2. MODELLING INTERNATIONAL TRANSMISSIONS: A GVAR APPROACH
 Top of page
 Abstract
 1. INTRODUCTION
 2. MODELLING INTERNATIONAL TRANSMISSIONS: A GVAR APPROACH
 3. THE GVAR MODEL (1979–2003)
 4. PAIRWISE CROSSSECTION CORRELATIONS: VARIABLES AND RESIDUALS
 5. ROBUSTNESS OF THE GVAR RESULTS TO TIMEVARYING WEIGHTS
 6. GENERALIZED IMPULSE RESPONSE FUNCTIONS
 7. IDENTIFICATION OF SHOCKS USING THE GVAR MODEL
 8. CONCLUDING REMARKS
 . APPENDIX
 Acknowledgements
 REFERENCES
 Supporting Information
One of the most striking features of the business cycles across countries are the patterns of comovement of output, inflation, interest rates and real equity prices. These comovements have become more pronounced over the past two decades owing to increased economic and financial integration, with important implications for macroeconomic policy spillovers across countries. The extent of comovement of real GDP across countries has been empirically investigated, both by considering bivariate correlation of real GDP across countries and by decomposing the variations of real GDP into common and countryspecific shocks. Multivariate and multicountry analyses have also been undertaken in the context of G7 economies. For example, Gregory et al. (1997) using Kalman filtering and dynamic factor analysis provide a decomposition of aggregate output, consumption and investment for G7 countries. Other similar decompositions have also been attempted by Canova and Marrinan (1998), Lumsdaine and Prasad (2003) and Kose et al. (2003).1
There are clearly many channels through which the international transmissions of business cycles can take place. In particular, they could be due to common observed global shocks (such as changes in oil prices), they could arise as a result of global unobserved factors (such as the diffusion of technological progress), or could be due to specific national or sectoral shocks.
Unobserved factor models with a large number of macroeconomic variables have recently gained popularity with the work of Stock and Watson (2002a). A related literature on dynamic factor models has also been developed by Forni and Reichlin (1998) and Forni et al. (2000). The factor models, estimated using principal components, are generally used to summarize by a small set of factors the empirical content of a large number of variables. Although unobserved factor models have important applications in forecasting, the identification of factors is often problematic, especially when we wish to give them an economic interpretation.2 It is also likely that even when all such ‘common’ factors are taken into account, there will be important residual interdependencies due to policy and trade spillover effects that remain to be explained. Therefore, a fairly detailed global framework would be needed if we are to investigate the relative importance of such diverse sources of comovements in the world economy, and their impacts on the euro area. For this purpose we make use of the global vector autoregressive model (GVAR) recently developed by PSW.
To motivate the GVAR model for the analysis of the international transmission mechanisms and to relate it to the unobserved factor models, suppose there are N + 1 countries (or regions) in the global economy, indexed by i = 0, 1, …, N, where country 0 serves as the numeraire country (which we take as the USA, but could be any other country). The aim is to model a number of countryspecific macroeconomic variables such as real GDP, inflation, interest rates and exchange rates collected in the vector x_{it}, over time, t = 1, 2…, T, and across the N + 1 countries. Given the general nature of interdependencies that might exist in the world economy, it is clearly desirable that all the countryspecific variables x_{it}, i = 0, 1, …, N, and observed global factors (such as oil prices) are treated endogenously. The following general factor model provides a good starting point and allows us also to relate the GVAR approach to the more familiar factor models used in the literature primarily for the analysis of G7 economies.
Denote the observed global factors by the m_{d} × 1 vector d_{t}, and the unobserved global factors by the m_{f} × 1 vector f_{t}, and assume that3
 (1)
where Γ_{i} = (Γ_{id}, Γ_{if}) is the k_{i} × m, matrix of factor loadings, m = m_{d} + m_{f}, ξ_{it} is a k_{i} × 1 vector representing the countryspecific effects involving lagged values of x_{it} or countryspecific dummy variables capturing major institutional and political upheavals, and δ_{i0} and δ_{i1} are the coefficients of the deterministics, here intercepts and linear trends. Other deterministics, such as seasonal dummies, can also be included in the model. The vector of observed global variables could include international variables such as oil or other commodity prices, world expenditure on R&D, or other indicators of global technology such as the number of international patents registered in the USA.
Unit root and cointegration properties of x_{it}, i = 0, 1, …, N, can be accommodated by allowing the global factors, h_{t} = (d′_{t}, f′_{t})′, and/or the countryspecific factors, ξ_{it}, to have unit roots. More specifically, we assume that
 (2)
 (3)
where L is the lag operator and
 (4)
The coefficient matrices, Λ_{ℓ} and Ψ_{iℓ}, i = 0, 1, …, N, are absolute summable, so that var(Δf_{t}) and var(Δξ_{it}) are bounded and positive definite, and [Ψ_{i}(L)]^{−1} exists. In particular we require that
 (5)
where K is a fixed bounded matrix.
First differencing (1) and using (3) we have
Using the approximation
we obtain the following approximate VAR(p_{i}) model:
 (6)
Without the unobserved common factors, f_{t}, the model for the ith country decouples from the rest of the country models and each country model can be estimated separately using the econometric techniques developed in Harbo et al. (1998) and Pesaran et al. (2000) with d_{t} treated as weakly exogenous. With the unobserved common factors included, the model is quite complex and its econometric analysis using Kalman filtering techniques would be quite involved unless N is very small. When N is relatively large a simple, yet effective, alternative would be to follow Pesaran (2006) and proxy f_{t} in terms of the crosssection averages of countryspecific variables, x_{it}, and the observed common effects, d_{t}. To see how this procedure could be justified in the present more complicated context, initially assume k_{i} = k and use the same set of weights, w_{j}, j = 0, 1, …, N, to aggregate the countryspecific relations defined by (1) to obtain
or
 (7)
Also, note from (3) that
 (8)
But using Lemma A.1 in Pesaran (2006), it is easily seen that for each t the lefthand side of (8) will converge to zero in quadratic mean as N∞, if (5) holds, the countryspecific shocks, v_{jt}, are independently distributed across j, and if the weights, w_{j}, satisfy the atomistic conditions
 (9)
where K is a fixed constant. Under these conditions (for each t), , and hence , where ξ* is a timeinvariant random variable. Using this result in (7) and assuming that the k × m_{f} average factor loading coefficient matrix, , has full column rank (with k ≥ m_{f}) we obtain
which justifies using the observable vector as proxies for the unobserved common factors.4 Substituting this result in (6), for N sufficiently large we have
 (10)
where , , and are given in terms of δ_{i0}, δ_{i1}, Γ_{id}, Γ_{if}, , , , and .
In practice, the number of countries, N + 1, may not be sufficiently large, and the individual countries not equally important in the global economy. The countryspecific shocks might also be crosssectionally correlated due to spatial or contagion effects that are not totally eliminated by the common factors, d_{t} and f_{t}. Finally, k_{i}, the number of countryspecific variables, need not be the same across i. For example, some markets may not exist or might not be sufficiently developed in some of the countries. Even if we focus on the same set of variables to model across countries, there will be one less exchange rate than there are countries in the global model. The GVAR framework developed in PSW addresses these considerations by using countryspecific weights, w_{ij}. Specifically, instead of using the same in all country models PSW use
 (11)
in the ith country model. The weights, w_{ij}, j = 0, 1, …, N could be used to capture the importance of country j for country ith economy. Geographical patterns of trade provide an obvious source of information for this purpose and could also be effective in mopping up some of the remaining spatial dependencies. The weights could also be allowed to be timevarying so long as they are predetermined. This could be particularly important in the case of rapidly expanding emerging economies with their fastchanging trade relations with the rest of the world. The use of the countryspecific weights also allows a simple solution to the problem of k_{i}, the number of countryspecific variables, being different across i. It would be sufficient to attach zero weights to the missing variable in country i, with the remaining weights being rebalanced to add up to unity.
With the above considerations in mind, the GVAR counterpart of (10) may now be written more generally as the individual country VARX*(p_{i}, q_{i}) models:
 (12)
for i = 0, 1, …, N, where for estimation purposes Φ_{i}(L, p_{i}), ϒ_{i}(L, q_{i}) and Λ_{i}(L, q_{i}) can be treated as unrestricted. For the empirical implementation that will follow, for each country model we consider at most a VARX*(2, 2) specification which in its error correction form may be written as5
 (13)
where , ζ_{i, t−1} = (z′_{i, t−1}, d′_{t−1})′, α_{i} is a k_{i} × r_{i} matrix of rank r_{i} and β_{i} is a matrix of rank r_{i}. By partitioning β_{i} as β_{i} = (β′_{ix}, β′_{ix*}, β′_{id})′ conformable to , the r_{i} error correction terms defined by (13) can now be written as
 (14)
that clearly allows for the possibility of cointegration both within x_{it} and between x_{it} and and consequently across x_{it} and x_{jt} for i ≠ j.
The number of cointegrating relations, r_{i}, and the vector of cointegrating relations for each country model can be consistently estimated separately, by treating d_{t} and as weakly exogeniety I(1) with respect to the long run parameters of the conditional model (13). Note that this assumption is compatible with a certain degree of weak dependence across u_{it}, as discussed in PSW. Following Johansen (1992) and Granger and Lin (1995), the weak exogeneity assumption in the context of cointegrating models implies no longrun feedback from x_{it} to , without necessarily ruling out lagged shortrun feedback between the two sets of variables. In this case is said to be ‘longrun forcing’ for x_{it}, and implies that the error correction terms of the individual country VECMs do not enter in the marginal model of . As discussed in more detail in Section (3.4), the weak exogeneity of these variables can then be tested in the context of each of the countryspecific models. Furthermore, conditional on a given estimate of β_{i}, it is possible to show that the remaining parameters of the conditional model (namely c_{i0}, α_{i}, ϒ_{i0}, Λ_{i0}, ϒ_{i1}, Γ_{i}) can be consistently estimated either indirectly from a VAR in z_{it}, or directly by OLS regressions of Δx_{it} on intercepts, the error correction terms, , Δd_{t}, Δd_{t−1}, and Δz_{i, t−1}. The two approaches will yield identical estimates.6
Once the individual country models are estimated, all the endogenous variables of the global economy, collected in the k × 1 vector x_{t} = (x′_{0t}, x′_{1t}, …, x′_{Nt})′, need to be solved simultaneously. PSW show how this can be done in the case where p_{i} = q_{i} = 1. In the present more general context we first rewrite (12) as
 (15)
where
Let p = max(p_{0}, p_{1}, …, p_{N}, q_{0}, q_{1}, …, q_{N}) and construct A_{i}(L, p) from A_{i}(L, p_{i}, q_{i}) by augmenting the p − p_{i} or p − q_{i} additional terms in powers of L by zeros. Also note that
 (16)
where W_{i} is a matrix, defined by the country specific weights, w_{ji}.
With the above notations (15) can be written equivalently as A_{i}(L, p)W_{i}x_{t} = φ_{it}, i = 0, 1, …, N, and then stacked to yield the VAR(p) model in x_{t}:
 (17)
where
 (18)
The GVAR(p) model (17) can now be solved recursively, and used for forecasting or generalized impulse response analysis in the usual manner. The issue of structural impulse response analysis poses special problems in the context of the GVAR model and will be dealt with in Section 7.
3. THE GVAR MODEL (1979–2003)
 Top of page
 Abstract
 1. INTRODUCTION
 2. MODELLING INTERNATIONAL TRANSMISSIONS: A GVAR APPROACH
 3. THE GVAR MODEL (1979–2003)
 4. PAIRWISE CROSSSECTION CORRELATIONS: VARIABLES AND RESIDUALS
 5. ROBUSTNESS OF THE GVAR RESULTS TO TIMEVARYING WEIGHTS
 6. GENERALIZED IMPULSE RESPONSE FUNCTIONS
 7. IDENTIFICATION OF SHOCKS USING THE GVAR MODEL
 8. CONCLUDING REMARKS
 . APPENDIX
 Acknowledgements
 REFERENCES
 Supporting Information
The version of the GVAR model developed in this paper covers 33 countries, where 8 of the 11 countries that originally joined the euro on 1 January 1999 are grouped together, and the remaining 25 countries are modelled individually (see Table I). The present GVAR model, therefore, contains 26 countries/regions. The original PSW model contained 11 countries/regions based on 25 countries. With increased country coverage, the countries in the present GVAR model account for 90% of world output as compared to 80% covered by the 11 countries/regions in PSW. Data sources are provided in Supplement A, available from the authors on request.
Table I. Countries and regions in the GVAR modelUSA  Euro area  Latin America 
China  Germany  Brazil 
Japan  France  Mexico 
UK  Argentina  
Italy  Chile  
Spain   
Other developed economies  Netherlands  Peru 
Canada  Belgium  
Australia  Austria  
New Zealand   Finland 
Rest of Asia   
Rest of W. Europe  Rest of the world  
Korea  Sweden  India 
Indonesia  Switzerland  South Africa 
Thailand  Norway  Turkey 
Philippines   Saudi Arabia 
Malaysia   
Singapore   
The models are estimated over the period 1979(2)–2003(4). This considerably extends the 11 country/region models estimated in PSW over the shorter period 1979(2)–1999(4), most notably including the first years of EMU. The variables included in the current version of the GVAR differ also from those considered by PSW. In order to capture more fully the possible effects of bond markets on output and inflation, we now include, wherever possible, both a short rate (), as well as a long rate of interest (). However, given the data limitations and problems associated with compiling comparable money supply measures, we have decided against the inclusion of real money balances in the current version. Other variables included are real output (y_{it}), the rate of inflation, (π_{it} = p_{it} − p_{i, t−1}), the real exchange rate (e_{it} − p_{it}), and real equity prices (q_{it}), when available. More specifically
 (19)
where GDP_{it} is the nominal Gross Domestic Product, CPI_{it} the consumer price index, EQ_{it} the nominal equity price index, E_{it} the exchange rate in terms of US dollars, is the short rate, and the long rate of interest, for country i during the period t.
The time series data for the euro area was constructed by crosssection weighted averages of y_{it}, π_{it}, q_{it}, , , over Germany, France, Italy, Spain, Netherlands, Belgium, Austria and Finland, using the average Purchasing Power Parity GDP weights, also computed over the 1999–2001 period.8
With the exception of the US model, all models include the countryspecific foreign variables, , , , , and the log of oil prices (), as weakly exogenous in the sense discussed above. In the case of the US model, oil prices are included as an endogenous variable, with , , and as weakly exogenous. Given the importance of the US financial variables in the global economy, the USspecific foreign financial variables, , and , were not included in the US model as they are unlikely to be longrun forcing with respect to the US domestic financial variables. The USspecific foreign output and inflation variables, and , were, however, included in the US model (which were not included by PSW) in order to capture the possible second round effects of external shocks on the USA. Given the importance of the USA for the global economy, initially it was thought that the inclusion of and as weakly exogenous in the US model might result in the violation of the weak exogeneity assumption. However, as reported below this turns out not to be the case.
In this paper, as the focus is mainly on the impact of external shocks on the euro area economy, from now on we shall concentrate the presentation of the results on countries/regions with special relevance to the euro area: USA, China, Japan, euro area, UK and the rest of Western Europe. A more detailed set of results are available in Supplement B, available from the authors on request.
3.1. Trade and Aggregation Weights
The trade shares used to construct the countryspecific foreign variables are given in the 26 × 26 trade share matrix provided in Supplement A. Table II presents the trade shares for our eight focus economies (seven countries plus the euro area itself, composed of eight countries), with a ‘Rest’ category showing the trade shares with the remaining 10 countries in our sample. First considering the euro area, we can see that the USA, the UK and the rest of Western Europe have a similar share in euro area trade (around 1/5) accounting together for almost twothirds of total euro area trade. Other important information that emerges from the trade matrix includes the very high share of the euro area in the trade of the UK and the rest of Western Europe (more than half of the trade relationships of these countries are with euro area countries). Hence, these countries are key in the transmission of shocks to the euro area via a third market, or through secondround effects.
Table II. Trade weights based on direction of trade statisticsCountry/region  USA  Euro area  China  Japan  UK  Rest of W. Europe  Resta 

 Sweden  Switz.  Norway  


USA  0.000  0.155  0.073  0.124  0.052  0.008  0.012  0.004  0.570 
Euro area  0.227  0.000  0.056  0.072  0.238  0.057  0.090  0.028  0.230 
China  0.236  0.164  0.000  0.248  0.029  0.010  0.007  0.003  0.304 
Japan  0.319  0.132  0.128  0.000  0.032  0.007  0.009  0.003  0.369 
UK  0.180  0.537  0.020  0.042  0.000  0.027  0.028  0.023  0.146 
Sweden  0.104  0.517  0.025  0.035  0.115  0.000  0.017  0.099  0.089 
Switz.  0.113  0.670  0.015  0.039  0.066  0.015  0.000  0.004  0.079 
Norway  0.090  0.449  0.020  0.030  0.181  0.132  0.008  0.000  0.091 
Although we estimate models at a country level (the euro area being considered here as a single economy), we also wish to derive regional responses to shocks. Hence, for the rest of Western Europe (and also for rest of Asia, Latin America, other developed countries and rest of the world), we will aggregate impulse response functions by using weights based on the PPP valuation of country GDPs, which are thought to be more reliable than weights based on US dollar GDPs.
3.2. Unit Root Tests
Although the GVAR methodology can be applied to stationary and/or integrated variables, here we follow PSW and assume that the variables included in the countryspecific models are integrated of order one (or I(1)). This allows us to distinguish between shortrun and longrun relations and interpret the longrun relations as cointegrating. Therefore, we begin by examining the integration properties of the individual series under consideration. In view of the widely accepted poor power performance of traditional Dickey–Fuller (DF) tests, we report unit root tstatistics based on weighted symmetric estimation of ADF type regressions introduced by Park and Fuller (1995). These tests, henceforth WS, exploit the time reversibility of stationary autoregressive processes in order to increase their power performance. Leybourne et al. (2004) and Pantula et al. (1994) provide evidence of superior performance of the WS test statistic compared to the standard ADF test or the GLSADF test proposed by Elliot et al. (1996). The lag length employed in the WS unit root tests is selected by the Akaike Information Criterion (AIC) based on standard ADF regressions. The results of the WS statistics for the level, first differences and the second differences of all the countryspecific domestic and foreign variables in the GVAR model can be found in Supplement A.
Real output, interest rates (short and long), exchange rates and real equity prices (domestic and foreign) are I(1) across the focus countries, with two notable exceptions. First, real output in the UK appears borderline I(0)/I(1) according to the WS statistics, although ADF tests indicate that UK real output is I(1). Second, e* in the US model is an I(2) variable. As in PSW, we deal with this problem by including (e − p) instead of the nominal exchange rate variable, e, in the different countryspecific models. Unit root tests applied to (e − p) and (e*− p*) indicate that these variables are I(1) in all cases. Finally, consumer price indices turn out to be I(2), so that inflation (Δp and Δp*) appears to be I(1) across all countries. The test results also generally support the unit root hypothesis in the case of the variables for the remaining countries except for (e − p) for Canada.
3.3. Specification and Estimation of the CountrySpecific Models
We begin the modelling exercise under the assumption that the countryspecific foreign variables are weakly exogenous I(1) variables, and that the parameters of the individual models are stable over time. The latter allows us to estimate and test the longrun properties of the different countryspecific models separately and consistently. Both assumptions are needed for an initial implementation of the GVAR model, and their validity will be examined in what follows.
We do not impose the same specification across the countryspecific models. For the euro area, Japan, the UK and countries belonging to the rest of Western Europe, we include real output (y), inflation rate (Δp), shortterm interest rate (ρ^{S}), longterm interest rate (ρ^{L}), real equity prices (q) and real exchange rate (e − p) as endogenous variables and foreign real output (y*), foreign inflation (Δp*), foreign real equity prices (q*), foreign interest rates (short, ρ^{*S}; and long, ρ^{*L}) and oil prices (p^{o}) as weakly exogenous variables. In the case of China, owing to data constraints, real equity prices and longterm interest rates are excluded from the set of endogenous variables. The US model contains y, Δp, ρ^{S}, ρ^{L}, q and oil prices (p^{o}), as the endogenous variables. The US dollar exchange rate is determined outside the US model. As in PSW, the only exchange rate included in the US model is the foreign real exchange rate variable, (), which is treated as weakly exogenous. The inclusion of oil prices in the US model as endogenous allows the evolution of the global macroeconomic variables to influence oil prices, a feature that was absent from the PSW version, which treated oil prices as weakly exogenous in all countryspecific models. Furthermore, unlike the PSW version, the present specification includes USspecific foreign real output () and foreign inflation () as weakly exogenous variables. This allows for the US model to be more fully integrated in the world economy and hence to take a more satisfactory account of second round effects in the global economic system as a whole. It is, of course, important that the weak exogeneity of these variables in the US model are tested, and this is done below.
Once the variables to be included in the different country models are specified, the corresponding cointegrating VAR models are estimated and the rank of their cointegrating space determined. Initially we select the order of the individual country VARX*(p_{i}, q_{i}) models. It should be noted that p_{i}, the lag order of the domestic variables, and q_{i}, the lag order of the foreign (‘star’) variables in the VARX* models, need not be the same. In the empirical analysis that follows we entertain the case where the lag order of the domestic variables, p_{i}, is selected according to the Akaike information criterion. Owing to data limitations, the lag order of the foreign variables, q_{i}, is set equal to one in all countries with the exception of the USA and the euro area. For the same reason, we do not allow p_{maxi} or q_{maxi} to be greater than two. We then proceed with the cointegration analysis, where the countryspecific models are estimated subject to reduced rank restrictions. To this end, the error correction forms of the individual country equations given by (12) are derived.9
The orders of the VARX* models, the number of cointegration relationships and diagnostic test results for all the models are provided in Supplement B. In Table III we give the lag orders and the number of cointegrating relations for the set of focus countries. For most countries a VARX*(2, 1) specification seemed to be satisfactory. For the USA and the euro area, however, a VARX*(2, 2) was favoured by the AIC. As regards the number of cointegrating relationships, we find 4 for Japan, 3 for the UK, Sweden and Switzerland, 2 for the euro area, Norway and the USA and 1 for China. The cointegration results are based on the trace statistic (at the 95% critical value level), which is known to yield better small sample power results compared to the maximal eigenvalue statistic.
Table III. VARX* order and number of cointegration relationships in the countryspecific models  VARX*(p_{i}, q_{i})  # Cointegrating 

  relationships 

Country  p_{i}  q_{i}  

USA  2  2  2 
Euro area  2  2  2 
China  2  1  1 
Japan  1  1  4 
UK  2  1  3 
Sweden  2  1  3 
Switzerland  1  1  3 
Norway  2  1  2 
3.4. Testing Weak Exogeneity
As noted earlier, the main assumption underlying our estimation strategy is the weak exogeneity of with respect to the longrun parameters of the conditional model defined by (13). Here we provide a formal test of this assumption for the countryspecific foreign variables (the ‘star’ variables) and the oil prices.
Weak exogeneity is tested along the lines described in Johansen (1992) and Harbo et al. (1998). This involves a test of the joint significance of the estimated error correction terms in auxiliary equations for the countryspecific foreign variables, . In particular, for each lth element of the following regression is carried out:
where , j = 1, 2, …, r_{i} are the estimated error correction terms corresponding to the r_{i} cointegrating relations found for the ith country model and . Note that in the case of the USA the term is implicitly included in . The test for weak exogeneity is an Ftest of the joint hypothesis that γ_{ij, l} = 0, j = 1, 2, …, r_{i} in the above regression. The lag orders s_{i} and n_{i}, need not be the same as the orders p_{i} and q_{i} of the underlying countryspecific VARX* models. We carried out two sets of experiments, one set using the lag orders of the underlying VARX* models given in Table III, and in another set of experiments we set s_{i} = p_{i} and n_{i} = 2 for all countries. In both cases the exogeneity hypothesis could not be rejected for most of the variables being considered. Under the former specification of the lag orders 8 out of 153 cases were found to be significant at the 5% level, while under the latter only 5 out of 153 exogeneity tests turned out to be statistically significant.10 The test results for this case are summarized in Table IV.
Table IV. Fstatistics for testing the weak exogeneity of the countryspecific foreign variables and oil pricesCountry   Foreign variables 

  y*  Δp*  q*  ρ^{*S}  ρ^{*L}  p^{o}  e*− p* 


USA  F(2, 75)  0.30  1.89  —  —  —  —  1.83 
Euro area  F(2, 67)  0.06  0.00  2.25  0.20  1.98  2.04  — 
China  F(1, 72)  1.66  0.48  1.30  1.00  1.30  0.19  — 
Japan  F(4, 71)  1.36  1.38  0.32  0.46  0.73  1.68  — 
UK  F(3, 66)  2.98†  0.63  0.07  1.11  1.34  0.57  — 
Sweden  F(3, 66)  2.52  0.81  0.16  0.40  0.40  0.90  — 
Switzerland  F(3, 72)  0.40  0.27  0.42  0.90  0.04  0.36  — 
Norway  F(2, 67)  0.95  0.57  0.41  0.14  0.87  0.28  — 
3.5. Testing for Structural Breaks
The possibility of structural breaks is one of the fundamental problems facing econometric modelling. The problem is likely to be particularly acute in the case of emerging economies that are subject to significant political and social changes. The GVAR model is clearly not immune to this problem. Unfortunately, despite the great deal of recent research in this area, there is little known about how best to model breaks. Even if insample breaks are identified using Bayesian or classical procedures, there are insurmountable difficulties in allowing for the possibility of future breaks in forecasting and policy analysis. See, for example, Stock and Watson (1996), Clements and Hendry (1998, 1999) and Pesaran et al. (2006).
However, the fact that countryspecific models within the GVAR framework are specified conditional on foreign variables should help in alleviating the structural problem somewhat. For example, suppose that univariate equity return equations are subject to breaks roughly around the same time in different economies. This could arise, for example, due to a stock market crash in the USA with strong spillover effects to the rest of the world. However, since equity return equations in the countryspecific models are specified conditional on the US equity returns, they need not be subject to similar breaks, and in this example the structural break problem could be confined to the US model. This phenomenon is related to the concept of ‘cobreaking’ introduced in macroeconometric modelling by Hendry (1996), and examined further by Hendry and Mizon (1998). The structure of the GVAR can readily accommodate cobreaking and suggests that the VARX* models that underlie the GVAR might be more robust to the possibility of structural breaks as compared to reducedform singleequation models considered, for example, by Stock and Watson (1996).
In the context of cointegrated models, structural stability is relevant for both the longrun coefficients and the shortrun coefficients, as well as the error variances.11 As our interest is in exploring the transmission mechanisms of the US and the euro area, we will not consider the stability of the longrun coefficients and rather focus on the structural stability of the shortrun coefficients. In fact, given the limited number of time series data available, a meaningful test of the stability of the longrun coefficients might not be feasible. Also to render the structural stability tests of the shortrun coefficients invariant to exact identification of the longrun relations, we consider structural stability tests that are based on the residuals of the individual equations of the countryspecific error correction models. It is well known that these residuals only depend on the rank of the cointegrating vectors and do not depend on the way the cointegrating relations are exactly identified. Fluctuation tests based on successive parameter estimates which reject the null of parameter constancy when the estimates fluctuate too much, such as those proposed by Ploberger et al. (1989), will not be invariant to the identification of the longrun parameters and will not be considered here.
Among the tests included in our analysis are Ploberger and Krämer's (1992) maximal OLS cumulative sum (CUSUM) statistic, denoted by and its mean square variant PK_{msq}. The statistic is similar to the CUSUM test suggested by Brown et al. (1975), although the latter is based on recursive rather than OLS residuals. Also considered are tests for parameter constancy against nonstationary alternatives proposed by Nyblom (1989), denoted by ℜ, as well as sequential Waldtype tests of a onetime structural change at an unknown change point. The latter include the Wald form of Quandt's (1960) likelihood ratio statistic (QLR), the mean Wald statistic (MW) of Hansen (1992) and Andrews and Ploberger (1994) and the Andrews and Ploberger (1994) Wald statistic based on the exponential average (APW). The heteroskedasticityrobust version of the above tests is also presented.
Table V summarizes the results of the tests by variable at the 5% significance level. The critical values of the tests, computed under the null of parameter stability, are calculated using the sieve bootstrap samples obtained from the solution of the GVAR(p) model given by (17).12 Note that the critical values employed in Stock and Watson (1996) are for the case of predetermined regressors and are therefore not applicable in the GVAR context.
Table V. Number of rejections of the null of parameter constancy per variable across the countryspecific models at the 5% levelAlternative  Domestic variables  Numbers(%) 

test statistics   

 y  Δp  q  e − p  ρ^{S}  ρ^{L}  


 0(0.0)  2(7.7)  3(15.8)  1(4.0)  0(0.0)  1(8.3)  7(5.2) 
PK_{msq}  1(3.9)  1(3.9)  3(15.8)  0(0.0)  1(4.0)  1(8.3)  7(5.2) 
ℜ  0(0.0)  6(23.1)  4(21.1)  2(8.0)  7(28.0)  5(41.7)  24(17.91) 
robustN  1(3.9)  1(3.9)  3(15.8)  2(8.0)  3(12.0)  3(25.0)  13(9.7) 
QLR  11(42.3)  9(34.6)  8(42.1)  9(36.0)  15(60.0)  7(58.3)  59(44.0) 
robustQLR  1(3.9)  2(7.7)  5(26.3)  2(8.0)  2(8.0)  1(8.3)  13(9.7) 
MW  1(3.9)  8(30.8)  6(31.6)  6(24.0)  8(32.0)  6(50.0)  35(26.1) 
robustMW  2(7.7)  2(7.7)  2(10.5)  2(8.0)  2(8.0)  1(8.3)  11(8.2) 
APW  11(42.3)  10(38.5)  6(31.6)  10(40.0)  15(60.0)  6(50.0)  58(43.3) 
robustAPW  2(7.7)  1(3.9)  4(21.1)  2(8.0)  2(8.0)  1(8.3)  12(9.0) 
The results vary across the tests and to a lesser extent across the variables. For example, using the PK tests (both versions) the null hypothesis of parameter stability is rejected at most 7 out of the possible maximum number of 134 cases, with the rejections spread quite evenly across the variables. Turning to the other three tests (ℜ, QLR and APW) the outcomes very much depend on whether heteroskedasticityrobust versions of these tests are used. The results for the robust version are in line with those of the PK tests, although the rate of rejections are now in the range 9–10% rather than the 4–5% obtained in the case of the PK tests. Once possible changes in error variances are allowed for, the parameter coefficients seem to have been reasonably stable. At least based on the available tests, there is little statistical evidence with which to reject the hypothesis of coefficient stability in the case of 90% of the equations comprising the GVAR model. The nonrobust versions of the ℜ, QLR and APW tests, however, show a relatively large number of rejections, particularly the latter two tests that lead to rejection of the joint null hypothesis (coefficient and error variance stability) in the case of 60 (QLR) and 59 (APW) out of the 134 cases. In view of the test outcomes for the robust versions of these tests, the main reason for the rejection seem to be breaks in error variances and not the parameter coefficients. This conclusion is in line with many recent studies that find statistically significant evidence of changing volatility as documented, among others, by Stock and Watson (2002b), Artis et al. (2004) and Cecchetti et al. (2005).
Overall, not surprisingly there is evidence of structural instability but this seems to be mainly confined to error variances. We deal with the problem of possibly changing error variances by using robust standard errors when investigating the impact effects of the foreign variables, and base our analysis of impulse responses on the bootstrap means and confidence bounds rather than the point estimates.
3.6. Contemporaneous Effects of Foreign Variables on their Domestic Counterparts
Table VI presents the contemporaneous effects of foreign variables on their domestic counterparts together with robust tratios computed using White's heteroskedasticityconsistent variance estimator. These estimates can be interpreted as impact elasticities between domestic and foreign variables. Most of these elasticities are significant and have a positive sign, as expected. They are particularly informative as regards the international linkages between the domestic and foreign variables. Focusing on the euro area, we can see that a 1% change in foreign real output in a given quarter leads to an increase of 0.5% in euro area real output within the same quarter. Similar foreign output elasticities are obtained across the different regions.
Table VI. Contemporaneous effects of foreign variables on their domestic counterpartsCountry  Domestic variables 

 y  Δp  q  ρ^{S}  ρ^{L} 


USA  0.54  0.06  —  —  — 
 [3.12]  [0.87]    
Euro area  0.53  0.25  1.15  0.09  0.63 
 [4.03]  [3.31]  [8.90]  [3.84]  [7.86] 
China  − 0.10  0.61  —  0.12  — 
 [−0.66]  [2.30]   [2.27]  
Japan  0.50  − 0.04  0.67  − 0.05  0.48 
 [3.47]  [−0.38]  [5.53]  [−0.89]  [4.84] 
UK  0.33  − 0.15  0.84  0.27  0.67 
 [2.33]  [−0.64]  [13.28]  [1.48]  [4.85] 
Sweden  1.19  1.23  1.15  1.25  0.96 
 [3.38]  [6.19]  [11.60]  [3.56]  [5.75] 
Switzerland  0.47  0.52  0.70  0.16  0.41 
 [3.81]  [3.68]  [2.17]  [3.10]  [5.88] 
Norway  0.80  1.11  1.03  0.15  0.56 
 [2.05]  [6.84]  [8.62]  [0.85]  [3.43] 
We can also observe a high elasticity between longterm interest rates, ρ^{L} and ρ^{*L}, implying relatively strong comovements between euro area and foreign bond markets. More importantly, the contemporaneous elasticity of real equity prices is significant and slightly above one. Hence, the euro area stock markets would seem to overreact to foreign stock price changes, although the extent of overreaction is not very large. Similar results are also obtained for Sweden and Norway. Contemporaneous financial linkages are likely to be very strong amongst the European economies through the equity and the bond market channels.
In contrast, we find rather low elasticities for inflation. For the euro area the foreign inflation elasticity is 0.25, suggesting that in the short run the euro area prices are not much affected by changes in foreign prices. The same is also true for the USA. For the remaining focus countries, with the exception of Japan and the UK, foreign inflation effects are much larger and are statistically significant.
Another interesting feature of the results are the very weak linkages that seem to exist across shortterm interest rates (Sweden being an exception) and the high, significant relationships across longterm rates. This clearly shows a much stronger relation between bond markets than between monetary policy reactions.
4. PAIRWISE CROSSSECTION CORRELATIONS: VARIABLES AND RESIDUALS
 Top of page
 Abstract
 1. INTRODUCTION
 2. MODELLING INTERNATIONAL TRANSMISSIONS: A GVAR APPROACH
 3. THE GVAR MODEL (1979–2003)
 4. PAIRWISE CROSSSECTION CORRELATIONS: VARIABLES AND RESIDUALS
 5. ROBUSTNESS OF THE GVAR RESULTS TO TIMEVARYING WEIGHTS
 6. GENERALIZED IMPULSE RESPONSE FUNCTIONS
 7. IDENTIFICATION OF SHOCKS USING THE GVAR MODEL
 8. CONCLUDING REMARKS
 . APPENDIX
 Acknowledgements
 REFERENCES
 Supporting Information
One of the key assumptions of the GVAR modelling approach is that the ‘idiosyncratic’ shocks of the individual country models should be crosssectionally ‘weakly correlated’, so that , with N∞, and as a result the weak exogeneity of the foreign variables is ensured. Direct tests of weak exogeneity assumptions discussed above indirectly support the view that the idiosyncratic shocks could only be weakly correlated. In this section we provide direct evidence on the extent to which this is likely to be true. The basic idea is similar to the crosssection dependence test proposed in Pesaran (2004). By conditioning the countryspecific models on weakly exogenous foreign variables, viewed as proxies for the ‘common’ global factors, it is reasonable to expect that the degree of correlation of the remaining shocks across countries/regions will be modest. These residual interdependencies, as mentioned in Section 2, could reflect policy and trade spillover effects.
As a simple diagnostic of the extent to which the countryspecific foreign variables have been effective in reducing the crosssection correlation of the variables in the GVAR model, we have computed average pairwise crosssection correlations for the levels and first differences of the endogenous variables of the model, as well as those of the associated residuals over the estimation period, 1979–2003. We also computed average pairwise crosssection correlations of the residuals obtained after reestimating all of the individual countryspecific models over the same period excluding the foreign (star) variables, including oil as endogenous in all the country models.13 The results for all variables are summarized in Table VII.
Table VII. Average pairwise crosssection correlations of all variables and associated model's residualsCountry  Real output  Inflation 

 Levels  1st diff.  VAR Residuals  VARX* Residuals  Levels  1st diff.  VAR Residuals  VARX* Residuals 


USA  0.96  0.16  0.05  − 0.05  0.39  0.13  0.16  0.02 
Euro area  0.96  0.14  0.12  − 0.01  0.39  0.13  0.14  0.00 
China  0.96  0.01  − 0.01  − 0.02  0.04  0.04  0.05  0.01 
Japan  0.92  0.03  − 0.03  − 0.08  0.30  0.01  0.04  0.02 
UK  0.95  0.11  0.07  0.01  0.34  0.03  0.10  0.02 
Sweden  0.96  0.08  0.07  0.02  0.37  0.06  0.11  − 0.01 
Switz.  0.93  0.13  0.08  0.01  0.30  0.09  0.11  0.04 
Norway  0.96  0.09  0.06  0.01  0.32  0.08  0.11  0.02 
USA  0.58  0.40  0.34  − 0.01  —  —  —  — 
Euro area  0.58  0.43  0.39  − 0.08  0.62  0.30  0.27  0.28 
China  —  —  —  —  − 0.19  0.07  0.05  0.03 
Japan  0.35  0.31  0.21  − 0.09  0.59  0.23  0.18  0.15 
UK  0.61  0.41  0.37  − 0.03  0.61  0.27  0.22  0.19 
Sweden  0.57  0.39  0.36  − 0.02  0.59  0.28  0.21  0.20 
Switz.  0.54  0.26  0.19  − 0.05  0.62  0.27  0.25  0.27 
Norway  0.61  0.37  0.33  0.02  0.61  0.31  0.27  0.27 
USA  0.39  0.09  0.04  0.00  0.75  0.38  0.30  − 0.04 
Euro area  0.49  0.15  0.08  0.02  0.78  0.44  0.34  − 0.05 
China  0.32  0.03  0.01  − 0.02  —  —  —  — 
Japan  0.48  0.05  0.03  − 0.01  0.78  0.27  0.26  − 0.04 
UK  0.52  0.12  0.09  0.00  0.79  0.37  0.29  − 0.01 
Sweden  0.46  0.03  0.04  − 0.01  0.81  0.36  0.28  0.06 
Switz.  0.33  0.09  0.08  0.00  0.61  0.36  0.31  0.02 
Norway  0.40  0.02  0.02  0.00  0.73  0.26  0.19  0.03 
The average crosssection correlations are generally high for the level of the endogenous variables and fall as first differences of these variables are considered. The results vary widely across variables and less so across countries, with inflation and real exchange rate for China being the exceptions. Output levels, sharing common trends, show the highest degree of crosssection correlations, of around 92–96%. This is followed by longterm interest rates (61–81%), real equity prices (35–61%), and shortterm interest rates (32–52%). The effect of first differencing on crosssection correlations differ widely over variables as well as countries, and is most pronounced in the case of the output series. Average crosssection correlations of output changes, Δy_{it}, range between 1% for China to 16% for the USA, as compared to crosssection correlations of output levels of 96% for both of these economies. Similar outcomes are also observed in the case of inflation and shortterm interest rates. By comparison, first differencing of equity prices and longterm interest rates has only limited effect on crosssection correlations. For example, the average crosssection correlations of equity prices fall from 35–61% to 26–43% as one moves from levels of equity prices to their first differences. Overall, there is significant evidence of crosscountry correlations for the variables in the GVAR model, although the extent of this correlation depends on the variable, whether it is transformed to stationarity by first differencing, and the country.
Turning to the crosssection correlation of the residuals from the VARX* models (including domestic and foreign star variables), it is quite striking that except for real exchange rates these correlations are very small and do not depend on the choice of the variable or country. This is particularly apparent in the case of the equity and bond markets where the crosssection correlation of the residuals ranges between − 8% and + 6%, as compared to the values in the range 35% and 81% (or 26% and 44%) if crosssection correlations of the levels (or first differences) are considered. The model has clearly been successful in capturing the common effects driving bond and equity markets. The real exchange rate variable presents an important exception which requires further consideration.
With regard to the crosssection correlations of the residuals from the individual country models that include only the domestic variables, their value appears to lie between that of the firstdifferenced variables and the residuals from the VARX* models. Exceptions are noted in the case of inflation, where the correlations of the residuals from the individual country models excluding the star variables are slightly higher than those based on the firstdifferenced variables, and for the real exchange rates where the correlations of the residuals from the VARX* models and VAR models (excluding the star variables) are virtually identical.
Overall, the crosssection correlation results show the importance of countryspecific variables in dealing with often significant dependencies that exist across macroeconomic variables. Although these results do not constitute a formal statistical test of the importance of the foreign variables in the GVAR model, they do provide an important indication of their usefulness in modelling global interdependencies. The results also show that once countryspecific models are formulated conditional on foreign variables, there remains only a modest degree of correlations across the shocks from different regions.
8. CONCLUDING REMARKS
 Top of page
 Abstract
 1. INTRODUCTION
 2. MODELLING INTERNATIONAL TRANSMISSIONS: A GVAR APPROACH
 3. THE GVAR MODEL (1979–2003)
 4. PAIRWISE CROSSSECTION CORRELATIONS: VARIABLES AND RESIDUALS
 5. ROBUSTNESS OF THE GVAR RESULTS TO TIMEVARYING WEIGHTS
 6. GENERALIZED IMPULSE RESPONSE FUNCTIONS
 7. IDENTIFICATION OF SHOCKS USING THE GVAR MODEL
 8. CONCLUDING REMARKS
 . APPENDIX
 Acknowledgements
 REFERENCES
 Supporting Information
This paper updates and extends the GVAR model of Pesaran et al. (2004) in a number of directions, provides an unobserved common factor interpretation of the countryspecific foreign variables included in the GVAR, addresses the issue of structural stability and shows how the model can be used for structural impulse response analysis. It also extends the geographical coverage from 11 country/regions to 26 countries with the euro area being treated as a single economy, updates the estimation period to the end of 2003 (from end of 1999 previously), adds the longterm interest rate in countryspecific models, and includes oil prices as an endogenous variable in the US model rather than treating it as a global exogenous variable. Also, the US model now allows for feedback effects from changes in output and inflation outside the US variables. The current version, therefore, captures more fully the interactions in the world economy and includes new channels of transmissions via bond markets, the feedback effects on oil prices from the global economy, and the changes in output and inflation from the rest of the world to the US economy.
Although the new GVAR model can be used for many different purposes, in this paper we have focused on its shortterm and longterm implications of external shocks for the euro area economy. We provide impact effects of external changes in interest rates (shortterm and longterm rates), inflation, output, real equity prices, real exchange rates and oil prices on the euro area and present the time profiles of these shocks using both generalized and structural impulse response functions.
The key to the GVAR modelling is the systematic inclusion of the countryspecific foreign variables in the individual country models in order to deal with the common factor dependencies that exist in the world economy. The average pairwise crosssection correlations computed for the endogenous variables, their first differences and the residuals from the GVAR model show that very little crosssection correlations remain once the effects of foreign variables have been taken into account. This is in line with the results of the tests of weak exogeneity of the foreign variables also reported in the paper. Considering the problem of structural breaks, we have found that structural instability is mainly confined to error variances and does not seem to adversely affect the coefficient estimates. To this end, we use robust standard errors when investigating the impact of the foreign variables and we base the analysis of the impulse responses on the bootstrap means and confidence bounds rather than point estimates.
In addition to generalized impulse response functions, we also consider structural identification of shocks in the global economy, and emphasize that, unlike the GIRFs, the results of structural impulse responses in general depend on the order in which different countries are included in the GVAR model. It is partly for this reason that in our structural impulse response analysis we focus on identification of shocks to the US economy, which we order as the first economy in the GVAR model. In particular, we consider the shortterm and longterm effects of a US monetary policy shock on the euro area.
From a policy analysis perspective, a number of interesting results emerge. The simulations clearly show that financial shocks are transmitted relatively rapidly, and often get amplified as they travel from the USA to the euro area. Equity and bond markets seem to be far more synchronous as compared to real output, inflation and shortterm interest rates. While the impact of an oil price shock on inflation is statistically significant, the impact on output remains limited despite some deterioration in the financing conditions through a tightening of monetary policy, an increase in longterm interest rates and a decrease in real equity prices.
Our analysis of monetary policy shocks has shown that the transmission of a change in US monetary policy to the euro area is limited and statistically insignificant. This result has been confirmed both from the GIRFs of a shock to US shortterm interest rates and from the IRFs of a monetary policy shock irrespective of the chosen ordering.
The model also highlights the importance of secondround effects of the shocks. A shock in the USA can be amplified because the USA will also be affected over time through the return impacts of output and inflation shocks in the rest of the world. The euro area in turn reacts to the US shocks directly as well as indirectly through the impact of the US shocks on euro area trade partners and so on. The transmission of shocks does not take place only through trade, but also as importantly through the impacts on financial variables with subsequent spillover effects on real variables. The GVAR presents a complicated, yet simpletofollow, spatiotemporal structure for the analysis of the world economy. To be sure, it can be modified and extended further. But it is hoped that the present version makes a further step towards the development of a transparent and coherent framework for the analysis of global interdependencies.