The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Economic Quarterly— Volume 102, Number 3— Third Quarter 2016— Pages 169–192
From Stylized to
Quantitative Spatial Models
of Cities
Sonya Ravindranath Waddell and Pierre-Daniel Sarte
1.
INTRODUCTION
Understanding how and why factors of production locate within and
around urban areas has been compelling social scientists for at least
150 years. Within mainstream economics, urban economists have been
developing modern theories of city systems at least since the 1960s.
However, modeling spatial interactions is highly complex, and, therefore, the theoretical literature on economic geography has necessarily
focused on stylized settings. For example, a model may have a central
business district— where …rms are assumed to be located— surrounded
by a symmetric circle or on a symmetric line. As the population grows,
the scarcity of land prevents consumers (who are also workers) from all
settling close to the center, so people move out to where commuting
costs are higher but housing costs are lower.
In the models of new economic geography (NEG), urban economists have incorporated advances developed in industrial organization,
international trade, and economic growth to remove technical barriers to modeling cities. The …eld of NEG was initiated primarily by
three authors: Fujita (1988), Krugman (1991), and Venables (1996),
who all use general equilibrium models with some version of monopolistic competition. The NEG models have been useful in helping to
pin down preferences, technology, and endowments and have provided
We thank Daniel Schwam and Daniel Ober-Reynolds for helpful research assistance.
The views expressed herein are those of the authors and do not necessarily represent the views of the Federal Reserve Bank of Richmond or the Federal Reserve
System. We thank Urvi Neelakantan, Santiago Pinto, and John Weinberg for helpful
comments and suggestions.
DOI: http://doi.org/10.21144/eq1020301
170
Federal Reserve Bank of Richmond Economic Quarterly
some fundamental theoretical explanation for the uneven distribution
of economic activity, for multiple equilibria in location choices, and for
a small (possibly temporary) asymmetric shock across sites to generate
a large permanent imbalance in the distribution of economic activities.
However, these models have also imposed structure that is not necessarily evident in the data, and the limitation of the analysis to stylized
spatial settings has not enabled an empirical literature that could directly corroborate the theory. In other words, the stylized models have
only guided empirical estimation in a way that is divorced from the
structure of those models, resulting in empirical research that has been
devoid of strong structural interpretations.
More recently, the introduction of quantitative models of international trade (in particular Eaton and Kortum [2002]) have served to
develop a framework that connects closely to the observed data. This
research does not aim to provide a fundamental explanation for the
agglomeration of economic activity but instead aims to provide an empirically relevant quantitative model. This article describes the progression from a simple canonical model of NEG to its counterpart in
the quantitative spatial framework. Section 2 engages the literature
to develop and understand the progression from the stylized models of
the NEG literature to the quantitative spatial models. Section 3 walks
through a version of the stylized model, with a linear monocentric city.
Section 4 introduces its counterpart as a quantitative spatial model as
was laid out in Redding and Rossi-Hansberg (forthcoming). Section 5
provides an example of how the spatial model can be matched to detailed microdata that describe actual interactions in the city. Section
6 concludes.
2.
LITERATURE REVIEW
The standard monocentric model of cities came out of a history of
work to model spatial allocations. The prototype for understanding
how factors of production distribute themselves across land, and how
prices govern that distribution, was developed by Johann Heinrich von
Thünen in the mid-nineteenth century to describe the pattern of agricultural activities in preindustrial Germany. Von Thünen’s model includes an exogenously located marketplace in which all transactions regarding …nal goods must occur and the di¤erences in land rent and use
are determined predominantly by transport costs (Fujita and Thisse
2002). The von Thünen model was both formalized mathematically
and enhanced in the second half of the twentieth century— including
the formalization of bid-rent curves by William Alonso in his basic urban land model. This basic urban model includes a monocentric city
Waddell & Sarte: From Stylized to Quantitative Spatial Models of Cities171
with a center, home to the central business district (CBD), where all
jobs are located. The space surrounding the CBD is assumed to be homogenous with only one spatial characteristic: its distance to the CBD.
Both the work of von Thünen and that of Alonso depended upon the
monocentricity of production activities— i.e., the models rely on one
CBD (or market) with surrounding land used for residential (or agricultural) purposes.
Although many early models assumed the existence of the CBD,
later work formalized mechanisms for the agglomeration forces that
create concentrations of economic activity. The models of NEG, as
summarized in Fujita et al. (1999), Fujita and Thisse (2002), and Ottaviano and Thisse (2004), create the framework to explain the imbalance
in the distribution of economic activity and better understand how a
small shock can generate that imbalance. These NEG models went a
long way toward overcoming the fundamental problem that kept economic geography and location theory at the periphery of mainstream
economic theory for so long: regional specialization and trade cannot
arise in the competitive equilibrium of an economy with homogenous
space. This spatial impossibility theorem is discussed more thoroughly
in Ottaviano and Thisse (2004) and articulated mathematically in Fujita and Thisse (2002).
Important ideas underlie the development of the NEG models.
These ideas (as described in Ottaviano and Thisse [2004]) include that
the distribution of economic activity is the outcome of a trade-o¤ between various forms of increasing returns and di¤erent mobility costs;
price competition, high transport costs, and land use foster the dispersion of production and consumption, and, therefore, …rms are likely to
cluster in large metropolitan areas when they sell di¤erentiated products and transport costs are low. Cities provide a wide array of …nal goods and specialized labor markets that make them attractive to
consumers/workers, and agglomeration is the outcome of cumulative
processes involving both the supply and demand sides. The contribution of NEG was to link those ideas together in a general equilibrium
framework with imperfect competition. Some of the earliest work in
NEG came from Krugman (1991), who developed a model that showed
that the emergence of an industrialized “core”and an agricultural “periphery” pattern depends on transportation costs, economies of scale,
and the share of manufacturing in national income (i.e., in consumption expenditures). More speci…cally, in his model, lower transportation costs, a higher manufacturing share, or stronger economies of
scale will result in the concentration of manufacturing in the region
that gets a head start compared to other regions. Venables (1996)
wrote a model where imperfect competition and transport costs create
172
Federal Reserve Bank of Richmond Economic Quarterly
forward and backward linkages between industries in di¤erent locations. He …nds that even without labor mobility, agglomeration can be
generated through the location decisions of …rms in industries that are
linked through an input-output structure. The models above develop
an argument for agglomeration into a single center of activity. However, other NEG models, most notably Fujita and Ogawa (1982) and
Lucas and Rossi-Hansberg (2002), introduced nonmonocentric models
where businesses and housing can be located anywhere in the city. The
latter models constitute a …rst step toward building frameworks that
more accurately capture the heterogeneity in economic activity across
space.
Unfortunately, although the theoretical work on NEG has been relatively rich, the empirical research has been comparatively less rich; establishing causality and controlling for confounding factors has proved
challenging in the empirical realm. One challenge, as articulated by
Redding and Rossi-Hansberg (forthcoming), is that the complexity of
the theoretical models has limited the analysis to stylized spatial settings, such as a few locations, a circle, or a line, and the resulting empirical research has been primarily reduced form in nature. As a result,
it is di¢ cult to provide a structural interpretation of the estimated coe¢ cients, and the empirical models cannot either withstand the Lucas
critique (coe¢ cients might change with di¤erent policy interventions)
or necessarily generalize to more realistic environments.
Empirical work, such as the spatial model laid out in Section 4, has
been instructed by another …eld of economics. Developments in the
international trade literature have o¤ered mechanisms for better modeling the distribution of economic activity across urban areas. Eaton
and Kortum (2002) developed a model of international trade that captures both the comparative advantage that encourages trade and the
geographic barriers that inhibit it (e.g., transport costs, tari¤s and quotas, challenges negotiating trade deals, etc.). They use the model to
solve for the the world trading equilibrium and examine its response to
policies.
This framework from the trade literature— combined with the availability of increasingly more granular data— enabled the emergence of
new quantitative spatial models in urban economics in which one can
carry out general equilibrium counterfactual policy exercises. In addition to o¤ering methodological insights and a mechanism for policy
analysis, these quantitative spatial models have made substantive contributions that borrow from, and contribute to, the theoretical literature. For example, Redding and Sturm (2008) provide evidence for a
causal relationship between market access and the spatial distribution
of economic activity. They show that the division of Germany after
Waddell & Sarte: From Stylized to Quantitative Spatial Models of Cities173
World War II led to a sharp decline in population growth in West German cities close to the new border relative to other West German cities
and that this decline was more pronounced for small cities than for
large cities. As another example, models such as those developed in
Ahlfeldt et al. (2015) and Monte et al. (2016), allow for heterogenous
gradients of economic activity within cities that can be matched directly to microdata and that can only be approximated in models such
as Fujita and Ogawa (1982) and Lucas and Rossi-Hansberg (2002).
The next section walks through a canonical monocentric urban
model and highlights key features that made that model attractive
for thinking about the distribution of economic activity across space.
In particular, this urban model allows many of a city’s features to be
endogenous, including its size, population, employment, wages, and
commercial land rents. In addition, at di¤erent locations within the
city, residential population, residential prices, and the consumption of
housing services can also be endogenous. In this model, as in the average city, production is concentrated at the center, where the CBD
is located, rent gradients decline with distance from the CBD, and
population density tends to decrease away from the city center.
3.
A STYLIZED MODEL OF CITIES
We consider a linear monocentric city with locations de…ned on the
interval [ B; B], where ` denotes the distance from the city center.
Each location ` is endowed with one unit of land available either for
residential housing or production. This analysis focuses on residential
localization decisions, i.e., the decisions of households rather than …rms.
The Central Business District
All production takes place at the city center, ` = 0, which de…nes the
CBD. Production per unit of land is given by
Y = A(L)L ;
(1)
where L denotes labor input and A(L) denotes a production externality. For simplicity, let A(L) = AL , < 1
< 1, and denote the
wage paid to workers by w. This condition ensures that labor demand,
L, is decreasing in the wage, w. There exists a unit mass of …rms (assuming …rms are small and do not internalize the externality) where
the representative …rm solves
max A(L)L
L
wL:
174
Federal Reserve Bank of Richmond Economic Quarterly
It follows that
A(L)L
1
=w,L=
1
A
w
1
:
(2)
We assume a competitive market with free entry so that in equilibrium …rms obtain zero pro…ts. Therefore, the commercial bid rent
faced by …rms in the business district is
+
b
q = (1
)A 1
1
1
w
:
(3)
Residential Areas
Workers live in the city at di¤erent locations, ` 2 [ B; B]nf0g, and
commute to the city center. Workers who reside at ` consume goods,
c(`), housing services, h(`), and experience a commuting cost, (`) 2
[1; 1), that reduces the utility derived from housing and increases with
distance from the CBD. In particular, the utility of a worker commut1
h(`)
,
ing from location ` to the CBD is given by s c(`)
(1 ) (`)
where 2 (0; 1) and s is a service amenity conferred by the city. This
approach to modeling commuting costs departs somewhat from the
more traditional approach of assuming that disposable income (thus
consumption of housing and nonhousing goods) declines with distance
from the CBD. In this case, similar to Ahlfeldt et al. (2015), commuting costs enter the utility function multiplicatively, which, as they note,
is isomorphic to a formulation in terms of a reduction in e¤ective units
of labor. Commuting costs are then ultimately proportional to wages
in the indirect utility function.
Conditional on living at location `, a worker then solves
u(`) = max s
c(`)
c(`);h(`)
h(`)
(1
) (`)
1
;
2 (0; 1)
subject to c(`) + q (`)h(`) = w;
r
where q r (`) is the price of a unit of residential housing services at
location `. Hence, we have that
c(`) = w;
(1
)w
h(`) =
;
r
q (`)
(4)
(5)
Waddell & Sarte: From Stylized to Quantitative Spatial Models of Cities175
and
1
w
(`)q r (`)
u(`) = s [w]
= sw [ (`)q r (`)]
1
:
(6)
The Residential Market
Let u denote the utility available to workers from residing in alternative
cities. To the extent that workers can move to or from another city and
are free to reside at any location within the city, it must be the case
that in equilibrium u(`) = u 8` 2 [ B; B]. Therefore, from equation
(6), we have that, for any location `,
sw [ (`)q r (`)]
1
r
= sw [ (B)qB
]
1
;
(7)
r is the price of land at the boundary of the city de…ned by the
where qB
opportunity cost of land at that location, such as an agricultural land
rent. Rewriting equation (7) gives residential land rents at di¤erent
locations within the city,
q r (`) =
(B) r
q ;
(`) B
(8)
where (B)
1 8` 2 [ B; B], since (`) increases with distance from
(`)
the city center. Thus, residential land rents are highest near the CBD
and decrease toward the boundaries of the city as commuting becomes
more expensive. However, as seen from equation (5), total housing
expenditures in this framework are constant across all locations in the
city since q r (`)h(`) = (1 )w, where (1 ) then represents the income
share of housing expenditures.
Recall that each location ` 2 [ B; B] is endowed with one unit of
land available for housing. Let R(`) denote the residential population
living at `. We assume that all available land in the city is fully developed and used by residents. Then, equilibrium in the housing sector
requires that
R(`)h(`) = 1.
(9)
176
Federal Reserve Bank of Richmond Economic Quarterly
In addition, the residential population living at di¤erent locations ` in
the city must sum up to the supply of labor working in the CBD,
ZB
R(`)d` = L
B
)
ZB
1
d` = L
h(`)
(10)
B
Solving for the City Equilibrium
We now describe the city equilibrium, …rst solving for equilibrium wages
as a function of the model parameters, from which all other city allocations immediately follow.
Given equations (5) and (8), equation (10) becomes
ZB
1
d` =
h(`)
B
ZB
r
(B)qB
q r (`)
d` =
(1
)w
(1
)w
B
ZB
1
d` = L;
(`)
(11)
B
which de…nes the boundaries of the city, B(L; w); as a function of its
population and wages given the model’s parameters.
Consider for instance the simple symmetric case where (`) = e j`j
ZB
1
B
so that (0) = 1 and (B) = e > 1. Then,
(`) d` gives
B
ZB
1
d` =
(`)
B
ZB
e
j`j
ZB
d` = 2 e
`
d` = 2(
`
e
jB
0)=
2
(1
e
B
);
0
B
so that equation (11) becomes
2
e
B q r (1
B
(1
e
)w
B)
=
2 r
qB (e B
)e
(1
B
1)
)w
=1+
=L
(1
)wL
:
r
2qB
Using the labor demand equation in equation (2), conditional on
the model parameters, the boundaries of the city may then alternatively
Waddell & Sarte: From Stylized to Quantitative Spatial Models of Cities177
be expressed in terms of wages only,
e
B
=1+
(1
1
A
w
(1
)w
=1+
r
2qB
)(A ) 1
r
2qB
1
1
w1
:
Using this last expression, we can solve for equilibrium wages in
the city as a function of the model parameters only. Speci…cally, note
that residents’utility at the boundary is given by
# 1
"
1
w1
(1
)(A ) 1
r
u = sw qB +
;
(12)
2
r ; u).
which de…nes w = w(s; ; ; A; ; ; qB
Proposition 1: There exists a unique w that solves equation (12).
"
# 1
r +
Proof: De…ne f (w) = sw qB
(1
1
)(A ) 1
w1
. Then
2
limw!0 f (w) = 0, limw!1 f (w) = 1, and, since f (w) is continuous
in w, there exists w such that u = f (w ). Moreover, since f (w) is
strictly increasing in w, w is unique.
Given w , all other allocations in the city then immediately follow.
In particular, as mentioned in the proposition, given parameter restrictions, the RHS of equation (12) is increasing in w so that w then increases with u. Thus, as the reservation utility from living elsewhere, u,
1
increases, the city population, L =
A
w
1
, falls as residents leave
1
)(A ) 1
r
2qB
1
w
),
the city, and its boundaries, B = log(1 +
shrink.
The stylized model described above is rich enough to allow for many
of a city’s features to be endogenous, including its size, population, employment, wages, and commercial land rents. In addition, at di¤erent
locations within the city, residential population, residential prices, and
the consumption of housing services can also be endogenous. These
allocations are such that there exists a very direct link between commuting costs to the CBD and residential prices. Speci…cally, taking
equation (8) and using the functional form for commuting costs described above, we can derive a simple expression for the elasticity of
residential prices with respect to commuting costs.
1
(1
Proposition 2: The elasticity of residential prices with respect to
commuting costs, "qr ; , is given by (B j`j).
178
Federal Reserve Bank of Richmond Economic Quarterly
Proof:
q r (`) =
"q r ; =
@q r (`)
= (B
@
q r (`)
e
e
B
qr
j`j B
j`j)e
=e
(B j`j) r
qB
(B j`j) r
qB
e
(B j`j) q r
B
= (B
j`j):
The proposition above highlights the e¤ect of commuting costs on
prices; speci…cally, this e¤ect is mitigated as we move away from the
employment center and is zero at the boundary. Intuitively, away from
the city center, residential prices become increasingly pinned down by
the agricultural land rent rather than economic activity near the center.
Despite its richness, the stylized model we have just described imposes a number of restrictions on the structure of the city, including its monocentric nature with all production being concentrated in
the CBD. Furthermore, residential prices decline monotonically as one
moves away from the city center, and there exists a general symmetry
and an evenness in allocations and prices across space. This smooth
and symmetric aspect of the city is illustrated in Figure 1. In that
…gure, residential population is highest near the CBD, where the commute is relatively cheap, and decreases monotonically away from the
center with the fewest workers living near the boundaries of the city.
In practice, of course, economic activity is more unevenly distributed across space. For example, Figure 2 shows that the city of Richmond, Virginia, has multiple employment clusters, one indeed in the
center of the city but two others to the south and west.
This activity re‡ects a balance of agglomeration forces (e.g., production externalities) and dispersion forces (e.g., commuting costs) that
play out in intricate and interrelated ways across space and that lead
to substantial variations in allocations and prices across a city. For example, production may take place in di¤erent parts of the city so that
cities with multiple production centers are not uncommon. In fact,
some productive activity potentially takes place at every location in the
city. Moreover, residential prices, even if they tend to fall away from a
central point in the city, seldom fall monotonically with distance from
that center. Instead, residential rents can exhibit substantial variation
across locations within the city. This variation re‡ects the potential
complexity of linkages within the city where, for example, the resident
population at a given location may depend on the entire distribution of
wages o¤ered across the city. Thus, in the next section, we show how
Waddell & Sarte: From Stylized to Quantitative Spatial Models of Cities179
Figure 1 Allocations and Prices in the Monocentric Model
to modify the stylized model presented in this section in a way that
can quantitatively account for the spatial allocations and prices it is
meant to study.
4.
A QUANTITATIVE SPATIAL MODEL OF CITIES
In this section, we show how to adapt the stylized model of the previous section to allow for the heterogeneity in spatial allocations and
prices that is typically observed in cities. In doing so, we preserve the
basic assumptions on preferences, technology, and endowments of our
stylized model to keep the frameworks comparable. Instead of thinking
of the city as located on an interval [ B; B], we will think of the city
as composed of J distinct locations, indexed by j 2 f1; :::; Jg (or i). In
the mapping to data, these locations may represent city blocks, census
tracts, or counties. It is this key change that will allow us to ensure
that the model is at least able to match given observed spatial allocations of, for example, resident population, land rents, employment, or
wages across locations in a city. Any subsequent counterfactual exer-
180
Federal Reserve Bank of Richmond Economic Quarterly
Figure 2 Density of Primary Jobs
cise involving a change to some exogenous aspect of the city is then
grounded in a model that is able to exactly replicate uneven spatial
observations that re‡ect, at least in part, complex linkages between
decisions involving where to reside and where to work within the city.
For example, the model would enable us to understand the e¤ect of
a new urban policy, such as one that provides housing assistance or
subsidized transportation.
Waddell & Sarte: From Stylized to Quantitative Spatial Models of Cities181
In a model where every location could potentially be used for both
residential and production purposes, a central component of a quantitative spatial model is the representation and matching of distinct
pairwise commuting ‡ows from any location j in the city to any other
location i. This step will rely on an approach developed by Eaton and
Kortum (2002) in modeling trade ‡ows between locations. As in the
model developed in the previous section, this analysis will focus on residential localization decisions, i.e., the decisions of households rather
than …rms. Unlike in the previous model, the commercial bid rent
schedule is nondegenerate and re‡ects variations in productivity and
wages across locations.
Firms
Production per unit of land in the business district of each location i
is given by
Yi = A(Li )Li ;
(13)
analogously to equation (1), where Li denotes labor input and A(Li )
denotes a production externality that we assume is local (so only employment in i a¤ects the productivity of businesses in i). For simplicity,
let A(Li ) = Ai Li , < 1
< 1, and denote the wage paid to workers in location i by wi . There exists a unit mass of …rms (assuming
that …rms are small and do not internalize the externality) where the
representative …rm solves
max A(Li )Li
Li
wi Li :
It follows that
A(Li )Li
1
= w , Li =
1
Ai
wi
1
:
(14)
As in the previous model, …rms operate in a competitive market with
free entry and thus obtain zero pro…ts in equilibrium. The implied
commercial bid rent schedule faced by …rms in the business district is
qib
+
1
= (1
1
1
)Ai
wi
:
(15)
Note the similarities between equations (14) and (15), and the analogous equations in the previous section, equations (2) and (3).
182
Federal Reserve Bank of Richmond Economic Quarterly
Residents
In each location j of the city, there exists a residential area composed of
a continuum of residents who commute to the business areas of di¤erent
locations i for work. These residents di¤er in their preferences for where
to work in the city according to a random idiosyncratic component
s. Unlike the previous model where s was a city amenity distributed
uniformly across locations, in this model, s is an individual-speci…c
preference component. Conditional on living in a particular location
j, this preference component captures the idea that residents of j may
have idiosyncratic reasons for commuting to di¤erent locations i in the
city. We model the idiosyncratic preference component associated with
residing in location j and working in location i as scaling the utility of
the residents of region j, where s is drawn from a Fréchet distribution
speci…c to that particular commute,
Fij (s) = e
ij s
;
> 0;
ij
> 0:
(16)
Residents of j who commute to i incur an associated cost, ij 2
[1; 1), that, analogous to the previous section, reduces the utility derived from housing. Thus, conditional on living in j and working in i,
the problem of a resident having drawn idiosyncratic utility s is given
by
uij (s) =
subject to cij (s) +
j
r
qj hij (s)
max
s
cij (s);hij (s)
cij (s)
j
j
hij (s)
(1
j)
1
;
ij
2 (0; 1)
= wi ;
where qjr is the price of a unit of residential housing services at location
j. Hence, we have that
cij (s) =
hij (s) =
j wi ;
(17)
(1
j )wi
qjr
;
(18)
and
uij (s) = s [wi ]
= swi
j
"
r
ij qj
wi
r
ij qj
j
#1
1
:
j
j
(19)
Note the similarities between equations (17), (18), and (19) and the
analogous equations in the previous sections, equations (4), (5), and
(6).
Waddell & Sarte: From Stylized to Quantitative Spatial Models of Cities183
Aggregation
The setup we have just described allows for a considerable degree of
heterogeneity within the city compared to the stylized model presented
earlier. In particular, all locations allow for simultaneous use by both
businesses and residents (mixed use), individuals living in any location
may commute to any other location for work, and commute costs between any two locations are speci…c to that pair of locations, so that
it is possible to take into account the particular geographical makeup
or road infrastructure of a city. However, having allowed for this high
level of heterogeneity in the city, it becomes important to be able to
aggregate economic activity at the level of a location, such as a census
tract for practical purposes. The steps in this subsection address this
question.
Distribution of Utility
Since residents of j who work in i have di¤erent preferences s, drawn
from equation (16), for commuting to that location, it follows that
0 h
i1 j 1
r
u
q
ij j
C
B
Gij (u) = Pr(uij < u) = Fij @
A;
wi
or
ij u
Gij (u) = e
;
ij
=
r (
ij qj
ij wi
j
1)
:
(20)
Each resident of j chooses to commute to the location i that o¤ers
maximum utility of all possible locations. Therefore,
Y
Gj (u) = Pr(maxfuij g < u) =
Pr(uij < u)
i
=
Y
e
i
ij u
:
i
Thus, it follows that
Gj (u) = e
ju
;
j
=
X
ij :
(21)
i
In other words, the distribution of resident utility in each location j
of the city is itself a Fréchet distribution. The expected utility from
residing in j is then given by
!1
X
1
(
1)
1
j
qjr j
;
(22)
uj =
ij wi ij
i
184
Federal Reserve Bank of Richmond Economic Quarterly
where (:) is the Gamma function.1 The expected utility from living in
j, therefore, is a weighted average of the utility gained from commuting
to the di¤erent business areas (raised to the ). Observe that in contrast
to utility in the stylized model, equation (6), the expected utility from
living in location j of the city now involves not only the price of housing
at that location, but also information about the entire city, including
the entire distribution of wages and associated commuting costs, since
residents of j can in principle commute to any other location i to work.
Commuting Patterns
Let ij represent the proportion of residents living at location j and
commuting to location i. Commuting patterns can then be described
by the following relationship,
Rij =
ij Rj ;
where Rij and Rj are, respectively, the number of residents commuting
from j to i and the total number of residents living at j. In particular,
ij
= Pr uij > maxfunj g :
n6=i
ij u
From equation (20), we have that Gij (u) = e
u ( +1) ij e ij u . It follows that
ij
=
Z1
u
( +1)
ij e
0
ij u
e j (u)du;
G
so that gij (u) =
e j (u) is de…ned as in equation (21) but with e j =
where G
which also implies that j = e j +
this expression reduces to
=X
X
nj ,
n6=j
ij . In Appendix B, we show that
( j 1)
ij
( j 1)
ij wi ij
ij wi
ij
(23)
:
(24)
i
In other words, the proportion of residents living in j and commuting to
i for work depends on wages earned in i adjusted for commuting costs
when coming from j, relative to a weighted average of wages earned
elsewhere adjusted for the corresponding commute (raised to the ).
1
A derivation of this result is given in Appendix A.
Waddell & Sarte: From Stylized to Quantitative Spatial Models of Cities185
The residential price at location j does not a¤ect commuting patterns
from j to i since it is speci…c to j and is faced by any resident
who
X
wants to live at j regardless of commute. By construction,
=
1.
ij
i
The Residential Market
Recall that hij (s) = hij represents housing consumption for those living
in j and commuting to i. It follows that average housing per resident
at location j, hj , is given by
X
hj =
ij hij
i
=
j)
(1
qjr
X
ij wi :
i
As in the stylized model of the previous section, we assume that each
location is endowed with one unit of land available for housing and that
this land is fully developed.2 In equilibrium, therefore, the residential
market must satisfy Rj hj = 1 similarly to equation (9) or
qjr
X
Rj =
:
(25)
(1
ij wi
j)
i
As in the previous section, let u denote the utility available to individuals from residing in alternative cities. To the extent that workers
can move to or from another city, and are free to reside at any location
within the city, it must be the case that in equilibrium uj = u 8j.
Therefore, we have that for any location,
!1
X
1
( j 1)
r j 1
qj
u=
ij wi ij
)
qjr
=
"
u
1
#
i
1
j 1
X
i
ij wi
( j 1)
ij
!
1
(1
j)
:
(26)
Comparing the residential price at location j, qjr , with its simpler analog
in the stylized model in equation (8), it is clear that the quantitative
spatial model allows residential prices to be determined by many more
2
Owens et al. (2017) present a more ‡exible model in which residential land in
any one location may be vacant, partially developed with some areas left for developers
to build on, or fully developed. In that model, a coordination problem arises between
developers and residents (no one wants to be the …rst mover) that potentially traps
neighborhoods in an equilibrium where they remain vacant.
186
Federal Reserve Bank of Richmond Economic Quarterly
factors, including the distribution of wages across all locations in the
city as well as all commuting costs. It is this richness that allows
for spatial variation in allocations and prices across locations in the
city that is unavailable in the more stylized framework of the previous
section.
The City Labor Market
Since ij Rj denotes the number of residents living in location j who
commute to the business area of location i for work, labor market
clearing in the city requires that
Li =
J
X
ij Rj ;
j=1
or alternatively
Ai
wi
1
1
=
J
X
ij Rj :
(27)
j=1
Solving for the City Equilibrium
We represent the parameters of the quantitative spatial model in a
vector, P = ( ; ; ; u; j ; Ai ; ij ; ij ). Conditional on P, equations
(24), (25), (26), and (27) then make up a system of J 2 + 3J equations
in the same number of unknowns, ij (P), Rj (P), qjr (P), and wi (P).
Importantly, the equilibrium allocations in this model allow for considerably more heterogeneity than in the stylized model of the previous
section. Since they are speci…c to locations within the city, equilibrium
allocations of the quantitative spatial model such as commuting patterns, ij (P), or equilibrium prices, such as residential prices, qjr (P),
and wages, wi (P), may be directly matched to their data counterpart
at the block or census tract level. In contrast, equilibrium allocations
of the stylized model in the previous section could only be indexed by
distance, `, from a central point in the city. The next section addresses
this last point in more detail.
Unlike the conventional monocentric model of the previous section,
equilibrium existence and uniqueness are more challenging to prove in a
quantitative spatial framework. However, Appendix C summarizes the
key equations needed to compute the model equilibrium and provides
an algorithm that yields the corresponding numerical solution given
the model’s parameters, P. Moreover, despite its added complexity, the
quantitative spatial model retains some degree of analytical tractability.
Waddell & Sarte: From Stylized to Quantitative Spatial Models of Cities187
For instance, as in the monocentric model of the previous section, we
can derive a simple expression for the elasticity of residential prices
with respect to commuting costs.
Proposition 3: The elasticity of residential prices with respect to
commuting costs, "qjr ; ij , is given by
ij .
Proof: We have that
qjr
1
j 1
u
=
1
(
)
Then
@qjr
=
@ ij
u
1
!
1
j 1
1
(1
j)
X
ij wi
( j 1)
ij
i
=
P
!
i
ij wi
1
(1
"qjr ;
ij
@qjr
=
@ ij
ij
qjr
=
ij wi
(
P
i
1)
j
( j 1) 1
ij
( j 1)
ij wi ij
ij wi
1 P
( j 1) 1 r
qj
ij
( j 1)
ij wi ij
1
(1
ij
qjr
j)
.
1
j)
i
It follows that
( j 1)
ij
=
ij wi
qjr
ij :
This …nding is intuitive. A 1 percent increase in commute costs
between any two locations, ij , lowers residential prices by the share
of residents a¤ected by that commute. In this relatively simple spatial
environment, even if it allows for more ‡exibility than the monocentric
setup, the relationship is exact. More importantly, unlike the analogous
elasticity in the more stylized model, the share of residents is itself an
endogenous outcome that depends on all of the city’s characteristics,
P, and thus will move along with the entire distribution of wages and
population across locations in any policy experiment.
5.
MATCHING THE QUANTITATIVE SPATIAL
MODEL TO URBAN MICRODATA
As elaborated upon in earlier sections, it is now possible to model cities
by matching these types of quantitative spatial models to available microdata. For the purpose of the discussion below, the parameters in P
fall into two broad classi…cations: citywide parameters and locationor neighborhood-speci…c parameters. The parameters in P that are
( j 1)
ij
1
188
Federal Reserve Bank of Richmond Economic Quarterly
not location speci…c have generally accepted values in the literature.
For example, Monte et al. (2016) estimate (the parameter that governs the shape of the distribution of the idiosyncratic preference, s, of
commuting from i to j) to be 4.43. Ciccone and Hall (1996) estimate
(the production externality) to be 0.06, and Ahlfeldt et al. (2015)
estimate (the parameter that de…nes the relationship between labor
and output, separate from the externality) to be 0.80, while u is a
normalizing constant. The parameters that are location-speci…c potentially present a greater computational challenge since there are many
of them. For example, in a city with 1; 000 census tracts, there are
1; 000; 000 ij ’s. Other location-speci…c parameters, such as pairwise
commuting costs, ij , may be directly calibrated to data on distances
or commuting times.
The Longitudinal Employer-Household Dynamics Origin-Destination
Employment Statistics provide reliable data on cities at the census tract
level, including commuting patterns ( ij ), resident population (Rj ),
employment (Li ), and wages (wi ). Other detailed data on cities are
also available; for example, residential prices (qjr ) are available from
CoreLogic or county assessors’o¢ ces. In general, such data show considerable unevenness across space within a city.
We now describe how, in our simple framework, the location-speci…c
parameters of our quantitative spatial model may be calibrated to exactly match, in equilibrium, all pairwise commuting patterns, ij , the
exact distribution of population across space, Rj , and thus also the
distribution of employment, Li , and the exact distribution of wages in
the city, wi , in a given benchmark period. In particular, we choose
the parameters of the model ( ij ; j ; Ai ) to match the observations
( ij ; Rj ; wi ) as equilibrium outcomes. In this way, counterfactual exercises involving a change to some exogenous aspect of the city, or a
change in urban policy, are rooted in a model that, as a benchmark, is
able to exactly match basic observed allocations and prices in the city
as equilibrium outcomes.
Recall that commuting patterns, ij , are given by equation (24),
( j 1)
ij
( j 1)
ij wi ij
ij wi
ij
=X
;
i
where ij 2 [1; 1). If ij = 0, then either ij = 0 or ij ! 1.
Commuting patterns can be alternatively expressed in terms of the
Head and Ries (2001) index,
ij
jj
=
( j 1)
ij
( j 1)
jj wj jj
ij wi
:
Waddell & Sarte: From Stylized to Quantitative Spatial Models of Cities189
Then, conditional on = 4:43 and values for j , the preference parameters, ij , can then be chosen to be consistent with commuting
patterns,
(1 j )
wj
ij
jj
;
(28)
wi
min
jj
where min is a lower bound on jj .3 Since ij may be directly inferred
from commuting costs data, this approach to obtaining ij to match
commuting patterns presumes we are also able to match wages, wi , as
part of the model inversion. We show below that this can indeed be
done through the choice of location-speci…c productivities, Ai . With
this in mind, we …rst choose j so as to match the distribution of
resident population across space, Rj , conditional on ij and wi .
The number of residents in location j is given by
qjr
P
:
(29)
Rj =
(1
j)
i ij wi
ij
=
ij
Using equations (26) and (28) with min = 1, and the normalization
jj
= 1, for all locations j, equation (29) simpli…es to
jj
1
Rj ( j ) =
Notice that as
wj
u
! 0+ , Rj ( j ) !
!
1
1
j
j)
(1
(P 1 )wj
1
P
i
ij wi
:
(30)
and as j ! 1 , Rj ( j ) !
( 1 )wj
1. Therefore, one may choose u so that Rj > u P ij wi for all j and
i
numerically solve the expression in equation (30) to obtain a set of j
that exactly matches the distribution of Rj , conditional on ij and wi .
Since the distribution of j then depends on u, and j represents the
share of income spent on housing in a given census tract j, we choose
u so that the mean of j is 0:76 as in Ahlfeldt et al. (2015).
Since commuting patterns ij can be exactly matched, given j and
wages wi , through the choice of ij in equation (28), it remains only
to ensure that the model is consistent with the spatial distribution
of wages in an equilibrium benchmark version of the model. Using
equation (26), we can write the city labor market clearing condition,
equation (27) as
j
Ai
wi
3
Since
X
i
ij
u
i
1
1
=
ij wi
J
X
ij Rj ;
j=1
= 1, one needs to also normalize the
ij ’s,
for example,
jj
jj
= 1 8j.
190
Federal Reserve Bank of Richmond Economic Quarterly
in which case we can simply choose location-speci…c productivities,
Ai , to ensure that equilibrium benchmark wages exactly match the
distribution of wages in the city,
8
J
wi <X
Ai =
:
j=1
ij Rj
91
=
;
:
(31)
Observe that on the righthand side of equation (31), we are free to use
the data on commuting patterns, ij , and residential population, Rj ,
since those are matched by construction through the choices of ij and
j in equations (28) and (30).
6.
CONCLUSION
The development of the new quantitative equilibrium models has initiated a more robust and realistic framework with which to model cities.
This framework will enable urban economists to provide empirically
driven insight into future theoretical or structural work on how cities
grow, shrink, and change. By o¤ering a more accurate grounding for
empirical models, it will also allow for more robust counterfactual policy exercises that can inform practitioners and policymakers regarding
strategies for urban development.
Waddell & Sarte: From Stylized to Quantitative Spatial Models of Cities191
REFERENCES
Ahlfeldt, Gabriel M., Stephen J. Redding, Daniel M. Sturm, and
Nikolaus Wolf. 2015. “The Economics of Density: Evidence from
the Berlin Wall.” Econometrica 83 (November): 2127–89.
Ciccone, Antonio, and Robert E. Hall. 1996. “Productivity and the
Density of Economic Activity.” American Economic Review 86
(March): 54–70.
Eaton, Jonathan, and Samuel Kortum. 2002. “Technology,
Geography, and Trade.” Econometrica 70 (September): 1741–79.
Fujita, Masahisa. 1988. “A Monopolistic Competition Model of
Spatial Agglomeration: Di¤erentiated Product Approach.”
Regional Science and Urban Economics 18 (February): 87–124.
Fujita, Masahisa, Paul Krugman, and Anthony J. Venables. 1999. The
Spatial Economy: Cities, Regions, and International Trade.
Cambridge, Mass.: MIT Press.
Fujita, Masahisa, and Hideaki Ogawa. 1982. “Multiple Equilibria and
Structural Transition of Non-Monocentric Urban Con…gurations.”
Regional Science and Urban Economics 12 (May): 161–96.
Fujita, Masahisa, and Jacques-François Thisse. 2002. Economics of
Agglomeration: Cities, Industrial Location, and Regional Growth.
Cambridge: Cambridge University Press.
Krugman, Paul. 1991. “Increasing Returns and Economic
Geography.” Journal of Political Economy 99 (June): 483–99.
Lucas, Robert E., and Esteban Rossi-Hansberg. 2002. “On the
Internal Structure of Cities.” Econometrica 70 (July): 1445–76.
Monte, Ferdinando, Stephen J. Redding, and Esteban Rossi-Hansberg.
2016. “Commuting, Migration, and Local Employment
Elasticities.” Princeton University Working Paper (October).
Ottaviano, Gianmarco, and Jacques-François Thisse. 2004.
“Agglomeration and Economic Geography.” In Handbook of
Regional and Urban Economics, vol. 4, ed. J. Vernon Henderson
and Jacques-François Thisse. North Holland: Elsevier: 2563–608.
Owens, Raymond III, Esteban Rossi-Hansberg, and Pierre-Daniel
Sarte. 2017. “Rethinking Detroit.” Working Paper 23146.
Cambridge, Mass.: National Bureau of Economic Research.
(February).
192
Federal Reserve Bank of Richmond Economic Quarterly
Redding, Stephen J., and Esteban Rossi-Hansberg. Forthcoming.
“Quantitative Spatial Economics.” Annual Review of Economics.
Redding, Stephen J., and Daniel M. Sturm. 2008. “The Costs of
Remoteness: Evidence from German Division and Reuni…cation.”
American Economic Review 98 (December): 1766–97.
Venables, Anthony J. 1996. “Equilibrium Locations of Vertically
Linked Industries.” International Economic Review 37 (May):
341–59.
Waddell & Sarte: From Stylized to Quantitative Spatial Models of Cities193
APPENDIX:
APPENDIX A
Under the maintained assumptions, expected utility at location j is
given by
uj =
Z1
ju
e
ju
du:
ju
( +1)
0
Consider the change in variables,
y=
ju
; dy =
du:
Then, we have that
uj =
Z1
1
j
y
1
e
y
1
dy =
1
j
;
0
from which equation (22) in the text follows.
APPENDIX:
APPENDIX B
From equation (23), we have that
ij
=
Z1
u
( +1)
ij e
ij u
=
Z1
u
( +1)
ij e
ju
e
0
eju
du
du:
0
Consider the change of variables,
y=
ju
; dy =
ju
( +1)
du:
194
Federal Reserve Bank of Richmond Economic Quarterly
It then follows that
=
ij
ij
Z1
( +1)
u
e
ju
du
0
ij
=
j
ij
=
Z1
e
y
dy
0
;
j
where
j
recall
= qjr
(
j
1)
that
X
=
ij
( j 1)
ij
ij wi
ij wi
h
r
ij qj
i(
j
1)
and
. Equation (24) in the text directly
i
follows.
APPENDIX:
APPENDIX C
The Basic Set of Equations and Unknowns
S
Let LD
i and Li represent, respectively, labor demand and labor supply in location i. Given a benchmark or counterfactual set of parameters, P, each endogenous variable in the model can ultimately be
expressed as depending only on a vector of wages across all locations,
w = (w1 ; :::; wJ )0 , and P,
ij (w)
qjr (w)
=P
=
Rj (w; ; q) =
"
(1
i
( j 1)
ij
( j 1)
ij wi ij
ij wi
u
1
#
qjr
P
j)
i
1
Ai
=
wi
X
S
Li ( ; R) =
ij Rj :
LD
i (w)
j
1
j 1
;
X
i
ij wi
1
;
;
ij wi
( j 1)
ij
!
1
(1
j)
;
Waddell & Sarte: From Stylized to Quantitative Spatial Models of Cities195
Then, …nding an equilibrium of the model is equivalent to …nding a
S
vector of wages that clears the labor market; LD
i (w) = Li (w) for all
locations i = 1; :::; J. Put another way, the task is to …nd a vector
w 2 RJ+ such that
(LD
i
Ai
wi
LSi )(w ) =
1
X
1
j
( j 1)
ij wi ij
P
( j 1)
i ij wi ij
1
j 1
u
(
1
ij wi
i
)
(1
P
j)
P
i P
( j 1)
ij
( j 1)
ij wi ij
ij wi
i
1
( j 1)
ij
(1
wi
=0
Several algorithms exist to numerically solve nonlinear system of equations, and MATLAB’s fsolve function handles this particular system
well.
Numerical Algorithm
Some quantitative spatial models can result in systems whose features
(such as nondi¤erentiability in the presence of thresholds or binding
constraints on available land) make traditional algorithms di¢ cult to
apply. In such cases, a simple “guess-and-iterate” method can be constructed to calculate solutions. We outline such a method here as it
applies to our model.
1. Choose a tolerance level " > 0 and guess a vector of wages, wn .
2. Calculate the implied matrix of ‡ows:
3. Calculate
qjr (wn ) =
the
1
j 1
u
(
1
)
ij (w n ) =
implied
P
i
( j 1)
ij wn;i ij
( j 1)
ij wn;i ij
P
( j 1)
i ij wn;i ij
.
prices:
1
(1
j)
.
4. Using the prices and ‡ows calculated in steps two and three, calqjr (wn )
culate the implied number of residents: Rj (wn ) = (1 ) P
.
ij (w n )wi
j
i
5. Using the residents calculated in step four, calculate
the implied
P
S
labor supply in each labor market: Li (wn ) = j ij (wn )Rj (wn ).
6. Calculate the implied labor demand in each labor market: LD
i (w n ) =
1
Ai
wi
1
:
j)
196
Federal Reserve Bank of Richmond Economic Quarterly
7. At the vector of wages, wn , calculate the implied excess demand
for labor in each market: LD
LSi (wn ):
i (w n )
P
S
8. If the aggregate labor market fails to clear, i jLD
i (w n ) Li (w n )j >
", then update the vector of wages as follows:
wn+1 = wn +
LD
i (w n )
LSi (wn ) ;
for some > 0. This updating rule raises wages in markets
where there is excess demand for labor or reduces it where there
is excess supply.
P
S
9. Repeat steps two through eight until i jLD
":
i (w n ) Li (w n )j
Economic Quarterly— Volume 102, Number 3— Third Quarter 2016— Pages 197–223
Beveridge Curve Shifts and
Time-Varying Parameter
VARs
Thomas A. Lubik, Christian Matthes, and Andrew Owens
A
t …rst glance, many macroeconomic time series exhibit some
form of nonlinearity. For instance, output growth and in‡ation show less volatility in the 1980s and 1990s than during
the Great In‡ation period of the 1970s, an observation that has been
labeled the Great Moderation. Over the business cycle, the unemployment rate exhibits an asymmetric sawtooth pattern whereby it rises
rapidly during downturns and declines only gradually during a recovery.
Many price variables, such as exchange rates or commodity prices, appear stable for a long period followed by sudden level shifts. The literature has studied various speci…c forms of nonlinearity— such as structural breaks, time-varying volatility, or business cycle asymmetries—
using sophisticated time-series methods ranging from threshold and
Markov switching to vector-autoregressions (VARs) with time-varying
parameters and stochastic volatility. The result as to whether there is
nonlinearity in the data has been mixed.1 A key issue in this
John Bailey Jones, Daniel Ober-Reynolds, Zhu Wang, and John Weinberg provided
helpful comments that improved the quality and exposition of the article. We also
wish to thank conference participants at the 2015 Meeting of the Society for Computational Economics in Taipei and the 2015 Symposium of the Society for Nonlinear Dynamics and Econometrics in Oslo for useful feedback and suggestions. The
views expressed in this paper are those of the authors and should not necessarily
be interpreted as those of the Federal Reserve Bank of Richmond or the Federal
Reserve System. Lubik: Research Department, Federal Reserve Bank of Richmond.
P.O. Box 27622, Richmond, VA 23261. Email: thomas.lubik@rich.frb.org. Matthes:
Research Department, Federal Reserve Bank of Richmond. P.O. Box 27622, Richmond, VA 23261. Email: christian.matthes@rich.frb.org. Owens: Department of
Economics, University of Rochester. 280 Hutchison Road, Box 270156, Rochester,
NY 14627. Email: owensaa@gmail.com.
1
See, for instance, Hamilton (1989), Primiceri (2005), Sims and Zha (2006), or
Amir-Ahmadi, Matthes, and Wang (2016), who use a wide range of empirical methodolgies, data sets, and sample periods.
DOI: http://doi.org/10.21144/eq1020302
198
Federal Reserve Bank of Richmond Economic Quarterly
literature is that tests for nonlinearity tend to have low power against
linear alternatives.
Against this background, time-varying parameter vectorautoregressions (TVP-VARs) with stochastic volatility have emerged
as a promising framework to analyze a wide range of underlying nonlinearities.2 In this class of models, the coe¢ cients of the time-series
representation for economic data are allowed to vary over time. The
idea is that this feature approximates the underlying nonlinearity in
the data-generating process to a satisfactory degree and in a parsimonious manner. For instance, a structural break in a deep parameter, or
a switch in regimes, could be captured by a shock to the innovation in
a random-walk VAR coe¢ cient. Since TVP-VARs o¤er this ‡exibility,
that is, since they can be understood as approximations to a wide range
of underlying nonlinear behavior, they have become increasingly popular in recent years as empirical modeling devices.3 TVP-VARs are estimated almost exclusively using Bayesian methods. This is necessitated
by the fact that, as with any model that features many parameters,
the use of prior information is crucial to deliver sensible estimates. In
TVP-VARs the choice of priors is of special importance because, with
standard sample sizes, they have a substantial impact on how much of
the variation in observables is attributed to stochastic volatility versus
time variation in other coe¢ cients.4 At the same time, there is a growing sense, e.g., Lubik and Matthes (2015), that the conclusions drawn
from the TVP-VAR literature warrant skepticism. More speci…cally,
TVP-VARs often …nd not much time variation in the lag coe¢ cients.
Instead, they attribute the variation seen in the data to movements in
volatilities as the right incidence of shocks can in principle capture a
range of time-series patterns.
The purpose of this article is to investigate the extent to which an
inherently nonlinear TVP-VAR with stochastic volatility does, in fact,
pick up nonlinear features in the underlying data. We do so by applying the TVP-VAR methodology to data generated from a simple (but
nonlinear) search and matching model that is designed to generate
endogenous shifts in parameters. We thus ask whether a TVP-VAR
is capable of detecting the resulting nonlinearity in Beveridge curve
dynamics. We follow standard procedure and prescriptions in the literature to specify the TVP-VAR and to choose the prior. The results
from these benchmark exercises show that the concerns about proper
2
See Lubik and Matthes (2015) for an introduction and survey of TVP-VARs.
See Canova, Ferroni, and Matthes (2015) for a discussion of these issues.
4
This point is demonstrated by means of a simple example in Lubik and Matthes
(2015).
3
Lubik, Matthes, and Owens: Beveridge Curve Shifts
199
Figure 1 The Beveridge Curve over the Great Recession
attributions of the sources of nonlinearity are warranted. We attempt
to resolve some of these concerns by means of an alternative strategy
in choosing priors with only partial success. While these …ndings are
largely negative and are also highly conditional on the chosen theoretical model environment, we argue that they serve as a cautionary tale
when conducting and interpreting TVP-VAR studies.
Our chosen framework to analyze these issues is the labor market
regularity captured by the so-called Beveridge curve. It describes the
joint behavior of unemployment and vacancies over the business cycle and is often seen as indicative of the state of the labor market.
The Beveridge curve depicts a negative relationship between these two
variables, whereby movements along this curve re‡ect expansions and
recessions. The behavior of the curve over the course of the Great
Recession and its aftermath has attracted much interest in the literature (e.g., Barlevy 2011; Lubik 2013; or Şahin et al. 2014). Figure
1 shows the Beveridge curve for data over the Great Recession period. The unemployment-vacancies relationship is often represented
by a scatter plot of the two series against each other, resulting in a
downward-sloping curve. For purposes of illustration, in Figure 1 we
…tted a regression line to data from 2001 up to September 2008, which
200
Federal Reserve Bank of Richmond Economic Quarterly
cluster tightly around the Beveridge curve. At the onset of the Great
Recession, the unemployment rate rises rapidly and vacancies fall. In
the graph, the data points start moving o¤ the normal curve and appear to settle at a location above their normal, or expected, level. In
other words, during the Great Recession, the Beveridge curve appears
to have shifted outward in a discrete manner, which could be indicative
of a structural break in a labor market parameter.
More generally, the Beveridge curve over the last sixty years reveals
a substantial degree of nonlinearity (see Benati and Lubik 2014). There
are discernible inward and outward shifts, tilts, and even the occasional
slope reversal over short periods. This is not necessarily prima facie
evidence of the presence of nonlinearities since these patterns can be rationalized through the right incidence of various shocks (e.g., Blanchard
and Diamond 1989; Barlevy 2011; or Lubik 2013). It is nevertheless
suggestive of underlying structural changes in the labor market. We
take this observation as a starting point for our investigation into the
practice of TVP-VAR estimation.
We develop a simple search and matching model of the labor market, where we allow for endogenous threshold switching in a key parameter, namely in the e¢ ciency of a match between employer and job
seeker. This match e¢ ciency is captured by a level parameter in the
matching function and summarizes the e¢ cacy of the labor market.
We assume that it can take on two values, which indicate di¤erent but
parallel locations of the Beveridge curve. A high level of match e¢ ciency translates into a location of the Beveridge curve closer to the
origin, whereby a lower level shifts it outward. Under high e¢ ciency,
employers need to open fewer vacancies for the same number of job
seekers to …ll a desired number of positions. The economy switches between the two e¢ ciency parameters when a threshold embedded in the
model is crossed endogenously. We assume that this threshold is given
by a low level of output that we associate with a weak labor market
performance. This threshold can be reached with a sequence of bad
and persistent productivity draws; that is, in this case, the labor market exhibits damage to the extent that the Beveridge curve shifts only
when a recession is deep and drawn out. In terms of the behavior of
the model, this threshold switch implies nonlinearity in the dynamics
of the economic variables.
In order to study the implications of this speci…c form of nonlinearity for empirical modeling, we solve the full nonlinear model and
simulate data on unemployment and vacancies. We then estimate a
Bayesian TVP-VAR with stochastic volatility on these data and assess
how well the nonlinear atheoretical time-series model captures the underlying nonlinearity in the model. Given a standard initialization and
Lubik, Matthes, and Owens: Beveridge Curve Shifts
201
choice of priors, the evidence suggests that the TVP-VAR attributes almost all of the changes in the simulated data to changes in the reducedform innovation variances. We argue that this raises doubts as to the
validity of TVP-VAR models with standard priors in detecting shifts.
In order to address this shortcoming, we suggest an approach that tries
to elicit priors for the TVP-VAR, but it is only moderately successful.
In order to better capture the time variation in parameters, researchers
will need to adapt the priors to the question at hand in more sophisticated ways. One possibility that delivers better performance is to
estimate the hyperparameters associated with the parameters governing the amount of time variation in the model (Amir-Ahmadi, Matthes,
and Wang 2017).
The contribution of this article is twofold. First, using simulated
data, we study to what extent a generic TVP-VAR with stochastic
volatility deals with a speci…c form of nonlinearity in these underlying
data. Our results suggest that some …ndings of this literature should be
regarded with skepticism since they attribute too much of this nonlinearity to time variation in the shocks rather than to structural breaks
in the underlying model parameters. Second, and of independent interest, we demonstrate how Beveridge curve shifts can be explained
conceptually via an endogenous mechanism that moves the economy
between a high-performing and a low-performing labor market. This
mechanism can thus be used to address issues like hysteresis, where
temporary shocks, such as business cycle shocks, can have permanent
e¤ects.
The article is structured as follows. In the next section, we lay out
our simple modeling framework of the standard search and matching
model and describe how we introduce the threshold-switching mechanism that leads to nonlinearity in the model. The second section
describes how we calibrate the model. In this section, we also describe
simulation results from the model and discuss the TVP-VAR that we
use to estimate the simulated data. Section 3 presents the estimation
results and details the shortcomings of the TVP-VAR approach in this
environment, while Section 4 introduces an alternative method to elicit
priors for the TVP-VAR. The …nal section concludes.
1.
A STRUCTURAL MODEL OF BEVERIDGE
CURVE SHIFTS
We now describe the simple structural labor market framework that we
use to model the Beveridge curve. We hereby draw heavily from the
existing literature, most prominently Shimer (2005). The speci…cation
of the model follows Lubik (2013). Our working assumption is that
202
Federal Reserve Bank of Richmond Economic Quarterly
the Beveridge curve has experienced a structural shift, as seen by the
evolution of unemployment and vacancies in Figure 1. We model the
structural break in terms of a threshold-switching process: when a target variable, aggregate output in our case, hits a threshold in terms of
deviations from its long-run level, it triggers a shift in a structural labor
market parameter. The idea is to capture the observation that Beveridge curve shifts appear to occur during strong and deep recessions
and expansions (see Benati and Lubik 2014).
We assume that in our model economy time is discrete and the time
period is a quarter. The labor market in this economy is characterized
by search and matching frictions, which help rationalize the existence
of equilibrium unemployment. Speci…cally, a job, that is, a relationship
between a worker and a …rm for the purpose of engaging in production,
is the outcome of a matching process. New jobs M are generated by
combining unemployed job seekers U with job openings (vacancies)
V . This process can be represented by a constant returns matching
function, Mt = m(st )Ut Vt1 , where 0 < < 1 is the match elasticity.
m(st ) > 0 is the match e¢ ciency that captures the ease with which the
unemployed are transformed into workers.
We assume that match e¢ ciency is subject to structural shifts.
Speci…cally, the level parameter in the matching function m(st ) can
switch between two values, st 2 fsH ; sL g, with m(sL ) < m(sH ). In our
framework, the switch is generated endogenously by a trigger mechanism, in contrast to the exogenous regime changes in Markov-switching
models. We implement this trigger by tying it to the severity of the
business cycles. Whenever GDP deviates too much from its current
target level, the labor market experiences a structural shift in terms
of a change in the matching e¢ ciency. As Lubik (2013) argues, Beveridge curve shifts are most parsimoniously and plausibly modeled by
a change in this one parameter. More speci…cally, one can show that
declines in match e¢ ciency are associated with outward shifts of the
curve.
For the purposes of capturing Beveridge curve dynamics, we assume
that the threshold mechanism is attached to aggregate output. More
speci…cally, we assume that match e¢ ciency mt = m(st ) follows a
threshold process:
mt =
m(sH ) if Yt Y
; where m(sH ) > m(sL ).
m(sL ) if Yt < Y
(1)
Yt is aggregate output and Y is the threshold at which the labor market experiences a structural shift. In the simple search and matching
framework, we assume linear production so that Yt is given by:
Yt = At Nt ;
(2)
Lubik, Matthes, and Owens: Beveridge Curve Shifts
203
where Nt is the stock of employed workers, and At is an aggregate
productivity process that obeys the law of motion:
log At = (1
A ) log A
+
A log At 1
+ "A;t ;
(3)
where 0 < A < 1 and "A;t N (0; 2A ). We normalize the mean of the
process A to a value of unity without loss of generality.
The dynamics of the model are such that sequences of low and
persistent productivity draws— in other words, a recession— will occasionally move aggregate output below the threshold Y . This damages
the labor market in the sense that match e¢ ciency declines and the
Beveridge curve shifts outward. This shift is persistent because of the
persistence in the productivity process and the inherent persistence of
employment in the search and matching framework. Once the recession abates, the labor market recovers in terms of a switch back to a
“normal”level of match e¢ ciency. In that sense, our framework shares
similarities with the “plucking”model of recessions, where the economy
is plucked away occasionally from its normal evolution due to a deep
recession but then transitions back over time.
The dynamics of employment are governed by the following relationship:
i
h
1
(4)
V
Nt = (1
)
N
+
m(s
)U
t
1
t
1
t
t 1 t 1 :
This is a stock-‡ow identity that relates the stock of employed workers
N to the ‡ow of new hires, M = mU V 1 , into employment. The
timing assumption is such that variations in match e¢ ciency do not
a¤ect employment contemporaneously. Unemployment is de…ned as:
Ut = 1
Nt ;
(5)
where the labor force is normalized to 1. In‡ows to unemployment
arise from exogenous job destruction at rate 0 < < 1. We assume
that the separation rate follows the process:
log
t
= (1
) log
+
log
t 1
+"
;t ;
(6)
where 0 <
< 1 and " ;t N (0; 2 ).
The matching function can be used to de…ne the job-matching rate,
i.e., the probability that a …rm is matched with a worker:
Mt
= mt t ;
q( t ) =
(7)
Vt
where t = Vt =Ut is labor market tightness. From the perspective of an
individual …rm, the aggregate match probability q( t ) is exogenous, and
hence new hires are linear in number of vacancies posted for individual
…rms: Mit = q( t )Vit .
204
Federal Reserve Bank of Richmond Economic Quarterly
A …rm chooses the optimal number of vacancies Vt to be posted
and its employment level Nt by maximizing the intertemporal pro…t
function:5
E0
1
X
t
[At Nt
wt Nt
Vt ] ;
(8)
t=0
subject to the employment accumulation equation (4). Pro…ts are discounted at rate 0 < < 1. Wages paid to the workers are w, while
> 0 is a …rm’s …xed cost of opening a vacancy. The …rst order
conditions are:
Nt :
Vt :
= At wt + Et (1
= q( t ) Et (1
t+1 )
t
t+1 ) t+1 ;
(9)
(10)
t+1 ;
where t is the multiplier on the employment equation. Combining
these two …rst-order conditions results in the job-creation condition:
q( t )
= Et (1
t+1 )
At+1
wt+1 +
q(
t+1 )
;
(11)
which captures the trade-o¤ faced by the …rm. The marginal, e¤ective
cost of posting a vacancy, q( t ) , that is, the per-vacancy cost adjusted for the probability that the position is …lled, is weighed against
the discounted bene…t from the match. The latter consists of the surplus generated by the production process net of wage payments to the
workers plus the bene…t of not having to post a vacancy again in the
next period.
Wages are determined based on the Nash bargaining solution: surpluses accruing to the matched parties are split according to a rule
that maximizes the weighted average of the respective surpluses. We
relegate the full discussion of the derivation to the Appendix (see also,
Lubik 2013). The resulting wage equation is:
wt = (At +
t)
+ (1
)b:
(12)
Wage payments are a weighted average of the worker’s marginal product At , of which the worker can appropriate a fraction , and the outside option b, of which the …rm obtains the portion (1
). Moreover,
the presence of …xed vacancy posting costs leads to a hold-up problem
where the worker extracts an additional
t from the …rm.
5
For ease of exposition and notation, we will drop the …rm-speci…c subscripts and
discuss the problem of a representative optimizing …rm with the understanding that
…rms are ex-ante heterogeneous in this framework.
Lubik, Matthes, and Owens: Beveridge Curve Shifts
205
We can substitute the wage equation and the job-matching rate
into the job-creation condition to obtain:
mt
t
= Et (1
t+1 )
(1
) (At+1
b)
t+1
+
mt+1
t+1
:
(13)
Firms are more willing to post vacancies if productivity shocks increase
the wedge to the outside option of the worker; they are less willing if
there are expected separations as this will reduce the present value of
a hired worker.
In our simulation and empirical analysis, we make use of the simple structure of the model. The dynamics can be fully described by
two equations, the employment accumulation equation (4) and the
job-creation condition (13), after convenient substitutions. Intuition
for why an outward shift of the Beveridge curve is generated by a fall
in match e¢ ciency can be gleaned from equation (4) and the logic of
the matching function. At any given unemployment rate, …rms would
need to post more vacancies to achieve a target hiring quota since the
matching process is now less e¢ cient.6 However, there is also a countervailing e¤ect, namely through the in‡uence of match e¢ ciency on
…rms’ vacancy posting decisions. A fall in m raises e¤ective vacancy
posting costs as captured by the left-hand side of the job-creation condition (13). This implies that vacancies are increasing in match e¢ ciency.
The overall e¤ect of a change in m therefore depends on the interaction
of these two margins. As Lubik (2013) shows, the stock-‡ow identity of
the law of motion has to hold in equilibrium, so that the …rst e¤ect via
the matching function dominates and shifts the Beveridge curve outward for a smaller m, whereas the e¤ect via the job-creation margin
generates movements along this equilibrium relationship. We now turn
to a discussion of our solution and simulation approach.
2.
SIMULATION AND ESTIMATION
Calibration
We calibrate our model to representative parameter values in the literature. Our benchmark calibration rests on the parameter estimates
6
Formally, this can also be seen from the steady-state representation of the employment equation (4), which describes an equilibrium locus of combinations of U and
V such that in‡ows and out‡ows to (un)employment are balanced:
V =
1
m1
1
1
U
1
U
1
1
U:
206
Federal Reserve Bank of Richmond Economic Quarterly
Table 1 Calibration
Parameter
Value
Source
Separation Rate
0.036
Shimer (2005); Monthly JOLTS Data
Match Elasticity
Match E¢ ciency mH
Match E¢ ciency mL
0.49
0.90
0.70
Beveridge Curve Estimation: Lubik (2013)
Beveridge Curve Estimation: Lubik (2013)
Beveridge Curve Estimation: Lubik (2013)
Bene…t b
Bargaining
Job Creation Cost
0.90
0.49
0.18
Hagedorn and Manovskii (2008)
Hosios-Condition: =
Imputed from Steady-State Sample Means
V = 2:6% and U = 5:2%.
Discount Factor
Productivity A
Threshold Value Y
0.99
1.00
0.91
Annual Real Interest Rate
Normalized
Cumulative Decline in U.S. GDP 2008–10
AR(1) Coe¢ cient A
AR(1) Coe¢ cient s
StD Productivity A
StD Separation Rate
0.95
0.95
0.01
0.01
Standard
Standard
Standard
Standard
s
Value
Value
Value
Value
in Lubik (2013) for the period 2000–08, after which a potential shift
in the Beveridge curve appears evident from the data (see Figure 1).
The calibrated parameters are reported in Table 1. We set the mean of
the separation rate to a value of 0:036. This follows the value reported
in Shimer (2005) for monthly data. We choose the match e¢ ciency in
the high state mH = 0:90 and in the low state mH = 0:70 based on
the estimate in Lubik (2013). The match elasticity is set to = 0:49.
These values broadly determine the slope and the location of the Beveridge curve in a scatter plot of vacancies and unemployment. We set
the discount factor = 0:99 and choose the bargaining parameter by
imposing the Hosios-condition for social e¢ ciency, = = 0:49. As
mentioned before, we normalize the mean of the level of productivity
to A = 1. Next, we assume that the outside option of the worker makes
up 90 percent of the productivity level, b=A = 0:9. The calibration is
therefore close to that of Hagedorn and Manovskii (2008), who argue
that a high outside option for the worker is needed to match the cyclical properties of the data. The job-creation condition can then be used
to back out the cost parameter for a given level of unemployment
and vacancies. We compute these from the sample averages for the
period 2000–08, V = 2:6 percent and U = 5:2 percent. This implies
= 0:18. Finally, we set the threshold value for Y = 0:91 to approximate the cumulative decline in U.S. GDP over the course of the Great
Lubik, Matthes, and Owens: Beveridge Curve Shifts
207
Figure 2 Policy Functions
Recession of 2007–09. We set the persistence parameter of the technology process and the separation rate to 0:95 and the standard deviations
of the respective innovations to 0:01.
Model Simulation and Discussion
We solve the search and matching model with threshold switching in
match e¢ ciency fully nonlinearly by means of the monotone mapping
algorithm. The algorithm computes an approximation of the …rm’s decision rule, which determines the number of vacancy postings given the
economy’s state variables: employment Nt , the exogenous productivity
shock At , and the separation rate process t . The algorithm is detailed
in the Appendix.
In order to understand the underlying dynamics of the model before
we turn to the estimation exercise, we compute the policy functions under the baseline calibration for given realizations of the shocks. The key
driving force behind the shifts in the Beveridge curve are movements
208
Federal Reserve Bank of Richmond Economic Quarterly
in productivity. An adverse enough realization of productivity At can
drive output below the threshold value, which then generates a switch
to a lower match e¢ ciency. However, equilibrium outcomes across the
threshold and within the distinct regions depend on the subtle interplay
between the state variables. To give a sense of the nonlinearities present
in our model, we plot the policy function for vacancies in Figure 2. For
the purposes of this exercise, we hold the productivity shock …xed at its
unconditional mean A = 1. Vacancies are graphed against the model’s
sole endogenous state variable, namely the level of employment. We
plot this relationship for di¤erent realizations of the separation rate .
Figure 2 shows the key aspect of the model. The policy function
has two distinct regions that coincide with the two distinct states of
the labor market. For given productivity, the policy function is discontinuous at the implied threshold level, N = Y = 0:91. To the right
of the threshold, the labor market is in its normal state with match
e¢ ciency m(sH ), and to the left, it has su¤ered from a deterioration of
the latter. We also note that vacancies are decreasing in employment.
When employment is high (and unemployment low), few vacancies are
being posted because the vacancy-unemployment ratio is high and
the labor market is tight. That is, the …rm’s probability of …nding a
worker is low relative to the costs of hiring him. When employment
is low, the labor market is awash with job seekers, so …rms can more
easily recoup the implicit hiring costs. We also note that the vacancy
policy function is increasing in the productivity shock.
The policy function for the high match e¢ ciency case tends to lie to
the right and above the respective function in the low e¢ ciency regime.
Other things being equal, a lower match e¢ ciency reduces the …rm’s
hiring probability and thereby the incentive to post vacancies relative
to the high e¢ ciency scenario. An interesting pattern emerges when we
additionally vary the policy function across separation rates. We …nd
that the higher the separation rate, the higher the vacancy postings
for given productivity and employment. More separations mean higher
churn, so for given employment, more vacancies need to be posted. The
di¤erences between the policy functions, however, are quantitatively
small for the low e¢ ciency case and almost nonexistent under high
e¢ ciency.7 What is interesting is that the relationship between the
separation rate and levels of match e¢ ciency appears nonlinear in its
e¤ect on vacancy postings.
7
This is consistent with the empirical …nding in Lubik (2009) and the assumption
and interpretation in Shimer (2005) that movements in the separation rate are not key
drivers of labor market ‡uctuations.
Lubik, Matthes, and Owens: Beveridge Curve Shifts
209
Table 2 Selected Moments
Sample
Sample
Sample
Sample
Sample
1
2
3
4
5
(V )
0.53
0.49
0.43
0.39
0.35
(U )
1.88
2.10
1.22
0.57
1.09
(V =U )
0.32
0.39
0.41
0.41
0.66
(V; U )
-0.35
-0.51
-0.47
-0.43
-0.69
We now simulate the model for 590 periods under the benchmark
calibration. We discard the …rst 450 periods as burn-in. We are thus
left with a sample of size 140, of which we will use the …rst forty observations as a training sample in the estimation of the VAR. This leaves
us with 100 periods, or twenty-…ve years, of data for the actual estimation. Table 2 reports moments for …ve representative samples. We
present this as a …rst pass for whether our regime-switching framework
can potentially capture salient labor market facts. The last column
shows the correlation between unemployment and vacancies, which is
considerably negative and ranges from -0.35 to -0.69 but is below the
correlation found in U.S. data. Nevertheless, the model can replicate
to some extent the strongly negative comovement between these two
labor market variables.
The model is less successful in terms of volatilities. The …rst two
columns of Table 2 report the standard deviations of vacancies and unemployment relative to the standard deviation of (labor) productivity
At = Yt =Nt as in Shimer (2005). Vacancies are roughly half as volatile
as productivity, while unemployment is twice as volatile for samples 1
and 2. The standard deviation drops considerably in sample 4, while
samples 3 and 5 show the volatilities of the driving process and the
endogenous variables as roughly equal. The low volatility of vacancies
is also re‡ected in that of labor market tightness V =U . Our framework
thus falls prey to the critique espoused in Shimer (2005), namely that
the basic search and matching model has di¢ culty replicating the observed high volatility of unemployment and vacancies. As Lubik (2009)
shows, this can be remedied by additional shocks to the model such as
the exogenous variations in the separation rate, but this comes at the
price of reducing the correlation between U and V since a shock to
separations moves unemployment and vacancies in the same direction.
Figure 3 shows data plots of the …ve simulation samples, including
the training sample, in the same order as presented in Table 2. Each
row in the graph represents one simulation. The panels on the left show
time series plots of unemployment (in red) and vacancies (in blue).
The middle column shows the same data as a scatter plot in order to
210
Federal Reserve Bank of Richmond Economic Quarterly
Figure 3 Simulated Data
highlight shifts in the Beveridge curve that are potentially induced by
the mechanism in our framework. The last column shows aggregate
output Yt for each simulation along with its threshold value for the
regime switch. The graphs con…rm that the simulated model reproduces the negative correlation between unemployment and vacancies;
that is, the model generates a Beveridge curve. What is notable visually from the middle column of Figure 3 is that there are generally
two separate clusters of data (with the exception of the sample in the
fourth row). On the face of it, this lends support to the mechanism in
our framework as it can replicate the shift patterns seen in actual data.
This outcome is not preordained, however, as is evident from sample
4. In this simulation, the threshold is never reached despite values of
output persistently below its mean for extended periods. The model
economy su¤ers from a recession, but not one that is deep enough to do
damage to the labor market.8 Consequently, a stable Beveridge curve
8
This is consistent with the interpretation of Benati and Lubik (2014) that most
shifts of the Beveridge curve during recessions are too small to be plausibly and statis-
Lubik, Matthes, and Owens: Beveridge Curve Shifts
211
pattern arises over the full simulation period. We also note that this
sample stands out in Table 2 because of the low volatilities of unemployment and vacancies.9
In the other sample paths, output falls below the threshold for
lengthy periods. For instance, in the …rst row, the initial productivity
draw pushes output below the threshold and keeps it there for …fty periods. During this period, there are two opposing forces at play. First,
the productivity process is mean-reverting; that is, eventually there will
be enough positive innovations to push productivity above its mean and
thereby drag output back above the threshold.10 The strength of this
e¤ect depends on the degree of persistence in productivity. If it is high
enough, very large negative draws can have staying power to keep the
economy below the threshold. Low persistence, on the other hand,
leads to faster mean reversion. The second, endogenous force works
against this pattern. If the economy is below the threshold, vacancy
postings are lower than they otherwise would be (see Figure 2 and the
discussion above). Consequently, matching with lower match e¢ ciency
reinforces the threshold switch. The observed Beveridge curve shift
would thus be consistent with the hypothesis that prolonged periods
of high unemployment are generated by mismatch in the labor market
(see Şahin et al. 2014).
To summarize, we show that the simple model with an endogenous threshold switch can qualitatively and, with some quali…cations,
quantitatively replicate the business cycle patterns of key labor market
variables. More importantly for our purposes, we demonstrate that our
model can generate structural shifts in the Beveridge curve. This raises
two questions. First, are these shifts large enough to be statistically different from a standard adjustment pattern, such as a counterclockwise
loop as discussed in Blanchard and Diamond (1989)? Using di¤erent
methodologies and sample periods, Lubik (2013) and Benati and Lubik
(2014) answered this question in the negative. In this paper, we ask a
second question, namely whether the shifts are even detectable as such
in a ‡exible time-series framework.
tically judged as structural. They are thus more consistent with the counterclockwise
loop identi…ed by Blanchard and Diamond (1989). Yet, a few recessions, notably the
most recent one, fall outside this pattern.
9
What drives this pattern, that is, the lack of a Beveridge curve shift, is the combination of not-large-enough random shocks in the simulation and a lack of adjustment
dynamics to the new (conditional) steady state associated with lower match e¢ ciency.
10
An alternative speci…cation would have productivity also obey a threshold switch,
so that the e¤ect of very bad recessions would be much more protracted. This would
render the model closer to the implications of a Markov-switching model such as Hamilton (1989).
212
Federal Reserve Bank of Richmond Economic Quarterly
A TVP-VAR for the Simulated Data
Given the simulated data, we now turn to assessing whether statistical
approaches can uncover the underlying shifts in the Beveridge curve.
For this purpose, we rely on a TVP-VAR with stochastic volatility,
which has proved to be a ‡exible and useful tool to study nonlinear
behavior in aggregate time series. It has recently been applied to the
question of Beveridge curve shifts by Benati and Lubik (2014). Our
speci…c time-series model builds on Cogley and Sargent (2005) and
Primiceri (2005). The exposition below follows Lubik and Matthes
(2015), who provide further details on the implementation.
We stack the unemployment rate Ut and the vacancy rate Vt in a
column vector yt , which we assume is determined by the following law
of motion:
yt =
t+
L
X
Aj;t yt
j
+ et :
(14)
j=1
t is a drift term that can contain deterministic and stochastic components. The Aj;t are conformable coe¢ cient matrices that contain
time-varying parameters. et is a vector of residuals. Most of the literature on TVP-VARs that use quarterly data pick the lag length in
the reduced-form speci…cation as L = 2. We follow this convention
since we use a quarterly calibration for our matching model. We de…ne
I (1; yt0 1 :::; yt0 L ) to provide a concise representation of the
Xt0
dynamics of yt . We thus rewrite equation (14) as:
yt = Xt0
t
+ et :
(15)
We assume that the law of motion for the time-varying parameters
in the coe¢ cient matrices Aj;t is given by:
t
=
t 1
+ ut ;
(16)
where ut is a zero mean i.i.d. Gaussian process. To characterize stochastic volatility, we assume that the covariance matrix of the onestep-ahead forecast error et can be decomposed using two matrices
such that:
et =
1
t
t "t ;
(17)
where the standardized residuals are distributed as "t N (0; I). t is
a lower triangular matrix with ones on the main diagonal and representative non…xed element it . t is a diagonal matrix with representative
non…xed element jt . The dynamics of the non…xed elements of t and
t are given by:
Lubik, Matthes, and Owens: Beveridge Curve Shifts
log
i
t
j
t
=
i
t 1
= log
+
j
t 1
i
t:
+
213
(18)
j
t:
(19)
We assume that all these innovations are normally distributed with
covariance matrix V . In order to provide some structure for the estimation, we restrict the joint behavior of the innovations as follows (see
Primiceri 2005):
20
13 2
3
"t
I 0 0 0
6B ut C7 6 0 Q 0 0 7
B
C7 6
7
V = V ar 6
(20)
4@ t A5 = 4 0 0 S 0 5 :
0 0 0 W
t
S is further restricted to be block diagonal, which simpli…es inference.
We use a Gibbs-sampling algorithm to generate draws from the posterior. The implementation of the Gibbs-sampling approach used for
Bayesian inference follows Del Negro and Primiceri (2013).
A key choice for TVP-VAR modeling is how to set the prior. In
order to achieve sharp inference, given the multiple sources of variation
in TVP-VAR models, a researcher needs to impose restrictions on the
relationship between the covariance matrices of the parameters. The
trade-o¤, however, is that a too restrictive prior may not leave room
for the time variation to appear. In our benchmark, we impose a typical choice of prior as recommended in, for instance, Primiceri (2005).
Speci…cally, we assume the following:
Q
IW (
W
S
IW (
IW (
2
Q
2
W
2
S
40 V (
OLS ); 40);
2 I; 2);
2 V ( OLS ); 2);
(21)
(22)
(23)
where IW denotes the Inverted Wishart distribution priors for all other
parameters are the same as in Primiceri (2005). For the prior hyperparameters Q ; W , and S, we use the values Q = 0:01, W = 0:01,
and S = 0:1. We will discuss alternative prior choices below.
3.
ESTIMATION RESULTS
We report estimation results for our benchmark TVP-VAR on simulated unemployment and vacancies data in Figures 4 and 5. In each
…gure, we report posterior mean estimates from the …ve representative
data samples discussed in the previous section. Since we specify a twovariable VAR with two lags, we report eight series overall for the lag
coe¢ cients, two series for the variances, and one for the covariance.
214
Federal Reserve Bank of Richmond Economic Quarterly
Figure 4 Posterior Means of VAR Coe cients
Figure 4 shows the median posterior estimates of the coe¢ cients in the
lagged matrices Aj;t in (14) for each sample and over the entire sampling horizon. Figure 5 shows additional estimated statistics. The left
column of Figure 5 reports the estimated o¤-diagonal elements of the
covariance matrix of the one-step-ahead forecast errors, while the middle column depicts the posterior means of the diagonal elements, that
is, the variances. We also report the implied regression coe¢ cients of a
period-by-period population regression of unemployment on vacancies
for each sample.
Lubik, Matthes, and Owens: Beveridge Curve Shifts
215
Figure 5 Summary of Benchmark Results: Estimated
Posterior Means
The results are almost unequivocal. Across all simulations, the
TVP-VAR attributes the shifts in the simulated Beveridge curve to
changes in the forecast error variance only. While both volatilities and
contemporaneous correlations change with shifts in the underlying series, all lag coe¢ cients are estimated to be unvarying and e¤ectively
constant (see Figure 4). The estimates for the individual samples show
that when there appears to be a shift in the Beveridge curve it is associated with a gradual drift in the coe¢ cients of the variance-covariance
matrix. Consider as a baseline case the simulated sample in the fourth
row of Figures 3 and 5. As discussed before, this sample path includes
declines in output that never cross the threshold and therefore do not
lead to Beveridge curve shifts. The TVP-VAR produces essentially
constant variances of the shock innovations and an implied population
regression coe¢ cient (i.e., a Beveridge curve slope) that is fairly constant at -0.4. There is some variation in the covariance, which rises
from -1.2 to -1.0 before retreating again. This seems commensurate
with the increase in unemployment and the fall in vacancy postings as
216
Federal Reserve Bank of Richmond Economic Quarterly
the economy enters a downturn in the …rst half of the simulated sample. The resulting pattern is that of a movement along the Beveridge
curve but not a shift.
These patterns are noticeably di¤erent when we consider sample
paths that include movements of output across the threshold. First,
the population regression coe¢ cients exhibit more variation and are
smaller (in absolute value) over the full sample period compared to
those of a sample path that does include a switch. Along a given Beveridge curve, unemployment and vacancies move in opposite directions.
But in the transition between the two Beveridge curves, unemployment
and vacancies tend to move in the same direction as vacancy postings
rise in order to counteract the lower match e¢ ciency. Shifts in the Beveridge curve are associated with shifts in the elements of the covariance
matrix. In particular, periods of high volatility and positive covariation
are associated with unemployment-vacancy combinations arising from
low match e¢ ciency. As discussed above, because of the constant mean
and the mean-reversion of the productivity process, large and persistent enough negative shocks are required to push output below the
threshold. These shocks also induce high volatility in unemployment
and vacancies. The TVP-VAR then attributes this increased volatility to time-variation in the innovation covariance matrix. The positive
correlation in the innovations thus mirrors the lower implied regression
coe¢ cient.
To summarize our …ndings, we posit that an econometrician who
attempts to discover shifts in the Beveridge curve using a standard
TVP-VAR would come to an erroneous conclusion. What appears in
the data as a parallel shift in the curve is interpreted by the TVP-VAR
as the outcome of time-variation in the variance-covariance matrix of
the shocks. Large shocks drive the labor market variables away from
their present location. Given the inherent persistence in the search
and matching model, this would then cluster temporally close data
points in a pattern that indicated a shift.11 In the logic of the search
and matching model, this outcome would be consistent with a higher
incidence and severity of shocks that primarily a¤ect the matching
process and transitional labor dynamics as captured in equation (4)
(see Barlevy 2011; Lubik 2013).
11
Incidentally, this reasoning is consistent with the argument in Lubik (2013) that
the degree of estimation, parameter, and model uncertainty in the empirical model is
large enough that it would be di¢ cult to distinguish statistically between competing
hypotheses, especially when compared to the relatively short time span of a Beveridge
curve cycle in the data. On the other hand, Benati and Lubik (2014) impose further
restrictions and utilize longer samples to show that a few Beveridge curve cycles do
allow for sharper inference, including the Great Recession.
Lubik, Matthes, and Owens: Beveridge Curve Shifts
217
However, and to reiterate this point, the underlying data are generated from a model where the presence of a structural shift for lengthy
periods of time is quite noticeable. The TVP-VAR thus attributes these
shifts erroneously to changes in volatility. This observation is consistent
with many studies using TVP-VARs that tend not to …nd substantial
changes in the lag coe¢ cient matrices, but rather apportion excess
volatility and breaks in behavior to stochastic volatility. Our …nding is
also reminiscent of the critique by Benati and Surico (2009) of Sims and
Zha’s (2006) argument that the switch from the Great In‡ation of the
1970s to the Great Moderation of the 1980s and beyond was not driven
by a break in policy but by a decline in the volatility of the shocks.
By means of a simulation study, Benati and Surico (2009) show that
a regime-switching VAR cannot recover a break in policy coe¢ cients
in the underlying model. Instead, it erroneously attributes the change
in reduced-form behavior to changes in the innovation variance, in a
manner similar to our results.
This naturally leads to the deeper question of why the TVP-VAR
we use is not capable of picking up these shifts seen in the theoretical
model. TVP-VARs are a very ‡exible modeling framework that, in
theory, can certainly capture substantial shifts in parameters. At the
same time, they also possess many moving parts, and the contribution
of each to the ultimate estimation result is not trivial to disentangle. One important aspect is certainly the length of the sample over
which the model is estimated. It is well-known that inference under
heteroskedasticity (or time variation in the innovation covariance matrix) is quite problematic in short sample (e.g., Toyoda 1974). For that
reason, TVP-VARs generally perform better in longer samples, as in
Amir-Ahmadi, Matthes, and Wang (2016). A second aspect is that in
models with many parameters, the choice of priors can be very important. In particular, priors in TVP-VARs encode a particular view of
how much of the variation in the data is due to changes in parameters,
changes in volatilities, or additive shocks. The following section shows
one alternative to the standard practice that could be used to elicit
priors. With standard priors, we would need drastic and sudden shifts
in the data to have the estimated coe¢ cients move substantially. Our
search and matching model can generate those shifts, but they would
arguably not be regarded as realistic for many developed economies.
4.
ELICITING PRIORS FOR A TVP-VAR
A key element of TVP-VAR modeling is the choice of the prior on the
time-varying components. In our benchmark speci…cation, we follow
the generally accepted practice in the literature going back to Primiceri
218
Federal Reserve Bank of Richmond Economic Quarterly
(2005). However, for data with considerably di¤erent properties than
those commonly used in the literature or to which TVP-VAR models
have been applied, our chosen values might not capture a researcher’s
prior view on time variation in the data set at hand. More speci…cally,
the prior on the lag coe¢ cient matrices Aj;t may be too tight in our
framework, so the true underlying time variation in the reduced-form
coe¢ cients is instead forced into the covariance matrix. We therefore
consider an alternative that is based on a prior predictive analysis.12
Our alternative approach proceeds as follows. We …rst estimate
…xed-coe¢ cient VARs on rolling samples of the same length as our
training sample (forty periods) to get paths for the time-varying coe¢ cients and volatilities. In a separate exercise, we then simulate paths for
those coe¢ cients based on the benchmark priors described above. The
hyperparameters of the alternative prior are chosen to match a set of
moments from the paths of the time-varying coe¢ cients and volatilities
obtained from the rolling window estimation. We choose the average
volatilities of the three sets of time-varying coe¢ cients and volatilities. For each set of values that govern the tightness of the prior
distribution on the covariance matrix, we run twenty-…ve simulations
to generate paths of the same length as the paths from our rolling window estimation and average over the moments obtained in those simulations. We then pick the vector of coe¢ cients that minimizes the
quadratic distance between the moments from the simulations and the
rolling window estimation. The di¤erence in the moments obtained by
simulation and the rolling window estimation is rescaled by their value
obtained in the rolling window estimation. This avoids one set of moments dominating our calculation since the coe¢ cients have di¤erent
scales. We use a grid of values for the parameters. As lower bounds
for the grid, we impose the values used by Primiceri (2005) since we
are worried about not capturing enough time variation. Upper bounds
are roughly ten times the values chosen by Primiceri (2005).
Figure 6 shows the resulting values for the prior hyperparameters
for our full set of twenty-…ve simulated samples. The horizontal green
line shows our benchmark values. As it turns out, there are, in fact,
substantial di¤erences between the values chosen by Primiceri (2005)
and the values implied by our approach. In the case of the innovation
variance in the law of motion for the time-varying parameters in the
coe¢ cient matrices Aj;t , equation (16), there are only two samples for
which our prior choice deviates from the one chosen by this approach;
12
An alternative would be to directly estimate the prior hyperparameters with the
rest of the parameters of the model. A Gibbs sampler to do this is described in AmirAhmadi, Matthes, and Wang (2017).
Lubik, Matthes, and Owens: Beveridge Curve Shifts
219
Figure 6 Eliciting Priors: Values of Prior Hyperparameters
the deviation is only Q = 0:03 when compared to our benchmark
choice of Q = 0:01. The hyperparameter scaling the variance of the
innovation in the triangular decomposition matrix of the forecast-errors
covariance matrix shows more deviations. They can be larger by a
factor of up to …ve, but this is not consistent across each simulation.
The largest di¤erence to our benchmark choice can be found for the
innovations on the process of the error variances. As can be seen from
the bottom graph in Figure 6, the hyperparameter W is larger by an
order of magnitude.
This raises the question of whether this alternative prior choice has
an e¤ect on the implications derived from the TVP-VAR. We therefore
reestimate our model with the much wider priors chosen by the procedure described above. The estimation results are reported in Figure 7,
which can be compared directly with Figure 5. We …nd that the parameter estimates using the alternative prior are almost identical to those
obtained using the benchmark speci…cation. The estimated entries of
the VAR companion form matrix (not reported) are also virtually identical to the benchmark case. Our conclusion that the Beveridge curve
220
Federal Reserve Bank of Richmond Economic Quarterly
Figure 7 Summary of Results: Estimated Posterior Means
from Alternative Choice of Hyperparameters
shifts in the simulated data are erroneously attributed by the TVPVAR to time variation in the covariance matrix of the one-step-ahead
prediction errors therefore remains intact. In order to get substantial
di¤erences in estimated parameters, the prior hyperparameters need
to be increased dramatically (e.g. Q = 1). For our application, the
benchmark values consistent with the existing literature therefore seem
to be a good choice as far as a naive exercise— that is, without knowledge of the underlying dynamics— is concerned.
5.
CONCLUSION
This article makes a simple point. TVP-VARs appear to be predisposed
to capture time variation in the underlying data by means of changes
in the innovation terms and not via movements in lag coe¢ cients. We
arrived at this conclusion by means of a simulation study where we
generate a speci…c form of nonlinearity that would imply time variation
in the data. This conclusion holds for a standard choice of priors as well
Lubik, Matthes, and Owens: Beveridge Curve Shifts
221
as an alternative set of priors that we obtain from a prior predictive
analysis.
Naturally, the results derived in this article are model dependent
and should therefore be taken with a grain of salt. As our model
analysis shows, the degree of nonlinearity in the policy function or in
the simulated data does not appear to be, heuristically speaking, large.
It thus may very well be that the posterior sampler in the Bayesian estimation attributes this type of variation in the data to residual shocks,
just as a …xed-coe¢ cient VAR would. What supports this argument is
that during times of economic upheaval, chie‡y the Great Depression
period, TVP-VARs do tend to exhibit considerable time variation in
the lag coe¢ cients (Benati and Lubik 2014; Amir-Ahmadi, Matthes,
and Wang 2016). That said, we argue that the basic point still applies as to the interpretability of TVP-VAR results. At the very least,
researchers should consider a more careful approach to prior selection.
In addition, and independently of the TVP-VAR angle, we propose in this article a modeling framework that conceptualizes structural changes in the labor market and links them to business cycle
movements. The mechanism works via an endogenous regime shift in a
key labor market parameter, whereby the shift is driven by the interaction of shocks and the intrinsic dynamics of the model. In the case of
a simple labor market model, we show that a deep and long recession
that originates in adverse productivity realizations can be prolonged by
deterioration in the labor market matching process. This mechanism
thus o¤ers a convenient setup for studying the behavior of the labor
market over the business cycle.
222
Federal Reserve Bank of Richmond Economic Quarterly
REFERENCES
Amir-Ahmadi, Pooyan, Christian Matthes, and Mu-Chun Wang.
2016. “Drifts and Volatilities under Measurement Error: Assessing
Monetary Policy Shocks over the Last Century.” Quantitative
Economics 7 (July): 591–611.
Amir-Ahmadi, Pooyan, Christian Matthes, and Mu-Chun Wang.
2017. “Estimating Prior Hyperparameters in VAR Models with
Time-Varying Parameters and Stochastic Volatility.” Manuscript.
Barlevy, Gadi. 2011. “Evaluating the Role of Labor Market Mismatch
in Rising Unemployment.” Federal Reserve Bank of Chicago
Economic Perspectives 35 (Third Quarter): 82–96.
Benati, Luca, and Thomas A. Lubik. 2014. “The Time-Varying
Beveridge Curve.” In Advances in Non-linear Economic Modeling:
Theory and Applications, edited by Frauke Schleer-van Gellecom.
Heidelberg: Springer Berlin Heidelberg, 167–204.
Benati, Luca, and Paolo Surico. 2009. “VAR Analysis and the Great
Moderation.” American Economic Review 99 (September):
1636–52.
Blanchard, Olivier J., and Peter A. Diamond. 1989. “The Beveridge
Curve.” Brookings Papers on Economic Activity (No. 1): 1–76.
Canova, Fabio, Filippo Ferroni, and Christian Matthes. 2015.
“Approximating Time Varying Structural Models With Time
Invariant Structures.” Federal Reserve Bank of Richmond
Working Paper 15-10 (September).
Cogley, Timothy, and Thomas J. Sargent. 2005. “Drift and
Volatilities: Monetary Policies and Outcomes in the Post WWII
U.S.” Review of Economic Dynamics 8 (April): 262–302.
Del Negro, Marco, and Giorgio Primiceri. 2013. “Time-Varying
Structural Vector Autoregressions and Monetary Policy: A
Corrigendum.” Federal Reserve Bank of New York Sta¤ Reports
619 (May).
Hagedorn, Marcus, and Iourii Manovskii. 2008. “The Cyclical
Behavior of Equilibrium Unemployment and Vacancies Revisited.”
American Economic Review 98 (September): 1692–706.
Hamilton, James D. 1989. “A New Approach to the Economic
Analysis of Nonstationary Time Series and the Business Cycle.”
Econometrica 57 (March): 357-84.
Lubik, Matthes, and Owens: Beveridge Curve Shifts
223
Lubik, Thomas A. 2009. “Estimating a Search and Matching Model of
the Aggregate Labor Market.” Federal Reserve Bank of Richmond
Economic Quarterly 95 (Spring): 101–20.
Lubik, Thomas A. 2013. “The Shifting and Twisting Beveridge
Curve: An Aggregate Perspective.” Federal Reserve Bank of
Richmond Working Paper 13-16 (October).
Lubik, Thomas A., and Christian Matthes. 2015. “Time-Varying
Parameter Vector Autoregressions: Speci…cation, Estimation, and
an Application.” Federal Reserve Bank of Richmond Economic
Quarterly 101 (Fourth Quarter): 323–52.
Primiceri, Giorgio. 2005. “Time Varying Structural Vector
Autoregressions and Monetary Policy.” Review of Economic
Studies 72 (July): 821–52.
Şahin, Ayşegül, Joseph Song, Giorgio Topa, and Giovanni L. Violante.
2014. “Mismatch Unemployment.” American Economic Review
104 (November): 3529–64.
Shimer, Robert. 2005. “The Cyclical Behavior of Equilibrium
Unemployment and Vacancies.” American Economic Review 95
(March): 25–49.
Sims, Christopher A., and Tao Zha. 2006. “Were There Regime
Switches in U.S. Monetary Policy?” American Economic Review
96 (March): 54–81.
Toyoda, Toshihisa. 1974. “Use of the Chow Test Under
Heteroscedasticity.” Econometrica 42 (May): 601–8.
224
Federal Reserve Bank of Richmond Economic Quarterly
APPENDIX:
DERIVATION OF THE WAGE SCHEDULE
The wage that …rms pay to workers is derived as the outcome of a Nash
bargaining process. Denoting the workers’ weight in the bargaining
process as 2 [0; 1], this implies the sharing rule:
Wt
Ut =
Jt ;
1
(A1)
where Wt is the asset value of employment, Ut is the value of being
unemployed, and Jt is, as before, the value of the marginal worker to
the …rm. In models with one-worker …rms, the net surplus of a …rm is
given by Jt Vt ; with Vt the value of a vacant job. By free entry, Vt
is then assumed to be driven to zero. The value of employment to a
worker is described by the following Bellman equation:
Wt = wt + Et [(1
t+1 )Wt+1
+
t+1 Ut+1 ]:
(A2)
Workers receive the wage wt and transition into unemployment in the
next period with probability s. The value of searching for a job, when
currently unemployed, is:
Ut = b + Et [ t (1
t+1 )Wt+1
+ (1
t (1
t+1 ))Ut+1 ]:
(A3)
An unemployed searcher receives bene…ts b and transitions into employment with probability t (1
t+1 ). It is adjusted for the probability
that a completed match gets dissolved before production begins next
period. Substituting the asset equations into the sharing rule (A1),
results, after some algebra, in the wage equation found in the text:
Wt = (At +
t)
+ (1
)b:
(A4)
Lubik, Matthes, and Owens: Beveridge Curve Shifts
APPENDIX:
225
MODEL SOLUTION
We solve the simple search and matching model fully nonlinearly by
means of the monotone mapping algorithm. The algorithm computes
an approximation of the decision rule b
hV (Nt ; At ; t ), which determines
the number of vacancy postings given the state variables: employment Nt , the exogenous productivity shock At , and the separation rate
process t . The algorithm contains the following steps:
1. Specify a threshold switching value Y and discretize the state
space S. Formulate an initial guess for the decision rule: b
hV0 (N; A; )
8fN; A; g 2 S.
2. Compute a residual function R(Vt ; fNt ; At ;
lowing:
t g)
based on the fol-
(a) Yt is calculated and mt is given according to the threshold
process (1).
(b) Calculate next period’s employment from (4).
(c) Expected values of next-period values in the …rm’s …rst-order
condition appear as:
Et
"
) At A e"At
(1
b
t+1
t+1
( 1 VN
) + ( mt+1 )( 1 VN
)
t+1
t+1
Yt+1
= Xt ;
which can be approximated with the truncated distribution:
bt =
X
where:
R"
"
(";
2
")
R"
(Nt+1 ; At )
d" +
(";
(Nt+1 ; At )
"
) At A e"At
(Nt+1 ; At ) = (1
(
2
")
(Nt+1 ; At )
d";
(Nt+1 ; At )
b
b
hV0 (Nt+1 ; At A e"At )
)+
1 Nt+1
b
hV (Nt+1 ; At A e"At )
+( )( 0
) ;
ml
1 Nt+1
b
(Nt+1 ; At ) = At A e"At Nt+1
hV0 (Nt+1 ; At A e"At ):
Estimate this expression with a trapezoid rule. Linear interpolation is used in the implementation of the decision rule.
#
226
Federal Reserve Bank of Richmond Economic Quarterly
(d) Given the expectations, the residual function is:
R(Vt ; fNt ; At ;
t g)
= (1
t)
b
Yt X
mt
Vt
1 Nt
;
which can be interpreted as the absolute value of the di¤erence between the right-hand side and left-hand side of the
…rm’s …rst-order condition.
3. This residual function is minimized over Vt for every triple
fNi ; Aj ; k g in S. The decision rule is then updated based on:
b
hV2 (Ni ; Aj ;
k)
= arg minfR(Vt ; fNi ; Aj ;
k g)g
8fni ; Aj ;
4. The algorithm is repeated until:
max b
hVk+1 (Ni ; Aj ;
k)
b
hVk (Ni ; Aj ;
k)
< ":
kg
2 S:
Economic Quarterly— Volume 102, Number 3— Third Quarter 2016— Pages 227–252
The Heterogeneous
Business-Cycle Behavior of
Industrial Production
Jackson Evert and Felipe Schwartzman
I
ndustry-level data can provide a window into the sources of business cycles as well as propagation mechanisms. This is because
depending on what determines those, one might expect di¤erent
industries to behave di¤erently. One notable example of the use of
industry-level data for that purpose is Gertler and Gilchrist (1994),
who pointed to the relatively larger impact of monetary shocks in industries with relatively smaller sized …rms as evidence for the role of
…nancial frictions in propagating those shocks. Another example is
Bils et al.’s (2013) comparison of markup ‡uctuations in durable vs.
nondurable sectors as a means to assess whether demand ‡uctuations
could cause ‡uctuations in markups.
The use of industry-level variation can also provide advantages over
the use of even more disaggregated …rm-level data. First, since industries are to a large extent de…ned by the nature of their products,
di¤erences between industries are more plausibly determined by stable
di¤erences in technology and preferences than di¤erences across …rms
within an industry. Second, because industry-level data already allow
for some aggregation, they capture at least part of the general equilibrium e¤ects that are likely to be important at the aggregate level.
Third, industry-level data are more readily available, allowing for a
useful …rst pass before acquiring harder-to-obtain …rm-level data. The
clear disadvantage is that because industries are di¤erent from one
We thank Bob Hetzel, Christian Matthes, Daniel Schwam, and John Weinberg for
helpful comments and suggestions. The views expressed in this article are those of
the authors and do not necessarily represent those of the Federal Reserve Bank
of Richmond or the Federal Reserve System. All errors are our own. E-mail:
felipe.schwartzman@rich.frb.org.
DOI: http://doi.org/10.21144/eq1020303
228
Federal Reserve Bank of Richmond Economic Quarterly
another along several dimensions, one needs to be always concerned
about the possibility that industry variation is driven by some omitted
characteristic. Thus, any work using industry-level data must incorporate extensive controls.
The purpose of this article is to present some stylized facts for how
the business-cycle behavior of sectoral output di¤ers with sectoral characteristics. Those stylized facts can be informative either as a means to
determine sources of ‡uctuations and transmission channels or as indications of important sources of sectoral heterogeneity that ought to be
controlled for in any study that attempts to uncover those sources and
channels. We construct these stylized facts by …rst calculating standard business-cycle statistics such as relative volatility and correlation
with GDP for each of the seventy-two sectors for which industrial production data are available separately. With those statistics in hand,
we can then ask which industry-level characteristics are most likely to
predict how these moments vary.
The measures of sectoral characteristics we focus on fall into four
categories. The …rst category includes determinants of the demand for
products in di¤erent sectors. Those may be informative about the role
of ‡uctuations in the composition of demand for di¤erent types of products on business cycles. For example, the extent to which sectors that
have the government as a main customer ‡uctuate more or less with
aggregate GDP provides some information about the role of government consumption in business cycles (Ramey [2011] provides a recent
review of the literature). The second category includes determinants of
production costs. Those can provide a window into the role of cost ‡uctuations in business cycles. For example, a wide literature has pointed
to energy cost ‡uctuations as an important driver of business cycles
(see Hamilton [2003] for a seminal example). Variables in the two categories, demand and cost, can provide information about the role of the
integration of di¤erent industries in production chains. This can help
shed light on theories of business-cycle propagation that emphasize the
input-output structure of the economy, such as Acemoglu et al. (2012).
The third category includes measures of pricing distortions, including
measures of market power and of price stickiness. Those can shed light
on theories of business cycles that emphasize markup ‡uctuations as a
key propagation mechanism (Rotemberg and Woodford [1999] provide
a review). The fourth category includes …rm-level characteristics that
the literature has pointed to as correlated with sensitivity to …nancial
frictions. Those are relevant for theories of business cycles that emphasize …nancial shocks and …nancial frictions (Bernanke and Gertler
1989; and Kiyotaki and Moore 1997). Those di¤erent categories are
Evert & Schwartzman: Behavior of Industrial Production
229
constructed in order to obtain a wide scope of cross-industry di¤erences
that the existing literature has pointed out as potentially important.
Some of the most salient …ndings are as follows:
1) Industries that are more oriented toward the production of consumer goods, which produce goods that are nondurable, and that
produce necessities tend to be less volatile and less correlated with
business cycles than other industries. Furthermore, they also tend to
lead them. A similar pattern is present in …rms that intensively use
agricultural inputs.
2) Industries that are more oriented toward the production of goods
consumed by the government are less correlated with business cycles
relative to other industries and tend to lag business cycles. At the same
time, industries that are more oriented toward the private sector tend
to lead business cycles.
3) Industries in which nominal prices change infrequently tend to
lag business cycles.
4) Industries whose characteristics are likely to be correlated with
sensitivity to …nancial frictions are likely to lag business cycles, whereas
those that are less likely to be exposed to those frictions tend to lead
them.
5) The position of di¤erent industries in the production chain matters. Industries that are highly integrated in the production chain either
by being intensive in the use of intermediate inputs or by dedicating a
large fraction of their output to intermediate inputs are more likely to
lead GDP.
The …rst section provides a more careful description and justi…cation of the methodology. The subsequent section represents the core of
the paper. First, it presents a description of how the di¤erent moments
are distributed across sectors. Then, in four subsections we provide
more detail on the …ndings for each of the four categories described
above and provide some discussion of those …ndings in light of existing
literature. After those, we perform a multivariate analysis to account
for the fact that industry characteristics might be correlated among
themselves. The last section summarizes the results. In the Appendix,
we present a detailed description of how we constructed the various
measures of industry characteristics.
1.
METHODOLOGICAL DETAILS
In this section, and in all sections that follow, we will examine statistics
for detrended time series. The detrending process follows Hodrick and
Prescott (1997) and involves …tting a curve through the time series
that strikes a balance between staying close to the data and remaining
230
Federal Reserve Bank of Richmond Economic Quarterly
relatively smooth.1 This trade-o¤ is controlled by a parameter that, in
one extreme, makes the estimated trend perfectly smooth and, hence,
linear and, on the other extreme, leads to an estimated trend that
is identical to the data. The commonly used parameter for quarterly
data is 1600. The detrended series is then the log di¤erence between the
series and the estimated trend. In what follows we refer to a moment
as being a “business-cycle” moment whenever it is constructed using
HP-…ltered time series.
In order to gather a better understanding of how di¤erent moments
provide di¤erent information about the comovement of sectoral output
and business cycles, consider …rst the following model of detrended
sectoral output in which, for simplicity, we abstract from dynamics:
Yi;t =
R
X
i;r r;t ;
r=1
where Yi;t is output in sector i, r;t are the values at time t of each of
R shocks potentially a¤ecting all sectors, and i;r is the sensitivity of
sectoral output to each of the aggregate shocks. Shocks r;t are uncorrelated with one another, i.e., cov( r;t ; r0 ;t ) = 0 for all r 6= r0 and all t.
Note that this speci…cation is quite ‡exible, since we do not restrict R
to be a small number relative to the number of sectors. In particular,
the shocks r;t can include idiosyncratic shocks, i.e., shocks that a¤ect
only one sector. It also accommodates setups in which shocks that affect primarily one sector also a¤ect other sectors through input-output
linkages, etc.2 For simplicity, assume that detrended aggregate output
can be approximated as a simple average of sectoral output, so that
Yt =
I
X
Yi;t
i=1
I
:
The simplest moment of interest is the business-cycle variance of
sectoral output relative to that of aggregate GDP. If we normalize the
variance of the aggregate shocks r;t to 1, this is
1
As a robustness test, we also generated the tables using a Band-Pass …lter (see
Baxter and King [1999] for details on that kind of …ltering). They are available upon
request.
2
See Acemoglu et al. (2012) for analytical and quantitative explorations. We refer
the reader to these papers for further details. For the purposes of this essay, one can
accommodate that view by reinterpreting some of the aggregate shocks as shocks that
a¤ect primarily particular sectors but do not “wash out” in aggregate due to linkages.
Evert & Schwartzman: Behavior of Industrial Production
v
u
PR
std(Yi;t ) u
r=1
t
= PR P
I
std(Yt )
r=1 ( i=1
231
2
i;r
2
i;r =I)
or, more compactly,
v
u PR
std(Yi;t ) u
= t Pr=1
R
std(Yt )
2
i;r
2 ;
r=1 r
PI
where r
i=1 i;r =I is the average sensitivity of sector i to aggregate
shock r. In this benchmark case, the relative variance of a sector is large
2
if 2i;r is on average large relative to r . Note that this measure does
not allow us to distinguish whether the large relative variance stems
from a relatively large sensitivity to shocks that are also important
for other sectors (i.e., i;r > r >> 0) or from a high sensitivity to a
shock that is not relevant for other sectors (i.e., i;r > r ' 0). The
latter case would correspond to a case in which sector-speci…c shocks
are very large for individual sectors as compared to aggregate shocks
but “wash-out” in aggregate.
The correlation of industrial output with GDP provides an alternative view on the cyclical sensitivity of a sector. If business cycles
were predominantly caused by a single common shock to all sectors,
with sector-speci…c shocks playing a very small role, one would expect
the correlation of all sectoral output with aggregate GDP to be very
close to one. Contrariwise, if sectoral shocks play a disproportionate
role in individual sector output, one would expect the correlation of
that sector with GDP to be relatively smaller. Similarly, one may …nd
small correlations if output in a given sector is driven by an aggregate
shock that is not the main driving force of aggregate business cycles.
In terms of our simple model with I ! 1, the correlation between any
given sector and aggregate output is
corr(Yi;t ; Yt ) = P
( r
P
r
i;r r
P
2
i;r ) ( r
2
r)
:
If i;r and r have mean zero, the correlation between Yi;t and Yt
would be simply given by the correlation between between i;r and r .
More generally, it is an increasing function of that correlation. Thus,
the correlation between sectoral output and aggregate output measures
the extent to which the two are driven by the same shocks.
Note that it is possible for the output of a given industry to be at the
same time much more volatile than aggregate output and to have a low
232
Federal Reserve Bank of Richmond Economic Quarterly
contemporaneous correlation. This would happen if such an industry’s
output is largely determined by idiosyncratic shocks, which have little
e¤ect on the output of other industries. Conversely, an industry might
be less volatile than aggregate output but also highly correlated if it
is mostly driven by the same shock that drives other industries but is
comparatively less sensitive to those.
Finally, apart from relative variances and correlation with GDP, we
also provide statistics for the correlation of sectoral output and leads
and lags of output. Interpreting those requires a dynamic model. This
is a straightforward generalization of the model described above, in
which industrial output depends on shocks that occurred in the past:
Yi;t =
1 X
R
X
i;r;s r;t s ;
s=0 r=1
where we now also impose that cov( i;t ; j;t s ) = 0 8i; j; s; that is, we
impose that shocks are i.i.d., with all persistence a function of i;r;s .
The model above is fairly general, as it corresponds to a moving average
representation of a vector-valued time-series model (see, for example,
Hamilton [1994] for a detailed discussion).
Note that under this more general framework, it is possible for two
variables to be contemporaneously uncorrelated even if they are driven
by the same shock, so long as that occurs at di¤erent lags. For example, if Yi;t = 1;t and Yi ;t = 1;t 1 , those two processes will have
zero contemporaneous correlation. However, the correlation of Yi;t and
Yi ;t+1 will be equal to one. More generally, examining lead and lagged
correlations may provide us with some indication of whether certain
industries are more likely to respond more sluggishly with shocks than
overall GDP, a fact that is likely to be re‡ected in relatively low contemporaneous correlations by relatively high correlations with lagged
output. Conversely, examining correlations with leads and lags of output may provide us a sense of variables that react more rapidly to
shocks, thus forecasting output.
2.
THE CROSS-SECTORAL DISTRIBUTION OF
BUSINESS-CYCLE MOMENTS
Table 1 shows some descriptive statistics for the distribution of various
business-cycle moments across sectors. The …rst observation is that in
all sectors, business-cycle variance is larger than that of aggregate output, and for the median sector it is four times as large. This observation
is consistent with the notion that output in individual sectors is largely
Evert & Schwartzman: Behavior of Industrial Production
233
Table 1 Summary Statistics
Std. Dev.
t-8
t-6
t-4
t-3
t-2
t-1
t
t+1
t+2
t+3
t+4
t+6
t+8
Variance
3.38
0.03
0.03
0.04
0.04
0.05
0.06
0.07
0.06
0.05
0.04
0.04
0.03
0.03
Mean
3.93
-0.21
-0.07
0.15
0.28
0.40
0.49
0.53
0.47
0.36
0.24
0.14
0.00
-0.11
Median
4.03
-0.23
-0.08
0.14
0.26
0.40
0.54
0.63
0.53
0.39
0.25
0.14
0.00
-0.13
25th Percentile
2.54
-0.31
-0.21
0.02
0.13
0.23
0.33
0.32
0.30
0.18
0.08
0.00
-0.16
-0.24
75th Percentile
4.80
-0.12
0.04
0.27
0.45
0.57
0.69
0.74
0.69
0.54
0.41
0.28
0.15
0.02
Note: The cells refer to descriptive statistics of moments across industries. For
each industry, we calculate a standard deviation and correlations with leads and
lags of output. We then report statistics summarizing the cross-industry distribution of those moments.
driven by idiosyncratic shocks that are to a large extent averaged out
in aggregate.
The second observation is that the correlation of sectoral output
with aggregate GDP is mostly positive (animal food manufacturing
and dairy product manufacturing being the only sectors with a negative
correlation). The median sector has a correlation of 0.63 with GDP,
and 75 percent of the sectors have a correlation of more than 0.32.
Third, the median correlation with leads and lags of GDP declines
as the number of leads or lags increase in a fairly symmetrical fashion. At six-quarter leads and lags, the median sector has a correlation
with output that is fairly close to zero. In the next subsection, we
will describe how those business-cycle moments correlate with various
measures of industry characteristics.
Demand
We start our investigation of stylized facts by examining how businesscycle moments depend on determinants of sectoral demand. There is
no a priori reason why the demand for di¤erent products should vary
in the same way with business cycles. In fact, sectoral variation in
sensitivity to di¤erent demand components can provide a way to test
theories of propagation of demand shocks. For example, Bils et al.
234
Federal Reserve Bank of Richmond Economic Quarterly
(2013) use cross-industry variation in sensitivity of demand as a means
to assess the ability of demand shocks to lead to markup variations.
It is a well-known stylized fact of business cycles that consumption
of nondurable goods varies less than output over the business cycle,
whereas the demand for durable consumer goods and investment goods
varies more than output. This suggests that sectors whose production
is more dedicated to consumption ought to experience relatively lower
business-cycle variation. We check whether this simple prediction is
true by constructing for each sector a measure of the importance of
household consumption in its output. Roughly speaking, it corresponds
to the share of the output of each industry that is purchased by households as consumer goods (see Appendix for a detailed discussion of
how this and other measures are constructed). As we can observe, the
prediction is born out by the data, with consumption-oriented sectors
exhibiting lower business-cycle variance, although the negative correlation is relatively small in absolute value. Interestingly, however, the
correlation of sectoral output with the business cycle also declines with
its orientation toward household consumption. This suggests that compared to other sectors, sectors oriented toward household consumption
are more likely to be driven by shocks other than the ones determining
overall GDP. Interestingly, the pattern disappears and, in fact, reverses
itself once one compares business-cycle ‡uctuations at the sectoral level
with that of future GDP. It appears that, relatively speaking, household consumption-oriented sectors tend to lead business cycles. This
may imply some ability on the part of households to forecast business
cycle shocks and adjust their consumption accordingly early on.
Bils et al. (2013) focus on durability of the goods produced in
di¤erent sectors as a major source of variation in sensitivity to demand
shocks. Demand for durable goods is particularly sensitive to shocks
because stocks of durables are much larger than purchases in any given
period, so large changes in those purchases are necessary in order to
change the stock in use. More concretely, suppose a car depreciates at a
rate , and aggregate household demand for cars is given by Xcar;t . For
simplicity of exposition, suppose demand follows an exogenous process.
Then, if demand for cars increases by 1 percent, this requires increasing
the stock of cars in circulation by 1 percent. However, if we take a stable
demand for cars as a baseline, households must increase their purchase
of cars from Xcar;t (the amount that they need to purchase in order
to make up for depreciation) to ( + 0:01)Xcar;t , an increase of 1=
percent. Thus, if cars depreciate at a rate of 5 percent per quarter, this
implies an increase in car purchases of 20 percent. Consistently with
those calculations, output volatility does seem to be tightly linked to
the durability of the good produced in a given sector.
Evert & Schwartzman: Behavior of Industrial Production
Figure 1 Demand Correlates
235
236
Household Share
Government Share
Construction Share
Export Share
Intermediate Share
Inputs Sold
Engel Curve
Durability
Std. Dev.
-0.19
-0.04
0.02
0.36
-0.15
-0.27
0.45
0.62
t-8
0.35
-0.27
0.00
-0.4
-0.06
0.05
-0.44
-0.44
t-4
0.16
-0.37
0.34
-0.34
0.19
0.1
0.01
-0.03
t-3
0.04
-0.39
0.41
-0.26
0.28
0.08
0.17
0.11
t-2
-0.1
-0.38
0.46
-0.16
0.34
0.04
0.34
0.25
t-1
-0.23
-0.34
0.48
-0.05
0.39
0.02
0.5
0.36
t
-0.34
-0.26
0.46
0.05
0.40
-0.01
0.63
0.45
t+1
-0.41
-0.18
0.4
0.15
0.36
0.04
0.68
0.48
t+2
-0.43
-0.06
0.32
0.22
0.27
-0.05
0.64
0.45
t+3
-0.38
0.09
0.23
0.27
0.14
-0.06
0.54
0.38
t+4
-0.28
0.24
0.15
0.26
0.02
-0.04
0.37
0.26
t+8
0.03
0.38
-0.17
0.15
-0.23
-0.04
-0.22
-0.12
Note: Table reports the correlations between industry characteristics and business-cycle moments (either relative
volatility or business-cycle correlation for various industry leads/lags).
Federal Reserve Bank of Richmond Economic Quarterly
Table 2 Demand Correlates
Evert & Schwartzman: Behavior of Industrial Production
237
The …ndings for the correlation between depreciation and various
moments largely resemble those for household consumption orientation,
with sectors producing more durable goods being more contemporaneously correlated with GDP and less-durable sectors leading aggregate
GDP. The main di¤erence between the two measures is that durability
is a much better predictor of the relative volatility of di¤erent sectors
than consumption orientation.
Another household-demand-related dimension that one might expect to be predictive of the sensitivity of output in di¤erent sectors
to business-cycle variations is the income elasticity of demand for that
good (or the slope of the Engel Curve). Bils et al. (2013) estimate this
elasticity using cross-sectional data. Using their estimates, we …nd that
sectors with steeper Engel Curves are also more volatile and more correlated with output. The result is interesting in that it suggests that
business-cycle variation in national income has a qualitatively similar impact on household demand composition as variation in income
across households at a given point in time. It is also noteworthy that
necessary goods (i.e., those with low income elasticity) are particularly
good predictors of business cycles. Those goods also tend to be more
household-oriented and have higher depreciation rates. Interestingly,
the magnitude of the correlations between Engel coe¢ cients and output
correlations stands out when compared to the other metrics.
Given the focus of much of business-cycle analysis on the role of
…scal shocks, one further demand-side related metric of interest is
orientation of a given sector toward government consumption. That
metric is especially interesting since it provides a window into the role of
…scal shocks in driving sectoral output. Sectors oriented toward government consumption do not appear to be more or less volatile than other
sectors. However, they are less contemporaneously correlated with
business cycles, as one would expect if government purchasing decisions
were largely disconnected from broader economic conditions. Interestingly, however, they become more correlated with lags, implying that
the impact of shocks a¤ecting output in most sectors only a¤ect those
that are oriented toward government consumption with delay.
The orientation of individual industries toward construction
provides a further dimension of industry demand that is likely to be
informative about theories of the business cycle. We …nd that those
sectors do tend to be more correlated with business cycles, in line with
theories that have gained prominence after the Great Recession, consistent with housing demand playing a prominent role in driving businesscycle ‡uctuations. Furthermore, they appear to lead business cycles
slightly.
238
Federal Reserve Bank of Richmond Economic Quarterly
A further source of industry-level variation is motivated by recent
work on the interplay between industry-level and aggregate dynamics,
which has emphasized the importance of input-output linkages in the
propagation of shocks. This suggests that it could be interesting to investigate whether industry-level business-cycle moments correlate with
a measure of how “upstream” an industry is, meaning what fraction
of its output is sold as inputs to other industries. We …nd that such
industries, while not more or less volatile than others, tend to be more
correlated with business cycles. They also are slightly more correlated
with future output than with past output, hinting at timing delays
between the production of intermediate inputs and …nal outputs.3
Finally, we investigate the extent to which the foreign orientation
of a sector makes it more or less correlated with business cycles. We
…nd that sectors that are less export-oriented tend to lead the business
cycle relative to sectors that are more export-oriented. Thus, exportoriented sectors appear to be more insulated from business-cycle shocks
in early stages.
Cost
We now turn to measures capturing the intensity of use of di¤erent inputs in production. We start by focusing on those input categories that
are likely to have the most volatile prices, including energy, food, and
mining. To the extent that industries that are intensive in those inputs
are correlated with business cycles, this may indicate that shocks to
the supply of these inputs may help drive business-cycle ‡uctuations.
Of those three, the one that appears to have the most predictive power
over industry-level business-cycle statistics is the fraction of agricultural inputs used in production. However, rather than implying that
3
Following Acemoglu et al. (2012), we also examine the role, if any, of heterogeneity in industry “degree,” as measured by the fraction of industry intermediate input
production in total production of intermediate inputs in the economy. For that measure,
we did not …nd that this has any predictive impact on business-cycle moments.
Evert & Schwartzman: Behavior of Industrial Production
239
Figure 2 Cost Correlates
Note: Figures report the correlations between industry characteristics and
business-cycle moments (either relative volatility or business-cycle correlation for
various industry leads/lags).
agricultural cost shocks drive business cycles, the main …nding is that
industries intensive in agricultural inputs appear to be more disconnected from business cycles, with contemporaneous correlations being
smaller the more agricultural inputs are used. Interestingly, however,
their volatility is also relatively smaller. Industries with agricultural
inputs also tend to lead business cycles, in a pattern reminiscent of low
Engel elasticity sectors. This occurs in part because sectors that use
agricultural goods in production are in part producing exactly such
necessities. The multivariate analysis in Section 2.5 should help us
disentangle those e¤ects.
240
Energy Inputs
Agricultural Inputs
Mining Inputs
Intermediate Inputs
Imported Inputs
Imp. Share of Inputs
Capital Share
Std. Dev.
-0.24
-0.36
0.04
0.00
0.30
0.32
-0.18
t-8
0.01
0.43
-0.01
0.39
-0.01
-0.17
0.07
t-4
-0.02
-0.03
0.07
0.38
0.15
-0.01
-0.25
t-3
-0.04
-0.18
0.11
0.32
0.15
0.02
-0.3
t-2
-0.08
-0.31
0.15
0.24
0.16
0.05
-0.33
t-1
-0.1
-0.44
0.17
0.14
0.16
0.09
-0.34
t
-0.12
-0.53
0.19
0.02
0.16
0.13
-0.33
t+1
-0.12
-0.55
0.16
-0.12
0.13
0.15
-0.31
t+2
-0.1
-0.52
0.13
-0.26
0.07
0.15
-0.25
t+3
-0.07
-0.44
0.1
-0.36
0.01
0.13
-0.16
t+4
-0.02
-0.3
0.06
-0.39
-0.04
0.1
-0.05
t+8
0.12
0.13
-0.03
-0.32
-0.22
-0.1
0.21
Federal Reserve Bank of Richmond Economic Quarterly
Table 3 Cost Correlates
Evert & Schwartzman: Behavior of Industrial Production
241
Comparatively speaking, sectors with high intensity in energy and
mining inputs do not seem to be more or less correlated with business
cycles than other sectors. The low correlation with energy intensity is
somewhat surprising in light of the common notion that energy shocks
are an important source of business-cycle ‡uctuations. We also examine
what happens when we eliminate the three industries with the highest
use of energy inputs, since those have a level of energy use that is
much higher than the others and are themselves involved in energy
production. Eliminating those sectors does not increase the extent to
which business-cycle correlations are associated with energy use.4
We also investigate whether capital intensity and intermediate input intensity are predictive of business-cycle correlations. Capitalintensive sectors appear to be less correlated with business cycles contemporaneously but more correlated after eight quarters. This suggests
a sluggish response of those sectors to business-cycle shocks in line with
capital-adjustment costs and planning lags.
Furthermore, we examine the correlation between the fraction of
intermediate inputs in total output and business cycles. We …nd that
sectors that use more intermediate inputs are no more or less correlated
with business cycles than sectors that use fewer intermediate inputs.
However, they do tend to lead business cycles, whereas sectors that use
proportionately less intermediate inputs tend to lag business cycles.
Lastly, we investigate how the use of imported inputs a¤ects businesscycle moments. We …nd that sectors with a high share of imported
inputs are also relatively more volatile. This is in line with the notion
that the price of imported inputs is likely to be more volatile since part
of that is tied to exchange rate ‡uctuations. At the same time, we …nd
that the share of imported inputs is not predictive of business-cycle
correlations.
Goods Market Pricing Distortions
The third category of industry characteristics that we examine are those
capturing goods market distortions. One measure attempts to capture
the competitive pressures faced by …rms in di¤erent industries, the idea
being that …rms in more concentrated industries have more scope to
vary their markups over the business cycle. The second one is a measure of nominal stickiness based on microeconomic price data. Bils et
al. (2014) have defended time-varying goods market distortions as a
4
For brevity, we do not report the numerical results for these exercises. The removed sectors are i) electric power generation, transmission and distribution; ii) oil and
gas extraction; iii) natural gas distribution; and iv) petroleum and coal manufacturing.
242
Federal Reserve Bank of Richmond Economic Quarterly
Figure 3 Goods Market Distortion Correlates
Note: Figures report the correlations between industry characteristics and
business-cycle moments (either relative volatility or business-cycle correlation for
various industry leads/lags).
key element in business-cycle propagation. As for nominal rigidities,
they of course underlie a large literature on monetary policy and business cycles. To measure those, we use the average frequency of price
adjustment as measured in the CPI data by Nakamura and Steinsson
(2008).
We …rst examine how market concentration in di¤erent industries is
related to their business-cycle behavior. We measure market concentration by the share of the top four …rms in each industry. This provides a
measure of the potential role for goods market pricing distortion under
the assumption that …rms in more concentrated industries have more
scope for markup variation. We …nd that …rms in more concentrated
industries are also less cyclical.
Four-Firm Concentration
Eight-Firm Concentration
20-Firm Concentration
50-Firm Concentration
Price Stickiness
Std. Dev.
0.11
0.07
0.02
-0.01
-0.08
t-8
0.09
0.12
0.13
0.12
0.2
t-4
-0.21
-0.17
-0.15
-0.14
-0.01
t-3
-0.28
-0.25
-0.24
-0.21
-0.09
t-2
-0.32
-0.3
-0.29
-0.27
-0.18
t-1
-0.33
-0.33
-0.34
-0.32
-0.25
t
-0.36
-0.36
-0.38
-0.37
-0.30
t+1
-0.38
-0.4
-0.41
-0.41
-0.33
t+2
-0.35
-0.38
-0.39
-0.39
-0.34
t+3
-0.25
-0.28
-0.3
-0.33
-0.31
t+4
-0.11
-0.15
-0.17
-0.21
-0.24
t+8
0.23
0.18
0.18
0.16
0.11
Evert & Schwartzman: Behavior of Industrial Production
Table 4 Goods Market Distortion Correlates
243
244
Federal Reserve Bank of Richmond Economic Quarterly
We then examine the correlation of business-cycle statistics with
the average frequency of price changes. The data indicate that industries with less sticky prices (higher frequency of price adjustment)
are less correlated with business cycles. This is in line with the view
that nominal rigidities play a role in the propagation of business-cycle
shocks.
Financial Sensitivity
The last category we measure includes the industry characteristics that
are likely to be correlated with their sensitivity to …nancial shocks.
The most prominent one is average …rm size, proposed by Gertler and
Gilchrist (1994), under the idea that smaller …rms are more likely to
be …nancially constrained. We also examine …rm age and a …nancial
frictions index proposed by Hadlock and Pierce (2010) using both size
and age. Two further measures of …nancial sensitivity are external
…nancial dependence, proposed by Rajan and Zingales (1998) to study
the role of …nancial development in growth, and the inventory/sales
ratio, used by Schwartzman (2014), Raddatz (2006), and others to
study the impact of …nancial shocks in less-developed economies.
We …nd that industries with smaller …rms (and, presumably, facing
higher …nancial frictions) tend to lag business cycles by about three
quarters, but even there, the correlation is relatively moderate. On
the other hand, older …rms (which presumably face lower …nancial frictions) tend to lead business cycles. The net e¤ect is that the size-age
index implies that industries in which …nancial constraints are less severe lead business cycles, whereas those where they are more severe
lag business cycles. This pattern does not suggest a simple story of
…nancial frictions amplifying business cycles, but it does suggest some
possibly interesting implications for the role of …nancial frictions in
their propagation. Of course, this interpretation presumes that susceptibility to …nancial constraints is the major di¤erence between …rms
of di¤erent ages and sizes. Presumably, those characteristics might be
correlated with many other aspects of …rm behavior.
A similar pattern is apparent when we use external …nancial dependence as a measure of sensitivity to …nancial conditions. External
…nancial dependence is equal to one minus the median ratio between
cash ‡ow and capital expenditures for …rms within an industry. It
measures how much …rms need to raise over and above their internally
generated cash ‡ow in order to …nance their typical investment. We
…nd that ‡uctuations in industries in which …rms are more dependent
on external …nance are more likely to lag ‡uctuations in output.
Evert & Schwartzman: Behavior of Industrial Production
245
Figure 4 Financial Correlates
Note: Figures report the correlations between industry characteristics and
business-cycle moments (either relative volatility or business-cycle correlation for
various industry leads/lags).
246
Assets
Age
Size-Age Index
Ext. Fin. Ratio
Cash Flow
Capital Exp.
Invt.-Sales Ratio
Std. Dev.
-0.13
-0.12
0.12
-0.08
-0.08
0.01
0.25
t-8
0.12
0.18
-0.25
-0.08
0.24
0.26
-0.28
t-4
-0.02
0.22
-0.16
-0.12
-0.14
0.01
-0.07
t-3
-0.03
0.25
-0.15
-0.14
-0.2
-0.06
0.07
t-2
-0.04
0.26
-0.12
-0.14
-0.23
-0.1
0.2
t-1
-0.07
0.25
-0.08
-0.14
-0.26
-0.15
0.32
t
-0.11
0.20
-0.02
-0.13
-0.27
-0.20
0.42
t+1
-0.15
0.13
0.06
-0.11
-0.29
-0.26
0.48
t+2
-0.18
0.07
0.12
-0.07
-0.26
-0.29
0.47
t+3
-0.2
-0.01
0.16
-0.04
-0.2
-0.28
0.4
t+4
-0.17
-0.06
0.16
0.02
-0.12
-0.22
0.26
t+8
0.00
-0.22
0.15
0.21
0.13
-0.03
-0.14
Federal Reserve Bank of Richmond Economic Quarterly
Table 5 Financial Correlates
Evert & Schwartzman: Behavior of Industrial Production
247
The …nal industry characteristic we examine is the inventory/sales
ratio. In contrast to the other measures, the business-cycle correlation
for …rms with a high inventory/sales ratio is fairly large. It is also
particularly pronounced contemporaneously, although the peak occurs
at one- or two-quarter lags.
Multivariate Analysis
The analysis so far is based o¤ the comparison of business-cycle moments across industries taking one industry characteristic at a time.
To disentangle those, we turn now to multivariate analysis, i.e., we
run a simple OLS regression with the di¤erent business-cycle statistics as a dependent variable and all the industry characteristics that
we explored on the right-hand side. Here we use the measure of energy intensity after excluding the four outlying sectors. This sharpens
the interpretation of the results since, as pointed out by Bils et al.
(2013), those very high energy intensity sectors are also sectors with
very ‡exible prices, leading to a strong multicolinearity between energy intensity and frequency of price changes. This multicolinearity
problem is eliminated once we exclude those outliers. Tables 6 and 7
present the results for the di¤erent statistics, with coe¢ cients that are
signi…cant at a 10 percent level marked in bold. Before running the
regression, all right-hand-side variables were normalized by their standard deviation, so the coe¢ cients can be interpreted as the e¤ect of a
one standard deviation change in the value of those regressors on the
various business-cycle statistics. Focusing on these statistically significant coe¢ cients, we obtain the following results, which are robust to
the introduction of multivariate controls:
1) Volatility is higher in sectors with durable goods, imported inputs,
and high frequency of price adjustment.
The …ndings for durable goods and imported inputs conform to
the …ndings from the univariate analysis above. The correlation with
frequency of price adjustment only emerges in the context of the multivariate analysis. It conforms to the notion that, all else constant, …rms
in industries that are subject to more variable shocks will choose to
adjust prices more frequently.
2) The sectors least correlated with aggregate GDP are those producing necessities (low Engel elasticity), those that have their production
oriented toward government consumption, and those that intensively
use agricultural and mining inputs. Sectors oriented toward the production of intermediate inputs are more correlated with output.
248
Federal Reserve Bank of Richmond Economic Quarterly
Table 6 Regression Coe cients (1)
Four-Firm
Concentration
Ratio
Durability
Energy Inputs
Ext. Fin. Ratio
Household
Share
Government
Share
Construction
Share
Inv.-Sales Ratio
Median Assets
Median Age
Engel Curve
Agricultural
Inputs
Mining Inputs
Intermediate
Inputs
Imported Inputs
Capital Share
Price Stickiness
Std.
Dev.
t-8
t-6
t-4
t-3
t-2
t-1
-0.117
0.655
-1.663
-0.126
0.002
-0.051
-0.065
-0.005
-0.006
-0.068
-0.056
-0.009
-0.032
-0.058
-0.079
-0.024
-0.043
-0.053
-0.099
-0.029
-0.044
-0.046
-0.108
-0.03
-0.034
-0.033
-0.093
-0.032
-0.137
0.033
0.049
0.048
0.035
0.016
-0.01
-0.382
-0.037
-0.060
-0.083
-0.089
-0.093
-0.090
-0.406
0.077
0.397
-0.357
0.004
0.015
-0.043
-0.022
0.021
-0.002
0.03
-0.058
-0.033
0.031
0.044
0.045
-0.061
-0.018
0.025
0.063
0.058
-0.047
-0.001
0.027
0.071
0.073
-0.031
0.016
0.029
0.083
0.086
-0.01
0.021
0.03
0.099
-0.373
0.443
0.026
0.125
0.023
0.087
-0.035
0.048
-0.062
0.018
-0.080
-0.015
-0.094
-0.047
-0.128
0.980
-0.034
0.971
0.036
0.037
0.015
0.037
0.057
0.056
-0.004
0.038
0.078
0.056
-0.005
0.057
0.078
0.056
-0.007
0.074
0.073
0.058
-0.008
0.076
0.062
0.051
-0.017
0.07
Note: Tables report OLS coe¢ cients for business-cycle moments against the set
of industry characteristics. Coe¢ cients signi…cant at the 10 percent level are in
bold. Each column is a separate regression.
The multivariate analysis suggests that the low correlation of sectors intensive in agricultural inputs is not a simple artifact of those
sectors also being oriented toward household consumption.
The last two facts concern the dynamic relationships between sectoral output and aggregate output:
3) Sectors that are oriented toward the private sector (have a low
government share), that sell a large fraction of their output as intermediate inputs, use fewer agricultural inputs, use intermediate inputs
intensively, adjust prices frequently, and are not dependent on external
…nance tend to lead business cycles.
and
4) Sectors that are government-oriented, sell a small fraction of
their output as intermediate inputs, are not intensive in mining inputs, adjust prices less frequently, and are more dependent on external
…nance tend to lag business cycles.
Evert & Schwartzman: Behavior of Industrial Production
249
Table 7 Regression Coe cients (2)
4-…rm
Concentration
Ratio
Durability
Energy Inputs
Ext. Fin. Ratio
Household
Share
Government
Share
Construction
Share
Inv. Sales Ratio
Median Assets
Median Age
Engel Curve
Agricultural
Inputs
Mining Inputs
Intermediate
Inputs
Imported Inputs
Capital Share
Price Stickiness
t
t+1
t+2
t+3
t+4
t+6
t+8
-0.031
-0.021
-0.067
-0.028
-0.035
-0.013
-0.018
-0.026
-0.027
-0.007
0.036
-0.018
-0.01
0.001
0.087
-0.008
0.012
0.015
0.136
0.008
0.046
0.046
0.182
0.03
0.051
0.054
0.150
0.035
-0.034
-0.058
-0.068
-0.061
-0.042
-0.005
0.002
-0.075
-0.054
-0.03
-0.001
0.029
0.061
0.063
0.082
0.018
0.024
0.026
0.108
0.058
0.037
0.028
0.013
0.095
0.033
0.045
0.022
0.005
0.063
0.013
0.043
0.008
0.003
0.034
0.001
0.036
-0.003
0.005
0.009
-0.011
0.02
-0.027
0.006
-0.036
-0.01
0.011
-0.026
-0.01
-0.06
-0.086
-0.086
-0.065
-0.131
-0.039
-0.162
-0.012
-0.177
0.011
-0.172
0.022
-0.117
0.044
-0.069
0.036
0.050
-0.019
0.048
-0.002
0.04
-0.019
0.005
-0.039
0.032
-0.019
-0.048
-0.059
0.018
-0.015
-0.095
-0.059
-0.002
-0.012
-0.135
-0.049
-0.055
-0.009
-0.146
-0.061
-0.085
-0.012
-0.082
Note: Tables report OLS coe¢ cients for business-cycle moments against the set
of industry characteristics. Coe¢ cients signi…cant at the 10% level are in bold.
Each column is a separate regression.
Those two latter sets of facts add some interesting details to the
…rst two. For example, it becomes clear that having demand oriented
toward government consumption does not insulate a sector’s output
from business cycles but rather leads it to react with a lag. It is also
interesting to note that sectors that are very integrated in the production chain (in the sense of using intermediate inputs intensively) tend
to lead business cycles, whereas those that do not use as many intermediate inputs tend to lag. The relatively low correlation of sectors with
high …nancial dependence also hides the fact that they respond with a
lag. Finally, the regressions also point to an early response of ‡exible
price sectors and a delayed response of sticky price sectors.
3.
CONCLUSION
We asked a simple question: How do business-cycle statistics vary with
sectoral characteristics? Some of the answers were predictable, others
250
Federal Reserve Bank of Richmond Economic Quarterly
less so. The results highlight the promise and pitfalls of using industrylevel data to identify driving forces and propagation mechanisms in
business cycles. On the one hand, the results help focus the analysis
on channels that are more likely to be relevant and take away from
others that do not appear so relevant. For example, the analysis points
to pricing and …nancial frictions as channels worth investigating but
provides very little evidence of a prominent role for oil shocks. On the
other hand, the results highlight the need to interpret results with care,
since di¤erences in business-cycle behavior between industries may be
dominated by di¤erences in durability or demand composition that may
be correlated with other characteristics of interest.
Evert & Schwartzman: Behavior of Industrial Production
251
REFERENCES
Acemoglu, Daron, Vasco M. Carvalho, Asuman Ozdaglar, and Alireza
Tahbaz-Salehi. 2012. “The Network Origins of Aggregate
Fluctuations.” Econometrica 80 (September): 1977–2016.
Baxter, Marianne, and Robert G. King. 1999. “Measuring Business
Cycles: Approximate Band-Pass Filters for Economic Time
Series.” Review of Economics and Statistics 81 (November):
575–93.
Bernanke, Ben, and Mark Gertler. 1989. “Agency Costs, Net Worth,
and Business Fluctuations.” American Economic Review 79
(March): 14–31.
Bils, Mark, Peter J. Klenow, and Benjamin A. Malin. 2013. “Testing
for Keynesian Labor Demand.” In NBER Macroeconomics
Annual, vol. 27, eds. Daron Acemoglu, Jonathan Parker, and
Michael Woodford. Chicago: University of Chicago Press, 311–49.
Bils, Mark, Peter J. Klenow, and Benjamin A. Malin. 2014.
“Resurrecting the Role of the Product Market Wedge in
Recessions.” Working Paper 20555. Cambridge, Mass.: National
Bureau of Economic Research. (October).
Gertler, Mark, and Simon Gilchrist. 1994. “Monetary Policy, Business
Cycles, and the Behavior of Small Manufacturing Firms.”
Quarterly Journal of Economics 109 (May): 309–40.
Hadlock, Charles J., and Joshua R. Pierce. 2010. “New Evidence on
Measuring Financial Constraints: Moving Beyond the KZ Index.”
Review of Financial Studies 23 (May): 1909–40.
Hamilton, James D. 1994. Time Series Analysis. Princeton: Princeton
University Press.
Hamilton, James D. 2003. “What is an Oil Shock?” Journal of
Econometrics 113 (April): 363–98.
Hodrick, Robert J., and Edward C. Prescott. 1997. “Postwar U.S.
Business Cycles: An Empirical Investigation.” Journal of Money,
Credit, and Banking 29 (February): 1–16.
Kiyotaki, Nobuhiro, and John Moore. 1997. “Credit Cycles.” Journal
of Political Economy 105 (April): 211–48.
Nakamura, Emi, and Jón Steinsson. 2008. “Five Facts about Prices: A
Reevaluation of Menu Cost Models.” Quarterly Journal of
Economics 123 (November): 1415–64.
252
Federal Reserve Bank of Richmond Economic Quarterly
Prescott, Edward C. 1986. “Theory Ahead of Business-Cycle
Measurement.” Carnegie-Rochester Conference Series on Public
Policy 25 (Autumn): 11–44
Raddatz, Claudio. 2006. “Liquidity Needs and Vulnerability to
Financial Underdevelopment.” Journal of Financial Economics 80
(June): 677–722.
Rajan, Raghuram G., and Luigi Zingales. 1998. “Financial
Dependence and Growth.” American Economic Review 88 (June):
559–86.
Ramey, Valerie A. 2011. “Can Government Purchases Stimulate the
Economy?” Journal of Economic Literature 49 (September):
673–85.
Rotemberg, Julio J., and Michael Woodford. 1999. “The Cyclical
Behavior of Prices and Costs.” In Handbook of Macroeconomics,
vol. 1, part B, eds. John B. Taylor and Michael Woodford.
Amsterdam: North-Holland: 1051–135.
Schwartzman, Felipe. 2014. “Time to Produce and Emerging Market
Crises.” Journal of Monetary Economics 68 (November): 37–52.
Evert & Schwartzman: Behavior of Industrial Production
253
APPENDIX
A.1 List of Industries
Our industrial classi…cation is primarily based on the four-digit 2007
NAICS codes, with certain four-digit industries consolidated into a single category to facilitate the construction of either the PCE/industry
crosswalk or the industry controls. The full list of industries used is
displayed in Table 8.
A.2 PCE/Industry Crosswalk
We use the 2007 PCE Bridge Table published by the Bureau of Economic Analysis to match PCE expenditure categories to industries.
The Bridge Table contains consumer spending levels by PCE category
and commodity pairs. For each pair, the level of total spending going
to producers, wholesalers, retailers, and transport is provided.
Also included in the Bridge Table …le is a concordance of commodity categories with NAICS codes. This allows the commodities to
be matched with our industry groups. However, this concordance is
less granular in many cases than our industry classi…cation— these industries are pooled for the purposes of constructing the PCE/industry
crosswalk. In addition, while the analysis in the paper focuses predominantly on manufacturing and related sectors, for the purpose of
constructing this crosswalk, it is important to capture all sectors of the
economy in order to construct a more detailed PCE/industry crosswalk.
For this purpose, we make use of all the commodities and industries
present in the Bridge Table.
Using this commodity/NAICS concordance with the expenditure
data in the bridge table, we obtain expenditure estimates for producer
margins by PCE category and industry. No observations for the wholesale, retail, or transport industries exist, as their expenditure is contained in the corresponding wholesale, retail, and transport margins for
each PCE/industry pair. To create observations for these industries,
we total the entire margin across a given PCE category and use this
total as the value for that PCE/industry category pair. For instance,
we total all wholesale margins across the “auto leasing”PCE category,
and this is taken as the value for the wholesale/auto leasing pair. We
do this for each PCE category and for wholesale, retail, and transport.
For these, we sum the total wholesale margin across all industries
for a given PCE category and construct an additional observation designating the total as the expenditure for a given PCE category and
254
Federal Reserve Bank of Richmond Economic Quarterly
wholesale industry pair. We repeat this process for all PCE categories
and do the same for retail and transport as well.
Given total consumer expenditure broken down by PCE category/
industry pairs, we construct a crosswalk between the two categories
using expenditure share weights. This allows the translation of some
set of values at the PCE level to the industry level, or vice versa. For
each PCE/industry pair, the crosswalk contains two weights; one is the
proportion of the total industry expenditure that is also from the PCE
category, and the other is the proportion of total PCE category expenditure that is also from the industry. The former is used to translate
PCE-level data to the industry level and the latter from the industry
to the PCE level.
As an example of how this occurs, consider a set of data at the
industry level with one value per industry. This dataset is merged with
the crosswalk so that now each PCE category/industry pair contains
both the expenditure-share weights and the industry-level data value.
The PCE-level data are then estimated as the weighted average for the
PCE category across all industries. This provides an estimate of the
PCE value by imputing the data from the constituent industries that
make up the PCE category. Using the other weight that exists for each
PCE category/industry pair, the same process can occur in reverse,
with PCE data translated to the industry level.
Note that, as stated above, some industries do not have a unique
commodity code in the original Bridge Table and were thus pooled
for the construction of the crosswalk. For these industry groups, the
crosswalk will provide a single value for the group rather than a separate
value for each industry. In these cases, we assume that all industries
share this value in common.
A.3 Controls
A.3.1 Concentration Ratios
Industry-concentration data are taken from the 2007 Economic Census.
For each 2007 NAICS industry at the six-digit level, the census contains the percentage of total industry sales from the largest four, eight,
twenty, and …fty …rms, along with total industry revenue. We match
each six-digit NAICS category to the industry in which it is contained
and take the revenue-weighted mean across all six-digit NAICS within
the industry as the concentration ratio for that industry. This provides
a four-…rm, eight-…rm, twenty-…rm, and …fty-…rm concentration ratio
for each of our industries.
Evert & Schwartzman: Behavior of Industrial Production
255
For robustness, we construct additional concentration measures
from the same data: in addition to taking the revenue-weighted mean,
we also take both the median and the maximum concentration ratio
across six-digit NAICS industries. This leaves us with twelve values,
corresponding to either four, eight, twenty, or …fty-…rm concentration
ratios, and to either the mean, median, or maximum across subindustries.
A.3.2 Durability
The BEA publishes depreciation/durability estimates for consumer
durables, equipment, and structures. We match each PCE category
to a durable good, equipment, or structure category if a corresponding
category exists. We then take the service life estimate published by the
BEA as a measure of the durability of the item. Nondurable goods are
assigned a durability of zero. Values are then translated to the industry
level using the PCE category/industry crosswalk.
A.3.3 Inputs
From the 2007 Benchmark Input/Output Use Table, we calculate the
exposure of an industry to energy, agriculture, mining, as well as the
industry’s use of intermediate inputs. Using the commodity/NAICS
crosswalk provided with the Use Table, we match each commodity to
its corresponding industry and aggregate the Use Table to our industry
classi…cation. Where the provided concordance is not granular enough
for our industry classi…cation, we pool industries and assign the corresponding values to all industries in the group.
Energy Inputs: We take the proportion of total intermediate inputs that are from (1) electrical power generation, (2) oil and gas
extraction, (3) natural gas distribution, and (4) petroleum and coal
products manufacturing as a measure of each industry’s energy exposure.
Agricultural Inputs: We take the proportion of total intermediate
inputs that are from (1) crop production, (2) animal production and
aquaculture, and (3) support activities for agriculture and forestry as
a measure of each industry’s exposure to agriculture.
Mining Inputs: We take the proportion of total intermediate inputs
that are from (1) metal ore mining and (2) nonmetallic mineral mining
and quarrying as a measure of each industry’s exposure to mining.
256
Federal Reserve Bank of Richmond Economic Quarterly
Total Intermediate Inputs: We construct a measure of the total
intermediate inputs used by the industry by taking the ratio of all
industry inputs to the industry’s output.
A.3.4 Capital Share
Also from the Use Table, we estimate the relative intensity of capital
as opposed to labor in each sector. As for the input measures, we
…rst aggregate the Use Table to our industrial classi…cation. To do so,
we compute the ratio of gross operating surplus over the sum of gross
operating surplus and compensation to employees.
A.3.5 Output Shares
Again from the Use Table, we estimate several measures related to the
destination of each industry’s output. As before, we aggregate the Use
Table to our industry classi…cation.
Household Output: We calculate the household share as the proportion of industry output that goes to PCE.
Government Output: We calculate the government share as the
total output sold to all federal, state, and local government categories
listed in the Use Table as a ratio to total industry output.
Construction Output: We calculate the construction share as the
proportion of each industry’s output that is purchased by the construction sector.
Total Intermediate Output: We construct the proportion of total
industry output that was used as an intermediate inputs by any other
industry. For robustness, we also take the raw number of intermediate
inputs sold without normalizing by industry output.
A.3.6 Imports and Exports
The Use Table also contains information on imports and exports by
industry and can therefore also be used to calculate several measures
describing the international linkages of each sector.
Import Penetration: For each industry, we take the value of industry outputs that are imported into the United States and divide by
total industry production plus imports minus exports. This provides
Evert & Schwartzman: Behavior of Industrial Production
257
the share of each industry’s …nal goods sold domestically that were
produced internationally.
Exports: We calculate the export ratio as the share of industry output that is exported.
Imported Inputs: To measure the level of input connections to foreign markets, we calculate the ratio between imported intermediate
inputs to total industry output.
Imported Share of Inputs: As an alternative measure of the input connections to foreign markets, we calculate the ratio of the total
industry inputs that are imported.
A.3.7 External Financing Ratio, Cash Flow,
and Capital Expenditure
Using capital expenditure and cash ‡ow by …rm and year from Compustat for 1979 through 2015, we can construct the external …nancing
ratio as in Rajan and Zingales (1998), as one minus the ratio between
cash ‡ow to capital expenditure. Matching each …rm to an industry, we
take the median capital expenditure value across …rms for each industry and year. Then, we take the median again across years to obtain a
single value for each industry. The same procedure is used to obtain a
median cash ‡ow and median capital expenditure value for each industry. Rajan and Zingales (1998) describe the construction of the cash
‡ow variable in greater detail.
A.3.8 Inventory Sales Ratio
From Compustat we take …rm-level data on annual inventories and total
sales from 1979 through 2015. From this, we normalize inventories by
total sales for each …rm. Matching …rms to industries, we then take the
median value for each industry and year and then select the median
across years as the …nal industry value.
A.3.9 Size-Age Index
To construct measures of industry-speci…c …nancial constraints, we follow Hadlock and Pierce (2010), who show that an index that is linear
in …rm age and quadratic in …rm asset size can capture the degree
of …rm …nancing constraints. Speci…cally, the index is calculated as
:737 size + :043 size2 :04 age. We calculate this index for each
…rm and year between 1979 and 2015. Matching …rms to industries,
258
Federal Reserve Bank of Richmond Economic Quarterly
we take the median for each industry/year pair and again for each
industry. We do the same for asset size and age separately.
A.3.10 Luxury Goods
We construct two measures of the degree to which the outputs of each
industry are luxury goods. First, we use BLS data from the Consumer
Expenditure Survey, which details the consumption expenditures for
various goods by income decile. Matching these expenditure categories
with PCE categories, we construct estimates of expenditures for each
PCE category for the fourth and sixth income deciles and take the ratio
of these values as an estimate of the luxury status of a PCE category.
We then use the PCE/industry crosswalk to map these values to the
industry level.
As an alternate measure of the income elasticity of industry output,
we also take the Engel Curve slopes estimated by Bils et al. (2013).
They estimate these Engel Curve values for PCE categories, which we
map into the industry level using our PCE/industry crosswalk.
A.3.11 Price Stickiness
To capture the frequency of price changes within an industry, we take
the price-adjustment durations estimated by Nakamura and Steinsson
(2008). The estimates are provided at the Entry Line Item (ELI) level.
By using the ELI/PCE crosswalk provided by the BLS, we can transfer
these ELI-level duration values to the PCE classi…cation. For each PCE
category, we assign the average of the duration values for the set of
ELIs with which the PCE category is matched. Following this, we can
match PCE-level values to the industry level using the PCE/industry
crosswalk.
Evert & Schwartzman: Behavior of Industrial Production
259
Table 8 Industries
Industry
Oil and gas extraction
Coal mining
Metal ore mining
Nonmetallic mineral mining and quarrying
Support activities for mining
Electric power generation, transmission, distribution
Natural gas distribution
Animal food manufacturing
Grain and oilseed milling
Fruit and vegetable preserving and specialty food manufacturing
Dairy product manufacturing
Animal slaughtering and processing
Bakeries and tortilla manufacturing
Other food manufacturing
Beverage manufacturing
Tobacco manufacturing
Textile mills and textile product mills
Apparel, leather, and allied manufacturing
Sawmills and wood preservation
Veneer, plywood, engineered wood product manufacturing
Other wood product manufacturing
Pulp, paper, and paperboard mills
Converted paper product manufacturing
Printing and related support activities
Petroleum and coal products manufacturing
Basic chemical manufacturing
Resin, synthetic rubber, arti…cial synthetic …bers and
…laments manufacturing
Pesticide, fertilizer, other agricultural chemical manufacturing
Pharmaceutical and medicine manufacturing
Paint, coating, and adhesive manufacturing
Soap, cleaning compound, and toilet paper manufacturing
Plastics product manufacturing
Rubber product manufacturing
Clay product and refractory manufacturing
Glass and glass product manufacturing
Cement and concrete product manufacturing
Lime, gypsum and other nonmetallic mineral
product manufacturing
Alumina and aluminum production and processing
Nonferrous metal (except aluminum) production and processing
Foundries
Forging and stamping
Cutlery and handtool manufacturing
Architectural, construction, and mining machinery manufacturing
Hardware manufacturing
Spring and wire product manufacturing
Machine shops, turned product, screw, nut, bolt manufacturing
Coating, engraving, heat treating, and allied activities
Other fabricated metal product manufacturing
Agricultural, construction, and mining machinery manufacturing
Industrial machinery manufacturing
2007 NAICS
2112121
2122
2123
2132211
2212
3111
3112
3114
3115
3116
3118
3119
3121
3122
313-, 314315-, 3163211
3212
3219
3221
3222
3233243251
3252
3253
3254
3255
3256
3261
3262
3271
3272
3273
3274, 3279
3313
3314
3315
3321
3322
3323
3325
3326
3327
3328
3329
3331
3332
260
Federal Reserve Bank of Richmond Economic Quarterly
Table 8 (Continued) Industries
Ventilation, heating, air conditioning, and commerical refrigeration
equipment manufacturing
Metalworking machinery manufacturing
Engine, turbine, power transmission equipment manufacturing
Computer and peripheral equipment manufacturing
Communications equipment manufacturing
Audio and video equipment manufacturing
Semiconductor & other electronic component manufacturing
Navigational, measuring, electromedical, and control
instruments manufacturing
Electric lighting equipment manufacturing
Household appliance manufacturing
Electrical equipment manufacturing
Other electrical equipment and component manufacturing
Motor vehicle manufacturing
Motor vehicle body and trailer manufacturing
Motor vehicle parts manufacturing
Aerospace product and parts manufacturing
Railroad rolling stock manufacturing
Ship and boat building
Other transportation equipment manufacturing
Household and institutional furniture and kitchen
cabinet manufacturing
Medical equipment and supplies manufacturing
Newspaper, periodical, book, and directory publishers
3334
3335
3336
3341
3342
3343
3344
3345
3351
3352
3353
3359
3361
3362
3363
3364
3365
3366
3369
3371
3391
5111
@FedFRASER