international business cycles: an unobserved components approach

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International Business Cycles: An Unobserved Components Approach

Sinchan Mitra Department of Economics

Washington UniversitySt. Louis, MO 63130

Tara M. SinclairDepartment of Economics

The George Washington UniversityWashington, DC 20005

January 31, 2006

JEL Classifications: C32, E32, 057

Keywords: International Business Cycles, Unobserved Components, G-7, International Trends

Abstract

We develop a multivariate correlated unobserved components model to investigate the

interaction between the permanent and transitory movements in output of the G-7 countries. Ourmodel provides estimates of the two unobserved components of GDP for each of the G-7countries by pooling cross-country evidence to analyze characteristics of their business cycles.Our empirical framework also enables us to assess the relative importance of permanent versustransitory shocks in driving output growth rate correlations across the G-7 countries, providingan alternative approach to dynamic factor models for analyzing international co-movements. Ourresults suggest that GDP in all the G-7 countries have variable stochastic permanent componentswith innovations that are negatively correlated with their respective transitory movements. Wefind only weak evidence for a world business cycle, with region-specific effects being moreimportant in explaining the pattern of correlations across the G-7 economies.

Corresponding author: [email protected]. The authors wish to thank Gaetano Antinolfi, Steve Fazzari,James Morley, Bruce Petersen, Ed Greenberg, Jean Imbs, Jahangir Hossain, Fred Joutz, and the participants of theApplied Time Series Research Group at Washington University in St. Louis and the Macro-International Seminar atGeorge Washington University. All remaining errors are our own.


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1 Introduction

The debate about what causes economic fluctuations has long been at the center of

macroeconomic research. One critical issue is whether the business cycle is wholly transitory, or

whether it might be real in the sense that it is characterized primarily by permanent rather than

transitory movements. 1 This research has generally focused on the US, but there has been some

work using international evidence as well (Mills and Wang 2002). Another issue that has

received increased attention in recent years is the linkage of economic activity across countries.

Recent research on international business cycles has documented international co-movements in

a wide array of macroeconomic variables (e.g. Backus, Kydland, and Kehoe (1992), Kose, Otrok

and Whiteman (2003), Gregory, Head, and Raynauld (1997)).

In this paper we apply a multivariate correlated unobserved components model to

investigate the interaction between the output of G-7 countries. Our main objective is to assess

the relative importance of permanent versus transitory disturbances as sources of variation in real

GDP. Diebold and Rudebusch (1996) suggest that one could usefully pool cross-country

evidence to ascertain business cycle characteristics as there are commonalities in the behavior of

real quantities across countries. Thus, in our empirical framework the series are modeled jointly.

This provides three distinct benefits. First, because we use variation in all countries to identify

the parameters for any single economy, the use of international data will improve the efficiency

of the estimates. Second, cross-country comparisons allow us to assess the generality of our

results. Third, we use our multivariate modeling framework to estimate the correlations between

the permanent and transitory innovations across countries. This enables us to analyze the extent

1 For a discussion of this debate, see Hamilton (2005), Morley, Nelson, and Zivot (2003), Kim, Piger, and Startz(2001), and Sinclair (2005).


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and nature of co-movement among the G-7 countries, thus contributing to the study of

international business cycles.

Following the work of Nelson and Plosser (1982), many empirical macroeconomists have

argued that real GDP or other macroeconomic variables might contain a unit root so that a

shock to the level of output in any period shifts the entire future path of output (Diebold and

Rudebusch 1996). However, in spite of the extensive literature on this issue, no clear consensus

has emerged on whether real output or other macroeconomic variables are unit root processes

and several researchers have concluded that the question is inherently unanswerable on the basis

of a finite sample of observations.2

A related line of research, to which we seek to contribute,directs attention away from testing for the existence of permanent components in output to

assessing the quantitative importance of permanent shocks for explaining variations in aggregate

output 3. Following Morley, Nelson, and Zivot (2003, hereafter MNZ), our model allows for

explicit interaction between the permanent (trend) and transitory (cyclical) components rather

than assuming that the correlation between innovations to the two unobserved components is

zero as has been done in many previous papers. We consider this to be a useful approach as a

key step in understanding the co-movements in aggregate economic variables is learning about

the link between changes in long-run economic trends and cyclical swings (Stock and Watson

1988).

2 For any unit root process there exists a stationary process that will be impossible to distinguish from the unit rootrepresentation for any given sample size T and the converse proposition is also true. See Hamilton (1989).3 These two questions are distinct as it is possible that a time series can contain a unit root but an innovation todayhas only a small effect on long-run forecasts of output (Campbell and Mankiw 1987).


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Unlike dynamic factor models, our empirical framework does not impose a common

dynamic factor structure on all countries prior to estimation. 4 Our model places fewer restrictions

on the correlation matrix of the permanent and transitory innovations across countries and is

therefore more suitable for understanding the extent of linkages between them. Additionally, our

approach does not require a prior transformation of the GDP series, through detrending or first-

differencing, which has been the common practice in the literature. In our unobserved

components framework, real GDP is modeled as containing a permanent component,

representing the stochastic trend, and a transitory component, representing the stationary

movements of the series. Thus, our approach explicitly allows cross-country correlations to bedriven by correlations between permanent movements as well as transitory movements.

To summarize our results, we find that all the G-7 countries have stochastic permanent

components and negative correlation between innovations to the permanent and transitory

components. For all countries, innovations to the permanent component are found to play a

significant role in explaining aggregate fluctuations even at short term horizons associated with

the business cycle. Additionally, innovations to the permanent and transitory components are

generally positively correlated across countries, but the correlations are often stronger among

countries within a particular geographic region (North America, Europe) than among countries

across these regions. Japan is found to be more closely correlated with the European countries in

our study than with the U.S. The correlation between the Euro zone countries and the U.S are

found to be quite low. We find only weak evidence for a world business cycle with region

specific and country-specific effects being more significant in explaining output fluctuations.

4 The assumption of a single common world factor leads a researcher to attribute all cross-dynamics acrosseconomies to the world shock (See Stock and Watson 2003).


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The rest of the paper proceeds as follows. In Section 2, we review the literature and

develop our empirical approach in greater detail. In Section 3 we present the multivariate

correlated unobserved components model and discuss the data used for estimation. In Section 4

we discuss the results, and in Section 5 we conclude and offer some suggestions for future

extensions of this work.

2 Literature Review

A large body of theoretical and empirical research now investigates the nature of

interdependence among countries in the global economy. One strand of this literature assesses

the degree to which macroeconomic fluctuations arise from common international shocks asopposed to country-specific factors. These papers most often apply dynamic factor models to

detrended macroeconomic data for a number of countries. Examples include Gregory, Head, and

Raynauld (1997), Kose, Otrok, and Whiteman (2003), Monfort, Renne, Ruffer, and Vitale

(2003), Helbling and Bayoumi (2003). Most of these studies assume that there are international

factors common to the cyclical fluctuations of all countries and seek to estimate these factors and

examine their relative importance in explaining country-specific movements. Findings in these

papers vary as to which component is most important, be it world, country-specific,

idiosyncratic, or regional.

Ahmed, Ickes, Wang, and Yoo (1993) use a vector autoregression (VAR) framework to

study international business cycles. They find that supply shocks, particularly country-specific

ones, are very important in generating international business cycles. Mills and Wang (2002) use

Friedmans Plucking model (1964,1993) cast into the uncorrelated unobserved components

framework as formulated by Kim and Nelson (1999a,b) to investigate asymmetries in business

cycles for G-7 countries. Using log real GDP data, they find that during expansionary phases,


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output seems to be driven mostly by permanent shocks, but during recessions and high-growth

recoveries, transitory shocks are the dominant factor. Therefore, macroeconomic models

emphasizing demand shocks may be more appropriate for explaining recessions and recoveries.

They estimate their model for individual countries and do not consider correlations among them.

Cheung and Westermann (2002) apply the cointegration test of Johansen (1991) and the

common cycle tests developed by Vahid and Engel (1993, 1997) to study the interaction of

annual real per capita GDP data in G-7 countries. They find evidence of six common non-

stationary processes and one common business cycle behind G-7 output dynamics. They find that

the relative variability and the correlation of the trend and cycle component are different acrosscountries.

Most of the studies of international business cycles discussed so far (except Cheung and

Westermann who use cointegration analysis) detrend the series that they use in their analysis

through first-differencing or through the use of H-P filter or Baxter-King filter. Detrending is

thought to be necessary as the macroeconomic variables studied (income, industrial production,

etc) may contain a non-stationary component. However, detrending nonstationary time series

using these filters might generate spurious cyclical behavior (Harvey and Jaegar (1993), Cogley

and Nason (1995), Murray (2003)). Our unobserved components approach obviates the

necessity of the prior transformation of the series by explicitly modeling both the permanent and

transitory unobserved components.

Until recently, time series models of output have primarily reflected the thinking that the

unobserved components of major macroeconomic time series are uncorrelated. 5 Clark (1987),

5 One major exception to this is the Beveridge and Nelson decomposition (1981) which does not make any particularassumption about the correlation between the components. The innovations in the estimated components of the


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Stock and Watson (1988), and others supposed that some assumptions about the correlations

between the unobserved components were necessary for identification of the model. 6 If the

correlation parameters cannot be estimated, a multivariate modeling framework does not provide

any additional information. Therefore, the assumptions of the literature led to univariate

detrending of nonstationary macroeconomic variables in a first stage, before modeling

international co-movements. MNZ (2003) and Sinclair (2005) find, however, that in some cases

we can estimate an unobserved components (UC) model without restricting the correlations

between the unobserved components. 7 We extend their correlated unobserved components model

to a multivariate framework and use it to model international output co-movements.MNZ (2003) find that when the correlation between shocks to the trend component and

the cyclical component is a free parameter to be estimated, it is found to be significantly negative

for US data, rejecting the restriction implied by the assumption of zero correlation. MNZ

interpret this negative correlation as being suggestive of the dominance of real shocks in the

macroeconomy, with much of the movement of the transitory component being explained as

adjustment dynamics to permanent shocks. Sinclair (2005) models the output and unemployment

rate jointly and also finds a negative correlation between innovations to the stochastic trend

component and the transitory component. However, in a recent paper, Perron and Wada (2005,

hereafter PW) argue that the MNZ model suffers from an important misspecification which

Beveridge and Nelson decomposition are perfectly negatively correlated. However, the implied correlation betweenthe true components is not restricted. It is possible to solve for this correlation in the univariate case, as shown byMorley, Nelson, and Zivot (2003). The multivariate case has been discussed by Morley (2005), Schleicher (2003),and by Sinclair (2005).6 A popular assumption is that the shocks to the trend function and the cyclical component are uncorrelated.7 MNZ (2003) showed that in the univariate case it is possible to identify the model provided p q+2 where thecycle is modeled as an ARMA (p, q) process. In the multivariate case, the requirement for identification is p 3/2+1/2n where the transitory components are modeled as AR (p) processes. (see Morley (2005), Engel and Morley(2001), Sinclair (2005)).


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biases their results. In particular, PW argues that these results are artifacts created by the neglect

of a change in the slope of the trend function in real GDP in 1973.They find, that once this

change is accounted for, the correlation between the trend and the cycle is zero as the former is

non-stochastic.

An additional objective of our paper is to thus to explore this debate further by testing the

results of MNZ and PW for G-7 output data. Estimating the correlation parameter between the

innovations to the trend function and the cyclical component is important as it helps us in

distinguishing between alternative macroeconomic theories. Some macroeconomic theories

suggest a positive correlation between innovations to the within-series components. For example,a primarily temporary increase in investment may lead to both permanent and transitory

increases in output, thereby suggesting a positive correlation between permanent and transitory

innovations. Hysteresis may also imply a positive correlation between the innovations where, for

instance, a temporary increase in the unemployment rate may partially persist and become

permanent. (e.g. Blanchard and Summers 1986). Some real business cycle theories, however,

imply a negative correlation between innovations to the permanent and transitory components of

output (e.g., the time-to-build hypothesis of Kydland and Prescott 1982).

Our research design seeks to improve upon the existing empirical research in several

ways. First, we study economic growth and fluctuations in a unified framework rather than

focusing on only one of these aspects of economic activity. Second, both common shocks to the

trend component and common shocks to the cyclical component may cause output co-

movements among countries, but most researchers focus only on the cyclical component of

common economic fluctuations among countries as defined by detrended data. Our analysis

allows for correlations between innovations to the trend component and innovations to the


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cyclical component across countries enabling us to capture more accurately the factors driving

international co-movements. Third, we allow for different correlations between each country, so

we are able to study pair-wise relationships, but within a model which uses all countries to

identify these relationships. Finally, we do not make any prior assumptions about the existence

of a common world factor and allow a more general correlation structure across countries.

3.1 The Model

The output for each country can be represented as the sum of a trend component and a

cycle component. The trend ( ), also called the permanent component, is the steady-state

level after removing all temporary movements. The cycle (c), also called the transitory

component, embodies all temporary movements and is assumed to be stationary:

countryeachfor7to1, =+= ic y it it it (1)

A random walk for each of the trend components allows for permanent movements in the series.

We also allow for a drift ( ) in the trend:

it it iit ++= 1 (2)

According to Perron and Wada (2005, hereafter PW), including a structural break in the

trend can be very important for proper estimates of the variability of the permanent component.

Following their suggestion of 1973:1, and also the extensive literature on the productivity

slowdown in the G-7 countries at about that time (Ben-David and Papell (1998), Bai, Lumsdaine,

and Stock (1998)), we allow for a structural break in the drift such that we have:

it it iit ++= 11 for t = 1 to 52 (2a)

it it iit ++= 12 for t = 53 to end of sample, (2b)


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where quarter t = 53 is 1973:1. We also allow an additional structural break in the drift term for

Japan in 1990:1, based on the results of likelihood ratio tests. We will discuss this further in the

results section.

Following Sinclair (2005), Morley, Nelson, and Zivot (2003), Clark (1987 and 1989), and

Watson (1986), we model each transitory component as an autoregressive process of order two

(AR(2)): 8

it it iit iit ccc ++= 2211 (3)

We assume the innovations ( it , and it ) are normally distributed random variables with mean

zero and a general covariance matrix (allowing possible correlation between any of the

innovations to the unobserved components).

As shown in Appendix 1, the model can be cast into state-space form. Then it is possible

to use the Kalman filter for maximum likelihood estimation of the parameters and the permanent

and transitory components. 9 The proof of identification for the model is available in Sinclair

(2005).

3.2 The Data

We apply the model of Section 3.1 to GDP data for the G-7 countries, namely Japan,

Italy, Germany, France, Canada, the U.K., and the U.S. The data are quarterly observations on

real GDP from 1960:1 to 2003:4. 10.We are also interested in knowing approximate business

cycle dates for these countries to study the movement of permanent and transitory components

8Univariate specification tests were also performed which suggested that an AR(2) model for each individualcountry would be appropriate. Note that this suggests an ARIMA(2,1,2) for the reduced form for each country.9See chapter 3 of Kim and Nelson (1999) or chapter 4 of Harvey (1993) for a discussion of the implementation ofthe Kalman filter. The estimation was done in GAUSS.10 We thank Christopher Otrok for providing us with the data. He obtained the data from OECD Quarterly NationalAccounts and the IFS.


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over the cycle. For the U.S., we use the NBER classification of the business cycle.

Unfortunately, for other countries there is no generally accepted classification of reference cycles

or classical business cycles. Camacho, Perez-Quiros, and Saiz (2005) employ the widely used

Bry and Boschan (1971) business cycle dating procedure to compute classical business cycle

dates for the OECD economies. 11 In their dating procedure the peaks and troughs for the U.S.

business cycle match NBER recession dates fairly accurately. We use their classification of

reference cycles for Japan and the European economies in our empirical analysis. Shading in

Figures 1 and 2 corresponds to these business cycle dates for each country.

4 The Results

Table 1 and 2 reports the maximum likelihood estimates of our correlated unobserved

components model with a structural break in the drift term in the first quarter of 1973 for all

countries, and an additional structural break for Japan in the first quarter of 1990. Figure 1

presents the estimated permanent component along with the GDP series for each country. Figure

2 presents the estimated transitory component of GDP for each country.

4.1 The Permanent-Transitory Relationships

The correlations between the permanent and transitory innovations are found to be

negative for each country. The diagonal entries in Table 1c range from -0.62 for the UK to -0.92

for France. 12 These findings provide strong confirmation for the MNZ result that shows a

negative correlation between innovations to the two components of GDP for US data. MNZ,

Stock and Watson (1988), and Sinclair (2005) interpret negative correlation between the

11 These authors develop an algorithm that isolates the local minima and maxima in a series, subject to reasonableconstraints on the length and amplitude of expansions and contractions.12 We performed a likelihood-ratio test with the null hypotheses that all the parameters in the estimated correlationmatrix are zero. We reject the null with a p-value of less than 0.001.


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permanent and transitory innovations as arising from real shocks which shift permanent GDP

today, but with slow adjustment of actual GDP to the steady-state level. This implies a negative

contemporaneous correlation since this positive trend shock is associated with a negative shock

to the transitory component of output. Blanchard and Quah (1989) suggest that this adjustment

time arises from supply shocks combined with nominal rigidities, such as imperfectly flexible

prices. Real business cycle theories explain fluctuations as arising from real shocks which

require more than one period for the construction of new productive capital, such as those of

Prescott (1987) and Kydland and Prescott (1982). In these theories, permanent movements are

the main drivers of the economy, with transitory movements in the series arise primarily fromadjustment to permanent changes. We thus interpret the negative correlation between the

permanent and transitory innovations as suggestive of the dominance of real shocks with the

series taking time to adjust. This interpretation is supported by the variable stochastic permanent

component estimated for each of the countries. 13 Another possibility is that since this model

requires transitory movements to be symmetric, the model may be primarily picking up what

happens in expansions (due to significantly more data on expansions). Recessions may still arise

from largely transitory shocks, but the model may not pick up these shocks due to the dominance

of expansions in the data (see Sinclair 2005).

13

There are other theories which can potentially explain the negative correlation between the innovations to the twocomponents of GDP. One example would be the creative destruction hypotheses. According to this view,recessions are times of cleansing when outdated or unprofitable techniques are pruned out of the productive system.A related idea is the pit-stop view of recessions according to which recessions are times when productivityimproving activities are undertaken because of their temporarily low opportunity costs. (See Caballero andHammour 1994). Both of these theories would imply a negative correlation between innovations to the within-seriescomponents. However, the dynamic behavior of our estimated permanent and transitory components duringexpansions and recessions suggest that a large part of the transitory movements in the series arise from adjustmentto permanent changes, rather than vice-versa.


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4.2 The Permanent Component

Since Nelson and Plosser (1982), many empirical macroeconomists have argued that

real GDP contains a nonstationary component. However, non-stationarity in the real GDP series

could be generated by low-frequency, high-magnitude shocks, or by high-frequency, low-

magnitude shocks. If the permanent shocks are occasional large shifts, perhaps ones we can

identify ex-post, then it suggests very different type of macroeconomic fluctuations than the

period-by-period shocks of the random walk with drift. Bradley and Jansen (1995) examine

output data for G-7 countries and find that real GDP for Japan, Italy, and the UK seem to be

characterized by infrequent breaks, such that subsamples are stationary, whereas the remainingcountries are better characterized by period-by-period shocks. In our study, however, we find

that including break dates that we identify ex-post does not remove the random-walk component

for any of the G-7 countries. 14

Our results thus differ significantly from those of Perron and Wada (2005). First of all,

even though we include a break in 1973, we still find variable stochastic permanent components

for all seven countries, including the US. PW instead find that for the univariate US case a

single break in the drift term in 1973 results in an estimate of the permanent component that is

non-stochastic except for a few periods around 1973. However, once we model the countries

jointly in a multivariate framework, we no longer find that the break in the drift term captures all

or even most of the variability in the permanent component. In fact, the standard deviation for

the innovation to the permanent component exceeds the standard deviation for the innovation to

the transitory component for five of the seven countries (Japan, Italy, France, Canada, and the

14 It might be better to use a structural break test such as Zivot and Andrews (1992) or Bai and Perron (1998), butas a first approximation we wanted to consider the large breaks suggested by historical evidence and considered by

previous authors with whom we wish to compare our results.


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US). This lends additional support to our interpretation of permanent shocks driving the G-7

economies.

4.3 The Cross-Country Relationships

We find that correlations between innovations to the permanent and transitory

components are both important in driving international co-movements. Table 1 lists the

estimated correlations between the permanent and transitory innovations across countries. We

should emphasize here that we directly estimate the correlation between the innovations rather

than estimating the components and then computing their correlation as has been done in many

previous papers. We would much rather study the estimate of the correlation than the correlationof the estimates due to concerns about measurement error.

High positive correlation exists between the US and the U.K. and the US and Canada.

The core European countries, notably France and Germany, and France and Italy are also

strongly positively correlated. However, the correlations between the Euro zone countries and

the US or Canada (with the exception of Italy with Canada) are quite small in magnitude. 15 On

the whole, our results appear to support the hypothesis of two relatively coherent economic

clubs: one consisting of the core European countries and another English-speaking group

consisting of U.S., U.K., and Canada (Stock and Watson (2003), Artis and Zhang (1997, 1999),

Helbling and Bayoumi (2003), Luginbuhl and Koopman (2003)). The small magnitude of the

correlation between the Eurozone countries and the US might seem surprising as these

economies share strong bilateral trading and financial linkages. However, the cross-correlation of

their growth rates derived at rolling 5-year windows for the period 1970-2000 are found to be

15 This pattern is confirmed by the analysis of subgroups of countries in our sample. Results for the subgroups areavailable on request.


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stable but low-valued. (0.245 between U.S and the Euro area, McAdam 2003). Canova and

Nicolo (2000) also find that output cycles across the G-7 countries are driven by different

sources of structural disturbances. Even when countries are hit by similar shocks (the oil-price

shocks and the consequent recessions), the effects may vary considerably across countries. For

example, although each G-7 country experienced a recession following the first oil price shock,

the duration of the recession varied widely across these countries (Figure 2). An explanation for

this might lie in significant differences between these economies in the degree of unemployment

persistence, nature of their factor markets, and other institutional features (Leon-Ledesma

(2002), Blanchard and Wolfers (2000)).We also find that Japan is closely related to the European group. The correlation between

permanent innovations is particularly strong between Japan, Germany, and Italy, and to a lesser

degree France. This could be partly explained by economic effects arising from post-war trend

growth common to all these countries. Productivity catch-up in relation to the United States is

recognized as an important ingredient in the post-war growth of the European economies and

Japan (Madison 1991). The productivity data reveal that the countries which grew fastest during

the period 1951-1988 (Japan, Italy, Germany, and Austria) are also those which were most

damaged during the war and which grew fastest during the early reconstruction phase.

Additionally, the productivity gap of these countries with respect to United States was to an

extent caused by war-time destruction. Finally, Germany and Japan are often considered to be

outliers in analyses of productivity convergence across countries 1950-1988.(Smolney 2000).

The U.K. permanent and transitory innovations are found to be closely related only to the

U.S., and to a much lesser degree, Germany. Thus, our results seem to suggest a U.K.

idiosyncrasy (Artis 2003) in the sense that the U.K.s economic experience of shocks is


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growth thereafter (Lincoln 2001). The decade of the 1990s in Japan saw three recessions

followed by a succession of incomplete recoveries. The bursting of the asset price bubble can be

taken as a turning point for the Japanese economy with indicators such as GDP growth and

investment deteriorating sharply after that in the face of rising bad loans of the Japanese financial

institutions. The Nikkei 225 stock price index reached its 38915 peak on the last business day of

1989 and then tumbled to barely above 20000 by 1 st October 1990, declining by almost 50% in 9

months. The first quarter of 1990 thus appears to be a reasonable choice for a break date. The

results are robust to which quarter we pick in 1990.This break date was found to be statistically

significant for Japan, with a likelihood ratio test statistic of 8.868 with only one additional parameter, leading to a p-value of 0.003.

It is often argued that a structural break could have been induced in the German GDP

series by the German unification of 1990 (Hoppner (2001), Brugermann, and Lutkepohl (2004)).

From that time onward, many German series refer to the unified Germany whereas data prior to

the unification often refer to West Germany only because reliable data for East Germany are not

available. Though the date of formal unification was 3 rd October 1990, in the OECD data for

real GDP, figures for unified Germany are reported from the first quarter of 1991.We introduce a

structural break in the drift term in the first quarter of 1991 to account for a possible structural

break. For Germany, however, the likelihood ratio test statistic, after including Japans additional

break, was only 0.612, and was found to be insignificant.

Italy appears to have the least persistent transitory component ( 12+ 22=0.26) and France

appears to have the most persistent transitory component ( 12+ 22=0.84). The remaining

countries have persistence (i.e. the sum of the AR coefficients) ranging from 0.68 for the US to

0.80 for Germany. None of these results appear to be outside the range of previous estimates. It


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should also be noted that the autoregressive process in the transitory component does not have

complex roots for some of the countries in our sample. We may expect the cycle to be

periodic, but there is nothing in the model that requires it. Our estimated transitory component is

simply the stationary part of the data, as identified based on the model presented in Section 3.

The exceptions to the general result that permanent innovations are positively correlated

across countries are: Japan with Canada and the UK, Italy with the UK and the US, and Germany

with Canada. Of these, only Japan with Canada, Italy with the UK, and Germany with Canada

appear significant. Interestingly, these same three pairings are also the significant exceptions to

positive cross-correlations among the transitory components. The negative signs of thesecorrelations look puzzling at first glance. Standard theoretical models of economic interaction

between countries would suggest a positive correlation, as a permanent shock (for example, a

productivity shock) in one country is expected to have a positive impact on the GDP of its

trading partners; both, contemporaneously and in later quarters, though the degree of such

positive association might vary. One possible interpretation of the results is to consider a

preference shock in one country, which shifts demand to non-tradable goods, thereby reducing

the exports of its trading partners. We can also think of a favorable terms of trade shock in one

country with respect to its trading partner which could explain the negative correlation. For the

permanent-transitory cross-correlations, most are negative, but there are many small (but

significant) positive correlations, with the three largest ones again being Japan with Canada, Italy

with U.K., and Germany with Canada.

4.4 Comparing the Results

Table 3 presents our results for the US (from the multivariate model) along with the

results of PW, MNZ, and several other papers that have estimated unobserved components


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models for US GDP. Although the time periods of the data vary across the different papers, we

can still see a pattern emerge. The less restrictive models, particularly the multivariate ones, find

both negative correlation between the permanent and transitory innovations, and also find a

variable permanent component.

Table 4 presents our results for Canada (from the multivariate model) along with

Basisthas (2005) estimates of the MNZ and PW model for Canada. Basistha also estimates a

bivariate model exploiting the Phillips Curve relationship, but his results are not directly

comparable to ours due to the additional structure added to his bivariate model. His bivariate

model does find similar results to ours in that he still finds a fairly variable stochastic trend evenafter accounting for the drift break. He also finds a negative correlation between the innovations

to the permanent and transitory components of Canadian GDP.

4.5 A Possible Interpretation of the Results

If we believe that the US trend is an important driver for trend movements other countries

(as suggested in Papanyan and Juselius 2005), we may wish to see how much a 1% change in US

GDP is reflected in other countries GDP.

A simple model would suggest the following:

t US i i += . (4)

where is an independent normal random variable which captures the imperfect correlation

between the two components and i represents the percentage increase in country is permanent

component that corresponds to a 1% increase in the US permanent component. To determine the

value of i, recall that the model assumes that i and US are jointly normally distributed and t is

an independent normal random variable. This implies that = cov( i ,US )/var( US ). Similarly, if

we want to consider innovations to the transitory components, where we believe that all


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transitory innovations to the US also affect the other countries, but there also exists additional

transitory shocks to the other countries, we could write:

t US i i += . (5)

Table 5 presents these results. It must be noted that it is a strong assumption that all US

innovations, whether permanent or transitory, affect the other G-7 countries. This is an

assumption made only for this brief section and Table 5, but not more generally in the estimates

presented in Tables 1 and 2. Since the standard deviations of the innovations for the permanent

components do not vary widely, there is little difference in the estimates of lambda and the

estimates of the correlations, but they do lead to slightly different interpretations. If our

assumption, that all US permanent innovations affect all other G-7 countries, is correct, then a

1% increase in the US permanent component leads to a 0.24% increase for Japan, a 0.04%

decrease for Italy (although this appears insignificant), a 0.21% increase for Germany, a 0.09%

increase for France, a 0.30% increase for Canada, and a 0.77% increase for the UK. For the

transitory innovations, the results are more distinguishable from the correlations. In this case, a

1% increase in the US transitory component leads to a 0.16% increase for Japan, a 0.31%

increase for Italy, a 0.31% increase for Germany, a 0.08% increase for France, a 1.00% increase

for Canada, and a 1.54% increase for the UK.

5 Conclusions and Extensions

In this paper we estimated a multivariate correlated unobserved components model for

the G-7 countries. The model examines the correlations between permanent innovations and

transitory movements within countries and across countries for this period. We also allow for a

structural break in the first quarter of 1973 in order to address the critique of Perron and Wada

(2005) who suggest that for a univariate model of the US, a structural break in 1973:1 captures


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Appendix 1: State Space Form

We can cast the model into state-space form and apply the Kalman filter for maximum

likelihood estimation of the parameters and the permanent and transitory components. For the

permanent components the Kalman filter requires initial values. This model uses a diffuse prior

for the initial values, but the results are robust to instead estimating them as parameters.

Observation Equation: [ ]=1

0

t

t

t

t

c

c I I y

,

State Equation: ++=

t

t

t

t

t

t

t

t

I

I

cc

I

I

cc

000

0

000

00

00

2

1

1

21

1

where yt , t , c t , , t , and t are each vectors with 7 rows, 1 and 2 are each diagonal 7 x 7

matrices, and the identity and zero matrices are each 7 x 7. 17 Note that when we allow structural

breaks in the drift term, this replaces with 1 for observations 1 to b and then 2 for

observations b to T , where b is the break date.

Variance-Covariance Matrix: [ ] =

t t

t

t E ,

where is the 7 x 7 variance-covariance matrix for the innovations to the permanent

components, is the 7x 7 variance-covariance matrix for the innovations to the transitory

components, and ' = represent the cross-covariance terms between the permanent and

transitory innovations.

17 Note that including a structural break in simply means estimating two state equations for the two subsamples but only allowing to change between the two equations.


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Table 1. Maximum Likelihood Estimates

Correlation Parameters (SE), Likelihood: -595.57

Table 1.a.: Permanent Innovations ( )

Std. Dev. Japan Italy Germany France Canada UK US

Japan 1.22(0.11) 1

Italy1.19

(0.05)0.53

(0.06) 1

Germany 1.15(0.11)

0.79(0.07)

0.13(0.11) 1

France 0.83(0.06)

0.43(0.10)

0.43(0.05)

0.70(0.09) 1

Canada 1.19(0.09)

-0.27(0.06)

0.55(0.08)

-0.45(0.10)

0.14(0.07) 1

UK0.98

(0.15)-0.05

(0.02)-0.43

(0.11)0.27

(0.12)0.13

(0.07)0.07

(0.13) 1

US 1.00(0.07)

0.20(0.07)

-0.03(0.07)

0.18(0.05)

0.11(0.07)

0.25(0.08)

0.79(0.04) 1

Table 1.b.: Transitory Innovations ( )

Std. Dev. Japan Italy Germany France Canada UK US

Japan 1.04(0.10) 1

Italy 0.90(0.08)-0.14

(0.11) 1

Germany1.62

(0.09)0.70

(0.06)-0.12

(0.08) 1

France0.36

(0.06)0.35

(0.09)0.57

(0.06)0.25

(0.08) 1

Canada0.86

(0.10)-0.38

(0.11)0.49

(0.12)-0.34

(0.06)0.23

(0.09) 1

UK 1.11(0.10) 0.24(0.09) -0.28(0.14) 0.26(0.11) -0.14(0.08) 0.19(0.15) 1

US0.55

(0.05)0.08

(0.11)0.19

(0.07)0.11

(0.10)0.12

(0.11)0.63

(0.07)0.76

(0.09) 1


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Table 1.c.: Permanent and Transitory Innovations Cross-Correlations ( )

PERMANENTJapan Italy Germany France Canada UK US

Japan-0.73(0.06)

-0.28(0.10)

-0.67(0.05)

-0.35(0.10)

0.14(0.08)

-0.30(0.08)

-0.34(0.08)

Italy -0.13(0.07)

-0.81(0.03)

0.15(0.09)

-0.34(0.05)

-0.81(0.07)

0.18(0.15)

-0.14(0.10)

Germany -0.65(0.07)

-0.04(0.08)

-0.75(0.04)

-0.32(0.08)

0.31(0.06)

-0.41(0.07)

-0.23(0.06)

France -0.38(0.09)

-0.56(0.04)

-0.53(0.11)

-0.92(0.02)

-0.47(0.06)

-0.28(0.06)

-0.37(0.08)

Canada 0.68(0.07)

-0.04(0.12)

0.59(0.08)

0.02(0.11)

-0.83(0.05)

-0.27(0.13)

-0.18(0.10)

UK 0.15(0.07)

0.43(0.11)

0.08(0.13)

0.26(0.08)

0.04(0.12)

-0.62(0.09)

-0.64(0.09)

TRANSITORY

US 0.28(0.07)

0.17(0.03)

0.28(0.07)

0.20(0.09)

-0.50(0.07)

-0.75(0.11)

-0.79(0.05)

Table 2: Maximum Likelihood Estimates Drift Terms and AR Parameters

Country Drift 1 ( 1i ) Estimate (SE)

Drift 2 ( 2i )| 3 ( 3i )Estimate(s) (SE)

1st AR parameter ( 1t )Estimate (SE)

2nd AR parameter ( 2t ) Estimate (SE)

Japan 2.22(0.17) 0.80

(0.12) 0.58

(0.12) 0.88

(0.08) -0.19(0.07)

Italy 1.26(0.17) 0.57

(0.11) 0.40

(0.08) -0.14(0.04)

Germany 1.01(0.18) 0.50

(0.11) 0.59

(0.06) 0.21

(0.05)

France 1.31(0.17) 0.53

(0.08) 0.27

(0.06) 0.57

(0.07)

Canada 1.36(0.16) 0.74

(0.11) 1.27

(0.03) -0.54(0.04)

UK 0.75(0.14) 0.57

(0.09) 0.75

(0.10) 0.03

(0.11)

US 1.08(0.14) 0.76

(0.09) 1.30

(0.02) -0.62(0.02)


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Table 3: Comparison of GDP Results for the US

Parameter Estimate(SE)PW: UCUR73

(SE)MNZ(SE)

Sinclair 2005(SE)

Morley 2003(SE)

Clark 1987(SE) 18

US 1.00

(0.07)0.104

(0.005)1.2368

(0.1518)1.5689

(0.2334)1.23

(0.15) 0.64

US 0.55(0.05)0.843

(0.010)0.7485

(0.1614)1.1406

(0.3407)0.74

(0.16) 0.74

US US -0.79(0.05)

0.088(0.006)

-0.9063(0.0728)

-0.8666(0.0500)

-0.906(0.072)

Restricted to be 0

1US 1.08

(0.14)0.952

(0.026)0.8156

(0.0865)0.8570

(0.0364)0.81

(0.08) Variable19

2US 0.76

(0.09) 0.66420 N/A N/A N/A Variable 21

1US 1.30

(0.02)1.328

(0.022)1.3419

(0.1456)0.7576

(0.0698)1.34

(0.14) 1.53

2US -0.62(0.02)

-0.418(0.028)

-0.7060(0.0822)

-0.2047(0.1008)

-0.70(0.08) -0.59

Table 4: Comparison of GDP Results for Canada

Parameter Estimate(SE)

Basisthas PW: UCUR73(SE)

Basisthas MNZ(SE)

C 1.19

(0.09)0.05

(0.48)1.49

(0.35)

C 0.86

(0.10)0.74

(0.47)1.96

(0.50)

C C -0.83(0.05)

1.00(0.00)

-0.99(0.01)

1C 1.36

(0.16)0.88

(0.11)1.28

(0.07)

2C 0.74

(0.11) N/A 0.71 22

1C 1.27

(0.03)1.22

(0.19)0.80

(0.25)

2C -0.54(0.04)

-0.28(0.19)

-0.04(0.06)

18 In the original paper, Clark (1987) considers ln(GNP). We have multiplied the results by 100, however the GNPestimates may be somewhat different than the GDP results of the other papers.

19 Clark allows for a variable growth rate for the permanent component, but finds the variance to be 0.000120Perron and Wada estimate the model t t bt T t d ++>+= 1)( and find d = -0.288 (SE: 0.046).21 Clark allows for a variable growth rate for the permanent component, but finds the variance to be 0.0001.22Following Perron and Wada, Basistha estimates the model t t bt T t d ++>+= 1)( and finds d = -0.57(SE: 0.09).


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Table 5: Response to 1% Shock to US Innovation

Table 5.a.: Permanent Innovations

Country Correlationwith US

Japan 0.20 0.24Italy -0.03 -0.04

Germany 0.18 0.21France 0.11 0.09Canada 0.25 0.30

UK 0.79 0.77

Table 5.b.: Transitory Innovations

Country Correlationwith US

Japan 0.08 0.16Italy 0.19 0.31

Germany 0.11 0.31France 0.12 0.08Canada 0.63 1.00

UK 0.76 1.54


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Figure 1 : GDP and the Estimate of the Permanent Component

Panel 1: Japan

640

680

720

760

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60 65 70 75 80 85 90 95 00

LOGGD P100 PERMC OMP

Panel 2: Italy

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60 65 70 75 80 85 90 95 00



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Panel 3: Germany

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Panel 4: France

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LOGGDP100 PERMCOMP


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Panel 5: Canada

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60 65 70 75 80 85 90 95 00

LOGGDP100 PERMCOMP

Panel 6: UK

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Panel 7: US

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940

60 65 70 75 80 85 90 95 00

LOGGDP100 PERMCOMP


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Figure 2: Estimate of the Transitory Component

Panel 1: Japan

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-1

0

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60 65 70 75 80 85 90 95 00

C1

Panel 2: Italy

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C2


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Panel 3: Germany

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Panel 7: US

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C7

international business cycles: an unobserved components approach

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