Industrial Connectedness and Business Cycle Comovements

ECONOMIC RESEARCH

FEDERAL RESERVE BANK OF ST. LOUIS

WORKING PAPER SERIES

Industrial Connectedness and Business Cycle Comovements

Authors	Amy Guisinger, Michael T. Owyang, and Daniel Soques
Working Paper Number	2020-052A
Creation Date	December 2020
Citable Link	https://doi.org/10.20955/wp.2020.052
	Guisinger, A., Owyang, M.T., Soques, D., 2020; Industrial Connectedness and
Suggested Citation	Business Cycle Comovements, Federal Reserve Bank of St. Louis Working Paper
	2020-052.URL https://doi.org/10.20955/wp.2020.052

Federal Reserve Bank of St. Louis, Research Division, P.O. Box 442, St. Louis, MO 63166

The views expressed in this paper are those of the author(s) and do not necessarily reflect the views of the Federal Reserve System, the Board of Governors, or the regional Federal Reserve Banks. Federal Reserve Bank of St. Louis Working Papers are preliminary materials circulated to stimulate discussion and critical comment.

Industrial Connectedness❸ and✯Business❹ Cycle➜

Comovements

Amy Y. Guisinger, Michael T. Owyang, Daniel Soques

December 30, 2020

ABSTRACT

The effect of economic shocks on business cycles fluctuations may vary across industries. For example, shocks that originate in a single industry may propagate elsewhere, either up or down stream in the production chain. Thus, industries that are more connected may be more vulnerable to industry-specific economic shocks. However, any model of industrial connectedness must account for the fact that much of the inter-industry correlation will be driven by national shocks. In light of this, we develop a panel Markov-switching model for industry-level data that incorporates a number of features relevant for sub-national analysis. First, we model industry-level trends to differentiate between cyclical downturns and secular decline in an industry. Second, we incorporate a national-level business cycle that industries may or may not attach to. Third, we model comovement off of the national-level cycle as factors that affect clusters of industries. We find that there are industry groupings that comove because their production networks are intra- sectoral and industry groupings that lack inter or intra-sectoral classification, but most industries move together.

Keywords: cluster analysis, Markov-switching

• JEL Codes: C32; E32

The authors benefited from conversations with Julieta Caunedo, Jeremy Piger, and Garey Ramey, comments from seminar participants at the University of Oregon, and conference participants at 2011 SNDE, 2016 SNDE, 2016 Midwest Macroeconomics Meeting, and 2020 CFE. Kristie M. Engemann, Charles S. Gascon, Kate Vermann, Hannah G. Shell, and Julie K. Bennett provided research assistance. The views expressed here are the authors' alone and do not reflect the opinions of the Federal Reserve Bank❸of St. Louis or the Federal Reserve System.

❹Department of Economics. Lafayette College.

Corresponding author. Research Division. Federal Reserve Bank of St. Louis. michael➜ [e-mail]

Department of Economics and Finance. University of North Carolina Wilmington.

1 INTRODUCTION

Recent study of the aggregate business cycle is increasingly focusing on the interaction between its disaggregate components-state (Owyang et al., 2005; Leiva-Leon,2017), regional (Hamilton and Owyang, 2012), and even city (Owyang et al., 2008). A portion of this literature has investigated the interaction and comovements of industries (Murphy et al., 1989; Cooper and Haltiwanger, 1990).1 Previous studies have found that there is comovement both within and across sectors (Christiano et al., 1998; Hornstein, 2000) that is linked to aggregate (Chang and Hwang, 2015) and state (Carlino and DeFina, 2004) business cycles.2

While much of the literature debates over the importance of sectoral versus aggregate shocks in explaining the volatility across the business cycle, there is seemingly little agreement on what constitutes a sector.3 For example, Garin et al. (2018) estimates a principal component from the 12 sectors that make up the bulk of aggregate IP index and Li and Martin (2019) estimates a factor model from real output for 16 nonfarm private sectors.4 On the other hand, Foerster et al. (2011) considers aggregate and industry factors for 117 industries, roughly corresponding to four-digit NAICS industries. The former papers suggest two levels of comovement: one at the aggregate level and one at the NAICS sectoral level. The latter paper restricts comovements to the aggregate level, albeit with multiple factors that could represent sectors.

We reinvestigate industrial comovements allowing a variety of possible correlation structures at different levels of disaggregation. We consider a model with three types of shocks: (i) an aggregate binary shock that affects all industries at once; (ii) cluster shocks that affect only subsets of industries; and (iii) an industry shock. One of the

1See also Kim and Kim (2006) for a relatively recent survey.

2The role of this comovement may be changing. For example, Comin and Philippon (2005) attribute the decline in aggregate volatility around the Great Moderation to a decline in the synchronization of industries and Camacho and Leiva-Leon (2019) find that industries have unique business cycles from the aggregate economy.

1. Foerster et al. (2011), Garin et al. (2018), and Li and Martin (2019) find that common shocks are more important than industry or sector-levelshocks when explaining aggregate volatility. However, Atalay (2017) finds industry-specificshocks can account for about half of aggregate volatility.
2. While the Garin et al. (2018) sectors do not exactly align with two-digit NAICS sectors, the level of aggregation is similar. The Li and Martin (2019) sectors are NAICS two-digit sectors with a few additional lower-level sectors.

1

primary differences between our model and those employed by the rest of the literature on industrial comovement is that we will determine cluster membership-the sets of industries that comove-endogenously.5

Instead of defining a sector as a NAICS two-digit industry, we allow common fluc- tuations across NAICS four-digit industries to form clusters. Thus, we define industrial groupings that maximize the explained volatility attributed to the cluster level, which is our analog to a sector. Moreover, because we do not limit our examination to within- sector comovements, our model may suggest whether industry demand effects or production networks are more important for determining these industry comovements. If clusters form mostly within industrial sectors, cycles might be thought of as being influ- enced by demand shocks common to similar industries. On the other hand, if clusters form across industries but within production networks, cycles may appear to be influenced by disruptions in the supply chain.6

Examining the comovement of industries differs substantially than examining state or regional comovement [as in Hamilton and Owyang (2012) or Gonz´alez-Astudillo (2019)]. Because the latter are diversified economies in a common currency zone, they tend to have positive growth rates during expansions and negative growth rates during recessions. This stylized fact allows routine identification of the business cycle phases: Expansions occur during periods of positive growth and recessions occur during periods of negative growth. This identification, however, is not as useful at the industry level because some industries experience long periods of secular declines. Thus, it is important to characterize both the trend and the cycle terms simultaneously.

We consider industrial production of 82 four-digit NAICS industries over the period 1972 - 2019 to determine (i) whether comovements occur, (ii) whether they are a pervasive feature of the U.S. business cycle, and (iii) whether they are limited to industries within a single sector or whether they are determined by industries' production streams. Each industry has a non-deterministic trend, an aggregate recession regime that affects

5Our model has a similar hierarchical structure to Kose et al. (2003) (henceforth KOW) but with endogenously-determined clusters as in Francis et al. (2017).

6Production networks may be important for the propagation of technology shocks (Holly and Petrella,

2012) and monetary policy (Ozdagli and Weber, 2017).

2

all industries, a connected or clustered component representing the comovement across industries, and an idiosyncratic autoregressive component.

The previous literature has focused on two types of shocks: aggregate shocks as affecting all industries simultaneously and idiosyncratic shocks that are specific to industries (Foerster et al., 2011; Atalay, 2017). Here, we consider an intermediate layer of co- movement analogous to the two-digit NAICS sectoral level (Garin et al., 2018; Li and Martin, 2019) but with endogenously defined groupings. While the sectoral characterization assumes that demand shocks create comovement within a sector or that supply chains are intra-sectoral, our model nests that definition by having a national business cycle and idiosyncratic industry shocks, similar to the previous literature, and allowing for industries to cluster endogenously. This allows for industries to be related to other industries within their own sector, but also allows for inter-sectoral supply chains or demand shocks of complementary goods. This is akin to Lee (2010), who found that the synchronization of international business cycles due to intra-industry trade flows rather than inter-industry and Acemoglu et al. (2012), who found that sector-specific shocks can propogate to other sectors and lead to aggregate fluctuations.

Similar to Camacho and Leiva-Leon (2019), who use industry employment data, we find that industries cluster into a few groups that are often similar to industrial subsectors but are sometimes different. A number of industries experience significant periods of secular decline, albeit with different timings. Apart from the aggregate business cycle phases, industries cluster together. Some of these clusters reflect supply chains that are isolated within the sector (motor vehicles); others are final good industries with similar demand elements (agricultural products). We find that for most industries the aggregate regime accounts for most of the cyclical variance, similar to Foerster et al. (2011), Garin et al. (2018), and Li and Martin (2019). However there are industries that behave similarly to the findings of Atalay (2017) where the "sectoral" or cluster grouping accounts for a larger portion of the variance.

The balance of the paper is laid out in the following order: Section 2 presents the clustered factor model with aggregate Markov-switching. Section 3 outlines the estima-

3

tion technique via the Gibbs sampler. Section 3.9 describes the disaggregated industrial production data and industry-specific characteristics. Sections 4 discusses the results. Section 5 concludes the paper.

2 MODEL

Evaluating industrial connectedness requires a model with a number of features. Each industry's IP time series has: (i) a trend component; (ii) a national cycle that moves all or most industries at once; (iii) a "connected" component-which industries are connected is endogenously determined-that moves industries together OUTSIDE of the national cycle; and (iv) an idiosyncratic cycle that accounts for possible AR dynamics outside of the industrial clusters.

The model is a variation of the hierarchical factor model first proposed in Kose et al. (2003) with endogenous clusters (similar to regions) as in Francis et al. (2017) and idiosyncratic trends. Let YNT denote the log level of industrial production for industry n = 1, ..., N in month t = 1, ..., T that has two unobserved components:

YNT = τNT + cNT,

where τNT and cNT represent a permanent trend and temporary cycle, respectively. We assume that shocks to the cycle and trend components are uncorrelated. The following subsections describe the various components.

2.1 THE TRENDS

Our motivation for estimating the model in levels and including a trend component stems from the fact that-unlike diversified economies in other studies-some industries experience secular declines. However, because the timing of the declines can vary across industries, imposing a linear or deterministic trend may not be flexible enough.7

7There are a variety of methods to detrend data [see Canova (1998) for an overview]. An alternative to UC is using the Hodrick-Prescott (HP) filter to prefilter the data. However, the HP-filter can introduce spurious cycles into the data (Cogley and Nason, 1995) and the conditions that would make the HP-filter

4

Instead, we adopt an unobserved components (UC) framework that simultaneously estimates the trend and cycle. Each industry trend is a random walk with possibly time-varying drift:

τNT = δNT + τNT−1 + eNT,

(1)

where eNT are normally-distributed permanent innovations, eNT ∼ N (0, σEN2 ). We assume that the shocks to trend eNT are independent across time [i.e., E(eNTeNS) = 0 ∀t = s and ∀n] and across industries [i.e., E(eMTeNT) = 0 ∀m = n]. To capture industries in secular decline, the sign of the drift parameter, δNT, is unrestricted. In the baseline model, we assume the drift term is constant, δNT = δN.

2.2 The Cycles

In many univariate applications, the stationary cyclical component is assumed to follow a simple autoregressive process. Because we are interested in industrial connectedness apart from the national cycle, we further decompose the industry cycle into two parts:

cNT = z˜NT + vNT,

where z˜NT represents the connected component and vNT represents the idiosyncratic com- ponent.

2.2.1 Connectedness

The connected component is correlated across industries and embodies both national and industrial cluster components. Instead of simply examining the correlation structure of the shocks to the various industries, comovement is captured by assuming that comoving industries have a common connected component, albeit scaled across industries. Using the language of Francis et al. (2017), we model the common connected component as a "cluster factor": Industries that belong to the same cluster are attached to the same cluster factor, subject to an industry-specific factor loading.

optimal are rare (Hamilton, 2018).

5

Formally, define a cluster indicator, γnk, where γnk = 1 if industry n belongs to cluster

• ∈ {1, ..., K} and γnk = 0 otherwise. Here, K << N represents the total number of

clusters. Each industry n belongs to a single cluster, so that k γnk = 1. Then, we can write the connected component, z˜nt, as:

K

z˜nt = αn γnkzkt, k=1

where zkt is the time-t value of kth cluster factor and αn > 0 is industry n's factor loading. The cluster factor must capture both the national business cycle and common comove- ment between industries within the cluster. We assume that zkt consists of an AR(pz )

process with a regime-switching intercept:

zkt = µkt + φk(L)zkt−1 + ukt

where the roots of φk(L) = φk1L + ... + φkpZ LpZ lie strictly outside of the unit circle and E(u2kt) = σuk2 . We assume the innovations to the cluster factors are uncorrelated across clusters and we normalize σuk2 = 1 ∀ k = 1, .., K to identify the scale of each z. Further, we assume no correlation between components [i.e., E(emtunt) = 0 ∀m, n].8

The regime-switching intercept depends on an aggregate, discrete state variable, St ∈ {0, 1}, where

µkt = µk0 + µkSt.

The drift term in the trend equation (1) requires us to normalize µk0; following Kim and Nelson (1999), we impose µk0 = 0 and µk< 0. Thus, St = 1 captures an aggregate downturn and St = 0 captures an aggregate expansion. We assume that St follows a first- order Markov-process with constant transition probabilities πji = Pr [St = j|St−1 = i] which are compiled in a transition matrix Π.

8While the trend and cycle innovations in a UC model can be correlated (Morley et al., 2003), the correlated version produces a volatile trend and a cycle that is mostly noise. Harvey and Koopman (2000) find that the uncorrelated UC model is appropriate for a wide array of applications.

6

2.2.2 Idiosyncratic

The idiosyncratic component, vnt, captures the residual industry-specific dynamics and follows an AR(pv):

vnt = ρn(L)vnt−1 + ηnt,

where the roots of ρn(L) = ρn1L + ... + ρnpV LpV lie strictly outside of the unit circle, E(ηnt2 ) = σηn2 , and E [ηntηmt] = 0 for n = m and for all t. The latter restriction assumes that any correlation across the cycles are attributable only to the connected component.

2.3 Model Overview

Define YT = [Y1t, ..., YN t]′; define τt and vt similarly. Define zt = [z1t, ..., zKt]′ and as the matrix of cluster memberships with representative element γnk ∈ {0, 1} defined above. The stacked system can be summarized as:

YT = τt + α ⊙ zt + vt,

where ⊙ is the Hadamard product and the stacked components are defined by:

τt = δ + τt−1 + et,

zt = µSt + Φ (L) zt−1 + ut,

and

vt = ρ (L) vt−1 + ηt.

The stacked parameter vectors collect the parameters across industries or clusters. The system can be written as a state space in either levels or differences.

Our model has the flavor of Friedman's plucking model [see Friedman (1964, 1993) and Dupraz et al. (2020)], where expansions are periods when the economy is near trend

7

and recessions are periods when the cycle is "plucked" downward from trend. While the timing of the national downturns are common across the industries, how these downturns affect each industry is idiosyncratic, determined by their common cluster parameter, µ, and the industry-specific factor loadings, α.

3 ESTIMATION

We estimate the model using the Gibbs sampler [see Gelfand and Smith (1990); Casella and George (1992); Carter and Kohn (1994)]. This Markov Chain Monte Carlo technique separates the latent variables and parameters into blocks to be drawn from their conditional posterior distributions, given the data and other latent variables and parameters.

The latent variables include the trend components, τ T = [τ1, ..., τT ]; the cluster fac- tors, zT = [z1, ..., zT ]; the idiosyncratic cycle components, vT = [v1, ..., vT ]; and the aggregate regime series, ST = [S1, ..., ST ]. Let Ψ represent the full set of parameters that includes the drift parameters, δ; the trend innovation variances, Σe = diag [σe21, ..., σeN2 ]; the common recession magnitudes, µ; the regime process transition matrix, Π; the cluster membership indicators, ; the factor loadings, α; the cluster AR coefficients, Φ; the

idiosyncratic AR parameters, ρ; and the idiosyncratic variances, Ση = diag ση21, ..., σηN2 . Recall that we normalized the cluster factor variances, Σu = diag [σu21, ..., σuK2 ] = IK . In what follows, Ψ−x represents the full set of parameters excluding x.

We set the lag lengths for each of the cluster factors and idiosyncratic components to 2. We run the sampler for 5,000 iterations after an initial burn-in of 5,000 iterations.

3.1 PRIORS

The drift parameter, the recession depth, and all AR coefficients have normal prior distri- butions. All innovation variances have inverse Gamma priors. The transition probabilities for the aggregate regime process are assumed to have a Dirichlet prior distribution.

The composition of each cluster is determined endogenously by the similarity in the movements in the Ynt's across industries. We can incorporate additional information by

8

assuming a multinomial logistic prior for the cluster membership indicator, γnk. Suppose there exists a vector, Xnk, of variables that may influence whether a series n belongs to cluster k. We assess the prior probability that series n belongs to cluster k as:

	 exp	X′	βk /	1 +	exp X′	βk	k = 2, ..., K
Pr [γnk = 1|Xnk] =			nk			nk		, (2)
		1/	1 +	exp	′	βk		k = 1
		Xnk
	

for n = 1, ..., N and where we have normalized β1 = 0. Note also that the vector, Xnk, need not be composed of the same variables for each cluster k but that the covariates in each Xnk must be time invariant and industry specific (i.e., not functions of the composition of the clusters or computed relative to another industry m = n). That is, we cannot include in Xnk variables such as the value of inputs flowing from one industry to another nor the value of inputs flowing from one industry to all other industries in a cluster. As in Hamilton and Owyang (2012) and Francis et al. (2017), we think of the prior hyperparameters, βk's, as population parameters signifying the clusters' relationships.

Table 1: PRIOR DISTRIBUTIONS FOR ESTIMATION

Parameter	Prior Distribution	Hyperparameter
πi	D(¯π1i, π¯2i)	π¯ji = 1 for j = 1, 2 and i = 1, 2
µk	N (m0, M0)	m0 = −2, M0 = 1
φk	N (f0, F0)	f0 = [0.9, 0]′, F0 = (0.1)2 × I2
αn	N (a0, A0)	a0 = 1, A0 = 1
βk	N (b0, B0)	b0 = 03, B0 = diag(0.5, 3, 3)
δn	N (d0, D0)	d0 = 1, D0 = 1
		¯		¯		¯
			ζ	en
			υen
σen	IG			,				υ¯en = 6, ζen = 4
2		2
ρn	N (r0, R0)	r0 = [0.9, 0]′, P0 = (0.1)2 × I2
		υ¯ηn	¯		¯
σηn	IG	ζηn
	,				υ¯ηn = 6, ζηn = 4
2		2

Table 1 shows the prior distributions for the model parameters.

3.2 DRAW τ T , zT , vT |Ψ, ST , YT

The latent variables are drawn using the smoothing sampler of Durbin and Koopman

(2002) with the correction outlined by Jaroci´nski (2015). To implement the smoothed

9

sampler, we must cast the model in its state space form. The measurement equation is

Yt = Hξt,

where H =

˜

˜

′

′

′

′

′

′

.

IN , IN , , 0N ×N ,

0N ×K ,

= α⊙, and the state vector is ξt = [τt

, vt

, zt

, vt−1

, zt−1

]

The state equation is

ξt = Mt + F ξt−1 + ωt

where Mt = [0N′

×1, 0N′

×1, Stµ′, 0N′

×1, 0K′

×1]



IN

0N ×(N +K)

0N ×(N +K)



F =

 0

(N +K)×N

Ψ

1

Ψ

2



,













 0(N +K)×N I(N + K) 0(N +K)×(N +K)



Ψp = diag(ρ1p, ..., ρN p, φ1p, ..., φKp), and ωt = [e′t, ηt′ , u′t, 0′(N +K)×1]′.

The sampler requires a distribution for the initial state vector ξ0 ∼ N (ξ0|0, P0|0) [see Durbin and Koopman (2012)]. The initial mean and variance-covariance terms for the stationary series in ξ0 (v and z) are set to the unconditional mean and variance-covariance terms [see (Hamilton, 1994, p. 378)]. Because the trend, τ , is nonstationary, the initial values are set using a diffuse prior, where the trend element in ξ0|0 is set to the initial observation Y0 and the corresponding diagonal elements of P0|0 to 107.

3.3 Draw ST , Π |Ψ−Π , τ T , zT , vT , YT

The draws for the cluster factor dynamics are standard steps for an AR process with Markov-switching (Kim and Nelson, 1999). Conditional on zT , µ, and Π, the state

vector ST can be drawn using the filter outlined by Hamilton (1989).	We initialize
the filter with the steady-state probabilities implied by Π: p(S0 = 0) =	1−π11	and
2−π00−π11
p(S0 = 1) =	1−π00	. The filter is run forward for t = 1, ..., T to obtain
	2−π00−π11

p(St|zt, ..., z1) = f (zt|St, zt−1, ..., z1)p(St|zt−1, ..., z1), f (zt|zt−1, ..., z1)

10

where
f (zt|St, zt−1, ..., z1) ∝ exp (−0.5 t=1 [zt − µStΦ(L)zt−1]′[zt − µSt − Φ(L)zt−1])	,
	T
	X
	1
	Xi
p(St|zt−1, ..., z1) =	p(St|St−1	= i)p(St−1 = i|zt−1, ..., z1),
	=0
1
X
f (zt|zt−1, ..., z1) =	f (zt|St = j, zt−1, ..., z1)p(St = j|zt−1, ..., z1).

j=0

We then draw the terminal state ST from p(ST |zT , ..., z1), which is provided by the last iteration of the forward filter. The remaining states, ST −1, ..., S1, are drawn recursively:

p(St|St+1 = j) ∝ p(St+1 = j|St)p(St|zt, ..., z1)

using a backward smoother (Chib, 1996).

Each of the columns πi = [π1i, π2i]′ of the transition matrix Π are drawn from their conditional posterior distribution given by

• i ∼ D(¯π1i + N1i, π¯2i + N2i),

where Nji is the number of transitions from state i to state j in ST and the prior is the Dirichlet distribution πi ∼ D(¯π1i, π¯2i).

3.4 Draw µ, Φ|Ψ−µ,Φ, τ T , zT , vT , ST , YT

Drawing the parameters governing the cluster dynamics is straightforward, conditional on knowing the state vector, ST , and cluster factors, zT . We first draw µk for each cluster, then the AR parameters, Φk = [φk1, φk2]′. Let µk ∼ N (m0, M0) be the prior distribution for the "plucking" parameter. We can then draw µk from its posterior distribution using a rejection sampler to ensure the identification is satisfied:

µk ∼ N (m1, M1)

11

where

m1 = M1(M0−1m0 + Xµ′ Yµ,k),

M1 = (A−0 1 + Xµ′ Xµ),

′

Xµ = [SpZ +1, ..., ST ]′, and Yµ,k = zk,pZ +1 − p φkpzk,pZ +1−p, ..., zk,T − p φkpzk,T −p .

Given the prior Φk ∼ N (f 0, F 0), we draw the cluster AR parameters using a rejection sampler to ensure stationarity from

Φk ∼ N (f 1, F 1),

where

• 1 = F 1(F −0 1f 0 + XΦ′ kYΦ,k),

• 1 = (F −0 1 + XΦ′ kXΦk),

X		k =		zk,pZ	· · ·	zk,1		,
	 ..		..	
	Φ		 .		.	
							
							
			 zk,T −1	· · ·	zk,T −pZ	

and YΦk = [zk,pZ +1 − µkSpZ +1, ..., zk,T − µkST ]′.

3.5 Draw , α, β|Ψ−α,β,, τ T , zT , vT , ST , YT , x

The cluster membership indicators in could be drawn industry-by-industry from each respective conditional posterior distribution p(γn|Y n) following (Fr¨uhwirth-Schnatter,2006, Ch. 3). However, we found this method mixed poorly due to the restriction on the minimum cluster size.9 Therefore, we opted to draw the membership indicators for each cluster γk using a Metropolis-within-Gibbs step. At each Gibbs iteration, we propose a new cluster membership matrix ∗ which differs from the previous iteration's draw [i−1] by a single industry n. That is, with equal probability we either add or take away an

9We set the minimum number of industries in a cluster to 3. This assumption ensures that a cluster does not capture movements isolated to one or two industries, but rather more pervasive movements across a number of industries as intended.

12

industry from cluster k to get ∗.10 The proposal is then accepted with probability

		A = min "1, p([i−1]	|β, x) f (Y |[i−1]) q(∗|[i−1]) #
			p(∗|β, x)		f (Y |∗) q([i−1]|∗)
where p(|β, x) is the prior given by (2),
f (Y |) ∝ exp	(−0.5 t=1 (ΔYt − δ −	vt − α ⊗ zt)′Σe−1(ΔYt − δ − vt − α ⊗ zt)) ,
		T
		X
and q [i]|[i−1]	is the proposal distribution. Because the proposal distribution is sym-

metric, the last term in the acceptance probability is equal 1.

After proposing the cluster membership indicators in , the draw for α is straightforward and can be carried out industry-by-industry. Assuming a prior distribution of αn ∼ N (a0, A0), each factor loading is drawn using a rejection sampler from its posterior

distribution given by

αn ∼ N (a1, A1),

where

a1 = A1(A−0 1a0 + Xα,n′ Yα,n),

A1 = (A0−1 + Xα,n′ Xα,n),

Xα,n = [

K γNK

zK1

K γNK

zKT

′

YN1

−δN− vN1

YNT −δN− vNT

′

, ...,

]

, and Yα,n = [

, ...,

]

.

σEN

σEN

σEN

σEN

Given the cluster membership indicators in , the draw of the prior hyperparameters

• follow from the data augmentation technique presented in Section 3.2 of Fr¨uhwirth- Schnatter and Fr¨uhwirth (2010). To identify the parameters in the multinomial logistic, we set β1 = 0 implying Cluster 1 is the baseline category. Thus, βk for k = 2, ..., K can be interpreted as the change in the log-odds ratio relative to Cluster 1.

10Note that if γ[KI−1] is already at the minimum cluster size then we only allow for the addition of a random industry to cluster K.

13

3.6 Draw δ, Σe|Ψ−δ,ΣE , τ T , zT , vT , ST , YT

We draw the drift parameters, δn, and the variance terms in ΣE industry-by-industry. Given the prior δn ∼ N (d0, D0), the posterior distribution is then:

δn ∼ N (d1, D1),

where

D1 = (D0−1 + Xδ,n′ Xδ,n),

d1 = D1(D0−1d0 + Xδ,n′ Yδ,n),

Xδ,n =	−1	1T −1, and Yδ,n	= [	τN2		τNT	′	. To allow for secular declines in certain
σen	σEN	, ...,	σEN	]

industries, we place no restrictions on the sign of δn.

	¯
Given the prior, σen2	∼ IG(	υ¯EN	, ζEN2 ), we draw σen2 from its posterior distribution given
2
by
			σen2 ∼ IG(	υen	,	ζen	),
				2
				2
	¯	T	2	.
where υen = υ¯en + T − 1 and ζen = ζen +	t=2 (Δτnt − δn)

3.7 Draw ρ, Ση|Ψ−ρ,Ση , τ T , zT , vT , ST , YT

Given the prior, ρn ∼ N (r0, R0), we draw ρn using a rejection sampler to ensure station- arity from its posterior distribution:

ρn ∼ N (r1, R1),

where

r1 = R1(R0−1r0 + Xρ,n′ Yρ,n),

R1 = (R0−1 + Xρ,n′ Xρ,n),

14

			vn,pv		· · ·			vn,1		
		σηn			σηn
Xρ,n =	 ..		..			,
	 .		.		
										
		vn,T −1		vn,T −pv	
				· · ·				
		σηn			σηn

vn,pv +1

vn,T

′

and Yρn =

, ...,

. We implement a rejection sampler to ensure all of the roots

σηn

σηn

for any draw ρn(L) lie outside the unit circle, and thus vn remains stationary.

υ¯ηn

¯

Let σηn2

∼ IG(

,

ζηn

) be the prior for σηn. The posterior distribution is then given

2

2

by

2

υηn

ζηn

)

σηn ∼ IG(

,

2

2

¯

T

2

.

where υηn = υ¯ηn + T − pv and ζηn = ζηn +

t=pv +1 (vnt − ρn(L)vnt−1)

3.8 Choosing the Number of Clusters K

We treat the number of clusters, K, to include in the model outlined in Section 2 as

• model selection issue. One option is to compute marginal likelihoods p(Y |K) for alternative values of K. Marginal likelihoods in our context are not directly available from the MCMC output and approximation can be computationally intensive. As shown by Kass and Raftery (1995), Bayesian Information Criterion (BIC) provides an asymptotic approximation of the marginal likelihood and is relatively easy to compute. Therefore, we compare BIC(K) across models with cluster sizes K = 2, ..., 10. Alternative criterion, including Akaike Information Criterion (AIC) and Deviance Information Criterion (DIC) corroborate our findings of the optimal cluster size.11

3.9 Data

Macroeconomists often use the growth rate of real gross domestic product (GDP) to represent business cycle fluctuations. The NBER Business Cycle Dating Committee also considers other variables such as employment and industrial production (IP). Some studies have found a recent divergence between the cycles identified by output variables (e.g., GDP, IP) and the cycles identified by employment. Following the three recessions

11See Spiegelhalter et al. (2002).

15

of 1990-91, 2001, and the Great Recession, employment has recovered more slowly than output [see, for example, Jaimovich and Siu (2020)]. These jobless recoveries can skew the timings of the turning points if labor market variables alone are used. Because industry- level GDP is available only at a quarterly frequency back to 2005 and employment can be problematic for identifying turning points, we use monthly, seasonally-adjusted, industry- level IP. These data are available from the Board of Governors of the Federal Reserve System.12

The monthly industrial production index is disaggregated to the four-digit NAICS industry level. Our sample includes 82 industries covering the time period 1972:01 to 2019:12. During the early-1990s, the standard industry classification changed from the SIC to the NAICS. The two industry classification systems differ in both the number of industries and how they are classified. Broad industrial sectors (one-digit) remain essentially unchanged but the three-digit industries cannot be easily merged. While the SIC classification provides a longer subsample (back to 1948), the NAICS contains industries (e.g., information technologies) that have recently risen in importance.

For the prior on cluster membership, we require a set of industry-specific characteristics that are time invariant and can represent extra- or intra-sectoral connectedness. These data will help determine whether comovement is determined by the connectedness within the broader (two-digit) sector industries or an industry's connection to its supply chain. Industry comovement could result from shocks that diffuse within the sector. Con- versely, comovement could result from economy-wide shocks that affect industries with large supply chain dependencies.

We evaluate whether comovement is determined by an industry's connectedness within its supply chain or sectoral demand shocks that affect similar industries simultaneously. The clusters membership indicator determines which industries comove; the prior on this indicator is logistic, parameterized by covariates that may influence cluster composition. The data used to populate the prior are industry-specific, time invariant, and not a function of the other industries in a cluster.

12https://www.federalreserve.gov/releases/g17/

16

To measure connectedness within a supply chain, we compute the dollar value of

commodities flowing from to other industries within or out of its broad sector, where the

broad sector is defined as the two-digit NAICS code. We use the raw "make"- "use"tables

from the Bureau of Economic Analysis.13

The "make" table collects WNCT, the dollar value of commodity C produced by industry

N, into a N × C output matrix WT for year T. The "use" table collects UCNT, the dollar

value of commodity C used as an input by industry N, into a similar C × N input matrix

UT for year T. The input-output matrix, then, is IOT = [WT ⊘ (JN WT)] UT where JN is a

N × N matrix of ones and ⊘ represents Hadamard division. We normalize the input-

output matrix by the total dollar flow into a given industry to get each industry's relative

˜	14
intermediate importance: IOT = IOT ⊘ (JN IOT).

Figure 1: CONNCECTIVENESS MEASURES. This histogram shows the two measures of industry connectiveness as constructed from the input-output matrices. INTRA-IO measures the percentage of total output of a given industry that is used as an input to industries within its sector (i.e, industries which have the same two digit NAICS code). INTER-IO measures the percentage of total output of a given industry that is used as an input to industries outside its sector.

15

IntraIO

InterIO

10

Frequency

5

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Proportion

Our measure intra-industry connectedness (INTRA-IO) is the sum the proportion of a

given industry's output used as an input into other industries with the same two-digit

1. https://www/bea.gov/industry/input-output-accounts-data
2. See Appendix 4 of Caunedo (2020) for more details about these computations.

17

NAICS code. Conversely, our measure of inter-industry connectedness (INTER-IO) is the sum of the proportion of a given industry's output used as an input into industries outside its two-digit NAICS code. Because the covariates entering the multinomial logistic prior must be time invariant, we take the average of each connectedness measure across all of the years for which data is available, 1997 - 2019. Figure 1 shows a histogram of the average connectedness measures for each industry.

4 Results

The proposal density for the cluster membership draw of the sampler is designed to prevent the creation of an empty cluster. While we could set the minimum size of the sampler equal to 1, we chose not to allow degenerate clusters (i.e., clusters with only a single member). We instead set the minimum number of industries in a cluster equal to 3. This means that there is a tension between the minimum size of a cluster and the optimal number of clusters. Based on our selection criteria, the optimal number of clusters is 6. Note that the value of the metrics do not necessarily decrease monotonically moving away from the optimal number of clusters. That is, the runner-up number of clusters is 3 instead of either 5 or 7. The results that follow are based on 6 clusters.

4.1 Industry Trends

One important component of our model is the industry-level trend. Estimating each industry's trend allows us to identify sectors that may be in secular decline over large portions of the sample period. Of particular interest are industries that experience both increasing and decreasing trends over the period. These periods could affect the identi- fication of the aggregate regimes and issues surrounding them might not be resolved by differencing.

18

Table 2: PARAMETER ESTIMATES. This table shows median posterior estimates of each industry's trend drift δN; magnitude of average aggregate recession αNγNK µK ; and the variance decomposition of the cycle component cNT into the aggregate regime ST and cluster-specific shock uKT.

Industry Name	NAICS	δN	αNγNK µK	V DCS	V DCU
Veneer, Plywood, and Engineered Wood Product	3212	0.09	-2.5	0.59	0.07
Other Wood Product	3219	0.04	-0.99	0.5	0.06
Clay Product and Refractory	3271	-0.07	-2.73	0.83	0.1
Glass and Glass Product	3272	0.02	-0.02	0	0.02
Cement and Concrete Product	3273	0.03	-2.6	0.86	0.1
Lime and Gypsum Product	3274	0.06	-2.58	0.73	0.09
Other Nonmetallic Mineral Product	3279	0.13	-2.53	0.87	0.1
Iron and Steel Products	3311	-0.03	-4.69	0.71	0.08
Alumina and Aluminum Production and Processing	3313	0.02	-3.03	0.83	0.1
Nonferrous Metal	3314	-0.04	-1.99	0.62	0.07
Foundries	3315	-0.06	-3.14	0.86	0.1
Hardware	3325	-0.08	-2.33	0.83	0.1
Machine Shops, Turned Product, and Screw, Nut, and	3327	0.17	-2.71	0.84	0.1
Other Fabricated Metal Product	3329	0.03	-2.15	0.86	0.1
Agriculture, Construction, and Mining Machinery	3331	0.04	-2.57	0.52	0.06
Industrial Machinery	3332	0.02	-2.18	0.56	0.07
Commercial and Service Industry Machinery and Othe	3333	0.2	-2.06	0.86	0.1
Ventilation, Heating, Air-Conditioning, and Commer	3334	0.03	-3.12	0.75	0.09
Metalworking Machinery	3335	-0.01	-2.49	0.72	0.09
Engine, Turbine, and Power Transmission Equipment	3336	0.02	-2.53	0.83	0.09
Computer and Peripheral Equipment	3341	1.43	-1.73	0.74	0.09
Communications Equipment	3342	0.73	-1.55	0.77	0.09
Audio and Video Equipment	3343	0.07	-3.72	0.65	0.08
Semiconductor and Other Electronic Component	3344	1.42	-2.76	0.86	0.1
Navigational, Measuring, Electromedical, and Contr	3345	0.36	-1.31	0.83	0.1
Electric Lighting Equipment	3351	-0.03	-2.4	0.79	0.09
Household Appliance	3352	0.05	-2.41	0.67	0.08
Electrical Equipment	3353	-0.02	-2.06	0.82	0.1
Other Electrical Equipment and Component	3359	0.12	-2.9	0.87	0.1
Motor Vehicle	3361	0.21	-13.03	0.59	0.38
Motor Vehicle Body and Trailer	3362	0.07	-2.92	0.46	0.3
Motor Vehicle Parts	3363	0.18	-5.26	0.57	0.36
Aerospace Product and Parts	3364	0.12	-0.64	0	0.17
Ship and Boat Building	3366	0.06	-0.57	0.02	0.14
Office and Other Furniture	3372	0.1	-1.89	0.83	0.1
Logging	1133	0.05	-1.42	0.31	0.04
Sawmills and Wood Preservation	3211	0.06	-1.91	0.56	0.07
Forging and Stamping	3321	0.07	-2.9	0.87	0.1
Cutlery and Handtool	3322	-0.09	-1.98	0.84	0.1
Architectural and Structural Metals	3323	0.08	-2.16	0.85	0.1

The first column of Tables 2 and 3 shows the NAICS four-digit industry code, and the second column shows the estimated drift terms for each industry in our sample. Eighteen of the 82 industries have negative mean drift terms. Three of these industries- tobacco, newspapers, and railroad rolling stock, which consists of train cars, etc-are not

19

Table 3: PARAMETER ESTIMATES (CONTINUED). This table shows median posterior estimates of each industry's trend drift δN; magnitude of average aggregate recession αNγNK µK ; and the variance decomposition of the cycle component cNT into the aggregate regime ST and cluster-specificshock uKT.

Industry Name	NAICS	δN	αNγNK µK	V DCS	V DCU
Spring and Wire Product	3326	0	-2.85	0.85	0.1
Coating, Engraving, Heat Treating, and Allied Acti	3328	0.2	-2.6	0.86	0.1
Railroad Rolling Stock	3365	-0.02	-1.94	0.85	0.1
Other Transportation Equipment	3369	0.23	-1.12	0.32	0.1
Household and Institutional Furniture and Kitchen	3371	0.02	-3.11	0.88	0.1
Medical Equipment and Supplies	3391	0.31	-1.08	0.8	0.09
Animal Food	3111	0.24	-0.13	0.02	0.58
Grain and Oilseed Milling	3112	0.15	-0.09	0.02	0.48
Sugar and Confectionery Product	3113	0.08	-1.29	0.48	0.06
Fruit and Vegetable Preserving and Specialty Food	3114	0.08	-0.06	0	0.12
Dairy Product	3115	0.09	-0.03	0	0.06
Animal Slaughtering and Processing	3116	0.19	-0.31	0.11	0.03
Other Food	3119	0.21	-0.45	0.19	0.04
Beverage	3121	0.15	-0.12	0.03	0.03
Tobacco	3122	-0.17	-0.1	0.01	0.12
Fabric Mills	3132	-0.11	-2.05	0.86	0.1
Textile and Fabric Finishing and Fabric Coating Mi	3133	-0.16	-2.63	0.86	0.1
Textile Furnishings Mills	3141	-0.07	-2.63	0.87	0.1
Other Textile Product Mills	3149	0.01	-2.68	0.77	0.09
Apparel	315	-0.38	-1.94	0.87	0.1
Leather and Allied Product	316	-0.31	-1.82	0.8	0.09
Pulp, Paper, and Paperboard Mills	3221	0.01	-1.47	0.69	0.08
Converted Paper Product	3222	0.05	-1.5	0.79	0.09
Printing and Related Support Activities	323	0.06	-1.21	0.87	0.1
Petroleum and Coal Products	324	0.07	-0.65	0.02	0.59
Basic Chemical	3251	0.06	-1.13	0.02	0.44
Resin, Synthetic Rubber, and Artificial and Synthe	3252	0.1	-2.91	0.7	0.08
Pesticide, Fertilizer, and Other Agricultural Chem	3253	0.13	-0.95	0.78	0.09
Pharmaceutical and Medicine	3254	0.22	-0.31	0.44	0.06
Paint, Coating, and Adhesive	3255	0.08	-1.6	0.73	0.09
Soap, Cleaning Compound, and Toilet Preparation	3256	0.13	-1.11	0.5	0.06
Plastics Product	3261	0.25	-2.66	0.87	0.1
Rubber Product	3262	0	-2.45	0.71	0.08
Newspaper, Periodical, Book, and Directory Publish	5111	-0.11	-0.9	0.79	0.09
Bakeries and Tortilla	3118	0.03	-0.43	0.49	0.06
Oil and Gas Extraction	211	0.09	-0.71	0.03	0.78
Metal Ore Mining	2122	0.06	-2.42	0.47	0.06
Nonmetallic Mineral Mining and Quarrying	2123	0.05	-2.23	0.81	0.1
Support Activities for Mining	213	-0.08	-0.58	0.04	0.01
Electric Power Generation, Transmission, and Distr	2211	0.18	-0.27	0.1	0.03
Natural Gas Distribution	2212	0.01	-0.34	0	0.27
Coal Mining	2121	0	-0.02	0	0

surprising as demand for these products have been declining for various reasons. Iron and steel and their related sectors also have negative drift. Apparel and leather may both have declining trends because of foreign competition. Other textile subsectors have

20

very small, positive drift terms. On the other hand, some high tech industries-computer peripherals, semiconductors, and communications have steep trends suggested by large, positive drift terms.

As we suggested above, not all of these industries display a secular decline over the entire subsample. For example, newspapers and apparel-both of which we will discuss in more detail below-increase or are constant for the beginning portion of the sample but then decline over the last 20-25 years. In both of these cases (and others), neither a linear trend nor differencing would remove the proper trend in the data to allow us to properly identify the cyclical features.

4.2 Aggregate Recessions

In our model, comovement is influenced by two components: the aggregate recession indicator, ST, and the common cluster cycle, ZKT. The aggregate recession indicator drives comovement across all industries, while the cluster cycle drives comovement between a subset of industries. Because the aggregate recession indicator is common to all of the industries, ST is well identified.

Figure 2: Aggregate Recession. This figure shows the mean probability of recession for the entire panel of industries (i.e., ST = 1). The gray bars reflect official NBER U.S. recession dates.

1
0.8
0.6
0.4
0.2
0
1970	1975	1980	1985	1990	1995	2000	2005	2010	2015	2020

Figure 2 plots the posterior regime probabilities for ST-the means across Gibbs iterations of ST-along with the NBER recessions shaded in grey. With the exception of the 1990-91 and 2001 events, the aggregate regime variable indicates a recession with posterior probability 1 for all of the NBER recessions in the sample period. The 1990-91

21

event is identified at an 80-percent posterior probability threshold. The 2001 event is not identified at all.15 The aggregate regime variable has only one false positive at the beginning of the sample, which may be a product of the initialization of the sampler and/or the poor end-of-sample properties of the filter. The dataset does not include the recession starting in 1969.

While industry-level IP appears to identify the ends of recessions, using an aggregation of the large cross-section of industries does not identify the beginnings of recessions as accurately. Industries recover simultaneously but fall into recession at different times. This is similar to Chang and Hwang (2015) who finds that industry trough dates are clustered around NBER turning points, but that peaks are more dispersed. This result motivates our approach of using both an aggregate regime and cluster cycles that allow groups of industries to experience business cycle fluctuations outside the aggregate cycle.

Figure 3: Recession Depth. This histogram shows the posterior median for each in- dustry's recession depth αNγNK µK . We omit motor vehicles from the histogram due to its extreme value (-13.03).

Frequency

14

12

10

8

6

4

2

0

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

In addition to the timing of the aggregate recessions, our model identifies the magnitude of the industry response to these events. Recall that the depth of the recession for

15We estimated the model using the latest vintage of data. Data revisions since the NBER declared these turning points have been positive, making an aggregate recession less likely. Similarly, the dynamic factor Markov switching model of Chauvet and Piger (2008) estimates a peak probability of recession of only 44% during this period.

22

each industry is characterized by αNγNK µK . The third column of Tables 2 and 3 show the magnitude of aggregate recessions-the mean of αNγNK µK across Gibbs iterations-for the cross-section of industries.16 Figure 3 plots a histogram of the depth of recessions for the various industries.

In general, raw materials experience some of the largest declines (between the high 2 to over 4 percentage points) in IP during recessions. Other durable goods industries- e.g., household furniture and kitchen cabinets, motor vehicles, and motor vehicle related industries-also experience large declines during recessions, possibly because consumers delay these purchases during tough times. Conversely, inelastic demand products such as food and food products-with the exception of sugar and confectionery products, which may be thought of as more elastic-and energy related industries are some of the industries least affected by the aggregate downturn.

4.3 Cluster Composition

While the aggregate regime affects all of the industries at once, cross-industry comove- ment in a subset of industries is also determined by membership in a cluster. Industries in the same cluster attach to a single cluster factor zKT. While two industries in the same cluster can have different trends and idiosyncratic cycles, they share a common cluster factor.

Tables 4 and 5 show the posterior probabilities for the cluster membership indicator. For many of the industries, membership is well identified-the posterior membership probability for one cluster is equal to 1. Only a few industries are not assigned to a single cluster with high probability. For example, coal mining is not assigned to a cluster with more than 50% posterior probability; however, coal mining is not assigned to any other cluster with more than 20% posterior probability. Beverages are assigned to one cluster with 51% posterior probability; however, beverages are also assigned to a different cluster with more than 40% posterior probability. This type of ambiguity is not unusual in clustering models of this sort.

16Because the clusters are fairly well identified, the difference between the mean and the mode of depth of recession is small.

23

Table 4: Cluster Membership. This table displays the posterior probability of cluster membership for each industry. These probabilities are computed as the average of the draws of γNK across all Gibbs iterations.

				Cluster (K)
Industry Name	NAICS	1	2	3	4	5	6
Veneer, Plywood, and Engineered Wood Product	3212	0	0	0	1	0	0
Other Wood Product	3219	0	0	0	1	0	0
Clay Product and Refractory	3271	0	0	0	1	0	0
Glass and Glass Product	3272	0.41	0	0.07	0.19	0.32	0.01
Cement and Concrete Product	3273	0	0	0	1	0	0
Lime and Gypsum Product	3274	0	0	0	1	0	0
Other Nonmetallic Mineral Product	3279	0	0	0	1	0	0
Iron and Steel Products	3311	0	0	0	1	0	0
Alumina and Aluminum Production and Processing	3313	0	0	0	1	0	0
Nonferrous Metal	3314	0	0	0	1	0	0
Foundries	3315	0	0	0	1	0	0
Hardware	3325	0	0	0	1	0	0
Machine Shops, Turned Product, and Screw, Nut, and	3327	0	0	0	1	0	0
Other Fabricated Metal Product	3329	0	0	0	1	0	0
Agriculture, Construction, and Mining Machinery	3331	0	0	0	1	0	0
Industrial Machinery	3332	0	0	0	1	0	0
Commercial and Service Industry Machinery and Othe	3333	0	0	0	1	0	0
Ventilation, Heating, Air-Conditioning, and Commer	3334	0	0	0	1	0	0
Metalworking Machinery	3335	0	0	0	1	0	0
Engine, Turbine, and Power Transmission Equipment	3336	0	0	0	1	0	0
Computer and Peripheral Equipment	3341	0	0	0	1	0	0
Communications Equipment	3342	0	0	0	1	0	0
Audio and Video Equipment	3343	0	0	0	1	0	0
Semiconductor and Other Electronic Component	3344	0	0	0	1	0	0
Navigational, Measuring, Electromedical, and Contr	3345	0	0	0	1	0	0
Electric Lighting Equipment	3351	0	0	0	1	0	0
Household Appliance	3352	0	0	0	1	0	0
Electrical Equipment	3353	0	0	0	1	0	0
Other Electrical Equipment and Component	3359	0	0	0	1	0	0
Motor Vehicle	3361	0	1	0	0	0	0
Motor Vehicle Body and Trailer	3362	0	1	0	0	0	0
Motor Vehicle Parts	3363	0	1	0	0	0	0
Aerospace Product and Parts	3364	0	0	0	0	0	1
Ship and Boat Building	3366	0	0	0	0.29	0	0.71
Office and Other Furniture	3372	0	0	0	1	0	0
Logging	1133	0	0	0	1	0	0
Sawmills and Wood Preservation	3211	0	0	0	1	0	0
Forging and Stamping	3321	0	0	0	1	0	0
Cutlery and Handtool	3322	0	0	0	1	0	0
Architectural and Structural Metals	3323	0	0	0	1	0	0

24

Table 5: Cluster Membership (continued). This table displays the posterior probability of cluster membership for each industry. These probabilities are computed as the average of the draws of γNK across all Gibbs iterations.

				Cluster (K)
Industry Name	NAICS	1	2	3	4	5	6
Spring and Wire Product	3326	0	0	0	1	0	0
Coating, Engraving, Heat Treating, and Allied Acti	3328	0	0	0	1	0	0
Railroad Rolling Stock	3365	0	0	0	1	0	0
Other Transportation Equipment	3369	0.38	0	0	0.62	0	0
Household and Institutional Furniture and Kitchen	3371	0	0	0	1	0	0
Medical Equipment and Supplies	3391	0	0	0	1	0	0
Animal Food	3111	0	0	1	0	0	0
Grain and Oilseed Milling	3112	0	0	1	0	0	0
Sugar and Confectionery Product	3113	0	0	0	1	0	0
Fruit and Vegetable Preserving and Specialty Food	3114	0	0	0	0	1	0
Dairy Product	3115	0	0	0.01	0.02	0.91	0.06
Animal Slaughtering and Processing	3116	0.2	0	0	0.69	0.1	0.01
Other Food	3119	0.2	0	0	0.8	0	0
Beverage	3121	0.51	0	0.05	0.45	0	0
Tobacco	3122	0	0	0.72	0.18	0.1	0
Fabric Mills	3132	0	0	0	1	0	0
Textile and Fabric Finishing and Fabric Coating Mi	3133	0	0	0	1	0	0
Textile Furnishings Mills	3141	0	0	0	1	0	0
Other Textile Product Mills	3149	0	0	0	1	0	0
Apparel	315	0	0	0	1	0	0
Leather and Allied Product	316	0	0	0	1	0	0
Pulp, Paper, and Paperboard Mills	3221	0	0	0	1	0	0
Converted Paper Product	3222	0	0	0	1	0	0
Printing and Related Support Activities	323	0	0	0	1	0	0
Petroleum and Coal Products	324	0	0	0	0	0	1
Basic Chemical	3251	0	0	0	0	0	1
Resin, Synthetic Rubber, and Artificial and Synthe	3252	0	0	0	1	0	0
Pesticide, Fertilizer, and Other Agricultural Chem	3253	0	0	0	0.84	0.06	0.09
Pharmaceutical and Medicine	3254	0.13	0	0.01	0.83	0.03	0
Paint, Coating, and Adhesive	3255	0	0	0	1	0	0
Soap, Cleaning Compound, and Toilet Preparation	3256	0	0	0	1	0	0
Plastics Product	3261	0	0	0	1	0	0
Rubber Product	3262	0	0	0	1	0	0
Newspaper, Periodical, Book, and Directory Publish	5111	0	0	0	1	0	0
Bakeries and Tortilla	3118	0	0	0	1	0	0
Oil and Gas Extraction	211	0	0	0	0	0	1
Metal Ore Mining	2122	0	0	0	1	0	0
Nonmetallic Mineral Mining and Quarrying	2123	0	0	0	1	0	0
Support Activities for Mining	213	0.09	0.01	0.01	0.84	0.05	0
Electric Power Generation, Transmission, and Distr	2211	0.27	0.08	0	0.6	0.05	0
Natural Gas Distribution	2212	0.8	0	0.09	0.1	0	0

25

Figure 4: Cluster Cycles. This figure shows the median posterior draw of the factor ZKT for each cluster K. The gray bars reflect official NBER U.S. recession dates.

Cluster 1 Cycle Factor

2
0
-2
-4
1970	1975	1980	1985	1990	1995	2000	2005	2010	2015	2020

Cluster 2 Cycle Factor

10

0

-10

-20
1970	1975	1980	1985	1990	1995	2000	2005	2010	2015	2020

Cluster 3 Cycle Factor

5

0

-5
1970	1975	1980	1985	1990	1995	2000	2005	2010	2015	2020

Cluster 4 Cycle Factor

20
0
-20
1970	1975	1980	1985	1990	1995	2000	2005	2010	2015	2020
2													Cluster 5 Cycle Factor
0
-2
1970	1975	1980	1985	1990	1995	2000	2005	2010	2015	2020

Cluster 6 Cycle Factor

5

0

-5

-10
1970	1975	1980	1985	1990	1995	2000	2005	2010	2015	2020

26

The number of industries in each of the 6 clusters is not balanced. Cluster 4 is by far the largest cluster; the other clusters are relatively small, approaching-if not equal to-the minimum cluster size. These results suggest that the majority of the manufacturing economy have a common component to their cycles. Figure 4 shows the 6 cluster cycles. The fourth panel is the cluster cycle associated with the largest group of industries. Unsurprisingly, this factor declines during NBER recessions and-for the most part-rises during expansions. This cyclical pattern is not entirely attributable to the aggregate regime, which does not tend to switch at the beginning of NBER recessions. It does, however, suggest that the majority of industries follow the NBER cycle.

Apart from these, a few clusters of industries have common cycles that vary from the NBER cycle. Many of these also fall-to varying degrees-during NBER recessions. However, some-e.g., a collection of energy sectors assigned to Cluster 6-experience fluctuations different from the aggregate but common to the group. One cluster appears entirely acyclical.

4.4 Marginal Effects

The preceding section reviewed how the industries cluster; here, we investigate what factors might influence these clusters. The prior is logistic-populated by the cluster covariates, xNK , and the prior hyperparameters, βK -which are estimated jointly with the model parameters. This allows us to evaluate how changing an element of xNK affects the prior probability of cluster membership.

Table 6 shows the marginal effects-the changes in the cluster inclusion prior probabilities produced by a change in the specified element of xNK from one-standard-deviation below its mean to one-standard-deviation above its mean. In this calculation, all other elements of xNK are fixed at their respective means. The two cluster covariates evaluate the effects of differences in an industry's supply chain. An increase in INTRA-IO increases the percentage of that industry's production network that lies within its two-digit sector. An increase in INTER-IO, on the other hand, increases the percentage of the industry's production network that lies outside its two-digit sector.

27

Table 6: Marginal Effects of the Connectiveness Measures. This table shows the marginal effects of each connectiveness measure on the prior probability of cluster membership. The marginal effects are computed as the difference in prior probability when the covariate is one standard deviation above its mean minus the prior probability when the covariate is one standard deviation below its mean. The numbers indicated the posterior median and bold indicates the 68% highest posterior density interval does not include 0.

	Intra-IO	Inter-IO
Cluster 1	-0.09	0.05
Cluster 2	0.43	0.17
Cluster 3	-0.20	-0.14
Cluster 4	-0.02	-0.02
Cluster 5	-0.20	-0.23
Cluster 6	0.10	0.20

An increase in INTRA-IO raises the prior probability of belonging to Cluster 2, which

consists of industries in the production network for motor vehicles. This result is con-

sistent with Acemoglu et al. (2012) and Carvalho (2014), who use the 2008 automotive

bailout to illustrate the interconnectedness stemming from an overlap of suppliers and

dealers. Any failure up or down the supply chain can lead to severe production lags for

another company and drive comovements within the production network.

Increase INTER-IO raises the prior probability of belong to Cluster 6, which is mostly

composed of energy production industries, who have widespread production networks.

Clusters 3 and 5 are agricultural final goods; thus, shifting their supply chains to more

within-sector or more outside-sector both lower the probability of belonging to these

clusters.

4.5 Variance Decompositions

To determine whether aggregate, "sectoral", or industry shocks are relatively more impor-

tant, we compute the percentage of the variance of the cycle component, cNT, attributed

to the aggregate regime, ST, the shock to the cluster factor, uKT, and shock to the id-

iosyncratic component, ηNT. We calculate the implied variance decomposition based on

the model parameters [see Kose et al. (2003), Jackson et al. (2016)] and treat the binary

variable, ST, as a shock.

28

The final two columns of Tables 2 and 3 show the results of the variance decomposi- tions. These columns show the percentage of the total variance of the cyclical component attributed to the aggregate regime and the cluster factor shock; the remaining portion is attributed to the idiosyncratic component.

As one might surmise, the aggregate regime accounts for the majority of the cycle variance for most of the industries in the largest industry group, Cluster 4. This result is again consistent with previous studies who argue for the importance of the national shock relative to sectoral shocks. For most of the industries in our sample, this hypothesis is true.

For most of the industries allocated to the other groups, the cluster-or "sectoral"- level-factor explains a relatively larger portion of the variance of the cyclical component. The exceptions are the members in Cluster 1, which are acyclical industries. Recall that these clusters are generally smaller than a two-digit NAICS sector. The relative importance of the cluster factor for these industries suggests that, when properly defined, "sectoral" shocks can still explain industry fluctuations.

4.6 Industry Examples

Our model combines clustering and aggregate regime components that can summarize a large panel of data with enough flexibility to differentiate individual idiosyncratic series. Here, we consider a few of the industries that have particularly interesting historical time series. Figure 5 show the decompositions of four industries with various features in their trends and cycles.

The top-left panel shows the trend and cycle for the motor vehicle industry. This industry exhibits textbook characteristics of the plucking model: The trend drifts up and the cycle shows dramatic declines during NBER recessions.

The top-right panel shows apparel, an industry we mentioned in the discussion above. Apparel has a flat trend from the beginning of the sample through the mid-1990s, when it began a secular decline. The timing of the decline is subsequent to the ratification of NAFTA in 1993 and just predates the normalization of trade relations with China in

29

Figure 5: Industry Examples of Trend-Cycle Decomposition. This figure shows actual IP as well as the median posterior estimate for the trend component, τNT, and cycle component, cNT, for four select industries. The gray bars reflect official NBER U.S. recession dates.

									Motor Vehicle																		Apparel
500																			40	650																			15
480																			20																				10
460																				600
																			0																				5
440
																			-20	550																			0
420
400																			-40																				-5
																			500
380
																			-60																				-10
360																				450
340																			-80																				-15
320																			-100	400																			-20
1970	1975	1980	1985	1990	1995	2000		2005	2010	2015	2020	1970	1975	1980	1985	1990	1995	2000	2005	2010	2015	2020

Newspaper, Periodical, Book, and Directory Publish										Oil and Gas Extraction
540																				8		520																			4
520																				6																					2
																					500
																				4																					0
500
																						480																			-2
																				2
480																																									-4
																				0		460
460																																									-6
																				-2
																						440																			-8
440																				-4
																																								-10
420																						420
																			-6																					-12
400																				-8		400																			-14
1970	1975	1980	1985	1990	1995	2000	2005	2010	2015	2020		1970	1975	1980	1985	1990	1995	2000	2005	2010	2015	2020
																		Actual IP	Trend	Cycle

2000. Apparel is highly cyclical, consistent with clothing being a good that consumers delay purchasing until the economy recovers.

The bottom-left panel shows another industry we mentioned above is newspapers, periodicals, books, and directory publishing. The industry has an upward trend from the beginning of the sample until the late 1980s; the trend flattens out until around 1998 and then begins a secular decline. The year 1998 is important for the newspaper industry as it marks the first time a significant news story broke first over the internet.17

The bottom-right panel shows the trend and cycle for oil and gas extraction. While there are cyclical features, the two most dramatic movements for the industry occur during large declines in oil prices in September 2005 and September 2008. In September 2005 there was a peak in oil prices followed by a decline over the next two months. This peak falls between a peaceful transition of power in Saudi Arabia (August) and their inclusion in the World Trade Organization (November). The decline in September 2008

17The Drudge report broke news of the Bill Clinton/Monica Lewinsky scandal in 1998.

30

lies between a precipitous decline of oil prices during the Great Recession, which saw a peak of ✩133.88 per barrel in June and a trough of ✩39.09 per barrel in February 2009.

This industry also displays unique features in its trend. At the beginning of the sample, the industry experiences a slight secular decline; around 2006, when production from shale oil reserves increased dramatically, the trend in oil and gas extraction reversed course, rising for the balance of the sample.

5 CONCLUSION

One focus of the analysis of industrial business cycles has been the relative importance of aggregate, sectoral, and industry-specific shocks. A confounding problem that has gone somewhat unstudied is how to define sectors, the level of aggregation below the nation but above NAICS four-digit industries. Past studies tended to consider two-digit industries with the idea that comovement is driven by either demand shocks across common sector goods or supply shocks in production networks contained within the sector.

We develop a model that allows us to relax such restrictions by defining clusters of comoving industries based on their business cycle characteristics. We can then determine whether the comoving industries are related by intra- or inter-sectoral production networks.

We find that there are particular industry groupings that comove because their production networks are intra-sectoral. In particular, a group of motor vehicle production industries that share a production network tend to comove. On the other hand, a group of final goods agricultural industries that do not have prevalent intra- or inter-sectoral production networks comove. Finally, a large group of industries that belong to multiple sectors also comove. This last group may be consistent with stylized facts established previously in the literature that argue that aggregate shocks have become relatively more important than sectoral shocks. This latter result may suggest that, when the sectors are properly defined, subnational industry shocks are still relatively important.

31

Bibliography

Daron Acemoglu, Vasco M Carvalho, Asuman Ozdaglar, and Alireza Tahbaz-Salehi. The network origins of aggregate fluctuations. Econometrica, 80(5):1977-2016, 2012.

Enghin Atalay. How important are sectoral shocks? American Economic Journal: Macroeconomics, 9(4):254-80, 2017.

Maximo Camacho and Danilo Leiva-Leon.The propagation of industrial business cycles. Macroeconomic Dynamics, 23(1):144-177,2019.

Fabio Canova. Detrending and business cycle facts. Journal of Monetary Economics, 41 (3):475-512, 1998.

Gerald A Carlino and Robert H DeFina. How strong is co-movement in employment over the business cycle? evidence from state/sector data. Journal of Urban Economics, 55 (2):298-315, 2004.

Chris K Carter and Robert Kohn. On gibbs sampling for state space models. Biometrika, 81(3):541-553, 1994.

Vasco M Carvalho. From micro to macro via production networks. Journal of Economic Perspectives, 28(4):23-48, 2014.

George Casella and Edward I George. Explaining the gibbs sampler. The American Statistician, 46(3):167-174, 1992.

Yongsung Chang and Sunoong Hwang. Asymmetric phase shifts in us industrial production cycles. Review of Economics and Statistics, 97(1):116-133,2015.

Marcelle Chauvet and Jeremy Piger. A comparison of the real-timeperformance of business cycle dating methods. Journal of Business & Economic Statistics, 26(1): 42-49,2008.

Siddhartha Chib. Calculating posterior distributions and modal estimates in markov mixture models. Journal of Econometrics, 75(1):79-97,1996.

32

Lawrence J Christiano, Terry J Fitzgerald, et al. The business cycle: it's still a puzzle. Economic Perspectives-Federal Reserve Bank Of Chicago, 22:56-83, 1998.

Timothy Cogley and James M Nason. Effects of the hodrick-prescott filter on trend and difference stationary time series implications for business cycle research. Journal of Economic Dynamics and Control, 19(1-2):253-278, 1995.

Diego Comin and Thomas Philippon. The rise in firm-level volatility: Causes and conse- quences. NBER Macroeconomics Annual, 20:167-201, 2005.

Russell Cooper and John Haltiwanger. Inventories and the propagation of sectoral shocks. The American Economic Review, pages 170-190, 1990.

St´ephane Dupraz, Emi Nakamura, and J´on Steinsson. A plucking model of business cycles. Technical report, National Bureau of Economic Research, August 2020.

James Durbin and Siem Jan Koopman. A simple and efficient simulation smoother for state space time series analysis. Biometrika, 89(3):603-616, 2002.

James Durbin and Siem Jan Koopman. Time Series Analysis by State Space Methods. Oxford University Press, 2012.

Andrew T Foerster, Pierre-Daniel G Sarte, and Mark W Watson. Sectoral versus aggregate shocks: A structural factor analysis of industrial production. Journal of Political Economy, 119(1):1-38, 2011.

Neville Francis, Michael T Owyang, and Ozge Savascin. An endogenously clustered factor approach to international business cycles. Journal of Applied Econometrics, 32 (7):1261-1276, 2017.

Milton Friedman. The optimum quantity of money and other essays. Monetary Studies of the National Bureau, the National Bureau Enters its 45th Year, 44th Annual Report, 7-25, 1964.

Milton Friedman. The "plucking model" of business fluctuations revisited. Economic Inquiry, 31(2):171-177, 1993.

33

Sylvia Fr¨uhwirth-Schnatter.Finite Mixture and Markov Switching Models. Springer Science & Business Media, New York, 2006.

Sylvia Fr¨uhwirth-Schnatter and Rudolf Fr¨uhwirth. Data augmentation and mcmc for binary and multinomial logit models. In Statistical Modelling and Regression Structures, pages 111-132. Springer, 2010.

Julio Garin, Michael J Pries, and Eric R Sims. The relative importance of aggregate and sectoral shocks and the changing nature of economic fluctuations. American Economic Journal: Macroeconomics, 10(1):119-48, 2018.

Alan E Gelfand and Adrian FM Smith. Sampling-basedapproaches to calculating marginal densities. Journal of the American Statistical Association, 85(410):398-409,1990.

Manuel Gonz´alez-Astudillo. Estimating the us output gap with state-level data. Journal of Applied Econometrics, 34(5):795-810, 2019.

James D. Hamilton. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica, 57(2):357-384, 1989.

James D. Hamilton. Time Series Analysis. Princeton University Press, Princeton, NJ, 1994.

James D Hamilton. Why you should never use the hodrick-prescott filter. Review of Economics and Statistics, 100(5):831-843, 2018.

James D Hamilton and Michael T Owyang. The propagation of regional recessions. Review of Economics and Statistics, 94(4):935-947,2012.

Andrew Harvey and Siem Jan Koopman. Signal extraction and the formulation of unobserved components models. The Econometrics Journal, 3(1):84-107, 2000.

Sean Holly and Ivan Petrella. Factor demand linkages, technology shocks, and the business cycle. Review of Economics and Statistics, 94(4):948-963,2012.

34

Andreas Hornstein. The business cycle and industry comovement. FRB Richmond Economic Quarterly, 86(1):27-48, 2000.

Laura E Jackson, M Ayhan Kose, Christopher Otrok, and Michael T Owyang. Specifi- cation and estimation of bayesian dynamic factor models: A monte carlo analysis with an application to global house price comovement. In Eric Hillebrand and Siem Jan Koopman, editors, Dynamic Factor Models - Advances in Econometrics, volume 35, pages 361-400. Emerald Group Publishing Limited, 2016.

Nir Jaimovich and Henry E Siu. Job polarization and jobless recoveries. Review of Economics and Statistics, 102(1):129-147, 2020.

Marek Jaroci´nski. A note on implementing the durbin and koopman simulation smoother. Computational Statistics & Data Analysis, 91:1-3,2015.

Robert E Kass and Adrian E Raftery. Bayes factors. Journal of the American Statistical Association, 90(430):773-795, 1995.

Chang-Jin Kim and Charles R Nelson. State-SpaceModels with Regime Switching. The MIT Press, Cambridge, MA, 1999.

Kunhong Kim and Young Sik Kim. How important is the intermediate input channel in explaining sectoral employment comovement over the business cycle? Review of Economic Dynamics, 9(4):659-682, 2006.

M Ayhan Kose, Christopher Otrok, and Charles H Whiteman. International business cycles: World, region, and country-specific factors. American Economic Review, 93(4): 1216-1239, 2003.

Jim Lee. Trade integration and business cycle comovement: Evidence from the us. The International Trade Journal, 24(4):361-388,2010.

Danilo Leiva-Leon. Measuring business cycles intra-synchronization in us: A regime- switching interdependence framework. Oxford Bulletin of Economics and Statistics, 79 (4):513-545, 2017.

35

Nan Li and Vance L Martin. Real sectoral spillovers: A dynamic factor analysis of the great recession. Journal of Monetary Economics, 107:77-95, 2019.

James C Morley, Charles R Nelson, and Eric Zivot. Why are the beveridge-nelson and unobserved-components decompositions of gdp so different? Review of Economics and Statistics, 85(2):235-243, 2003.

Kevin M Murphy, Andrei Shleifer, and Robert W Vishny. Industrialization and the big push. Journal of Political Economy, 97(5):1003-1026, 1989.

Michael T Owyang, Jeremy Piger, and Howard J Wall. Business cycle phases in us states. Review of Economics and Statistics, 87(4):604-616, 2005.

Michael T Owyang, Jeremy M Piger, Howard J Wall, and Christopher H Wheeler. The economic performance of cities: A markov-switching approach. Journal of Urban Economics , 64(3):538-550, 2008.

Ali Ozdagli and Michael Weber. Monetary policy through production networks: Evidence from the stock market. Technical report, National Bureau of Economic Research, 2017.

David J Spiegelhalter, Nicola G Best, Bradley P Carlin, and Angelika Van Der Linde. Bayesian measures of model complexity and fit. Journal of the royal statistical society: Series b (statistical methodology), 64(4):583-639, 2002.

36