Virginie Coudert

Cécile Couharde

Carl Grekou

Valérie Mignon

Heterogeneity within the euro area: New insights into an old story Heterogeneity within the euro area: New insights into an old story 1

Keywords: JEL Classication: F33, F45, E5, C38 Euro area, Equilibrium Exchange Rates, Cluster analysis, Factor analysis, Macroeconomic imbalances

We assess cross-country heterogeneity within the eurozone and its evolution over time by measuring the distances between the equilibrium exchange rates' paths of member countries. These equilibrium paths are derived from the minimization of currency misalignments, by matching real exchange rates with their economic fundamentals. Using cluster and factor analyses, we identify two distinct groups of countries in the run-up to the European Monetary Union (EMU), Greece being clearly an outlier at that time. Comparing the results with more recent periods, we nd evidence of rising dissimilarities between these two sets of countries, as well as within the groups themselves. Overall, our ndings illustrate the building-up of macroeconomic imbalances within the eurozone before the 2008 crisis and the fragmentation between its member countries that followed.

Introduction

The 2008 nancial crisis and the sovereign crisis that followed in Europe have revived debates about the European Monetary Union (EMU). The most lingering question underlying most issues is to know if member states are similar enough to share the same 1 Corresponding author : Valérie Mignon, EconomiX-CNRS, University of Paris Nanterre, 200 avenue de la République, 92001 Nanterre cedex, France. Email: valerie.mignon@parisnanterre.fr.

We are very grateful to Anne-Laure Delatte for valuable comments and suggestions. This paper reects the opinions of the authors and does not necessarily express the views of the institutions to which they belong. currency in the long run. Whatever the answer, drastic steps had to be taken rapidly in 2012 to avoid a fragmentation of the eurozone. Monetary policy has largely contributed to alleviate the diverging pressures, notably though the quantitative easing strategy, enlargement of collateral and the public sector purchase program (PSPP). Fiscal policy has also been more tightened in the peripheral countries in order to restore the sustainability of public nances. Besides, banking supervision has been strengthened through the banking union. On the whole, the functioning of the euro area has been improved compared to the pre-2008 period.

Despite all these advances, the key question remains to know if the countries are reasonably similar to benet from sharing the same currency and if dierences have been ironed out since the launch of the euro. As pointed out by [START_REF] Lane | The Real Eects of European Monetary Union[END_REF], this issue was identied as a major challenge for the success of the euro from its beginning. It led to numerous contributions in the wake of the Maastricht Treaty that largely focused on studying convergence in prices and business cycles within the euro area. In particular, earlier empirical studies typically relied on the optimum currency area (OCA) literature [START_REF] Mundell | A Theory of Optimum Currency Areas[END_REF][START_REF] Mckinnon | Optimum Currency Area[END_REF][START_REF] Kenen | The Theory of Optimum Currency Areas: An Eclectic View in[END_REF], in order to examine whether future eurozone members meet the OCA criteria that could allow them to be less vulnerable to shocks and then to undergo low stabilization costs in joining the EMU. 2 The ndings of this literature usually tended to be pessimistic about the possibility for European countries to form a viable monetary union. In particular, in their seminal article, [START_REF] Bayoumi | Shocking Aspects of European Monetary Integration[END_REF] highlighted the existence of a coreperiphery pattern in the run-up to the EMU. Using pre-EMU data to estimate the degree of supply shocks synchronization, they showed that, over this period, there was a core (Germany, France, Belgium, the Netherlands, and Denmark) where shocks were highly correlated, and a periphery (Greece, Ireland, Italy, Portugal, Spain, and UK) where synchronization was signicantly lower. [START_REF] Bayoumi | Shocking Aspects of European Monetary Integration[END_REF] suggested that, if persistent, this pattern would be detrimental to the EMU project.

As the single currency seemed to perform successfully until the 2008 crisis despite these bleak predictions, some observers criticized the previous analyses on the ground that they ignored the complex nature of monetary unication and its endogenous character. In particular, [START_REF] Frankel | The Endogeneity of the Optimum Currency Area Criteria[END_REF] presented empirical evidence that countries with more bilateral trade will feature higher business cycle correlations. As monetary unication induced eurozone members to trade more with each other (see [START_REF] Baldwin | The euro's trade eects, Working Paper Series[END_REF] 2 For a survey, see Bayoumi andEichengreen (1994), De Grauwe (1997), [START_REF] Lafrance | Optimal Currency Areas: A Review of the Recent Literature[END_REF], [START_REF] Alesina | Optimal Currency Areas[END_REF].

for a survey), the process should be matched by an increase in business cycle synchronization among countries. It would then become easier for euro-area members to meet the OCA criteria. The shock caused by the 2007-08 nancial collapse, followed by the European sovereign debt crisis, has however raised new doubts about the ability of the single currency to work well in a region with huge economic and political diversity (see [START_REF] Stiglitz | The Fundamental Flaws in the Euro Zone Framework, in The Euro and the Crisis: Perspectives for the eurozone as a Monetary and Budgetary Union, Nazaré da Costa Cabral[END_REF]. It has also given a new dimension to this debate by highlighting the building-up of unsustainable macroeconomic imbalances within the eurozone.

The objective of this paper is to revisit such issue of sharing a common currency by taking stock of the consensus reached after the 2008 crisis that highlighted the accumulation of macroeconomic imbalances in the run-up to most nancial crises. Specically, we examine between-country disparities in terms of equilibrium exchange rates, i.e., exchange rates that would prevent currency misalignments resulting from unsustainable macroeconomic disequilibria within the member countries. We consider the oldest members of the EMU, namely the ten founding members and Greece in order to have a long historical record our sample spanning from 1980 to 2016. First, we assess the equilibrium exchange rate paths for the considered euro-area members. Second, we rely on a cluster analysis to partition the euro-area countries into homogeneous groups or clusters in order to measure how equilibrium exchange rate trajectories have been distant across euro-area members. Third, we try to identify the characteristics of EMU members that explain the formation of such clusters. In these two last parts, we split the sample into several sub-periods in order to investigate the dynamics of these clusters over time thanks to the identication of their main underlying factors, before and after the monetary union, as well as before and after the 2008 crisis.

Previewing our main results, we nd that that the pre-euro conguration of the eurozone can be partitioned into three groups of countries. On the one hand, Belgium, France, Germany, Ireland, and the Netherlands form the most homogenous group; on the other hand, Austria, Finland, Italy, and Spain constitute the second group. We also nd evidence of two outliers, namely Portugal and Greece. Indeed, these countries exhibit the most divergent equilibrium exchange rate paths; Greece being the most idiosyncratic country. The comparison with the post-euro period reveals that member states have not moved structurally closer to each other. On the contrary, we nd that (i) disparities have increased across and within countries' groups, and (ii) Greece has moved away from the other member states, becoming more peripheral over time. Only Spain seems to gradually converge towards the core countries. Overall, our ndings point out the disparities within the euro area and their evolution through time.

The rest of the paper is structured as follows. Section 2 describes our methodological framework to assess equilibrium exchange rates. Section 3 presents the partition of the euro area based on cluster analysis, while that issued from factor analysis is analyzed in Section 4. In both sections, we make the distinction between the period before and after the implementation of the EMU to get evidence of the build-up of macroeconomic imbalances within the euro area. Finally, Section 5 concludes.

Equilibrium exchange rates within the euro area

The relevance of currency misalignments inside the monetary union

There is common sense among economists and policymakers that currency misalignmentsi.e., departures of real exchange rates from their equilibrium levelsare likely to cause substantial losses in economic eciency and social welfare. This conviction is substantiated by the new Keynesian literature in which the equilibrium exchange rate corresponds to a real exchange rate that allocates resources eciently and thus maximizes social welfare. For example, [START_REF] Engel | Currency Misalignments and Optimal Monetary Policy: A Reexamination[END_REF] argues that any violation of the law of one price is inecient and, in turn, leads to a reduction in world welfare. This literature also widely recognizes that in a world of imperfect markets, the oating exchange rates cannot fully adjust to an ecient level. As a consequence, minimizing currency misalignments may be a goal for monetary policy, along with domestic objectives regarding ination and the output gap. If the equilibrium exchange rate matters for eciency and social welfare, it also plays a key role in allowing any economy to reach simultaneously both its internal and external balances according to more traditional Keynesian open-economy models. Following this literature (see [START_REF] Driver | Concepts of equilibrium exchange rates, Bank of England working paper[END_REF], the equilibrium real exchange rate is compatible with the steady state of an economy that is characterized by @iA an output gap close to zero or equilibrium in the non-tradable goods sector (internal balance), and @iiA consistent relations between net foreign assets and current account balances (external balance). An optimal mix of consistent policies can make the real exchange rate converge towards its equilibrium, thus wiping out external and internal imbalances.

3

In the context of a monetary union, currency misalignments are especially detrimental because the nominal parity can no longer be adjusted. The only possible adjustment con-3 It is also worthwhile noting that the implications of minimizing currency misalignments are in line with the European Commission Reection paper on the deepening of the EMU (May 31, 2017) especially regarding the rst of the four guiding principles ("Jobs, growth, social fairness, economic convergence and nancial stability should be the main objectives of our Economic and Monetary Union"; see Page 18). cerns prices. This is particularly painful when one member country has an overvalued real exchange rate, as it has no choice but to reduce its relative prices by cutting spending and limiting wages. Before the nancial crisis, there was a fairly widespread consensus that the single currency will bring about prosperity and these potential shortcomings will be more than compensated by other benets, such as low-ination and credibility (Alesina and Barro, 2002). Another prominent argument advocated by the new Keynesian open economy literature was that @iA instead of acting as shock absorbers, nominal exchange rates act as sources of shocks, and @iiA monetary unication, by eliminating excess volatility of nominal variables, was then superior to a exible exchange rate regime [START_REF] Neumeyer | Currencies and the Allocation of Risk: The Welfare Eects of a Monetary Union[END_REF][START_REF] Devereux | Expectations and Exchange Rate Policy[END_REF]. Lastly, imbalances in current account positions within the euro area rose for good reasons, as they were mainly driven by productivity dierentials and catching-up developments [START_REF] Blanchard | Current Account Decits in the Euro Area: The End of the Feldstein-Horioka Puzzle[END_REF]. In short, there were widespread expectations that the increased nancial integration would play an important role in the adjustment process. However, macroeconomic imbalances initially considered as good imbalances turned out to be bad imbalances [START_REF] Belke | Current Account Imbalances in the Euro Area: Does Catching Up Explain the Development?[END_REF]. They also brought about severe misalignments of real exchange rates within the euro area [START_REF] Coudert | On Currency Misalignments within the Euro Area[END_REF]. Moreover, these imbalances as well as low interest rates and credit boom paved the way to the sovereign debt crisis in several Southern member states. As currency misalignments as well as imbalances are especially harmful in a monetary union, we consider an analysis that emphasizes how the exchange rates have departed from their equilibrium paths. 4

Deriving nominal equilibrium exchange rate paths

A preliminary step in our methodology is to reconcile the real approach of the equilibrium exchange rates with the nominal nature of a monetary union. Another is to switch from the multilateral approach of the equilibrium exchange rates to the xed bilateral parities implied by a single currency.

Regarding the transition from real to nominal, on one side, equilibrium exchange rates are generally dened in real terms; this is also true for currency misalignments that measure the gap between the observed exchange rates and their equilibrium levels. On the other side, a monetary union only deals with nominal parities, by xing them irreversibly 4 [START_REF] Couharde | Revisiting the theory of optimum currency areas: Is the CFA franc zone sustainable?[END_REF] have also developed an approach based on the use of equilibrium exchange rates to assess the sustainability of the CFA zone; a sustainable currency area being dened as a zone in which real exchange rates do not deviate persistently from their equilibrium paths. Similarly, [START_REF] Coulibaly | Optimality of a monetary union: New evidence from exchange rate misalignments in West Africa[END_REF] fall into this strand of the literature by addressing the optimality of monetary unions in West Africa through the analysis of exchange rate misalignments. between members, but this does not prevent the real exchange rates to evolve dierently across countries along with the relative prices. Regarding the multilateral versus bilateral approach, the equilibrium exchange rate is generally assessed towards a whole set of trade partners, whereas the monetary union dened xed bilateral parities between member countries.

To sum up, we have to convert our multilateral real equilibrium exchange rates into nominal bilateral parities. Various methods exist to deal with this issue. For example, [START_REF] Alberola | Global equilibrium exchange rates: euro, dollar, ins, outs, and other major currencies in a panel cointegration framework[END_REF] propose a framework that allows determining nominal equilibrium exchange rates that are consistent at the global level. Their approach consists in inverting the weighted matrix of eective equilibrium exchange rates to deduce bilateral equilibrium values. However, as only (N I) bilateral exchange rates can be derived from N eective rates, one currencycorresponding to the chosen numerairehas to be discarded. This amounts to assuming that the misalignment of the selected currency (the rest of the world) is the mirror image of the misalignment of all other currencies.

Instead of imposing such assumption, we adopt here another approach in which nominal exchange rates that are consistent with minimized currency misalignments follow an equilibrium path independent of each other.

By denition, the real eective exchange rate of country i, REER i ;t , is calculated as the weighted average of country i's real bilateral exchange rates against each of its N trading partners j:

REER i ;t a N j =1 RER w i j;t i j;t (1)
where RER i j;t a NER i j;t ¢ P i ;t P j;t is an index of the real bilateral exchange rate of the country i's currency vis-à-vis the currency of the trading partner j in period t. NER i j;t is the index of the nominal bilateral exchange rate between the currency of country i and the currency of its trade partner j in period t (number of units of currency j per currency i), and P i ;t (resp. P j;t ) stands for the price index of country i (resp. j). N denotes the number of trade partners, and w i j;t stands is for country i's the trade-based weights associated to the for all its partners j. Note that an increase in REER i ;t and NER i j;t denotes an appreciation of currency i. , where NER i ;t denotes the exchange rate of country i against the numeraire. We choose the Special Drawing Rights (SDR) as the numeraire, as in [START_REF] Housklova | Panel cointegration approaches to estimating equilibrium real exchange rates separating the wheat from the cha[END_REF] for instance. [START_REF] Frankel | Estimation of De Facto Exchange Rate Regimes: Synthesis of the Techniques for Inferring Flexibility and Basket Weights[END_REF] advocated for this choice of numeraire because monetary authorities generally do not monitor their exchange rate towards a single currency, but have rather to focus on several key currencies. Other authors use the dollar, the Swiss franc or several numeraires [START_REF] Frankel | Yen Bloc or Dollar Bloc? Exchange Rate Policies of the East Asian Economies[END_REF][START_REF] Bénassy-Quéré | Exchange Rate Regimes and Policies: An Empirical Analysis[END_REF], although this choice generally does not aect their results. Although in a bilateral set-up the choice of the numeraire will not qualitatively aect the estimates, the derivation of bilateral misalignments from eective misalignments leads to assessments that are aected by the eective misalignment of the numeraire currency at all points in time [START_REF] Housklova | Panel cointegration approaches to estimating equilibrium real exchange rates separating the wheat from the cha[END_REF]. The introduction of the SDR avoids this problem.

Furthermore, the use of the SDR allows us @iA to dene the value of each currency independently of the others, and, in turn, @iiA to derive an equilibrium exchange rate path specic to each country. Recalling that the equilibrium exchange rate of country i is independent from country j's exchange rate level, Equation (3) can thus be rewritten where S £ i SDR;t is the equilibrium nominal exchange rate of the currency of country i vis-àvis the SDR and S j SDR;t denotes the nominal exchange rate of the currency of country j vis-à-vis the SDR.

Equation ( 4) can be rewritten as:

REER £ i ;t a S £ i SDR;t ¢ I N j =1 S w i j;t j SDR;t ¢ i ;t a S £ i SDR;t ¢ I i ;t ¢ i ;t (5)
where i ;t a N j =1 S w i j;t j SDR;t corresponds to the weighted average of the nominal exchange rate of the N trade partners vis-à-vis the SDR.

The equilibrium value of the currency of country i vis-à-vis the SDR that minimizes currency misalignments (i.e., that equalizes REER i ;t and REER £ i ;t ) is then given by:

S £ i SDR;t a REER £ i ;t ¢ i ;t i ;t (6)
The time series of this equilibrium bilateral exchange rate allows us determining the paths of equilibrium parities S £ i SDR;t against the numeraire.

Assessing equilibrium exchange rates

There are three approaches of equilibrium real exchange rates: @iA the fundamental equilibrium exchange rate (FEER; [START_REF] Williamson | Estimates of FEERs, in Estimating Equilibrium Exchange Rates[END_REF] approach also referred to as the macroeconomic approach, @iiA the external sustainability approach, and @iiiA the behavioral equilibrium exchange rate (BEER; [START_REF] Clark | Exchange rates and economic fundamentals: a methodological comparison of BEERs and FEERs[END_REF] approach. In the FEER framework, the equilibrium real exchange rate corresponds to the exchange rate level that allows the current accountprojected over the medium term at prevailing exchange ratesto converge towards an estimated equilibrium current account, or a current account target. In the external sustainability approach, the equilibrium exchange rate aims at stabilizing the ratio of net foreign assets (NFA) to GDP at an appropriate level. These two approaches share a common drawback since they require dening the long-run equilibrium paths of economies. This exercise involves making assumptions on the long-run values of the economic fundamentals (such as current account norms in the FEER approach, and the appropriate ratio of NFA to GDP), which may be viewed as quite arbitrary. On the contrary, the BEER approach directly estimates an equilibrium real exchange as a function of medium-to long-term fundamentals, taking into account both internal and external balances without any ad hoc judgments. In this paper, we rely on such BEER framework for these reasons.

We consider the stock-ow model of exchange rate determination in the long run originally proposed by Faruqee (1995)followed by [START_REF] Alberola | Global equilibrium exchange rates: euro, dollar, ins, outs, and other major currencies in a panel cointegration framework[END_REF] and [START_REF] Alberola | Misalignment, liabilities dollarization and exchange rate adjustment in Latin America[END_REF]which is particularly suitable for describing advanced economies. This model emphasizes the net foreign asset position and the relative sectoral productivity (i.e., the Balassa-Samuelson eect) as important drivers of the real eective exchange rate.

Following [START_REF] Clark | Exchange rates and economic fundamentals: a methodological comparison of BEERs and FEERs[END_REF] among others, we augment this benchmark specication by including the terms of trade as an additional fundamental variable to account for real shocks.

5 These fundamentals are particularly relevant for the euro-area countries as they fairly reect the main sources of macroeconomic imbalances within the EMU that have been stressed: ination rate dierentials exceeding what could be explained by Balassa-Samuelson eects [START_REF] Belke | Current Account Imbalances in the Euro Area: Does Catching Up Explain the Development?[END_REF], presence of large stock imbalances [START_REF] Lane | Capital Flows in the Euro Area[END_REF], and nally real shocks paving the way for asymmetric shocks. The full specication of our model is then the following:

reer i ;t a i C 1 rprod i ;t C 2 nf a i ;t C 3 tot i ;t C " i ;t (7)
where reer i ;t is the real eective exchange rate (in logarithm), rprod i ;t stands for the relative productivity (in logarithm), nf a i ;t is the net foreign asset position (as share of GDP), and tot i ;t denotes relative terms of trade, expressed in logarithm. i are the country-xed eects and " i ;t is an error term. A positive relationship between the real eective exchange rate and each of the fundamentals is expected. Once the coecients estimated, the equilibrium real exchange rate reer £ i ;t will be calculated as the tted value of reer i ;t in Equation ( 7). Corresponding currency misalignments, Mis i ;t , are then given by: Mis i ;t a reer i ;t reer £ i ;t .

Data and results

Our panel consists of the following eleven eurozone countries: Austria, Belgium, Finland, France, Germany, Ireland, Italy, the Netherlands, Portugal and Spain; i.e., the founding members of the euro area in 1999 plus Greece which joined the union in 2001.

6

The data are annual and cover the 1980-2016 period.

7 To assess equilibrium real ex- change rates, we collect real exchange rate indexes from the EQCHANGE database provided by the CEPII.

8 These indexes correspond to the real eective exchange rates vis-à-vis 186 trading partners computed using time-varying weights representative of trade ows (5-year windows).

9 These REER indexes are dened so that an increase corresponds to a real appreciation of the domestic currency. We use the same trade partners and weights for the calculation of the relative productivity, proxied here by the relative real GDP per capita (in PPP terms). 10 The net foreign asset positions are extracted from the [START_REF] Lane | The external wealth of nations mark II: revised and extended estimates of foreign assets and liabilities, 19702004[END_REF] database (extended to 2014) and updated using information on national current accounts provided by the IMF databases (International Financial Statistics and World Economic Outlook). The terms of trade series are taken from the World Bank's WDI (World Development Indicators) database.

We rely on the cross-sectionally augmented pooled mean group procedure (CPMG; see [START_REF] Pesaran | Estimation and Inference in Large Heterogeneous Panels with a Multifactor Error Structure[END_REF][START_REF] Binder | International investment positions and exchange rate dynamics: a dynamic panel analysis[END_REF][START_REF] De | Commodity Price Volatility and the Sources of Growth[END_REF] to estimate the long-run relationship between the REER and its fundamentals, described by Equation ( 7). This method has the advantage to provide consistent estimates of a longrun relationship in presence of cross-sectional dependencies. Indeed, CPMG augments the pooled mean group procedure (PMG; see [START_REF] Pesaran | Pooled Mean Group Estimation of Dynamic Heterogeneous Panels[END_REF] with cross-sectional averages of the variables, therefore accounting for unobservable common factors. Furthermore, the CPMG procedure, compared to other long-run estimation methods (e.g. dynamic OLS, fully-modied OLS), better accounts for potential heterogeneity between countries as it allows short-run dynamics heterogeneity for each member of the panel.

11

The CPMG procedure appears therefore as the most adequate to capture not only interdependencies between countries, but also each country particularities (e.g. resilience to shocks). However, as a condition for the eciency of the CPMG estimator is homogeneity of the long-run parameters across countries, we also rely on the cross-sectionally augmented mean group estimator (CMG) and test the long-run slope homogeneity hypothesis.

12 The tted values of real eective exchange rates given by the estimation of Equation (7) provide the equilibrium real exchange rates.

13 Then we calculate the nominal equilibrium rates against the SDR (S £ i SDR;t ) from Equation (6). To this end, we deate the equilibrium real eective exchange rate series by the weighted relative prices (¨i ;t ) that are used for the computation of REER indexes in the EQCHANGE database. 14

Finally, equilibrium real exchange rates are adjusted for movements in the nominal bilateral exchange rates of trading partners vis-à-vis the SDR (variable i ;t constructed as a weighted average). 15

3. Evaluating the distance between euro-area countries: a cluster analysis

The purpose of this section is @iA to identify relatively homogeneous groups of euromembers based on their respective equilibrium exchange rate path, and @iiA to examine whether the subsequent partition of the euro area has evolved over time, specically since the launch of the single currency. To assess the size of dissimilarities across euro-12 The CMG procedure provides consistent estimates of the averages of long-run coecients, although they are inecient if homogeneity is present. Under long-run slope homogeneity, the CPMG estimates are consistent and ecient [START_REF] De | Commodity Price Volatility and the Sources of Growth[END_REF]. 14 Weighted relative prices are computed as a geometric average of the ratios of the Consumer Price Index (CPI) of a country to the CPI indexes of its main trade partners. The CPI series are from the IMF and OECD databases (see [START_REF] Couharde | EQCHANGE: A World Database on Eective Exchange Rates and Equilibrium Real Eective Exchange Rates[END_REF].

15 Nominal bilateral exchange rates vis-à-vis the SDR are extracted from the IMF database. area countries, we implement a cluster analysis, based on the hierarchical ascendant classication (HAC). This method allows us partitioning the eurozone into relatively homogeneous groups of countries without imposing any reference group or leading country as in business cycle synchronization analyses. Moreover, it provides further information on the level of heterogeneity in the euro area, by evidencing interrelationships within and between the dierent groups of economies.

The method

The HAC procedure begins by estimating the dissimilarities between any pair of objects (here the dissimilarities between the optimal exchange rate paths for any pair of euroarea countries) using an appropriate metric (i.e., a measure of distance between pairs of objects). Here we use the standard Euclidian distance. 16 Let X i ;t be the optimal exchange rate for country i at period t (t a T 1 ; : : : ; T N ), the dissimilarity coecient dened by the Euclidean distance between the optimal exchange rate of country i and country j is: d@X i ; X j A a jjX i ; X j jj a

T N t =T 1 X i ;t X j;t 2 (8)
where d@X i ; X j A or jjX i ; X j jj denotes the Euclidean distance between X i and X j .

Using distance information, pairs of objects are grouped into clusters that are further linked to other objects to create bigger clusters. The algorithm stops when all the objects are linked. This agglomeration is realized using another metric that measures the distance between two clusters and therefore determines the borders of the homogeneous groups.

We adopt here the following four agglomerative methods: @iA Ward's linkage, @iiA single-linkage, @iiiA complete-linkage, and @ivA average-linkage. Let A and B be two clusters with, respectively, n A and n B as the number of objects, and A and B as the centroids. The Ward's method, which is the most commonly used procedure, consists in joining two clusters that result in the minimum increase in the sum of squared errors (so the loss of within-cluster inertia is minimum). The within-cluster sum of squares is dened as the sum of the squares of the distances between all objects in the cluster and the centroid of the cluster. The Ward's method relies on the distance d W between the 16 While other measures exist (Squared Euclidean distance, Manhattan distance . . . ), the Euclidian distance is the most common metric. Since we are working in a continuous space where all dimensions are properly scaled and relevant (due to the numeraire currency), the Euclidean distance is the best choice for the distance function. In addition, it does not suer from sensitivity to outliers and is a real metric. centroids of the two clusters X A and X B :

d W @A; BA a Pn A n B @n A C n B A k X A X B k 2 (9)
The single-linkage, also called nearest neighbor, considers the smallest distance d S between objects in the two clusters. On the contrary, the complete-linkage or furthest neighbor, as it name suggests, concentrates on the largest distance d C between objects in two clusters. Finally, the average-linkage method uses the average distance d A between all pairs of objects in any two clusters. 17 These inter-cluster distances are expressed as follows:

d S @A; BA a min d X Ai ; X Bj (10)

d C @A; BA a max d X Ai ; X Bj (11) d A @A; BA a I n A n B n A i =1 n B j =1 d X Ai ; X Bj ( 12
)
where i a I; : : : ; n A (resp. j a I; : : : ; n B ) designates the i t h (resp. j t h ) object in cluster A (resp. B).

We use the HAC analysis to measure the distance between the euro-area countries regarding the path of their equilibrium exchange rates. First, we consider the period before the monetary union 1980-1998. Then, we study how the clustering of countries has evolved over time. To do so, we perform the same cluster analysis over dierent time periods in order to track structural changes. This investigation aims at capturing any change in dissimilarities across the euro area. On the whole, we retain three periods all starting in 1980: @iA before the monetary union: 1980-1998; @iiA before the global nancial crisis: 1980-2006; and @iiiA the whole period 1980-2016.

Cluster analysis before the launch of the euro

The groups identied by applying the HAC analysis before the launch of the euro are shown in the four dendrograms reported in Figure 1. These "cluster trees" indicate the order in which successive aggregations were made (and therefore the optimal groupings).

The vertical axis of the dendrograms represents the distance or dissimilarity between the 17 For more details regarding these measures, see [START_REF] Kaufman | Finding Groups in Data: An Introduction to Cluster Analysis[END_REF].

objects (i.e., the countries' equilibrium exchange rate paths), while the horizontal axis displays the dierent countries. The dissimilarity measures are captured by the heights of the links.

As can be seen, the four dierent methods release consistent information regarding dissimilarities in exchange rate paths across the eurozone members. From these dendrograms, we clearly identify two clusters of countries (two branches that occur at about the same vertical distance). A rst cluster can be identied as the core countries, it includes Germany, Belgium, the Netherlands, France and Ireland. A second group of homogeneous countries is made of Austria, Spain, Italy and Finland. Portugal and Greece are fused separately at much higher distances compared to the other countries and can be considered as two outliers. 18

The division of the euro area into several groups of countries is quite in line with the existing literature while the composition of the core group may dier depending on research undertaken.

19 For instance, applying clustering techniques to a set of OCA vari- ables, [START_REF] Artis | Core and Periphery in EMU: A Cluster Analysis[END_REF] also nd evidence in support of three groups of countries:

those belonging to the core (Germany, France, Austria, Belgium and the Netherlands), those part of a Northern periphery (Denmark, Ireland, the UK, Switzerland, Sweden, Norway and Finland) and those belonging to a Southern periphery (Spain, Italy, Portugal and Greece). Also relying on OCA theory but using a modied Blanchard-Quah decomposition within the aggregate supply aggregate demand setup, [START_REF] Bayoumi | Shocking Aspects of European Monetary Integration[END_REF] identify (i) a core composed of Germany, France, Belgium and the Netherlands, and (ii) a periphery including Greece, Ireland, Italy, Portugal and Spain.

Three sets of countries are also obtained by [START_REF] Bayoumi | Ever Closer to Heaven? An optimum-currency area index for European countries[END_REF], distinguishing the economies in terms of readiness for EMU: Germany, Austria, Belgium and the Netherlands which exhibit a high level of readiness; Finland and France which experienced little convergence; and Italy, Greece, Portugal and Spain which are gradually converging.

Looking back on the debates at the introduction of the euro, our results thus conrm that dissimilarities across the euro area candidates persisted until the eve of the EMU.

18 This can be conrmed by the dissimilarity matrices reported in Appendix C for the 1980-1998 (Table C.1) and 1980-2016 (Table C.2) periods: as shown, Greece and, to a lesser extent, Portugal are the two countries exhibiting the highest values.

19 Part of these dierences can be related to the denition of membership of the core. While the approach suggested by [START_REF] Bayoumi | Shocking Aspects of European Monetary Integration[END_REF] denes core countries as those whose aggregate supply and demand shocks are relatively highly correlated with Germany, the clustering approach separates the most similar countries into several clusters, without assuming a representative core country. In this latter approach, core countries are then dened as those that are theoretically suitable for a common currency, i.e., those forming the most homogeneous group.

If we restrict our sample to the countries that adopted the euro in 1999 (i.e., excluding Greece), two groups of countries are clearly identied, with Portugal as an outlier. From this point of view, our results are consistent with the ndings of the OCA literature that counselled against the pursuit of further and deeper monetary integration within Europe at that time. The partition of the euro area into dierent groups of relatively homogenous countries called into question @iA the eectiveness of Maastricht criteria in making these countries converge before the launch of the euro, and @iiA the desirability of a unique monetary policy for all the economies composing these groups. Our results also reect the argument advanced by [START_REF] Eichengreen | European Monetary Unication[END_REF] and [START_REF] Feldstein | The Political Economy of the European Economic and Monetary Union: Political Sources of an Economic Liability[END_REF] that much of the driving force for monetary union was political, as in economic terms the eurozone project would have been postponed or designed dierently to allow for some exibilities between the core and the periphery.

The evolution in the groupings of countries

So far, our ndings cannot refute the argument that dissimilarities across the potential eurozone member countries were due to insucient nancial integration and would be swept o by monetary union. We now extend our analysis by including the period after the launch of the single currency in order to analyze the evolution of cluster memberships over time. The corresponding results are displayed in Figure 2. 20 The conguration of the eurozone shares the same features as before, with a set of core countries and a second group of economies. These two groups are fused at the same distance, suggesting that the monetary union has not reduced dissimilarities between these two clusters of countries. The peripheral countries seem also more fragmented in sub-groupings. In particular, Italy now exhibits slightly distinct features from the other members, and is depicted as a singleton inside the peripheral group. This is also the case for Portugal, which is linked to the peripheral set in a single element group. Greece still appears as 20 To save space, we only report the results obtained with the Ward method. The three other approaches lead to similar results, which are available upon request to the authors.

On the whole, these ndings suggest that the conguration of the eurozone since the launch of the single currency has become more fragmented; dissimilarities between groups of countries have augmented. The peripheral countries that are aggregated together exhibit some increased distinct features. The comparison between the graphs before and after the 2007-08 collapse shows that the adjustment that followed the nancial crisis has not changed the deal between countries. The clusters have not been brought closer despite all the steps that have been taken to counter imbalances.

Identication of heterogeneous patterns: a factor analysis

Apart from the partition of countries by itself, it is also interesting to analyze which variables have mostly explained the formation of such clusters. For this purpose, we now develop a factor analysis in order @iA to identify the common features shared by euroarea countries belonging to the same group, and @iiA to check if the results are similar to those issued from the cluster analysis. Accordingly, we collect data on several key variables that are more prone to reect macroeconomic imbalances. Then, we perform a factor analysis to identify the structural economic dierences between the EMU members emphasized by the cluster analysis.

Method and selected indicators

We use factor analysis to select the main relevant indicators, i.e., the variables that underlie the formation of clusters. Specically, being a multivariate explorative analysis tool, factor analysis gathers together several variables with similar patterns and containing most of the information into a few interpretable unobserved (underlying) variables, called factors. Thus, as a technique of data reduction, the aim is to reduce the dimension of the observations by grouping p observed variables into a lower number, say k, of factors. To this end, the p variables are modeled as a linear combination of the potential factors (i.e., latent unobserved variables that are reected in the behavior of the observed variables) plus an error term. In doing so, factor analysis is a useful tool to detect the structure of the relationships between the variables. Specically, let us assume we have a set of p observable random variables (Y 1 ; : : : ; Y p ). From these p observed variables, factor analysis aims at identifying k common factors which linearly reconstruct the original variables as follows:

Y i j a Z i 1 1j C Z i 2 2j C ::: C Z i k k j C u i j (13)
where Y i j is the value of the i t h observation of the j t h variable (j a I; : : : ; p), Z i l is the value of the i t h observation of the l t h common factor (l a I; : : : ; k), the coecients l j denote the factor loadings (l a I; : : : ; k), and the error term u i j is the unique factor of the j t h variable.

The selection of indicators must meet the need for both comprehensiveness and parsimony in order to set out clear factors that can be easily interpreted. We select indicators among a set of fundamental variables often linked to the formation of imbalances within the euro area. This set includes the current account balance, consumer price, ination, public debt, GDP per capita, real growth, output gap, unemployment rate and unit labor cost, to which we add the currency misalignments that we have calculated. Table A.1 in the appendix details the source of these series. This set of variables is large enough to account for the usual OCA criteria as well as the economies' internal and external balances and their dynamics. For example, ination obviously accounts for price stability and debt-to-GDP ratios measure the soundness and sustainability of public nances:

these two variables, as well as unemployment rates are able to gauge the internal equilibrium of the economies. 21 Regarding the external balance, the current account-to-GDP ratio encompasses various key determinants according to the usual medium to long-term specications such as net foreign asset position, scal position, output gap, population growth rate, dependency ratio, openness, etc. (see e.g. [START_REF] Chinn | Medium-term Determinants of Current Accounts in Industrial and Developing Countries: an Empirical Exploration[END_REF][START_REF] Gruber | Explaining the Global Pattern of Current Account Imbalances[END_REF][START_REF] Cheung | Structural and Cyclical Factors behind Current-Account Balances[END_REF]; it has thus the advantage of parsimony by synthetizing them in a sole series.

The selected variables should also meet considerations/rules regarding the functioning of the euro area, although rules much changed over the long period 1980-2016 that we consider. For example the Maastricht criteria were important before the monetary union, then the stability and growth pact (SGP) and the six-pack legislation, including the MIP, in the aftermath of the 2008 crisis.

22 We therefore take stock of the macroeco- nomic imbalance procedure (MIP) that the European Commission has been using since 2011 in order to deal with imbalances in the member countries. Most of the 14 headline 21 Beyond all the technical aspects and rules regarding the limits of the variables mainly dened by the Maastricht Treaty (i.e., an ination rate that should be lower than a reference valuedened as the average of the ination rates in the 3 eurozone member states with the lowest ination plus 1.5 percentage points; a debt ratio that should not exceed 60%) or SGP rules (the unemployment three-year moving average should be lower than 10%), factor analysis ignores such rigidity in the criteria and simply maps out the countries on the basis of their proximity regarding the variable levels, thus potentially underlining clusters of countries.

indicators that are monitored in this procedure are taken into account in our analysis.

Indeed, both the 3-year moving average of the current account (% of GDP) and the net international investment position (% of GDP) indicators are considered either @iA directly through the current account, or @iiA through currency misalignments thanks to the net foreign asset position. As well, the 3-year percentage change of the real eective exchange rate is also accounted for via the currency misalignments.

23

The detailed results of the factor analysis are presented in Appendix B.2; 1980-1998 (respectively 1999-2016) period, the other tables in Appendix B.2 concerning the rotated analysis. 24 Each variable is assigned to the factor in which it has the highest loading. Once @iA variables have been assigned to the common factors and @iiA the factors and their loadings have been estimated (right-hand side of Equation ( 13)), the factors must be interpreted. To this end, the factor loadings have to be examined. B.2.1.3). Table B .2.1.4 shows that the rst factor (Factor 1) has a high positive correlation with ination and the debt-to-GDP ratio, and a high negative correlation with the current account balance.

Factor analysis before monetary union

Factor 1 thus principally opposes @iA on the left side, the current account balance and, @iiA on the right side, the debt-to-GDP ratio and ination. Accordingly, Factor 1 may be interpreted as the Finance/Wealth axis or reecting the Financial position: on the left side, countries with good/sustainable foreign position; on the right side, countries facing external imbalances characterized by a high ination rate and a high debt-to-GDP ratio, which result in current account decits. The second factor, Factor 2, correlates strongly 23 Furthermore, we consider the country total debt (government plus private sector) while in the MIP scoreboard a distinction is made regarding the debt indicators. By the same token, we use the total economy unemployment rate instead of the dierent types/horizons of unemployment.

24 Recall that factor loadings could be rotated to make easier the interpretation of the factors. Indeed, rotation consists in expressing the factors so that loadings on a quite low number of variables are as large as possible, while being as small as possible for the remaining variables. Here, we retain the usual orthogonal varimax rotation [START_REF] Kaiser | The varimax criterion for analytic rotation in factor analysis[END_REF] which maximizes the variance of the squared loadings within factors.

and positively with the indicators for unemployment and misalignments, and negatively with economic growth. Factor 2 may thus appropriately reect internal imbalances (Table B.2.1.4).

Regarding methodological aspects, our results are satisfying as shown by the low values of uniqueness in Table B.2.1.4. Indeed, recall that uniqueness measures the percentage of variance for the considered variable that is not explained by the common factors. Uniqueness could represent measurement error, which is likely if it takes a high value, typically larger than 0.6. The values we obtain being quite low (except for ination), our retained variables are well explained by the identied factors.

The results regarding the rst two factors are synthesized in Figure 3. Specically, the top chart, i.e., the factor loadings plot, displays the position of each variable in the Factor 1 -Factor 2 plane. Its aim is to identify clusters of variables with similar loadings.

The bottom chart ( Scores) of Figure 3 displays the score of individual countries on each factor, the values being provided in Table B.2.1.6. The closer the country is to a variable, the more important is the score of the country regarding this variable.

An interesting observation, which is in line with the results of the cluster analysis, is that Greece and, to a lesser extent, Portugal score very high with respect to external imbalances. Their position appears far from that of any other country (except Italy).

With the HAC analysis, Greece and Portugal were viewed as outliers among the eurozone countries in the sense that they were the last countries that merged into the nal cluster that included all other members. This is reected by their extreme position in the scores plot, @iA close to Debt and Ination suggesting the highest debt and ination levelson average, and @iiA far away from the other variables (as the current account position), reecting recurrent current account decits. The positions of Greece and Portugal as outliers are therefore conrmed by their poor performance reected in high nancing requirements before the launch of euro area. 25

Another interesting nding which also corroborates the results of the cluster analysis is that some core countries score rather low on the factor of external imbalances: France, Germany, the Netherlands and Belgium. As showed by the bottom chart of Figure 3, these countries form a cluster at the left-center of the scores plot, indicating that they had low ination rates and debt-to-GDP ratios before the EMU, which resulted in lower current imbalances and/or current surpluses. Other countries either score high on the factor of external imbalances, such as Italy, or score low on this factor, but 25 The bad performance of Greece is conrmed by the highest score displayed by this country when con- sidering the third factor (see Table B.2.1.6).

Figure 3 Factor analysis results (1980)(1981)(1982)(1983)(1984)(1985)(1986)(1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998) at the expense of higher internal imbalances, as Spain and Finland. Although Austria appears close to the core group, Factor 2 which displays a strong correlation with misalignmentsmakes this country apart from the group composed by France, Germany, the Netherlands and Belgium, as highlighted by the cluster analysis. 26 Only the position of Ireland gives a picture that is less clear-cut than the one delivered by the cluster analysis. Indeed, whereas Ireland is merged into the core group when the clustering method is used, its high unemployment rate makes it move away from the core in the factor analysis. This gap is partly explained by the highestnegativescore regarding the third factor displayed by Ireland. As this third factor is not represented in Figure 3, this may aect the global pattern. Despite these facts, its position issued from the factor analysis remains compatible with our previous ndings. Indeed, the factor analysis being performed using data on the total unemployment rate, its result simply highlights that Ireland was one of the countries exhibiting the highest unemployment rate before the monetary union. Meanwhile, Ireland also had a quite highestimatedNAIRU (Non-Accelerating Ination Rate of Unemployment) which makes the aforementioned rate of unemployment sustainable, i.e., reecting a situation of near internal balance.

Overall, this rst factor analysis proves to be very informative and comes in support to the ndings of the HAC analysis. Indeed, both approaches suggest that Belgium, France, Germany, the Netherlands, and Ireland form the core group. While the picture for the rst four aforementioned countries is clear-cut from both the cluster and factor analyses, the inclusion of Irelandindicated by the HAC approachis also found relevant by the factor analysis, except for the unemployment gure. The other economies appear quite dissimilar; with persistent imbalances, the participation to the monetary union together with these core countries would imply costs. This is particularly true for Greece and, to a lesser extent, for Portugal. 27

Factor analysis after the launch of the euro

Let us now turn to the 1999-2016 period to examine how the observed trends have evolved since the launch of the single currency, and whether they remain consistent with those derived from the cluster analysis.

As for the previous pre-EMU period, two principal factors explain most (about 90%) of the total variance (Tables B.2.2.1 and B.2.2.3), while reecting dierent patterns (Table B.2.2.4). The rst factor may now be interpreted as reecting virtuous countries: this axis opposes the debt-to-GDP ratio on the left side, and growth on the right side. For its part, Factor 2 can be seen as ranking countries facing macroeconomic imbalances:

it displays negative and important correlation with the current account balance, and is positively correlated with unemployment and ination. Hence, the higher the country's score on this axis is, the larger its macroeconomic imbalances.

28 Thus, the two principal factors covering the EMU period point to a stronger divergence across countries inside the eurozone. This result is in accordance with the time-increasing dissimilarities across euro-area members revealed by the cluster analysis.

As regards scores of individual countries on each factor (Table B.2.2.6), it appears from the bottom chart of Figure 4 that the overall economic situation of Greece and, to a lesser extent, of Portugal deteriorated, compared to the previous period. Indeed, these two countries have the highest scores on the macroeconomic imbalances' factor.

They have accumulated more imbalances in the rst decade of the euro than before because of abundant capital ows and low interest ratesas reected by the current account decit, the unemployment rate, their zero-growth rate on average and their considerable overvaluation. This nding is in line with the continuous divergence path of Greece and Portugal revealed by the cluster analysis. Furthermore, Belgium, France, Germany, and the Netherlands appear very close to each other: they have negative scores regarding Factor 1, and their performancesespecially for the German and Dutch economiesremain better than most of the other countries. Ireland is still quite close to this core group according to the second factor.

29 Finland is located near Austria which, as for the pre-EMU period, is still very close to the core group regarding the rst two factors. The main dierence making it apart from this group comes from the third factor for which Austria displays a positive score, while the core countries score negatively. As this third factor is mainly related to currency misalignments, this corroborates our ndings from the cluster analysis. The situation of Spain has been catching up with those of the core countries thanks to the improvement in its debt-to-GDP ratio, while being still quite distant regarding Factor 2 with remaining macroeconomic imbalances at play. 30

Turning to Italy, it still faces macroeconomic imbalances but, most importantly, exhibits the second highest negative score regarding Factor 1, which tends to distance 29 Again, the two-dimensional plane representation blurs the perception since its does not allow for the third dimension (i.e., Factor 3 ). Ireland distorts the core group countries' cloud towards the right because of its high growth rate (the highest in average).

30 As for Ireland, Spain exhibits high values of both unemployment rate and estimated NAIRU. Since the factor analysis is performed using the unemployment levels, the position of the Spanish economy on the graph does not reect inconsistency between the core group and Spain, but simply the use of the total unemployment data.

Figure 4 Factor analysis results (1980-1998 and 1999-2016) its economy from those of the core, as Portugal. Once again, these ndings are in line with the results of the HAC analysis.

Overall, the observations made on economic performances of the eurozone countries over the 1999-2016 period are not far from the predictions based on the cluster analysis.

This conrms the relevance of our methodology: by separating the most similar equilibrium exchange rate paths into several clusters, we are able to analyze the extent of dissimilarities across the eurozone members in terms of macroeconomic imbalances and show how they have increased over time.

Conclusion

This paper analyzes the degree of dissimilarity among eurozone members by focusing on their equilibrium exchange rate paths. Since a country entering a single currency area gives up its own exchange rate, its equilibrium exchange rate path has rather to be in line with that of the other members in order to prevent unsustainable internal and external imbalances that could undermine the smooth functioning of the currency area. We thus rely on these equilibrium exchange rate trajectories to assess the similarities/dissimilarities of countries inside the euro area.

By applying our methodology to the eurozone project from 1980 to 1998, we are able to identify disparities among the member countries, distinguishing two groups of economies and two outlier countries. On the one hand, Belgium, France, Germany, Ireland, and the Netherlands form the most homogenous group; Austria, Finland, and Spain constitute the second group to which Italy can also be linked. Portugal and Greece are found to be the countries exhibiting very dierent equilibrium exchange rate paths;

Greece being the most idiosyncratic country, because of its structural weaknesses regarding the nancing of its economy.

Extending the period further reveals that countries did not move structurally closer to each other. Our ndings point to increasing disparities across and within countries' groups over time. This may not be surprising when considering historical examples of monetary unions. Indeed, according to [START_REF] Rocko | How Long Did It Take the United States to Become an Optimal Currency Area?[END_REF], it took decades to the United States to become an OCA, and this happened only after strong institutional steps such as the introduction of interregional transfers. The success of the euro during its rst decade of existence may have create the appearance of a convergence process driven by abundant capital ows that fueled consumption and real estate demand in the peripheral countries without however increasing their production capacities. Since the 2008 crisis that undermined this debt-driven growth process, many steps have been taken in order to correct macroeconomic imbalances. However, too little has been done concerning a common scal policy. It is now time to recognize that OCA conditions will not arise spontaneously by the mere eect of the single currency. Large reforms of the euro governance are still needed to ensure the optimality of the single currency. Besides, real convergence should be also dealt with, because it was not fostered automatically by monetary union as expected. It is necessary to revive the catching-up process in the Southern members, for example by channeling more EU funds to the development of specic industries, in order to maintain social cohesion inside the area. Note: LR test: independent vs. saturated: chi2(15) = 20.82 Prob>chi2 = 0.1426. The proportions and cumulative proportions columns are computed using the sum of all eigenvalues as the divisor. This explains cumulative greater than 1. Notes: N. obs. = 19; retained factors = 3; N. of params = 15. LR test: independent vs. saturated: chi2(15) = 41.81 Prob>chi2 = 0.0002. The proportions and cumulative proportions columns are computed using the sum of all eigenvalues as the divisor. This explains cumulative greater than 1. Notes: N. obs. = 19; retained factors = 3; N. of params = 15. LR test: independent vs. saturated: chi2(15) = 41.81 Prob>chi2 = 0.0002. The proportions and cumulative proportions columns are computed using the sum of all eigenvalues as the divisor. This explains cumulative greater than 1. As can be seen, when we impose two clusters (top left chart), Greece, in line with our previous results, appears as an outlier. For k = 3 (i.e., 3 clusters), Austria, Belgium, France, Germany, Ireland and the Netherlands form the core countries group; Finland, Italy, Portugal and Spain constitute a second group while Greece still appears as an outlier. For k = 4, the obtained clusters are the same except that Portugal now appears

  13 Figure A.1 in Appendix A displays the calculated real eective exchange rates and estimated equilibrium real exchange rates. Corresponding misalignments are reported in Figure A.2.

  an outlier, neither linked to core nor to peripheral members. It has even become more distinct from the other countries over time as the distance to the other clusters has increased. These latter results are in line with those of[START_REF] Wortmann | One Size Fits Some: A Reassessment of EMU's Coreperiphery Framework[END_REF] and[START_REF] Ahlborn | The coreperiphery pattern of European business cycles: A fuzzy clustering approach[END_REF] who apply dierent cluster algorithms respectively to the Macroeconomic Imbalance Procedure (MIP) indicators and to output gap series. They also nd evidence in support of a strengthening over time of the coreperiphery structure of the euro area.

  Let us start with the pre-EMU, 1980-1998 period. As shown in Table B.2.1.1 in Appendix B, only the rst three factors are retained as the eigenvalues associated with the other factors are negative. The rst two factors are the most meaningful, explaining the major part (around 75%) of the total variance (Table

Figure A. 1

 1 Figure A.1 Real eective exchange rates: calculated and equilibrium levels Note: The real exchange rate indexes are in logarithmic scale. An increase (resp. a decrease) indicates an appreciation (resp. a depreciation).

Figure A. 2

 2 Figure A.2 Currency misalignments (in percentage)Note: a positive (resp. negative) value corresponds to an overvaluation (resp. undervaluation).

  The k-means procedureIn contrast with the HAC analysis used in the core of the paper, the k-means procedure belongs to the second family of classication techniques. It proceeds by partitioning n objects into k clusters, each object belonging to the cluster with the nearest mean. The number of clusters, k, is specied ex ante by the user, and cluster centers are iteratively estimated from the data. The objective of the k-means algorithm is to minimize total intra-cluster variance, or, the squared error function: the objective function; k and n indicate respectively the ex ante number of clusters and the number of objects, k : k denotes the distance function; c j corresponds to the centroid of cluster j and x i refers to the object i.The resultsIn applying the k-means procedure, we begin by investigating the number k of clusters that would lead to a clear partitioning of the objects. More specically, we run the algorithm considering k = 2, 3, and 4. While this could be seemed ad hoc, it should be noted that these cases are sucient to confront the k-means-based results with the HAC-based results. The clusters obtained relying on the k-means procedure are presented in Figure D.1.

Table A .

 A 2 in Appendix A presents the CPMG and CMG estimates, as well as the Hausman test statistic examining panel heterogeneity. According to the test statistic, the long-run homogeneity restriction is not rejected for individual parameters and jointly in all regressions. We therefore focus on the CPMG estimates. Considering the whole period, results reported in TableA.2 appear consistent with the theory since the coecients have the expected signs. Indeed, the real eective exchange rate appreciates in the long run with the increase in the relative real GDP per capita (in PPP terms), the improvement in the net foreign asset position and in the terms of trade.

Table B

 B

	.1 giving some descriptive statistics on the variables. Specically, Tables B.2.1.1 and
	B.2.1.2 (respectively B.2.2.1 and B.2.2.2) are related to the results of the unrotated
	analysis for the

Table 1

 1 Estimation of the long-run relationship

	Dependent variable:			d.reer
			CPMG				CMG
	Estimation procedure:	Coef.	Std. Err.	Coef.	Std. Err.
	Long-run dynamic
	rprod	0.242 £	0.128	0.170	0.268
	nfa	0.189 ££	0.093	0.448	0.340
	tot	0.867 £££	0.230	1.198	£££	0.349
	Short-run dynamic
	ec.	-0.092	£££	0.021	-0.202	£££	0.025
	d.rprod	-0.195	£££	0.074	-0.215 ££	0.110
	L.drprod	-0.033	0.036	-0.084	0.054
	d.nfa	-0.098 £	0.059	-0.078	0.062
	L.dnfa	-0.040	0.036	-0.039	£	0.024
	d.tot	0.318 £££	0.111	0.257	0.158
	L.dtot	0.253 ££	0.104	0.186	££	0.084
	Constant	0.051 £££	0.012	-0.331	0.397
	Specication tests
	Joint Hausman test a 2 @QA			2.41 [p.value = 0.4919]
	No Obs. / No. countries	451/11	451/11	451/11	451/11
	Notes: The estimation is done on the 1976-2016 period. Symbols ***, **, and * denote signicance at the 1%, 5%, and 10% statistical level. d (resp. "L:") is the dierence operator (resp. the lag operator); "e c :" is the error correction term. a: Null of long-run homogeneity

Table B

 B

	B.2. The results
	B.2.1. Period: 1980-1998
	Table B.2.1.1 Factor analysis (principal factors, unrotated) .1 Average data Period 1 CA Ination Debt Unemp. Growth Mis Austria -0.011 0.033 0.704 0.036 0.023 -0.052 (0.017) (0.018) (0.102) (0.009) (0.001) (0.217) Belgium 0.019 0.036 1.059 0.089 0.021 -0.017 (0.031) (0.026) (0.182) (0.014) (0.001) (0.048) Finland -0.008 0.049 0.450 0.088 0.026 0.024 (0.033) (0.035) (0.213) (0.047) (0.001) (0.143) France 0.002 0.048 0.682 0.087 0.021 -0.009 (0.014) (0.043) (0.183) (0.013) (0.001) (0.077) Factor Eigenvalue Dierence Proportion Cumulative 0.83544 0.6564 0.6564 Factor 1 1.79817 0.46947 0.3514 1.0078 Factor 2 0.96273 0.49570 0.1801 1.1879 Factor 3 0.49326 0.24153 -0.0009 1.1870 Factor 4 -0.00245 0.02430 -0.0891 1.0979 Factor 5 -0.24398 . -0.0979 1.0000 Factor 6 -0.26828 Notes: N. obs. = 14; retained factors = 3; N. of params = 15. LR test: i nd e pe nd e nt vs. s at ur at e d : chi2(15) = 20.82; Prob>chi2 = 0.1426. The proportions and cumulative proportions columns are computed using the sum of all eigenvalues as the divisor. This explains cumulative greater than 1.	Period 2 CA Ination Debt Unemp. Growth Mis 0.018 0.019 0.732 0.047 0.016 0.202 (0.016) (0.008) (0.125) (0.007) (0.001) (0.038) 0.018 0.019 1.013 0.079 0.017 0.046 (0.024) (0.011) (0.102) (0.006) (0.001) (0.057) 0.029 0.016 0.453 0.085 0.016 0.092 (0.035) (0.011) (0.115) (0.010) (0.001) (0.076) 0.001 0.014 0.735 0.091 0.014 0.074 (0.017) (0.009) (0.186) (0.009) (0.001) (0.089)
	Germany	0.006 (0.022)	0.028 (0.019)	0.656 (0.078)	0.074 (0.016)	0.021 (0.001)	-0.004 (0.067)	0.043 (0.032)	0.014 (0.008)	0.684 (0.098)	0.077 (0.021)	0.013 (0.001)	0.072 (0.089)
	Greece	-0.030 (0.016)	0.160 (0.062)	1.170 (0.277)	0.079 (0.021)	0.013 (0.001)	-0.075 (0.086)	-0.076 (0.048)	0.022 (0.019)	1.284 (0.311)	0.143 (0.071)	0.003 (0.001)	0.191 (0.103)
	Ireland	-0.018 (0.047)	0.061 (0.061)	0.568 (0.186)	0.162 (0.036)	0.048 (0.001)	0.079 (0.062)	-0.010 (0.041)	0.021 (0.020)	0.589 (0.377)	0.080 (0.041)	0.051 (0.001)	0.110 (0.159)
	Italy	-0.004 (0.021)	0.080 (0.058)	1.096 (0.146)	0.091 (0.014)	0.020 (0.001)	-0.051 (0.064)	-0.006 (0.017)	0.018 (0.011)	1.112 (0.158)	0.092 (0.020)	0.003 (0.001)	0.178 (0.093)
	Netherlands 0.032 (0.015)	0.027 (0.018)	0.571 (0.116)	0.082 (0.013)	0.025 (0.001)	0.004 (0.077)	0.065 (0.027)	0.019 (0.013)	0.555 (0.099)	0.044 (0.014)	0.015 (0.001)	0.060 (0.097)
	Portugal	-0.028 (0.044)	0.122 (0.082)	0.726 (0.092)	0.069 (0.018)	0.031 (0.001)	-0.139 (0.441)	-0.072 (0.047)	0.021 (0.015)	0.832 (0.330)	0.089 (0.039)	0.006 (0.001)	0.406 (0.104)
	Spain	-0.014 (0.015)	0.075 (0.042)	0.574 (0.173)	0.155 (0.033)	0.026 (0.001)	0.006 (0.100)	-0.040 (0.037)	0.022 (0.015)	0.610 (0.261)	0.157 (0.061)	0.019 (0.001)	0.097 (0.096)

Note: Period 1 (resp. 2) corresponds to the

1980-1998 (1999-2016)

period. Standard deviations are in parentheses.

Table B

 B

	.2.1.2 Factor loadings (pattern matrix) and unique variances
	Variable CA Ination Debt Unemployment 0.4086 0.3856 0.4125 Factor 1 Factor 2 Factor 3 -0.6271 0.3602 -0.3132 0.5780 -0.2241 -0.0505 0.7208 -0.2922 -0.1900 -0.5052 -0.3400 0.4288 Growth Mis 0.3596 0.6581 0.0497	Uniqueness 0.3790 0.6132 0.3590 0.5142 0.4453 0.4352
	Table B.2.1.3 Factor analysis (principal factors, rotation: ortho-
	gonal varimax (Kaiser o ))
	Factor Variance Dierence Proportion	Cumulative
	Factor 1 1.49477 0.56347 Factor 2 0.93130 0.10323 Factor 3 0.82808 .	0.5456 0.3400 0.3023	0.5456 0.8856 1.1879

Table B

 B

	Table B.2.1.5 Factor rotation matrix
	Variable Factor 1 Factor 2 Factor 3 Factor 1 0.8091 0.3799 0.4484 Factor 2 -0.5601 0.7294 0.3928 Factor 3 0.1779 0.5690 -0.8029
	Table B.2.1.6 Country scores on the factors
	Country	Scores Factor 1 Factor 2 Factor 3
	-0.1314 0.0287 -0.3933 -0.1974 -0.4121 1.4614 0.0413 0.6358 Netherlands -0.8481 0.2555 -0.3485 Austria -0.0055 Belgium 0.6349 Finland 0.3027 France 0.2030 Germany -0.2746 Greece 1.5273 Ireland -0.0812 Italy 0.7534 -0.3659 Portugal Spain 0.1033 1.1802	0.2882 0.8777 -0.0379 0.4071 0.4288 1.0653 -0.9380 0.7774 0.3726 -0.2106 -0.1085
	B.2.2. Period: 1999-2016
	Table B.2.2.1 Factor analysis (principal factors, unrotated)
	Factor Eigenvalue Dierence Proportion Cumulative 0.51829 0.5975 0.5975 Factor 1 1.93655 1.05895 0.4376 1.0350 Factor 2 1.41826 0.43996 0.1109 1.1459 Factor 3 0.35931 0.07513 -0.0249 1.1210 Factor 4 -0.08065 0.08059 -0.0481 1.0729 Factor 5 -0.15578 Factor 6 -0.23637 . -0.0729 1.0000
	.2.1.4 Rotated factor loadings (pattern matrix) and unique
	variances
	Variable	Factor 1 Factor 2 Factor 3	Uniqueness
	-0.7648 0.5841 0.7130 Unemployment 0.1879 0.6712 0.0035 -0.1537 0.1117 CA 0.0274 0.2117 Ination -0.0475 0.3610 Debt -0.1420 -0.1959 -0.7044 Growth Mis -0.0688 0.6448 0.3798	0.3790 0.6132 0.3590 0.5142 0.4453 0.4352

Table B .

 B 2.2.2 Factor loadings (pattern matrix) and unique variances

	Table B.2.2.3 Factor analysis (principal factors, rotation: ortho-
	gonal varimax (Kaiser o ))
	Factor Variance Dierence Proportion	Cumulative
	Factor 1 1.73215 0.13224 Factor 2 1.59991 1.21785 Factor 3 0.38206 .	0.5344 0.4936 0.1179	0.5344 1.0280 1.1459
	Variable	Factor 1 Factor 2 Factor 3	Uniqueness
	-0.4918 0.7703 -0.6201 Unemployment 0.4892 0.6023 -0.2388 -0.6788 -0.1358 CA 0.0588 -0.0419 Ination 0.6047 0.0221 Debt 0.6910 -0.4311 0.1423 Growth Mis -0.0046 0.1996 0.5112	0.2789 0.4014 0.2494 0.3410 0.3164 0.6988

Table B

 B This appendix is devoted to the presentation of the alternative clusters based on the k-means procedure. It begins with a brief description of the method before presenting the results.

	D. The k-means clustering
	.2.2.4 Rotated factor loadings (pattern matrix) and unique
	variances
	Variable CA Ination Debt Unemployment -0.0015 0.7980 -0.1493 Factor 1 Factor 2 Factor 3 0.0115 -0.8167 -0.2325 0.5684 0.5237 -0.0349 -0.8545 0.0890 0.1122 0.8202 0.0714 0.0757 Growth Mis -0.0777 0.0948 0.5350	Uniqueness 0.2789 0.4014 0.2494 0.3410 0.3164 0.6988
	Table B.2.2.5 Factor rotation matrix
	Variable Factor 1 Factor 2 Factor 3
	Factor 1 0.7893 0.6140 -0.0028 Factor 2 -0.6071 0.7810 0.1465 Factor 3 0.0921 -0.1140 0.9892
	Table B.2.2.6 Country scores on the factors
	Country	Scores Factor 1 Factor 2 Factor 3
	-0.4028 -0.8397 -0.0808 -0.6286 -0.5415 -1.6878 0.7229 -1.3998 Netherlands -0.1176 -1.4009 -0.4481 -0.8675 0.2877 Austria -0.3891 -0.1807 Belgium -0.6958 -0.5925 Finland -0.3139 -0.2989 France -0.9036 -0.5632 Germany 1.4056 0.0155 Greece -0.2253 0.5022 Ireland -0.0008 -0.0857 Italy -0.8742 0.5663 0.7849 Portugal Spain -0.2410 1.0243 -0.7789

For the sake of robustness, we have also estimated a simpler specication including only productivity dierential and net foreign asset position as determinants of real exchange rates; the results remain similar and are available upon request.

Luxembourg and the other countries are excluded due to data availability issues during the 1980s and 1990s.

See Table A.1 in Appendix A for details regarding the data.

http://www.cepii.fr/CEPII/fr/bdd_modele/presentation.asp?id=34.

The use of time-varying weights is important to move closer to the reality by capturing trade dynamics.See[START_REF] Couharde | EQCHANGE: A World Database on Eective Exchange Rates and Equilibrium Real Eective Exchange Rates[END_REF] for details regarding the EQCHANGE database.

Due to the lack of sectoral data on output and employment for traded and non-traded goods sectors, most empirical studies test the Balassa-Samuelson hypothesis by relating the real exchange rate to the real GDP per capita (in PPP terms) dierential which is a common measure for productivity. We follow here the same approach.

Further note that all the mentioned procedures impose long-run slope homogeneity but, in the case of the CPMG procedure, this hypothesis can be tested (Hausman-test type).

It should be noted that for the sake of consistency and uniformity of the analysis, the selection of the variables must satisfy both ex ante prospective and ex post evaluation exercises. This explains why we did not restrict the analysis to the sole convergence criteria.

It should be noted that the position of Austria is not straightforward as this country is found to be in the core group over the 1980-1998 period when we use the k-means procedure as an alternative clustering method (see Appendix D).

In contrast with Greece, the analysis indicates that Portugal only suers from competitiveness problems. This holds for Spain, but to a lesser extent.

Note that the third factor is principally dened by currency misalignmentswhile opposing them with the current account balance. Factor 3 can thus be seen as the competitiveness factor: the higher the overvaluations, the lower the trade performances.

Appendices

A. Data and cointegrating regression estimation results

Greece (GRE) also as an outlier, between Greece and the peripheral countries.

Overall, except Austria, the obtained clusters are consistent with those of the HAC analysis.