Assessing Interdependence Among Countries’ Fundamentals and its Implications for Exchange Rate Misalignment Estimates: An Empirical Exercise Based on GVAR^*

Abstract

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 theoretical models. This study seeks to contribute to this literature by showing that the global vector autoregressions model (GVAR) proposed by Pesaran, Schuermann, and Weiner (2004) can add relevant information to the literature on measuring exchange rate misalignment. Our empirical exercise suggests that the estimate 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.

Keywords
Real effective exchange rate; cointegration; global VAR

Resumo

Taxas de câmbio é um preço macroeconômico importante e as mudanças nessa taxa afetam a atividade econômica, o nível de preços, as taxas de juros e os fluxos comerciais. Metodologias foram desenvolvidas em estudos empíricos de desalinhamento cambial para avaliar se a taxa de câmbio real efetiva real está sobre ou sub valorizada. Existe uma literatura sobre os determinantes das taxas de câmbio reais de longo prazo e sobre estratégias empíricas para implementar as normas de equilíbrio obtidas a partir de modelos teóricos. A contribuição deste é mostrar que o modelo de global vetorial autoregressivo (GVAR) proposto por Pesaran, Schuermann, and Weiner (2004) pode agregar informações relevantes à mensuração do desalinhamento da taxa de câmbio. O nosso exercício empírico sugere que o desalinhamento da taxa de câmbio estimada obtido a partir do GVAR pode ser bastante diferente daquele que utiliza as técnicas tradicionais de séries temporais, que tratam os países como unidades independentes. As diferenças entre as duas abordagens são mais pronunciadas para os países pequenos e em desenvolvimento. Nossos resultados também sugerem uma forte interdependência entre os países da Zona do Euro.

1. Introduction

The exchange rate is an important macroeconomic price and changes in these rates affect economic activity, prices, interest rates, and trade flows. Large changes in an exchange rate always generate debate on whether the movements are “excessive”, reflect “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 defined as the difference between a measure of the real exchange rate and some equilibrium norm. Discussions on exchange rate misalignment can be divided into two levels. The first 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 effective 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 different countries’ models. Hossfeld (2010) reviews exchange rate misalignment literature, and evaluates the benefits 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, Schuermann, and Weiner (2004) 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 five sections. The first is this introduction. Section 2 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. Section 3 presents the global vector autoregressive model and explains how to adapt Gonzalo and Granger’s methodology to this framework. Section 4 describes the results of an empirical exercise that models real effective 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. section 5 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. A short review of exchange rate misalignment literature

The literature on real exchange rates is extensive (Froot & Rogoff, 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 confirm the validity of PPP for tradable goods, although the adjustment towards equilibrium is quite slow. Ahmad and Craighead (2011) 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 effects 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 (1989,1991), who analyzes the causes and consequences of exchange rate misalignment, and Dornbusch (1976), who developed the classic flexible 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 difference 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 Rogoff (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 (2013) 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 (2013).

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 significant 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 and Williamson (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 different targets. In addition, this approach focuses on flows, not stocks.

Faruqee (1995) incorporates the evolution of stocks and constructed a model that allows flows 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, Cervero, Lopez, and Ubide (1999).

Kubota (2009) includes a representative agent who maximizes intertemporal consumption and accumulates 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 nontradable 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 efforts into measuring the exchange rate misalignment in several of its member countries (IMF, 2013). 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 effective 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 differential, and factors that may affect saving, investment, current account, capital flows, 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, financial home bias, population growth, expected GDP growth over the next five 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 differential, 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 fixed effects. To guarantee multilateral consistency in the results, the exchange rate misalignment must be adjusted.

Given the results of the estimation, the “Total REER Gap” can be calculated as the sum of the regression residual and the “Total Policy Gap”. The policy gap is a measure of a cyclical gap (over a benchmark) in six policy areas: fiscal balance, capital controls, social spending, foreign exchange market intervention, financial policies, and monetary policy. The gap is calculated as the difference between the actual level of the variable and its “desirable” level, multiplied by the value of the estimated coefficient. The “desirable” levels are supplied by the desk of each IMF country.

3. Methodologies to calculate exchange rate misalignment

3.1 Traditional time series approach

The analysis starts with an estimation of a vector error correction model (VECM), as suggested by Johansen (1988, 1995). The model is given by equation (1):

(1) $\Delta {\mathit{x}}_{\mathit{it}}={\mathit{\alpha }}_{\mathit{i}}{{\mathit{\beta }}^{\prime }}_{\mathit{i}}{\mathit{x}}_{\mathit{i},\mathit{t}-1}+{\Gamma }_{\mathit{i}1}\Delta {\mathit{x}}_{\mathit{i},\mathit{t}-1}+\cdots +{\Gamma }_{\mathit{i},\mathit{k}}\Delta {\mathit{x}}_{\mathit{i},\mathit{t}-\mathit{k}+1}+{\Phi }_{\mathit{i}}{\mathit{D}}_{\mathit{t}}+{\epsilon }_{\mathit{it}},$

where 𝜀_it are not correlated random errors. The vector x_it 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 𝜃 = {𝛼_i, 𝛽_i, Γij, … Γ_{i, k-1}, Φ_i} is the 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. The decomposition of Gonzalo and Granger (1995) is widely used in exchange rate misalignment empirical literature.³ In their decomposition, the transitory components do not cause changes in the permanent component in the long term. In other words, exchange rate misalignment - defined as the transitory component of the real exchange rate in a multivariate equation system - does not contain relevant information for predicting the changes of the permanent components in the long term.

Using the parameters from (1), it is possible to calculate the transitory component (T_it) and the permanent component (P_it) from the following equations:

(2) ${\mathit{P}}_{\mathit{it}}={\mathit{\beta }}_{\mathit{i}}{\left({{\mathit{\alpha }}^{\prime }}_{\mathit{i}\perp }{\mathit{\beta }}_{\mathit{i}\perp }\right)}^{-1}{{\mathit{\alpha }}^{\prime }}_{\mathit{i}\perp }{\mathit{x}}_{\mathit{it},}$

(3) ${\mathit{T}}_{\mathit{it}}={\mathit{\alpha }}_{\mathit{i}}\left({{\mathit{\beta }}^{\prime }}_{\mathit{i}}{\mathit{x}}_{\mathit{it}}\right).$

The estimate of exchange rate misalignment is the component associated with the position of the real exchange rate in vector x_it. Assuming that the real exchange rate is in the first 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:

(4) ${\mathit{mis}}_{\mathit{it}}\equiv \left[10\cdots 0\right]{\mathit{\alpha }}_{\mathit{i}}{\left({{\mathit{\beta }}^{\prime }}_{\mathit{i}}{\mathit{\alpha }}_{\mathit{i}}\right)}^{-1}\left({{\mathit{\beta }}^{\prime }}_{\mathit{i}}{\mathit{x}}_{\mathit{it}}-𝔼\left[{{\mathit{\beta }}^{\prime }}_{\mathit{i}}{\mathit{x}}_{\mathit{it}}\right]\right).$

3.2 Motivation for a global model

tte severity of the U.S. economic crisis in 2008 brought the fear of a strong negative contagion to the rest of the world. ^e 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 inflationary 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. ttere is much discussion about the extension of these effects and whether they are deleterious.

We construct a global model to assess the magnitude of effects, 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.

3.2.1 GVAR model

In this study, we apply the GVAR methodology to ascertain whether there is any external factor affecting 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 first is related to domestic fundamentals and the second to global factors. tte GVAR explains the source of external influences on the domestic economy by including external variables in VARX.⁴ External variables are usually assumed to be weakly exogenous for each country, as defined in Engle, Hendry, and Richard (1983) and Hendry (1995).

In general, the GVAR can be described as a two-step approach. In the first step, a specific model for each country is estimated using the variables of the country and the trading partners weighted average of external variables. tten, all individual models are stacked and grouped into a system of equations, which are solved. Once this is done, the model provides options for different 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 specification with first-order dynamics, as represented by VAR(1,1). We consider a set of N countries and the VAR(1,1) below:

(5) ${\mathit{x}}_{\mathit{it}}={\mathit{a}}_{\mathit{i},0}+{\mathit{a}}_{\mathit{i},1}\mathit{t}+{\Phi }_{\mathit{i}}{\mathit{x}}_{\mathit{i},\mathit{t}-1}+{\Lambda }_{\mathit{i},0}{\mathit{x}}_{\mathit{i},\mathit{t}}^{*}+{\Lambda }_{\mathit{i},1}{\mathit{x}}_{\mathit{i},\mathit{t}-1}^{*}+{\epsilon }_{\mathit{it}},$

where x_it is a vector of k_t × 1 specific variables for each country; x^*_it is a vector of k^*_i × 1 foreign variables, i = 1,2,..., N, t = 1,2,..., T; Λ_i,0 and Λ_i,1 are respectively matrices with parameters of the contemporaneous and lagged terms; 𝑎_i,0 is a vector containing the constant; and 𝑎_i,1 is the coefficient associated with the time trend. The term 𝜀_it is a vector of idiosyncratic shocks for each country.

We construct the external variables using the trade weights, 𝑤,

(6) ${\mathit{x}}_{\mathit{it}}^{*}=\sum _{\mathit{j}=1}^{\mathit{N}}{\mathit{w}}_{\mathit{i},\mathit{j}}{\mathit{x}}_{\mathit{jt}},$

where $\mathrm{where}{\sum }_{\mathit{j}=1}^{\mathit{N}}{\mathit{w}}_{\mathit{ij}}=1\mathrm{with}\mathit{j}=1,\dots \mathit{N}\mathrm{and}{\mathit{w}}_{\mathit{ii}}=0\mathrm{with}\mathit{i}=1,\dots ,\mathit{N}$ with j = 1,..., N and 𝑤_ii = 0 with i = 1,..., N. From (5), we can see that the domestic variable, x_it, depends on the external variable x^*_it. The system from equation (5) needs to be solved for all domestic variables.

We can obtain the global VAR defining ${\mathit{z}}_{\mathit{it}}=\left(\begin{array}{c}{\mathit{x}}_{\mathit{it}}\hfill \\ {\mathit{x}}_{\mathit{it}}^{*}\hfill \end{array}\right)$ and we rewrite (5) as

(7) ${\mathit{A}}_{\mathit{i}}{\mathit{z}}_{\mathit{it}}={\mathit{a}}_{\mathit{i}0}+{\mathit{a}}_{\mathit{i}1}\mathit{t}+{\mathit{B}}_{\mathit{i}}{\mathit{z}}_{\mathit{i},\mathit{t}}+{\epsilon }_{\mathit{it}},$

where A_i = (I_ki,-Λ_i0), and B_i = (0_;-,A_;1). The terms A_i and B_i have dimension k_i × (k_i + k^*_i), and A_i has full column rank, k_i.

We can define the external variables as

(8) ${\mathit{z}}_{\mathit{it}}={\mathit{W}}_{\mathit{i}}{\mathit{z}}_{\mathit{t}},$

where W_i is a weight matrix with dimension $\left({\mathit{k}}_{\mathit{i}}+{\mathit{k}}_{\mathit{i}}^{*}\right)\times \mathit{k},\mathrm{where}\mathit{k}={\sum }_{\mathit{i}=1}^{\mathit{N}}{\mathit{k}}_{\mathit{i}}$. The matrix W_i reflects the relationship between countries and allows us to unify the model into a complete global model. Using (8) and (7), we obtain

(9) ${\mathit{A}}_{\mathit{i}}{\mathit{W}}_{\mathit{i}}{\mathit{z}}_{\mathit{t}}={\mathit{a}}_{\mathit{i}0}+{\mathit{a}}_{\mathit{i}1}\mathit{t}+{\mathit{B}}_{\mathit{i}}{\mathit{W}}_{\mathit{i}}{\mathit{z}}_{\mathit{t}-1}+{\epsilon }_{\mathit{it}},$

where A_iW_i and B_iW_i have dimension k_i × k. Finally, the stacked equations can be written as a GVAR(1):

(10) $\begin{array}{c}{\mathit{Gz}}_t={\mathit{a}}_0+{\mathit{a}}_1\mathit{t}+{\mathit{Hz}}_{t-1}+{\epsilon }_t,\\ \mathrm{where}{\mathit{a}}_{\mathit{o}}=\left[\begin{array}{c}{\mathit{a}}_{10}\\ {\mathit{a}}_{20}\\ ⋮\\ {\mathit{a}}_{\mathit{N}0}\end{array}\right],{\mathit{a}}_1=\left[\begin{array}{c}{\mathit{a}}_{11}\\ {\mathit{a}}_{21}\\ ⋮\\ {\mathit{a}}_{\mathit{N}1}\end{array}\right],{\epsilon }_{\mathit{t}}=\left[\begin{array}{c}{\epsilon }_{1\mathit{t}}\\ {\epsilon }_{2\mathit{t}}\\ ⋮\\ {\epsilon }_{\mathit{Nt}}\end{array}\right],\mathit{G}=\left[\begin{array}{c}{\mathit{A}}_1{\mathit{W}}_1\\ {\mathit{A}}_2{\mathit{W}}_2\\ ⋮\\ {\mathit{A}}_{\mathit{N}}{\mathit{W}}_{\mathit{N}}\end{array}\right],\mathit{H}=\left[\begin{array}{c}{\mathit{B}}_1{\mathit{W}}_1\\ {\mathit{B}}_2{\mathit{W}}_2\\ ⋮\\ {\mathit{B}}_{\mathit{N}}{\mathit{W}}_{\mathit{N}}\end{array}\right].\end{array}$

Assuming that G is not singular and has dimension kxk, the reduced form of (10) can be rewritten as:

(11) $\begin{array}{c}{\mathit{z}}_{\mathit{t}}={\mathit{b}}_0+{\mathit{b}}_1\mathit{t}+{\mathit{Fz}}_{\mathit{t}-1}+{\mathit{\upsilon }}_{\mathit{t}},\\ \mathrm{where}\mathit{F}={\mathit{G}}^{-1}\mathit{H},{\mathit{b}}_0={\mathit{G}}^{-1}{\mathit{a}}_0,{\mathit{b}}_1={\mathit{G}}^{-1}{\mathit{a}}_1\mathrm{and}{\mathit{\upsilon }}_{\mathit{t}}={\mathit{G}}^{-1}{\epsilon }_{\mathit{t}}.\end{array}$

After estimating the models for each country separately from (5), we can solve the global model in (11) to obtain recursively the future values of all endogenous variables (z_t).

3.2.2 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). Both are equivalent, but the GVECM allows us to deal with permanent and transitory decompositions.

Assuming the following GVECM and stacking the models, we obtain that

(12) $\begin{array}{c}\Delta {\mathit{X}}_{\mathit{t}}=\mathit{A}{{\mathit{BZ}}^{\prime }}_{\mathit{t}-1}+{\tilde{\Gamma }}_1\Delta {\mathit{Z}}_{\mathit{t}-1}+\cdots +{\tilde{\Gamma }}_{\mathit{k}-1}\Delta {\mathit{Z}}_{\mathit{t}-\mathit{k}+1}+\tilde{\Phi }{\mathit{D}}_{\mathit{t}}+{\tilde{\Gamma }}_{0,1}\Delta {\overline{\mathit{X}}}_{\mathit{t}}+{\tilde{\epsilon }}_{\mathit{t}},\hfill \\ \mathrm{where}\hfill \\ \mathit{A}=\left[\begin{array}{ccccc}{\tilde{\mathit{\alpha }}}_1& 0& \cdots & 0& 0\\ 0& {\tilde{\mathit{\alpha }}}_2& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & {\tilde{\mathit{\alpha }}}_{\mathit{N}-1}& 0\\ 0& 0& \cdots & 0& {\tilde{\mathit{\alpha }}}_{\mathit{N}}\end{array}\right];\mathit{B}=\left[\begin{array}{ccccc}{\tilde{\mathit{\beta }}}_1& 0& \cdots & 0& 0\\ 0& {\tilde{\mathit{\beta }}}_2& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & {\tilde{\mathit{\beta }}}_{\mathit{N}-1}& 0\\ 0& 0& \cdots & 0& {\tilde{\mathit{\beta }}}_{\mathit{N}}\end{array}\right];\hfill \\ {\tilde{\epsilon }}_{\mathit{t}}\mathrm{are}\mathrm{random}\mathrm{errors},\mathrm{not}\mathrm{time}\mathrm{correlated};\hfill \\ {{\mathit{X}}^{\prime }}_{\mathit{t}}={\left[{{\mathit{X}}^{\prime }}_{1,\mathit{t}}{{\mathit{X}}^{\prime }}_{2,\mathit{t}}\cdots {{\mathit{X}}^{\prime }}_{\mathit{N}-1,\mathit{t}}{{\mathit{X}}^{\prime }}_{\mathit{N},\mathit{t}}\right]}^{\prime };\hfill \\ {{\mathit{Z}}^{\prime }}_{\mathit{it}-1}={\left[{\mathit{x}}_{\mathit{i},\mathit{t}-1}^1{\overline{\mathit{x}}}_{\mathit{i},\mathit{t}-1}^1\cdots \cdots {\mathit{x}}_{\mathit{i},\mathit{t}-1}^{\mathit{p}}{\overline{\mathit{x}}}_{\mathit{i},\mathit{t}-1}^{\mathit{p}}\right]}^{\prime };\hfill \\ \mathrm{and}{\overline{\mathit{x}}}_{\mathit{i},\mathit{t}-1}^{\mathit{j}}\mathrm{is}\mathrm{the}\mathrm{average}\mathrm{of}\mathrm{variable}{\mathit{x}}^{\mathit{j}}\mathrm{for}\mathrm{country}\mathit{i}.\hfill \\ U\sin g{\mathit{Z}}_{\mathit{t}}=\mathit{W}{\mathit{X}}_{\mathit{t}},\mathrm{we}\mathrm{can}\mathrm{now}\mathrm{write}\mathrm{the}\mathrm{Global}\mathrm{VECM}\mathrm{as}\hfill \end{array}$

(13) $\Delta {\mathit{X}}_{\mathit{t}}=\mathit{A}{\mathit{BW}}^{\prime }{\mathit{X}}_{\mathit{t}-1}+{\tilde{\Gamma }}_1\Delta {\mathit{Z}}_{\mathit{t}-1}+\cdots +{\tilde{\Gamma }}_{\mathit{k}-1}\Delta {\mathit{Z}}_{\mathit{t}-\mathit{k}+1}+\tilde{\Phi }{\mathit{D}}_{\mathit{t}}+{\tilde{\Gamma }}_{0,1}\Delta {\overline{\mathit{X}}}_{\mathit{t}}+{\tilde{\epsilon }}_{\mathit{t}}.$

Defining ${\Gamma }_{0,1}^{*}\mathit{W}\Delta {\mathit{X}}_{\mathit{t}}\equiv {\tilde{\Gamma }}_{0,1}\Delta {\overline{\mathit{X}}}_{\mathit{t}}$, and after some algebra, we obtain equation (14):

(14) $\begin{array}{c}\left[\mathit{I}-{\Gamma }_{0,1}^{*}\mathit{W}\right]\Delta {\mathit{X}}_{\mathit{t}}=\mathit{A}{\mathit{BW}}^{\prime }{\mathit{X}}_{\mathit{t}-1}+{\tilde{\Gamma }}_1\Delta {\mathit{Z}}_{\mathit{t}-1}+\cdots +{\tilde{\Gamma }}_{\mathit{k}-1}\Delta {\mathit{Z}}_{\mathit{t}-\mathit{k}+1}+\tilde{\Phi }{\mathit{D}}_{\mathit{t}}+{\tilde{\epsilon }}_{\mathit{t}}.\hfill \\ \mathrm{Now},\mathrm{assume}\mathrm{that}\mathrm{we}\mathrm{can}\mathrm{calculate}\mathrm{the}\mathrm{inverse}\mathrm{of}\mathrm{matrix}\left[\mathit{I}-{\Gamma }_{0,1}^{*}\mathit{W}\right],\mathrm{we}\mathrm{define}\hfill \\ \mathit{A}*={\left[\mathit{I}-{\Gamma }_{0,1}^{*}\mathit{W}\right]}^{-1}\mathit{A},\hfill \\ {\tilde{\Gamma }}_{\mathit{l}}^{*}={\left[\mathit{I}-{\Gamma }_{0,1}^{*}\mathit{W}\right]}^{-1}{\tilde{\Gamma }}_{\mathit{l}},\mathit{l}=1,\dots ,\mathit{k}-1,\hfill \\ \mathrm{and}\hfill \\ {\tilde{\epsilon }}_{\mathit{t}}^{*}={\left[\mathit{I}-{\Gamma }_{0,1}^{*}\mathit{W}\right]}_{{\tilde{\epsilon }}_{\mathit{t}}}^{-1}.\hfill \end{array}$

Then, we can solve the global model, yielding the solution to the global VECM, as shown in (15):

(15) $\Delta {\mathit{X}}_{\mathit{t}}=\mathit{A}*{{\mathit{BWX}}^{\prime }}_{\mathit{t}-1}+{\tilde{\Gamma }}_1^{*}\Delta {\mathit{Z}}_{\mathit{t}-1}+\cdots +{\tilde{\Gamma }}_{\mathit{k}-1}^{*}\Delta {\mathit{Z}}_{\mathit{t}-\mathit{k}+1}+\tilde{\Phi }*{\mathit{D}}_{\mathit{t}}+{\tilde{\epsilon }}_{\mathit{t}}^{*}.$

So we obtain the transitory component as

(16) ${\mathit{T}}_{\mathit{t}}^{\mathrm{GVAR}}=\mathit{A}*{\left({\mathit{BWA}}^{\prime }*\right)}^{-1}{{\mathit{BWX}}^{\prime }}_{\mathit{t}}-𝔼\left[\mathit{A}*{\left({\mathit{BWA}}^{\prime }*\right)}^{-1}{{\mathit{BWX}}^{\prime }}_{\mathit{t}}\right].$

The permanent component is defined as the difference between the actual values of the series and the transitory component given in (16). The matrix given by (17) contains the weights that each cointegrated relationship will contribute to the transitory component:

(17) ${\mathit{LF}}^{\mathit{GVAR}}=\mathit{A}*{\left({\mathit{BWA}}^{\prime }*\right)}^{-1}.$

The exchange rate misalignment can be calculated for country i by picking the country’s real exchange rate in vector X_t,

(18) ${\mathit{mis}}_{\mathit{i},\mathit{t}}^{\mathit{GVAR}}\equiv \left[0\cdots 0_{\mathit{p}\left(\mathit{i}-1\right)}10\dots 0\right]{\mathit{T}}_{\mathit{t}}^{\mathit{GVAR}}.$

In the following section, we estimate the exchange rate misalignment of equations (4) and (18), and we compare the misalignment measures.

4. Results

4.1 Dataset

The database of this study is annual and covers the period from 1970 to 2012. The foreign trade figures were collected from the IMF Direction of Trade Statistics (DOTS-IMF). We use these weights to calculate the external variables in the GVAR model. We collect the real effective exchange rate from the IMF’s International Financial Statistics (IFS-IMF). We obtain the values of net foreign assets from Lane and Milesi-Ferretti (2007) and the IFS-IMF. The full sample consists of 33 countries, but we only analyze countries with data for all series and years during 1970 to 2012. This restricts the sample to 27 countries. Finally, we opt to work with end-of-period figures to avoid problems caused by data aggregation. In certain contexts, the temporal aggregation can cause significant distortions (Ghysels & Miller, 2015; Taylor, 2001).

4.2 Is there evidence of cointegration between real exchange rates and fundamen tals?

To assess the existence of cointegration between variables, we run the following cointegration tests: Engle and Granger (1987), Shin (1994), and Johansen (1988). The first two tests are univariate whereas the third is multivariate. The null hypothesis of the first test is absence of cointegration and alternative hypothesis is existence of cointegration. Second test is a confirmatory procedure whre the null hypothesis is existence of cointegration and alternative hypothesis is no cointegration. Trace and maximum eigenvalue tests proposed by Johansen if and how many cointegration relationships may exist.

Table 1 shows the results of the Engle and Granger (1987) and Shin (1994) cointegration tests, while Table 2 shows the results of the Johansen test for the sample countries. The null hypothesis of no cointegration is not rejected for many countries when looking at the cointegration of Engle and Granger or Johansen’s tests. However, this conclusion is not corroborated by Shin’s test. We do not reject the null hypothesis of cointegration at the 1 % level of significance for most countries when both domestic and external variables are used. The rejection of the null hypothesis of cointegration is more frequent when we investigate only the relationship between real exchange rate and domestic fundamentals. These results suggests that external factors may explain real exchange rates in the long term for some countries. Since no formal rejection of the null hypothesis of cointegration occurs at the 1 % level when all variables are used, we choose to work with the hypothesis that there exists one cointegrated relationship between the variables in specific countries’ models. Another point to justify this choice is the cost of keeping irrelevant variable in econometric model compared to exclude a relevant variable. Omitting a relevant variable can induce bias in estimates of remaining parameters.

Table 1
Results of univariate cointegration test.

	Engle and Granger test					Shin cointegration test
Country\Lag	0	1	2		N. of regressors	1	2	3
Australia	-1.33	-1.38	-3.27			0.276	0.149	0.131
Austria	-2.29	-2.99	-3.40			0.287	0.167	0.161
Belgium	-2.97	-3.86	-3.84			0.925	0.510	0.380
Brazil	-2.41	-2.51	-3.06			0.300	0.069	0.070
Canada	-2.23	-2.16	-2.63			0.171	0.178	0.055
China	-1.57	-1.61	-2.45			0.367	0.372	0.199
Colombia	-3.35	-2.51	-2.50			0.821	0.814	0.092
Denmark	-3.49	-3.58	-3.55			0.163	0.306	0.289
Finland	-1.53	-1.92	-2.01			0.054	0.066	0.066
France	-2.05	-3.16	-3.10			0.576	0.290	0.272
Germany	-2.11	-2.66	-2.68			0.788	0.312	0.355
Greece	-2.29	-2.23	-2.33			0.350	0.087	0.088
India	-1.26	-1.86	-1.84			0.202	0.237	0.146
Indonesia	-1.28	-1.64	-1.93			0.344	0.059	0.053
Ireland	-1.91	-2.90	-3.09			1.441	0.703	0.619
Italy	-3.37	-3.48	-3.94			0.929	0.472	0.101
Japan	-2.33	-3.63	-3.55			0.082	0.108	0.074
Mexico	-2.74	-2.74	-2.97			0.417	0.179	0.068
Netherlands	-2.78	-3.28	-3.80			0.307	0.349	0.089
Singapore	-1.98	-2.96	-2.98			0.128	0.392	0.126
South Africa	-2.96	-2.35	-2.59			0.365	0.389	0.384
Spain	-3.08	-3.29	-3.27			0.244	0.170	0.070
Sweden	-1.31	-2.28	-2.28			0.167	0.093	0.085
United Kingdom	-2.87	-2.85	-2.58			1.192	0.234	0.245
United States	-3.04	-3.04	-3.31			0.200	0.223	0.164
Uruguay	-2.48	-2.50	-2.46			0.116	0.172	0.148
Turkey	-1.43	-1.97	-1.96			0.390	0.159	0.138
Critical Values					Critical Values
10%	-3.07	-3.45	-3.83
5%	-3.37	-3.77	-4.11		5%	0.314	0.221	0.159
1%	-3.96	-4.31	-4.73		1%	0.533	0.380	0.271

Table 2
Results of multivariate cointegration test.

	Countries Trace Statistics: Null Hypothesis
	N. of vectors	Absence	One vector
Australia	1	40.21	10.14
Austria	1	30.01	12.93
Belgium	0	15.88	3.21
Brazil	1	35.69	12.60
Canada	0	11.20	4.58
China	1	39.88	8.38
Colombia	0	26.91	12.70
Denmark	0	24.09	5.30
Finland	0	14.84	5.42
France	0	21.10	2.48
Germany	0	14.94	2.64
Greece	0	17.70	5.46
India	1	28.66	5.90
Indonesia	0	25.60	3.30
Ireland	0	15.78	2.99
Italy	0	18.21	5.73
Japan	0	26.89	6.55
Mexico	0	27.25	5.93
Netherlands	0	24.22	7.22
Singapore	0	17.37	6.40
South Africa	1	32.31	10.03
Spain	0	23.35	9.63
Sweden	0	22.55	6.32
United Kingdom	0	21.30	6.17
United States	0	23.25	8.83
Uruguay	0	16.05	5.16
Turkey	1	28.11	6.66

The specification may be improved by adding other variables. For example, a variable that controls the possible Balassa-Samuelson effect may alter the results towards finding stronger evidence of cointegration.⁵

4.3 Is there evidence of global effects?

This section attempts to answer the question of whether the model with external factors is better than the model without these factors. We compare eight different specifications. We consider models with complete interdependence, in other words, that have external factors, in the short and long term dynamics, similar to the equation (12). There are models in which interdependence is allowed only in the long term. In another specification, we allow interdependence only on short-term dynamics. Finally, there are models in which no interdependence is allowed. We estimate the models also allowing a structure with or without

common cycles.⁶ We compare these eight models using the Schwarz (SC), Hannan-Quinn (HQ), and Akaike information criteria (AIC).

Table 3 presents a detailed description of the models and all information criteria. Options 3 and 4 consist of models with no interdependence. The evidence in favor of interdependence varies between countries. According to all information criteria, there is no evidence of interdependence for the following countries: Australia, Ireland, India, Netherlands, and Turkey. For other countries, there is evidence of interdependence in the short and/or long term. Thus, we can conclude that statistical evidence corroborates the hypothesis of interdependence for a large group of countries. The next step in the analysis is to assess the relevance of global and domestic factors to exchange rate misalignment estimate.

Table 3
Results of the tests for interdependence.

		External Variables in the Model
Models	Common Features	Short Run	Long Run	Interdependence
1	No	Yes	Yes	Yes, Both types
2	Yes	Yes	Yes	Yes, Both types
3	No	No	No	No
4	Yes	No	No	No
5	No	Yes	No	Yes, Short Run
6	Yes	Yes	No	Yes, Short Run
7	No	No	Yes	Yes, Long Run
8	Yes	No	Yes	Yes, Long Run
	Best Model - Information Criteria
Country	SC	HQ	AIC
Australia	8	8	2
Austria	6	6	6
Belgium	6	6	1
Brazil	8	8	8
Canada	4	4	5
China	8	8	8
Colombia	6	6	2
Denmark	4	4	2
Finland	3	3	2
France	8	1	1
Germany	4	4	4
Greece	4	4	4
India	8	8	8
Ireland	6	6	6
Italy	8	8	8
Japan	6	1	1
Mexico	4	8	8
Netherlands	2	2	2
Singapore	4	4	5
South Africa	8	2	2
Spain	4	4	2
Sweden	6	2	2
United Kingdom	6	5	5
United States	6	6	5
Uruguay	4	6	6
Turkey	6	2	2

4.4 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 estimate loading factor given by (17) for real exchange rate equation. The value in line i and column 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 columns 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 effect in eurozone countries. Brazil is an example of the opposite case. Here, the Brazilian exchange rate misalignment causes minor effects on all countries other than Uruguay. Although Brazil is a large economy, its global share is relatively small. Intuitively, the Brazilian economy is affected by others countries’ disequilibrium, but its own disequilibrium does not affect others countries. The United States exchange rate misalignment may generate quite small effects on eurozone countries.

Table 4
Loading factor for calculating exchange rate misalignment from the GVAR model.

	Error Correction Mechanism
	AUS𝔞	AUT	BEL	BRA	CAN	CHN	COL	DEN	FIN	FRA	DEU	GRC	IND	IRL
Australia	0.55	0.00	0.04	-0.01	-0.04	0.01	0.00	0.02	0.00	0.24	0.08	0.01	-0.01	0.03
Austria	0.00	0.13	-0.01	0.00	0.01	0.00	0.00	-0.04	0.00	-0.17	0.34	0.01	0.00	0.01
Belgium	0.01	0.00	0.97	0.00	0.01	0.00	0.00	-0.09	0.00	-0.32	0.31	0.01	0.02	0.04
Brazil	0.02	-0.01	-0.16	1.26	0.24	-0.04	0.02	-0.18	0.00	-1.94	-0.02	0.00	0.03	0.09
Canada	0.00	0.00	0.00	0.00	1.02	0.00	0.00	0.00	0.00	-0.03	-0.01	0.00	0.00	0.00
China	-0.05	0.00	0.00	-0.06	-0.07	-0.22	0.00	-0.02	0.00	0.34	-0.09	0.00	-0.05	0.04
Colombia	0.01	0.00	0.06	0.06	0.19	-0.01	0.53	-0.04	0.00	-0.36	0.36	0.01	0.02	0.01
Denmark	0.00	0.03	0.25	0.02	0.05	-0.01	0.00	0.92	-0.03	1.46	1.02	0.08	0.01	0.00
Finland	0.01	0.04	0.47	0.03	0.08	-0.01	0.00	-0.37	0.00	2.83	1.11	0.12	0.02	0.07
France	0.02	0.08	1.11	0.06	0.16	-0.02	0.00	-0.41	-0.01	7.01	2.55	0.28	0.05	0.13
Germany	0.00	0.01	0.10	0.01	0.02	0.00	0.00	-0.03	0.00	0.29	1.20	0.02	0.00	0.01
Greece	0.00	0.00	-0.04	0.00	0.00	0.00	0.00	0.01	0.00	-0.33	0.00	0.05	0.00	-0.01
India	0.05	0.00	-0.01	0.04	0.18	-0.03	0.01	0.02	0.00	-1.57	-0.13	-0.01	1.09	-0.18
Ireland	-0.04	-0.02	-0.40	-0.03	-0.09	0.01	0.00	-0.10	0.01	-1.72	-0.89	-0.07	-0.04	0.83
Italy	-0.02	-0.02	-0.87	-0.07	-0.11	0.02	0.00	0.23	0.01	-5.04	-2.04	-0.05	-0.04	0.01
Japan	-0.02	0.00	-0.03	-0.02	-0.14	0.04	0.00	0.04	0.00	-0.08	-0.12	-0.02	-0.01	-0.01
Mexico	-0.01	0.00	0.00	-0.01	-0.12	0.00	0.00	0.00	0.00	0.03	0.02	0.00	0.00	0.02
Netherlands	0.01	0.00	0.06	0.00	0.00	0.00	0.00	-0.10	0.00	-0.39	0.32	0.01	0.00	0.04
Singapore	0.02	0.00	0.01	0.00	0.01	0.00	0.00	0.00	0.00	0.03	0.02	0.00	0.01	0.00
Spain	0.00	-0.07	-0.94	-0.06	-0.11	0.02	-0.01	0.34	0.01	-6.16	-2.10	-0.21	-0.03	0.03
Sweden	0.02	0.03	0.51	0.03	0.09	-0.01	0.00	-1.68	-0.06	3.00	0.60	0.11	0.03	0.15
UK	-0.05	-0.03	-0.43	-0.03	-0.15	0.01	0.00	0.18	0.01	-1.30	-0.99	-0.09	-0.05	-0.27
USA	0.01	0.00	0.01	0.01	0.13	0.00	0.00	-0.01	0.00	0.02	0.01	0.00	0.00	-0.01
Uruguay	0.02	0.01	0.29	-0.72	0.06	-0.02	0.00	0.08	-0.01	1.24	0.77	0.15	0.04	0.11
South Africa	-0.02	0.01	0.07	-0.01	0.00	0.05	0.00	-0.03	0.00	1.11	0.07	0.02	-0.10	0.05
Indonesia	-0.05	0.00	0.00	0.00	0.02	0.01	0.00	-0.03	0.00	0.25	0.02	0.01	-0.06	0.02
Turkey	0.01	0.02	-0.04	0.00	0.04	-0.01	0.00	-0.05	0.00	-0.75	0.62	0.08	0.01	0.05
	ITA	JAP	MEX	NLD	SGP	ESP	SWE	GBR	USA	URY	ZAF	IDN	TUR
Australia	0.04	0.02	-0.02	0.02	0.05	0.02	0.00	-0.03	-0.08	0.00	0.00	0.00	0.00
Austria	0.05	0.00	0.00	0.02	0.00	0.00	0.00	-0.01	0.01	0.00	0.00	0.00	0.00
Belgium	-0.05	0.01	0.00	0.30	0.00	0.00	0.00	-0.05	0.03	0.00	0.00	0.00	0.00
Brazil	-0.35	0.16	0.23	0.54	-0.02	-0.02	0.00	-0.09	0.39	0.05	0.01	0.00	0.00
Canada	0.00	0.02	0.01	0.00	0.00	0.00	0.00	0.00	0.11	0.00	0.00	0.00	0.00
China	-0.06	-0.96	-0.05	-0.03	0.06	0.01	0.00	-0.03	-0.29	0.00	-0.01	0.01	0.00
Colombia	0.02	0.15	0.14	0.14	-0.01	0.07	0.00	0.00	0.66	0.00	0.00	0.00	0.00
Denmark	0.36	0.06	0.03	0.32	0.01	0.17	-0.03	-0.03	0.07	0.00	0.00	0.00	0.00
Finland	0.52	0.02	0.04	0.42	0.02	0.29	0.03	-0.09	0.11	0.00	0.01	0.00	0.00
France	1.27	0.06	0.09	0.90	0.04	0.68	0.00	-0.18	0.23	0.00	0.02	-0.01	0.01
Germany	0.12	0.00	0.01	0.09	0.00	0.05	0.00	-0.01	0.03	0.00	0.00	0.00	0.00
Greece	0.09	0.00	0.00	-0.02	0.00	-0.03	0.00	0.01	0.00	0.00	0.00	0.00	0.00
India	-0.04	0.87	0.06	0.03	0.18	-0.13	0.01	0.12	0.57	0.00	0.02	-0.02	0.00
Ireland	-0.48	-0.07	-0.04	-0.20	0.01	-0.22	0.02	0.25	-0.14	0.00	-0.03	0.00	0.00
Italy	0.16	-0.05	-0.08	-0.68	-0.03	-0.55	0.00	0.04	-0.20	0.00	-0.02	0.00	0.00
Japan	-0.01	1.16	-0.07	-0.06	-0.15	-0.02	0.00	0.01	-0.22	0.00	0.00	0.01	0.00
Mexico	0.00	-0.01	1.42	0.03	0.04	0.00	0.00	-0.02	0.07	0.00	0.00	0.00	0.00
Netherlands	-0.07	0.00	0.00	0.87	0.00	-0.01	0.00	-0.04	0.01	0.00	0.00	0.00	0.00
Singapore	0.01	-0.01	0.01	0.01	1.13	0.00	0.00	0.00	0.03	0.00	0.00	-0.01	0.00
Spain	-1.49	-0.04	-0.10	-0.70	-0.05	0.39	-0.01	0.02	-0.17	-0.01	-0.01	0.00	0.00
Sweden	0.42	-0.04	0.05	0.38	0.03	0.28	0.02	-0.16	0.13	0.00	0.02	0.00	0.00
UK	-0.39	-0.09	-0.06	-0.43	-0.03	-0.23	0.00	0.16	-0.24	0.00	-0.03	0.00	0.00
USA	0.01	0.03	0.06	0.00	-0.01	0.00	0.00	0.01	0.43	0.00	0.00	0.00	0.00
Uruguay	0.20	0.10	0.04	0.24	0.01	0.15	0.00	-0.11	0.20	0.94	0.01	0.00	0.00
South Africa	0.15	-0.22	0.00	-0.04	-0.01	0.04	0.00	-0.02	-0.02	0.00	0.41	0.00	0.00
Indonesia	-0.01	-1.10	0.01	0.00	0.05	0.00	0.00	-0.02	-0.11	0.00	0.00	-0.16	0.00
Turkey	-0.01	0.01	0.01	0.06	0.01	0.00	0.00	-0.06	0.07	0.00	0.00	0.00	0.02

• ^𝔞
  ISO alpha-3 country codes: AUS (Australia), AUT (Austria), BEL (Belgium), BRA (Brazil), CAN (Canada), CHN (China), COL (Colombia), DEN (Denmark), FIN (Finland), FRA (France), DEU (Germany), GRC (Greece), IND (India), IRL (IRELAND), ITA (Italy), JPN (Japan), MEX (Mexico), NLD (Netherlands), SGP (Singapore), ESP (Spain), SWE (Sweden), GBR (United Kingdom), USA (United States of America), URY (Uruguay), ZAF (South Africa), IDN (Indonesia), TUR (Turkey).

Table 5
Comparing exchanges rate misalignment estimates.

	Percentage of case where:
	Sign of misalignment is the same (%)	Magnitude of misalignment is the same𝔞 (%)
Australia	58.14	76.74
Austria	48.84	86.05
Belgium	65.12	81.40
Brazil	74.42	60.47
Canada	79.07	51.16
China	62.79	25.58
Colombia	67.44	76.74
Denmark	69.77	48.84
Finland	55.81	39.53
France	67.44	48.84
Germany	93.02	79.07
Greece	76.74	55.81
India	60.47	44.19
Ireland	60.47	51.16
Italy	51.16	34.88
Japan	60.47	32.56
Mexico	81.40	74.42
Netherlands	65.12	83.72
Singapore	88.37	90.70
Spain	62.79	27.91
Sweden	74.42	46.51
United Kingdom	62.79	41.86
United States	79.07	86.05
Uruguay	81.40	60.47
South Africa	55.81	41.86
Indonesia	41.86	58.14
Turkey	55.81	44.19
Total	66.67	57.36

• ^𝔞
  If both estimates give a value of misalignment above 10% in modulus or below 10% in modulus simultaneously.

Table 6
Coefficient estimates for each country model used to solve the GVAR.

Country	Loading Matrix			Cointegrated Vectors						Contemporaneous Effect
	α i1	α i2		βi,RER	βi,NFA	βi,RERw	βi,NFAw	βi,Constant		C i,11	C i,12	C i,21	C i,22
Australia	-0.45	0.27		1	-1.34	0.71	-2.07	-8.21		-0.55	0.55	0.54	-0.66
Austria	-0.07	-0.37		1	1.31	0.23	-2.25	-5.43		0.90	-0.10	-0.04	0.06
Belgium	-0.39	-0.03		1	-0.13	-0.86	-0.77	-0.60		1.11	-0.23	-0.29	0.47
Brazil	-0.89	-0.18		1	-1.19	0.85	-3.21	-9.18		-0.25	2.74	0.82	-0.44
Canada	-0.16	-0.03		1	0.00	-0.37	0.11	-2.79		-0.02	1.22	0.10	-0.49
China	0.06	0.04		1	-8.88	-18.54	14.25	80.15		0.42	0.10	-0.39	0.41
Colombia	-0.08	0.01		1	-6.45	4.47	0.48	-26.80		0.39	1.58	0.10	-0.14
Denmark	-0.05	0.29		1	0.10	-2.26	1.03	6.12		-0.01	0.46	-1.09	-0.78
Finland	0.00	0.41		1	-1.27	-2.48	7.74	6.92		0.62	0.47	0.41	5.56
France	-0.82	0.59		1	0.18	-1.49	0.18	2.24		1.15	-0.62	0.42	-0.52
Germany	-0.45	-0.18		1	-0.15	-0.80	0.14	-0.89		0.52	-0.96	0.33	0.15
Greece	0.00	-0.31		1	0.36	-1.84	-1.81	4.28		0.61	-0.25	-0.49	-0.69
India	-0.32	-0.03		1	-1.42	-7.74	0.49	31.01		1.87	0.63	0.09	-0.27
Ireland	-0.52	0.17		1	0.30	-0.68	2.44	-1.19		-0.44	-0.90	0.75	-4.21
Italy	-0.46	0.14		1	0.09	2.25	-2.56	-15.09		-0.17	0.62	-0.39	-0.01
Japan	-0.48	-0.11		1	0.14	1.41	0.70	-11.19		-1.36	-1.91	-0.19	-0.53
Mexico	-1.05	-0.40		1	-0.83	-0.41	0.84	-2.95		0.98	0.42	0.06	-0.13
Netherlands	-0.21	-0.55		1	0.00	-0.56	-1.85	-1.94		1.33	-0.05	-1.37	1.72
Singapore	-0.20	-0.45		1	-0.05	-0.51	0.08	-2.19		0.64	0.24	-0.80	-2.65
Spain	-0.36	-0.17		1	0.16	2.97	-4.67	-18.32		0.29	0.82	-0.88	1.40
Sweden	-0.04	-0.27		1	2.07	-8.08	-1.09	32.92		0.59	-0.23	2.26	0.13
UK	-0.04	0.09		1	-3.56	9.64	-12.90	-50.20		-2.27	1.22	-0.10	-2.19
USA	-0.12	-0.06		1	2.35	-2.59	3.40	7.75		-0.25	0.46	0.59	-0.68
Uruguay	-0.42	0.02		1	-1.85	2.16	-3.16	-15.47		-0.60	-0.53	0.09	1.19
South Africa	-0.41	0.14		1	-4.45	-7.39	-1.89	28.86		1.98	-3.85	-0.72	-0.32
Indonesia	0.05	0.07		1	-5.11	-10.19	4.05	39.97		-1.60	-0.59	-1.94	-3.51
Turkey	-0.02	-0.05		1	17.13	-53.64	9.75	249.77		1.10	0.88	1.42	-1.75

Table 5 compares the results of the exchange rate misalignment using the traditional and GVAR methodologies. In general, the estimate misalignment tends to have the same sign for almost 67% of the sample. There are 1,161 (27 × 43) estimate 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 estimate misalignments for both models have the same sign and absolute value above 10%, or different signs but absolute an below 10%. In the 57% of cases, these criteria were satisfied. However, in about 43% of cases, the estimates are not the same. The results in Table 5 suggest that quite different 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 partners. Table 6 shows the estimates of all parameters necessary to solve the GVAR.

Table 7 provides information about the source of the differences between the two methodologies. First, the coefficient of the NFA variable changes quite significantly for many countries. This may explain part of the change in the results and highlights the importance of investigating the main drivers of real exchange rates in the long run. However, there are countries where the coefficient hardly changes at all, but the external variables cause changes in the magnitude of the exchange misalignment estimate. Brazil is a good example. Although there is a minor change in the NFA coefficient when external variables are included in the model, they add relevant information to the long run level of the exchange rate. The misalignment of the exchange rate for both models is different after introducing foreign variables.

Table 7
Comparing the cointegrated coefficient of the GVAR and traditional methodologies.

	Cointegrated Relationship											Difference between NFA long run coefficients in both models
	GVAR						Traditional Methodologhy
	RER	NFA	RERw	NFAw	Constant		RER	NFA	RERw	NFAw	Constant
Australia	1	-1.3	0.7	-2.1	-8.2		1	-0.9	0	0	-4.9	-0.4
Austria	1	1.3	0.2	-2.2	-5.4		1	-0.1	0	0	-4.6	1.4
Belgium	1	-0.1	-0.9	-0.8	-0.6		1	0.0	0	0	-4.6	-0.2
Brazil	1	-1.2	0.9	-3.2	-9.2		1	-1.3	0	0	-5.1	0.1
Canada	1	0.0	-0.4	0.1	-2.8		1	1.0	0	0	-4.3	-1.0
China	1	-8.9	-18.5	14.2	80.2		1	-2.1	0	0	-4.5	-6.8
Colombia	1	-6.5	4.5	0.5	-26.8		1	-5.0	0	0	-5.5	-1.4
Denmark	1	0.1	-2.3	1.0	6.1		1	-0.2	0	0	-4.6	0.3
Finland	1	-1.3	-2.5	7.7	6.9		1	-0.3	0	0	-4.8	-0.9
France	1	0.2	-1.5	0.2	2.2		1	-0.3	0	0	-4.6	0.5
Germany	1	-0.2	-0.8	0.1	-0.9		1	-0.1	0	0	-4.6	-0.1
Greece	1	0.4	-1.8	-1.8	4.3		1	0.0	0	0	-4.3	0.3
India	1	-1.4	-7.7	0.5	31.0		1	-8.1	0	0	-5.6	6.7
Ireland	1	0.3	-0.7	2.4	-1.2		1	0.2	0	0	-4.4	0.1
Italy	1	0.1	2.3	-2.6	-15.1		1	0.2	0	0	-4.6	-0.1
Japan	1	0.1	1.4	0.7	-11.2		1	0.3	0	0	-4.5	-0.1
Mexico	1	-0.8	-0.4	0.8	-3.0		1	-0.4	0	0	-4.7	-0.4
Netherlands	1	0.0	-0.6	-1.8	-1.9		1	-0.2	0	0	-4.6	0.2
Singapore	1	-0.1	-0.5	0.1	-2.2		1	0.0	0	0	-4.6	0.0
Spain	1	0.2	3.0	-4.7	-18.3		1	0.1	0	0	-4.4	0.0
Sweden	1	2.1	-8.1	-1.1	32.9		1	-1.0	0	0	-4.9	3.1
United Kingdom	1	-3.6	9.6	-12.9	-50.2		1	-0.2	0	0	-4.8	-3.3
United States	1	2.4	-2.6	3.4	7.8		1	-0.5	0	0	-4.7	2.8
Uruguay	1	-1.9	2.2	-3.2	-15.5		1	-1.3	0	0	-4.9	-0.5
South Africa	1	-4.5	-7.4	-1.9	28.9		1	1.5	0	0	-4.5	-5.9
Indonesia	1	-5.1	-10.2	4.1	40.0		1	-4.0	0	0	-6.6	-1.1
Turkey	1	17.1	-53.6	9.7	249.8		1	0.2	0	0	-4.2	16.9

We show exchange rate misalignment estimates from traditional and GVAR methodologies in Figures 1 and 2. The case of Brazil and USA are shown as examples. For the USA, estimates of real exchange rate misalignment by traditional and GVAR methodologies differ in magnitude for most of the sample according to Figure 1. The amplitude of misalignment is lower by GVAR model. The Brazilian estimates of misalignment by traditional and GVAR methodologies are quite similar until 2008, according to Figure 2. In the end of the sample, particularly after the financial crisis of 2008, the results are different for both of the countries.

Figura 1
Real exchange rate misalignment estimate – GVAR versus traditional methodology – United States.

Figura 2
Real exchange rate misalignment estimate – GVAR versus traditional methodology – Brazil.

In a recent paper, Ericsson and Reisman (2012) proposes a refinement of the GVAR approach using a model selection procedure called Autometrics.⁷ This refinement allows one to search for points of instability and structural changes in the modeling process. It is also possible to incorporate a wider range of information sets in each country model. However, this must occur within a rigorous general-to-specific econometric modeling approach in the spirit of the London School of Economics (LSE) tradition developed by David Hendry. This reduces the possibility of criticism that a time series analysis can only manage a few real exchange rate determinants at a time, as compared to panel models. By using the GVAR refinement proposed by Ericsson and Reisman (2012), it is possible, for example, to match the merits of the IMF approach, with its wide range of variables, to a panel with the merits of a global model. One can opt to model different set fundamentals for each country.

The interdependence among countries is introduced in GVAR by adding trade weighted foreign variables in the model of each specific country. We opt to use Autometrics search algorithm to check if these variables explain the dynamics of each country models. The traditional methodology can be extended in many directions. The GVAR is one possible way but not the only one. We decide to construct indicators functions (impulse, step and time trend dummies) to allow for structural change in mean of the process. We also run principal components analysis on all first differences of the logarithm of real effective exchange rate and net foreign assets and use the factors as possible regressors. The rationality for this relies on possible existence of another global factor not well captured by GVAR averages variables.

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 first step the general unrestricted model (GUM) is given by the following equation:

(19) $\begin{array}{c}\Delta {\mathit{x}}_{\mathit{i},\mathit{t}}=\sum _{\mathit{h}=1}^{\mathit{q}}{\mathit{\theta }}_{1\mathit{h}}^1{\mathit{V}}_{1\mathit{h}}^{\mathit{i}}+\sum _{\mathit{h}=1}^{{\mathit{T}}_{\mathit{i}}}{\mathit{\theta }}_{2\mathit{h}}^1\mathit{IS}\left(\mathit{Th}\right)+\sum _{\mathit{h}=1}^{{\mathit{T}}_{\mathit{i}}}{\mathit{\theta }}_{2\mathit{h}}^1\mathit{DS}\left({\mathit{T}}_{\mathit{h}}\right)+\sum _{\mathit{i}=1}^{{\mathit{T}}_{\mathit{i}}}{\mathit{\theta }}_{3\mathit{h}}^1\mathit{DT}\left({\mathit{T}}_{\mathit{h}}\right)+{\mathit{u}}_{\mathit{i}}^1,\hfill \\ \mathrm{where}\hfill \\ \mathit{IS}\left({\mathit{T}}_{\mathit{i}}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathit{t}={\mathit{T}}_{\mathit{i}},\hfill \\ 0,\mathrm{if}\mathit{t}\ne {\mathit{T}}_{\mathit{i}};\hfill \end{array}\right.\mathit{DS}\left({\mathit{T}}_{\mathit{i}}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathit{t}\ge {\mathit{T}}_{\mathit{i}},\hfill \\ 0,\mathrm{if}\mathit{t}<{\mathit{T}}_{\mathit{i}};\hfill \end{array}\right.\mathit{DT}\left({\mathit{T}}_{\mathit{i}}\right)=\left\{\begin{array}{c}\mathit{t},\mathrm{if}\mathit{t}\ge {\mathit{T}}_{\mathit{i}},\hfill \\ 0,\mathrm{if}\mathit{t}<{\mathit{T}}_{\mathit{i}};\hfill \end{array}\right.\hfill \end{array}$

and Vⁱ = Vⁱ₁, Vⁱ₂,..., Vⁱ_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 final model in the first step and broader set of variables:

(20) $\begin{array}{c}\hfill \Delta {\mathit{x}}_{\mathit{i},\mathit{t}}=\sum _{\mathit{h}=1}^{\mathit{q}}{\mathit{\theta }}_{1\mathit{i}}^2{\mathit{FM}}_{\mathit{it}}+\sum _{\mathit{h}=1}^{\mathit{N}}\sum _{\mathit{j}=1}^{{\mathit{k}}_{\mathit{i}}}{\mathit{\theta }}_{2_{\mathit{ij}}}^2\Delta {\mathit{x}}_{\mathit{h},\mathit{t}-\mathit{j}}+\sum _{\mathit{h}=1}^{\mathit{N}}\sum _{\mathit{j}=1}^{{\mathit{k}}_{\mathit{i}}}{\mathit{\theta }}_{3\mathit{ij}}^2\Delta {\mathit{x}}_{\mathit{h},\mathit{t}-\mathit{j}}^{*}\\ \hfill +\sum _{\mathit{h}=1,\mathit{h}\ne \mathit{j}}^{\mathit{N}}{\mathit{\theta }}_{4\mathit{ij}}^2{\mathit{ecm}}_{\mathit{h},\mathit{t}-1}+\sum _{\mathit{h}=1}^{2*\mathit{N}}\sum _{\mathit{j}=1}^{{\mathit{k}}_{\mathit{i}}}{\mathit{\theta }}_{3\mathit{ij}}^2{\mathit{PC}}_{\mathit{h},\mathit{t}-\mathit{j}}+{\mathit{u}}_{\mathit{i}}^2,\end{array}$

where FM denotes the variables in the final 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 defined previously in the paper.

The GUM of the first step in our exercise contains a total of TOT₁ = T*3 + 2*6 + 2 + 1 variables. In our application TOT₁ equals 75 = 30 * 3 + 12 + 2 + 1. The GUM in the second step will contain at least TOT₂ variables (2 * [N + (N - 1)] + 2 * [N + (N - 1)] + (N - 1) + 2 * N). In our example the number of countries is 27 and TOT₂ equals 292. The GUM upper limit will be 292 + 75 = 367. The Autometrics will retain under the null that all coefficients are zero an average of irrelevant regressors given by TOT = (TOT₁ + TOT₂) times the chosen nominal size. If we set the nominal size to 0.0001 (minute), Autometrics will retain 0.0367 irrelevant regressors on average and about 0.367 irrelevant regressors if the nominal size is set to 0.001 (tiny).

Autometrics allows us to check whether the GVAR approach is a congruent statistical model and a rival model that contains broader information set can encompass the GVAR. If the final model contains only the variables of the traditional approach it is reasonable to conclude that neither GVAR nor the broader GUM are worth taking.

Table 8 provides information on each country final model chosen by Autometrics search algorithm. In some case there is evidence of structural change (e.g. Australia, Greece, Uruguay, Turkey). In these cases, a closer look to other domestic variables can be an alternative. For some countries our evidence suggests that the GVAR can not capture all possible sources of interdependence. For example, in the case of Ireland, a closer link to the dynamics of United Kingdom’s real effective exchange was revealed. However the foreign NFA weigthed variable seems to explain the dynamics in United Kingdom and United States cases quite well. These countries have the leading financial centers of the world, and GVAR is probably capturing this interdependence channel. In many countries the final model selected by Autometrics contains the foreign trade weighted variable. This results suggest that GVAR improves of our understanding on the dynamics real exchange rate and net foreign asset. The principal components variables were selected only in some few models. For some countries, the results of Autometrics using minute instead of tiny significance level seems more intuitive (e.g. Brazil or Germany). Spain is the only country that Autometrics results are not easy to rationalize.

Table 8
Summary of Autometrics results.

	Type of search	Variables in the final model						Evidence of structural change	Are specification tests good?
Countries		ECM	RER	Weighted RER	NFA	Weighted NFA	Factors		At 5%	At 1%
Australia	Tiny	O	No	No	No	No	No	DT, DI, DS	No	Yes
	Minute	O	No	No	No	No	No	DI	Yes	Yes
Austria	Tiny	O	No	O	No	No	No	DI	No	Yes
	Minute	O	No	No	No	No	No	No	Yes	Yes
Belgium	Tiny	O	No	No	No	No	No	No	No	Yes
	Minute	O	No	No	No	No	No	No	No	Yes
Brazil	Tiny	O	No	India, USA	No	No	NFA	DI	Yes	Yes
	Minute	O	No	No	No	No	No	DI	No	Yes
Canada	Tiny	O	No	No	No	O	No	No	No	Yes
	Minute	No	No	No	No	No	No	No	Yes	Yes
China	Tiny	O, Sweden	No	No	Germany	No	NFA	No	Yes	Yes
	Minute	O	No	No	No	No	No	No	Yes	Yes
Colombia	Tiny	O	USA	O, Mexico	No	No	No	No	Yes	Yes
	Minute	O	No	No	No	No	No	No	Yes	Yes
Denmark	Tiny	O	No	No	No	No	No	No	Yes	Yes
	Minute	O	No	No	No	No	No	No	Yes	Yes
Finland	Tiny	O	No	No	O, Ireland	No	No	DT	No	Yes
	Minute	O	No	No	No	No	No	DT	No	No
France	Tiny	O	No	O	No	No	NFA	No	Yes	Yes
	Minute	O	No	No	No	No	No	No	No	Yes
Germany	Tiny	O, South Africa	No	No	No	No	No	No	Yes	Yes
	Minute	O	No	No	No	No	No	No	Yes	Yes
Greece	Tiny	No	No	No	No	No	No	DT, DI, DS	No	Yes
	Minute	O	No	No	No	No	No	No	No	No
India	Tiny	O	No	No	No	No	No	No	No	Yes
	Minute	O	No	No	No	No	No	No	No	Yes
Ireland	Tiny	O	UK	No	No	O, Singapore	NFA	No	Yes	Yes
	Minute	O	UK	O	No	O	No	No	Yes	Yes
Italy	Tiny	O, South Africa	No	No	No	No	NFA	No	Yes	Yes
	Minute	O	No	No	No	No	No	No	Yes	Yes
Japan	Tiny	O	Spain	O	No	O	No	No	Yes	Yes
	Minute	O	No	O	No	O	No	No	No	Yes
Mexico	Tiny	O	No	O	No	O	No	No	No	Yes
	Minute	O	O	No	No	No	No	No	No	No
Netherlands	Tiny	O	No	O	No	No	No	No	No	No
	Minute	O	No	O	No	No	No	DI	No	Yes
Singapore	Tiny	No	No	No	No	No	No	No	No	Yes
	Minute	No	No	No	No	No	No	No	No	Yes
Spain	Tiny	O, Australia	USA	No	No	No	No	No	No	Yes
	Minute	O, Australia, Turkey, Uruguay	USA	Colombia	No	No	No	No	No	Yes
Sweden	Tiny	O	No	O	No	No	No	No	No	Yes
	Minute	O	No	No	No	No	No	No	No	No
UK	Tiny	O	No	No	No	O	No	No	No	Yes
	Minute	O	No	No	No	O	No	No	No	Yes
USA	Tiny	O	No	No	No	O	No	No	No	Yes
	Minute	O	No	No	No	No	No	No	Yes	Yes
Uruguay	Tiny	O	No	No	No	No	No	DT, DI	No	No
	Minute	O	No	No	No	No	No	DT	No	No
South Africa	Tiny	O	No	No	No	No	No	No	Yes	Yes
	Minute	O	No	No	No	No	No	No	Yes	Yes
Turkey	Tiny	No	No	No	No	No	No	DT, DI, DS	No	Yes
	Minute	No	No	No	No	No	No	DT, DI, DS	No	Yes

• Notes: "O" refers to own country, DT is break time trend dummy, DI is an impulse indicator, and DS is a step dummy.

4.5 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 flexible 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 specific units, not to assess the relevance of a group of variables in explaining real exchange rate movements and their average effect. A GVAR model can reconcile the merits of the two approaches, allowing us to map directly the effect 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 Proietti (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 effect among eurozone countries.

Even the IMF approach does not directly tackle the question of interdependence between countries. However, the IMF approach does have the benefit of considering a wide range of fundamentals, and incorporates the role of policy gaps in determining the misalignment.

5. Final remarks

In this study, we estimated a global VAR to investigate the interdependence hypothesis among countries in terms of their real effective exchange rates and fundamentals. We were

able to find 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 effects to exchange rate misalignment estimates. Our findings show that the effects are greater for small or developing countries, because they tend to be more affected 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 effect on the real exchange rate of other countries. However, their exchange rate misalignment estimates are only marginally affected 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 effective exchange rates, which are almost not affected by others countries’ variables.

Finally, possible extensions to our approach include improving on country-specific 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 significance of external factors but their relative importance in relation to domestic factors.

• *
  The authors acknowledge the comments from the participants of the first International Annual Applied Econometric Conference, fourth Emerging Market Group Conference, fourteenth OxMetrics User Conference. The authors also acknowledge Joseph Gagnon, Luis Catão, Irineu Evangelista, PedroValls, Megumi Kubota, Cesar Calderon, Aluisiode Lima Campos andVeraThorstensen for helpful comments on a very preliminary version of this work. All remaining errors are the responsibility of the authors.
• 1
  A full description of the methodology, data, and routines are available at http://www.imf.org/external/np/spr/2013/esr/
• 2
  Chicago Board Options Exchange Market Volatility Index.
• 3
  For example, Alberola et al. (1999) and Kubota (2009).
• 4
  VARX is the vector autoregression (VAR) model that contains exogenous variables.
• 5
  In this study, we could have analyzed a broader set of information using variables to control for the Balassa-Samuelson effect, similar to Kubota (2009) and Alberola et al. (1999). However, the number of countries in the sample would have been further reduced. We opt 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 effect reduces the sample in both the temporal and cross-sectional dimensions.
• 6
  See Hecq, Palm, and Urbain (2000) and Hecq, Palm, and Urbain (2002) for a common cycle definition, a discussion, and its relationship to permanent and transitory decomposition.
• 7
  See Doornik (2009).

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Publication Dates

• Publication in this collection
  Oct-Dec 2018

History

• Received
  04 July 2017
• Accepted
  06 Mar 2018