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Multilateral Comovement in a New Keynesian
World: A Little Trade Goes a Long Way

WP 22-10

Paul Ho
Federal Reserve Bank of Richmond
Pierre-Daniel Sarte
Federal Reserve Bank of Richmond
Felipe Schwartzman
Federal Reserve Bank of Richmond

Multilateral Comovement in a New Keynesian World:
A Little Trade Goes a Long Way∗
Paul Ho

Pierre-Daniel Sarte

Felipe Schwartzman

Federal Reserve Bank of Richmond
November 16, 2022

Abstract
We study how international linkages and nominal price rigidities jointly shape the dynamics of
inflation and output across multiple large economies. We describe how these features produce
a global system of Phillips curves explicitly connected by multilateral trade relationships. In
equilibrium, disturbances abroad propagate to domestic variables not only directly, through
pairwise trade between countries, but also indirectly through third-country effects arising from
the network structure of trade. The combined propagation mechanisms imply that countryspecific shocks alone explain almost 90 percent of the observed average pairwise comovement in
output growth between countries. These idiosyncratic shocks also explain more than 1/2 the
cross-country comovement in inflation, and between output and inflation. We estimate that a
European inflationary shock results in significant U.S. inflation accompanied by lower output,
and that these responses transpire almost entirely from the network effects of trade. In addition,
a tightening of U.S. monetary policy generates a percentage decline in output globally that is
comparable to 1/2 the domestic response.

Keywords: International Comovement, Multilateral Trade, New Keynesian Phillips Curve
JEL Codes: E31, E32, F41, F44

∗

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 David Baqaee, Zhen Huo, Ricardo Reis, and
Stephanie Schmitt-Grohé, for discussions. We also thank participants at the Richmond Fed-UVA-Duke Research
Workshop and the Marvin Goodfriend Memorial Conference, as well as seminar participants at Pennsylvania State
University and the Brazilian Macroeconomics Group for their comments. Nick Garvey, Colton Lapp, James Lee, and
Erin Henry provided outstanding research assistance.
Paul Ho: paul.ho@rich.frb.org.
Pierre-Daniel Sarte: pierre.sarte@rich.frb.org.
Felipe Schwartzman felipe.schwartzman@rich.frb.org.

1

1

Introduction

Inflation and output growth exhibit substantial comovement between countries. Remarkably, the
correlation between inflation and output growth at business cycle frequencies, a hallmark of the
New Keynesian framework, is at times as pronounced across countries as it is within countries.
However, given that large economies including the U.S. or the European Union (E.U.) purchase
goods and services mostly from themselves, with shares of domestic purchases exceeding 80 percent
of total purchases, the quantitative study of inflation dynamics has largely proceeded within the
context of closed-economy models. Does it then follow that international linkages have only a
small role in explaining this comovement, and that global shocks are the main drivers of common
variations in output growth or inflation? In this paper, we develop a structural framework featuring
multiple large open economies with nominal rigidities that suggests otherwise. In other words, a
little trade goes a long way in explaining global comovement across countries.
The structural framework we develop brings together nominal rigidities in the tradition of New
Keynesian models (e.g., Christiano, Eichenbaum, and Evans (2005); Smets and Wouters (2007);
Justiniano, Primiceri, and Tambalotti (2010); Del Negro, Giannoni, and Schorfheide (2015)) with
the literature on multilateral trade (e.g., Eaton and Kortum (2002), Dekle, Eaton, and Kortum
(2007)). The network structure of trade, in particular, plays a critical role in increasing the significance of foreign shocks, including foreign demand shocks, for the domestic economy. These network
effects arise by way of both backward trade linkages, summarizing a country’s import network, and
forward trade linkages, summarizing its export network. The end result is that, despite relatively
low observed trade shares across countries, country-specific shocks generate sizeable cross-country
comovement in output growth, in inflation, and between output growth and inflation.1
International linkages between countries subject to nominal rigidities imply important departures from the conventional closed-economy New Keynesian Phillips curve (NKPC). First, whereas
marginal costs are only a function of domestic input costs in a closed economy, they now also
depend on foreign input costs and exchange rates of all trading partner countries. Second, these
input costs, and notably wages, in part reflect demand conditions abroad.
Both these departures from the conventional NKPC imply a global system of Phillips curves
explicitly connected by weights reflecting multilateral trade relationships. This system relates
inflation in one country to variations in output growth, consumption, and exchange rates in every
other country. The coefficients in this system of Phillips curves, therefore, provide a structural
interpretation for the weights used in empirical work relating domestic inflation to foreign real
activity (e.g., Bianchi and Civelli (2015)). In equilibrium, cross-country variations in output growth,
consumption, and exchange rates reflect the effects of foreign disturbances not only directly, through
pairwise trade between countries, but also indirectly through third-country effects. It is these third1

In that sense, our analysis generalizes the seminal contributions of Clarida, Gali, and Gertler (2002), and Galí
and Monacelli (2005), to a setting with multiple large countries with network implications from trade for global
comovement across countries.

2

country effects, in particular, that help generate sizeable comovement between domestic and foreign
variables.
The literature has studied various sources of international comovement. The importance of
global factors for the comovement in real activity is explored empirically by Kose, Otrok, and
Whiteman (2003), Stock and Watson (2005), and Del Negro and Otrok (2008), among others.
In Ciccarelli and Mojon (2010), Mumtaz and Surico (2012), and Andrés, Schmitt-Grohé, and
Uribe (2017), global factors also account for some of the observed comovement in inflation across
countries. Finally, Mumtaz, Simonelli, and Surico (2011) and Ferroni and Klaus (2015) establish
empirically that global factors play another role, connecting nominal and real variables. Aside from
global factors, the comovement in variables between countries can arise more structurally from the
endogenous propagation of idiosyncratic country shocks through trade, as explored recently with
respect to real activity in Huo, Levchenko, and Pandalai-Nayar (2019), as well as Boeckelmann,
Imbs, and Pauwels (2022), or through spillovers in global financial markets as in Imbs, Mumtaz,
Ravn, and Rey (2005), Imbs (2006), or Itskhoki and Mukhin (2021).
This paper merges these different strands of literature within a single structural framework
that highlights their interplay in a parsimonious set of equations. In the absence of trade, this set
of equations collapses to the canonical closed-economy New Keynesian model country by country.
More generally, however, New Keynesian forces interact with the effects of foreign wages and
exchange rates on marginal costs, of trade flows on local demand, and of financial flows on the
trade balance and exchange rates.
We use our structural model to quantify the role of both global factors and international linkages for the cross-country comovement in output growth, in inflation, and between output growth
and inflation. Our findings derive from a full-information Bayesian estimation of all structural
parameters (e.g., Justiniano and Primiceri (2008); Schmitt-Grohé and Uribe (2012); Herbst and
Schorfheide (2015); Bianchi and Melosi (2017)). This approach allows us to estimate the crosscountry effects of shocks while accounting for the network effects of trade, financial flows and other
general equilibrium channels. Our data cover five world regions, namely Canada, China, the E.U.,
Japan, and the U.S., and we allow for heterogeneous structural parameters, such as in the degree of
nominal rigidities across countries. Estimating a New Keynesian model with multiple large countries poses two key computational challenges. First, there are 2.5 times as many parameters and
about double the number of equations relative to Smets and Wouters (2007).2 Second, steady state
calculations are more involved than typical closed-economy New Keynesian models, requiring an
iterative procedure that can take half the time needed to compute the likelihood for a given set of
parameters. We show that recent state-of-the-art adaptive sequential Monte Carlo algorithms (Cai,
Del Negro, Herbst, Matlin, Sarfati, and Schorfheide (2021)) can help overcome those challenges.
The multi-country New Keynesian framework we develop implies that country-specific shocks
2
This particular curse of dimensionality is noted by Guerron-Quintana (2013) as a significant obstacle to the
estimation of New Keynesian models with more than two countries.

3

alone explain almost 90 percent of the observed average pairwise cross-country correlation in output
growth. These idiosyncratic shocks also explain more than 1/2 the cross-country comovement in
inflation, as well as between output and inflation. Our findings, therefore, highlight how trade
linkages, given nominal rigidities, are critical in propagating country specific shocks globally both
directly and indirectly.
Consider, for example, the effects of a European stimulative monetary shock on the U.S. Since
demand increases in the E.U., and E.U. consumption also consists of foreign goods, U.S. exports
to the E.U. rise, and so does U.S. production (i.e., a forward linkage). To the extent that higher
demand also increases E.U. production and its associated costs, including wages, the composition
of E.U. expenditures shifts towards U.S. goods, further increasing U.S. output. Importantly, these
direct effects on U.S. exports and production are complemented by indirect effects through third
countries. For example, suppose that Japan (the third country in this example) also trades with
the E.U. Then, higher E.U. wages lead Japan to substitute away from E.U.-produced goods and
towards imports from the U.S. (i.e., a backward linkage). Finally, the propagation of the initial
stimulative European shock continues further afield as higher U.S. export activity and production
raise U.S. wages, thus prompting higher U.S. imports from, say, China. Higher U.S. wages also
lead China to redirect some of its imports from the U.S. back towards the E.U., raising E.U. wages
and thus creating a new round of trade from the E.U. and so on.
We use our model to quantitatively explore two applications related to the domestic effects of
foreign shocks in the post-pandemic world. First, we estimate that an inflationary supply shock in
Europe spills over into a significant rise in U.S. inflation and fall in U.S. output, and that these
responses almost entirely transpire from third-country network effects. Second, we find that a
tightening of U.S. monetary policy results in a percentage output decline globally that is comparable
to 1/2 the domestic response.
This paper is organized as follows. Section 2 reviews some motivating data. Section 3 presents
the economic environment. Section 4 highlights key equilibrium conditions that describe the interplay between sticky prices and multilateral trade. Section 5 discusses the model solution in
linearized form. Section 6 explores the mechanisms underlying global spillovers from independent
country-specific shocks. Section 7 describes our quantitative findings. Section 8 concludes. A
detailed online Appendix contains derivations of all results discussed in the text, a comprehensive description of the data and statistical methods, discussions of departures from our benchmark
assumptions, and additional figures and tables complementing our main analysis.

2

GDP Growth and Inflation Across Countries

We begin by documenting some key stylized facts that motivate the paper using data for five world
regions, namely Canada, China, the E.U., Japan and the United States, over the period 2004Q2
through 2019Q4. These regions cover roughly 62 percent of world GDP over that period.

4

Figure 1: Comovement in Real GDP and Inflation Within and Across Regions
GDP Growth

5

Inflation

10
5

Canada

0

0

China
E.U.

-5

Japan
U.S.

-5

-10
2006 2008 2010 2012 2014 2016 2018

2006 2008 2010 2012 2014 2016 2018

Correlation Between GDP Growth and Inflation Across Countries
0.4

Correlation

Canada Inflation

0.2

China Inflation
E.U. Inflation

0

Japan Inflation
U.S. Inflation

-0.2
Canada
GDP Growth

China
GDP Growth

E.U.
GDP Growth

Japan
GDP Growth

U.S.
GDP Growth

Notes: In upper panels, GDP growth rates are quarter-over-quarter and deflated by actual inflation, while inflation
rates are quarter-over-quarter and annualized. In lower panel, each bar corresponds to correlation between GDP
growth in country i with inflation in country j. Filled bars correspond to cross-country correlations (i.e., i ̸= j),
empty bars correspond to own-country correlations (i.e., i = j) without trade. Each group of bars corresponds to a
country i’s GDP growth and each color corresponds to a country j’s inflation.

The top left-hand panel of Figure 1 illustrates the behavior of real GDP growth in each region.
While there are differences across regions, particularly with respect to the level of GDP growth in
China, the panel shows positive comovement in real activity across countries. The mean pairwise
correlation in GDP growth across countries is around 0.4, with Europe and Japan exhibiting the
largest correlation in real activity at 0.6.
The top right-hand panel of Figure 1 similarly shows that inflation rates tend to move together
across countries, albeit with a somewhat lower mean pairwise correlation of 0.25. The degree of
comovement in inflation between any two countries, however, can be quite large, as in the case of
Canada and the U.S. with a correlation of 0.8.
Finally, while both of these observations have been well documented for a wide range of countries
and time periods, the bottom panel of Figure 1 shows a less known fact - inflation and output comove
positively not only within countries but also across countries. Interestingly, the correlation between
real activity and inflation may be as large or larger across countries than it is within countries. For
instance, the correlation of E.U. output growth with E.U. inflation is around 0.2 but well above 0.3
5

with U.S. inflation and almost 0.4 with Chinese inflation. Similarly, Canadian real GDP growth
comoves as much with Canadian inflation as it does with U.S. inflation. Overall, mean cross-country
and own-country correlations are of similar magnitude, at 0.25 and 0.24, respectively. The notion
of a Phillips curve, therefore, if pertinent within countries is potentially equally relevant across
countries.
Two main possible mechanisms can underlie the comovement between countries depicted in
Figure 1. One relates to global factors in productivity, policy, or other disturbances. Another is
spillovers across countries that arise from the endogenous propagation of country-specific shocks
through international linkages. How does the interplay between global factors and the propagation
of country-specific shocks shape the dynamics of output and inflation across countries? What do
these interactions imply for the comovement shown in Figure 1? This paper presents a quantitative
framework aimed at addressing these questions.

3

Economic Environment

This section presents a New Keynesian world economy with international trade in goods and imperfectly integrated asset markets. There are N countries indexed by n = 1, ..., N . As in the
prototypical New Keynesian closed-economy model, each country has three types of agents: firms
that produce various types of outputs, households that consume final goods aggregated from these
outputs, and a monetary authority that sets policy according to a Taylor rule. However, we depart
from the closed-economy literature by introducing multilateral trade in goods and financial assets.
We model the interplay between sticky prices and multilateral trade by having two layers of intermediate inputs rather than the single layer of monopolistically competitive firms in the standard
model. The first layer consists of different varieties that may be purchased from different countries.
We refer to this first layer of intermediate varieties as tradable goods. The second layer consists of
intermediate inputs that are subject to nominal rigidities, as in Calvo (1983), and assembled into
final goods. We refer to this second layer of intermediate goods as sticky-price goods.
Frictions in global financial markets determine the extent to which countries can run trade
deficits or surpluses, thereby influencing the behavior of exchange rates and the way that shocks spill
over across countries. Thus, we introduce trade in financial assets by allowing households not only
to purchase and sell bonds domestically but also to trade state-contingent assets internationally.
Securities traded in global financial markets are subject to portfolio adjustment costs and exogenous
disturbances that, in equilibrium, influence the behavior of exchange rates.
Finally, a monetary authority in each country sets policy according to a country-specific Taylor
rule. To the degree that some coordination exists between different central banks, we allow for
common factors across countries’ monetary policy shocks.
Multilateral trade in goods and global financial markets means that disturbances abroad spill
over to domestic variables through the network effects of trade. We show that these departures

6

imply a global Phillips curve for vectors of output, inflation, consumption, and real exchange rates
that are interconnected across countries.

3.1

Households

The representative household in country n has preferences given by
∞
∑

{
βt

t=0

n
L1+η
(Cn,t /An,t )1−σn − 1
n,t
− ωn
1 − σn
1 + ηn

}
, 0 < β < 1, σn , ωn , ηn > 0,

(1)

where Ln,t denotes labor supplied by the household, Cn,t is a bundle of consumption goods, and
An,t summarizes country n’s technical progress. As in Lubik and Schorfheide (2005), the presence
of the term An,t in equation (1) ensures that each country evolves along a balanced growth path in
the long run, allowing hours worked, Ln,t , to remain stationary even if the intertemporal elasticity
of substitution, 1/σn , differs from 1.
Households in country n face the budget constraint
[
(
)]
Vt+1 Sn,t+1
Bn,t
Et [Vt+1 Sn,t+1 ]
1
+ Et Pn,t An,t Θn
+ ξn,t
+
Ern,t
Ern,t Pn,t An,t τ
Rn,t
1
= Wn,t Ln,t + Πn,t +
Sn,t + Bn,t−1 ,
Ern,t

Pn,t Cn,t +

(2)

where Pn,t is the nominal price of the consumption bundle (i.e., the price of final goods), Wn,t
denotes nominal wages, and Πn,t represents nominal firm profits. All prices in country n are
expressed in units of the local currency. The term Bn,t denotes the quantity of a bond traded
domestically by households at price 1/Rn,t at date t which pays 1 unit of the local currency in the
following period.
The budget constraint (2) differs from the usual closed-economy household budget constraint in
that countries trade a state-contingent security internationally, Sn,t+1 , subject to portfolio adjustment costs, Θn (·). The security is purchased at date t by country n at price Vt+1 , expressed in the
currency of a reference country r. It pays off 1 unit of the reference country currency, translated
into local currency units by the exchange rate Ern,t in the budget constraint (2).3
The function Θn (·) reflects a cost incurred by country n when adjusting its financial position,
similar to Schmitt-Grohé and Uribe (2003). We assume that Θn (0) = Θ′n (0) = 0 and Θ′′n (·) = τ > 0
so that portfolio adjustment costs increase at an increasing rate the larger country n’s financial
position is in absolute value. The curvature parameter τ captures the degree of financial frictions
in global financial markets: τ → 0 corresponds to perfect capital markets, while τ → ∞ represents
financial autarky. The function Θn varies with two objects. First, portfolio adjustment costs
incurred by country n depend on
3
1

Vt+1 Sn,t+1
Ern,t Pn,t An,t ,

the real value of state-contingent securities purchased

Thus, if reference country r uses, say, the Euro and country n is the U.S., Ern,t is expressed in €/$. Moreover,
Sn,t then represents a payoff in dollars from Sn,t shares of the state-contingent security purchased at date t − 1.

Ern,t

7

in global financial markets in units of local goods (i.e., the consumption bundle, Cn,t ), normalized
by An,t to account for country n’s rate of technical progress. Second, the term

1
τ ξn,t

captures

exogenous disturbances that affect country n’s ability to trade in international financial markets.
Substantively, these adjustment costs, Θn , and the shocks, ξn,t , ensure that we match exchange
rate movements when estimating the model. These frictions imply deviations from Uncovered
Interest Parity (UIP) that are an important feature of the data. As emphasized by Clarida, Gali,
and Gertler (2002) and Galí and Monacelli (2005), exchange rates generally affect the behavior of
domestic inflation in an international setting. Those deviations, therefore, become part of global
inflation dynamics.

3.2

Firms

The production side consists of final goods and two layers of intermediate goods. As in the conventional New Keynesian framework, final goods are produced using intermediate inputs subject
to nominal rigidities. We thus refer to this layer of intermediate goods as sticky-price goods. However, sticky-price goods are now produced using another layer of intermediate varieties that can be
purchased from other countries.
We denote final goods in each country by Cn,t . These goods may be used by

Final Goods:

households
for consumption proper
[ either (
)] or to pay for portfolio rebalancing costs, so that Cn,t =
Vt+1 Sn,t+1
1
Cn,t + Et An,t Θn Ern,t Pn,t An,t + τ ξn,t in the household’s budget constraint (2).
Final goods are assembled by firms using sticky-price intermediate goods, yn,t (j), with the
technology

[∫
Cn,t =

1

yn,t (j)

εn,t −1
εn,t

] ε εn,t−1
n,t

dj

, εn,t > 1.

0

We index the elasticity of substitution between intermediate varieties by t to account for potential
time-varying markups, denoted by µt =

εt
εt −1

> 1. Firms that produce final goods are competitive

and, given the constant-returns-to-scale technology, make zero profits. Given a price Pn,t (j) for
variety j, cost minimization implies the demand function
(
yn,t (j) =

where Pn,t =

[∫

1
1−εn,t dj
0 Pn,t (j)

]

1
1−εn,t

Pn,t (j)
Pn,t

)−εn,t

such that Pn,t Cn,t =

Non-Tradable Sticky-Price Intermediate Goods:

Cn,t ,
∫1
0

(3)

Pn,t (j)yn,t (j)dj.

In country n, each variety j of sticky-price

goods is produced using a composite input bundle, Qn,t (j), according to
yn,t (j) = An,t Qn,t (j).

8

(4)

This input bundle consists of different varieties (described below) that are traded across countries.
Crucially, shocks affecting production costs abroad will then affect firm choices at home. We denote
Q
Q
the price index associated with the composite bundle by Pn,t
so that Pn,t
Qn,t (j) = M Cn,t yn,t (j),
Q
where M Cn,t = Pn,t
/An,t is the nominal marginal cost faced by sticky-price goods producers. Their

real marginal cost is then given by
mcn,t =

Q
Pn,t
,
Pn,t An,t

(5)

which, in contrast to its closed-economy counterpart, can reflect foreign production costs through
Q
the price index of tradable goods, Pn,t
.

As in Calvo (1983), in each period a firm producing sticky-price goods in country n is unable
to choose its price optimally with probability θn . It then sets its price using the indexation rule,
Pn,t (j) = (1 + πn )Pn,t−1 (j), where πn denotes country n’s long-run or steady-state inflation rate.4
∗ , that
With probability 1 − θn , the firm is able to reset its price and optimally chooses a price, Pn,t

maximizes its expected present discounted value of future profits,
[
Et

∞
∑
k=0

where β

Λn,t+k
Λn,t

Λn,t+k k
β t+k
θ
Λn,t

(

∗ (1 + π )k
Pn,t
n
Pn,t+k

)−εn,t (

∗ (1 + π )k
Pn,t
n
− mcn,t+k
Pn,t+k

)

]
Cn,t+k ,

(6)

is the discount factor from t to t + k.

The implied first-order conditions indicate that under constant markups, µ =
nominal price chosen by sticky-price producers at date t,

∗ ,
Pn,t

ε
ε−1

> 1, the

is proportional to a weighted average

of their expected future nominal marginal costs. Inflation, therefore, will tend to rise when those
marginal costs are expected to increase. Importantly, because domestic marginal costs depend on
conditions abroad, foreign shocks will affect domestic prices and inflation.
Tradable Intermediate Goods:

The second layer of intermediate goods consists of tradables

whose production follows, for the most part, the framework originally laid out in Eaton and Kortum
(2002). Tradable goods come in varieties that differ in their production technology. In particular,
varieties are indexed by a vector of country-specific productivity parameters, z = {z1 , z2 , ..., zN },
where each element zn reflects the productivity of firms producing that variety in country n. Formally, the production of tradable variety z in country n is given by
qn,t (z) = zn,t ℓn,t (z),

(7)

4
In general, this indexation rule may also include lagged inflation, denoted 1 + πn,t−1 = Pn,t−1 /Pn,t−2 . In that
case, firms that are unable to choose their price set Pn,t (j) = (1 + πn,t−1 )ϱn (1 + πn )1−ϱn Pn,t−1 (j), ϱn ∈ [0, 1], where
ϱn and 1 − ϱn indicate the degree of indexation to lagged inflation and long-run inflation respectively. In Section D
of the Appendix, we show that this more general formulation introduces greater inflation persistence in the set of
interconnected country-specific Phillips curves that arises in this setting.

9

where ℓn,t (z) denotes local labor used in the production of that variety.5
Producers of sticky-price goods in country n can acquire any tradable variety z necessary to
produce those goods from any country including their own. From here on, we assume that stickyprice goods j ∈ [0, 1] are symmetric and abstract from the j index, Qn,t (j) = Qn,t . Thus, ∀j ∈ [0, 1],
the demand for tradable variety z in country n is
Qn,t (z) =

∑

Qnn′ ,t (z),

n′

where Qnn′ ,t (z) denotes traded varieties produced by country n′ and used by country n. These
tradable varieties are then aggregated into a single composite tradable bundle according to a CES
γ
] γ−1
[∫
γ−1
γ
aggregator, Qn,t =
Qn,t (z)
dΦ(z)
, γ > 1, which is then used as an input by sticky-price
firms in equation (4). The function Φ denotes the joint c.d.f. from which z is drawn.
As in Eaton and Kortum (2002), the elements of z are jointly drawn from a Fréchet distribution
{ ∑
}
with shape parameter φ, Φ(z) = exp − n (zn )−φ . This distributional assumption ensures
analytically tractable aggregate import demand functions (discussed in detail below). The shape
parameter, φ, governs how much a given variety z is likely to differ in productivity across countries.
Low values of φ imply large cross-country variations in productivity. In contrast, as φ → ∞, the
level of productivity associated with tradable varieties becomes common across countries.
Sticky-price producers in country n acquire each variety z from the country that supplies it
most cheaply given its production costs and bilateral trade costs. The technology in equation (7)
implies nominal production costs in country n′ of Wn′ ,t /zn′ ,t , where Wn′ t is the nominal wage in n′
in local currency. The cost of acquiring tradable varieties from abroad also includes an allowance
for shipping. In particular, goods imported by country n from country n′ are subject to iceberg
shipping costs such that only a fraction 1/κnn′ ∈ (0, 1) of those goods make it to their destination.
This implies that the price paid for variety z in country n, Pn,t (z), satisfies
Pn,t (z) = min
κnn′ Enn′ ,t
′
n

Wn′ ,t
.
zn′ ,t

Consistent with our definition of nominal exchange rates above, Enn′ ,t converts prices quoted in the
∑
currency of country n′ into that of country n.6 It also follows that qn,t (z) = n′ κn′ n Qn′ n,t (z) since
actual output of any tradable variety z, qn,t (z), must account for losses in transit in meeting the
demand for that variety.
5

We follow the convention adopted by Eaton and Kortum (2002) in indexing varieties by their productivity vectors,
z, rather than an arbitrary label i ∈ [0, 1]. Observe that any differences in production and prices across varieties
reflect only productivity differences. Therefore, denoting the output of variety i by qn,t (i), and letting z(i) represent
its corresponding productivity vector, it follows that qn,t (i) = qn,t (z(i)). Thus, one need only refer to qn,t (z).
6
Observe that no-arbitrage in currency markets implies that Enn′ ,t = Enn′′ ,t En′′ n′ ,t .

10

3.3

Monetary Authority

As in Taylor (1993, 1999), monetary policy in country n is described by the standard monetary
policy rule (e.g., Smets and Wouters (2007); Justiniano et al. (2010); Del Negro et al. (2015)),
Rn,t =

ρn
∗1−ρn νn,t
Rn,t−1
Rn,t
e

where

∗
Rn,t

(
= Rn

1 + πn,t
1 + πn

)ϕn,π (

Yn,t
Y n,t

)ϕn,Y
,

(8)

Rn,t is the nominal interest rate, denoted by Rn in the steady state, Y n,t denotes trend output,7
and νn,t is an exogenous monetary policy shock. With ϕn,π > 0 and ϕn,Y > 0, the monetary
authority increases nominal interest rates in response to high inflation or output relative to their
long-run paths. We allow the monetary policy rule to differ across countries to account for different
responses to inflation or the output gap. Unlike closed-economy settings, we also allow the shock,
νn,t , to be correlated across countries to account for potential coordination in policy.8

4

Equilibrium Conditions in a World with Trade and Sticky Prices

4.1

Some Key Equilibrium Relationships

We now highlight several key equilibrium conditions that describe the role of trade in an environment with nominal rigidities. Details of all derivations are provided in the Appendix.
Aggregate Import Shares:

The economic environment we have just presented yields tractable

expressions that describe aggregate trade shares as functions of their underlying determinants.
In the notation introduced above, κnn′ Qnn′ ,t represents real imports of variety z by country
n. Letting pn′ ,t = Wn′ ,t /zn′ ,t denote the price charged by country n′ for variety z quoted in local
currency, Enn′ ,t pn′ ,t (z)κnn′ Qnn′ ,t is the value of country n’s imports from n′ in n’s currency. Then,
∫
Xnn′ ,t =

Enn′ ,t pn′ ,t (z)κnn′ Qnn′ ,t (z)dΦ(z)

represents expenditures by country n on all traded intermediate varieties from country n′ . It follows
∑
that Xn,t = n′ Xnn′ ,t represents country n’s total expenditures on varieties from abroad.
Under the maintained assumptions, each country purchases all tradable varieties. It acquires
any one variety z from a single other country (frequently itself). The choice of whom to purchase
a given variety from depends on the production cost of that variety in different countries as well as
shipping costs. The assumption that z is Fréchet distributed ensures that the total value of trade
7

In our model, trend output is proportional to the stochastic trend At .
Similar to Cúrdia, Del Negro, and Greenwald (2014) and Leeper, Traum, and Walker (2017), we abstract from
the zero lower bound, keeping the system log-linear. To the extent that the zero lower bound amplifies the effects
of demand shocks, it would also reinforce the effect of ‘third countries’ in trade and their implications for global
comovement.
8

11

between any country pair is a smooth function of foreign marginal production costs. In particular,
the share of country n’s expenditures on varieties from country n′ at date t, ϖnn′ ,t , is given by
ϖnn′ ,t

(
)−φ
κnn′ Enn′ ,t Wn′ ,t
Xnn′ ,t
=∑ (
=
)−φ .
Xn,t
n′′ κnn′′ Enn′′ ,t Wn′′ ,t

(9)

Thus, the share of country n’s expenditures on imports from country n′ reflects the costs of acquiring
varieties from n′ , including production costs, Wn′ ,t and bilateral shipping costs, κn,n′ , relative to
all other countries (all expressed in n’s currency).
We briefly highlight three important features of the matrix of import shares, ϖt = {ϖnn′ ,t }.
First, ϖt has the properties of stochastic matrix in the sense that it is a real square matrix where
each row sums to one, a feature that will prove useful below. Second, the larger the Fréchet shape
parameter φ, the more import shares depend on relative wages and exchange rates. Put another
way, when φ is large, different varieties are produced with similar productivities in all countries. In
that case, small shifts in wages or exchange rates lead to large changes in the number of varieties
traded between country pairs. In contrast, limφ→0 ϖnn′ t = 1/N ∀n, n′ . Finally, equation (9) implies
that, conditional on wages and exchange rates, trade costs, κnn′ , can be directly inferred from longrun import shares, ϖnn′ . This feature makes it possible to condition the quantitative analysis on
those shares directly rather than trade costs.
Marginal Cost and the Price of the Input Bundle:

Recall that the marginal cost faced

Q
by producers of sticky-price goods in country n depends on the price of their input bundle, Pn,t
,

in equation (5). The assumption in Eaton and Kortum (2002) that z is drawn from a Fréchet
distribution with shape parameter φ implies that the price of the input bundle is
(
Q
Pn,t

= Γ (Υ)

1
1−γ

∑(

κnn′ Enn′ ,t Wn′ ,t

)−φ

)− 1

φ

,

(10)

n′

where Γ(Υ) is the Gamma function evaluated at Υ = 1 +

1−γ
φ .

Equation (10) illustrates how

production costs, Wn′ t , trade costs, κnn′ , and exchange rates, Enn′ ,t , across different countries
Q
all factor into the price of the input bundle, Pn,t
, paid by country n firms. This mechanism

is analogous to how these same factors determine import shares in equation (9). In that sense,
equation (10) captures the role of backward trade linkages, or alternatively country n’s import
network, in determining domestic marginal production costs. Because these multilateral trade
linkages then directly impact real profits of country n firms, they also affect their pricing decisions.
Aggregate Market Clearing for Traded Intermediate Goods:

Given perfectly competitive

markets, the total cost of producing tradable varieties in country n, which is to say its wage bill,
∫
Wn,t Ln,t = Wn,t ℓn,t (z)dΦ(z), exhausts the value of varieties it produces and exports to foreign

12

countries. In equilibrium, these exports must equate to the demand for country n’s domestic
product by economies abroad. Foreign demand for n’s goods in turn amounts to expenditures
abroad on country n’s varieties, or Enn′ ,t Xn′ ,t for trading partner n′ (in n’s currency), weighted by
their import share in country n’s goods, ϖn′ n,t . It follows that
Wn,t Ln,t =

∑

ϖn′ n,t Enn′ ,t Xn′ ,t .

(11)

n′

Equation (11), therefore, highlights how forward trade linkages, or alternatively country n’s export
network, help determine the value of domestic output.

4.2

Steady State

As in much of the New Keynesian closed-economy literature, our objective is in part to estimate
our multi-country framework approximated around its steady state and explore its quantitative
implications. However, unlike much of the related literature, the New Keynesian steady state we
study possesses unique features that arise from international trade linkages. We discuss salient
implications of these features here and refer the reader to the Appendix for details.
Country-Specific Balanced Growth Paths: In departing from the closed-economy environment, we focus on a world balanced growth path that has two features. First, steady state growth
rates for variables such as consumption, real wages, and inflation are country-specific. Second, the
world steady-state growth path nevertheless features constant country shares in the world economy
so that no country’s share of the world economy grows or shrinks indefinitely.
For a given economy n, the derivation of country-specific balanced growth paths for real quantities and steady-state inflation follows that of closed-economy models. In particular, real quantities
grow at rate gn = E [gn,t ], which denotes the long-run mean of technical progress, gn,t =

An,t
An,t−1 .

In addition, prices increase at the rate 1 + πn , which also determines the steady state interest rate
through the Fisher equation, 1 + πn =

βRn
gn .

The existence of a global balanced growth path where no country eventually dominates or
becomes irrelevant in the world economy is shown in the Appendix. This requires, among other
P

,
conditions, that the real exchange rate for each country n relative to the reference country, Ern,t Pn,t
r,t
grow in the long run at a rate that reflects the difference in technical progress between country n
and the reference country.9
9

The derivations of these paths, and the conditions that must hold for global balanced growth in that environment,
open up applications beyond nominal rigidities, for example with respect to the study of long-run global change. See
Müller, Stock, and Watson (2020) for a statistical analysis of the joint long-run evolution of GDP per capita for 113
countries over 118 years from 1900 to 2017. From a structural modeling standpoint, the assumption that varieties z
are drawn from a Fréchet distribution in much of the trade literature is not innocuous. In some instances, countryspecific balanced growth paths within a stable world economy, where country shares are constant in the long run,
will not be feasible given constant trade costs, 1/κnn′ ∈ (0, 1). See Section B of the Appendix.

13

Trade Balances:

While the model features transition dynamics whereby different countries can

run trade deficits or surpluses, the form of portfolio adjustment costs does not ultimately pin down
countries’ long-run net asset positions. In other words, analogous to the static framework of Eaton
and Kortum (2002), the environment does not provide a theory of countries’ steady-state trade
deficits or surpluses. As such, we adopt a benchmark defined by balanced trade in steady state.
However, even over periods as long as our sample (16 years), a country’s net asset position can
differ from zero on average. We address this observation following the approach first proposed by
Dekle, Eaton, and Kortum (2007) in a static environment but now applied to the steady state of
a dynamic setting. Specifically, we first write a variant of the economic environment whose steady
state can be matched to observed average regional trade imbalances over our entire sample, 2004Q2–
2019Q4. This can be done by allowing for exogenous frictions in international asset markets even in
the long-run, that is ξn ̸= 0. Then, for the set of parameters implied by a world with long-run trade
imbalances, we calculate counterfactual steady-state allocations and prices corresponding to zero
trade imbalances. More specifically, for the implied set of parameters, we compute counterfactual
steady-state allocations consistent with ξn = 0 ∀n, in which case real net asset positions are zero
for all countries, Sn = 0 ∀n. Analogously to Dekle et al. (2007), the resulting counterfactual world
steady-state allocations without imbalances can then be used to produce a linearized model for
estimation and analysis using standard linear rational expectations toolkits.

5

The Linearized Global New Keynesian Model

This section describes the solution to our model economy, casting some of its equations in terms
of expressions familiar from international trade. We use these expressions to provide intuition for
the mechanisms and forces underlying the New Keynesian world economy.
To obtain intuitive expressions, we follow the New Keynesian literature and log-linearize the
deflated-detrended equilibrium equations around their steady state. Thus, all variables in the
expressions below refer to their appropriately deflated-detrended counterparts.10 We use a ‘hat’
over variables to denote log-deviations from steady-state (i.e., x
bn,t = log(xn,t ) − log(xn )), and
use dx to denote level deviations from steady state, (i.e., dxn,t = xn,t − xn ). We simplify these
expressions further by assuming that the demand faced by sticky-price firms is subsidized at rate
1 + τn , set to the inverse of the markup to free our discussion from steady-state distortions arising
from monopolistic pricing.11 Thus, the long-run real marginal cost of sticky price goods is 1 in all
countries (i.e., mcn = 1 ∀n). In the quantitative analysis, however, we allow for pricing distortions.
We let En,t ≡ Ern,t for notational convenience and, without loss of generality, take country 1 to be
the reference country, (i.e., r = 1). Under no arbitrage in currency markets, E1,t = Er,t = Err,t = 1.
10

See Sections B.1 and B.2 of the Appendix for details of the model normalization with respect to country-specific
inflation, 1 + πn , and technical progress, gn .
(
)−εn,t
(
)
Π
(j)
Pn,t (j)
Pn,t (j)
11
Specifically, firms’ profits are then given by Pn,t
=
C
(1
+
τ
)
−
mc
, where the
n,t
n
n,t
Pn,t
Pn,t
n,t
subsidy to sticky-price firms is financed by lump sum taxes, Tn,t .

14

Similarly, we define the trend-adjusted real exchange rate, which converts country n final goods
A

P

n,t
into units of country 1 final goods, as en,t = En,t An,t
. Finally, while we allow for country1,t P1,t

specific preference and technology parameters in the estimation, σn , ηn , ωn , θn , etc., we set these
parameters to be the same in this section to simplify notation. To that end, we also express all
equations in vector form, stacking country-specific variables into vectors x
bt = (b
x1,t , ..., x
bN,t ).
The interaction between nominal rigidities, international trade, and international financial flows
can be summarized by two main blocks of equations. The first block describes conventional New
Keynesian mechanisms in all countries. The second block captures forces underlying international
trade in goods and financial assets. Crucially, these two blocks are linked through the marginal
cost of sticky price goods, a central element of the New Keynesian Phillips curve (NKPC), that
now reflects export prices of trading partner countries and the fundamentals that drive them.

5.1

The New Keynesian Block

The New Keynesian block reflects intertemporal mechanisms at home along with a description of
monetary policy and domestic labor supply,
(
)
bt = Et C
bt+1 − 1 R
bt − Et π
C
bt+1 − Et gbt+1 ,
σ
(1 − θ)(1 − βθ)
π
bt =
(mc
ct + µ
bt ) + β(Et [b
πt+1 ]),
θ
b t = ρR
bt−1 + (1 − ρ)(ϕπ π
R
bt + ϕY Ybt ) + νbt ,
bt + σC
bt ,
w
bt = η L

(12)
(13)
(14)
(15)

bt is a vector that contains consumption deviations from steady state for each country,
where C
bt = (C
b1,t , ..., C
bN,t ), and similarly for π
bt , etc. Equations (12)-(14) replicate familiar expressions
C
bt , R
from the New Keynesian literature in vector form: equation (12) is a standard household Euler
equation for every country, equation (13) describes a vectorized NKPC that highlights the role
of marginal cost, markup shocks, and expected inflation, and equation (14) describes a monetary
policy rule for each country. Finally, equation (15) describes labor supply in all countries with its
attendant income and substitution effects. The model includes three exogenous shocks per country:
shocks to technical progress, gbt , markup shocks, µ
bt , and monetary policy shocks, νbt .
In the absence of trade, the New Keynesian block reduces to a set of conventional closed-economy
three-equation New Keynesian models (e.g., An and Schorfheide (2007); Galí (2008)), thus yielding
isolated New Keynesian economies (i.e., autarky). In particular, absent trade, the real marginal
cost of sticky-price goods, mc
c t , is determined by payments to domestic inputs, in this case wages,
b
b
bt , the marginal
w
bt = η Lt +σ Ct . Since, absent trade, linearized output, Ybt , equates to consumption, C
cost faced by domestic firms ultimately reflects a domestic output gap, mc
c t = (η + σ)Ybt .

When countries are instead engaged in multilateral trade, the marginal cost of sticky-price
goods no longer equates to payments for domestic inputs. This marginal cost now reflects global

15

and country-specific forces operating through adjustments in countries’ trade balances and net asset
positions along with global market clearing. These international forces are described in the second
block of equations.

5.2

The International Block

The second block of equations relates to relationships that arise under international trade. These
include i) a set of equations describing the marginal cost of producing sticky-price goods in each
country as a function of cost conditions abroad, ii) a set of equations describing market clearing
conditions for traded goods, iii) a set of equations describing the balance of payments in each
country, and iv) a set of equations describing the evolution of countries’ financial assets.
i) Marginal Cost of Producing Sticky-Price Goods:
Unlike a closed economy where marginal costs are determined only by the price of domestic inputs,
marginal costs generally depend on costs in all trading partner countries. In particular, equation
(10) implies that
ebt + mc
c t = ϖ(b
et + w
bt ),

(16)

where ebt , mc
c t , and w
bt are vectors of exchange rates, marginal costs, and wages, respectively, in loglinearized form. The left-hand side of equation (16) is the marginal cost of producing sticky-price
goods for each country, expressed in real units of the reference country’s goods. This is determined
by wages in each country, expressed in the reference country’s goods, weighted by the matrix of
import shares, ϖ = {ϖnn′ }. The influence that country n′ has on country n increases in proportion
to how much n imports from n′ , ϖnn′ . As such, ϖnn′ captures the role of backward trade linkages
in determining marginal cost.
ii) Market Clearing for Traded Goods:
There must be N market clearing conditions for traded goods, one for each country. These can
be expressed in vector form in a way that describes the makeup of the real trade balance. From
equation (11),
b
Yb − C
| t {z }t
real trade balance

=

bt −
(ψ T − I)C
|
{z
}
relative demand
in export markets

(1 + φ)
| {z }
price elasticity
of trade

(

)
I − ψ T ϖ)(b
et + w
bt ,
{z
}
|

(17)

relative price
at destination

where ψ = {ψn′ n } denotes the matrix of (steady state) export shares from country n to n′ .12
bt , captures real quantities produced and consumed,
The left-hand side of the equation, Ybt − C
defining the real trade balance. Deviations in countries’ real detrended GDP from their steady state
b t . Naturally, the real trade balance depends on the relative demand in export markets, the
are Ybt = L
12

Formally, ψnn′ =

∑

n′′

ϖnn′ en mcn Cn
ϖn′′ n′ en′′ mcn′′ Cn′′

.

16

first term on the right-hand side of equation (17). It moves towards a surplus when the consumption
bt , increases. Here, each foreign country’s consumption of home-produced
of home goods abroad, ψ T C
goods is weighted by the share of exports it purchases from the home country. Conversely, the real
bt , increases.
trade balance moves towards a deficit when domestic consumption, I C
The real trade balance also depends on the relative price of exports in foreign markets, the
second term on the right-hand side of equation (17). The term compares the price of exports in
each country (in real units of the reference country), ebt + w
bt , with the corresponding prices of goods
produced in other countries weighted by ψ T ϖ. As such, the term (I − ψ T ϖ)(b
et + w
bt ) represents
the substitution from countries with higher costs to those with lower costs.
Importantly, this substitution means that the trade balance in country n depends on country n′
not only through economic conditions in country n′ , but also through conditions in other countries
that trade with n′ . The import matrix, ϖ, indicates that in response to an increase in wages in
a ‘third’ country n′′ , country n′ will substitute toward country n (and thereby increase country
n’s trade balance according to how much n′ imports from n). Country n′ reacts more strongly to
real wages in places from which it imports heavily, substituting away from these ‘third’ countries
and towards country n proportionally to its import shares with n′′ . Concretely, suppose that real
wages in China increase, ebn′′ ,t + w
bn′′ ,t > 0. All else equal, Japan reacts by substituting toward U.S.
exports and away from China in proportion to its import share from China (captured in ϖ). This
shift in trade increases the U.S. trade balance in proportion to its exports towards Japan (captured
in ψ T ). Equation (17) thus emphasizes the role of both backward trade linkages, ϖ, and forward
linkages, ψ T , in driving the effects of foreign prices on the trade balance.13
The Fréchet parameter, φ, governs the scale of real trade adjustments to relative export prices.
In other words, φ ultimately plays the role of a price elasticity of trade. Larger values of φ imply
less dispersion in the productivity of any given variety across countries. In that case, firms are
more sensitive to a given relative price change and substitute inputs more readily across countries
resulting in larger trade effects.
iii) Balance of Payments:
Rearranging countries’ household budget constraints (2), linearized around a steady state with no
monopolistic trade distortions, yields the following expression relating the trade balance and the
terms of trade to financial flows
13

These third-country effects stand in contrast to much of the open economy New Keynesian literature modeling
either two large countries or individual small open economies. Guerron-Quintana (2013) and Aggarwal, Auclert,
Rognlie, and Straub (2022) go a step further by considering multiple small open economies, but Guerron-Quintana
(2013) acknowledges that such models abstract from the multiple layers of interactions across countries that are
central in an environment with multiple large countries and highlighted herein.

17

[

]
dst+1
dst
.
+ (I − ϖ)(b
et + w
bt ) = V g1 (1 + π1 )Et
−
|
{z
}
eC
eC
|
{z
}
real trade balance
terms of trade
b
Yb − C
| t {z }t

× s.s. import shares

(18)

financial flows

The left-hand side of the equation is a vector of trade balances for each country in local units. It
bt , and a
can be decomposed into a real trade balance of quantities produced and consumed, Ybt − C
measure of the terms of trade for each country, scaled by import shares, (I − ϖ)(b
et + w
bt ).14 The
terms of trade captures the wedge between production costs (which are a function of local factor
prices) and consumption costs (which are a function of foreign factor prices and exchange rates).
As emphasized by Kehoe and Ruhl (2008), while terms of trade fluctuations do not affect real GDP,
they do affect welfare. Importantly, the equation highlights that the terms of trade also affect the
domestic value of the trade balance that needs to be covered by financial flows.
The right-hand side captures financial flows, normalized by consumption. Keeping track of financial flows becomes necessary when international financial markets are subject to frictions. These
frictions imply that adjustments to countries’ net asset positions are costly and so distort consumption allocations across time and states of nature. To the extent that portfolio rebalancing costs
affect financial flows, the balance-of-payments equation implies that they also affect the behavior
of the real trade balance and the terms of trade. In the limiting case, where portfolio adjustment
costs become infinitely high, trade balance holds period by period and creates a tight link between
consumption, output, and exchange rates.
iv) Financial Asset Allocations:
Finally, around a steady state with balanced trade, the Euler equation governing the evolution of
countries’ financial assets satisfies
cumulative financial
asset position

b1,t − C
bt
C
| {z }

=

relative consumption

1
σ

ebt
|{z}
real
exchange rate

where dSt =

τ
+β
σ
|

t
∑

z(

}|

1
dSt − dS1,t
e

){

cumulative portfolio
adjustment shocks

+
{z

z
}|
{
dΞt − dΞ1,t

,
}

(19)

asset pricing wedge

dsk ; dΞt =

t−1
1 ∑
dξk .
σ
k=−∞

k=−∞

Absent frictions in global financial markets, defined by τ = 0 and dξt−k = 0 ∀k, equation (19)
reduces to the expression first emphasized by Backus and Smith (1993). In that case, the (log)
b1,t − C
bt , reflects variations in their real exchange
ratio of consumption between any two countries, C
14

The terms of trade is the ratio of a country’s export prices to import prices (in common units,∑here the reference
1
country’s final goods). Thus country n’s linearized terms of trade are given by ebn,t + w
bn,t − 1−ϖ
ϖ ′ (b
e ′ +
(
)
(
) nn n′ ̸=n nn n ,t
∑
−1
1
w
bn′ ,t ) = 1−ϖnn ebn,t + w
bn,t − n′ ϖnn′ (b
en′ ,t + w
bn′ ,t ) , or in matrix form, diag(I − ϖ)
(I − ϖ)(b
en,t + w
bn,t ).

18

rate, (1/σ)b
et , (i.e., the relative price of consumption between the two countries). Put another way,
higher consumption at home goes hand in hand with a depreciating exchange rate (as consumption
goods abroad become more expensive).
However, as also noted by Backus and Smith (1993), the data do not support a clear connection
between countries’ relative consumption and their real exchange rate. This ‘consumption-exchange
rate disconnect puzzle’ is suggestive of potential financial frictions that drive a wedge between crosscountry differences in consumption and their corresponding real exchange rates.15 Here, we allow
for two dimensions in this wedge on the right-hand side of equation (19): one relates to exogenous
disturbances in financial markets (relative to the reference country), dΞt − dΞ1,t , and the other
relates to endogenous portfolio adjustments (relative to the reference country), dSt /e − dS1,t , where
dSt is a vector representing the history of countries’ net asset positions (in linearized form). This
history matters because financial frictions affect consumption growth through the Euler equation,
and thus have a permanent effect on consumption levels. For example, following a one-time financial
disturbance that lowers consumption in country n relative to the reference country, country n
households will from then on keep consumption smooth at a level lower than they otherwise would
have absent the shock.16

6

Foreign Spillovers to the Domestic Economy

Given the equilibrium conditions characterizing the interaction between nominal rigidities and
international linkages, we now combine the two blocks of equations to derive the implications for
cross-country correlations in output and inflation. First, we show that our model implies a set of
interconnected Phillips curves that characterize how each country’s inflation depends not only on
domestic factors but also on real activity and the exchange rates of its trading partners. Next,
we present a special case of our model whose analytical solution makes the role of third-country
network effects for cross-country comovement transparent.

6.1

The Global Phillips Curve

Implicit in the two blocks of equations above is a system of interrelated Phillips curves. This system
defines a global Phillips curve for vectors of variables pertaining to all countries, highlighting how
domestic inflation in each country is connected to real activity and exchange rates of trading
partners. The global Phillips curve thus provides a useful summary of how trade connects nominal
and real variables across countries, similar to how the conventional Phillips curve summarizes the
trade-off between output and inflation in a closed economy.
15
See Kollmann (1995), Benigno and Thoenissen (2008) and Corsetti, Dedola, and Leduc (2008), for examples of
papers that specifically address the Backus and Smith (1993) puzzle through financial frictions.
16
An interesting implication of this history dependence is that the model can imply very persistent real exchange
rate fluctuations in response to transitory financial or monetary shocks, in line with the ‘purchasing power parity
puzzle’ documented in Rogoff (1996).

19

In the NKPC for each country, described by equation (13), the marginal cost of producing stickyprice goods depends on multilateral trade considerations. It follows from the labor supply equation
(15) in the New Keynesian block and the expression for marginal cost (16) in the international
block that
bt ) + (ϖ − I)b
mc
c t = ϖ(η Ybt + σ C
et .
In other words, in equation (16), the marginal cost at home depends not only on domestic production costs (domestic wages in this case), but also on the real exchange rate and production costs of
imported goods (wages in trading partner countries). These in turn reflect real activity embodied
bt .
in the determination of labor supply abroad, w
bt = η Ybt + σ C
The global Phillips curve is then given, in vector form, by
[
]
bt ) + (ϖ − I)b
π
bt = κ ϖ(η Ybt + σ C
et + µ
bt + β(Et [b
πt+1 ]),
where κ =

(1−θ)(1−βθ)
θ

(20)

is the coefficient of inflation on the output gap in the conventional closed

economy model. In this context, however, the coefficient, κ, applies to an import-share weighted
bt ), and also applies to variations in
average of domestic and foreign production costs, ϖ(η Ybt + σ C
real exchange rates, (ϖ − I)b
et . Put another way, inflation in country n no longer moves one-for-one
with country n’s own output gap (conditional on expected inflation), but also depends on real
activity and exchange rates in trading partner countries. Specifically, the comovement between
inflation in country n and real activity and exchange rates in country n′ is proportional to country
n’s import share from country n′ , ϖnn′ .17
The global Phillips curve (20) imposes theoretical structure on the weights that different foreign
countries have in the determination of domestic inflation. The choice of appropriate weights is a
problem that Borio and Filardo (2007) and Bianchi and Civelli (2015), among others, had to
contend with in their study of global determinants of domestic inflation. The global Phillips curve
implies that conditional on international output and consumption data, import-weighted exchange
rates are appropriate.
To quantify the role of foreign spillovers for domestic inflation, one approach is to estimate
equation (20) using direct observations or proxies of the variables. However, in equilibrium, variations in exchange rates and output are ultimately determined by global or country-specific shocks
operating through adjustments in countries’ trade balances and asset positions as well as global
market clearing. Simply put, the variables on the right-hand side of (20) are endogenous. Real
17

The global Phillips curve as written in equation (20) depends not only on output in all countries, but also on
expenditures (here, consumption) and exchange rates. Therefore, it differs from more compact expressions developed
in frameworks with complete markets such as Galí and Monacelli (2005) and Clarida et al. (2002), where equilibrium
trade relationships and financial market equilibrium conditions may be used to substitute for consumption and
exchange rates. Importantly, our multi-country setup allows the expression to highlight the role of heterogeneous
trade linkages. Another difference is that our expression holds for total headline inflation, rather than just inflation in
domestically produced goods. This difference stems from our assumption that local sticky price producers use goods
produced by workers in all countries rather than local labor. Our framework also assumes local currency pricing by
construction.

20

exchange rates, in particular, will deviate from uncovered interest rate parity (UIP) according to
(
)
bt + Et [b
b1,t − Et [b
(Et ebt+1 − ebt ) = − R
πt+1 + gbt+1 ] + R
π1,t+1 + gb1,t+1 ]
(
)
1
− βτ Et
dst+1 − ds1,t+1 − (dξt − dξ1,t ),
e
where βτ Et

(1

e dst+1

(21)

)
− ds1,t+1 are costs arising from endogenous portfolio adjustments and (dξt −

dξ1,t ) are exogenous disturbances that affect countries’ trading in global financial markets. Therefore, in a generalized open economy setting, financial frictions create an additional challenge to
directly estimating New Keynesian Phillips curves. Moreover, in equilibrium, spillovers from output on the right-hand side of (20) are driven in part by third-country effects involving export shares
that do not directly appear in that equation, and that we illustrate explicitly in the next section.
These endogeneity issues add to existing complications in using limited-information approaches
to estimate closed-economy NKPCs, including ambiguous interpretations of the output gap (e.g.,
Galí and Gertler (1999); Neiss and Nelson (2005)) or measurements of inflation expectations (e.g.,
Roberts (1995); Sbordone (2005)). Mavroeidis, Plagborg-Møller, and Stock (2014) highlight and
summarize how these complications translate into a broad range of estimates in the literature.
Given these challenges, our quantitative analysis in Section 7 takes a full-information Bayesian
approach, utilizing the complete set of cross-equation restrictions from our model for identification.

6.2

Spillovers from Idiosyncratic Demand Shocks: An Analytical Example

We now provide further intuition on the nature of cross-country spillovers through an analytical
example. As in closed-economy models, this example highlights the importance of demand shocks
with nominal rigidities. To obtain closed-form solutions, we make two key assumptions. First, all
shocks are i.i.d. across countries and over time. In particular, all comovement in this example arises
from idiosyncratic country-specific shocks through endogenous spillovers rather than exogenous
global shocks. Second, following Clarida et al. (2002), we replace the monetary policy rule (8) with
a simple forward-looking inflation-centric rule in every country,
bt = ϕEt π
R
bt+1 + σνt .
For transparency, we let all elements of the monetary shocks, νt , have unit variances. However, for
the discussion in this section, this is without loss of generality.
These assumptions imply a solution for consumption of the form,
bt = −νt .
C
bn,t depends only on its own-country monetary policy shocks. Since these shocks are
That is, C
independent across countries in this example, consumption is also uncorrelated across countries,

21

bt C
b T ] = I. As expected, lower nominal rates in a given country, νn,t < 0 say, raises its conE[C
t
sumption relative to trend. In what follows, we delve into the mechanisms at work by focusing on
the limit case with no frictions in global financial markets, τ = 0 and dξt = 0 ∀t.
International Comovement from Demand Shocks: Under the above assumptions, analytical
expressions for output and inflation provide intuition for how the model reproduces the positive
correlations described in Section 2.
The solution for output is, in vector form,
1
−1
T
−1
T
Ybt = −A−1
Y Y ψ νt , where AY Y = (I − bψ ϖ) ,
a

(22)

(1+φ)η
with a = 1 + (1 + φ)η and b = 1+(1+φ)η
. Absent trade, ϖ = ψ T = I and equation (22) reduces
bt , so that each country’s production adjusts to meet its own local demand. With
to Yb = −νt = C

trade, the export matrix ψ T captures the first layer of trade spillovers: when demand in country
n′ increases through an expansionary monetary shock, νn′ ,t < 0, output in country n increases
in proportion to its exports to country n′ . The matrix A−1
Y Y summarizes subsequent rounds of
trade in the form of a Leontief inverse capturing the effects of backward linkages, ϖ, and forward
linkages, ψ T . The strength of these effects depends on model elasticities with respect to labor, 1/η,
and trade, φ. We discuss these successive rounds of trade below after deriving additional results
pertaining to the model solution.
The solution for inflation also reflects spillovers from foreign demand shocks,
π
bt = κηϖYbt − κσνt + κµt ,

(23)

which stands for the global Phillips curve (20) in this case. As before, domestic inflation comoves
with output abroad to the extent that countries are connected through the import matrix ϖ.
However, consumption increases in country n′ are also offset by a depreciating exchange rate, as
implied by equation (19) when τ = 0.18
The above expressions for output, (22), and inflation, (23), imply the following proposition:
Proposition 1. For any matrix M , let M ≥ 0 if and only if mij ≥ 0 ∀ij. Then, under the
maintained assumptions, E[Ybt Yb T ] ≥ 0, E[b
πt π
bT ] ≥ 0 and E[b
πt Yb T ] ≥ 0.
t

t

t

Proof. From equation (22), the variance-covariance matrix of output across countries is given by,
[
]
( −1 )T
T
E Ybt YbtT = A−1
.
Y Y ψ ψ AY Y
18

The net effect reduces to the term −κσνt , as the depreciating exchange rate increases inflation in country
n in response to an expansionary monetary shock. In a closed economy, (23) becomes the familiar expression,
bt = −νt as inflation now only depends on domestic conditions.
π
bt = κ(η + σ)Ybt + κµt , with Ybt = C

22

[
]
Since ψ T ≥ 0, it immediately follows that E Ybt YbtT ≥ 0 whenever A−1
Y Y ≥ 0.

T
To show that A−1
Y Y ≥ 0, recall that AY Y = a(I − bψ ϖ) where a and b are scalars such that

a = 1 + (1 + φ)η and b =

(1+φ)η
1+(1+φ)η .

Because the matrices of import and export shares, ϖ and ψ T

respectively, are stochastic matrices, so is their product, ψ T ϖ. It follows that ρ(ψ T ϖ) = 1 where
ρ(ψ T ϖ) = maxΛ(ψT ϖ) {|λ|} such that Λ(ψ T ϖ) = {λ|λ is an eigenvalue of ψ T ϖ}, i.e., the spectral
radius of ψ T ϖ is 1. In addition, since b < 1, the spectral radius of bψ T ϖ is strictly less than 1,
ρ(bψ T ϖ) < 1. We can then write A−1
Y Y as
T
−1
A−1
=
Y Y = {a(I − bψ ϖ)}

∞
1 ∑ m T m
b (ψ ϖ) .
a
m=0

[
]
bt Yb T ≥ 0.
Given that ψ T ϖ ≥ 0, A−1
≥
0
and
E
Y
t
YY
The solution for inflation above is given by π
bt = κηϖYbt − κσνt + κµt . Therefore, E[b
πt π
btT ] =
T
2 2
2
T
κηϖE[Ybt YbtT ]ϖT ηκ + 2κ2 ησϖA−1
Y Y ψ I + κ σ I + κ I since E[νt µt ] = 0. Moreover, given that
E[Ybt Yb T ] ≥ 0, it then follows that E[b
πt π
bT ] ≥ 0. It also follows that E[b
πt Yb T ] = κηϖE[Ybt Yb T ] +
t

T
κσψ(A−1
YY)

t

t

t

≥ 0.

The proposition states that under frictionless asset markets, demand shocks imply that output
comoves positively across countries, inflation comoves positively across countries, and inflation
comoves positively with output across countries, as in Figure 1. These comovement properties
derive from three channels: i) the direct effects of shocks on the demand for goods both domestically
and from abroad, ii) their effects on wages, and iii) their effects on capital flows.
Comovement arises first as spillovers from multilateral trade in goods. Consider the effects on
country n of an expansionary monetary shock in country n′ , νn′ ,t < 0, that raises consumption in n′ ,
bn′ ,t = −νn′ ,t > 0. From equation (22) and the Neumann series expression for A−1 in Proposition
C
YY
1, we can summarize the effects of this increase in demand on output growth as
)
1
1( T
Ybt = − ψ T νt −
bψ ϖ + b2 (ψ T ϖ)2 + b3 (ψ T ϖ)3 + ... νt ,
| a{z }
|a
{z
}
Direct Effects

(24)

Indirect Effects

which illustrates both direct and indirect spillovers. These direct and indirect effects then extend
to inflation via equation (23).
Since consumption in country n′ includes foreign goods, there is a direct positive demand
spillover onto its trading partners, including country n. The effect on country n is greater the
more country n exports to country n′ , as reflected in the matrix of export shares, ψ T . More generally, production in each country immediately increases in proportion to its exports to country n′
via forward linkages, −(1/a)ψ T νt > 0.
However, additional indirect effects also arise through repeated rounds of trade created by
import or export substitution in third countries, thus involving both backward and forward linkages,
23

the terms −(1/a)bm (ψ T ϖ)m νt > 0 on the right-hand-side of (24). These terms make the multiple
rounds of third-country effects captured in equation (17) even more transparent. Specifically, with
the rise in the demand for goods in country n′ (arising from its expansionary shock) comes an
increase in labor demand and thus wages in country n′ . Therefore, a ‘third’ country, say n′′ , that
imports from country n′ will substitute away from n′ towards other countries including n, more
so if a large fraction of imports in country n′′ originates from n′ in the import matrix, ϖ. This
effect induces a further increase in the demand for goods produced by country n, the extent of
which will be larger the more n exports to n′′ in the export matrix, ψ T . Finally, the propagation
of the initial stimulative shock in country n′ continues further afield as higher export activity and
production in country n raises its wages so that its trading partners, country n′′′ say, redirect some
of their imports from n back towards n′ . This redirection induces higher labor demand and wages
in country n′ thus generating new demand for foreign goods from n′ and creating a new round of
trade, −(1/a)bm+1 (ψ T ϖ)m+1 νt > 0, and so on.
The strength of these spillovers depend on the trade and labor elasticities, φ and 1/η, respectively, through a = 1 + (1 + φ)η and b =

(1+φ)η
1+(1+φ)η

in the solution for output growth and inflation.

The greater the Fréchet parameter, φ, the more sensitive countries are to changes in export prices
and the more easily they substitute across trading partners. The smaller the labor elasticity, 1/η,
the more wages increase in country n′ as the demand for its goods rises, thus magnifying the extent
of trade substitutions in third countries.
Finally, multilateral trade in assets also plays an important role. A positive demand shock in
country n′ increases its budget deficit in order for households to smooth consumption over time.
The demand for state-contingent assets in country n′ , therefore, falls and implies a depreciation of
its currency in equilibrium. Equation (19) indicates that the extent of this depreciation is smaller
the lower the intertemporal elasticity of substitution, 1/σ. As 1/σ falls, households are less willing
to smooth consumption intertemporally so that equilibrium requires a larger depreciation of the
exchange rate.
Besides the positive cross-country comovement in real activity or inflation implied by nominal
rigidities and trade, these features also mean that inflation at home comoves positively with real activity abroad, captured in part by the global Phillips curve (20). In this setting, higher employment
at home and abroad imposes a higher disutility of labor and higher wages everywhere. Therefore,
higher employment in any given country potentially induces higher inflation in all of its trading
partners.
Implications for Other Variables and Shocks: We end this section with two observations.
First, output is more correlated across countries than consumption. This observation is contrary
to that implied by international real business cycle models (IRBCs) with no trade costs (Backus,
Kehoe, and Kydland (1992)). Proposition 1 implies that this feature of IRBCs need not be the

24

case in a setting with multilateral trade costs and sticky prices.19 In such a setting, uncorrelated
demand shocks can generate orthogonal consumption movements that spill over into correlated
output fluctuations.
The second observation is that i.i.d. shocks to the growth rate of a country’s TFP do not spill
over to other countries in this example. This is because the model admits country-specific balanced
growth paths so that it can be written entirely in terms of detrended variables. Therefore, an i.i.d.
shock that changes the level of TFP permanently in a country will only scale other variables in
that country to the same extent (including bilateral exchange rates). In other words, a one-off
change in a country’s TFP generates offsetting wealth and substitution effects that keep its labor
unchanged along the balanced growth path. This shock also generates an offsetting real exchange
rate appreciation that keeps trade patterns from changing along that path, and keeps output in
other countries unchanged. These results are specific to growth rate shocks that do not persist.
If TFP were instead expected to keep rising after an initial unanticipated increase, consumption
decisions would then be affected. The effects of variations in consumption relative to trend will
then propagate across countries through the same channels highlighted in Proposition 1.

7

Quantitative Findings

To study the quantitative implications of our model, we calibrate the steady-state values of productivity growth, inflation, employment, and the wage bill to their averages over the period 2004Q2
through 2019Q4, as well as those of the trade matrix over the same period. We then estimate the
remaining parameters using Bayesian methods.

7.1

Shock Structure

Our model includes four types of shocks: TFP growth shocks, gn,t , markup shocks, µn,t , monetary
policy shocks, νn,t , and shocks to portfolio costs, ξn,t . The first three shocks are standard in the
New Keynesian literature and the last one ensures that we match the behavior of real exchange
rates, as discussed in Section 6, through the modified UIP condition, (21).
To allow for common shocks across countries, we consider a factor structure for the first three
shocks.20 In particular, for a generic shock xn,t , x ∈ {g, µ, ν}:
xn,t = λx,n xG
t + x̃n,t ,

(25)

G G
G G
xG
n,t = ρx xt−1 + σx ϵt ,

(26)

x̃n,t = ρ̃x,n x̃n,t−1 + σ̃x,n ϵ̃n,t .

(27)

19

Trade costs are necessary to generate real exchange rate fluctuations, allowing consumption to vary across countries even when τ = ξn,t = 0 (see equation (19)).
20
Shocks to portfolio costs only matter to the degree that they are country specific. Global shocks in international
financial markets cancel out in equation (19) unless they impact each country differently.

25

Thus, each shock consists of a global factor, xG
t , and an idiosyncratic country component, x̃n,t , both
of which follow AR(1) processes. Since λx,n and σxG cannot be separately identified, we normalize
σxG = 1/100. The sign of λx,n is identified through its prior.21
With this factor structure, comovement across countries can arise from either common global
shocks or spillovers from country-specific shocks through trade. Note that trade also affects the
propagation of global shocks.

7.2

Data and Calibrated Parameters

In the quantitative application, we rely on data for Canada, China, the E.U., Japan, and the U.S.
over the period 2004Q2 through 2019Q4. The latter four countries capture the world’s largest
economies, while Canada is included as a major trading partner of the U.S. Altogether, these
countries and regions represent on average 53 percent of world imports and 50 percent of world
exports. The time period used in the quantitative analysis allows us to have a balanced sample for
the Bayesian estimation and captures most of the period following China’s integration in the World
Trade Organization. Details pertaining to all data sources and construction as well as estimation
are included in Section G of the Appendix.
The calibrated parameters are summarized in Table 1. In the steady state, these parameters
reflect time series means for each country over the sample period. In calibrating the matrix of
import shares, we set its diagonal elements to the share of domestic goods in consumption for each
country. The off-diagonal elements, ϖnn′ , are proportional to import shares of country n from
country n′ for every country pair (n, n′ ), scaled so that each row sums to one (because the model
does not capture the entire world economy). The import trade matrix implies that most countries
purchase goods mostly from themselves.

7.3

Estimation

We estimate the remaining parameters of the model using Bayesian methods. The estimation relies
on observable variables given by per capita GDP growth, inflation, interest rates, and exchange
rates in the measurement equations for each country. GDP growth, inflation, and interest rates
are standard observables in the estimation of closed-economy New Keynesian models (e.g., An and
Schorfheide (2007); Aruoba, Cuba-Borda, and Schorfheide (2018)). Here, we keep to this conventional approach for the choice of observables but take into account data from multiple countries
simultaneously. In addition, we include exchange rates as observables to account for their role
in determining trade patterns and, as suggested by the global Phillips curve (20), in determining
variations in cross-country inflation.
21

The approach of using a prior for identification is similar to Baumeister and Hamilton (2015) and Matthes and
Schwartzman (2021), who apply the idea in the context of shock identification in vector autoregressions. Unlike those
applications, the sign of λx,n is purely a normalization without economic content, making the identification scheme
innocuous.

26

Table 1: Calibrated Parameters
Steady States
400 (gn − 1)
400πn
Ln
En Wn Ln

Productivity Growth
Inflation
Employment
Wage Bill

Canada
0.76
1.76
0.03
1.56

China
7.93
2.68
1.70
7.83

E.U.
0.80
1.55
0.22
12.04

Japan
0.70
0.31
0.12
5.14

U.S.
0.92
2.05
0.26
16.39

Notes: Productivity growth and inflation are annualized and calibrated based on quarterly averages of per capita
GDP growth and inflation, respectively, over the 2004Q2 to 2019Q4 sample period. Total employment, Ln , is in
trillions of hours, calibrated to the product of employment and average hours, using annual data from the Penn
World Table over the 2004 to 2019 sample period. The wage bill, En Wn Ln , is calibrated to average GDP in trillions
of current U.S. dollars, using annual data from the World Bank and OECD over the 2004 to 2019 sample period.

Import Trade Matrix (ϖ)

Canada
China
E.U.
Japan
U.S.

Canada
0.674
0.009
0.007
0.006
0.043

China
0.047
0.775
0.105
0.082
0.055

E.U.
0.036
0.074
0.779
0.031
0.040

Japan
0.014
0.086
0.029
0.843
0.019

U.S.
0.230
0.057
0.080
0.039
0.842

Notes: Import shares correspond to average Import Partner Shares over the 2004Q2 through 2019Q4 sample period,
using data from World Integrated Trade Solution compiled by the World Bank. Diagonal elements are calibrated
to match the share of domestically produced output. Off-diagonal elements are proportional to import shares from
respective countries, scaled so that each row sums to one.

Model parameter priors are summarized in Table 2. The prior for the price elasticity of trade,
φ, covers the range of estimates in the literature.22 We use a prior that is relatively flat for the
degree of adjustment costs in global financial markets, τ . Priors for the factor loadings are chosen
so that global factors account for approximately half of the variation of each shock at the prior
mode. As mentioned earlier, these priors also help normalize the sign of the factor loadings by
placing a small probability that factor loadings for a given shock are negative for a majority of the
shock realizations. The remaining priors are comparable to the existing literature (e.g., Lubik and
Schorfheide (2005); Adolfson, Laséen, Lindé, and Villani (2007); An and Schorfheide (2007)).
Accounting for international linkages within a multi-country sticky price framework raises two
computational challenges relative to existing work. First, with five countries, there are a total of
90 parameters to be estimated, with 63 equations characterizing the equilibrium.23 At the same
time, for quantitative purposes, the model we lay out is in some sense the smallest version of an
environment with nominal rigidities and multilateral trade that one would reasonably consider.
Second, a key related challenge in estimating the model is the calculation of the steady state,
22

Estimates of trade elasticities center around 4 but are as large as 17. See Footnote 44 in Caliendo and Parro
(2015) as well as section 4.2 in Head and Mayer (2014) for comprehensive summaries of estimates.
23
In comparison, the medium-scale New Keynesian model in Smets and Wouters (2007) has 36 parameters and 32
equations.

27

Table 2: Priors
Parameter
Global Parameters
100(β −1 − 1) Discount Rate
φ
Fréchet Parameter
τ
Financial Frictions
G
ρx,n
Global Shock Persistence
Country-Specific Parameters
σn
Inverse Intertemporal Elasticity of Substitution
ηn
Inverse Labor Supply Elasticity
θn
Calvo Price Stickiness
ρn
Taylor Rule: Persistence
ry,n
Taylor Rule: Output Coefficient
rπ,n
Taylor Rule: Inflation Coefficient
ρ̃x,n
Idiosyncratic Shock Persistence
100σ̃x,n
Idiosyncratic Shock Standard Deviation
λx,n
Factor Loading

Distribution

Mean

Std Dev

Gamma
Gamma
Gamma
Beta

0.20
4.00
5.00
0.50

0.10
2.00
3.00
0.20

Gamma
Gamma
Beta
Beta
Gamma
Gamma
Beta
Inverse Gamma
Normal

2.00
2.00
0.70
0.50
0.50
1.50
0.50
0.50
0.15

0.50
0.50
0.10
0.20
0.25
0.25
0.20
4.00
0.50

described in Section 4.2. Recall that in applying the approach developed by Dekle et al. (2007)
to the steady state of our dynamic setting, we first solve for a variant of the model whose steady
state matches observed average trade imbalances over our entire sample, then use the implied set
of parameters to calculate counterfactual steady-state allocations and prices corresponding to zero
trade imbalances. The latter step uses an iterative procedure that comprises up to half the time
needed to compute the model’s likelihood for a given set of parameters. This step, therefore,
potentially doubles the time taken to compute the likelihood relative to the typical steady-state
calculations in prototypical New Keynesian models.
To address these computational challenges, we estimate the model using the adaptive sequential
Monte Carlo (SMC) algorithm described in Cai et al. (2021). Relative to standard Markov Chain
Monte Carlo methods, SMC is more robust to irregular posteriors and improves computational efficiency by allowing for parallelization.24 The SMC algorithm constructs a sequence of intermediate
bridge distributions which begin with the prior and end with the posterior. In some cases, Cai et al.
(2021)’s adaptive approach to constructing the bridge distributions are key to obtaining accurate
results. The procedure allows us to begin with particle draws from the prior and gradually modify
those particles until they approximate the posterior.25 Details of the algorithm can be found in
24
These advantages have increasingly led to the use of SMC methods to estimate DSGE models (e.g., Creal (2007);
Herbst and Schorfheide (2014)). In our case, the estimation takes 17 hours on an Intel 1.8 GHz workstation with 16
cores using MATLAB.
25
At each stage, the distribution is represented by a set of particles where each particle consists of a parameter value
and a weight. With a large number of particles, the weighted average of particle values converges to expectations of
the bridge distribution. To generate particles from the stage n distribution, we modify the stage n − 1 particle values
and weights appropriately. See Herbst and Schorfheide (2015) for a textbook treatment on SMC methods.

28

Table 3: Posterior Means for Selected Parameters
Parameter
Global Parameters
φ Fréchet Parameter
τ
Financial Frictions

Mean Estimate
6.01
1.3 × 10−3

Country-Specific Parameters
σn Inverse Intertemporal Elasticity of Substitution
ηn Inverse Labor Supply Elasticity
θn Calvo Price Stickiness

Canada
2.47
2.52
0.83

China
2.54
1.83
0.73

E.U.
1.75
3.01
0.85

Japan
2.01
2.72
0.90

U.S.
2.89
1.88
0.74

Section G of the Appendix.

7.4

Posterior Parameter Estimates

Table 3 presents posterior means for selected parameters of interest. We plot the prior and posterior
distributions for all model parameters in Section H of the Appendix. Overall, posterior estimates
align with values used in both the New Keynesian and trade literatures.
We find a price elasticity of trade, φ, of 6, close to estimates typically used in trade though
away from the largest values in that literature. The posterior mean of τ is 0.001, which implies
low endogenous portfolio costs. In particular, the model at its posterior mean implies that UIP
deviations in equation (21) between the U.S. and Canada, China, the E.U., and Japan are primarily
driven by (differences in) exogenous financial shocks, dξt − dξ1,t , with standard deviations that are,
respectively, 9.4, 1.8, 14.7, and 10.9 times as large as that of the endogenous portfolio costs,
(
)
βτ Et 1e dst+1 − ds1,t+1 . The estimate is far in the left tail of our prior, with a c.d.f. of less than
0.001, suggesting that the data heavily favor such a small value.
The mean estimates of the inverse intertemporal elasticity of substitution, σn , lie between
1.75 and 2.89, which is consistent with existing estimates in the DSGE literature (e.g., Smets
and Wouters (2007) and Herbst and Schorfheide (2015) report posterior means of 1.38 and 2.83,
respectively, for the U.S.; Iiboshi, Shintani, and Ueda (2018) report a posterior mean of 1.55 for
Japan). Similarly, estimates of the inverse labor supply elasticity, ηn , lie between 1.83 and 3.01,
similar to existing estimates (e.g., Smets and Wouters (2007) and Justiniano et al. (2010) find point
estimates of 1.83 and 3.79, respectively, for the U.S.; Iiboshi et al. (2018) find a posterior mean of
3.92 for Japan). The posterior mean estimates for the Calvo price setting parameter, θn , exhibit
substantial heterogeneity across countries. The shorter average duration of prices in the U.S. and
China, less than 4 quarters, is similar to typical estimates in the New Keynesian literature. In
contrast, the average duration of prices in the E.U. and Japan is considerably longer, at more than
8 quarters, similar to those found within closed-economy settings for those countries (e.g., Sugo,
Ueda, et al. (2007) find a mean estimate of 0.88 for θn using data from Japan; Galí, Gertler, and
Lopez-Salido (2001) reports estimates of between 0.67 and 0.92 for Europe).
29

Table 4: Mean Cross-Country Correlations Under Different Counterfactual Models
Model Details

Data
1.
2.
3.
4.
5.

Global
Factors
Y
Y
N
N
N

Trade
Y
N
Y
Y
Y

φ
φ̂
φ̂
φ̂
0.03
φ̂

τ
τ̂
τ̂
τ̂
τ̂
106

θ
θ̂
θ̂
θ̂
θ̂
0.01θ̂

Mean Cross-Country Correlation
GDP Growth– Consump
GDP Growth Inflation
Inflation
Growth
0.39
0.45
0.25
0.18
0.39
0.45
0.25
−0.05
0.15
0.15
0.15
0.15
0.35
0.09
0.14
−0.02
0.28
−0.13
0.07
0.17
0.18
−0.14
0.07
0.19

Notes: ‘Data’ line presents correlations in the data; Line 1 presents correlations in full model with parameters set at
posterior mean; line 2 corresponds to model without trade; line 3 corresponds to model without global factors; line
4 corresponds to model with low trade elasticity, φ; line 5 corresponds to model with high portfolio costs, τ and low
price rigidity, θ. Notation ẑ denotes posterior mean of parameter z.

7.5

Multilateral Trade and Comovement Between Countries

We now consider a series of counterfactual exercises that highlight the quantitative role of different
model features. We use the Kalman smoother to extract the shocks implied by our model under the
posterior mean, then feed the same shocks into versions of the model with alternative assumptions
regarding trade in goods and financial assets, global factors, and price stickiness.
Cross-Country Correlations in GDP Growth and Inflation:

Our estimated model indi-

cates that absent trade, cross-country correlations in output growth and inflation are substantially
lower. This finding emerges despite the presence of global factors, is shared broadly across all
countries, and results primarily from third-country network effects.
The average correlation in GDP growth across countries is 0.39 in the data, but falls by about
60 percent to 0.15 without trade (line 2 of Table 4). Shutting down countries’ exposure to global
shocks rather than trade, the average cross-country correlation in output growth only falls to 0.35
(line 3 of Table 4). In other words, nearly 90 percent of the cross-country comovement in GDP
growth arises through country-specific shocks propagating through the trade network.
The left panel of Figure 2 further highlights the striking difference in the role of trade and
global factors in driving GDP growth cross-country correlations. In the counterfactual without
trade, GDP growth correlations fall in almost all cross-country pairs, with four pairs even yielding
negative correlations. In contrast, the markers for the counterfactual without global factors all lie
close to the 45◦ line, indicating that shutting down global factors has little effect on any of the
cross-country correlations. For example, the correlation in output growth between the U.S. and
the E.U. is 0.57 in the data, falls to −0.02 without trade, but remains as high as 0.53 with trade
but without global factors.
The average cross-country correlation in inflation is similarly driven to a large degree by trade.
The average correlation falls from 0.45 in the data by about two-thirds to 0.15 in the counterfactual

30

Figure 2: Cross-Country Correlations in Data and Counterfactuals
GDP Growth
Cross-Country Correlations

0.6

Inflation
Cross-Country Correlations

0.5

Counterfactual

Counterfactual

0.4
0.4
0.3
0.2
0.1

0.2
0
0.2

0.4

0.6

-0.2
Without Trade

0
0.2
-0.1

0.4

Data

0.6

Without Global Factors
-0.4

Data

Notes: Each marker indicates the correlation between a country pair’s GDP growth (left panel) or inflation (right
panel). Horizontal axis indicates correlation in the data; vertical axis indicates correlation in a counterfactual. Blue
circles correspond to the counterfactual without trade; red diamonds correspond to the counterfactual without global
factors.

without trade (line 2 of Table 4). However, here global factors play an equally important role, as
the mean correlation without global shocks falls to just 0.09 (line 3 of Table 4).
As indicated in the right panel of Figure 2, trade and global factors have varying relative
importance depending on the specific country pair. For instance, the correlation in inflation between
the U.S. and Canada, two major trading partners, falls from 0.81 in the data all the way down
to 0.17 without trade. Absent global factors but with trade, this correlation falls much less to
0.45. In contrast, global factors appear more important in driving the correlation in inflation
between the U.S. and Japan, who are not as tightly connected by trade. The correlation observed
in the data, 0.33, falls only to 0.27 without trade, but becomes negative at −0.24 without global
factors. Intuitively, equation (23) indicates that while inflation inherits the comovement properties
of output (i.e., through the global Phillips curve), it is also driven in each country by inflationary
supply shocks specific to that country (i.e., the markup shocks).
Finally, the comovement in GDP growth and inflation between countries also falls substantially
without trade. This finding is evident in Figure 3, which compares the pairwise correlations between
inflation and output across countries in the model without multilateral trade linkages (but with
global factors) to the observed correlations reported in Figure 1. These correlations fall by about
40 percent on average, from 0.25 in the data to 0.15 in the counterfactual without trade.
These findings indicate that trade plays a fundamental role in driving international comovement
despite countries trading mostly with themselves in Table 1 - a little trade goes a long way. The
intuition follows mainly from two features of multilateral trade and nominal rigidities. First, while
countries trade mostly with themselves, the multilateral nature of trade, and its implied network
of backward and forward international linkages, means that the comovement in output between
countries is substantially enhanced by indirect third-country effects reflected in the Leontief in-

31

Figure 3: Counterfactual Correlation Between GDP Growth and Inflation Without Trade
Correlation Between GDP Growth and Inflation Without Trade
0.4

Correlation

Canada Inflation

0.2

China Inflation
E.U. Inflation

0

Japan Inflation
U.S. Inflation

-0.2
Canada
GDP Growth

China
GDP Growth

E.U.
GDP Growth

Japan
GDP Growth

U.S.
GDP Growth

Notes: Each bar corresponds to correlation between GDP growth in country i with inflation in country j. Filled
bars correspond to cross-country correlations (i.e., i ̸= j) without trade, empty bars correspond to own-country
correlations (i.e., i = j) without trade, gray dotted bars correspond to correlations in the data. Each group of bars
corresponds to a country i’s GDP growth and each color corresponds to a country j’s inflation.

verse, (1/a)

∑∞
1

bm (ψ T ϖ)m , in equation (24), capturing the different rounds of trade generated by

disturbances to the economic environment. Second, the global Phillips curve described by equation
(20) makes clear that every country’s inflation rate potentially comoves with real activity in every
other country in a way that is proportional to their trade shares, reflected here in the elements
of ϖ. This relationship in turn implies that the cross-country comovement between output and
inflation depends in large part on the comovement in output between countries.
In line 4 of Table 4, we illustrate the quantitative importance of third-country indirect effects
by setting the price elasticity of trade, φ = 0.03, or 0.5 percent of its benchmark value, thereby
reducing the indirect effects from trade. Our analytical example in Section 6.2 provides intuition
for how decreasing φ diminishes the role of indirect effects. Observe in equation (24) that indirect
effects vanish in that example when b = 0 or φ = −1 (in which case a = 1). However, the
parameterization of the Fréchet distribution imposes that φ > 0, with the general model becoming
non-invertible near φ = 0. Thus, some indirect effects remain under the parameterization in line
4 of Table 4, yet we already find a notable fall in the comovement in output, the comovement in
inflation becoming counterfactually negative, and the cross-country comovement between inflation
and output declining close to 1/4 of its counterpart in the data, underscoring the material role of
third-country effects.26
The Consumption Correlation Puzzle:

The last column of line 1 in Table 4 indicates that our

model ‘overperforms’ in resolving the Backus et al. (1992) cross-country consumption correlation
puzzle. In a standard IRBC model, Backus et al. (1992) underscore that consumption growth should
be more correlated across countries than output growth. However, the opposite is true in the data.
26
These results suggest that in two-country models, as in Justiniano and Preston (2010), the ability of trade to
generate cross-country comovement is a lower bound given the absence of third-country effects.

32

For our sample of countries, the corresponding mean correlations are 0.39 for output growth and
0.18 for consumption growth. Our model matches observed GDP growth cross-country correlations
by construction, and yields an implied average consumption growth cross-country correlation that
is even lower than that observed in the data at −0.05.
Proposition 1 provides some intuition for why output growth in our model is substantially
more correlated across countries than is consumption growth. In particular, note that consumption
is uncorrelated across countries in that example, while output growth in each country comoves
positively with every other country. As indicated in equation (17), the reason is that even with an
idiosyncratic shock that affects consumption in only one country, trade nevertheless dictates that
the composition of production will be affected in the same direction in every country.
International Comovement, Flexible Prices and Financial Autarky: Lastly, we compare
our findings to those obtained under a neoclassical benchmark defined by flexible prices and financial
autarky. This version of the model is in the vein, for example, of recent work by Huo et al. (2019),
who focus on the role of sectoral linkages and multilateral trade in explaining the comovement
in real activity across multiple countries. Thus, line 5 of Table 4 sets the Calvo price stickiness
parameter, θ, to 1 percent of the posterior mean (for each country) in the estimated model, and
increases the financial frictions parameter, τ , to 106 .
As in Huo et al. (2019), absent global factors, this combination of parameters results in considerably lower comovement in real activity than in the data, with an average cross-country correlation
in output growth that is about 1/2 its observed value, 0.18 instead of 0.39. In addition to Huo
et al. (2019), however, this benchmark case also results in counterfactually negative cross-country
comovement in inflation, and a reduction in the cross-country comovement between output and
inflation of more than 70 percent.
In essence, two changes take place in this case. First, as prices become more flexible, demand
shocks cannot spread as easily across countries. In that case, demand shocks in individual countries
lead to higher inflation and a stronger interest rate response under the maintained monetary policy
rules. These higher rates then dampen the effects of the initial shocks. Second, under financial
autarky, countries no longer run trade surpluses or deficits in response to shocks. As a result, as
countries trade mostly with themselves, trade patterns remain largely unaffected by changes in
the terms of trade, which lowers the comovement in output growth across countries. In this case,
therefore, both the cross-country inflation correlations, and the cross-country correlations between
output and inflation, no longer reflect whatever comovement in output remains from trade.

7.6

The Domestic Effects of Foreign Disturbances

We conclude our set of exercises by exploring two scenarios pertinent to the post-pandemic world.
One concerns the effects of inflationary supply shocks in Europe, exacerbated by the war in Ukraine,
on U.S. variables. The other considers the effects of a tightening of U.S. monetary policy on
33

Figure 4: Response of U.S. Variables to an E.U. Markup Shock
Inflation

0.6

Output
0

0.4

-0.1

0.2

-0.2

0

-0.3
0

2

4

6

8

10

12

0

0

2

0.2

-2

0.1

-3

6

8

10

12

Consumption

0.3

-1

4

Baseline
Low Trade Elasticity

0
0

2

4

6

8

10

12

0

2

4

6

8

10

12

Notes: Impulse responses scaled to generate a 1 percent increase in E.U. inflation on impact. Solid black lines
correspond to estimated impulse response at posterior mean; gray shaded regions correspond to 68 percent posterior
credible intervals; red dashed lines correspond to impulse response at posterior mean with φ = 0.03. Horizontal axis
is in quarters; vertical axis is in percent. Inflation is annualized; real exchange rate normalized so that a decrease
corresponds to a depreciation in the U.S. dollar.

countries abroad.
Implications of European Inflationary Shocks for the U.S.:

In the context of New Keyne-

sian models, inflationary shocks driven by supply disturbances, such as shortages stemming from
factory lockdowns or disruptions in supply chains, arise through changes in the markup.27 In the
New Keynesian block of equations, (12) through (15), markup shocks directly affect inflation but
do not otherwise appear in the other equations. In other words, markup shocks essentially drive an
exogenous change in inflation that then pervades to the rest of the economy. Figure 4 illustrates
the effect of one such 1 percent increase in European inflation on key U.S. variables.
As supply disturbances increase inflation in the E.U., the European monetary policy rule prescribes raising interest rates, which lowers European consumption (shown in Section H of the
Appendix). This fall in consumption lowers the demand for foreign goods, including from the U.S.,
which induces a decline in U.S. output in Figure 4. However, lower consumption in Europe also
means, in equilibrium, that the U.S. dollar depreciates relative to the euro. In addition, the U.S.
central bank reacts to the decline in domestic output by inducing lower real rates which, in addition
27
Ramirez-Cuellar and Espinosa-Torres (2022) indeed document marked increases in firm-level markups in Compustat during and following the COVID-19 crisis.

34

Figure 5: Response of Foreign Variables to a Tightening of U.S. Monetary Policy
Output

Inflation

0

Interest Rate

0

0
-0.05

-0.1

Canada
China
E.U.
Japan

-0.05

-0.2

-0.1

-0.1
0

2

4

6

8

10

12

0

2

4

6

8

10

12

0

2

4

6

8

10

12

Notes: Impulse responses correspond to a 25 basis point increase in U.S. nominal interest rates on impact. Horizontal
axis is in quarters; vertical axis is in percent. Inflation and interest rates are annualized.

to the depreciation of the dollar (making U.S. goods less expensive relative to foreign goods), then
raises U.S. consumption. In equation (20), the fall in output lowers inflation while the increase
in consumption and the depreciation of the exchange rate stimulate it. The net effect is the U.S.
inheriting roughly 20 percent of the initial increase in European inflation.
To place the quantitative effect on U.S. inflation in the context of the recent global inflationary
episode, suppose that 3/4 of the 4.4 percent increase in European inflation from 2021Q4 to 2022Q1
stems from the supply shock associated with the Ukraine war and its effect on energy prices (here
modeled through an exogenous change in the markup that directly raises European inflation). Our
estimates in Figure 4 then suggest that such a shock would account for around 1/2 of the increase
in U.S. inflation over the same period.
We highlight two additional features of Figure 4. First, we estimate European markup shocks
to be relatively persistent, so that any inflation inherited from these shocks in the U.S. also tends
to persist. Second, the dashed red lines in the figure illustrate the response of U.S. variables under
attenuated third-country effects as in line 4 of Table 4, i.e., where φ is set to 0.5 percent of its
benchmark value. With attenuated third-country effects, the decline in U.S. output stemming from
lower demand from abroad is considerably smaller. This smaller decline in output in turn mitigates
the need from the U.S. central bank to lower real rates. The net effect is that U.S. inflation hardly
responds.
The Effects of a Tightening of U.S. Monetary Policy Abroad:

Following the steady rise

in post-pandemic U.S. inflation, the Federal Reserve has reacted with a tightening of its monetary
policy. In the New Keynesian world with multilateral trade relationships, this tightening of U.S.
monetary policy has effects abroad not only because the U.S. bilaterally trades with other countries
but also because its trading partners trade among themselves (i.e., through third-country effects).
Figure 5 then illustrates the global effects of a 25-basis-point increase in U.S. interest rates.
As the U.S. central bank raises domestic interest rates, U.S. consumption falls, which translates
into lower demand for foreign goods. Consequently, all of its trading partners see their output
decline in Figure 5, and the percentage decline in foreign output can reach 70 percent of the
35

domestic response (shown in Section H of the Appendix). These declines of between 0.14 to 0.22
percent, in response to a 25-basis-point U.S. interest rate increase, are especially notable given the
recent sequence interest rate hikes that have cumulatively totalled 3 percent over the first three
quarters of 2022.
In response to the fall in output globally, foreign central banks lower real rates so that, despite
inflation initially falling everywhere abroad, after two quarters Canada and China find themselves
with both higher inflation and lower output. This exercise, of course, takes as given our estimated
monetary policy rules for each country. In practice foreign central banks might choose to deviate
from these rules to avoid the outcomes depicted in Figure 5. Nevertheless, the figure establishes that
higher inflation and lower output abroad are not only feasible responses to tighter U.S. monetary
policy, but also represent the likely outcomes based on the historical behavior of the data. Thus,
key questions for future research are: what is the optimal monetary policy response at home to
disturbances abroad in a New Keynesian world featuring multiple large countries? How do central
banks of countries linked by multilateral trade internalize other banks’ actions in choosing their
monetary policy?

8

Concluding Remarks

This paper addresses the comovement between countries in output growth, in inflation, and between
output growth and inflation. The model we develop brings nominal rigidities in the tradition of
New Keynesian models together with multilateral trade. The result is an environment where
variables in different countries reflect the effects of foreign disturbances not only directly, through
pairwise trade between countries, but also indirectly through third-country effects arising from the
network structure of trade. The combined propagation mechanisms mean that despite relatively low
observed trade shares across countries, country-specific shocks alone generate sizeable comovement
between countries.
International linkages in a world with nominal rigidities are associated with important changes
to the standard closed-economy New Keynesian Phillips curve. These changes imply, outside autarky, a global system of Phillips curves explicitly connected by multilateral trade relationships. In
this system, inflation in any one country is related to variations in output growth and exchange rates
in all of its trading partners. Moreover, in equilibrium, variations in output growth and exchange
rates in turn reflect the effects of foreign disturbances operating through adjustments in countries’
trade balances and net asset positions along with global market clearing. Using Bayesian methods
to quantify the role of global factors and international linkages, we find that country-specific shocks
alone explain almost 90 percent of the observed average pairwise cross-country correlation in output growth. These idiosyncratic shocks also explain more than 1/2 the cross-country comovement
in inflation, as well as between output and inflation. In contrast, as nominal rigidities vanish,
multilateral trade generates only limited comovement in output between countries, counterfactu-

36

ally negative comovement in inflation, and almost no cross-country comovement between output
and inflation. Finally, we estimate that a European inflationary shock results in significant U.S.
inflation accompanied by lower output, and that these responses transpire almost entirely from
the network effects of trade. We also find that a tightening of U.S. monetary policy generates a
percentage decline in output globally that is comparable to 1/2 the domestic response.

37

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