Breaks in the Variability and Co-Movement of G-7 Economic Growth

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 786 December 2003 Breaks in the Variability and Co-Movement of G-7 Economic Growth Brian M. Doyle and Jon Faust NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/.

Breaks in the Variability and Co-Movement of G-7 Economic Growth Brian M. Doyle and Jon Faust* Abstract: This paper investigates breaks in the variability and co-movement of output, consumption, and investment in the G-7 economies. In contrast with most other papers on co-movement, we test for changes in co-movement allowing for breaks in mean and variance. Despite claims that rising integration among these economies has increased output correlations among them, we find no clear evidence of an increase in correlation of growth rates of output, consumption, or investment. This finding is true even for the United States and Canada, which have seen a tremendous increase in bilateral trade shares, and for the members of the euro area in the G-7. Keywords: international business cycles, economic integration. * Brian Doyle is an Economist and Jon Faust is an Assistant Director in the Division of International Finance of the Federal Reserve Board. We wish to thank Craig Burnside, Dale Henderson, Jim Stock, Eric van Wincoop, Mark Watson, Jonathan Wright, Kei-Mu Yi, and participants at the 2003 ASSA Meetings, the University of Virginia, Arizona State University and the Federal Reserve Board’s International Finance Workshop for helpful comments and Jonathan Halket for excellent research assistance. We would also like to thank Jorgen Elmeskov of the OECD for sending us his data, and David Bowman for his GAUSS code for interpolation. The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System.

1 Introduction Much recent research has focused on changes in the business cycle properties of economic activity in industrialized economies. It is now common wisdom that the standard deviation of U.S. economic growth fell by one-third or more in the mid- 1 1980s. As Doyle and Faust (2002) and many others have noted, growth abroad 2 also seems to have stabilized. While the source of the drop in variance is still unresolved, one important branch of research now focuses on whether cross-country linkages in growth have also shifted, perhaps in a way that can help rationalize the variance reduction. Economic integration has increased markedly since the 1970s, leading some observers to argue that economic growth will be more highly correlated. Speculation 3 on this topic peaked with the nearly simultaneous slowing of growth in the G-7 and other economies at the beginning of 2001. Discussions of the possibility of a 4 new era of more synchronized growth appeared in major news outlets and policy 5 publications. Thecontinuedprogresstoward economic integration inEurope,highlighted by monetary union in 1999, has further fueled these discussions—the major question being whether the euro area is becoming more nearly an optimal currency area. Changes in the standard deviation and correlation of growth are related, of course. Simplybydefinition,withallelseconstant,afallby1/3intheU.S.standard deviation implies a 50 percent increase in the correlation of the United States with every other country. All else was not constant, however. If the growth variation 1 See, for example, Ahmed, et al. (2004), Blanchard and Simon (2001), Kahn, McConnell and Perez-Quiros (2001), Kim and Nelson (1999), Kim, Nelson and Piger (2001), McConnell and Perez-Quiros (2000), Stock and Watson (2002), and Warnock and Warnock (2000). 2 Authors who looked for changes in the volatility of foreign activity include Daalsgaard, Elmeskov and Park (2002), Fritsche and Kouzine (2002), McConnell and Perez-Quiros (1998), Mills and Wang (2000), Simon (2001), and vanDijk, Osborn and Sensier (2002). 3 Canada, France, Germany,Italy,Japan, theUnited Kingdom, and theUnited States. 4 e.g., Washington Post, July 18, 2001, p. A1; the New York Times, August 20, 2001, p. A1, and November25, 2001, p. A12; and theEconomist, August 23, 2001, p. 22-24. 5 See the Fall 2001 and Spring 2002 editions of the International Monetary Fund’s World Economic Outlook, OECD Economic Outlook (December 2001, June 2002), and Doyle and Faust (2002). 1

thatdisappearedintheUnitedStatesandabroadwasidiosyncratic, thencorrelation should have risen. If the growth variation that disappeared was common, then covariance and correlation should have fallen. Thus, summarizing developments in growth variability and covariance could help shed light on the sourceof the changes. Thispapersummarizesthechangesinthevariability ofandco-movement among growth rates of G-7 countries. Various analyses suggest that the G-7 can interestingly be broken into 3 groups: Japan; the euro area countries; and Canada, the UnitedStates andtheUnitedKingdom,whichwerefertoastheEnglish(-speaking) 6 countries. Of course, Japan has behaved very differently than the other 6 nations in the 1990s. We set Japan aside for most of the analysis and focus on the G-6 and the euro and English subgroups. Our analysis differs from many others in two ways. Like some earlier work, we consider consumption and investment growth as well as GDP growth. This consideration potentially allows us to distinguish among some theories of the source of changes. What is most unique about our paper, however, is that we focus on formal tests for changes in various measures of co-movement. Up to now, there has been much work testing for breaks in variance, but the work on co-movement has been almostexclusivelydescriptiveofchanges, withoutformaltestsofsignificance. Virtually all existing work we are aware of either reports point estimates of co-movement across different sub-samples or estimates time-varying parameter models without 7 any test of the hypothesis of no change. While as a general rule, inference tools should be preferred to descriptive statistics, we demonstrate that there are particular reasons for care in interpreting point estimates without confidence measures in the current context. 6 That is, English is one of the official languages in thesecountries. 7 While much of this work is still in draft form and still changing, among the work falling into this category are Stock and Watson (2003a, 2003b), Bayoumi and Helbling (2003), Kose, Otrok and Whiteman (2003a, 2003b), Luginbuhl and Koopman (2003), Del Negro and Otrok (2003), HeathcoteandPerri(2003),Kose,PrasadandTerrones(2002),Monfortetal(2002),Carvalhoand Harvey(2002),Dalsgaardetal. (2002),AngeloniandDedola(1999)andArtisetal. (1997). Bordo andHelbling(2003)donotconsiderchangesafter1973whichisthemainfocusofthispaper. Stock and Watson (2003a) include one test that arguably sheds some light on the emergence of a higher co-movement among euro-area countries. 2

Weconsiderdatafrom1960totheendof2002andfocusmainlyonathree-break case, where the breaks are found to fall in the early 1970s, 1980s, and 1990s. Thus, the 4 regimes correspond roughly to the decades and we talk about them in that way. Our main conclusions are these: 1 The 1960s were generally a period of low volatility relative to the 1970s and low co-movement relative to later periods more generally. 2 Volatility of outputand consumption growth fell broadly over the periodsince about 1980. This GDP result is now standard in the literature. 3 There is no evidence of a change in co-movement between any two periods after the 1960s. This lack of evidence is not simply a case of point estimates favoring a rise, but standard errors being large. One could probably make the case for a fall in co-movement as easily as for a rise. We note three details of the final claim: (i) There is no evidence of a rise in correlation among the euro area countries in the sample (France, Germany, Italy) or among the English group (Canada, U.S., U.K.). (ii) From a welfare standpoint, one benefit of global integration might be better consumption risk sharing. We find virtually no statistically significant evidence of an increase in consumption growth correlation that is sometimes assumed to be reflective of better risk sharing. (iii) DespiteahugeincreaseintradebetweentheU.S.andCanadaaftertheCanada-U.S. Free Trade Agreement, we see no increase in correlation between the United States and Canada. Section 2 provides some background; section 3 gives some descriptive evidence illustrating our major points. Section 4 provides the formal inference that is our primary unique contribution. Section 5 has some robustness checks and section 6 concludes. 3

2 Background on co-movement of growth The average correlation of growth between pairs of G-7 economies from 1960Q2 to 2002Q4 is moderately positive at 0.24. The highest correlation is 0.48, between both France and Italy and Canada and the United States. The lowest correlation is between Italy and the United Kingdom at 0.07. This section presents some background that may help interpret both the level of correlation across countries and changes in those correlations. 2.1 A simple framework We start with a very simple exercise to illustrate the accounting relations among variance,covariance, andcorrelation. Home(h)andforeign(f)growtharedrivenby common shocks, ε c, that directly affect both countries, and by idiosyncratic shocks, ε h and ε f, that directly affect only one country. Writing growth as y, we have, y h = ε h+ε c +γy f y f = ε f +ε c+γy h For simplicity, the countries are treated symmetrically. We include the foreign country’s growth in the equation determining home country growth to summarize how linkages may transfer idiosyncratic shocks across borders. We focus on the case with 0 < γ < 1, with each idiosyncratic shock having a variance σ2 and with the x common shock having a variance σ2 . c The following facts are easy to confirm. The existence of common shocks and the cross-border effects of idiosyncratic shocks imply that the growth rates will be positively correlated. A decrease in the variance of the common shock (σ2 ) will c decrease the variance in each economy as well as decrease the covariance. Because covariance decreases proportionally more than variance, correlation also falls. Thus, if we explain a decrease in variance by a lower variance of common shocks, in this framework we would also see both lower covariance and lower correlation. 4

Afallinthevarianceof bothcountries’ idiosyncraticshocks reducesthevariance of each economy and again reduces covariance through spillovers. In this case, however, growth correlation rises. Intuitively, correlation is the share of variation that is shared in common. The fall in idiosyncratic variation reduces the total variation but increases the share of variation that is common. Finally, making linkages stronger through increasing γ, holding σ2 and σ2 conx c stant, raises variance, covariance, and correlation. Thus, a fall in the variance of eithercountryinthiscasemustbeduetoasmallerγ andwillhaveanassociated fall in covariance and correlation. Theproponents of theview that risingeconomic integration has increased correlation have a richer set of channels in mind. We provide a brief review of the theory evidence regarding increased linkages and co-movement. 2.2 Some evidence on increased economic linkages Therehave been substantial increases in trade and financial linkages among the G-7 countries in the last several decades. Each G-7 country, except Japan, has shown an increase in merchandise trade (exports plus imports) with its G-7 partners over the period since 1960 (figure 1). As a percentage of its own GDP, Canada’s trade with its G-7 partners almost tripled from just over 20 percent to more than 60 percent, 8 with much of therise coming after theCanada-U.S. Free Trade Agreement in 1989. The U.S. share rose from about 3 percent to about 9 percent over the period, and shares for each of the European G-7 nations have about doubled, reaching about 20 percent. Financial integration has also increased. For example, the share of foreign equities in U.S. equity portfolios rose from less than 2 percent in the early 1980s to almost 12 percent in 2001 (figure 2). The share of U.S. equities in foreign equity portfolios also rose markedly over the period. Other measures of international fi- 9 nancial market integration show a similar pattern of increase. 8 Most of Canada’s trade with the rest of the G-7 is with the United States (rising from 17 percent of GDPin 1960 to 56 percent in 2000). 9 See IMF, World Economic Outlook, October 2001. Chinn and Forbes (2003) also look at 5

2.3 The relationship between integration and co-movement Even in the simplestcases, theory makes remarkably few predictions aboutthe relationbetweenintegration andco-movement. Theeasiestcasetoanalyzeisthatoftwo economies moving from partial integration to a completely integrated, fully efficient equilibrium. Upon moving to complete integration, intertemporal marginal rates of substitution are, by definition, equalized. With additional strong assumptions such as a single good and log utility, complete integration implies that consumption growth will be perfectly correlated. Thus, the increase in integration raises consumption correlation. Weakening any of the assumptions can overturn this result. Ifwemovetogreater, butstillincomplete, integration, consumptioncorrelationmay increase or decrease. Further, the fact that consumption is an aggregate of many types of items, including durables, is problematic. There seems to be a presumption, however, that consumption correlation would increase across countries with increased integration. As for output and investment, even when moving to fully efficient integration, there are factors working to increase and to decrease correlation. Under autarky, outputand investment decisions areintimately linked with consumption smoothing. With integration, the trade and asset flows that facilitate complete consumption insurance allow output and investment decisions in the individual economies to be substantially delinked from current consumption decisions. Thus, for example, a country experiencingapositive country-specificproductivity shock canborrowfrom abroad—immediately raising consumption and investment by more than would be efficient under autarky. This borrowing can magnify the effect of the differential productivity shocks, decreasing output and investment correlations. Whether integration raises or lowers correlation through this channel can depend on whether there is horizontal or vertical integration of production (Kose and Yi (2001, 2002)). If the model allows a role for demand shocks, increased integration can increase the change in financial linkages over time. While some argue that equity returns have become more correlated across nations (OECD Economic Outlook, June 2002), others (e.g., Brooks and Del Negro, 2002) question thisresult. 6

output, consumption and investment correlations as demand shocks in one country fall partly on imported goods and are, thereby, transmitted to others. Some longerterm implications of integration may lower correlation. For example, integration may lead to specialization of production along the lines of comparative advantage. If there are asymmetric shocks by specialty, output and investment correlation can 10 onceagaindecreaseundergreaterintegration. Acommonintuitionisthatfinancial integrationwillraisecorrelations,butHeathcoteandPerri(2002)showthatfinancial 11 autarky can generate higher output correlations through terms of trade effects. In principle, one could build a dynamic stochastic general equilibrium (DSGE) model to sort out the relative magnitudes of these various effects, and there has been much progress in this regard, as evidenced by some of the articles cited above. There is reason to doubt, however, whether we have yet specified these models with sufficientdetail toresolve theempirical issue. DSGE modelswith flexibleprices and complete markets have difficulty generating anything close to the positive output 12 correlation found in the data. Even with nominal rigidities, which allow a role for demand shocks, the standard Mundell-Fleming model has a negative output correlation in responseto monetary policy shocks. Kollmann (2001) shows that in a new Keynesian open economy model one can generate positive output correlations 13 in response to both productivity and monetary shocks. Although theory does not resolve whether stronger links increase co-movement, empirical studies suggest that in the limit, at least, integration raises output correlation. Growth correlations of regions within a country are generally higher than thecorrelations of similarly situated regions across national boundaries. Since trade 10 Paul Krugman (1993) develops this argument. Kalemli-Ozcan, Sorensen and Yosha (2001) show that U.S. States and OECD countries with a more specialized production structure have output that is less correlated with other states or countries. Kalemli-Ozcan, Sorensen and Yosha (2003)provideevidencethatmorerisksharingleadstogreaterindustrialspecialization bothacross regions and across countries. 11 For other examples of ambiguous effects related to increased capital mobility, see Frankel (1988). 12 SeeBackus, Kehoe and Kydland(1992) and the surveyby Baxter (1995). 13 Whetherhematchesthedatadependson hisassumptionsabout theelasticity ofsubstitution between home and foreign goods. 7

and financial links are usually higher within countries, these studies suggest that 14 more integration raises correlation. Of course, the G-7 has not become fully integrated, and economic integration may not have risen enough over the past several decades to lead to a detectable change in correlation. We know of no clear evidence that changes of the magnitude 15 we have observed should significantly raise the correlation among G-7 economies. In light of this prior work, the goal of the remainder of the paper is to document any changes that have occurred. 3 Descriptive evidence In this section, we present a graphical summary of changes in the variability and 16 co-movement ofG-7growth. Figure3showsthefour-quarterGDP growthrates of the G-7 economies and a simple average of the growth rates in two sub-regions—the English group and the euro group. U.S. recessions are shaded. Japan and the euro economies both seem to have a slight downward trend in their growth rates. One can also see the overall positive correlation, with the movements being most similar around the time of U.S. recessions. It is also possible to see the much-discussed decline in the standard deviation growth in the United States and elsewhere. Figure 4 summarizes changes in the standard deviation of growth in the G-7. Each point on the figure shows the sample standard deviation of quarterly growth measured over the previous 5 years (20 quarters). The standard deviation for the English and euro panels is a simple average of growth rates in the subgroup. ThedeclineofthestandarddeviationfortheUnitedStates (dashed)isdramatic, 14 Thisresultseemstoholdwhencontrollingforsuchasfactorsassizeoftheeconomies,distance between the areas compared, and policy differences. See, for example, Bayoumi and Eichengreen (1993), and Clark and van Wincoop (2001). 15 Most estimates of the effect of small increases in trade intensity on output correlation are similarly small. See Canova and Dellas (1993), Frankel and Rose (1998), Anderson, Kwark and Vahid (1999), Imbs (1999, 2003), Clark and van Wincoop (2001), Otto, Voss and Willard (2001), Gruben,KooandMillis(2002)andCalderon,ChongandStein(2003). Calderon,ChongandStein find even smaller effects between developing and advanced economies and smaller again among developing economies. 16 Details about thedata are in AppendixA1. 8

fallingbymorethan1/3inarelatively shortperiodintheearly1980s. Thestandard deviation hasfallen moregenerally, withtheexception ofJapan, wherethestandard deviation rosemarkedlyalongwiththeeconomicproblemsof the1990s. Thedecline in France is quite small after the 1970s and Germany’s decline may have occurred 17 later than the others. Incontastwiththestandarddeviations,moving5-yearcorrelationsshownoclear change over the period (figure 5). Estimated correlations between G-7 economies fluctuatewidelyover thebusinesscycle, tendingtoreachpeaksafterU.S.recessions. The moving correlations show no clear break in behavior within the English group (panel A), the euro group (panel B), or elsewhere (panel C). Japan is again an exception: the correlation of Japan with the rest of the G-7 has fallen sharply and turned negative in the 1990s. The high levels of correlation at the end of our sample are consistent with the historical pattern of high correlation following U.S. recessions. Figure 6 presents the moving correlations for consumption growth. Once again, there are no obvious changes to be observed. With standard deviations falling and correlations either showing no clear change or falling, measured covariances must have fallen in the 1980s (figure 7). Our main focus is co-movement, and we can see that there is no dramatic break in correlation despite the aforementioned dramatic increased in trade and financial linkages across countries. Remember that Canada-G-7 trade tripled from 20 to 60 percentofGDPoverthesample. Similarly,theeuroareaeconomiesshownoincrease in correlation; indeed, Germany’s correlation with France shows a steady decline since the early 1970s. Even without formal testing, we can conclude that dramatic rises in certain integration measures have not led to similarly dramatic changes in correlation. The next section presents formal evidence on these questions. 17OurconclusionsconcerningGermanyshouldbetakenwithcautionbecauseofthemeasurement issues surrounding German reunification. See data appendix for an explanation of how we treat German reunification. 9

4 Formal Inference Inthissectionwepresentformalinferencesaboutchangesinthetimeseriesprocesses for GDP, consumption and investment. We drop Japan from the analysis and study only the rest of the G-7 (G-7x). While there are clearly breaks in the behavior of the Japanese aggregates and in their relation with the rest of the G-7, we believe that these are related to the special problems Japan has experienced. If dropping Japan biases our analysis, it biases it in favor of finding increased correlation. We present no new evidence about whether there exist breaks in the processes we study. The existence of breaks has been well-documented in earlier work, which we review briefly below. Given the existence of breaks, there are many interesting hypotheses regarding which features of the processes changed and which, if any, remained constant. For example, as noted above, there is strong evidence that the variance of output across the G-7x declined. As a result, correlation or covariance must have changed, and it would be of interest to know whether one of these stayed more or less the same. Similarly, as Ahmed et. al (2004) point out, certain explanations of the fall in variance suggest that most of the reduction would have come at business cycle frequencies, while in other explanations the reduction would have come evenly across all frequencies. In this latter case, the shape of the spectrum would remain unchanged across sub-samples. As noted in the introduction, most papers addressing co-movement have pre- 18 senteddescriptiveevidencelikethatintheprevioussectionwithoutformalinference. In the most common approach, the papers first establish in some way that breaks have occured. For example, they use appropriate techniques to establish and date the break in the variance of growth. Having chosen break dates, the papers go on to report point estimates of various parameters such as covariances or correlations before and after the breaks. Differences in these point estimates are then interpreted as evidence of changes in those parameters. An alternative approach is to estimate a time-varying parameter model for the relevant data and simply report 18 Seefootnote 7. 10

the estimates without any test of the null of no change in the relevant parameter. Of course, when we observe changes in parameter estimates it is important to ask whether those differences are large relative to what we might expect to see if no change had occured in the underlying parameter. In the next section, we show that bypassing inference may be especially risky in the current context. 4.1 Detecting changes in variance, covariance, and correlation Inference about breaks in correlation and covariance in the current context is fundamentally less precise than inference regarding breaks in variance. For example, if the covariance of GDP growth between the United States and the United Kingdom remained constant as variance fell in the early 1980s, it would be quite likely that the point estimates of covariance from the two sub-samples would differ greatly. We illustrate the generic point with a simple example. Suppose we have two samples, each drawn independently and identically from a bi-variate normal distribution with mean zero. We want to detect whether the variance-covariance matrices, Σi ,i = 1,2, differ across the samples. Suppose in fact thatΣ2 = αΣ1 whereα < 1,sothatthevariancesandcovariances fellproportionally while the correlation stayed fixed. One generally has more power to detect the change in variance than the change in covariance. One way to see this is to study limiting cases. If the correlation between the series is 1, then testing for a change in covariance and in variance is identical (as the variance and covariance are exactly the same) and equal power is attained with the two tests. If the correlation is zero, then there is no change in covariance between the two samples and, hence, no change can be detected. Thus, thepower of thecovariance testto detect α< 1 relative tothe power of thevariance test is zero. If the correlation of the two series is small, but positive, the variance and the covariancebothchangebythesamefactorofproportionality,butthepowertodetect the change in covariance relative to the power to detect the change in variance may 11

be quite small. Moreformally,asweconsidercorrelationsgoingfromonetozerointhisproblem, the relative power to detect the covariance change versus the variance change also goes fromoneto zero. For moderatelevels ofcorrelation, suchas theaverage output correlationof0.24amongG-7GDPgrowthoveroursample,therelativepowerofthe covariance test may be quite small. For example, in the simple example considered here, with correlation of 0.24 the Appendix shows it would take about 5 times as many observations in the covariance test to attain equal power as one has in the variance test. The bottom line is that strong statistical evidence of breaks in the variance of processes does not provide a sound basis for concluding that similarly large changes in the point estimates of covariance or correlation are also likely to be statistically significant. 4.2 Description of the inference procedures We consider breaks in the processes for GDP, consumption, and investment separately. The data are stated as annualized quarterly growth rates (400 times the logarithmic quarterly change). Details regarding the data, including the treatment ofsomeoutlierquartersand,inparticular,GermanunificationaregiveninAppendix A1. Forconcreteness,considertheGDPcase. Weassumethatthetime-seriesprocess for GDP growth of the G-7x can be adequately approximated by a vector autoregression with one lag and a constant, whereall the parameters (constant, slope, and shock variance-covariance) are allowed to break at a fixed number of dates. Allowing B breaks at the dates τ = {τ 1 ,...,τ B }, the parameters are θ = {θ 1 ,...,θ B+1 }, where θ i is all the parameters of the VAR process in the ith subsample. The key unique assumption we make is that we know the number of breaks, but not their dates. If we know the number of breaks, Bai (2000) suggests estimating the parameters by maximizing the Gaussian pseudo-likelihood for τ and θ. More 12

specifically, for any τ and θ, the log-likelihood is given by, B(cid:1)+1 L = L(Xi|θ i) i=1 where Xi is the data for the ith subsample when the observations are partitioned according to τ and L is the conventional Gaussian log likelihood given the stated data and parameter arguments. 19 We estimate τ and θ by maximizing L over all unique partitions τ and parameters θ. 20 Bai (2000) shows that so long as our conditioning assumption is correct, the asymptotic distribution of θˆ is the same as if the break dates were known and imposed a priori. 21 In particular, θˆ is asymptotically normal, θˆ i is uncorrelated with θˆ j if i(cid:1)= j and the variance-covariance matrix of the elements of θ i is just as if we were estimating a single VAR for the relevant subsample. We are assuming that some features of the VAR broke, and want to test that others remained constant. All the features in which we are interested—e.g. unconditional variances and covariances—can be written as scalar functions of the VAR parmeters, G(θ). We want to test H 0 :G(θ i) = G(θ j) i (cid:1)= j for various G. 22 Given that θ i and θ j are jointly asymptotically normal and the variance-covariancematrixoftheasymptoticdistributionisknown,anyconventional testing approach may do. The simplest approach would be to form a confidence 19 In practice we use theconditional likelihood, conditional on initial lags in thefirst segment. 20 We require that at least 20 percent of the sample lies between any two breaks or break and endpoint. 21 This point may seem to go against the intuition from the endogenous breakpoint literature. Thatliteratureisconcernedwithtestingforexistenceofabreak. Searchingoverbreakpointsaffects the asymptotic distribution of tests for the existence of a break. If the maintained hypothesis containsafixednumberofbreaks,searchingoverdatesdoesnot affecttheasymptoticdistribution of parameter estimates in this case. The reason for this stems from the fact that the estimated breaktimes(viewedasashareofthesample)convergemorerapidlythanthecoefficientestimates. Notethattheasymptotictheoryinvolvesthesizeofthechangeinthecoefficientsgoingtozero(at therightrate)withthesamplesize. Despitetheshrinkingbreak,thebreaktimesconvergerapidly. 22 This works for any H 0 of this form that is consistent with the conditioning assumption that something in θ changed. 13

interval for ∆ij = G(θ i)−G(θ j) as ∆ˆ plus and minus twice its asymptotic standard error computed, say, using the delta method. Some of the G(θ)s we are interested in are correlations, however, and conven- 23 tional asymptotic approaches—even standard bootstrap approaches are known to perform poorly in such cases. Thus, we compute confidence intervals for ∆ij using what in Hall’s (1992) language is an iterated other percentile bootstrap. (See the Appendix for details.) Now we discuss some strengths and weaknesses of our procedure. We consider separate systems for GDP, consumption, and investment in part to keep the size of our VARs small. Parsimony issues would become more important if we were to estimate an 18 variable VAR. Furthermore, theory gives reasons why economic integrationmightraiseconsumptioncorrelationwhiledecreasinginvestmentcorrelation. Thus, we opted to run separate systems for the 3 different aggregates. The procedures condition on breaks existing and investigate what, if anything, remained constant across sub-samples. An important choice, then, is the number of breaks. We perform the analysis for 1, 2 and 3 breaks, but report only the 3break case. We prefer the 3-break case for the following reasons. McConnell and Perez-Quiros (2000) and others give a strong case for a break in the variability of U.S. output growth in the early 1980s. Bai, Lumsdaine and Stock (1998) make a strong case for a break in growth around 1970. Finally tests for break dates in bivariate systems that include either Canada or Germany tend to find dates in the 24 early 1990s. We review below which conclusions are sensitive to the number of breaks. Finally, the only type of parameter change we consider is a small number of discrete breaks. Others have found evidence for other types of time variation in parameters (such as Stock and Watson (2003a) and Luginbuhl and Koopman (2003)). We believe that the discrete break framework remains an important baseline in this 23 Forexample, see Hall (1992) on theweakness of thepercentile-t bootstrap in thiscontext. 24 These break dates may be a result of reunification, in the case of Germany—despite our solutionforthejumpinGermangrowthrates(seedataappendix)—andperhapsinpartinfluenced by theCanada-U.S. Free Trade Agreement in thecase of Canada. 14

literature. This case still dominates discussion and we believe our results shed useful light on earlier results in the area. Further work on both approaches is surely warranted. 4.3 Results for the mean and variance Point estimates of the breaks. With 3 breaks, there are 4 subperiods. Our inference approach allows the data to pick the break dates. Since we estimate a different VAR for GDP, consumption, and investment, the chosen break dates differ across the three systems. In practice, there is little variation, and the 4 periods chosen correspond roughly to the 1960s, 1970s, 1980s, and 1990s (table 1). We will use these decade labels to describe the periods, but one should take care to remember 25 that the actual break dates do not fall exactly on the decade boundaries. Mean growth. We begin with a summary of the statistical significance and direction of breaks in mean growth. Table 2 reports whether the confidence interval for the change in mean between any two periods covers zero. If the 95 percent confidence interval lies entirely below zero, the table has a bold “D” for down; if the 90 percent confidence interval (but not the 95) is entirely below zero, there is a (plain) “D”; otherwise if the point estimate is below zero there is a ‘d’. For the analogous cases above zero, there is the appropriate “U”, “U”, or “u” entry for up. In all cases, we report the value in the later sub-sample minus the earlier, so up means the coefficient rose through time. The rows of the table labelled “All”, “Eng” and “Eur” report the test for a change in the average value of the parameter across the economies in the group. Onenotable feature of theresults is that mean growth of GDP andconsumption fellverygenerally between the1960s andboththe1970s and1980s. Thisresultisno surpise—the 1960s were a period of unusually strong growth; the 1970s and 1980s were not as strong. Some idea of the economic magnitude of the changes can be 25 Wedonotmeantoimplythatnothingturnsonwhetherwetakethebreakdatestobeexactly ondecadechangesorshiftedslightlyasinourprocedure. Someconclusionsmaybesensitive. This is because the last 3 U.S. recessions happen to fall near decade changes and shifting slightly can change which subsample certain recessions fall in. 15

gained by considering the point estimates of mean growth in the subperiods (table 3). A second important feature is the difference between the English and euro groups. There are no significant changes in mean growth in the English group between periods after the 1960s, and few for the individual countries in the group. In contrast there are many significant reductions through time in both GDP and consumption growth in the euro group. Consumption growth fell significantly between every pair of periods except the 1980s to 1990s in the euro group. Since our main concern is with second moments, it is worth noting that these changes inthemeancouldhaveimportantimplications forempiricalworkonsecond moments which does not allow for these mean breaks. Allowing for, say, only one mean break over the period could lead to spurious results regarding the variance in euro area countries. Standard Deviations. The tests for changes in standard deviations confirm the familiar results from the literature and the graphical evidence seen earlier (table 4, upper panel). The unconditional standard deviation of GDP and consumption growth perhaps grew from the 1960s to the 1970s, but fell very generally after that. In the point estimates, the standard deviation of growth of GDP fell on average by about one-half and that of consumption fell by about one-third. The results for investment are more mixed, except for the case of the United States where the standard deviation fell signifcantly. Generally, investment growth ismorevolatilethanconsumptionandGDPgrowth;thisvolatility reducesthepower to detect any sort of change, so we should expect not to find signficant changes in investment throughout the results. 26 Theconditionalstandarddeviation followed aboutthesamepattern astheunconditional variance (table 4, lower panel). The similarity of the unconditional and conditional standard deviation results suggest thepossibility that only the variancecovariance matrix of the shocks changed, leaving the slope coeficients of the VAR 26 Theconditionalstandarddeviation isthestandarddeviationoftheone-stepahead prediction error, or equivalently,thestandard deviation of the VARreduced form error. 16

27 unchanged. AnanalogousresulthasbeendocumentedbeforefortheUnitedStates 28 and other countries. We shed additional light on this question by asking whether there have been any changes in the shares of variance attributable to cycles in three regions of the frequencydomain: businesscyclefrequencies(8to32quarters)andhigherandlower frequencies. Table 5 shows tests of whether the share of variance in each of the 3 frequency bins changed. If the change in unconditional variance came exclusively from a change in the shock variance-covariance matrix, then the share of variance in each frequency bin would be unchanged. Except for comparisons with the 1960s in the U.K. and Canada, there are very few significant entries. There is little evidence that the reduction in unconditional variance was focused on one frequency bin more than others. These results are quite consistent with existing results in the literature. We now turn to the results for co-movement, an area where few testing results have been reported. 4.4 Results for co-movement For the correlation and covariance measures (tables 6 through 9), the all, English, and euro values represent the mean of the off-diagonal elements of the relevant covariance or correlation matrix. The English-euro measure is the mean of the offdiagonal elements of the matrix correspondingto elements with one country in each subgroup. From the group estimates for GDP, it appears that unconditional correlations have risen quite generally between the 1960s and any subsequent period. The result is similar, although somewhat less uniform for consumption. After the 1960s, there are no significant changes in GDP for the all, the English, or the euro groups and the point estimates are mixed. The average correlation between the English and euro group countries fell signficantly between the 1970s and 1980s, but shows no signifcant change thereafter. The statistically significant changes in consumption 27 Ahmedet al. (2004). 28 Stock and Watson (2003a). 17

after the 1960s in the country pairs are about equally split between ups and downs and are not clearly differentiated by sub-group. This result casts some doubt on the claim that euro and English groups have emerged with each group having strong internal co-movement but with correlation between the groups falling. The one bit of statistically significant evidence in line with thistheory is that theEnglish-eurocorrelation fellfrom the1970s to the1980s. The decline in the point estimate is from 0.38 to 0.16, about the same magnitude as the fall in correlation within the euro group itself, 0.50 to 0.39, however. The U.S.-Canada correlation shows no significant change during the period in which the trade share was growing sharply. The point estimate of the correlation change in GDP was down between the 1970s and 1980s and between the 1980s and 1990s. The conditional correlations show somewhat more evidence for a fall in correlation between periods after the 1960s. Several of the bi-variate correlations show significant declines in both GDP and consumption correlation. Interestingly, from a standpoint of consumption risk sharing, the average conditional correlation among all 6 nations fell in the point estimates from the 1970s to the 1980s and from the 1980s to the 1990s. These changes are not statistically significant, however. Given the strong evidence of declines in variance and little clear evidence of change in correlation, one might hope to find statistically significant evidence of declines in covariance. As we noted above, however, the data may simply not reveal whether correlation has risen or covariance has fallen. There is somewhat more evidence of declines in covariance for periods after the 1960s than there was for increases in correlation. By crude count considering GDP growth,thereare18significantdowns(table6)afterthe1960sincovariance,whereas therearenocomparablesignificantupsincorrelation (table8). Weknowthateither correlation, covariance or both changed as a result of the variance changes. As the statistical theory suggested, however, the evidence in favor of the proposition that eithercorrelationorcovariancechangedismuchstrongerthantheevidenceresolving 18

which one it was that changed. 5 Robustness Inthis section wereview some additional resultsthat shedlight on therobustnessof the3central conclusionslistedintheintroduction. Inshort,the1960s wereaperiod of low variance and correlation, variance fell after 1980s, and there is no significant evidence of a rise in co-movement in general, in the subgroups, in consumption, or between the U.S. and Canada. 5.1 Alternative correlation measures Since co-movement is our primary interest in this topic, we consider additionally 3 types of factor-model-based measures of co-movement. Each of our three types of measures is a standard “atheoretical” way to summarize how similarly variables move together. Our first measure, λ(N), is the sum of the largest N eigenvalues of any correlation matrix. This measure has a standard factor-theoretic or principal components interpretation. We want to choose factors f t where f t is N ×1, t = 1...T, and the factors can be serially correlated. We choose these factors to maximize the sum of the R2 s for regressions of the form, y jt = α j +β j (cid:2)f t+ε t j = 1,...,k where k is the number of countries in the group. In this sense, these are the N < k factors that best explain our k variables. If we choose the factors in this way, then λ(N)isthesumoftheR2 s. Ifλ(N)increases,thenthemovements ofourk variables can be better explained by our N < k factors. Thesecondmeasureismoredynamicandsomewhatmorefamiliarfromidentified VARwork. InanyVARsystem,oncetheshocksareorthogonalized,wecancalculate the variance share of any variable that can be attributed to any shock. Following Faust (1998), we can calculate the maximum variance share, summing shares across 19

allvariables,thatcouldbeattributedtoanyN orthogonalized shocks. Thismeasure can also be seen as a sum of R2 s. In particular, it maximizes the sum of R2 s across the k regressions, y jt = α j +β j(L) (cid:2)f t+ε t j = 1,...,k where β j(L) is a one-sided lag polynomial and f t is serially uncorrelated and lies in the space spanned by the shocks of the VAR. Our third measure is a fully dynamic version of the first. In this case, we only consider a single factor for computational ease. This factor is designed to maximize the sum of R2 s from the k regressions, y jt = α j +β j(L) (cid:2)f t+ε t where now β(L) is a two-sided filter. Our measure is the sum of R2 that maximizes this expression over all f. Brillinger (1981) discusses the calculation of this 29 quantity. Table 10 gives evidence on the breaks in these measures. The evidence is consistent with the earlier evidence for correlations: there is some evidence of an increase in correlation from the 1960s to later periods and very little evidence of further change thereafter. 5.2 Alternative versions of the results SofarwehavereportedresultsforVARswith3breaksestimated onrawgrowthrate data. We repeat all of this work for 12 total versions of the system. In particular, we consider conditioning on 1, 2, and 3 breaks, using per capita versions of all the variables, and using Hodrick-Prescott filtered versions of the variables. In the end, by considering all combinations of these options, we arrive at 12 sets of results. A 30 complete setofresults(about400pages) isavailable fromtheauthors. Thereader 29 While the first measure is given by eigenvalues of a correlation matrix, this measure is given by theintegral across thespectrum of themaximum eigenvalue of a coherence matrix. 30 Seehttp://patriot.net/∼faustj/jon or http://www.geocities.com/brian m doyle/yctabs.pdf. 20

will be relieved to know that we will not attempt to give a detailed account of all these results. We focus on the three broad conclusions stated in the introduction. The first conclusion is that systems with only one break give somewhat different results. As we have seen above, the changes from the 1960s to later periods often are more significant and different in character than the changes between any two later periods. Systems with one break cannot accommodate both the early-1970s changeandtheearly-1980s breakandgivesomewhatdifferentpicture. Solongaswe allow for two breaks (which the procedure places in the mid-1970s and mid-1980s), the general results come through: correlation rose after the 60s, but showed little change between subsequent periods. The per capita and Hodrick-Prescott filtered versions of the variables generally lead to the same break dates listed in table 1. The tables for per capita growth rates are virtually identical to the results reported above—population moves slowly and does not affect the general conclusions about either variability or co-movement. The tests on the HP filtered data generally show even weaker evidence of changes in co-movement after the 1960s. 6 Conclusions We find that the reduction in growth variation that has been documented for the United States seems to be present in almost all of the other G-7 countries. The exception is Japan, which in the 1990s is anomalous to most macroeconomic generalities regarding the G-7. Thereisnoclearevidencethatcorrelation hasincreasedwiththerisingeconomic integration over the sample period. In general, we cannot reject the hypothesis of no change in correlation. This conclusion holds even for Canada and the United States, which has seen a substantial increase in trade, and for the included euro area countries—Germany, France, and Italy. The result also holds for consumption growth rates, despite the thought that greater integration should lead to greater consumption insurance. 21

This result contrasts with some earlier claims in the literature. For the most partthose claims rest on changes in pointestimates with noattempt to do inference about whether the changes are statistically significant. We provide a theory-based reason to doubt whether that approach is reliable. Our formal tests suggest that the changes are not statistically significant. Appendix A1. Data We use quarterly real GDP, consumption and investment data from the first quarter of 1960 until the final quarter of 2002. For each country we use official nationalseriesasreportedbyHaver Analyticsfromthestartingpointoftherelevant series to the end of the sample. In cases where the current vintage of national 31 accounts data do not extend to 1960, we splice data from an older vintage official series as specified below. To splice the data, we use the quarterly growth rates from the earlier data along with the first level in the recent data to construct a new level series extending back to 1960Q1. We handle German reunification by taking the quarterly growth rates of West German GDP, investment and consumption for the period up to and including the first quarter of 1991, the quarter of reunification; growth rate data are for united Germany thereafter. To create a level series consistent with the units for united Germany, we use the same splicing method described above. A search for outliers in the growth rate data reveals two quarters for France, 1968Q2 and 1968Q3, where the GDP growth rate is more than six standard deviations from the mean. Of course, these quarters were associated with well-known strikes andgeneral unrestinFrance. We replacethe datafor thesequarters in GDP, consumption, and investment using a univariate EM algorithm—an AR(1) model is estimated for each series and the EM algorithm is used to replace the data for relevant quarters. TherearesomeotherquarterswithGDPgrowthrates3to4standarddeviations from the series-specific average, including 1973Q1, 1979Q2 (U.K.), 1974Q1, 1997Q2 (Japan) and 1970Q1 (Italy). The 1970Q1 Italy outlier falls where we splice two series, but the large change is explained by a general strike in 1969Q4. See the OECD Economic Survey, July 1970 for details. Sources of data by country. Canada: 1960 OECD data, 1961-2002 Statistics Canada via Haver Analytics. France: 1960-1969 OECD data, 1970-1977 Insitut 31 The ‘current’vintage was retrieved from Haveron May 8, 2003. 22

National de la Statistique et des Etudes Economiques (INSEE), undated historical vintage, via Haver Analytics; 1978-2002 current vintage from INSEE via Haver Analytics. Germany: for GDP, 1960-2002 Deutsche Bundesbank via Haver Analytics; forconsumptionandinvestment, 1960-1967 WestGermanydataspecifiedbelow, 1968-2002 DeutscheBundesbankviaHaver Analytics. WestGermany: 1960-1995Q3 Deutsche Bundesbank,undatedhistorical vintage, viaFederalReserve Board. Italy: 1960-1969 OECD data, 1970-2002 Istituto Nazionale di Statistica via Haver Analytics. Japan: 1960-2002 Economic Planning Agency via Haver Analytics. United Kingdom: forGDPandconsumption,1960-2002OfficeforNationalStatistics(ONS) via Haver Analytics; for investment, 1960-1964 OECD data, 1965-2002 ONS via Haver Analytics. United States: 1960-2002 Bureau of Economic Analysis via Haver Analytics. In all cases, the OECD data were kindly provided by Jorgen Elmeskov at the OECD. Appendix A2. Inferences about variance an covariance breaks In this Appendix we give local and nonlocal accounts of the relative difficulty of detecting variance and covariance breaks in the case described in the text. In particular, suppose we have one sample of size N drawn from data that are independently and identically distributed as a bi-variate N(0,Σ1). We have a second sample of the same size distributed as N(0,Σ2), where Σ2 = (1−k2 )Σ1—we replace the α in the text with (1−k2 ) for notational convenience. Suppose we test for a change in the variance of the first variable using the statistic, N(σˆ 2 (2)−σˆ 2 (1)) 2 t v = 2(σˆ 1 4 (2)+σˆ 1 4 (1)) . 1 1 where σˆj((cid:9)) is the sample standard deviation of the jth variable in the (cid:9)th sample. Usingstandardresults,thisstatisticisthedifferenceofthesamplevariancessquared over the asymptotic variance of this difference and is asymptotically χ2 under the (1) null hypothesis of k = 0. Since the sample moments are consistent, k4 N p → lim ∞ t v /N = 2((1−k2)2+1) . The analogous test for the change in covariance is, N(cˆ(2)−cˆ(1)) 2 t c = (σˆ 2 (2)σˆ 2 (2)+cˆ2(2))+(σˆ 2 (1)σˆ 2 (1)+cˆ2(1)) . 1 2 1 2 where cˆis the sample covariance. Following the results above, k4 ρ2 N p → lim ∞ t c /N = (1−k2)2+11+ρ2 , 23

where ρ is the population correlation of the two variables in each sample. Note that if |ρ| = 1 this limit is the same as in the variance case. We can characterize thenonlocalrelative power of thetwo tests usingtheresults on the approximate slope given by Geweke [1981]. Since these tests are both χ2 (1) under the null, the approximate slope for fixed k of each is given by the probability limits given above. As Geweke (Theorem 2) shows, as N gets large, the ratio of the number of observations required to attain an given power in the covariance-based test to the number required to obtain the same power with the variance test can be approximated by the inverse ratio of the approximate slopes. Thus, N c(x)/N v(x)≈ (1+ρ2 )/ρ2 where N c(x) is the number of observations required to attain power x using test c. NotethatattypicalcorrelationamongGDPgrowthratesofabout1/4, thisratio is 5: it takes 5 times as many observations to detect the covariance break as the variance break. Of course, this example is for the iid normal case and does not take the time series properties into account. The general point carries over, however. Appendix A3. Details on our inference approach Here we give a brief description of the iterated other percentile bootstrap used to create the reported confidence intervals. For details, see Hall (1992). First, we describe what in Hall’s terminology is called an other percentile bootstrap (OPB) confidence interval. This involves creating N 1 bootstrap samples—our parameteric method of generating samples is described below. For each such sample calculate the parameter of interest; call the estimate for the mth sample ∆ˆ(m) . The OPB confidence interval with nominal coverage 100(1 − 2k) percent (0 < k < 1/2) is given by the interval from the kth to the (1−k) th percentile of the ∆ˆ(m) s. This confidence interval is known to have poor coverage properties that can be substantially improved by iteration. For the iterated OPB confidence interval, the nominal 90 percent confidence interval is the κth to the (1−κ) th percentile of the ∆ˆ(m) s, where κ is chosen based on an iterated, or nested bootstrap. The additional round of bootstrapping is used to pick an adjusted nominal level, κ, that brings the coverage closer to the desired level of 90 percent. Tocalculateκ,foreachoftheN 1 samplesinthemainbootstrapdothefollowing. For concreteness we talk of the mth original sample. The parameter estimates in the mth sample are θˆ(m) = (θˆ(m),...,θˆ(m) ), where B is the number of breaks. 1 B+1 Draw N 2 samples from the distribution implied by the parameter θˆ(m) using the same parametric approach used in the main bootstrap. Calculate the parameter of interest, ∆ˆ(m,n) , n = 1,...,N 2. Based on N 2 values, we can calculate, for any k, the 100(1−2k) percent OPB confidence interval for ∆. Since we know the process generating the data in this case, we can record whether this interval covers the true value of ∆, ∆ˆm . To form a 90 percent iterated OPB confidence interval, we choose 24

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1. Trade (exports plus imports) of each G-7 country with the rest of the G-7 as a share of its own GDP, 1960-2000 Percent Canada 80 60 40 20 France 25 20 15 10 5 Germany 25 20 15 10 5 Italy 25 20 15 10 5 Japan 25 20 15 10 5 United Kingdom 25 20 15 10 5 United States 25 20 15 10 5 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 NOTE. The data are annual. Imports, exports, and GDP are in current U.S. dollars at current exchange rates. SOURCE. International Monetary Fund, Direction of Trade Statistics (various issues); Organisation for Economic Co-operation and Development. 31

2. Share of foreign equities in equity holdings of U.S. residents and share of U.S. equities in equity holdings of residents of foreign countries, 1980-2002:Q1 Percent 15 Foreign holdings of U.S. residents 12 9 6 3 U.S. holdings of foreign residents 1982 1986 1990 1994 1998 2002 SOURCE. International Finance Corporation; International Federation of Stock Exchanges; Federal Reserve Board. 32

3. 4-quarter real GDP growth of each G-7 country, 1961-2002 Percent Canada 15 10 5 0 _5 France 15 10 5 0 _5 Germany 15 10 5 0 _5 Italy 15 10 5 0 _5 Japan 15 10 5 0 _5 United Kingdom 15 10 5 0 _5 United States 15 10 5 0 _5 English 15 10 5 0 _5 Euro 15 10 5 0 _5 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 NOTE. Shaded bars are periods of recession in the United States as defined by the National Bureau of Economic Research (NBER). English refers to Canada, the United Kingdom and the United States. Euro denotes France, Germany and Italy. 33

4. Standard deviations of quarterly real GDP growth rates in each of the G-7 countries, rolling five-year periods, 1965-2002 Standard deviation Canada 10 8 6 4 2 United States France 10 8 6 4 2 Germany 10 8 6 4 2 Italy 10 8 6 4 2 Japan 10 8 6 4 2 United Kingdom 10 8 6 4 2 English 10 8 6 4 2 Euro 10 8 6 4 2 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 NOTE. Values for each quarter are calculated over the five years ending in that quarter. English refers to Canada, the United Kingdom and the United States. Euro denotes France, Germany and Italy. For description of shaded bars, see general note to figure 3. 34

5. Correlation of quarterly real GDP growth rates, selected country pairs, rolling five-year periods, 1965-2002 A: ‘English-speaking’ nations Correlation coefficient United States & United Kingdom 1 .5 0 _.5 United States & Canada 1 .5 0 _.5 United Kingdom & Canada 1 .5 0 _.5 B: Continental European nations Germany & France 1 .5 0 _.5 Germany & Italy 1 .5 0 _.5 France & Italy 1 .5 0 _.5 C: English & euro 1 .5 0 _.5 D: United States & Japan 1 .5 0 _.5 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 NOTE. See note to figure 4. 35

6. Correlation of quarterly real Consumption growth rates, selected country pairs, rolling five-year periods, 1965-2002 A: ‘English-speaking’ nations Correlation coefficient United States & United Kingdom 1 .5 0 _.5 United States & Canada 1 .5 0 _.5 United Kingdom & Canada 1 .5 0 _.5 B: Continental European nations Germany & France 1 .5 0 _.5 Germany & Italy 1 .5 0 _.5 France & Italy 1 .5 0 _.5 C: English & euro 1 .5 0 _.5 D: United States & Japan 1 .5 0 _.5 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 NOTE. See note to figure 4. 36

7. Covariance of quarterly real GDP growth rates, selected country pairs, rolling five-year periods, 1965-2002 A: ‘English-speaking’ nations Covariance United States & United Kingdom 20 10 0 _10 _20 United States & Canada 20 10 0 _10 _20 United Kingdom & Canada 20 10 0 _10 _20 B: Continental European nations Germany & France 20 10 0 _10 _20 Germany & Italy 20 10 0 _10 _20 France & Italy 20 10 0 _10 _20 C: English & euro 20 10 0 _10 _20 D: United States & Japan 20 10 0 _10 _20 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 NOTE. See note to figure 4. 37

Table 1: Break Dates for 3 breaks. 1 2 3 GDP: 1972Q2 1981Q1 1992Q2 Consumption: 1969Q2 1981Q1 1993Q1 Investment: 1974Q3 1983Q1 1993Q1 Table 2: Tests of changes in mean growth with 3 breaks. GDP Consumption Investment period 6 7 8 6 7 8 6 7 8 versus 7 8 9 8 9 9 7 8 9 8 9 9 7 8 9 8 9 9 All d DD dd d DDD d d u D d d uu u Eng d d d du u d d d d u u d d d uu u Eur d DD dd d DDD DD d D d D uu u US d d d du u d d d u u u d d d uu d UK d d u uu u d u U u u u D d u uu u Canada d D D dd u d DD d d u d d d du u Germany d DD ud d d D D d D d d u d ud d France DDD dd d d DD DD u DDD uu u Italy d DD dd d d DD d D u d d d uu u Notes: The table presents tests for changes in the mean between all pairs of the 4 sub-samples de(cid:12)ned by the 3 breaks. The subsamples correspond roughly to the decades (see table 1) and are labeled 6, 7, 8, 9, for the 60s, 70s, 80s, and 90s, respectively. Changes in the mean from an earlier to later period are denoted D, for down, or U, for up. Bold upper case letters indicate a change that is signi(cid:12)cant at the 5 percent level; plain upper case indicates signi(cid:12)cance at the 10 percent level; lower case simply signi(cid:12)es the sign of the change in the point estimate. Eng denotes Canada, the United Kingdom and the United States; Eur denotes France, Germany, and Italy. 38

Table 3: Value of selected statistics with 3 breaks. GDP Conusmption 6 7 8 9 6 7 8 9 Mean Growth All 4.46 2.83 2.38 2.33 4.54 3.44 2.39 2.66 Eng 3.93 2.78 2.40 3.11 3.93 3.07 2.78 3.48 Eur 4.99 2.88 2.37 1.56 5.16 3.81 2.00 1.83 Unconditional Standard Deviation All 4.82 4.92 3.48 2.14 3.87 4.64 3.48 2.38 Eng 4.11 5.24 3.72 1.92 4.06 5.13 3.52 1.82 Eur 5.53 4.61 3.25 2.35 3.68 4.16 3.43 2.94 Unconditional Correlation All 0.08 0.41 0.26 0.30 -0.03 0.21 0.11 0.12 Eng 0.02 0.43 0.42 0.36 0.05 0.24 0.36 0.20 Eur 0.11 0.50 0.39 0.36 -0.08 0.23 0.17 0.19 Eng-Eur 0.09 0.38 0.16 0.26 -0.04 0.19 0.00 0.08 Notes: For break dates see table 1. Also see notes to table 2. 39

Table 4: Tests of changes in standard deviation with 3 breaks. GDP Consumption Investment period 6 7 8 6 7 8 6 7 8 versus 7 8 9 8 9 9 7 8 9 8 9 9 7 8 9 8 9 9 Unconditional All u DD DD D U d D DD D d d D d D D Eng u d D DD D U d D DD D u u D d D D Eur d DD DD D u d D D D d DDD u D D US u d D d D D u d D d D D U d D DD D UK U d D DD D u DD DD D d u d u d d Canada d u D u D D U u D d D D u u d d D D Germany DDD d D D d u D u D D D d D u d D France d DD D d u u DD DD u d d d d d d Italy u DD DD d u d u d d u D DD d d d Conditional (one-step) All d DD DD D U d D DD D d DD d D D Eng u D D DD D U d D DD D u d D d D D Eur DDD d D D U d d D D d DDD u D d US u D D d D D u d D d D D u d D DD d UK u d D DD D u d D DD D d u d u u d Canada d d D u D D U u D DD D u u D d d D Germany D d D u D D u u d u D D D d D u d d France d DD d d d u D D DD u d DD d D d Italy D DD DD d U u U D d u d DD d d d Notes: For break dates see table 1. Also see notes to table 2. 40

Table 5: Tests of changes in unconditional variance partition by frequency with 3 breaks. GDP Consumption Investment period 6 7 8 6 7 8 6 7 8 versus 7 8 9 8 9 9 7 8 9 8 9 9 7 8 9 8 9 9 US lo u u u u d d d u d u d d u u u u d d US bc u u u u d d u u d u d d u u u u u d US hi d d d d u u d d u d u u d d d d u u UK lo u U U u u d u U u U u D u u u u u d UK bc u UU u U u u U u Uu D d u U u U u UK hi d DD d D u d D d Dd U u d D d D u Canada lo u U U u u d u U u u d d u u u u u d Canada bc u UU UU u u U u u d d u u u d u u Canada hi d DD DD d d D d d u u d d d d d d Germany lo u U U d u u d u d u u d u u u d d u Germany bc u u U d u u u u u u u d u d u d d u Germany hi d d D u d d u d d d d u d u d u u d France lo u u u d d u u u u u d d u U U u u d France bc u u u d u u u u u u d d uUU u U u France hi d d d u d d d d d d u u dDD D D u Italy lo U u u d d d D d d U u d d u u u u d Italy bc U u u D d u D d d Uu d d d u u u u Italy hi D d d U u d U u U Dd U u d d d d u Notes: Business cycle frequencies, denoted bc, are those with periods between 8 and32 quarters; highfrequencies(hi)are thosewithperiodsshorter than bc; low frequncies (lo) have longer than bc periods. See also the notes to tables 1 and 2. 41

Table 6: Tests of changes in unconditional correlation with 3 breaks. GDP Consumption Investment period 6 7 8 6 7 8 6 7 8 versus 7 8 9 8 9 9 7 8 9 8 9 9 7 8 9 8 9 9 All U U U d d u U u U d d u u u u u u d Eng U U U d d d u U u u d d u u u u u d Eur U U u d d d U u U d d u u U u u u d Eng-Eur U u u D d u U u u d d u d u u u u d US-UK U u u d d d u u u u d d U u U d u u US-Canada U U u u d d u u u u d d u u u d u u US-Germany u u u d d u u u u u u d d u d U d D US-France UUU d d u u d u d d u u d U d u U US-Italy u d d D d u U u u DD u D D d u u u UK-Canada u d u D d u d d d d d d u u d d d d UK-Germany U u u D D u U u U D d U u d u d u U UK-France d d d d d u u d d d d u D d D U U d UK-Italy U u u D d u u u d d d d u U d u d D Canada-Germany U u u d d d U u u d d u u d d d d u Canada-France u u d d d d u u U d u u u d u d d u Canada-Italy u u u u u u d u d u d d d u d u u d Germany-France u u u u u u d u u u u d u U u u d d Germany-Italy u U u u d d u u U d U U u U u u u D France-Italy u u u d d u u u d u d d u U u u u d Notes: For break dates see table 1. Also see notes to table 2. 42

Table 7: Tests of changes in conditional correlation with 3 breaks. GDP Consumption Investment period 6 7 8 6 7 8 6 7 8 versus 7 8 9 8 9 9 7 8 9 8 9 9 7 8 9 8 9 9 All U U u DD d U u u d d u d u u u u d Eng U U U d d u u u u u d d u u d u d d Eur u U u u d D u u u d d u u U u u d d Eng-Eur U u u DD d U u u d d u d d u u u u US-UK U u u d d u u u d d d d u u U d u u US-Canada U U U d d d u u u u d d d d u u u u US-Germany U u u d d u u u u u u u d u D U d D US-France UUU d d d u u u d d u u u U d u U US-Italy u u d d D d U u u D d u d d d d u u UK-Canada U u u D D u d d d u d d u u d d d d UK-Germany U d u D D u U d U D d U u DU D u U UK-France d D D D D u u d u D d u DD d u U u UK-Italy U u u DD d u d d d d d u u d u d d Canada-Germany UU u d D d U u u d d u u d u d d u Canada-France u d d d D d u U U u u u u d u d u u Canada-Italy u u u u u d u u U u u u d u u u u d Germany-France u d d d d d d u u u u d u U u u d d Germany-Italy u U u u d D d d u u U u u U u U d D France-Italy d u u u u d u u u u d d u U u u d D Notes: For break dates see table 1. Also see notes to table 2. 43

Table 8: Tests of changes in unconditional covariance with 3 breaks. GDP Consumption Investment period 6 7 8 6 7 8 6 7 8 versus 7 8 9 8 9 9 7 8 9 8 9 9 7 8 9 8 9 9 All U u d d D d Uu u DD d u u d d d d Eng U u u d D d u u u d D d u u u d d d Eur u u u d d d Uu U D d u d u d u d d Eng-Eur u d d DD d Uu u DD u d d d d d u US-UK U u u d D d u u u d d d U u U d d u US-Canada u u d u D D u u d u d d u u u d d u US-Germany u u u d d d u u u d d d d u d U u d US-France U U U d D d u d u d d u u u u d d u US-Italy u d d DD d Uu u DD u D d d d u u UK-Canada u d d D D u d d d d d d u u d d d d UK-Germany U u u D D d UuU D D U u d u d d u UK-France u DD d D d u d d DD d D d D u U d UK-Italy U u u D d u u u d d d d u u d u d d Canada-Germany u d d d D d Uu u DD u d DD d d d Canada-France u u d d d d u u u d u u u d u d d u Canada-Italy u u u d d d d u d u d d d u d u u d Germany-France u u u d d d d u u u u d u U u u d d Germany-Italy u u u d d D u uU d u u u u u u d D France-Italy u d d d d u u u d d d d u u u u u d Notes: For break dates see table 1. Also see notes to table 2. 44

Table 9: Tests of changes in conditional covariance with 3 breaks. GDP Consumption Investment period 6 7 8 6 7 8 6 7 8 versus 7 8 9 8 9 9 7 8 9 8 9 9 7 8 9 8 9 9 All u u u d D d uuu d d d u d d dd u Eng u u u d d d uud d d d u u u dd d Eur u u u d d D uuu d d u u u u ud d Eng-Eur u u d d D d uuu d d u d d d du u US-UK u u u d d d uud d d d u u u dd u US-Canada u u d d d d uud d d d d d u du u US-Germany u u u d d d uuu d d d d u d ud D US-France UUU d d d uuu d d u u u u du u US-Italy u d d d d d uuu d d u d d d du u UK-Canada U u u d D u ddd u d d u u d dd d UK-Germany u u u DD u uuu Dd U u d u du u UK-France u d d d D u udd d d u DDd uu u UK-Italy U u u d D d udd d d d u u d ud d Canada-Germany u u u d D d uuu d d d d d d dd u Canada-France u d d d d d uuu u u u u d u du u Canada-Italy u u u u d d uuu u u u u u u uu d Germany-France d d d d d d duu u u d u u u ud d Germany-Italy u u u d d D ddu u u u u U u ud d France-Italy d d d d d d uuu d d d u u u ud d Notes: For break dates see table 1. Also see notes to table 2. 45

Table 10: Tests of changes in other measures of comovement with 3 breaks. GDP Consumption Investment period 6 7 8 6 7 8 6 7 8 versus 7 8 9 8 9 9 7 8 9 8 9 9 7 8 9 8 9 9 Sum of the n largest eigenvalues All, (cid:21)(1) U u u d d u uuu dd d d ud u d d All, (cid:21)(2) U u u d d d uuu dd d d ud u d d Eng, (cid:21)(1) u u u u d d uuu ud d d uu u u d Eng, (cid:21)(2) U u u u d d uuu uu d u uu u u u Eur, (cid:21)(1) u u u d d d uuu dd u u uu u u d Eur, (cid:21)(2) d D d d u u ddu uu u d dd d d u Maximum variance share of N shocks All 1 u u d d d d uud dd d d du u u u All 2 u u d d d d uud ud d d dd u u u Eng 1 u u u d u u uud ud d d dd d d u Eng 2 u u u d d u ddd ud d d dd d u u Eur 1 d u d U d D dud ud D d dd u u d Eur 2 d u D u D D ddd ud D Ddd Uu d Dynamic factor All U u U D d u uuu dd d u ud u d D Eng u U u d d d uuu uu d d du u u u Eur U u U d d u udu du u u ud u d D Notes: For break dates see table 1. Also see notes to tables 2 and 5. 46