0.425 | 1.853* |  |  | 0.267 | 1.025**
| (0.421) | (0.951) |  |  | (0.293) | (0.488)
Across (demand) | 0.057 | 0.790*** |  |  | 0.183* | 0.768***
| (0.090) | (0.215) |  |  | (0.121) | (0.207)
Between (demand) | 0.309*** | 0.461*** |  |  | 0.524*** | 0.471**
| (0.091) | (0.121) |  |  | (0.146) | (0.199)
Self-origin (supply) |  |  | 0.310*** | 0.004 | 0.267 | 1.025**
|  |  | (0.109) | (0.185) | (0.293) | (0.488)
Across (supply) |  |  | -0.184* | 0.002 | 0.136 | 0.327*
|  |  | (0.101) | (0.139) | (0.193) | (0.258)
Between (supply) |  |  | 0.235** | 0.028 | -0.247* | -0.268
|  |  | (0.092) | (0.112) | (0.192) | (0.244)
Size |  | 0.183 |  | -0.406*** |  | -0.141
|  | (0.209) |  | (0.122) |  | (0.214)
Upstream centrality |  | 1.430 |  | 3.297* |  | 4.077*
|  | (3.309) |  | (2.975) |  | (3.421)
Downstream centrality |  | 5.582 |  | -5.585 |  | 0.810
|  | (8.923) |  | (6.799) |  | (8.718)
Durability |  | 5.285*** |  | 1.175 |  | 5.536***
|  | (1.493) |  | (0.847) |  | (1.677)
Vertical position |  | 12.20*** |  | 0.948 |  | 12.73***
|  | (3.184) |  | (1.236) |  | (4.096)
Constant | -1.223 | -1.809 | -7.991 | 1.81 | -10.389 | -0.263
Obs | 839 | 666 | 1003 | 828 | 839 | 666
Adj R2 | 0.239 | 0.211 | 0.043 | 0.08 | 0.248 | 0.215
Notes: This table reports panel regressions of realized industry variance on a
variety of characteristics including the average log variance of supply and
demand shocks, log concentration across trade partners, log concentration
between trade partners, log total employment (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. I
calculate network components as the average value of 1000 bootstrap samples
that randomly drop 10% of estimated pairwise non-zero correlations 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 sales data from Compustat. Concentration between and
across trade partners are calculated as in 4 and 5, where I calculate the
variance-covariance matrix of changes in total obligations from newly awarded
federal procurement contracts measured from FPDS. Following Ahern (2013), I
compute industry centrality as the eigenvector centrality of upstream and
downstream propagation adjacency matrices. I obtain durability classifications
from Gomes et al. (2009) and calculate vertical position of each industry as
in 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%, 5%, and 10% levels, respectively. Standard
error are clustered at the BEA 15 major industry level. Sample is at an annual
frequency from 1991 to 2011 for 479 BEA manufacturing industries.

Table 10: Network Determinants of Industry Variance (technological proximity)
Panel A: Market Return Variance
---
| (1) | (2) | (3) | (4) | (5) | (6)
Self-origin (demand) | 0.257*** | 0.352*** |  |  | 0.286*** | 0.308***
| (0.046) | (0.043) |  |  | (0.050) | (0.047)
Across (demand) | 0.102*** | 0.120*** |  |  | 0.124*** | 0.145***
| (0.011) | (0.013) |  |  | (0.011) | (0.013)
Between (demand) | 0.246*** | 0.117*** |  |  | 0.227*** | 0.058***
| (0.013) | (0.018) |  |  | (0.013) | (0.018)
Self-origin (supply) |  |  | 0.290*** | 0.258*** | 0.229*** | 0.419***
|  |  | (0.035) | (0.039) | (0.040) | (0.043)
Across (supply) |  |  | 0.009 | 0.032 | 0.223*** | -0.035
|  |  | (0.027) | (0.036) | (0.031) | (0.042)
Between (supply) |  |  | 0.190*** | 0.168*** | 0.284*** | 0.143**
|  |  | (0.031) | (0.043) | (0.034) | (0.051)
Size |  | -0.343*** |  | -0.312*** |  | -0.334***
|  | (0.031) |  | (0.027) |  | (0.030)
Upstream centrality |  | -0.755*** |  | -0.102* |  | -0.543***
|  | (0.108) |  | (0.087) |  | (0.104)
Downstream centrality |  | 3.933*** |  | 1.052*** |  | 3.632***
|  | (0.279) |  | (0.190) |  | (0.275)
Durability |  | -0.195* |  | -0.177* |  | -0.076
|  | (0.112) |  | (0.098) |  | (0.108)
Vertical position |  | 1.669*** |  | 0.130 |  | 2.160***
|  | (0.152) |  | (0.086) |  | (0.172)
Constant | -7.926 | -1.275 | -3.85 | -0.341 | -8.468 | -1.177
Obs | 1994 | 1467 | 1994 | 1467 | 1994 | 1467
Adj R2 | 0.593 | 0.642 | 0.127 | 0.279 | 0.61 | 0.665
Panel B: Cash Flow Variance
| (1) | (2) | (3) | (4) | (5) | (6)
Self-origin (demand) | 0.391*** | 0.521*** |  |  | 0.473*** | 0.548***
| (0.103) | (0.103) |  |  | (0.102) | (0.103)
Across (demand) | 0.065** | 0.061* |  |  | 0.050* | 0.049*
| (0.028) | (0.033) |  |  | (0.029) | (0.034)
Between (demand) | 0.253*** | 0.139*** |  |  | 0.172** | 0.217*
| (0.030) | (0.045) |  |  | (0.077) | (0.122)
Self-origin (supply) |  |  | -0.019 | -0.245*** | -0.162** | -0.185*
|  |  | (0.076) | (0.087) | (0.079) | (0.097)
Across (supply) |  |  | 0.175*** | 0.138* | 0.072 | 0.160*
|  |  | (0.067) | (0.100) | (0.068) | (0.102)
Between (supply) |  |  | 0.307*** | 0.228** | 0.271*** | 0.179***
|  |  | (0.076) | (0.119) | (0.031) | (0.048)
Size |  | -0.298*** |  | -0.332*** |  | -0.298***
|  | (0.070) |  | (0.066) |  | (0.071)
Upstream centrality |  | 0.113 |  | 0.355* |  | -0.068
|  | (0.267) |  | (0.276) |  | (0.280)
Downstream centrality |  | 3.217*** |  | 0.311 |  | 3.555***
|  | (0.665) |  | (0.599) |  | (0.680)
Durability |  | 0.574*** |  | 0.265* |  | 0.481**
|  | (0.210) |  | (0.211) |  | (0.218)
Vertical position |  | 1.922*** |  | -0.311 |  | 1.493***
|  | (0.323) |  | (0.208) |  | (0.373)
Constant | -10.991 | -1.043 | -5.631 | -0.101 | -11.776 | -1.152
Obs | 1994 | 1467 | 1994 | 1467 | 1994 | 1467
Adj R2 | 0.21 | 0.23 | 0.011 | 0.051 | 0.216 | 0.233
Notes: This table reports panel regressions of realized industry variance on a
variety of characteristics including the average log variance of supply and
demand shocks, log concentration across trade partners, log concentration
between trade partners, log total employment (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. I
calculate network components as the average value of 1000 bootstrap samples
that randomly drop 10% of estimated pairwise non-zero correlations. 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 sales data from Compustat. Concentration
between and across trade partners are calculated as in 4 and 5, where I
calculate the variance-covariance matrix of supply and demand shocks using the
weighted technological similarity scores following Bloom and Shankerman
(2013). Following Ahern (2013), I compute industry centrality as the
eigenvector centrality of upstream and downstream propagation adjacency
matrices. I obtain durability classifications from Gomes et al. (2009) and
calculate vertical position of each industry as in 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%, 5%, and 10% levels, respectively. Standard
errors are clustered at the BEA 15 major industry group level. Sample is at an
annual frequency from 1988 to 2018 for 479 BEA manufacturing industries.

Table 11: Network Determinants of Industry Variance (product similarity) Panel
A: Market Return Variance
---
| (1) | (2) | (3) | (4) | (5) | (6)
Self-origin (demand) | 0.136** | 0.354*** |  |  | 0.303*** | 0.321***
| (0.056) | (0.040) |  |  | (0.046) | (0.041)
Across (demand) | 0.103*** | 0.071*** |  |  | -0.065*** | 0.051***
| (0.013) | (0.011) |  |  | (0.013) | (0.012)
Between (demand) | 0.131*** | 0.081** |  |  | 0.384*** | 0.058***
| (0.013) | (0.012) |  |  | (0.014) | (0.016)
Self-origin (supply) |  |  | 0.096*** | 0.056* | -0.047 | 0.207***
|  |  | (0.027) | (0.033) | (0.033) | (0.034)
Across (supply) |  |  | 0.147*** | 0.137*** | 0.186*** | -0.000
|  |  | (0.015) | (0.020) | (0.020) | (0.021)
Between (supply) |  |  | -0.086*** | -0.106*** | -0.362*** | -0.071***
|  |  | (0.012) | (0.016) | (0.017) | (0.020)
Size |  | -0.361*** |  | -0.257*** |  | -0.310***
|  | (0.025) |  | (0.021) |  | (0.024)
Upstream centrality |  | -0.149*** |  | 0.031 |  | -0.176***
|  | (0.053) |  | (0.049) |  | (0.055)
Downstream centrality |  | 2.660*** |  | 0.811*** |  | 2.891***
|  | (0.174) |  | (0.117) |  | (0.180)
Durability |  | 0.216** |  | 0.265*** |  | 0.286***
|  | (0.102) |  | (0.096) |  | (0.098)
Vertical position |  | 3.414*** |  | 0.368*** |  | 3.336***
|  | (0.086) |  | (0.076) |  | (0.106)
Constant | -5.915 | -0.862 | -3.631 | -0.263 | -8.380 | -0.937
Obs | 3800 | 2430 | 3819 | 2430 | 3800 | 2430
Adj $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R^{2}$ | 0.359 | 0.625 | 0.037 | 0.175 | 0.488 | 0.639
Panel B: Cash Flow Variance
| (1) | (2) | (3) | (4) | (5) | (6)
Self-origin (demand) | 0.357*** | 0.549*** |  |  | 0.544*** | 0.558***
| (0.091) | (0.090) |  |  | (0.091) | (0.091)
Across (demand) | 0.112*** | 0.055** |  |  | -0.059** | 0.011
| (0.023) | (0.027) |  |  | (0.025) | (0.030)
Between (demand) | 0.118*** | 0.066** |  |  | 0.373*** | 0.108***
| (0.023) | (0.030) |  |  | (0.027) | (0.040)
Self-origin (supply) |  |  | 0.030 | -0.026 | -0.141* | 0.084*
|  |  | (0.060) | (0.073) | (0.062) | (0.078)
Across (supply) |  |  | 0.129*** | 0.199*** | 0.170*** | 0.091**
|  |  | (0.029) | (0.039) | (0.031) | (0.043)
Between (supply) |  |  | -0.074*** | -0.156*** | -0.347*** | -0.157***
|  |  | (0.023) | (0.032) | (0.028) | (0.041)
Size |  | -0.313*** |  | -0.242*** |  | -0.273***
|  | (0.050) |  | (0.048) |  | (0.051)
Upstream centrality |  | 0.185* |  | 0.327** |  | 0.090
|  | (0.135) |  | (0.132) |  | (0.138)
Downstream centrality |  | 3.213*** |  | 0.791*** |  | 3.587***
|  | (0.406) |  | (0.300) |  | (0.413)
Durability |  | 1.097*** |  | 1.104*** |  | 1.145***
|  | (0.171) |  | (0.171) |  | (0.170)
Vertical position |  | 3.495*** |  | 0.226 |  | 3.102***
|  | (0.193) |  | (0.164) |  | (0.251)
Constant | -10.060 | -1.041 | -5.694 | -0.256 | -12.580 | -1.162
Obs | 3800 | 2430 | 3819 | 2430 | 3800 | 2430
Adj $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R^{2}$ | 0.15 | 0.235 | 0.005 | 0.047 | 0.205 | 0.242
Notes: This table reports panel regressions of realized industry variance on a
variety of characteristics including the average log variance of supply and
demand shocks, log concentration across trade partners, log concentration
between trade partners, log total employment (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. I
calculate network components as the average value of 1000 bootstrap samples
that randomly drop 10% of estimated pairwise non-zero correlations. 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 sales data from Compustat. Concentration
between and across trade partners are calculated as in 4 and 5, where I
calculate the variance-covariance matrix of supply and demand shocks using the
product similarity distances from Hoberg and Phillips (2016). Following Ahern
(2013), I compute industry centrality as the eigenvector centrality of
upstream and downstream propagation adjacency matrices. I obtain durability
classifications from Gomes et al. (2009) and calculate vertical position of
each industry as in 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%, 5%, and 10% levels, respectively. Standard
error are clustered at the BEA 15 major industry group level. Sample is at an
annual frequency from 1988 to 2018 for 479 BEA manufacturing industries.

##  Appendix F Simulation Evidence

Proposition 2.3 claims that there is no matrix
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{W}$
with entries
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{ij}\in(0,1)$
such that Assumption 2.2 and (1) are satisfied. This implies that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Delta:=||(\mathbf{I}-\mathbf{W})\mathbf{\Sigma}_{y}(\mathbf{I}-\mathbf{W})-\mathbf{\Sigma}_{u}||$
should always be different from zero when
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{\Sigma}_{u}$
is diagonal. I test
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H_{0}:\Delta=0$
numerically as follows:

  1. 1.

Fix dimension
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n$,
number of iterations
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}S$,
and consider the following constrained optimization problem:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
f_{min}(\mathbf{\Sigma}_{u})$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle=\min_{w_{ij,i\neq
j}\in(0,1)}f(\mathbf{W};\mathbf{\Sigma}_{u})=\min_{w_{ij,i\neq
j}\in(0,1)}\bigg{|}\bigg{|}\mathbf{\Sigma}_{u}-(\mathbf{I}_{n}-\mathbf{W})^{-1}\mathbf{\Sigma}_{u}(\mathbf{I}_{n}-\mathbf{W}^{\top})^{-1}\bigg{|}\bigg{|}_{F},$
(49)

  2. 2.

Draw two random samples
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}S_{u}^{d}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}S_{u}^{*}$
for the variance-covariance matrix of residuals
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{\Sigma}_{u}$
such that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}S_{u}^{d}$
is diagonal and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}S_{u}^{*}$
is not. I sample the non-diagonal matrix as follows:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
S_{u}^{*}(s)$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\sim_{iid}\mathcal{W}^{-1}(\mathbf{\Psi}_{u},\nu),$

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{W}^{-1}(\mathbf{\Psi},\nu)$
denotes an Inverse-Wishart random variable with scale
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{\Psi}$
and degrees of freedom
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\nu>n-1$.
On the other hand, the diagonal matrix is given by: