Assessing Interdependence Among Countries' Fundamentals and Its Implications for Exchange Rate Misalignment Estimates: An Empirical Exercise Based on GVAR

Exchange rates are important macroeconomic prices and changes in these rates affect economic activity, prices, interest rates, and trade flows. Methodologies have been developed in empirical exchange rate misalignment studies to evaluate whether a real effective exchange is overvalued or undervalued. There is a vast body of literature on the determinants of long-term real exchange rates and on empirical strategies to implement the equilibrium norms obtained from the theoretical models. This study seeks to contribute to this literature by showing that the global vector autoregressions model (GVAR) proposed by Pesaran and co-authors can add relevant information to the literature on measuring exchange rate misalignment. Our empirical exercise suggests that the estimative exchange rate misalignment obtained from GVAR can be quite different to that using the traditional cointegrated time series techniques, which treat countries as detached entities. The differences between the two approaches are more pronounced for small and developing countries. Our results also suggest a strong interdependence among eurozone countries, as expected.


Introduction
The exchange rate is an important macroeconomic price and changes in these rates aect economic activity, prices, interest rates, and trade ows. Large changes in an exchange rate always generate debate on whether the movements are "excessive", reect "fundamentals", or are "rational". Empirical studies have developed models to assess the long-term determinants of real exchange rates. Empirical strategies are then formulated based on these models, using the doctrine of purchasing power parity (PPP), or based on a fundamentals analysis.
Many studies have attempted to construct more accurate estimates of the magnitude and sign of exchange rate misalignment. Exchange rate misalignment is dened as the dierence between a measure of the real exchange rate and some equilibrium norm. Discussions on exchange rate misalignment can be divided into two levels. The rst focuses on which is the best norm to use to evaluate exchange rate equilibrium. Economic models give a better understanding of the determinants of the real exchange rate. These models attempt to determine the best set of fundamentals that explain real eective exchange rates in the long run. The second level of debate revolves around the best empirical strategy to measure exchange rate equilibrium norms. This is an econometric debate.
Empirical studies also need to choose between a time series or panel approach. The time series approach has the advantage of allowing a particular structure to be estimated for each country. However, the approach does not allow a broader set of variables to be analyzed at the same time because the available macroeconomic samples are not long enough. Panel techniques allow analysts to enlarge the spectrum of variables, but at the cost of imposing untested similarities between the parameters of dierent countries' models. Hossfeld (2009) reviews exchange rate misalignment literature, and evaluates the benets and limits of the time series and panel approaches.
This study seeks to contribute to the current body of literature by showing that the global vector autoregressions model (GVAR) proposed by Pesaran et al. can be used to model the interdependence between countries.
In addition, the model can add relevant information to the literature on measuring exchange rate misalignment.
As far as the authors are aware, this approach has not been applied to exchange rate misalignment estimation. This paper is divided into ve sections. The rst is this introduction. The second section provides a brief review of current literature on exchange rate misalignment determinants and describes the challenges faced by empirical studies in trying to determine whether a country's exchange rate is overvalued or undervalued. The third section presents the global vector autoregressive model (GVAR) and explains how to adapt Gonzalo and Granger's methodology to this framework. The fourth section describes the results of an empirical exercise that models real eective exchange rates for a selected group of countries. Here, we also present a comparative analysis of the traditional time series approach and the GVAR approach to exchange rate misalignment. Our results suggest that the GVAR approach is worth considering. The fth section applies the limits and the merits of the GVAR approach to the exchange rate misalignment problem and suggests possible extensions to our work. This section also concludes the paper. 2 2 A short review of exchange rate misalignment literature The literature on real exchange rates is extensive (Froot and Rogo, 1995). The classical doctrine, and perhaps the oldest one on real exchange rate determinants is that of purchasing power parity (PPP). Reference to this theory can be found in classic studies. Recent studies conrm the validity of PPP for tradable goods, although the adjustment towards equilibrium is quite slow. Ahmad and Craighead (2010) obtained strong evidence of a mean reversion with a high half-life using a secular consumer price index dataset for the United States and United Kingdom. Their work investigates the point made by Taylor (2001) on the eects of temporal aggregation on PPP tests.
There is also much theoretical discussion on which variables drive the real exchange rate in the long term.
Older studies include those of Edwards (1987Edwards ( , 1991, who analyzes the causes and consequences of exchange rate misalignment, and Dornbusch (1976), who developed the classic exible exchange rate model approach under which monetary policy shocks cause deviations from PPP fundamentals.
The studies of Bilson (1979) and Mussa (1976) are also classics. These are key references for the monetary approach to exchange rates. Under this approach, the exchange rate would be primarily driven by two fundamentals: the dierence between domestic and foreign income, and the money supply. The approach assumes that PPP and uncovered interest parity (UIP) hold continuously, and that the demand for money is stable in all countries. However, the research by Meese and Rogo (1983) casts doubt on the explanatory power of this theory by showing that the predictions of this approach are not superior to a naive forecast model for exchange rates, such as a pure random walk. Rossi (2013b) shows that the random walk can be outperformed by an econometric model that uses information based on net foreign investment position. Predictability is most apparent when one or more of the following hold: the predictors are Taylor rule and net foreign assets fundamentals; the model is linear; and a small number of parameters are estimated (Rossi, 2013a). Stein (1995) formulated the natural exchange rate approach (NATREX). According to the author, the equilibrium exchange rate is one that is equal to the level of investment and savings generated by economic fundamentals. Williamson (1994) had a signicant impact on exchange rate misalignment theory. Here, the equilibrium exchange rate is the one that allows a country to sustain a desirable result in its external accounts. This is referred to as the fundamental real exchange rate approach (FRER). A more recent reference to this approach is that of Cline (2008). A limitation of this approach is that choosing the target of foreign accounts is highly arbitrary and subjective. As a result, the results may not be robust to dierent targets. In addition, this approach focuses on ows, not stocks. Faruqee (1995) incorporates the evolution of stocks and constructed a model that allows ows and stocks to interact. Thus, there must be a stable relationship between the real exchange rate and the net foreign asset position between residents and non-residents. This is referred to as the behavioral real exchange rate (BRER) approach. The model was subsequently extended by Alberola et al. (1999). Kubota's (2009) model includes a representative agent who maximizes intertemporal consumption and ac-cumulates capital. This study indicates that the real exchange rate is a function of terms of trade, net external position, and the relative productivity of the tradable and non-tradable sectors. This approach seeks to reduce the degree of subjectivity when estimating exchange rate misalignment. To this end, she establishes a link between the real exchange rate and a set of fundamentals derived from a theoretical model. She then decomposes the series of real exchange rates into transitory and permanent components using the time series econometric technique.
Recently, the International Monetary Fund (IMF) began to systematically disseminate its research eorts into measuring the exchange rate misalignment in several of its member countries. Two documents were recently released. These works are an important advance towards transparency. The codes and dataset used to calculate the exchange rate misalignment are available on the IMF website, and the results are easy to replicate. The methodology is also a step forward in incorporating the role of policy gaps in exchange rate misalignment estimates.
The External Balance Assessment (EBA) methodology, developed by the IMF's research department, is based on two panel estimations: one for the current account and one for the real eective exchange rate (REER) indices.
1 The basic idea is that the REER can be written as a function of the output gap, real interest rate dierential, and factors that may aect saving, investment, current account, capital ows, and changes in foreign currency reserves.
The explanatory variables included in the EBA model are the commodity terms of trade, trade openness, share of administered prices, VIX, 2 the share of own currency in world reserves, nancial home bias, population growth, expected GDP growth over the next ve years, productivity, and changes in foreign reserves. The following policy-related regressors are also included: health expenditure to GDP, foreign exchange interventions, real short-term interest rate dierential, private credit to GDP, and capital controls. Most of the variables described are relative to the country's trade partners. They use the same weights as the REER calculation and/or interact with capital account openness. In addition, some variables are lagged to control for endogeneity.
The sample data covers 40 countries over the period 1990-2010. The model includes countries xed eects. To guarantee multilateral consistency in the results, the exchange rate misalignment must be adjusted.  4 3 Methodologies to calculate exchange rate misalignment

Traditional time series approach
The analysis starts with an estimation of a vector error correction model (VECM), as suggested by Johansen (1988), Johansen (1995), and Juselius (2009). The model is given by equation (1): where ε t are not correlated random errors, and Ω i is dened as the covariance matrix of the errors. The vector x i,t contains the variables for the real exchange rate and the fundamentals (e.g., net foreign investment position, etc.) and has dimension p, D t contains deterministic terms, and set of parameters to be estimated.

The Gonzalo and Granger decomposition
Several decompositions have been proposed to decompose the series into transitory and permanent components.
In general, the decomposition takes the following form: The existence of this decomposition is not always guaranteed, because the matrix c i β i⊥ may not have full rank. Gonzalo and Granger (1995) proposed c i = α i⊥ . This representation always exists for a model with a VECM of zero order. Johansen (1995) suggests c i = α i⊥ (Γ i1 + ... + Γ i1 − I). This decomposition always exists, provided that there are variables in the system with an order of integration of at most one. Kasa (1992) proposes β i⊥ . Another possibility is to generate forecasts from the VECM estimated for each point.
The estimative of exchange rate misalignment is the component associated with the position of the real exchange rate in vector x i,t . Assuming that the real exchange rate is in the rst position of the vector, and using the value of the error correction mechanism centered on their own means, one can calculate the misalignment using the following equation:

Motivation for a global model
The severity of the U.S. economic crisis in 2008 brought the fear of a strong negative contagion to the rest of the world. The U.S. authorities have subsequently adopted an aggressive monetary policy with a strong reduction in nominal interest rates and monetary expansion, among other measures. Some analysts may argue that this policy could have generated strong pressure to depreciate the U.S. dollar against currencies whose domestic interest rates did not follow the same movement. Countries that did not follow such a reduction and opted to accumulate reserves to prevent the appreciation of their currency could have faced inationary pressures.
Some authors argue that the United States was using its monetary policy to depreciate its currency, thereby fostering aggregate demand to reduce the intensity and duration of the economic slowdown. This policy may have generated repercussions around the world. There is much discussion about the extension of these eects and whether they are deleterious.
A global model must be constructed to assess the magnitude of eects, similar to those discussed in the previous paragraph. In this context, the GVAR appears to be an interesting option, as the relevance and magnitude of global factors, vis-a-vis domestic components, can be explicitly and properly evaluated and tested.

GVAR model
In this study, we apply the GVAR methodology to ascertain whether there is any external factor aecting the real exchange rate in the long or short run for a group of selected countries. In this sense, the measure of exchange rate misalignment may have two components. The rst is related to domestic fundamentals and the 6 second to global factors. The GVAR explains the source of external inuences on the domestic economy by including external variables in VARX. 4 External variables are usually assumed to be weakly exogenous for each country, as dened in Engle et al. (1983) and Hendry (1994).
In general, the GVAR can be described as a two-step approach. In the rst step, a specic model for each country is estimated using the variables of the country and the trading partners weighted average of external variables. Then, all individual models are stacked and grouped into a system of equations, which are solved.
Once this is done, the model provides options for dierent types of analyses, such as the forecast evaluation and impulse response analysis. Overall, there is a set of individual models represented as VARX that are combined to obtain the GVAR.
Following the notation of Pesaran et al. (2004), we restrict our discussion to the specication with rst-order dynamics, as represented by VARX (1,1). Consider a set of N countries. In this case, it follows that where x it is a vector of k i x1 specic variables for each country, x * i,t is a vector of k * i x1 foreign variables, i = 1, 2, ..., N e t = 1, 2, ..., T , Λ i,0 and Λ i,1 are matrices with parameters of the contemporaneous and lagged terms, a i,0 is a vector containing the constant, and a i,1 is the coecient associated with the time trend. The term it is a vector of idiosyncratic shocks for each country.
It is assumed that , and where σ ii,ls = cov( ilt , ils ) and s, l denote the variables for each of the countries in analysis i, respectively.
The shocks, it , are assumed to be weakly correlated across countries.
From (6), we can see that the domestic variable, x it , depends on the external variable x * it . The system from equation (6) needs to be solved for all domestic variables, (i = 1, ..., N ).
The external variables are dened in (11) 4 VARX is the vector autoregression (VAR) model that contains exogenous variables.
where W i is a weight matrix with dimension (k i + k * i )xk. The matrix W i reects the relationships between countries and allows the analyst to unify the model into a complete global model. For a specic country, i = 1, the matrix W i takes the form of where I is an identity matrix with dimension k.
To obtain the global VAR, we dene z it =    x it x * it    and rewrite (6) as where The terms A i and B i have dimension k i x(k i + k * i ), andA i has full column rank, k i .
At this stage, the endogenous domestic variables are stacked in a global vector of dimension kx1 The specic models for each country can be rewritten as function of x t . Using (11) and (12), we obtain where A i W i and B i1 W i have dimension k i xk. Finally, the stacked equations can be written as a GVAR (1): where . .
Assuming that G is not singular and has dimension kxk, the reduced form of (15) can be rewritten as: After estimating the models for each country separately from (6), we can solve the global model in (16) to obtain recursively the future values of all endogenous variables, (x t ).

The GVAR and Gonzalo and Granger decomposition
For the remainder of the paper, we will rewrite the GVAR as the global vector error correction model (GVECM).
The two are equivalent, but using the GVECM helps us to deal with permanent and transitory decompositions.
Assume that the model given by (17) ts the data well, and that it is part of the GVECM. Then: where ε t are random errors, not time correlated, Ω i is the respective covariance matrix for each country, and Stacking the models, it is possible to obtain where We can now write the Global VECM as: Dening Γ * 0,1 W ∆X t ≡Γ 0,1 ∆X t , and after some algebra, we obtain equation (20): Now, assume that we can calculate the inverse of matrix [I − Γ * 0,1 W ] and dening A * = [I − Γ * 0,1 W ] −1 A.
Then, the global model can be solved, yielding the solution to the global VECM, as shown in (21): The transitory component is given by (22): The permanent component is dened as the dierence between the actual values of the series and the transitory component given in (22). The matrix given by (23) contains the weights that each cointegrated relationship will contribute to the transitory component: The exchange rate misalignment can be calculated for country i by picking the country's real exchange rate in vector X t : In the following section, both estimative from equations (5)  There is still the possibility that some important variable was omitted. For example, a variable that controls the possible Balassa-Samuelson eect may alter the results towards nding stronger evidence of cointegration. 5 5 In this study, we could have analyzed a broader set of information using variables to control for the Balassa-Samuelson eect, similar to Kubota (2009) and Alberola et al. (1998). However, the number of countries in the sample would have been further reduced. We opted to explore a longer sample with a wider number of countries rather than a restricted sample with more variables. The inclusion of a variable to control for the Balassa-Samuelson eect reduces the sample in both the temporal and cross-sectional dimensions.

Is there evidence of global eects?
This section attempts to answer the question of whether the model with external factors is better than the model without these factors. Eight dierent specications were compared. Models with complete interdependence, in other words, that have external factors, are placed in both the short-and long-run dynamics, similarly to equation (17). There are models in which interdependence is allowed only in the long term. Another specication, the interdependence, is allowed only in the short run part. Finally, there are models in which no interdependence is allowed. These models are estimated while allowing for a structure with and without common cycles. 6 We have a total of eight dierent models. The models are compared using the Schwarz, Hannan-Quinn, and Akaike information criteria.

Calculating exchange rate misalignment using the GVECM
This section describes the results of the GVECM estimation. Table 6 shows the estimated cointegrated vectors for each country. Table 4 shows the results of the estimative loading factor given by (23). The value in Line i and Row j represents the weight of the error correction mechanism of country j that will be used to calculate the misalignment for country i. For example, we can check that the United States and Germany rows contain many non-zero terms. This suggests strong linkages between these economies and others economies analyzed in the sample. In the case of Germany, there seems to be a strong eect in eurozone countries. Brazil is an example of the opposite case. Here, the Brazilian exchange misalignment causes minor eects on all countries other than Uruguay. Although Brazil is a large economy, its global share is relatively small. Intuitively, the Brazilian economy is aected by others countries' disequilibrium, but its own disequilibrium does not aect others countries. The United States exchange rate misalignment may generate quite small eects on eurozone countries. Table 5 compares the results of the exchange rate misalignment using the traditional and GVAR methodologies. In general, the estimative misalignment tends to have the same sign for almost 67% of the sample. There are 1161 (=27*43) estimative of exchange rate misalignments, across all countries and periods. For the United States, the results are virtually the same in terms of sign and magnitude. The overall picture does not change when the comparison is made using the magnitude rather than the sign of the exchange rate misalignment. We compute the proportion of each case out of the total, where the estimative misalignments for both models have the same sign and absolute value above 10%, or dierent signs but absolute an below 10%. In the 57% of cases, these criteria were satised. However, in about 43% of cases, the estimates are not the same. The results in Table 5 suggest that quite dierent results can be obtained from the GVAR. The dynamics of real exchange rates in these countries cannot be seen as detached from the rest of world, or at least from their main trading 6 See Hecq et al. (2000 and 2002) for a common cycle denition, a discussion, and its relationship to permanent and transitory decomposition.
13 partners. Table 6 shows the estimates of all parameters necessary to solve the GVAR.  Table 4: Loading factor for calculating exchange rate misalignment from the GVAR model.       In order to evaluate the GVAR we must compare it to a more general model. We opt to include others sources of interdependence and structural change in the mean parameters of the model. This general model nests the GVAR structure.
We run the autometrics in two steps.
In the rst step the general unrestricted model (GUM) is given by the following equation: 7 See Doornik (2009) 18 where q contains all GVAR's country i variables. The superscript denotes the step.
In the second step the general unrestricted model contains the variables of the nal model in the rst step and broader set of variables: where FM denotes the variables in the nal model of step 1 and PC denotes the principal components of all RER and NFA variables, ecm denotes error correction mechanism and the remainder variables were dened previously in the paper.
The principal components variables were selected only in some few models. For some countries, the results of autometrics using minute instead of tiny signicance level seems more intuitive (e.g. Brazil or Germany). Spain is the only country that autometrics results are not easy to rationalize.

Discussion, limitations and possible extensions
The previous discussion on the merits and limitations of the time series and panel approaches is addressed in this section, as well as whether the GVAR model can be a bridge linking both approaches. The time series approaches allow little room to introduce fundamentals because of the sample size available in macroeconomic datasets. The panel approach allows a more exible structure and the inclusion of a larger group of fundamentals in the analysis. However, this approach must limit heterogeneity to a manageable level. It is not clear to what extent this can lead to distortions, since the main goal is to make assertions on specic units, not to assess the relevance of a group of variables in explaining real exchange rate movements and their average eect. A GVAR model can reconcile the merits of the two approaches, allowing us to map directly the eect of trading partner shocks on a country.
In the same way, it is possible to adapt the decomposition of Gonzalo and Granger (1995) to the GVAR environment. The same can be done for a Beveridge and Nelson decomposition, as shown in the work of Proetti (1997), under a VECM framework. The development of a model for the eurozone is a natural extension of our work. A regional factor can be easily added to the GVAR to map directly the interdependence between countries in the region. To the best of our knowledge, this has not been done before. Although we did not do so, our GVAR model was able to capture a strong interdependence eect among eurozone countries.
Even the IMF approach does not directly tackle the question of interdependence between countries. However, the IMF approach does have the benet of considering a wide range of fundamentals, and incorporates the role of policy gaps in determining the misalignment.

Final remarks
In this study, we estimated a global VAR to investigate the interdependence hypothesis among countries in terms of their real eective exchange rates and fundamentals. We were able to nd evidence in favor of interdependence in both the short and long run for some countries. In only a few cases in the sample could the null hypothesis of no interdependence not be rejected.
We also discussed the impact that the GVAR may have on exchange rate misalignment estimates. Here, we adapted the Gonzalo and Granger decomposition to a GVAR framework and conducted an empirical exercise to try to explain the relevance of global eects to exchange rate misalignment estimates. Our ndings show that the eects are greater for small or developing countries, because they tend to be more aected by global economy conditions.
Our global model was also capable of detecting important linkages between eurozone countries, as expected.
The United States and Germany, two leading economies in the world, seem to have an eect on the real exchange rate of other countries. However, their exchange rate misalignment estimates are only marginally aected in terms of magnitude and sign when both models' estimates are compared. The reason for this has to do with the dynamics of their real eective exchange rates, which are almost not aected by others countries' variables.
Finally, possible extensions to our approach include improving on country-specic models by using recent advances in time series model selection to investigate the role of a broader set of domestic variables. We can access not only the statistical signicance of external factors but their relative importance in relation to domestic 21 factors.