Propagation of Output Fluctuations across Countries

Daniel G. Dassel

Department of Economics

University of Frankfurt

Zeppelinallee 29

60325 Frankfurt

Email: dgross@wiwi.uni-frankfurt.de

First Draft, January 2002

Abstract

This paper studies the international propagation of shocks through trade. A

structural VAR model is estimated that allows to distinguish between three

types of disturbances: a country-specific and a common supply shock and

a demand type disturbance. This decomposition enables us to study the

transmission of purely idiosyncratic shocks through bilateral trade flows. The

evidence suggests that a substantial fraction of fluctuations in bilateral trade

flows can be attributed to the transmission of idiosyncratic shocks though

other disturbances also appear to be important. It is shown that a flexible-

price two-country model fails to replicate the magnitude of fluctuations in

bilateral trade flows as observed in the data. The decomposition of the simu-

lated series suggests that the flexible-price model misses an intrinsic demand

component to generate the empirically observed fluctuations in exports and

imports. The approach may be conceived as a specification test of a particular

RBC model.

JEL Classification: E32, F41

Keywords:VAR, Transmission, Trade, RBC

1

1 Introduction

There exists an extensive empirical literature documenting the time series properties of

aggregate macroeconomic variables in an international context. For example, prominent

studies include Zimmermann (1991), Blackburn and Ravn (1991), (1992), Backus, Ke-

hoe and Kydland (1992), (1994), Brandner and Neusser (1992) and Baxter (1995). This

literature characterizes the dynamic properties of a set of variables in terms of standard

deviation, cross-correlation and autocorrelation of the cyclical components. The evidence

suggests the existence of an international business cycle in the sense that there appear to

be large and substantial co-movements in variables across countries (e.g. Gerlach (1988)).

The empirical studies typically do not reveal to what extent the co-movements result from

interdependencies in capital and goods markets, from coincidental structural1 innovations

or from a combination of both. A large body in the theoretical literature on international

business cycles examines models that incorporate both - goods and production interde-

pendencies and some correlation structure in the innovation processes - to replicate the

stylized facts to some degree. In doing so - as noted by Schmitt-Grohe (1998) - these

studies test a complex joint hypothesis involving the propagation mechanism and the

correlation structure of innovations. Following this strategy, it is impossible to analyze

the inherent propagation structure of the model separately from the correlation structure

of the innovation processes. Attributing the failure or success of a model to either the

propagation mechanism or the autocorrelation structure in the innovation process are

difficult to answer at best.

The discussion of the international propagation of idiosyncratic innovations has received

relatively little attention in the context of international business cycle theory. Kwark

(1999) suggests a multinational vector autoregression (VAR) model that enables him

to assess the impact of country-specific and global shocks on the output variability. In

Gross(2001), an identification scheme is suggested that dissociates country-specific and

common supply shocks in a bilateral context. Canova and Marrinan (1998) examine

various propagation mechanisms in a three-country model with different types of distur-

bances. However their strategy also involves a complex hypothesis in that they compare

impulse response functions (IRFs) derived from actual and artificial data using a semi-

structural identification scheme. Here, semi-structural is meant in the sense that the

authors analyze dynamic responses to uncorrelated shocks but the IRFs are hard to in-

1We will use the term ”structural” to denote innovations that are orthogonal to any other innovation.

In the theoretical and empirical literature, structural shocks are assigned an economic interpretation,

like technological, monetary or terms of trade shocks.

2

terpret. Their economic interpretation is not clear because the IRFs have been derived

from an a-theoretical Cholesky decomposition. In a similar spirit, Schmitt-Grohe (1998)

studies the transmission of output cycles from the US to Canada through various chan-

nels. She compares the dynamic responses of the model to those found in the data. Her

approach differs from Canova and Marrinan (1998) in that she narrows the focus on the

transmission of idiosyncratic innovations being transmitted unidirectional from the US

to Canada by cutting off all feedback effects. However her strategy to derive empirical

IRFs is also subject to some criticism. No account is given to the occurrence of common

shocks which may result in an overestimation of the transmission channels. Further,

the empirical identification of the propagation channels is a-theoretical and needs not be

consistent with the models in question.

With this paper, we attempt to contribute to this literature by quantifying the interna-

tional propagation of asymmetric innovations through trade in a structural VAR model.

The assumptions needed to just-identify the VAR are consistent with a class of interna-

tional real business cycle (RBC) models. It will be tested wether a version of a flexible

price open economy model - like the one analyzed by Backus, Kehoe and Kydland (1994)

- is able to replicate the empirical distribution of a set of variables when it is given the

same set of structural innovations as ”observed” in the empirical data. The analysis

is similar in spirit to Canova and Marrinan (1998) and Schmitt-Grohe (1998) although

our methodology differs with respect to quantifying the trade channel empirically and

evaluating the performance of the model.

Technically speaking, the actual data may be conceived as a mapping of the set of struc-

tural innovations. The rule by which the set of innovations is mapped into the actual

data is defined by the ”true” data generating process (DGP). Economic theorists attempt

to understand the ”true” DGP by building a model. The model can also be understood

as a mapping that is postulated by the researcher. It allows to transform a set of innova-

tions into artificial data. In order to evaluate the usefulness of the model, the researcher

performs various tests by comparing the actual with the artificial data. It has been a com-

mon practice in the literature to use different sets of structural shocks in the mappings,

like Canova and Marrinan (1998) and Cogley and Nason (1995). In this case, a particular

mapping may be refuted simply for the reason that the set of structural innovations is

different. This objection is of particular importance in RBC models. Their performance

hinges decisively on exogenous forces because their endogenous propagation mechanism is

typically weak (e.g. Cogley and Nason (1995)). Generally speaking, an ideal comparison

of the mappings requires that the set of structural innovations be equal. We draw on this

3

idea by simulating an international RBC model using structural innovations that have

been identified in the actual data. To assess the fit between the actual and the artificially

generated data, we compare the IRFs and provide some measures of fit. Similar to Cogley

and Nason (1995), our study may be regarded as specification test of a particular model.

The results confirm the hypothesis of the ”locomotive” character of the US economy.

Country-specific supply side disturbances to the US exert a sizable influence on Canadian

and German output implying an international transmission mechanism. The evidence

suggests that bilateral trade linkages are an important channel through which idiosyn-

cratic shocks are transmitted internationally though complementary channels seem to

exist as well. By decomposing the series into structural components, we can show that

inherent to both, bilateral exports and imports, there is a strong demand component.

It is mainly due to this feature that the baseline international RBC model we consider

cannot replicate the magnitude of changes in bilateral trade flows as observed in the data.

In the theoretical economy, bilateral exports and imports are largely supply-side deter-

mined. The implied wealth and substitution effects appear to be too weak to account for

a sizeable propagation of output cycles across countries through trade. This outcome is

based on a decomposition of the simulated time series.

This paper is organized as follows. In Section 2, we briefly present the theoretical model

that serves as benchmark for the empirical analysis. Empirical estimates of the trade

channel are presented in Section 3. In Section 4, we ask if the model can propagate

asymmetric innovations through trade as observed in the data. In a sensitivity analysis,

the robustness of our results is tested with respect to particular parameter values. Section

5 summarizes the principal findings and conclusions.

2 The Baseline International RBC Model

2.1 The Core Structure

As a benchmark, we consider a baseline international RBC model that follows closely

the work by Backus, Kehoe and Kydland (1994). In this world there exist two countries

i = 1, 2 where each country is represented by a single consumer that stands for a large

number of like agents. The consumers’ preferences are described by an expected lifetime

utility function of the form:

Ui = E0

[ ∞∑t=0

βtu (Cit, 1−Nit)

](1)

4

where Ci and Ni denote consumption and employment in country i. For this model, it is

assumed that preferences are adequately described by the period utility function:

u (Cit, 1−Nit) =1

γ

[Cµ

it (1−Nit)(1−µ)

]γ

. (2)

Each country specializes in the production of intermediate goods that are denoted by

A for country 1 and B for country 2. Agents in both countries have access to a Cobb-

Douglas production function using capital Ki and labor Ni. The shares of goods A

that are consumed by agents in country 1 and 2 are labelled as A1 and A2. A similar

notation is adopted for goods B2. Thus, A2 is equal to the volume of exports from

country 1 to country 2 whilst B1 is akin to total imports of country 1 from country 2,

both denominated in country 1’s currency. This gives rise to the resource constraints of

intermediate goods:

A1t + A2t = ZgtZ1tKθ1tN

(1−θ)1t

B1t + B2t = ZgtZ2tKθ2tN

(1−θ)2t .

(3)

Zi indicates the state of total factor productivity (TFP) in country i; similarly Zg rep-

resents the common component in the state of TFP3. θ is a production parameter that

defines the capital income share in intermediate output. Domestic and foreign interme-

diate goods are aggregated to final goods Qi using a CES function. At the final stage

of production, there is no need of capital and labor input. Final goods Qi are converted

into consumption Ci and investment Xi purchases. Additionally, a fraction of final goods

are absorbed by the government Gi. It will be assumed that government absorption fol-

lows a stochastic process. The resource constraints at the final goods stage can then be

formulated as:

Q1t =((1− w)1/ρ A

(ρ−1)/ρ1t + w1/ρB

(ρ−1)/ρ1t

)( ρρ−1)

= X1t + C1t + G1t

Q2t =(w1/ρA

(ρ−1)/ρ2t + (1− w)1/ρ B

(ρ−1)/ρ2t

)( ρρ−1)

= X2t + C2t + G2t

(4)

w is a weighting parameter indicating preferences for domestic goods if 0 < w < 12. For

w = 12, domestic and foreign goods are equally valued. The parameter ρ in the CES

function defines the intratemporal elasticity of substitution (IES) between domestic and

2From the perspective of country 1, A1 can be interpreted as the amount of domestic goods used

in the final production whilst B1 represents the set of importable goods in final production. A sim-

ilar interpretation applies to country 2 where B2 and A2 denote the domestic and importable goods

respectively.3We explain the motivation for introducing a common component in the process of TFP below.

5

foreign intermediates with ρ > 04. The stocks of capital are assumed to evolve according

to:

Kit+1 = (1− d) Kit + Xit − Φ(Kit+1, Kit) (5)

where d is the depreciation rate of physical capital. The term Φ(Kit+1, Kit) captures

the costs of adjusting the capital stock. The introduction of transaction costs helps to

avoid unrealistically high capital flows in response to technology shocks. Φ(Kit+1, Kit) is

assumed to take the functional form of 12φ (Kit+1 −Kit)

2 /Kit so that the costs increase

with the magnitude of the adjustment. For φ = 0, changes in the capital stock do not

incur any costs. The functional form adopted here is similar to Kollmann (2001). It

retains simplicity while linking adjustment costs to Tobins’ q.

2.2 The Exogenous Forces

Empirically, we are able to dissociate country-specific and global supply disturbances

which can be understood as innovations to TFP. However the literature on international

business cycle theory does not sharply discriminate between disturbances to domestic

and foreign TFP. It is common practice to assume that TFP follows an AR(1) process

like:

Zt = Γ× Zt−1 + et (6)

where Z ′t = [Z1t, Z2t] comprises TFP for country 1 and 2. Γ is a 2 × 2 matrix and

e′t = [e1t, e2t] stands for the innovations to TFP in country 1 and 2 where e ∼ N(0, Σ

).

In choosing numerical values for Γ and Σ, the large body of the existing literature has

focused on what came to be known as the Solow Residual (SR).

The SR is the fraction of a country’s output change that is not accounted for by adjust-

ments in the capital stock and fluctuations in employment. Pioneered by Solow (1957),

the SR for country i is computed by:

Zit = Yit − θKit − (1− θ) Nit (7)

which results from the application of the logarithm to a Cobb-Douglas production func-

tion on the right hand side of equation (3). As is apparent in (7), the SR (Z) is the

residual from a univariate decomposition of output (Y ) into capital (K) and employment

(N). θ is a parameter that governs the distribution of income amongst capital and labor.

4Formally, the intratemporal elasticity of substitution is defined as: ρ = − δ ln(a1/b1)δ ln(q1/q2)

where q1 is the

price of good A and q2 is the price of B. The ratio q1/q2 defines the terms of trade between the home

and the foreign country.

6

Note that this strategy of computing the SR is akin to treating the SR separately for

each country. This characteristic renders the SR somewhat dubious. Recalling that large

and substantial co-movements in macroeconomic variables have been documented, one

expects the SR to be mutually dependent across countries. In fact, the SR displays a

high serial and a substantial correlation across countries as documented in Zimmermann

(1997) amongst others.

Recognizing the interdependencies in the SR, Γ and Σ have been calibrated such that

TFP exerts a mutual influence across countries (non-zero off-diagonal elements in Γ) and

that innovations to TFP are correlated across countries (non-zero off-diagonal elements

in Σ). Given this specification of TFP, an innovation to TFP in country 1 is not a pure

structural disturbance in the sense that we attempt to identify it empirically because the

innovations to TFP are correlated and the processes of TFP are linked themselves by

non-zero AR (1) coefficients in Γ. Hence, even if there were no linkages between country

1 and 2, an innovation to TFP in country 1 that raises output in country 1 would be

accompanied by a less than proportionate increase in TFP and output in country 2. A

naive observer may fall victim to the observational equivalence and may falsely interpret

the co-movement in output as a result of a non-existent propagation mechanism.

In an attempt to prevent this type of misinterpretation, we introduce some changes in

the specification of the AR (1) process of TFP. First, we cut off the linkages in the vector

process of TFP by setting off-diagonal elements in Γ to zero. Second, a common com-

ponent of TFP is being introduced. This pays tribute to the fact that TFP measures

display large and substantial co-movements across countries5. With these assumptions,

equation (6) may be rewritten as:

Zt = Γ× Zt−1 + ezt (8)

where Z ′ = [Z1, Z2, Zg] and ez ∼ N(0, Σz

). Off-diagonal elements in Γ are assumed to be

zero. These adjustments give innovations to TFP an economic (structural) interpretation

in terms of pure country-specific or common disturbances.

There exists another type of disturbance in the model. Government expenditures are

also assumed to follow an AR (1) process given by:

Gt = Θ×Gt−1 + ept (9)

5Given that output follows an I (1) process and that output is co-integrated across countries, a

decomposition like in equation (7) implies that TFP is also co-integrated across countries. In such a

case, the common component to TFP represents the common trend. In fact this seems to be a very

realistic characterization of the US-Canada case.

7

where G′ = [G1, G2] and ep ∼ N(0, Σp

). Ruling out interdependencies in government

consumption across countries, Θ is restricted to be diagonal. Ideally, the covariance

matrices Σz and Σp have to be diagonal. Unfortunately, the empirical model is too

limited as it could fully assure the orthogonality in the innovations.

Let s denote the vector of state variables with s′ = [K1, K2, Z1, Z2, Zg, G1, G2]. The

competitive equilibrium consists of a set of decision rules for the vector of controls χ′ =

[C1 (s) , C2 (s) , X1 (s) , X2 (s) , N1 (s) , N2 (s) , A1 (s) , A2 (s) , B1 (s), B2 (s)] such that i)

agents in country 1 and 2 maximize lifetime utility (1), ii) intermediate firms rent capital

and labor until marginal costs equal marginal revenue, iii) the resource constraints (3) -

(4) are binding and iv) the transversality condition lims→∞ βsKit+s = 0 is met.

There is no analytical solution to this model. An approximate solution is obtained by

linearizing the equilibrium conditions around the steady state. The resulting system of

difference equations are then solved by applying the Blanchard-Kahn (1982) algorithm6.

The solution results in a first-order difference equation system like:

χt = Π× st

st = W × st−1 + M × et

(10)

with e′ = [ez, ep]. Due to the linearization, all variables are expressed as percentage

deviation from their steady state which is denoted by the hat. The equation system (10)

is henceforth referred to as the model being used in the numerical analysis of subsequent

sections. The dynamic properties of this model are well understood (see amongst others

Backus, Kehoe, Kydland (1994), (1995) or Ravn (1992)). Hence embarking merely into

another standard numerical exercise does not seem to be worthwhile. Instead we use

the model in an attempt to understand the international transmission of shocks through

bilateral trade flows quantitatively. Before we proceed, the empirical trade channel must

be quantified.

3 Empirical Estimates of the Trade Channel

Implicit to an empirical analysis of the trade channel as a propagation mechanism is the

necessity to isolate those movements in output and trade variables that emanate purely

from country-specific shocks. In other words, an evaluation of the propagation mechanism

6The solution of this set of difference equations, the impulse response function analysis and the

simulation studies have been performed using the software package Matlab. Programs are available

upon request.

8

must disregard those changes in variables that result from common disturbances which -

by their very meaning - are international and cannot be further propagated.

3.1 Specification, Estimation and Data Properties

One way to evaluate the propagation mechanism empirically requires the use of a vector

autoregression (VAR) model. An advantage of this strategy is that the VAR framework

fully exploits the dynamic nature of the theoretical model. However, the identification

of country-specific and common components involves exogenous restrictions that could

take the form of short run (Kwark (1999), Hoffmann (2000)) or long run constraints

like in Gross (2001). There, the suitability of long run constraints to dissociate country-

specific and global shocks in output has been demonstrated. We adhere to this strategy

here. The identification scheme is of particular importance as the identified structural

innovations are used in the numerical exercise below. It is noteworthy that the long

run restrictions we use are consistent with different versions of the model. In fact, the

constraints are related to crucial insights of the intertemporal approach to the current

account whose logic is inherent in a broad class of open-economy models. In doing so,

we follow Canova (2001) to exploit robust implications of theoretical models to test their

validity empirically.

The ”filter” we use to extract the structural innovations from the data is of general

nature and not confined to the particular model in question. Our approach is essentially

an extension of the analysis by Gross (2001) and includes beyond output also a trade

variable that characterizes the bilateral trade channel.

To be more specific, we assume that x′ =[(yh − yf ), yh, nxhf

]is a trivariate covariance

stationary vector process where each element has zero mean. yh and yf represent the

cyclical components of domestic and foreign output. nxhf stands for bilateral exports

(imports) between the home and foreign country7. Then, x has a vector moving average

representation like:

xt = C (L)× εt (11)

where ε′ =[εh, εg, εd

]is a vector of iid structural disturbances with ε ∼ N (0, I). The

elements of ε have a structural interpretation. εh represents an innovation to domestic

TFP whilst εg stands for a shock to worldwide TFP. Both, εh and εg are supply side

7In fact, we estimate the propagation mechanism separately for exports and imports. To be precise,

exports refer to bilateral exports from the home to the foreign country. Imports measure simply the

reverse flow of goods and services.

9

disturbances that potentially have permanent effects on the level of domestic and foreign

output and bilateral trade. The shock εd cannot be assigned a direct economic inter-

pretation consistent with the model. Conceivably εd represents some type of demand

shock. C (L) is a 3 × 3 matrix of polynomial lags with C (0) = I. The reduced form

representation of (11) is:

xt = D (L)× et (12)

where C (L) = D (L) × S−1. Formally, S is the mapping of the reduced-form residual

vector e into the structural innovations ε by:

et = S × εt. (13)

The reduced form residual vector e is characterized by e ∼ N (0, Σ). Equation (13)

implies that the reduced form variance-covariance matrix Σ defines S by:

Σ = S × S ′. (14)

In our three-dimensional system, (14) imposes six restrictions on S. To just-identify S,

additional three restrictions are needed.

As shown in Gross (2001), common shocks to TFP are predicted to have no long run

impact on the output differential if the asset market structure is incomplete and the per

capita stock of physical capital and employment are equal across countries in the steady

state8. Exploiting this feature yields the restriction:

c11 (1) s12 + c12 (1) s22 + c13 (1) s32 = 0. (15)

The assumption that εd be a demand shock results in two more restrictions. As suggested

by Blanchard and Quah (1989), demand shocks are assumed to have no long run impact

on the level of output. This restriction postulates some form of long run neutrality of

demand disturbances which is consistent with a variety of economic models. This insight

restricts the long run multipliers of εd to the level of output and the output differential

to zero. Formalizing these conditions yields:

c11 (1) s13 + c12 (1) s23 + c13 (1) s33 = 0

c21 (1) s13 + c22 (1) s23 + c23 (1) s33 = 0.(16)

8This assumption is consistent with an even broader class of international RBC models. As long as a

social planner weighs utility in each country equally, these assumptions carry over to international RBC

models with a complete asset market structure.

10

Once we have estimated the reduced form model (12), S can be identified from (14) -

(16). Given estimates of D (L) and S, we can finally map the reduced form MA process

into the structural model by:

C (L) = D (L)× S−1. (17)

We approximate the reduced form model (12) by a finite order VAR representation. The

VAR model is truncated at lag two. The lag length is chosen such that the model induces

white noiselike residuals without loosing too much degrees of freedom.

Since the elements of x are covariance stationary, all variables need to be detrended prior

to estimation. However, there exists no consensus in the literature with respect to the

appropriate detrending method. Additionally, as indicated by Canova (1998), the choice

of the detrending method induces different distributional properties on the underlying

time series structure and thus affects the description of the empirical evidence. In an

attempt to render theory and evidence as closely comparable as possible, we remove the

long run trend from the variables by applying the HP-filter. To the credit of this filter,

we avoid an a priori decision in favor of a deterministic or stochastic trend that is difficult

to sort out by current econometric tests.

Basically, the VAR model is applicable to analyze the trade channel between any con-

ceivable country-pair. Even if one narrows the focus to the G7 countries, there still

remain 42 country-pairs to be considered. Such a large scale analysis is certainly beyond

the scope of the present work. To limit the scope further, we restrict our attention to

the US-Canadian and the US-German country-pair. The US-Canadian case is a natural

point of departure because it has been previously analyzed in the context of dynamic

general equilibrium theory by Schmitt-Grohe (1998). It deserves interest from reasons

of its own since strong trade interdependencies characterize the US-Canadian economic

relationships. 75 (66) percent of total Canadian exports (imports) are accounted for by

the US. The US-German pair is chosen to shed some light on the transmission of eco-

nomic fluctuations from North America to Europe. Germany, as the biggest European

economy and the largest G7 economy in terms of openness to trade, is believed to be a

natural starting point for this analysis. These country pairs are interesting from another

perspective. Canadian exports (imports) to the US - measured as ratio to Canadian

output - are more than twice as high as the corresponding numbers for Germany9. The

9Canadian exports to (imports from) the US amount to 4.87 (3.90) percent of Canadian output. In

contrast, only 2.03 (1.52) percent of German output are exported to (imported from) the US. The shares

are computed as averages over the period 1980 - 1999.

11

comparison of Canada to Germany is expected to shed some light on the importance of

bilateral trade interdependencies.

Quarterly data of real GDP and real exchange rates for the US and Canada is taken from

the OECD’s Main Economic Indicators. Because real GDP for Germany is not available

over the full sample period, we use the equivalent data from the IFS tapes. Data on

bilateral exports and imports are retrieved from the OECD’s Quarterly Foreign Trade

Statistics10. Unless otherwise indicated, all variables are denominated in terms of the

home country’s currency. The sample period is 1961:1 - 1999:4. Prior to detrending, the

natural logarithm is taken from all variables.

3.2 Impulse Response Functions and Forecast Error Decompo-

sition

Unless otherwise indicated, the US is always treated as the ”home” economy whilst

Canada and Germany are alternatively referred to as ”foreign” countries. The analysis

is conducted repeatedly for exports and imports. The figures 1 - 4 summarize the mean

IRFs. The upper and lower 95 percent confidence limits are shown as dashed lines.

Several interesting features deserve attention.

There is evidence of an international transmission of output shocks from the US to

Canada and Germany thus confirming previous findings by Canova and Marrinan (1998)

and Schmitt-Grohe (1998). The evidence also suggests that the transmission occurs to

some extent through trade. An asymmetric supply side innovation in US output has a

significantly large and positive impact on Canadian and German output. The increases

in output are accompanied by more than proportionate jumps in bilateral exports and

imports. The shape of the dynamic responses in output and bilateral trade are similar

across country-pairs. The peak responses in output occur on impact (Germany) or after

the first quarter (Canada). Depending on the particular trade variable used in the VAR

estimation, a one percent supply side innovation in US output induces peak responses in

foreign output that vary between 2.2 (imports) and 2.4 (exports) percent in Canada and

1.3 (imports) and 2.3 (exports) percent in Germany. The picture shows similar patterns

for exports and imports. US exports to Canada reach a peak after three quarters with

7.2 percent; the corresponding statistics for Germany are one quarter and 5.9 percent.

Typically, the movements of exports exceed those of imports. US imports from Canada

rise merely by 4.6 percent at the peak after two quarters. Imports from Germany reach

10See appendix C for a detailed description of the data used in this analysis.

12

their maximum on impact with 3.7 percent. The persistence of the supply side

innovations ranges from six to eight quarters which implies an average duration of a

cycle of one and a half to two years after a supply shock to US output.

It is important to observe that the dynamic responses of output, exports and imports

to asymmetric and common supply side innovations are observationally equivalent. This

evidence makes a strong case for discriminating between country-specific and common

components.

A possible objection against our empirical model might be that the common components

in output, exports and imports are partially accounted for by a one-to-one transmission

of country-specific shocks. Admittedly we cannot rule out that part of the common

components may have been induced by ”perfect” transmission of asymmetric shocks.

Following this logic supports our case even stronger because we must then conclude that

the empirical trade channel is even more important than our estimates suggest. The

impact of demand shocks on output is typically of smaller size and not significantly

different from zero in any country. As far as fluctuations in output are concerned, they

seem to be of minor importance.

Another way to look at the impact of structural innovations on the elements of x involves

the decomposition of the forecast error. Let vx (n) denote the n-th step ahead forecast

error of x defined by

vx (n) = xt+n − Et [xt+n] =n−1∑L=0

C (L)× εt+n−L. (18)

There exists an analogous representation of (18) for each element of x. The forecast

error for each element of x has a country-specific, a common and a demand component.

Stating that vy,k (n) represents the forecast error in variable y, y ε x, that is entirely due

to innovations of type k, for k = h, g, d. Then

σ2y,k (n) =

(vy,k (n))2

(vy (n))2 (19)

defines the share of the forecast error variance in variable y that results from innovations

k at forecast horizon n. As is clear from the definition11,

σ2y,h (n) + σ2

y,g (n) + σ2y,d (n) = 1 (20)

for any y ε x and n ε R+. σ2y,k (n) can be interpreted as some measure that indicates the

relative importance of innovation k in the fluctuations of y. The upper part of table 1

11This result holds because the variance-covariance matrix of the structural innovations is assumed to

be diagonal, implying zero-covariances.

13

05

10

-1012

US

O

utp

ut

US Shock

05

10

-101234

Can

ad

ian

O

utp

ut

05

10

-3

-2

-101

Ou

tp

ut D

ifferen

tial

05

10

-505

10

Exp

orts

05

10

-0.50

0.51

1.52

Common Shock

05

10

-0.50

0.51

1.52

05

10

-0.50

0.51

05

10

-202468

05

10

-0.050

0.05

0.1

0.15

Demand Shock

Qu

arters

05

10

-0.050

0.05

0.1

0.15

Qu

arters

05

10

-0.050

0.05

0.1

0.15

Qu

arters

05

10

-1

-0.50

0.5

Qu

arters

Fig

ure

1:IR

Fs

for

US-C

anad

a(E

xpor

ts),

(-)A

ctual

IRF,(–

)Tw

oSta

ndar

dE

rror

s

14

05

10

-10123

US

O

utp

ut

US Shock

05

10

-101234

Can

ad

ian

O

utp

ut

05

10

-3

-2

-101

Ou

tp

ut D

ifferen

tial

05

10

-202468

Im

po

rts

05

10

-10123

Common Shock

05

10

-10123

05

10

-0.50

0.51

1.5

05

10

-202468

05

10

-0.1

-0.050

0.05

0.1

Demand Shock

Qu

arters

05

10

-0.2

-0.10

0.1

0.2

Qu

arters

05

10

-0.1

-0.050

0.05

0.1

0.15

Qu

arters

05

10

-2

-1.5

-1

-0.50

0.5

Qu

arters

Fig

ure

2:IR

Fs

for

US-C

anad

a(I

mpor

ts),

(-)A

ctual

IRF,(–

)Tw

oSta

ndar

dE

rror

s

15

05

10

-1

-0.50

0.51

1.5

US

O

utp

ut

US Shock

05

10

-101234

Germ

an

O

utp

ut

05

10

-3

-2

-101

Ou

tp

ut D

ifferen

tial

05

10

-505

10

Exp

orts

05

10

-0.50

0.51

1.5

Common Shock

05

10

-0.50

0.51

1.52

05

10

-1

-0.50

0.51

1.5

05

10

-202468

05

10

-0.6

-0.4

-0.20

0.2

0.4

Demand Shock

Qu

arters

05

10

-1

-0.50

0.5

Qu

arters

05

10

-0.4

-0.20

0.2

0.4

Qu

arters

05

10

-8

-6

-4

-202

Qu

arters

Fig

ure

3:IR

Fs

for

US-G

erm

any

(Expor

ts),

(-)A

ctual

IRF,(–

)Tw

oSta

ndar

dE

rror

s

16

05

10

-0.50

0.51

1.5

US

O

utp

ut

US Shock

05

10

-20246

Germ

an

O

utp

ut

05

10

-4

-202

Ou

tp

ut D

ifferen

tial

05

10

-202468

Im

po

rts

05

10

-0.50

0.51

1.52

Common Shock

05

10

-10123

05

10

-2

-1012

05

10

-505

10

05

10

-0.4

-0.20

0.2

0.4

Demand Shock

Qu

arters

05

10

-1

-0.50

0.5

Qu

arters

05

10

-0.20

0.2

0.4

0.6

Qu

arters

05

10

-6

-4

-202

Qu

arters

Fig

ure

4:IR

Fs

for

US-G

erm

any

(Im

por

ts),

(-)A

ctual

IRF,(–

)Tw

oSta

ndar

dE

rror

s

17

summarizes the estimates for Canada; the lower part reports the corresponding statistics

for Germany. Standard errors are reported in parenthesis. For each country, table 1

essentially reports estimates for two different regressions. The model (12) is estimated

separately for exports and imports to capture potential differences in the trade channel.

Therefore forecast errors for output are reported twice for each country. For briefness,

we neglect the result for the US here as they provide no additional information of the

transmission mechanism.

A few observations deserve some attention. Common disturbances are important, but

shocks to US output account for the largest share in the forecast error of Canadian

and German output at all horizons. Further, US output shocks also have the largest

explanatory power with respect to fluctuations in bilateral trade flows of Canada and

Germany - with one exception. US imports from Germany are mostly explained by

common rather the US output shocks. Comparing the impact of US shocks on the

Canadian and German economy reveals that the relative shares are consistently larger

for Canada. Demand shocks appear to have no significant impact on either variable.

3.3 Decomposition of Outputs, Exports and Imports

In the previous Section, we characterized the transmission mechanism by estimating

IRFs and by decomposing the forecast errors for different structural innovations. We

have argued that there is evidence of a substantial transmission of US shocks to Canada

and Germany and that the transmission involves trade in goods and services. To provide

another perspective on the propagation mechanism, this Section presents a decomposition

of the relevant time series that is implied by the structural model (11). One may think of

this experiment as identifying those movements in output and bilateral exports (imports)

that can be attributed to the transmission of idiosyncratic innovations. Formally, let xkj

be the j − th element of x that is purely accounted for by structural innovations of type

k, with k = h, g, d as before and j = 1, 2, 3. In other words, we asked how would the

j − th element of x look like if there had only been structural innovations of type k in

the past. Formally, the country-specific, common and demand components of the j − th

element of x are defined by

xhj =

∑3i=1 Cji × Si,1 × εh

xgj =

∑3i=1 Cji × Si,2 × εg

xdj =

∑3i=1 Cji × Si,3 × εd.

(21)

18

For

ecas

tErr

orVar

iance

Dec

ompo

sition

for

Can

ada

and

Ger

man

y

Hor

izon

Can

adia

nO

utp

ut

Expor

tsto

Can

ada

Can

adia

nO

utp

ut

Impor

tsto

Can

ada

εhεg

εdεh

εgεd

εhεg

εdεh

εgεd

1Q

uar

ter

0.94

0.06

0.00

0.63

0.30

0.07

0.90

0.10

0.00

0.54

0.39

0.07

(0.1

2)(0

.12)

(0.0

0)(0

.16)

(0.1

6)(0

.01)

(0.1

3)(0

.13)

(0.0

0)(0

.09)

(0.0

8)(0

.02)

5Q

uar

ters

0.90

0.10

0.00

0.78

0.21

0.00

0.86

0.14

0.00

0.59

0.38

0.03

(0.1

2)(0

.12)

(0.0

0)(0

.16)

(0.1

6)(0

.00)

(0.1

2)(0

.11)

(0.0

0)(0

.12)

(0.1

1)(0

.02)

10Q

uar

ters

0.91

0.09

0.00

0.78

0.21

0.01

0.87

0.13

0.00

0.59

0.38

0.03

(0.1

2)(0

.12)

(0.0

0)(0

.16)

(0.1

6)(0

.00)

(0.1

1)(0

.10)

(0.0

0)(0

.12)

(0.1

1)(0

.01)

Hor

izon

Ger

man

Outp

ut

Expor

tsto

Ger

man

yG

erm

anO

utp

ut

Impor

tsto

Ger

man

y

εhεg

εdεh

εgεd

εhεg

εdεh

εgεd

1Q

uar

ter

0.92

0.08

0.00

0.48

0.26

0.26

0.54

0.45

0.01

0.18

0.64

0.18

(0.1

4)(0

.13)

(0.0

4)(0

.11)

(0.1

3)(0

.12)

(0.1

7)(0

.15)

(0.0

4)(0

.06)

(0.1

2)(0

.10)

5Q

uar

ters

0.93

0.07

0.00

0.56

0.31

0.13

0.61

0.39

0.00

0.20

0.70

0.10

(0.1

2)(0

.11)

(0.0

4)(0

.14)

(0.1

4)(0

.10)

(0.1

2)(0

.11)

(0.0

2)(0

.07)

(0.1

1)(0

.09)

10Q

uar

ters

0.94

0.06

0.00

0.56

0.31

0.13

0.63

0.36

0.00

0.21

0.70

0.09

(0.1

2)(0

.11)

(0.0

3)(0

.14)

(0.1

3)(0

.11)

(0.1

1)(0

.10)

(0.0

3)(0

.06)

(0.1

1)(0

.09)

Tab

le1:

For

ecas

tE

rror

Shar

esar

ein

Per

cent

-Sta

ndar

dD

evia

tion

sar

ein

Par

enth

esis

19

E.g., xh3 denotes the series of bilateral exports (imports) that would have been obtained

in a world purely driven by shocks to US output. In order to evaluate the relative impor-

tance of each of these components, we compute the instant correlation coefficients with

respect to the actual time series. The sample correlations are interpreted as measures of

fit. High correlation coefficients are indicative for important determinants of the actual

series. The upper part of table 2 summarizes the correlation coefficients12 of the actual

series. Variables with a hat refer to simulated series. xk denotes the series x that one

would have obtained if only shocks of type k had been realized. The decomposition is

based on the same VAR model for both, the actual and simulated series. The last row

reports the contemporaneous correlations between the actual and simulated series. For

each country, results are reported separately for exports and imports. Which conclusions

could be drawn from statistics in the upper part of table 2?

A world that is purely driven by shocks to US output accounts reasonably for the actual

Canadian and German output series. The correlations coefficients between the actual

and decomposed series are 0.64 (exports) and 0.77 (imports) for Canada and 0.71 (ex-

ports) and 0.62 (imports) for Germany. The evidence lends support to the hypothesis of

a substantial transmission of US shocks abroad though it remains unclear to what extent

this occurs through export supply or import demand or alternatively, through a combi-

nation of both. While disturbances to US output seem to exert a significant influence

on US imports from Germany, there is no evidence suggesting that this is true for US

exports to Germany. Interestingly enough, for Canada, this observation is reversed. An

US output shock appears to have a sizeable impact on exports to Canada. In contrast,

US imports from Canada seem to be hardly affected by US shocks. This observation

might be accounted for by the supply determination of exports whilst changes in imports

tend to be triggered by demand factors. Further, it is not clear wether the fluctuations in

bilateral trade flows result primarily from US output shocks. A world that is restricted

to common supply and demand shocks tends to replicate the actual series better in terms

of matching sample correlations. In particular, we note the relatively high correlations

between the actual export (import) series and those that would have been observed in a

world that was purely driven by demand type disturbances.

Overall, we think the empirical evidence gives rise to the following conclusions. A sub-

stantial fraction of output fluctuations in Canada and Germany result from an interna-

12We chose to report only the absolute values of the correlation coefficients because the particular sign

depends upon the perception of innovations. As has been noted, the structural innovations are unique

up to a scalar that could be either positive or negative.

20

Fitnes

sbe

twee

nAct

ual

,Sim

ula

ted

and

Dec

ompo

sed

Tim

eSer

ies

Act

ual

Can

adia

nSer

ies

Act

ual

Ger

man

Ser

ies

Shoc

k( y

,yk)

( ex,e

xk)

( y,y

k)

( im,i

mk)

( y,y

k)

( ex,e

xk)

( y,y

k)

( im,i

mk)

k≡

h0.

640.

540.

770.

050.

710.

080.

620.

30

k≡

g0.

120.

010.

110.

290.

250.

310.

020.

33

k≡

d0.

240.

680.

260.

700.

190.

670.

280.

73

Sim

ula

ted

Can

adia

nSer

ies

Sim

ula

ted

Ger

man

Ser

ies

Shoc

k( y

,yk)

( ex,e

xk)

( y,y

k)

( im,i

mk)

( y,y

k)

( ex,e

xk)

( y,y

k)

( im,i

mk)

k≡

h0.

650.

630.

780.

920.

630.

600.

800.

93

k≡

g0.

160.

630.

500.

760.

540.

060.

230.

55

k≡

d0.

460.

080.

160.

110.

430.

260.

360.

09

Fit

nes

sbet

wee

nA

ctual

and

Sim

ula

ted

Ser

ies

Fit

nes

sbet

wee

nA

ctual

and

Sim

ula

ted

Ser

ies

corr

(x,x

)0.

860.

270.

860.

250.

730.

390.

730.

19

Tab

le2:

Con

tem

por

aneo

us

Cor

rela

tion

sbet

wee

nA

ctual

,Sim

ula

ted

and

Dec

ompos

edSer

ies

21

tional transmission of shocks to US output. This conclusion reinforces the ”locomotive”

character of the US economy. The Canadian economy seems to be more exposed to US

shocks than Germany though the transmission mechanism appears to be similar. The in-

ternational propagation of US shocks to Canada and Germany involves bilateral exports

and imports. However, it remains unclear to what extent trade linkages can account

for the propagation of output shocks. The decomposition of the forecast error variances

attributes bilateral trade flows a substantial role. In contrast, a comparison of the actual

series with those implied by the VAR model suggests that trade linkages are far less im-

portant. To the credit of this conclusion, the large differences in the size of bilateral trade

between Canada and Germany with respect to the US are not adequately reflected in the

dynamic responses of both economies to an US output shock. This latter interpretation

would be consistent with earlier findings by Schmitt-Grohe (1998) and Selover (1999).

4 Can a Model with Trade Interdependence Repro-

duce the Estimates?

In order to shed further light on the relative strength of the trade channel, we use a novel

strategy that builds on previous work by Canova and Marrinan (1998). Essentially, we

compare the empirical transmission mechanism - as characterized in the previous Section

- to a theoretical counterpart that is implied by equation (10). From the structure of the

model, we know that the theoretical transmission mechanism is entirely based on trade

linkages. The comparison is expected to qualify the relative strength of the trade channel

in the international propagation of US shocks to Canada and Germany.

4.1 The Experiment

Rather than simulating the model with artificially generated realizations of e, like Canova

and Marrinan (1998), we use the same set of structural innovations to compare the im-

plied transmission mechanism. This strategy is intended to narrow the gap between the

theoretical and empirical model in the sense that we solely compare the implied mapping

rules. This is of particular importance as the theoretical model itself displays only a weak

endogenous propagation mechanism. The structural innovations identified by the empir-

ical VAR model are taken to generate artificial data from the theoretical model outlined

in Section 2. That is, we set e = ε, where ε is obtained by (13). The IRFs are estimated

from the simulation data using the same VAR model and the same set of restrictions to

22

just-identify it as in Section 3. In one particular aspect, we follow Canova and Marrinan

(1998). The confidence bands around the actual IRFs are used as a ”window” that the

IRFs from the simulation data are expected to meet. To characterize the fit, the cumu-

lated absolute deviations between the actual and theoretical IRFs and, as a benchmark,

the cumulated absolute differences between the actual IRFs and the two-standard error

confidence bands are computed. A reasonable approximation is achieved if the cumulated

differences between the IRFs are smaller than those implied by the 95 percent confidence

limits.

The simulation is essentially a numerical experiment that requires to calibrate the model.

The relevant parameters of the model are the import share w, the preference parameters

γ, µ and β, the production parameters θ and ρ, parameters related to the evolution of

the capital stock δ and φ, and the elements of Γ and Θ governing the serial- and cross-

correlation in the exogenous forces. Given these parameters, the mapping rules Π, W

and M from (10) are fully identified. For the benchmark model, we stick for the majority

of parameters to those values chosen by Backus, Kehoe and Kydland (1994). As far as

φ, ρ, w and the elements of Γ are concerned, we slightly deviate from their parameteri-

zation and chose values such that the sample moments best fit the data properties. The

import share is set to .06 for Canada and .03 for Germany to reflect the bilateral trade

interdependencies. Since the dynamic properties of the model are hardly affected by the

import share, one could have well adhered to the share chosen by Backus, Kehoe and

Kydland (1994). The capital adjustment parameter φ governs the instantaneous impact

of a productivity shock on output. Unfortunately, no reliable estimates of φ are available,

nor does it seem that estimates are obtainable from the data. Therefore, φ is set such

that the long run impact of a shock to US productivity - implied by the IRFs - is equal to

empirical estimates13. ρ defines the extent to which importable goods are substitutable

by domestically-produced goods. As summarized by Hooper and Marquez (1995), em-

pirical estimates for ρ are typically in the range of 0 to 4. However, no precise point

estimates exist. Since ρ also influences the cross-country output correlation, we chose a

value such that i) ρ is in the interval [0, 4] and ii) the model mimics best the empirical

IRFs of foreign output. In order to explore the robustness of our results to changes in

these parameters, we perform a sensitivity analysis around these benchmark values.

It is common knowledge that detrended US output can be represented by an AR (2)

13In some sense, this is just a scaling of the model. It will be shown in the sensitivity analysis that φ

does not affect the dynamic properties of the IRFs. Also, the particular value chosen for φ is reasonable

in the sense that previous studies assumed comparable sizes. For example, Kollmann (2001) set φ to 8.

23

process14. Cogley and Nason (1992), (1993) show that models like those in (10) typically

require large autocorrelation coefficients in the exogenous forces in order to approximate

the AR (2) behavior of detrended output. It is for this reason that the diagonal elements

of Γ are set to .65 in the benchmark calibration. However, there appears to be some

conflict with our estimates of ε that we use in the simulation experiment. The elements

of ε represent those changes in the elements of x that cannot be accounted for by the

estimated lag structure. That is, ε essentially corresponds to changes in productivity

Z and some output demand G. Because we assume the equality ε = e, this correspon-

dence holds only if Γ and Θ are zero matrices. This can easily be verified by linearizing

equations (8) - (9). Another way to see this is to note that the elements of ε are es-

sentially white noise processes implying no serial correlation. The disparity arises from

the distinct lag structures of the theoretical and empirical model. Recall that the MA

representation of the VAR model has been truncated at two lags which corresponds to

an AR (2) representation of detrended output. The theoretical model (10) displays only

an AR (1) lag structure. To reconcile both representations, one could either truncate the

VAR model at lag one, or one could assume - counterfactually - some autocorrelation

structure in the innovation processes. Since a truncation at lag one amounts to a mis-

specification of the empirical model and would thus bias the empirical estimates, we have

chosen the latter alternative. As before, a sensitivity analysis is conducted for different

assumptions on the autocorrelation structure. Notwithstanding the particular choices of

AR (1) coefficients, the off-diagonal elements of Γ and Θ are assumed to be zero in all

experiments. Then the matrix M in equation (12) becomes lower block-diagonal ren-

dering each innovation independently from one another. There are no cross-linkages in

the innovation processes. Table 3 summarizes the current benchmark parameterization.

The figures in the line BKK refer to the original parameterization by Backus, Kehoe and

Kydland (1994). The elements of ε are unique up to a scalar. The normalization of the

variances to one restricts the absolute size of the scalar, but leaves room for the ”sign”.

That is, multiplying the elements of ε by minus one is an equivalent representation with

reversed signs. The ”sign” of each element of ε has been chosen such that an innovation

is interpreted as positive shock. For example, a country-specific shock to the US leads to

an increase in output rather than a contraction. Further, the span of x allows to identify

three orthogonal vectors of innovations. From estimates of (12) - (13), we are able to

identify structural innovations to US output and demand, and output innovations that

14For a discussion of output behavior, see Nelson and Plosser (1881) and McCallum (1989) amongst

others.

24

Model Calibration

β γ µ w θ φ δ ρ Γii Γij Γg Θii Θij

BKK .99 -1 .34 .10 .36 - .025 .66 .908 .088 - .950 0

Benchmark Economies

CAN .99 -1 .34 .06 .36 5.5 .025 1.9 .65 0 .65 -.15 0

GER .99 -1 .34 .03 .36 5.5 .025 1.9 .65 0 .65 -.15 0

Table 3: Model Calibration

are common across countries. In order to get some measures of innovations to Cana-

dian and German output and demand, the VAR model has been re-estimated reversing

home and foreign countries. This two-step approach is far from being optimal because

it does not assure that country-specific shocks to the US and Canada (Germany) are

orthogonal15. Indeed, estimates of the variance-covariance matrices suggest a positive

and significant correlation in the country-specific residuals. The shape of the IRFs must

in part be understood as reflecting the coincidence in the innovations.

4.2 Simulation Results

Given the parameterization and the assumptions on the autocorrelation structure, we

are in a position to simulate the model (10). The distributional properties of the data

are then analyzed using the VAR model of Section 3. The figures 5 - 8 display the

dynamic responses of domestic and foreign output, the output differential and bilateral

exports (imports). The figures also include the IRFs and the two-standard error con-

fidence bands from the actual data as presented in figures 1 - 4. This is intended to

facilitate the comparison. The summary statistics in table 4 and 5 characterize the fit of

the IRFs for both country pairs. The numbers in the line ”Data” refer to the cumulative

differences between the actual IRFs and the standard errors bands after 10 quarters.

We begin by summarizing the results for the US-Canada pair and consider first the

dynamic responses following a country-specific shock to US output. As far as US and

15There seems to be no tractable alternative to this two-step procedure. The simultaneous identi-

fication of idiosyncratic shocks to the US and Canada (Germany) would require a higher dimensional

structural VAR model which in turn seems to be difficult to just-identify.

25

05

10

-2

-1012

US

O

utp

ut

US Shock

05

10

-2024

Can

ad

ian

O

utp

ut

05

10

-3

-2

-101

Ou

tp

ut D

ifferen

tial

05

10

-505

10

Exp

orts

05

10

-0.50

0.51

1.52

Common Shock

05

10

-1012

05

10

-0.50

0.51

05

10

-202468

05

10

-0.050

0.05

0.1

0.15

Demand Shock

Qu

arters

05

10

-0.050

0.05

0.1

0.15

Qu

arters

05

10

-0.050

0.05

0.1

0.15

Qu

arters

05

10

-1

-0.50

0.5

Qu

arters

Fig

ure

5:IR

Fs

for

US-C

anad

a(E

xpor

ts),

(-)A

ctual

IRF,(+

)Sim

ula

ted

IRF,(–

)Tw

oSta

ndar

dE

rror

s

26

05

10

-10123

US

O

utp

ut

US Shock

05

10

-101234

Can

ad

ian

O

utp

ut

05

10

-3

-2

-101

Ou

tp

ut D

ifferen

tial

05

10

-202468

Im

po

rts

05

10

-10123 Common Shock

05

10

-10123

05

10

-0.50

0.51

1.5

05

10

-202468

05

10

-0.1

-0.050

0.05

0.1

Demand Shock

Qu

arters

05

10

-0.2

-0.10

0.1

0.2

Qu

arters

05

10

-0.1

-0.050

0.05

0.1

0.15

Qu

arters

05

10

-2

-1.5

-1

-0.50

0.5

Qu

arters

Fig

ure

6:IR

Fs

for

US-C

anad

a(I

mpor

ts),

(-)A

ctual

IRF,(+

)Sim

ula

ted

IRF,(–

)Tw

oSta

ndar

dE

rror

s

27

05

10

-1

-0.50

0.51

1.5

US

O

utp

ut

US Shock

05

10

-101234

Germ

an

O

utp

ut

05

10

-3

-2.5

-2

-1.5

-1

-0.50

0.5

Ou

tp

ut D

ifferen

tial

05

10

-202468

10

Exp

orts

05

10

-0.50

0.51

1.5

Common Shock

05

10

-0.50

0.51

1.52

05

10

-1

-0.50

0.51

1.5

05

10

-202468

05

10

-0.5

-0.4

-0.3

-0.2

-0.10

0.1

0.2

0.3

Demand Shock

Qu

arters

05

10

-0.8

-0.6

-0.4

-0.20

0.2

0.4

0.6

Qu

arters

05

10

-0.4

-0.3

-0.2

-0.10

0.1

0.2

0.3

0.4

Qu

arters

05

10

-7

-6

-5

-4

-3

-2

-101

Qu

arters

Fig

ure

7:IR

Fs

for

US-G

erm

any

(Expor

ts),

(-)A

ctual

IRF,(+

)Sim

ula

ted

IRF,(–

)Tw

oSta

ndar

dE

rror

s

28

05

10

-0.50

0.51

1.5

US

O

utp

ut

US Shock

05

10

-20246

Germ

an

O

utp

ut

05

10

-4

-202

Ou

tp

ut D

ifferen

tial

05

10

-202468

Im

po

rts

05

10

-0.50

0.51

1.52

Common Shock

05

10

-10123

05

10

-2

-1012

05

10

-505

10

05

10

-0.4

-0.20

0.2

0.4

Demand Shock

Qu

arters

05

10

-1

-0.50

0.5

Qu

arters

05

10

-0.20

0.2

0.4

0.6

Qu

arters

05

10

-6

-4

-202

Qu

arters

Fig

ure

8:IR

Fs

for

US-G

erm

any

(Im

por

ts),

(-)A

ctual

IRF,(+

)Sim

ula

ted

IRF,(–

)Tw

oSta

ndar

dE

rror

s

29

Canadian output are concerned, the predictions derived from the benchmark economy

are broadly consistent with the empirical evidence. The more than proportionate increase

in Canadian output can also be found in the simulation data. For most horizons, the

IRFs from the simulated data are inside the confidence region of the actual IRFs. This

observation is also reflected in the cumulative absolute errors of the simulated IRFs. The

theoretical predictions for Canadian output are off by 4 percent (2 percent) relative to

the steady state over 10 quarters while the confidence region implied by the data is 11

percent. This observation is independent from the particular trade variable - exports or

imports - used in the VAR model. It is noteworthy that these prediction errors emanate

almost entirely from the failure of replicating the on-impact dynamics16. In the medium

and the long run, the simulated IRFs nearly match those obtained from the actual data.

The benchmark economy is less consistent with the data with respect to the responses

of US exports to and US imports from Canada. The model correctly predicts that the

increase in exports exceeds the rise in imports17 following an US output shock. However

the benchmark economy underpredicts the export (import) responses in the short and

medium run. The differences between the actual and simulated IRFs accumulate over a

horizon of 10 quarters to 22 (14) percent for exports (imports) and are thus at the margin

of the confidence regions.

These observations are mirrored in the dynamic responses to a common output innova-

tion. Again, the benchmark economy accounts well for the responses in US and Canadian

output but it fails with regard to the dynamic responses of exports and imports. The

magnitude of the responses is too low over much of the horizon. Demand type distur-

bances have been identified empirically by assuming that they exert no long-run impact

on output. The model proves to be consistent with the data in this regard. The IRFs are

inside the confidence region over the entire horizon. The impact of a demand type dis-

turbance on exports and imports cannot be replicated by the benchmark economy. But

this failure comes at no surprise. In the theoretical model, government expenditures are

assumed to represent the demand type shocks. In contrast, the empirical model does not

identify exactly shocks to government expenditures. This semi-structural demand shock

can be understood as amalgam of a variety of shocks including monetary and terms-

of-trade disturbances. Thus, we cannot expect to understand the dynamic responses of

exports and imports in the actual data merely by shocks to domestic government expen-

16The upper panels of figure 5 and 6 show that the benchmark economy implies a decline in output

on impact for both, the US and Canada.17The export response reaches a peak after 2 quarters with an increase of 3.07 percent. The peak of

the import response occurs after 4 quarters with an increase of 2.34 percent.

30

Fit

ofth

eM

odel

for

the

Cou

ntr

yPai

rU

S-

Can

ada

Can

adia

nO

utp

ut

Expor

tsC

anad

ian

Outp

ut

Impor

ts

εhεg

εdεh

εgεd

εhεg

εdεh

εgεd

Dat

a0.

110.

040.

002

0.23

0.13

0.01

0.11

0.05

0.00

50.

130.

090.

01

Ben

chm

ark

Tabl

eIII

0.04

0.02

0.00

10.

220.

090.

020.

020.

020.

003

0.14

0.12

0.02

Exper

imen

tI

φ=

120.

080.

030.

002

0.27

0.10

0.02

0.02

0.01

0.00

30.

140.

110.

02

φ=

20.

220.

040.

002

0.36

0.12

0.02

0.03

0.04

0.00

40.

170.

140.

02

Exper

imen

tII

ρ=

2.5

0.08

0.04

0.00

40.

270.

100.

020.

050.

030.

005

0.17

0.14

0.03

ρ=

1.3

0.06

0.03

0.00

10.

210.

100.

020.

020.

020.

003

0.15

0.12

0.02

Exper

imen

tII

IΓ

ii=

00.

240.

030.

002

0.19

0.13

0.02

0.24

0.05

0.00

40.

110.

080.

02

Tab

le4:

Can

ada:

Cum

ula

tive

Diff

eren

ces

bet

wee

nA

ctual

and

Sim

ula

ted

IRFs

31

ditures.

The predictions of the benchmark economy are also consistent with the US-Germany pair

in several dimensions. To begin with, we consider the responses to an US supply shock.

The responses of US output implied by the benchmark economy are entirely consistent

with the data. The IRFs are inside the confidence regions at all horizons. The estimated

responses for German output and bilateral exports (imports) from (to) Germany depend

on the choice of the trade variable used in the VAR model. If exports are considered, a

large mismatch between the simulated and actual IRFs results. The short and medium

run dynamics of output and exports are heavily underpredicted by the model. The the-

oretical predictions for German output are off by 12 percent over 10 quarters while the

confidence region implied by the data is merely 10 percent. The mismatch for exports is

even larger with 26 percent of the steady state, coinciding with the confidence limits. The

picture changes considerably if imports replace exports in the VAR model. The responses

of German output can be reasonably replicated. Apart from the short run dynamics, this

is also true for US imports from Germany. Notably, the IRFs are inside the confidence

regions over the entire horizon. The differences in the IRFs accumulate to 3 percent for

output and 8 percent for imports which amount to only small fractions of the confidence

bands implied by the actual data.

As far as the dynamic responses to a common supply and a demand type disturbance

are concerned, the observations from the Canadian case still apply. From the data of the

benchmark economy, we derived responses for US and German output that reasonably

mimic those found in the actual data. The IRFs are largely in the confidence region

regardless of the particular choice of the trade variable. The accumulated mismatches

between the actual and simulated IRFs are smaller than the confidence limits.

The responses of exports and imports to a common supply side innovation stand in con-

trast to the actual data. In particular, the benchmark model fails to generate the short

run dynamics in bilateral exports and imports that are observable in the data. The mag-

nitude of the responses is by far too low as to account for the empirical evidence. The

cumulated differences between the actual and simulated IRFs are beyond the confidence

limits that proves the failure in this regard. The dynamic responses following a demand

type disturbance are typically insignificant and for most horizons inside the confidence

region of the actual IRFs. The on-impact dynamics of exports and imports due to an

innovation in demand cannot be replicated by the theoretical economy. The reason for

this mismatch has been explained for the Canadian case and it applies here as well.

Do we recognize structural differences in the pattern of the dynamic responses between

32

Fit

ofth

eM

odel

for

the

Cou

ntr

yPai

rU

S-

Ger

man

y

Ger

man

Outp

ut

Expor

tsG

erm

anO

utp

ut

Impor

ts

εhεg

εdεh

εgεd

εhεg

εdεh

εgεd

Dat

a0.

100.

050.

010.

260.

160.

060.

160.

060.

010.

260.

170.

06

Ben

chm

ark

Tabl

eIII

0.12

0.03

0.00

80.

260.

200.

090.

030.

030.

008

0.08

0.30

0.08

Exper

imen

tI

φ=

12.0

0.13

0.02

0.00

30.

240.

190.

080.

020.

020.

006

0.08

0.29

0.08

φ=

2.00

0.14

0.03

0.01

20.

320.

230.

090.

020.

040.

007

0.14

0.32

0.08

Exper

imen

tII

ρ=

2.50

0.13

0.02

0.00

30.

230.

200.

090.

020.

020.

007

0.27

0.17

0.08

ρ=

1.30

0.14

0.04

0.02

0.27

0.17

0.08

0.02

0.03

0.00

60.

110.

300.

08

Exper

imen

tII

IΓ

ii=

00.

120.

020.

004

0.26

0.19

0.08

0.07

0.04

0.00

50.

170.

250.

08

Tab

le5:

Ger

man

y:

Cum

ula

tive

Diff

eren

ces

bet

wee

nA

ctual

and

Sim

ula

ted

IRFs

33

Canada and Germany that would eventually make a case for trade interdependencies?

There are discrepancies in the responses, but they do not seem to be structural in nature.

Following an US output shock, the short run dynamics of US exports to Germany are

much less replicated by the benchmark economy when compared to Canada. Over the

entire horizon, the implied IRFs are at odds with the actual IRFs for both countries.

The magnitude of the failure is nearly the same: exports to Canada are off by 22 percent

whilst exports to Germany accumulate to a mismatch of 26 percent of the steady state

value. This observation carries over to the response of US exports following a common

supply shock. Another discrepancy can be recognized by comparing the on-impact dy-

namics of output to an US and a common shock. The predicted IRFs for Germany are

more consistent with the data than those obtained for Canada. As will be discussed in

the sensitivity analysis, the on-impact dynamics appear to be largely related to the serial

correlation in the innovation processes rather than to the core structure of the model.

Overall, the differences between Canada and Germany are in an order of magnitude that

is quantitatively small. Differences in trade interdependencies appear to be of minor

importance with respect to the dynamic properties of the benchmark economy.

Which conclusions could be drawn from this evidence? The international transmission

of idiosyncratic output shocks through trade cannot be fully understood by a flexible

price open economy model that emphasizes the supply side of an economy. The implied

wealth and substitution effects are typically too weak to induce fluctuations in bilateral

trade flows that we observe for the US, Canada and Germany, and that we suspect to

synchronize the international output cycles. To be concrete, we are not saying that trade

linkages are generally unimportant from a theoretical point of view, but they simply seem

to be not that important in a model economy that is supply-side driven. This observation

comes along with the failure of flexible price models to replicate the magnitude of relative

price changes that trigger the substitution effects.

To reinforce our arguments, we have decomposed the simulated data in analogy to the

actual data by equation (21). The results can be found in the lower part of table 2. The

bottom line of the same table summarizes the fit of the actual and simulated data by

means of sample correlations.

The benchmark economies achieve a much better fit for the output series. The contem-

poraneous correlation between the actual and simulated series are 0.86 for Canada and

0.73 for Germany. The corresponding statistics for exports (imports) are 0.27 (0.25) for

Canada and 0.39 (0.19) for Germany. This evidence essentially confirms our observations

from the analysis of the IRFs. To better understand these results, we may compare the

34

decomposition of the actual with the simulated data. From the actual data we know

that bilateral exports and imports typically display a strong demand component. The

sample correlation between actual exports (imports) and a series that one would have

obtained from pure innovations to demand are on average 0.70. In contrast to the em-

pirical evidence, demand components do not seem to be important in the benchmark

economies. Instead, simulated export and import series appear to be largely driven by

country-specific and common supply shocks whilst the impact of demand type innova-

tions is negligible. The average sample correlation between actual and demand-driven

series is insignificantly low with 0.13. Given the lack of a strong demand component

in the benchmark economies, the low predictive power with respect to bilateral exports

and imports comes as no surprise. Interestingly, the decomposition of simulated out-

put data is broadly consistent with the empirical evidence. The simulated output series

also appear to be primarily driven by supply shocks which explains why the benchmark

economies perform well in replicating the actual output series.

The linkage between the theoretical economies is entirely based upon intratemporal trade.

If this channel appears to be rather weak, the co-movements in output across countries

- as exemplified in the shape of the IRFs in figures 1 - 4 - cannot be fully understood

by this type of linkage. The on-impact dynamics of the simulated output series seem to

be much more the result of the correlation structure in the innovation processes. As the

off-diagonal elements in Θ and Γ are set to zero, the simultaneity of the domestic and

foreign output movements in the simulated data - as shown in figures 5 - 8 - seems to be

better understood by a strong contemporaneous correlation in the idiosyncratic supply

shocks across countries18 rather than a strong transmission mechanism through trade.

4.3 Sensitivity Analysis

We now run several experiments to test if the dynamic properties of the theoretical

economies survive modifications in those parameters that have been chosen by assumption

or with no reliable measures at hand. The summary statistics are reported in the lower

part of table 4 and table 5.

18Although we have attempted to dissociate country-specific and common disturbances in the VAR

model, we cannot rule out the possibility that idiosyncratic shocks are positively correlated across coun-

tries. The reason seems to be that the VAR model does not identify them as common shocks as long

as their permanent impact on output is equal across countries - which may not be the case under all

circumstances.

35

4.3.1 Experiment I - Capital Adjustment Costs

qt measures the marginal increase in lifetime utility from an additional unit of capital

Kt+1 at time t19. One can show that qt is related to the adjustment cost function Φ

by qt = 1 + Φ′ (It). By using the law of motion for capital (5), this relationship can be

approximated by

qt ≈ 1 + φIt

Kt

(22)

which holds up to a constant. The analogy to Tobin’s q theory is obvious. qt defines

also the ratio of a marginal unit of capital to its replacement costs. As is well known

from Tobin’s q theory, the representative agents will increase its capital stock as long as

qt exceeds the replacement costs at the margin. The parameter φ influences the size of

the replacement costs at the margin as is apparent from (22). For φ = 0, no adjustment

costs are incurred and investment changes on a one-to-one basis with q above one. This

explains why investment is highly volatile if no adjustment costs are assumed. For φ > 0,

investment generates adjustment costs φ and absorbs some fraction of the funds directed

to increase the stock of physical capital. As adjustment costs increase, the magnitude

of the investment responses are expected to decline for some given q. As output and

investment are highly correlated in the theoretical economies, we also suppose output to

behave less (more) volatile as φ rises (diminishes). Changes in φ affect in particular the

short and medium run dynamics of output and exports. For φ = 12, the IRFs display

smaller swings whilst for φ = 2, the amplitude of the output responses is typically larger.

A decline in adjustment cost tends to affect the predictions more decisively than an

increase in the sense that the cumulative differences between actual and simulated IRFs

are smaller for the latter experiment. The assumption of no adjustment costs (φ = 0)

implies excessive output responses that are quantitatively at odds with the actual IRFs.

However, the qualitative features of the IRFs are unaltered by changes in the adjustment

costs of physical capital.

4.3.2 Experiment II - Intratemporal Elasticity of Substitution

To see how ρ affects the dynamic properties of the model, note that the relative price of

domestic and foreign goods in equilibrium corresponds to their marginal rate of substi-

19Say that λt denotes the Lagrangian associated with the budget constraint in the decision problem

summarized by equations (1) - (5 ). Formally, qt is related to λt, by qt = λt/u′ (Ct) .

36

tution. For the home country, the terms of trade ToT are defined by

ToTt =1

w

(A1

B1

) 1ρ

. (23)

Given differences in the relative supply of domestic and foreign goods, ρ governs the am-

plitude of relative price movements to achieve equilibrium. In other words, ρ determines

the size of wealth and substitution effects implied by changes in the terms of trade. Hence,

the better domestic and foreign goods are substitutable - (ρ increases to larger values) -

the smaller the changes in relative prices are needed to clear intermediate goods markets.

Intuitively, an increase in ρ from 1.9 to 2.5 essentially reduces the reliance on importable

goods in the final stage of production. Hence the impact of a domestic supply shock on

foreign output declines as ρ, increases. This logic is reversed if ρ is assumed to be smaller

(ρ = 1.3). Overall the impact of changes in the intratemporal elasticity of substitution

on the dynamic properties of the model are quantitatively small. The implied wealth

and substitution effects seem to be hardly affected by ρ. A similar observation has been

made by Cole and Obstfeld (1991). The qualitative features of the model are robust to

small changes in ρ.

4.3.3 Experiment III - Serial Correlation of Shocks to TFP

The most restrictive and potentially counterfactual assumption has been made by in-

troducing large AR (1) coefficients in the processes of TFP (8). In this experiment, we

relax this assumption and set the the AR (1) coefficients to zero. This change will affect

foremost the short run dynamics of output and trade variables. As the results differ

across country pairs, we begin by summarizing the evidence for Germany. The theoret-

ical economy predicts almost no international transmission of US shocks to Germany.

The shock dies out quickly and the cycle induced by an US output shock is small and

short (below 3 quarters). All supply side innovations are captured in the common shock.

Since the model with no serial correlation better mimics the actual IRFs over the medium

and long run, the differences with respect to the benchmark economy are quantitatively

small. The cumulated differences between actual and simulated IRFs over 10 quarters

are unchanged for exports and slightly increase for imports.

Unlike in Germany, there are substantial differences in the IRFs for the US-Canada pair.

The largest differences to the benchmark economy occur in the IRFs following an US

shock. The simulated IRFs overshoot the actual IRFs on-impact by far. The responses

of output and imports are persistent and die out slowly over time whilst the increase in

exports vanishes quickly. The implied on-impact dynamics of exports (imports) following

37

an US output shock are in the range of the actual data. Overall, the cumulated differ-

ences in the output responses over 10 quarters are more than twice as large as in the

benchmark case. Inversely, the implied export and import responses achieve a slightly

better fit with the actual data. There is no evidence of significant qualitative differences

with respect to the benchmark economies for both, the Canadian and the German case.

5 Conclusion

In this paper, we have attempted to assess the extent to which idiosyncratic shocks are

transmitted internationally through variations in bilateral trade of goods and services.

The focus of this analysis is confined to the country pairs US-Canada and US-Germany.

It has been assumed that a country’s output is driven by a country-specific supply and

demand shock and a common supply shock. A VAR model has been estimated that

allows to identify those movements in Canadian and German output and bilateral ex-

ports (imports) that result from shocks to US output. Subsequently we have asked if a

highly stylized two-country model of the international business cycle is able to replicate

the magnitude of the dynamic responses of output and exports (imports) following an

US output shock as observed in the empirical data. Several conclusions arise from this

study.

Empirically, there is evidence of a significant international propagation of US output

fluctuations to Canada and Germany. Though Canada seems to be more exposed to

US output shocks than Germany, there are no qualitative differences. The international

transmission of US output shocks is accompanied by sizable variations in bilateral trade

flows hinting to the existence of a trade channel. Though we cannot rule out that com-

plementary transmission channels exist. The theoretical economies match the empirical

evidence qualitatively, but generally fail to replicate the magnitude of the dynamic re-

sponses in exports (imports) in the short run. The implied wealth and substitution effects

- generated by relative price changes - appear to be too small to induce variations in out-

put and exports (imports) that are observable in the data.

The qualitative implications of the model economies are robust to small variations in

selected parameters. However, changing capital adjustment costs or the serial correlation

in the innovation processes trigger substantial changes in the quantitative predictions of

the model. Empirically, exports and imports appear to be largely driven by demand type

disturbances. The analysis of the simulated export (import) series attributes supply side

innovations a leading role. The strong reliance upon supply side economics is counterfac-

38

tual and explains in part the model’s failure to replicate the dynamics of bilateral exports

and imports.

References

Backus, P. Kehoe and F. Kydland (1992), International real business cycles , Journal

of Political Economy 101, 745 - 775

Backus, P. Kehoe and F. Kydland (1994), Dynamics of the trade balance and the terms

of trade: The J-curve, American Economic Review 84, 84 - 103

Backus, P. Kehoe and F. Kydland (1995), International business cycles: Theory and ev-

idence, in: Cooley, T.F.,ed. Frontiers of Business Cycle Research, Princeton University

Press

Baxter, M. (1995), International trade and the business cycle, in: G. Grossman and K.

Rogoff, eds., Handbook of International Economics, vol. 3 (North-Holland, Amsterdam)

Baxter, M. and M.J. Crucini (1993), Explaining saving-investment correlations , Ameri-

can Economic Review 83, 416 - 436

Blackburn, K. and M. Ravn (1991), Contemporary macroeconomic fluctuation: An in-

ternational perspective, University of Southampton Working Paper 9106

Blackburn, K. and M. Ravn (1992), Business cycles in the UK: Facts and fictions, Econo-

mia 59, 383 - 401

Blanchard, O. and C. Kahn (1982), A solution algorithm of rational expectation equation

systems, Econometrica , 1305 - 1311

Blanchard, O. and D. Quah (1989), The dynamic effects of aggregate demand and supply

disturbances, American Economic Review 79, 655 - 673

Brandner, P. and K. Neusser (1992), Business cycles in open economies: Stylized facts

for Austria and Germany, Weltwirtschaftliches Archiv 128, 67 - 87

Canova, F. (1998), Detrending and business cycle facts, Journal of Monetary Economics,

41(3), 475 - 512

Canova, F. (2001), Validating DSGE models through VARs, mimeo

Canova, F. and J. Marrinan (1998), Sources and propagation of international output cy-

cles: common shocks or transmission ?, Journal of International Economics 46, 133 - 166

39

Cogley, T. and J.M. Nason (1993), Impulse dynamics and propagation mechanisms in a

real business cycle model, Economics Letters, 43, 77 - 81

Cogley, T. and J.M. Nason (1995), Output dynamics in a real business cycle model, Amer-

ican Economic Review, 85(3), 492 - 511

Gerlach, S. (1988), World business cycles under fixed and flexible exchange rates, Journal

of Money, Credit and Banking 21, 195 - 232

Gross, D.G. (2001), Identifying country-specific and global shocks in the business cycle,

CFS Working Paper

Hoffmann, M. (2000), The relative dynamics of investment and the current account in

G7 economies, forthcoming in The Economic Journal

Hooper, P. and J. Marquez (1995), Exchange rates, prices and external adjustments in

the United States and Japan, in Kenen, P.B., ed., Understanding interdependence: The

macroeconomics of the open economy, Princeton University Press 1995

Kollmann, R. (2001), Explaining international co-movements of output and asset returns:

The role of money and nominal rigidities”, Journal of Economic Dynamics and Control,

25(10), 1547-1583

Kwark, N.S. (1999), Sources of international business fluctuations: country-specific shocks

or worldwide shocks ?, Journal of International Economics 48, 367 - 385

Obstfeld, M. and H.L. Cole (1991), Commodity trade and international risk sharing,

Journal of Monetary Economics, 28, 3 - 24

Schmitt-Grohe, S. (1998), The international transmission of economic fluctuations: ef-

fects of U.S. business cycles on the Canadian economy, Journal of International Eco-

nomics 44, 257-287

Selover, D.D. (1999), International interdependence and business transmission in ASEAN,

Journal of the Japanese and International Economies 13, 220-253

Zimmermann, C. (1991), International business cycles among large and small countries,

manuscript

Zimmermann, C. (1997), Technology innovations and the volatility of output: An inter-

national perspective, CEPR Working-Paper

40