This paper develops an estimated multi-country open economy dynamic stochastic general equilibrium (DSGE) model with incomplete Exchange Rate Pass-Through (ERPT) for the Euro-area. It is designed to model global international linkages and to assess international transmission of shocks under an endogenous framework and incomplete ERPT assumption. On the one hand, we relax the small open economy framework (SOEF) but derive a canonical representation of the equilibrium conditions to maintain analytical tractability of the complex international transmission mechanism underlying the model. Namely, the model considers economies of di¤erent size that are open and endogenously related. On the other hand, in order to take into account international linkages, possible cointegration relationships within domestic variables and between domestic and foreign variables, and the role of common unobserved and observed global factors such as the oil price, we use the Global VAR model to estimate the steady state of observed endogenous variables of the multi-country DSGE model. Namely, steady states are computed as long-horizon forecasts from a reduced-form cointegrating GVAR model. ERPT analysis conducted from the estimated multi-country DSGE model for the Euro-area in relation with its …ve main trade partners which are the United Kingdom, the United States, China, Japan and Switzerland yields the following results. First, exchange rate volatility contributes to a large part of import price in ‡ation variation of the Euro-area in contrast to foreign mark-up shocks. Second, deviation from in ‡ation objective of the foreign trade partners contributes to another source of the Euro-area import price variability. Third, nominal rigidity induces a persistent but a lower impact of the exchange rate changes on import in ‡ation.

Introduction

This paper develops a multi-country open economy dynamic stochastic general equilibrium (DSGE) model with incomplete exchange rate pass-through (ERPT) assumption for the Euro-area. The latter assumption is introduced, following [START_REF] Monacelli | Monetary policy in a low pass-through environment[END_REF], by the means of nominal import price rigidity. That is, there exists a continuum of monopolistic importing …rms that buy homogeneous goods in international market and sell it to domestic consumers. Optimal mark-up problem of the importing …rms, namely by using staggered price setting framework à la Calvo, leads to deviation from the law of one price assumption and, hence, to incomplete ERPT. Hence, and as detailed in [START_REF] Burstein | International prices and exchange rates[END_REF], we assume a normative perspective in choosing this framework. 1 The main focus of this study is to model global international linkages under an endogenous framework and to highlight the important role of the incomplete ERPT assumption on the international transmission of shocks. For these purposes, …rst, we relax the small open economy framework (SOEF) and consider economies of di¤erent sizes that are open and endogenously related. 2 However, we maintain analytical tractability of the complex international transmission mechanism underlying the model by deriving a canonical representation of the model equilibrium solutions. Second, to model explicitly the direct and indirect link between domestic and foreign variables, but also the possible cointegration relationships within domestic variables and between domestic and foreign variables, we estimate the steady state of variables as the long-horizon forecasts from a reduced-form cointegrating Global VAR. This approach permits to give steady states an economic interpretation and, thus, to circumvent the "black box" paradigm, in the sens of [START_REF] Garratt | Permanent vs transitory components and economic fundamentals[END_REF], that arises from the use of pure statistical …ltering procedure. Moreover, the important role of the oil price that is considered as common observable global factor and the United States'…nancial variables in the world economy are handled through the speci…cation of the GVAR model. Third, structural parameters of the model are estimated using variables measured as deviation from their estimated GVAR steady states and Bayesian estimation methodology.

It is admitted that incomplete ERPT phenomenon modi…ed the traditional Keynesian international transmission channel such the expenditure swtiching e¤ects, consumption risk sharing and the law of one price assumption [START_REF] Betts | Exchange rate dynamics in a model of pricing-tomarket[END_REF], and Engel(2002b)), and the conduct of monetary policy [START_REF] Gagnon | Monetary policy and exchange rate pass-through[END_REF] and [START_REF] Monacelli | Monetary policy in a low pass-through environment[END_REF]). De- 1 One can cite among others Obstfeld and Rogo¤ (2000) and [START_REF] Smets | Openness, imperfect exchange rate pass-through and monetary policy[END_REF] for theoretical studies that support the normative issue. At the empirical side, one can cite the work of [START_REF] Campa | Exchange rate pass-through into import prices: A macro or micro phenomenon?[END_REF], [START_REF] Campa | Exchange rate pass-through to import prices in the euro area[END_REF], and [START_REF] Campa | Di¤erences in exchange rate passthrough in the euro area[END_REF]. Normative issue is commonly known as the Producer Currency Pricing (PCP) literature. 2 It is worth noting that it is common in the open economy DSGE model literature to assume domestic economy as small relative to the rest of the world. Among others, one can cite the work of [START_REF] Adolfson | Bayesian estimation of an open economy dsge model with incomplete pass-through[END_REF] and Christo¤el et al. (2008). Under the SOEF assumption, domestic variables have negligible e¤ects on foreign variables which are taken as exogenous and usually assessed using a vector autoregressive model. This assumption is mainly adopted, as is the case in [START_REF] Gali | Monetary policy and exchange rate volatility in a small open economy[END_REF], to have isomorphism between the closed and open economy version of the model equilibrium solutions. parture from the SOEF framework permits to handle some important features of the open economy model under the assumption of incomplete ERPT. First, the multi-country open economy model considers economies of di¤erent sizes that are open and endogenously related. As regards to ERPT analysis, this has stark implications. It permits to have endogenously foreign producers' cost-push shocks in the import price law of motion in contrast to other models in the literature that have exogenous shocks that measure a shift in the elasticity of substitution among imported goods. Second, it permits asymmetric trade ‡ows and diversi-…cation of foreign trade for each economy. For ERPT analysis, it renders the role of import market share and currency invoicing more relevant. On the one hand, economies are endogenously linked by international trade and the importance of a trade partner is measured by the means of the import and export share. Hence, the extent to which trade partners'variables in ‡uence domestic variables depends mainly on their respective market share. One the other hand, the multi-country speci…cation permits to handle the important role of currency invoicing in in ‡uencing the extent of ERPT under a multi-currency analysis. 3 It leads to a distinction that has to be made between pass-through that arises from bilateral and multilateral or e¤ective exchange rate movements. Moreover, it highlights the role of currency invoicing in international trade. Namely, we found that the Pound bilateral exchange rate drives to a larger extent the variability of the Euro-area import price in ‡ation and thus con…rms the important role played by the United Kingdom as trade partner.

Focusing our interpretation on the ERPT analysis, we …nd the following results. First, the variance decomposition analysis permits to conclude that exchange rate volatility contributes to a large part of the import price in ‡ation volatility. More precisely, it accounts for 70:83% upon the impact of the shock. Such evidence supports the relevance of the normative approach in contrast to the positive one that supports the declining value of the extent of ERPT over time. 4 Second, foreign mark-up shocks have a reduced impact on the variability of the import price in ‡ation. Along with the traditional arguments such as the presence of international competitive market and the weight attributed to foreign goods in the domestic consumption basket, it is mainly explained by the fact that foreign mark-up shocks' impact on foreign in ‡ation, and to a certain extent on foreign marginal cost, is limited and short-lived. This is again an argument in favor of the Producer Currency Pricing (PCP) behavior rather than the LCP-PTM at the export side. Third, we found that deviation from in ‡ation objective, namely in the foreign trade partners, contributes to another important part of the Euro-area import price variability. More precisely, it accounts for 12:47% of the volatility. If one assumes without loss of generality that foreign marginal cost follows the same dynamics as foreign in ‡ation, an assumption generally adopted in the empirical literature of pass-through, we found that the United Kingdom marginal cost variability contributes to a large part, more precisely 6:28%, in the volatility of the Euro-area import in ‡ation. Fourth, and most importantly, nominal rigidity induces a persistent but a lower impact of the exchange rate changes on import price in ‡ation. This has a stark implication for the conduct of the monetary policy. Reinforced by the presence of home consumption bias and the trade-o¤ between output and law of one price gap stabilization generated by the new independent channel of monetary policy arising from incomplete pass-through assumption, the monetary authority could pursue a stable in ‡ation target with less action, as far as ERPT is concerned. Namely, this …nding is supported by the low estimate value of the law of one price gap in the interest rate rule.

The rest of the paper is organized as follows: section 2 develops the multi-country open economy DSGE model; section 3 summarizes the equilibrium solutions of the model into the standard canonical representation; section 4 describes the estimation procedure and the data; section 5 reports the results; and section 6 concludes.

Multi-country open economy DSGE

The world economy is considered as a set of N open economy countries indexed by i 2 f1; : : : ; N g which are supposed to be di¤erent from one another with regard to their population size, degree of openness related to the share of imports and exports on gross domestic product and international portfolio diversi…cation in world asset market. There are four types of agents in each country i: a continuum of monopolistic competitive …rms in the unit interval [0; 1] indexed by f i , a continuum of monopolistic competitive importing …rms also, without loss of generality, in the unit interval [0; 1] indexed by l i , households h 2 [0; P i ] which are supposed to be a monopolistic supplier of their individual labor indexed by L i (h) and where P i is the number of population in country i, and …nally a central bank.

All goods are tradable. Following [START_REF] Monacelli | Monetary policy in a low pass-through environment[END_REF], we assume that the law of one price holds "at the dock " for any international trade. However, deviation from the law of one price is generated by the optimal pricing behavior of monopolistic importing …rms. For simplicity, and without loss of generality, each single …rm uses only labor as input factor to produce one type of goods which will be consumed at home as well as exported abroad. It has a monopoly power over its single good but faces competition from other substitutable goods produced by other home and foreign …rms. Each household consumes and supplies monopolistically a distinctive variety of its labor to home …rms. Retailers import goods and sell it without any transformation to domestic consumers. Finally, central bank conducts monetary policy using nominal interest rate as instrument.

Within each country i, we assume symmetry so that households and …rms share respectively the same preference and technology. However, across countries, technology and preference over di¤erent types of home and foreign goods may di¤er. It is assumed that households have access to a complete set of nominal contingent claims traded internationally.

Concerning notation, lower case letters indicate the natural logarithm of variables, while upper case letters indicate level. Hatted lower case letters indicate log-deviation of variables from their respective steady state values. 5

Nominal and real exchange rate

Among N countries around the world indexed by i 2 f1; : : : ; N g, country 1 is chosen, without loss of generality, as a numeraire. It is the Euro-area in this study. 6 Denote E ijt the nominal bilateral exchange rate between country i and j expressed as country i's currency price of one unit of foreign country j's currency. By choosing country 1 as a numeraire, bilateral exchange rate between country i and numeraire country E i1t is, henceforth, denoted E it . This allows to write the bilateral exchange rate E ijt in log-linear term as:

êijt = êit êjt
Therefore, e¤ective real exchange rate will be given by

q it = (ê it pit ) N X j=1 ij (ê jt pjt ) (1)
where pit represents the consumption-based price index (CPI) of country i. Parameter ij , where ij > 0 and P N j=1 ij = 1, in turn represents the share of imported goods from country j on the consumption basket of the representative household in country i.

It is worth noting that we assume home consumption bias which is formally represented by the following condition: max i6 =j ij < ii < 1 8i 2 f1; : : : ; N g and that each economy is open. 7

Terms of trade and in ‡ation

Bilateral terms of trade between country i and j is de…ned to be the price of foreign country j's imported goods in terms of domestic country i's goods. That is in log-linear deviation terms, ŝijt = pijt pd it

where pijt denotes the home-currency (country i) price index of goods imported from country j and pd it the domestic producer price index. Using the de…nition of country i's aggregate consumer price index [START_REF] Dées | Supply, demand and monetary policy shocks in a multi-country new keynesian model[END_REF] and that of import price index [START_REF] Engel | Expenditure switching and exchange rate policy[END_REF], e¤ective terms of trade is given by:

ŝit = pf it (1 ii )p d it = pit pd it (3)
In terms of in ‡ation, de…ned as the rate of change in price index, CPI, IPI and producer price index (PPI) are linked according to:

^ it = ^ f it + ii ^ d it (4)
where, for latter use, import in ‡ation index (IPI) is de…ned as:

^ f it = N X j=1 j6 =i ij (p ijt pijt 1 ) = N X j=1 j6 =i ij ^ jf it (5)

Incomplete pass-through

Following [START_REF] Monacelli | Monetary policy in a low pass-through environment[END_REF], let

ij;t = êijt + pd jt pijt (6)
be the bilateral law-of-one price gap (hereafter l.o.p gap) between country i and j that measures the di¤erence between foreign prices and domestic currency prices of imports, and

it = N X j=1 j6 =i ij ij;t = N X j=1 ij ij;t
be the e¤ective l.o.p gap. The domestic currency price of imports pijt is set by monopolistic importing …rms as a result of an optimal markup problem. Namely, import prices are set in a staggered fashion à la Calvo. Hence, incomplete pass-through arises due to a fraction of …rms that do not adjust their prices. That is, changes in the exchange rate are not entirely transmitted into domestic price of imports due to nominal price rigidity, implying a deviation from the law of one price assumption.

Using the de…nition of the e¤ective real exchange rate (1) and that of the e¤ective terms of trade (3), one obtains:

it = q it N X j=1 ij ŝjt (7)
This last equation indicates that there are two sources of deviation from the purchasing power parity assumption, the relative price variations captured by the e¤ective terms of trade and the deviation from the law of one price assumption captured by the e¤ective l.o.p gap. Therefore, under the assumption of Calvo price setting, nominal rigidity generates gradual and persistent deviation from the law of one price assumption, hence incomplete pass-through, which in turn yields a persistent deviation from the purchasing power parity assumption.

CPI and PPI index

Combining country i's aggregate CPI, IPI, PPI, home-currency price index of good from country j (21), and the de…nition of e¤ective l.o.p gag, yields a relationship between CPI and PPI index. That is,

pit êit = N X j=1 ij pd jt êjt it
which in matrix form yields:

êt pt = M êt pd t + t (8)
where pt = (p 1t ; : : : ; pNt ) 0 , êt = (0; ê2t ; : : : ; êNt ) 0 , pd t = pd 1t ; : : : ; pd N t 0 , t = ( 1t ; : : :

; N t ) 0 ,
and M is the (N xN ) import share matrix where its (i; j)-th element equals ij . Using this relationship, the real e¤ective exchange rate (1) can be written in function of relative prices and e¤ective l.o.p gap in matrix form as:

q t = (I N M) M êt pd t + (I N M) t (9)
where q t = (q 1t ; : : : ; q N t ) 0 and I N is an (N xN ) identity matrix.

International risk sharing condition

Under the assumption of complete asset markets, households share risk internationally. Consumption index between two countries is therefore linked for all t according to:8

(C it H it ) = # i (C jt H jt ) Q ijt ( 10
)
where # i is a constant that depends on initial conditions and H it = hC it 1 is an external habit taken as exogenous by households. Without loss of generality, we assume that:

# i = 1 8i 2 f1; : : : ; N g
Log-linearized version of the international risk sharing condition yields:

ĉit hĉ it 1 = ĉjt hĉ jt 1 + (1 h)q ijt (11)
That is, through its impact on the real exchange rate, equation ( 9) and [START_REF] Campa | Exchange rate pass-through into import prices: A macro or micro phenomenon?[END_REF] clearly show that the l.o.p gap generates a deviation from the consumption risk sharing condition. Therefore, incomplete exchange rate pass-through distorts the traditional Keynesian e¤ects of exchange rate changes such as the consumption risk sharing and the expenditure switching e¤ect. [START_REF] Betts | Exchange rate dynamics in a model of pricing-tomarket[END_REF] get to the same conclusion using the class of new open economy model (NOEM). Intuitively, nominal depreciation of the home currency i generates a competitive gain for home producers. This will increase domestic revenue and, hence, consumption. However, demand of imported goods is not altered by the depreciation due to import price nominal rigidity. At the extreme case of zero pass-through, foreign …rms will also bene…t from the increase in home revenue which yields an increase in the foreign revenue, a departure from the expenditure switching e¤ect. In the foreign country j, incomplete pass-through prevents decrease in the foreign price of home exported goods. Thus, consumption will not increase as is the case under complete pass-through. This will generate, combined with the rise in home consumption, a deviation from risk sharing condition. Moreover, the international risk sharing condition [START_REF] Campa | Exchange rate pass-through into import prices: A macro or micro phenomenon?[END_REF] indicates that rise in home consumption (country i) relative to foreign, due for instance to a rise in home output, necessitates a real exchange rate depreciation to attain equilibrium. The latter can be achieved through terms of trade deterioration (a rise in ŝit through a fall in the price of domestic goods pd it ) or deviation from the law of one price assumption (a rise in the l.o.p gap through a nominal exchange rate depreciation).

Uncovered interest parity

Under the assumption of complete asset markets, the uncovered interest parity condition between country i and j E t f êijt+1 g = rn it rn jt [START_REF] Campa | Exchange rate pass-through to import prices in the euro area[END_REF] is obtained using households optimal condition (25) and international risk sharing condition [START_REF] Campa | Exchange rate pass-through into import prices: A macro or micro phenomenon?[END_REF]. rn it denotes the nominal interest rate of country i.

Canonical representation

To conclude the presentation of the model, let us resume the equilibrium conditions into the standard canonical representation. This permits to have tractability in the interpretation and to see what our speci…cation brings compared to the standard small open economy framework. Typically, the latter can be represented, as in the closed economy setup, by three well know equations which are the new Keynesian Phillips curve, the dynamic investment-saving equation and the monetary policy rule. However, the inclusion of the incomplete pass-through assumption will add in this canonical representation another equation that characterizes the law of motion of the import price in ‡ation.

Phillips curve

Substituting the real marginal cost (49) into the domestic in ‡ation dynamics (40) yields the new keynesian hybrid Phillips curve:

^ d it = id 1 + id ^ d it 1 + 1 + id E t n ^ d it+1 o + ! i ŷit d i h 1 h ŷit 1 d i h 1 h it + d i ^ w it d i (1 + ) âit + d i " p it where ! i = d i (1 + (1 h)) = (1 h).
Hence, the producer price in ‡ation ^ d it is driven by its past value through in ‡ation indexation and its future discounted path. Moreover, it depends positively on the current value of output ŷit and negatively on productivity shock âit . A rise in output induces a rise in domestic goods in ‡ation through its positive impact on marginal cost whereas a rise in productivity shock induces a decrease in domestic goods in ‡ation through its negative impact on marginal cost, namely the labor productivity. Apart from these variables that typically characterize the new Keynesian Phillips curve, there are additional frictions that arise from the model speci…cation. First, the habit formation leads domestic goods in ‡ation to depend negatively on past output and real interest rate di¤erential it through international risk sharing condition and optimal allocation of expenditures. Second, labor market friction ^ w it induced by the households' monopolistic wage setting behavior has a positive impact on domestic goods in ‡ation. It acts as households' mark-up that raise the marginal cost of producing domestic goods using labor. Finally, domestic goods in ‡ation rises with the cost-push shock " p it .

Import price law of motion

Aggregating bilateral import price in ‡ation across foreign partners j and using import in ‡ation index (5) yield the aggregate import in ‡ation law of motion. That is,

^ f it = if 1 + if ^ f it 1 + 1 + if E t n ^ f it+1 o + f i it + " f p it ( 13
)
where it is the country i's e¤ective l.o.p gap de…ned in [START_REF] Betts | Exchange rate dynamics in a model of pricing-tomarket[END_REF] and " f p it = P N j=1;j6 =i ij " p jt . This last equation deserves some comments. At the aggregate level, import price in ‡ation varies positively with the e¤ective l.o.p gap and the e¤ective foreign cost-push shock. On the one hand, under the assumption that the law of one price holds at the dock, the foreign price of imported goods (ê ijt + pd jt ) can be interpreted as the importing …rm marginal cost. Therefore, nominal depreciation of the home currency i triggers an increase in marginal cost that takes the form of an increase in the l.o.p gap. The extent to which, however, this depreciation is passed-through import price in ‡ation depends on the degree of import price nominal rigidity iF contained in the coe¢ cient f i . On the other hand, the multi-country speci…cation allows the import price in ‡ation to depend on foreign cost-push shocks. However, the latter di¤er from the importing …rms markup shocks usually obtained in the open economy DSGE model incorporating import block and that are generally interpreted as a shift in the elasticity of substitution among imported goods. Here, " f p it represents a weighted average of trading partners cost-push or markup shocks.

It is worth noting that the impact on the aggregate import price in ‡ation of bilateral exchange rate movements and cost-push shocks between home country i and foreign country j will depend both on the degree of import price nominal rigidity iF and the share of imported goods ij . That is, bilateral analysis shows how market share plays an important role in the exchange rate pass-through issue.

Dynamic IS-equation

Substituting the link between consumption and output (53) into the Euler equation ( 25) yields the following dynamic IS-equation in matrix notation:

Mŷ t = 1 1 + h ME t fŷ t+1 g + h 1 + h Mŷ t 1 + 1 h 1 + h E t f t+1 g (14)
h 1 + h ME t f t+1 g 1 h 1 + h (r n t E t f^ t+1 g)
Thus, assumptions adopted in the model introduce new variables in the dynamics of the demand compared to the standard canonical representation. First, the introduction of the habit formation in the model leads output to depend negatively on the expected changes in the e¤ective real interest rate di¤erential. Second, expected value of the output and that of changes in the l.o.p gap are negatively linked. As will be explained below, this indicates a trade-o¤ between the stabilization of output gap and l.o.p gap that the monetary authority has to face due to the introduction of incomplete exchange rate pass-through assumption.

Monetary policy

To close the model, let us assume that monetary authority follows an instrument rule rather than optimizing a speci…c loss function. Monetary authority adjusts nominal interest rate in response to deviation of CPI in ‡ation from its target, to output gap ỹit = ŷit ŷn it and to l.o.p gap. Following [START_REF] Smets | An estimated dynamic stochastic general equilibrium model of the euro area[END_REF], policy maker adopts the following interest rate rule:

rn it = ir rn it 1 + (1 ir ) it + r i (^ it 1 it ) + r y i ỹit + r i it ( 15
)
+r i ^ it + r y i ỹit + " r it
with it being a persistent shock to the CPI in ‡ation objective which is assumed to follow a …rst-order autoregressive process it = i it 1 + i t where i t is an in ‡ation targeting shock, whereas " r it is a monetary policy shock.

To see how the introduction of the incomplete exchange rate pass-through assumption a¤ects the conduct of monetary policy, namely by the introduction of the e¤ective l.o.p gap in the interest rate rule, let us …rst discuss to what extent di¤erent speci…cations adopted in this model break the standard canonical representation obtained with the small open economy framework in [START_REF] Gali | Monetary policy and exchange rate volatility in a small open economy[END_REF]. First, some speci…cations adopted in the model in order to generate nominal and real rigidity break the standard canonical representation. Namely, the external habit formation leads to the presence of the real interest rate di¤erential it and the lagged value of the output ŷit 1 in the dynamic IS equation and the Phillips curve. The monopolistic wage setting behavior of the households in turn leads the labor market friction ^ w it to enter the Phillips curve. Second, the multi-country approach allows the model to depart from the small open economy assumption. Namely, there exists a number of trading partners that are no longer taken as exogenous for domestic economy. This speci…cation leads to the matrix representation of the dynamic IS equation where outputs are linked by the means of the import matrix share. However, it does not break the isomorphism between the closed and open economy model. The important implication of this assumption, namely for the exchange rate pass-through analysis, is that trading partners cost-push shocks weighted by their respective import market share enter endogenously the import price in ‡ation dynamics and hence that of CPI in ‡ation. Third, the assumption of incomplete exchange rate passthrough breaks the traditional aggregate demand and supply channel of monetary policy through the existence of independent channel by the means of the l.o.p gap variables in the dynamic IS and import in ‡ation (and hence the CPI in ‡ation) equation.

Suppose now that the monetary authority reacts to a contraction of the output gap by lowering interest rate. Through the dynamic IS equation, the real interest rate therefore decreases. It leads to a rise in output and hence, the output gap. However, lowering interest rate yields a nominal depreciation that in turn raises the l.o.p gap. In turn, if the monetary authority raises interest rate to stabilize a rise in the l.o.p gap, this leads to a contraction of the activity. Therefore, and as argued by [START_REF] Monacelli | Monetary policy in a low pass-through environment[END_REF], apart from the traditional trade o¤ faced by the monetary authority between stabilizing in ‡ation and output gap, this example illustrates how the assumption of the incomplete pass-through leads to another trade o¤ between stabilizing output and l.o.p gap. The introduction of the l.o.p gap variable in the interest rate rule aims at capturing this new trade-o¤ and treating endogenously the deviation from the law of one price.

Estimation

We estimate structural parameters of the model using Bayesian methodology which formal description can be found in [START_REF] An | Bayesian analysis of dsge models[END_REF]. It is worth noting that it is common in the literature to take the rest of the world as a …ctional exogenous economy in order to reduce the curse of dimensionality. The multi-country assumption therefore does not permit a shrinkage of the data in this study. Therefore, inference of the structural parameters is done by block in order to reduce the curse of dimensionality and to attain convergence more rapidly. Structural parameters are classi…ed into three blocks that are behavioral, nominal friction from wage and price setting problems, and monetary policy parameters. 9For each country i, the log-linearized version of the equilibrium conditions contain …ve observable endogenous variables such as output ŷit , nominal interest rate rn it , domestic price in ‡ation ^ d it , import price in ‡ation ^ f it and real wage c wr it , …ve non-observable endogenous variables such as the law of one price gap it , real interest rate di¤erential it , wage mark-up ^ w it and natural output ŷn it , and …nally two exogenous processes of bilateral nominal exchange rate êit (observable) and technology âit (non-observable). The stochastic behavior of the model is mainly driven by seven exogenous shocks such as exchange rate shock " e i t , idiosyncratic technology shock " a i t , monetary policy shock " r it , in ‡ation targeting shock i t , wage mark-up shock " w it , domestic price mark-up shock " p it and preference shock " g it . 10 4

.1 Data

The multi-country model that is estimated includes the Euro-area and its …ve main trade partners which are the United Kingdom, China, the United States, Japan and Switzerland.

In order to estimate the model, we use quarterly data for the period 1998Q2 to 2011Q2. Note that we do not use data constructed by Fagan et al. ( 2001) as is commonly the case when estimating DSGE model for Euro-area given that the last update of the Area-Wide Model database includes dataset that ends in 2009Q4. Data used in this study comes mainly from national sources and the International Monetary Fund database. 11 Estimation of the structural parameters are conducted using variables expressed in deviation from their respective steady states. In turn, steady states of observable variables are estimated as the long-horizon forecasts from a reduced-form cointegrating Global VAR. The use of GVAR to estimate the steady state is introduced by Dees It has the advantage of taking into account direct and indirect link between domestic and foreign variables, and possible cointegration within domestic variables and between domestic and foreign variables. This permits to give economic interpretation to the estimated steady state in contrast to pure statistical …ltering procedures, quali…ed by [START_REF] Garratt | Permanent vs transitory components and economic fundamentals[END_REF] as a "black box", that are commonly used in the literature. Moreover, the GVAR speci…cation permits to take into account the role of oil price that is considered as common observable global factor and the importance of Unites States …nancial variables in the world economy. 12 As in [START_REF] Smets | An estimated dynamic stochastic general equilibrium model of the euro area[END_REF] and Christo¤el et al. ( 2008), we proceed to some normalizations of structural shocks in the log-linearized version of the equilibrium solutions of the model before estimation. These are done in order to ease choice of the prior and to improve convergence of estimated structural parameters. First, we normalize coe¢ cients a¤ected to domestic price mark-up shock " p it in the Phillips curve and wage mark-up shock " w it in the real wage law of motion to be equal to unity. Second, we assume that foreign mark-up shock " f p it follows a …rst order autoregressive process and normalize its coe¢ cient to be equal to unity.

Priors

Some parameters are kept …xed throughout the estimation procedure. The discount factor is calibrated to be equal to 0:99 which implies a quarterly steady-state real interest rate of 1%. The optimal allocation of expenditures between goods produced in di¤erent countries [START_REF] Gagnon | Monetary policy and exchange rate pass-through[END_REF] gives the formal de…nition of the imported goods share. That is,

ij = P ijt C ijt P it C it and ii = P iit C iit P it C it P ijt C ijt
represents the nominal value of goods imported from country j that is proxied by the corresponding value of importation from country j. P iit C iit represents consumption of domestic produced goods and is proxied by the value of domestic production which is not exported. Finally, P it C it represents nominal consumption of country i and is assumed to be equal to P ijt C ijt + P iit C iit . The import matrix share is constructed by averaging the right hand side of the above expression during the period 1999-2011. This speci…cation permits to respect the restriction imposed for ij . In turn, the expression of the export share of goods C in the appendix reports prior distribution, mean and standard deviation or degree of freedom of the structural parameters. We assume that domestic price and wage nominal rigidity last in average one year. This is line with the …ndings of Alvarez et al. (2006). They provide a comprehensive revue of the literature concerning micro-data analysis of price stickiness for the Euro-area. More exactly, they report that Euro-area CPI and PPI price duration last respectively 13 and 10.8 months. In turn, we assume that price duration last 7 months for import prices. The fact that import price duration is assumed to be shorter than domestic price stems from the fact that exchange rate volatility contributes to frequent changes in import prices. Concerning the l.o.p gap coe¢ cient in the interest rate rule, we set a low value of 0:05 and a relatively narrow range of parameters with a standard deviation that amounts to 0:05 for the prior given the trade-o¤ that the monetary has to face between stabilizing output gap and l.o.p gap. Finally, prior mean of the volatility of exchange rate corresponds to the mean of the di¤erent currencies standard error obtained with an ordinary least square estimation of a …rst order autoregressive process.

{ ij in (

Results

Posterior estimates

Results obtained with the Metropolis-Hasting algorithm are presented in Table C in the appendix. We report posterior mode of the structural parameters for two di¤erent speci…cations of the model as regards to the methodology used for computing steady state of variables. The …rst speci…cation, labeled "model1" in the Table, uses deviation of variables from GVAR estimates of the steady state in order to estimate structural parameters of the model. In turn, the second speci…ation, labeled "model2" in the Table, uses variables detrended by a linear trend and demeaned. This is the common …ltering procedure used in the DSGE literature and, thus, will serve us as a benchmark throughout the analysis.

First, the estimate of the degree of habit formation is around 0:5. These estimated values are reasonable. For instance, Smet and Wouters (2003) found an estimate equal to 0:55. Walque et al. ( 2005) in turn have found a higher estimate equal to 0:74 for the Euro-area and 0:72 for the United States. The estimated value of the labor Frish elasticity is around 2 apart from that of the Euro-area and Japan for which it amounts respectively to 0:65 and 0:5.

Second, posterior mode of the persistence parameters amounts to around 0:9. However, estimates of the autoregressive coe¢ cient in the in ‡ation objective are more heterogeneous. Namely, it is lower for Japan and Switzerland, and amounts respectively to 0:70 and 0:63, but remains higher for the rest of countries.

Third, concerning nominal price rigidity, posterior estimates of Calvo parameters for domestic prices amount to around 0:8. These correspond to an average price duration of 1:5 year. Domestic in ‡ation indexation parameters are estimated to be around 0:4. These values are slightly higher compared to the estimated value of 0: [START_REF] Erceg | Sigma: A new open economy model for policy analysis[END_REF] It is worth noting however that nominal price rigidity is higher for the Euro area. Estimates of Calvo parameters for domestic and import prices amount respectively to 0:94 and 0:85. That is, higher persistence in the time varying in ‡ation objective does not permit to have a lower price stickiness. One plausible explanation is the following. To match real exchange rate volatility and persistence, Chari et al. (2002) argue that the model needs price stickiness and in ‡ation inertia by the means of Calvo and in ‡ation indexation parameters. However, and as is argued by Justiniano and Preston (2004), the introduction of the habit formation has a drastic implication to parameter estimates. Increase in the degree of habit formation permits to match real exchange rate volatility by an increase of the output volatility. Thus, and however, to match real exchange rate persistence, price stickiness and in ‡ation persistence have to increase. This increase may even overstate for that elasticities of substitution between domestic and foreign goods given by import share matrix coe¢ cients are …xed in our study. That is, high estimate of in ‡ation objective, in ‡ation indexation and price stickiness parameters is necessary in order for the model to generate persistence in real exchange rate in presence of external habit.

Fourth, concerning monetary policy parameters, interest rate response to l.o.p gap is low except for the United Kingdom. It indicates that monetary authority reacts less to the deviation from the law of one price. There are two main reasons behind this …nding. First, and as is explained above, stabilization of the l.o.p gap could be pursued but at the cost of the output gap deviation. That is, there is a trade-o¤ between l.o.p gap and output gap stabilization. Second, nominal rigidity, and as far as exchange rate pass-through is concerned, induces a persistent but low changes in import price in ‡ation following exchange rate changes that may stem from various shocks in the model. This will be further illustrated in the next section on impulse response function.

Impulse response

To start with the impulse response analysis, let us consider standard shocks that generate responses of nominal and real variables which are known to follow a given particular shape. This permits to have a general idea on the dynamic properties of the model. The dynamic e¤ects of interest rate rate shock, interpreted as an unexpected monetary policy shock are given in Figure E.1 in the appendix. Responses of output, real wage and domestic in ‡ation have the standard hump-shaped form as in closed economy model. However, response of real wage is much more persistent compared to those of output and in ‡ation where they reach a peak within one year. Impulse responses following an in ‡ation targeting shock are depicted in Figure E.2. It seems clear that a temporary and unexpected rise in in ‡ation target induce a persistent rise in domestic in ‡ation and real wage. The duration of the e¤ect on output is heterogeneous across countries where it is more persistent for the Euro-area. In Figure E.3, preference shock induces a hump-shaped rise in output and a rather persistent rise in domestic in ‡ation and real wage. Given the assumption of home consumption bias and …xed elasticity of substitution between home and foreign goods given by import share ij , preference shock has a reduced and non-persistent e¤ect on consumption and, hence, on output. Finally, impulse responses to a domestic mark-up shock are given in Figure E.4. It induces a decrease in output and real wage. In turn, domestic in ‡ation rise but the e¤ect is short-lived. These …ndings give a support to a certain isomorphism between the dynamics of closed and open economy model. Namely, concerning monetary policy shock, even if one could not match exactly the response obtained under an identi…ed structural VAR due to the presence of various shocks and new channel for monetary policy, the dynamics are very similar.

If we turn to exchange rate pass-through analysis. Impulse response of the Euro-area import price in ‡ation following an exchange rate shock is given in the following …gure.

It is worth noting that the extent of the response is di¤erent from exchange rate passthrough. The latter is driven by the degree of nominal rigidity but in ‡uence the extent of the import price in ‡ation impulse response. 13 The top panel of the …gure depicts the cumulative responses of the Euro-area aggregate import price changes following a bilateral nominal exchange rate shock, namely a depreciation of Euro which is formally de…ned as a decrease in êit . It is evident that depending on the currency, the extent of the e¤ect is di¤erent. Namely, the highest impact stems from the Pound whereas the lowest from the Swiss Franc. The bottom panel in turn depicts the cumulative response of the Euro-area import price following a shock to Euro currency that induces a depreciation of the e¤ective or multilateral nominal exchange rate. It is clear that the impact is higher than that of the bilateral shock. Therefore, these …ndings indicate that exchange rate pass-through analysis should make the distinction between bilateral and multilateral exchange rate changes. In this simple example, all bilateral nominal exchange rates are assumed to depreciate vis-à-vis the Euro. There are situations in which bilateral exchange rates move in opposite direction. In this case, separating bilateral e¤ects would permits to avoid aggregation downward bias in the extent of pass-through.

Moreover, the response of the import price in ‡ation is hump-shaped and reach a peak roughly after 2 years. This indicates that nominal frictions generate a persistent impact on import price in ‡ation. This is a key issue for the exchange rate channel of monetary policy analysis. As can be seen, import price in ‡ation rise by 0:04% upon impact following an e¤ective Euro depreciation. After 2 years, the cumulative sum of the impact amounts to 0:25%. That is, nominal import price rigidity, and hence incomplete exchange rate passthrough, generates a persistent but reduced impact of exchange rate changes on import price in ‡ation. Therefore, and combined with the weight attributed to import price in ‡ation into consumer price in ‡ation, the monetary authority could pursue a stable in ‡ation target with less action. The latter is reinforced by the presence of trade-o¤ between output gap and law of one price gap. These explain the low estimated value of monetary policy parameters r i that are a¤ected to the l.o.p gap. This …nding is in line with the extensive studies conducted by the In ‡ation Persistence Network (IPN) team in the European Central Bank.

Variance decomposition

Table D in the appendix reports conditional variance decomposition of the Euro-area aggregate import price. It permits to formally assess the contribution of di¤erent shocks to the variability of the variable of interest at di¤erent horizons. We choose to present decomposition of the import price in ‡ation volatility of the Euro-area for eight quarters. The …rst quarter gives the contribution of shocks upon impact and can be considered as a short run decomposition. In the analysis of exchange rate pass-through, de…ning long term as two years is su¢ cient. Apart from foreign mark-up shock, we only report shocks that explain at least one percent of the variance of the import price. If one refer to the traditional exchange rate pass-through equation of [START_REF] Goldberg | Goods prices and exchange rates: What have we learned[END_REF], movements in import price are explained by three major factors that are nominal exchange rate, mark-up and marginal cost. Other variables are also introduced to capture demand conditions, the degree of local market competition and market share. An important strand of the literature, which is the Pricing-to-Market (PTM) or Local Currency Pricing (LCP) literature, argues that incomplete pass-through arises due to adjustment of the mark-up following exchange rate movements in order to maintain market share. Results from variance decomposition presented below highlight some interesting …ndings.

First, if one consider decomposition upon the impact of shocks which is given in the …rst column, exchange rate movements contribute to 70:83% of the import price volatility. This con…rms the fact that import prices are characterized with lower price stickiness than domestic producer prices. Moreover, it is possible with the multi-country speci…cation to identify the role played by each nominal bilateral exchange rate vis-à-vis each trade partner. This leads to a distinction that has to be made between bilateral and multilateral (or e¤ective) exchange rate pass-through. As one can see, a large part of the Euro-area import price volatility explained by exchange rate movements comes from the Pound bilateral exchange rate. Upon the impact, the latter explains 45% of the import price volatility. This con…rms the important role played by the United Kingdom as trade partner for the Euro-area. Concerning the Chinese-Renminbi, the model reports that it explains 20% percent of the import price volatility upon impact. Nevertheless, care must be taken when interpreting this result. The fact that the Chinese-Renminbi explains such a large part of import price volatility comes more from its comovements with the US-Dollar than anything else. In fact, the Chinese authority allowed o¢ cially the use of the Chinese-Renminbi as invoicing currency in China's foreign trade only since 2005. Moreover, and despite this o¢ cial decision, Chinese …rms continue to price their goods generally in US-Dollar leading Chinese authorities to conduct in July 2009 a trial scheme where some enterprises are asked to invoice their trade in the Chinese-Renminbi as argued in Cui et al.( 2009). 14 The US-Dollar explain 3% of the Euro-area import price volatility whereas only 1% and 0:25% are explained respectively by the Japanese-Yen and Swiss-Franc.

Second, foreign mark-up shock plays a minor role in explaining the volatility of the Euroarea import price. Its in ‡uence is less than 1% for all countries considered and conditional on all quarters. Under the assumption that international market is competitive and the law of one price holds at the dock, mark-up shocks have a reduced in ‡uence in international price of exported goods. Combined with price setting behavior of the monopolistic importing …rms, they have a negligible impact on the volatility of import prices. This is an argument in favor of the Producer Currency Pricing (PCP) behavior rather than LCP-PTM at the export side.

Third, upon the impact, in ‡ation objective shock explained 12:47% of the import price volatility. It is common in the empirical literature of pass-through to proxy marginal cost with CPI in ‡ation of the trade partner. If one consider that the monetary authority follows an in ‡ation target, deviation from this objective will impact the CPI in ‡ation dynamics and hence the marginal cost. Therefore, one can interpret without lost of generality in ‡ation objective shock as marginal cost shock. As is shown in the Table, the United Kingdom in ‡ation objective shock explains 6:28% of the import price of the Euro-area volatility upon the impact. Under the context of regional interdependency, it is not surprising to have marginal cost dynamics in ‡uenced by that of the United Kingdom.

Fourth, although to a lesser extent, preference shock in ‡uences import price variability. On the one hand, the fact that domestic (Euro-area) shock has a negligible impact comes from the existence of home bias consumption. On the other hand, foreign preference shocks will have an in ‡uence to foreign in ‡ation dynamics. This will in ‡uence foreign marginal cost and, hence, import price.

Conclusion

This study estimates a multi-country open economy DSGE model under the assumption of incomplete exchange rate pass-through. It builds on the seminal paper of [START_REF] Monacelli | Monetary policy in a low pass-through environment[END_REF] but relaxes the small open economy framework in order to capture some important features and dynamics of an open economy model where countries are endogenously related and where deviation from the law of one price generated by nominal price rigidity is allowed. Moreover, within a context of a multi-country analysis, one important issue that arises is the existence of a long term relationship between variables within the economy or between domestic and foreign variables. In this study, we use Global (GVAR) to model long-run international linkage between countries and calculate the steady state by the means of the long-horizon forecast.

First, and as regards to exchange rate pass-through analysis, we found that exchange rate pass-through still remains when …rms adjust prices but its impact on the aggregate import prices is limited and delayed by the presence of nominal price rigidity. This model presents results in this direction, namely that exchange rate volatility accounts for a large part, more precisely 70%, of the Euro-area import price in ‡ation variability whereas foreign mark-up shocks have a reduced impact. Second, it is more convenient to make distinction between exchange rate pass-through that stems from bilateral and multilateral exchange rate movements. This permits to take into account explicitly the role of currency invoicing and namely the role of a vehicle currency in the extent of pass-through. Third, the presence of incomplete exchange rate pass-through has a stark implication in the conduct of the monetary policy. It induces a new channel of the monetary policy and at the same time a trade-o¤ between output gap and law of one price gap stabilization. Moreover, exchange rate changes have a reduced and persistent impact on import price in ‡ation due to the presence of nominal rigidity. Combined with the home consumption bias assumption, less action is needed for the monetary authority to achieve a certain in ‡ation target as far as exchange rate pass-through is concerned.

It is worth noting that the primary focus of this study is to obtain tractability of the complex mechanism underlying the model under the multi-country assumption. Nevertheless, this is achieved to a certain extent at the expense of some important features of the model such as the capital accumulation dynamics. This puts a pressure on the nominal rigidity parameters in order for the model to generate enough volatility to match the data. Though more complicated, it is interesting for future research to extend the model with more robust dynamics by introducing investment and various shocks such as the risk premium, equity premium and investment shocks. Moreover, using long horizon forecast of the GVAR to estimate steady state of variables permits to circumvent the "black box" property but, however, presents one major drawback. Indeed, steady state estimate values will depend on the GVAR speci…cation. Therefore, it is also interesting for future research to take into account structural breaks, to mention only the 2007 global crisis, in the estimation of the steady states.

A Model details A.1 Households

Households h within a country i share the same preference technology. They maximize a string of discounted future value of utilities given by:

E t 1 X k=0 k U i;t+k (h)
with period t felicity function de…ned as:

U i;t (h) = ln(C it H it ) L 1+ it (h) 1 + (16)
where C it and L it (h) represent respectively the consumption index and the individual labor supply (number of hours worked) of a representative home agent. H it is an external habit taken as exogenous by the household and de…ned as H it = hC it 1 . Positive parameter represents the inverse of the intertemporal elasticity of labor supply.

Given that all goods are tradable, consumption bundles are composed of goods produced by home …rms but also of goods imported from the rest of the world. We assume that consumption index is represented by the following Cobb-douglas preference:

C it = N Q j=1 C ij ijt N Q j=1 ij ij (17)
where C ijt represents country i individual households'consumption of goods imported from country j.

Following [START_REF] Smets | An estimated dynamic stochastic general equilibrium model of the euro area[END_REF], consumption of goods produced in country j is assumed to be a CES function de…ned as:

C ijt = Z 1 0 • C i;t (f j ) 1 1+ p jt df j 1+ p jt (18)
where • C i;t (f j ) is the consumption of individual good produced by an individual foreign …rm f j 2 [0; 1] with j 2 f1; : : : ; N g. The elasticity of substitution between varieties of products p jt is assumed to be random and is given by: ln(1

+ p jt ) = ln(1 + p j ) + " p jt
It will turn out that (1 + p jt ) = p0 j is the time-varying markup of prices over marginal costs at the intermediate goods level. Thus, " p jt is interpreted as a good markup or a cost-push shock with E

n " p jt o = 0 and E n (" p jt ) 2 o = 2 " p j .

A.1.1 Prices and demand

With consumption index preference [START_REF] De Bandt | Exchange rate pass-through to import prices in the euro-area: a multicurrency investigation[END_REF], the overall home-currency consumption-based 15 price index (CPI) is given by:

P it = N Y j=1 P ij ijt ( 19
)
whereas the aggregate import price index (IPI) is de…ned as:

P f it = N Y j=1 j6 =i P ij ijt ( 20
)
where P ijt denotes the home-currency (country i) price index of good imported from country j, which using the CES function ( 18) is de…ned as:

P ijt = Z 1 0 P ijt (f j ) 1= p jt df j p jt (21)
with P ijt (f j ) being the consumer price of good f j in country i.

The allocation of country i representative individual's demand across di¤erentiated goods f j produced within a country j is given by:

• C i;t (f j ) = P ijt (f j ) P ijt (1+ p jt )= p jt C ijt ( 22
)
and …nally, the optimal allocation of expenditures between goods produced in di¤erent countries j 2 f1; : : : ; N g is given by:

C ijt = ij P ijt P it 1 C it (23)
Using this last equation and the property of import share coe¢ cients P N j=1 ij = 1, one can derive total expenditure. That is,

P it C it = N X j=1 P ijt C ijt A.1.

Household' s optimization problem

Country i representative household has a budget constraint of the form:

P it C it + E t (Q it+1 B it+1 ) B it + W it (h)L it (h) + T it ( 24
)
where B it+1 is the nominal payo¤ in period t + 1 of portfolio held at the end of period t whereas Q it+1 is the asset market price of nominal bonds. W it (h) is the nominal wage and T it denotes a lump-sum taxe or transfer. Formally,

Q it+1 = 1 R n it = 1 1 + i it
where i it denotes country i's one period nominal interest rate that prevails on date t.

Representative household chooses a string of variables fC it+k ; L it+k (h); B it+k+1 g1 k=0 that maximize a string of discounted future value of its utilities given by:

E t 1 X k=0 k " ln(C it+k H it+k ) L 1+ it+k (h) 1 +
# subject to budget constraint [START_REF] Gali | Monetary policy and exchange rate volatility in a small open economy[END_REF]. First order conditions to this optimization problem with respect to C it+k and B it+k+1 yield the standard optimal stochastic Euler equation. In log-linear terms, it is given by:

ĉit = h 1 + h ĉit 1 + 1 1 + h E t fĉ it+1 g 1 h 1 + h (r n it E t f^ it+1 g) (25)

A.1.3 Wage setting problem

Households supply monopolistically a distinctive variety of labor to home …rms and set nominal wages in a staggered contracts fashion à la [START_REF] Calvo | Staggered prices in a utility-maximizing framework[END_REF]. That is, individual household h resets its nominal wage only after receiving a random price-change signal with constant probability 1 iW , so that W it (h) = W o it (h). However, whenever household is not allowed to adjust its contracts, wage is indexed to last period CPI in ‡ation 16 rate according to the following indexation rule:

W it (h) = P it 1 P it 2 iw W it 1 (h) = ( it 1 ) iw W t 1 (h) (26)
If iw = 0 there is no indexation, whereas if iw = 1 there is a perfect indexation of wage to past in ‡ation. Therefore, wages remain …xed on average 1= (1 iW ) periods. When setting nominal wage, household chooses W it (h) to maximize:

E t 1 X k=0 ( iW ) k " ln(C it+k H it+k ) L 1+ it+k (h) 1 +
# subject to budget constraint [START_REF] Gali | Monetary policy and exchange rate volatility in a small open economy[END_REF], labor demand [START_REF] Álvarez | Sticky prices in the euro area: A summary of new micro-evidence[END_REF] and the equality W it (h) = W o it (h)X w itk , where:

X w itk = iw it : iw it+1 : : iw it+k 1 for k 1 1 for k = 0
k iW is the probability that the wage W it (h) set at time t still holds k periods ahead. The …rst order condition to this optimization problem is standard. The log-linear version around steady state yields:

ŵo it (h) iw pit 1 (27) = (1 iW )E t ( 1 X k=0 ( iW ) k ( ŵit+k iw pit+k 1 ) 1 k w ^ w it+k + 1 k w " w it )
where k w = 1 + (1 + w i ) = w i and ^ w it denotes the labor market friction de…ned as the di¤erence between the real wage and the marginal rate of substitution. That is,

^ w it = ŵit pit lit 1 1 h (ĉ it hĉ it 1 ) (28)

A.1.4 Aggregate home wage dynamics

Given that, on the one hand, all individual households that adjust in period t choose the same wage W o it , and on the other hand, the average wage of households that do not adjust is simply the last period wage level W it 1 indexed with past CPI in ‡ation, we can rewrite the wage index (37) in terms of wage in ‡ation de…ned as w it = W it =W it 1 and in log-linear deviation around steady state. That is,

^ w it = iW iw ^ it 1 + (1 iW ) ( ŵo it ŵit 1 ) (29)
Therefore, using wage in ‡ation dynamics [START_REF] Marazzi | Declining exchange rate pass-through to u.s. import prices: The potential role of global factors[END_REF] and reset wage equation ( 27), we have an expression determining domestic country i's real wage law of motion as a function of labor market friction variable. That is,

c wr it = 1 1 + c wr it 1 + 1 + E t f c wr it+1 g + iw 1 + ^ it 1 ( 30
)
1 + iw 1 + ^ it + 1 + E t f^ it+1 g w i ^ w it + w i " w it where w i = (1 iW )(1 iW ) (1+ ) iW kw
. Note that by de…nition, c mc w t = ^ w it , where c mc w t is the (logdeviation from steady state) real marginal cost of supplying labor.

A.2 Domestic producers

In the domestic market i, there exists a continuum of monopolistic competitive …rms, indexed by f i 2 [0; 1]. Each …rm produces a single good using a variety of labor as the only input factor. Let Y it (f i ) denotes the output of di¤erentiated good of …rm f i and L it (h; f i ) its demand for labor input h. To obtain symmetry in the model, assume that the production function is given by:

Y it (f i ) = A it L it (f i ) (31)
with the composite labor L it (f i ) de…ned as: [START_REF] Obstfeld | New directions for stochastic open economy models[END_REF] where the degree of substitutability among di¤erent types of labor w it is assumed to be random and is given by: ln(1

L it (f i ) = " (P i ) w it 1+ w it Z P i 0 L it (h; f i ) 1 1+ w it dh # 1+ w it
+ w it ) = ln(1 + w i ) + " w it
In the ‡exible-wage economy, (1 + w it ) = w0 it is the real wage mark-up over the usual ratio of marginal disutility of labor to the marginal utility of consumption. Thus, " w it is interpreted as a wage markup shock with E f" w it g = 0 and E

n (" w it ) 2 o = 2 " w it .
We assume that the productivity follows a …rst order autoregressive process. That is,

âit = a i âit 1 + " a i t (33)
where a i < < 1 8i 2 f1; : : : ; N g and " a i t is the idiosyncratic productivity shock with E(" a i t ) = 0 and E(" 2 a i t ) = 2 "a i .

A.2.1 Cost minimization problem

Given the wage index W it , each …rm f i chooses L it (f i ) to minimize its production cost:

W it P d i;t L it (f i ) subject to A it L it (f i ) Y i (f i ) 0
where P d i;t is the aggregate domestic producer price index de…ned as: 17

P d i;t = Z 1 0 P d it (f i ) 1 p it df i p it (34)
The log-linear version of the …rst order condition of this minimization problem yields the following standard equation of real marginal cost expressed in terms of domestic producer price index P d i;t :

c mc it = ŵit pd i;t âit (35)
Note that real marginal cost will be common across domestic …rms because the right hand side of the equation does not depend on f i .

A.2.2 Labor demand

Given the optimal aggregate level of labor L(f i ) chosen in the cost minimization problem above, …rm f i chooses the best combination of the di¤erent types of labor when setting its labor demand,

L it (h) = 1 P i W it (h) W it 1+ w it w it L it (36)
Substituting into the expression of L it (f i ) in ( 32) yields the expression of the aggregate wage index:

W it = 1 P i Z P i 0 W it (h) 1 w it dh w it (37)

A.2.3 Price setting

Domestic …rm f i faces a downward slopping demand from domestic households and foreign households via retailers that import goods where the law of one price holds "at the dock ".

That is,

C it (f i ) = N X j=1 C jit (f i )P j (38)
where C jit (f i ) is the quantity of goods imported by country j's retailers that is produced by country i's …rm f i , and is de…ned as:

C jit (f i ) = P d it (f i ) P d it 1+ p it p it C jit (39)
where P d it (f i ) is the producer price of …rm f i , P d it is the aggregate producer price index of country i, and C jit is the quantity of goods imported by country j's retailers from country i.

Like wage setting optimization, it is assumed that prices are sticky and …rms set prices in a staggered fashion à la [START_REF] Calvo | Staggered prices in a utility-maximizing framework[END_REF]. A fraction 1 iH of randomly selected …rms set new prices P d;o it (f i ) each period, whereas a fraction iH of …rms keep their prices unchanged and simply adopt the following indexation rule:

P d it (f i ) = P d it 1 P d it 2 ! id P d it 1 (f i ) = d it 1 id P d it 1 (f i )
Note that, the probability of re-optimizing in any given period is independent of the time elapsed since …rms last reset price. Therefore, prices are sticky in average duration of (1 iH ) 1 and iH is interpreted as the index of price stickiness. Formally, individual …rm f i solves the following problem:

max P d;o it (f i ) 1 X k=0 k iH E t n Q it;t+k h P d;o it (f i )X d itk C it+k (f i ) it+k (C it+k (f i ))
io subject to the sequence of demand constraints (38). Q it;t+k is the stochastic discount factor for nominal payo¤s, it+k ( ) is the cost function and X d itk is de…ned as:

X d itk = d it id : d it+1 id : : d it+k 1 id for k 1 1 for k = 0
Log-linear deviation from steady state of the …rst order condition of this optimization problem is typically standard. It is straightforward to derive an expression determining domestic in ‡ation as a function of the deviation of marginal cost from its steady state value. That is,

^ d it = id 1 + id ^ d it 1 + 1 + id E t n ^ d it+1 o + d i ( c mc it + " p it ) (40)
where

d i = (1 iH )(1 iH ) (1+ id ) iH
and c mc it denotes log-deviation of real marginal cost, de ‡ated by producer price index P d i;t , from its steady-state value.

A.3 Importing …rms

There exist a continuum of local retailer …rms in country i indexed by l i 2 [0; 1] that import goods in international trade market where the law of one price holds "at the dock ". Nevertheless, retailers behave as a monopolistic …rm when setting their home currency price of imported goods. Hence, deviations from the law of one price occur due to the optimal markup problem that the importing …rms have to solve when setting prices. We assume that prices are sluggish and are set in a staggered fashion à la [START_REF] Calvo | Staggered prices in a utility-maximizing framework[END_REF]. Denote iF the fraction of local retailers that keep their local currency prices unchanged in period t even if nominal exchange rate ‡uctuates and simply follow the following indexation rule:

P ijt (f j ) = P ijt 1 P ijt 2 if P ijt 1 (f j ) = jf it 1 if P ijt 1 (f j )
where jf it 1 is the import in ‡ation that comes from country j. This behavior generates a deviation from the law of one price in the short run until price will be adjusted. In average, prices remain …xed during (1 iF ) 1 periods. 18 This is in line with a large strand of recent empirical studies concerning exchange rate pass-through. 19 Essentially, it is shown that passthrough estimates are incomplete in the short run but become, gradually, complete in the long run.

Formally, country i's local retailer l i (f j ) that import a single good f j from country j faces a downward slopping demand [START_REF] Erceg | Optimal monetary policy with staggered wage and price contracts[END_REF] from domestic households. Therefore, individual import …rm solves the following maximization problem:

max P o ijt (f j ) 1 X k=0 k iF E t n Q it;t+k h P o ijt (f j )X jf itk E ijt+k P d jt+k (f j ) i C ijt+k (l i (f j ))
o subject to the sequence of demand constraints given by:

C ijt+k (l i (f j )) = • C i;t+k (f j ) = P o ijt (f j )X jf itk P ijt+k ! 1+ p jt p jt C ijt+k where • C i;t (f j ) = C ijt (l i (f j ))
is the country i's individual consumption of good produced by an individual …rm f j and imported by local retailer l i (f j ) from country j and

X jf itk = ( jf it if : jf it+1 if : : jf it+k 1 if for k 1 1 for k = 0
Log-linear deviation from steady state of the …rst order condition of this optimization problem yields:

po ijt if pijt 1 (41) = (1 iF ) E t ( 1 X k=0 ( iF ) k h ij;t+k + pijt+k if pijt+k 1 + " p jt+k i )
where ijt , de…ned in [START_REF] Bailliu | Exchange rate pass-through and the in ‡ation environment in industrialized countries: An empirical investigation[END_REF], is the bilateral l.o.p gap between country i and j. By de…nition, it is the di¤erence between the price paid by importing …rms in the world market and the home currency price of the imported goods. Therefore, it stands clear from the last equation that the l.o.p gap acts as marginal cost. Moreover, country i's aggregate (log-deviation) import price index evolves according to:

pijt = iF if ^ jf it 1 + iF pijt 1 + (1 iF )p o ijt (42) 1 8
As pointed out by Obsteld and Rogo¤ (2000) [START_REF] Obstfeld | New directions for stochastic open economy models[END_REF], import price stickiness last in average 90 days. 1 9 Among others, one can cite Campa and Goldberg (2002) [START_REF] Campa | Exchange rate pass-through into import prices: A macro or micro phenomenon?[END_REF], [START_REF] Anderton | Extra-euro area manufacturing import prices and exchange rate passthrough[END_REF] and [START_REF] Bailliu | Exchange rate pass-through and the in ‡ation environment in industrialized countries: An empirical investigation[END_REF].

Using optimal condition (41), calvo import pricing law of motion (42), and the de…nition of import in ‡ation ^ jf it = pijt pijt 1 , we have an expression determining import in ‡ation as a function of bilateral l.o.p gap:

^ jf it = if 1 + if ^ jf it 1 + 1 + if E t n ^ jf it+1 o + f i ijt + " p jt where f i = (1 iF )(1 iF ) (1+ if ) iF .

A.4 Market clearing conditions A.4.1 Goods market clearing

Import goods market clear when the quantity imported by retailer …rms match exactly that consumed by households. That is, C ijt (f j ) = • C i;t (f j ) and C ijt = C ijt 8i; j 2 f1; : : : ; N g. Country i's goods market clear when aggregate demand of good f i by domestic and foreign retailers equals its production by domestic …rm. That is,

Y it (f i ) = N X j=1 C jit (f i )P j
Aggregating through domestic …rm f i by using a Dixit-Stiglitz aggregator analogous to that of consumption index [START_REF] Dees | Identi…cation of new keynesian phillips curves from a global perspective[END_REF], using the expression of individual demand (39) and domestic producer price index [START_REF] Smets | Openness, imperfect exchange rate pass-through and monetary policy[END_REF], one can derive the following aggregate good market clearing condition:

Y it = N X j=1 P j C jit (43)
Denote the share of country i's domestic product that is exported to country j as:

{ ijt = P j C jit Y it (44)
where P N j=1 { ijt = 1. Therefore, log-linearization of the aggregate good market clearing condition around steady state yields:

ŷit = N X j=1 { ij ĉjit
Using the optimal allocation of expenditures between goods produced in di¤erent countries j (23), the risk sharing condition [START_REF] Campa | Exchange rate pass-through into import prices: A macro or micro phenomenon?[END_REF], and the de…nition of e¤ective terms of trade (3), domestic output and consumption are linked according to:

ŷit hŷ it 1 = (ĉ it hĉ it 1 ) + (1 h)ŝ it + h it (45)
where it denotes country i's e¤ective real interest rate di¤erential de…ned as:

it = N X j=1 { ij h rn it 1 ^ d it rn jt 1 ^ jt i (46)
As can be seen here, the existence of the habit formation increases the volatility of the output via the real interest rate di¤erential. This permits the model to generate volatility in the real exchange rate which is an important characteristics of the data.

A.4.2 Labor market clearing

Labor market clears when the aggregate …rms'labor demand equals the aggregate households' labor supply:

Z 1 0 L it (f i )df i = Z P i 0 L it (h)dh
That is,

L it = L it (h)P i (47)

A.4.3 Real marginal cost

Up to a …rst order approximation, the relationship between aggregate output and employment is obtained using the labor market clearing condition, the production technology [START_REF] Obstfeld | Foundations of International Macroeconomics[END_REF] and the Dixit-Stiglitz aggregator index for domestic output, analogous to the one for consumption index [START_REF] Dees | Identi…cation of new keynesian phillips curves from a global perspective[END_REF]. That is, ŷit = âit + lit (48)

Substituting the labor market friction [START_REF] Gust | Trade integration, competition, and the decline in exchange-rate pass-through[END_REF] and the e¤ective terms of trade (3) into the expression of the real marginal cost [START_REF] Smets | An estimated dynamic stochastic general equilibrium model of the euro area[END_REF], and using the aggregate domestic output (48) and the good market clearing condition (45) yield:

c mc it = 1 1 h [(1 + (1 h)) ŷit hŷ it 1 h it ] (49)
+^ w it (1 + ) âit
Moreover, it is worth noting that under ‡exible price natural equilibrium, the real marginal cost is constant, there is no friction in the labor market and the exchange rate pass-through is complete. Therefore, the natural level of output evolves according to:

ŷn it = âit + h 1 + (1 h) ŷn it 1 âit (50)

A.4.4 Output and consumption

Using the link between CPI and PPI index [START_REF] Bouakez | Has exchange rate pass-through really declined? evidence from canada[END_REF] and good market clearing condition (45), one obtains in matrix notation an aggregate equation for the output. That is,

ŷt hŷ t 1 = ĉt hĉ t 1 + (1 h) (I N M) êt pd t (1 h) t + h t (51)
where ŷt = (ŷ 1t ; : : : ; ŷNt ) 0 and ĉt = (ĉ 1t ; : : : ; ĉNt ) 0 . Moreover, using the international risk sharing condition [START_REF] Campa | Exchange rate pass-through into import prices: A macro or micro phenomenon?[END_REF], relative consumption and e¤ective real exchange rate are linked according to:

(I N M) (ĉ t hĉ t 1 ) = (1 h) q t
which combined with the aggregate output equation above and using the expression of the effective real exchange rate (9) yield a simple relationship between relative output and domestic prices. That is,

(I N M) (ŷ t hŷ t 1 ) = (1 h) (I N M) êt pd t + h (I N M) t (52)
Finally, substituting the last equation into (51) yields a link between consumption and output given by:

ĉt hĉ t 1 = M (ŷ t hŷ t 1 ) + (1 h) t hM t (53)

B GVAR estimates of the steady states

Estimating the steady state using Global VAR in this paper steems from two important limits in the empirical and theoretical macro-modelling literature. One the one hand, macroeconometric models such as the Global VAR su¤er from the problem of identi…cation procedure of the stuctural shocks. On the other hand, macro-economic models, namely the estimated DSGE model, su¤er from the lack of economic interpretation of the statistical …ltering procedure, quali…ed by [START_REF] Garratt | Permanent vs transitory components and economic fundamentals[END_REF] Estimating structural parameters of the multi-country DSGE model, using variables expressed in deviation from their respective GVAR estimates of the steady states, permits to take into account the direct and indirect link between domestic and foreign variables, but also possible cointegration within domestic variables and between domestic and foreign variables. Moreover, structural analysis using the estimated multi-country DSGE model permits to circumvent the identi…cation problem procedure.

Formally, deviation of a given variable from its steady state, xt = x t x t , could be acheived using GVAR model by decomposing variables into their permanent x P t = x t and transitory component x T t = xt . In the case of stationary or trend-stationary variables, the steady state value x t is respectively approximated by a constant term and a linear trend. However, there are cases where variables under consideration are likely to contain stochastic trends. Therefore, a given variable can be decomposed into transitory and permanent components as x t = x T t + x P t where x P t can be further decomposed into deterministic and stochastic components, that is, x P t = x P d;t + x P s;t . The permanent deterministic component x P d;t is de…ned as:

x P d;t = c + t
where c is a constant and t a linear trend. The permanent stochastic component x P s;t is de…ned as long horizon forecast of x t . That is,

x P s;t = lim h !1 E t x t+h x P d;t+h = lim h !1 E t [x t+h c (t + h)]
where E t [ ] is the expectational operator conditional on all information available at the period t and lim h !1 E t x T t+h = 0. As shown in [START_REF] Dees | Identi…cation of new keynesian phillips curves from a global perspective[END_REF], the global error correction representation of the GVAR is given by the following equation:

G x t = a 0 [x t 1 (t 1)] + p 1 X i=1 i x t i + u t (54)
The (kxk) link matrix G is the contemporaneous coe¢ cients that contain trade weights ! ij that verify P N j=1 ! ij = 1 and ! ii = 0. Within the context of international interdependencies and variables co-movements, the role of observed and unobserved common factors is crucial. In this study, the oil price is chosen as observed common factor and is considered as an endogeneous variable for the United States. For the rest of countries, it is considered as "long run forcing". 21 Moreover, r n; it is not included as part of the United States speci…c foreign variables. Given the major role of the United States …nancial variables, r n; it is unlikely to be long run forcing. The global error correction representation of the GVAR in (54) can be re-arranged to become a VAR(p) speci…cation given by: 22

x t = b 0 + b 1 t + p X i=1 i x t i + t
from which the permanent stochastic component x P s;t is derived as:

x P s;t = C(1) t X j=1 j (55)
where

C i = C i 1 1 + C i 2 2 + + C i P P , C 0 = I k , C 1 = (I k 1 ), C i = 0 for i < 0,

  et al. (2009) and used by Dees et al. (2010) to estimate structural parameters of multi-country new Keynesian model.

  44) is approximated by the share of nominal export from a given trade partner relative to the nominal value of domestic production. Priors of structural parameters are set based on earlier studies that estimate DSGE model using Bayesian methods such as Smet and Wouters (2003), Adolfson et al. (2007), Walque et al. (2005) and Christo¤el et al. (2008). Each parameter is given the same prior for all countries. Table

  found by Adolson et al. (2005) and Walque et al. (2005), but are in line with the estimated value of 0:42 found by Smet and Wouters (2003) and Christo¤el et al. (2008). In turn, estimates of Calvo parameters for import prices are equal to around 0:7 which corresponds to an average duration of 3 quarters.

  as a "black box", that is used to compute deviations of variables from steady states. The use of GVAR to estimate the steady state is introduced by Dees et al.(2009) and used byDees et al. (2010) to estimate structural parameters of multi-country new Keynesian model. We follow this approach and use the longhorizon forecasts from a reduced-form cointegrating Global VAR to measure steady state of variables.

  20 It re ‡ects inter-country linkages by the means of bilateral trade. x t = (x 0 1t ; x 0 2t ; :::; x 0 it ) 0 is a global vector that contains individual country-speci…c vector of variables x it . In this study, x it = (y it ; r n it ; d it ; f it ; e it ; wr it ; p oil t ) 0 where y it is the logarithm of real output, r n it the nominal interest rate, d it the domestic price in ‡ation, f it the import price in ‡ation, e it the logarithm of the bilateral nominal exchange rate between country i and numeraire country, wr it the logarithm of the real wage and p oil t the logarithm of the oil price. It is worth noting that the link matrix G is constructed such that: for the United Kingdom, China, Japan and Switzerland, individual country VARX model is composed of vector of domestic endogeneous variables x d it = (y it ; r n it ; d it ; f it ; e it ; wr it ) 0 and foreign weakly exogeneous variables x it = (y it ; r n; it ; wr it ; p oil t ) 0 , EXij) where IMij and EXij represent respectively the value of country i's import and export from country j. for the Euro-area, x d it = (y it ; r n it ; d it ; f it ; wr it ) 0 and x it = (y it ; r n; it ; e it ; wr it ; p oil t ) 0 , for the United States, x d it = (y it ; r n it ; d it ; f it ; e it ; wr it ; p oil t ) 0 and x it = (y it ; e it ; wr it ) 0

	2 0 In this study, the trade weight is given by:
	!ij =	IMij + EXij j=1 (IMij + P N

  and C(1) = P 1 i=0 C i . It is worth noting that the expression (55) is the multivariate version of Beveridge-Nelson stochastic trend component. Therefore, if we denote z t = x t x P s;t , the transitory component or the deviation from the steady state x T

	Parameters Parameters Parameters	Prior distribution Prior distribution Prior distribution			Posterior mode Posterior mode Posterior mode
		type type type	mean mean mean	std std std	model1 model1 model1		10% 10% 10%	90% 90% 90%	model2 model2 model2
	Behavioral parameters Domestic prices in ‡ation indexation ( d ) Interest rate ( r )
	Habit formation (h) -Euro-area -Euro-area	beta beta	0:85 0:50	0:05 0:10	0.9899 0.4585	0.9884 0.3440	0.9899 0.5208	0.9827 0.3230
	-Euro-area -United-Kingdom -United-Kingdom	beta beta beta	0:60 0:85 0:50	0:05 0:05 0:10	0.4575 0.9640 0.4991	0.4276 0.9495 0.3851	0.4956 0.9735 0.6010	0.5093 0.9741 0.3822
	-United-Kingdom -China -China	beta beta beta	0:60 0:85 0:50	0:05 0:05 0:10	0.3746 0.9743 0.4304	0.3519 0.9676 0.3040	0.4098 0.9794 0.4967	0.4739 0.9799 0.3418
	-China -United-states -United-states	beta beta beta	0:60 0:85 0:50	0:05 0:05 0:10	0.5202 0.9583 0.3773	0.4852 0.9354 0.2772	0.5754 0.9650 0.4615	0.3458 0.9786 0.3368
	-United-states -Japan -Japan	beta beta beta	0:60 0:85 0:50	0:05 0:05 0:10	0.4863 0.9899 0.2352	0.4543 0.9870 0.1313	0.5335 0.9899 0.2831	0.5151 0.9899 0.2006
	-Japan -Switzerland -Switzerland	beta beta beta	0:60 0:85 0:50	0:05 0:05 0:10	0.4147 0.9075 0.3133	0.3796 0.9046 0.1889	0.5287 0.9338 0.4322	0.4451 0.9776 0.3169
	-Switzerland In ‡ation objective ( ) Import prices in ‡ation indexation ( f ) beta 0:60	0:05	0.5120		0.4879	0.5504	0.4843
	Frisch elasticity ( ) -Euro-area -Euro-area	beta beta	0:85 0:50	0:10 0:10	0.8941 0.7525	0.8630 0.6839	0.9403 0.8604	0.9035 0.5163
	-Euro-area -United-Kingdom -United-Kingdom	normal beta beta	2:00 0:85 0:50	0:75 0:10 0:10	0.6551 0.8962 0.5195	0.5000 0.8501 0.4270	1.4159 0.9265 0.6115	1.8936 0.5581 0.5936
	-United-Kingdom -China -China	normal beta beta	2:00 0:85 0:50	0:75 0:10 0:10	1.8925 0.8201 0.6404	0.8596 0.7415 0.4918	2.9805 0.8764 0.7184	1.0484 0.5899 0.6427
	-China -United-states -United-states	normal beta beta	2:00 0:85 0:50	0:75 0:10 0:10	1.5862 0.8604 0.5892	0.9010 0.7941 0.4984	2.9200 0.9145 0.7087	0.6222 0.9074 0.6193
	-United-states -Japan -Japan	normal beta beta	2:00 0:85 0:50	0:75 0:10 0:10	2.1938 0.7043 0.4465	1.2223 0.6534 0.3261	3.3953 0.8023 0.5411	2.1653 0.7943 0.6039
	-Japan -Switzerland -Switzerland	normal beta beta	2:00 0:85 0:50	0:75 0:10 0:10	0.5000 0.6377 0.6023	0.5002 0.5545 0.5145	2.2827 0.7207 0.6825	2.0317 0.7652 0.5900
	-Switzerland Wages in ‡ation indexation ( w ) normal Price and wage setting parameters 2:00 0:75 2.3387 1.8572	3.5137	2.5035
	Calvo domestic prices ( H ) -Euro-area beta	Autoregressive parameters 0:50 0:15 0.7004	0.5205	0.7658	0.3394
	Technology ( a ) -Euro-area -United-Kingdom	beta beta	0:75 0:50	0:05 0:15	0.9499 0.3355	0.9483 0.1844	0.9499 0.5240	0.7996 0.4742
	-Euro-area -United-Kingdom -China	beta beta beta	0:80 0:75 0:50	0:10 0:05 0:15	0.9387 0.8989 0.5785	0.7378 0.8711 0.3836	0.9735 0.9225 0.6689	0.8301 0.8464 0.5204
	-United-Kingdom -China -United-states	beta beta beta	0:80 0:75 0:50	0:10 0:05 0:15	0.7566 0.8573 0.5241	0.5698 0.8175 0.3407	0.8656 0.9185 0.7101	0.9224 0.6305 0.5740
	-China -United-states -Japan	beta beta beta	0:80 0:75 0:50	0:10 0:05 0:15	0.9050 0.6922 0.5034	0.8135 0.6253 0.3732	0.9659 0.7231 0.7127	0.9224 0.7964 0.6176
	-United-states -Japan -Switzerland	beta beta beta	0:80 0:75 0:50	0:10 0:05 0:15	0.7169 0.8001 0.61792	0.6250 0.7531 0.4728	0.8121 0.8360 0.7499	0.8396 0.8055 0.4391
	-Japan -Switzerland Steady state wage mark-up ( w ) beta beta 0:75 0:80	0:10 0:05	0.8264 0.8489	0.7605 0.8230	0.9265 0.8748	0.8833 0.8732
	-Switzerland Calvo import prices ( F ) beta -Euro-area normal	0:80 0:50	0:10 0:15	0.9448 0.5297		0.9046 0.4371	0.9732 0.7328	0.6712 0.4828
	Preference law of motion ( g ) -Euro-area beta -United-Kingdom normal	0:60 0:50	0:05 0:15	0.8573 0.6287	0.8365 0.4996	0.8649 0.8707	0.6211 0.5022
	-Euro-area -United-Kingdom -China	beta beta normal	0:85 0:60 0:50	0:05 0:05 0:15	0.8397 0.6998 0.5436	0.7375 0.6582 0.4030	0.8687 0.7587 0.6892	0.7512 0.6741 0.5488
	-United-Kingdom -China -United-states	beta beta normal	0:85 0:60 0:50	0:05 0:05 0:15	0.9007 0.6793 0.4106	0.7974 0.6095 0.4000	0.9424 0.7138 0.5545	0.8994 0.6207 0.5131
	-China -United-states -Japan	beta beta normal	0:85 0:60 0:50	0:05 0:05 0:15	0.8583 0.6504 0.6312	0.7283 0.5774 0.5361	0.9015 0.7073 0.7923	0.8677 0.6817 0.5148
	-United-states -Japan -Switzerland	beta beta normal	0:85 0:60 0:50	0:05 0:05 0:15	0.8616 0.6256 0.4000	0.7394 0.5787 0.4000	0.9076 0.6534 0.5576	0.8669 0.6676 0.4843
	-Japan -Switzerland	beta beta	0:85 0:60 Monetary policy parameters 0:05 0.8445 0.7039 0:05 0.7929 0.7304	0.9030 0.8258	0.7836 0.6325
	-Switzerland Calvo wages ( W ) Interest rate response to in ‡ation (r ) beta 0:85	0:05	0.8713		0.8044	0.9313	0.8629
	Exchange rate ( e ) -Euro-area -Euro-area	beta normal	0:75	0:05	0.8299	0.8075	0.8625	0.7684
	-Euro-area -United-Kingdom -United-Kingdom	beta beta normal	0:85 0:75	0:05 0:05	0.6388	0.5548	0.6602	0.7613
	-United-Kingdom -China -China	beta beta normal	0:85 0:75	0:05 0:05	0.7538 0.8155	0.6685 0.7604	0.8505 0.8560	0.8661 0.6816
	-China -United-states -United-states	beta beta normal	0:85 0:75	0:05 0:05	0.8348 0.7219	0.7746 0.6320	0.9081 0.7641	0.8939 0.7620
	-United-states -Japan -Japan	beta beta normal	0:85 0:75	0:05 0:05	0.8412 0.7823	0.7218 0.6951	0.8823 0.8139	0.8817 0.7583
	-Japan -Switzerland -Switzerland	beta beta normal	0:85 0:75	0:05 0:05	0.7880 0.8428	0.7106 0.8028	0.8794 0.8802	0.8839 0.7711
	-Switzerland	beta	0:85	0:05	0.7828		0.6773	0.8625	0.9039

t can be estimated as:

z t = ^ + ^ t + xT t

where xT t is the OLS residuals from the regression of z t on a constant and a linear trend t.

C Prior and Posterior distribution

  (" e i t ) i = U K 45.16 38.91 32.70 26.98 22.19 18.54 15.99 14.37 i = CH 20.27 18.66 16.64 14.43 12.29 10.46 9.05

	D Variance decomposition F Convergence diagnostics
	Parameters Parameters Parameters	Q1	Prior distribution Prior distribution Prior distribution Q2 Q3 Q4		Q5	Q6	Posterior mode Posterior mode Posterior mode Q7 Q8
	type Interest rate response to output gap (r y ) mean -Euro-area normal 0:125 type mean Preference shocks ( g ) -Euro-area Inv. gam. 0:20 type mean In ‡ation objective shocks ( ) -Euro-area Inv. gam. 2:00 Exchange rate shock 8.05 std model1 10% 0:05 0.1353 0.1191 std model1 10% 90% 90% 0.1541 2 6.7195 5.5190 10.1022 std model1 10% 90% 2 4.3174 2.7985 4.9419	model2 0.0906 model2 1.6386 model2 0.9779
	-United-Kingdom -United-Kingdom -United-Kingdom i = U S	normal Inv. gam. Inv. gam. 3.72 3.43	0:125 0:20 2:00 3.05	0:05 2 2 2.62	0.0981 0.1623 1.0157 2.22 1.87 0.7547 0.0742 0.0582 1.62	0.1179 0.37193 1.4099 1.44	0.1187 0.2825 1.1279
	-China -United-states -Japan -Switzerland Interest rate response to l.o.p gap (r ) normal normal normal normal 0:125 0:125 0:125 0:125 -Euro-area -United-Kingdom -China -United-states -Japan -Switzerland normal normal normal normal normal normal 0:05 0:05 0:05 0:05 0:05 0:05 Interest rate response to change in in ‡ation (r ) 0:05 0:05 0:05 0:05 0:05 0:05 0:05 0:05 0:05 0:05 -Euro-area -United-Kingdom -China -United-states -Japan -Switzerland normal normal normal normal normal normal 0:30 0:30 0:30 0:30 0:30 0:30 0:10 0:10 0:10 0:10 0:10 0:10 Interest rate response to change in output gap (r y ) -Euro-area -United-Kingdom -China -United-states normal normal normal normal 0:125 0:125 0:125 0:125 0:05 0:05 0:05 0:05 -China -United-states -Japan -Switzerland Inv. gam. Inv. gam. Inv. gam. Inv. gam. 0:20 0:20 0:20 0:20 2 2 2 2 Exchange rate shocks ( e ) -Euro-area -United-Kingdom -China -United-states -Japan -Switzerland Inv. gam. Inv. gam. Inv. gam. Inv. gam. Inv. gam. Inv. gam. 2:50 2:50 2:50 2:50 2:50 2 2 2 2 2 -China -United-states -Japan -Switzerland Inv. gam. Inv. gam. Inv. gam. Inv. gam. 2:00 2:00 2:00 2:00 2 2 2 2 0.9951 0.0269 0.0576 0.1174 0.1700 0.0556 0.1028 0.0040 0.0515 0.0120 0.0290 0.0100 0.0238 0.0100 0.0100 0.0100 0.0529 0.2934 0.1813 0.1115 0.1013 0.0917 0.0947 1.1420 0.0902 9.0875 6.7975 2.7054 5.0098 1.5544 1.0342 1.6114 2.9131 i = JP 1.43 1.19 0.97 0.77 0.62 0.51 0.6866 0.0100 0.0548 0.0995 0.1375 0.0339 0.0720 0.0000 0.0234 0.0000 0.0049 0.0100 0.0100 0.0100 0.0100 0.0100 0.0164 0.2752 0.1740 0.0919 0.0944 0.0506 0.0525 0.7263 0.0472 7.8376 5.9192 2.2998 4.2206 1.3322 0.6405 0.9625 1.8501 0.45 i = SW 0.25 0.22 0.19 0.16 0.13 0.11 0.10 Foreign mark-up shock (" p it ) i = U K 0.02 0.01 0.00 0.00 0.00 0.00 0.00 Monetary policy shocks ( "r ) -Euro-area -United-Kingdom -China -United-states -Japan -Switzerland Inv. gam. Inv. gam. Inv. gam. Inv. gam. Inv. gam. Inv. gam. 0:10 0:10 0:10 0:10 0:10 0:10 2 2 2 2 2 2 0.0451 0.1494 0.3134 0.1298 0.0408 0.0426 0.0249 0.0619 0.2444 0.0406 0.0311 0.0242 i = CH 0.14 0.06 0.04 0.03 0.02 0.02 0.02 i = U S 0.20 0.10 0.06 0.04 0.03 0.03 0.02 i = JP 0.00 0.00 0.00 0.00 0.00 0.00 0.00 i = SW 0.00 0.00 0.00 0.00 0.00 0.00 0.00 In ‡ation objective shock ( i t ) i = ZE 2.61 3.41 4.25 5.09 5.86 6.51 7.03 i = U K 6.28 8.24 10.34 12.42 14.32 15.88 17.07 17.91 0.0435 0.0836 0.1340 0.1695 0.0645 0.1257 0.0284 0.0590 0.0226 0.0731 0.0126 0.0295 0.0165 0.0144 0.0113 0.0623 0.3058 0.1980 0.1273 0.1247 0.29311 0.38445 2.7936 0.2320 10.6148 8.3449 3.1652 5.8833 1.8186 1.5685 1.3662 2.0682 3.1541 0.41 0.09 0.00 0.1038 0.2145 0.4061 0.2575 0.0656 0.0738 0.01 0.02 0.00 0.00 7.45 Domestic price mark-up shocks ( " p ) -Euro-area -United-Kingdom -China -United-states -Japan -Switzerland Inv. gam. Inv. gam. Inv. gam. Inv. gam. Inv. gam. Inv. gam. 2:00 2:00 2:00 2:00 2:00 2:00 2 2 2 2 2 2 2.4142 0.4559 1.5563 1.4994 0.5712 0.3855 2.0216 0.3983 1.3559 1.2582 0.4678 0.3336 2.9955 0.5677 1.9023 1.8912 0.7033 0.4628 Modi…ed Harmonic Mean estimator -3792 i = CH 0.98 1.28 1.61 1.93 2.22 2.47 2.65 2.79 i = U S 2.37 3.05 3.74 4.36 4.85 5.17 5.33 5.37 i = JP 0.03 0.04 0.05 0.06 0.07 0.08 0.08 0.09 i = SW 0.20 0.26 0.32 0.38 0.43 0.46 0.49 0.50 Preference shock (" g it ) i = ZE 2.05 2.81 3.73 4.74 5.76 6.67 7.40 7.91 i = U K 0.10 0.12 0.15 0.18 0.21 0.22 0.23 0.24 Import price mark-up shocks ( " f ) -Euro-area -United-Kingdom -China -United-states Inv. gam. Inv. gam. Inv. gam. Inv. gam. 2:50 2:50 2:50 2:50 2 2 2 2 11.1442 1.3906 4.2043 2.5290 9.3925 1.1153 3.6180 2.1019 12.4874 1.6688 5.2544 3.2342 i = CH 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 i = U S 0.03 0.03 0.04 0.04 0.05 0.05 0.05 0.05 i = JP 0.10 0.13 0.16 0.19 0.21 0.23 0.23 0.23 i = SW 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00	0.0887 0.0819 0.1533 0.1552 0.0354 0.0884 0.0043 0.0038 0.0901 0.0545 0.0100 0.0100 0.0100 0.0100 0.0100 0.0194 0.0667 0.1156 0.0617 0.0757 0.0928 0.0916 1.7004 0.0866 2.7678 4.1530 4.2323 5.0516 1.8986 1.0702 1.2465 1.5655 0.8552 0.0415 0.0345 0.0766 0.1946 0.0429 0.0455 0.6805 -3115 0.4824 0.8780 1.6442 0.5378 0.4009 4.0187 1.3077 4.1071 2.4997
	-Japan -Japan	normal Inv. gam.		0:125 2:50	0:05 2	0.0500 4.5901	0.0500 4.0441	0.05480 5.7329	0.0696 4.9456
	-Switzerland -Switzerland	normal Inv. gam.		0:125 2:50	0:05 2	0.1918 1.2265	0.1733 1.0173	0.1998 1.6920	0.1166 1.1021
	Wage mark-up shocks ( " w )	Standard deviations of shocks
	Idiosyncratic technology shocks ( "a ) -Euro-area Inv. gam. 0:25	2		10.5981	8.7066	11.8695	2.3589
	-Euro-area -United-Kingdom	Inv. gam. Inv. gam.		0:70 0:25	2	2	1.8049 1.8918 2.0569	1.7588 0.2050 1.6444	1.8380 2.1933 2.3217	1.6280 1.0510 1.0543
	-United-Kingdom -China	Inv. gam. Inv. gam.		0:70 0:25	2	2	1.8965 1.2717 3.3418	1.8361 0.9789 2.8671	1.9327 1.6249 4.1175	1.8672 1.2448 3.3540
	-China -United-states	Inv. gam. Inv. gam.		0:70 0:25	2	2	1.7320 3.7334 2.7771	1.6954 2.9439 2.4307	1.7927 4.9275 3.4218	1.7828 3.7574 2.7215
	-United-states -Japan	Inv. gam. Inv. gam.		0:70 0:25	2	2	1.6934 4.2720 3.0837	1.6586 3.2088 2.6995	1.7859 5.4352 3.8421	1.6265 0.3377 3.4512
	-Japan -Switzerland	Inv. gam. Inv. gam.		0:70 0:25	2	2	1.6217 2.2951 0.8285	1.5728 1.7501 0.7089	1.6552 2.7368 1.0344	1.6443 1.8148 0.6293
	-Switzerland	Inv. gam.		0:70		2	1.5968 1.4703	1.5777 1.1531	1.6353 1.8865	1.6779 0.8948

F.1 Multivariate diagnostic

See de Bandt and Raza…ndrabe (2014) for a comprehensive revue of the link between currency invoicing and ERPT.

See for example Marazzi and Sheets (2007), Bouakez and Rebei (2008) and Gust et al. (2010). Positive issue is commonly known as the Local Currency Pricing -Pricing to Market (LCP-PTM) literature.

For a variable Xt, xt = log(Xt) and xt = log(Xt) log( X) = xt x, where X is the steady state value of Xt.

The choice of numeraire country is irrelevant and does not alter the equilibrium of the model.

As shown by[START_REF] Chudik | Size, openness, and macroeconomic interdependence[END_REF], no group of countries is isolated from the rest of the world if and only ifrank(IN M) = N 1where IN is a N xN identity matrix and M is the (N xN ) import share matrix where its (i; j)-th element equals ij .

For a formal derivation of this risk sharing condition, see[START_REF] Gali | Monetary policy and exchange rate volatility in a small open economy[END_REF].

Estimation is conducted using Dynare. It is a software platform for handling DSGE model. See Adjemian et al. (2011). Initial value of structural parameters is obtained using a posterior sample of

100,000 draws. Thereafter, posterior sample of each block of parameters is generated in sequence using 20,000 draws.1 0 Introduction of the preference is standard and straightforward. For the sake of clarity, we report interested readers in a supplement available upon request for details.1 1 Details of the sources and data transformation are given in a supplement available upon request. Moreover, output, real wage and nominal exchange rate are expressed in 100 times logarithm of the variable. In turn, in ‡ation and interest rate are expressed in quarterly basis and in percentage.

[START_REF] Adolfson | Bayesian estimation of an open economy dsge model with incomplete pass-through[END_REF] See apppendix B for detailed and formal derivation of the long-horizon forecast and the GVAR speci…cation.

[START_REF] An | Bayesian analysis of dsge models[END_REF] A formal way to obtain the extent of exchange rate pass-through is to evaluate change in import price (not import price in ‡ation) in percentage of change in nominal exchange rate following an exchange rate shock. Moreover, it is worth noting that changes in exchange rate can stem from other shocks such as monetary policy (interest rate) shock, risk premium shock, ... .

[START_REF] Anderton | Extra-euro area manufacturing import prices and exchange rate passthrough[END_REF] See[START_REF] Bacchetta | A theory of the currency denomination of international trade[END_REF] and[START_REF] Goldberg | Vehicle currency use in international trade[END_REF] for the role of currency-invoicing in the extent of exchange rate pass-through and the role played by a vehicle currency in international trade.

[START_REF] Bacchetta | A theory of the currency denomination of international trade[END_REF] As well explained inObstfeld and Rogo¤ (1996), this is the price index de…ned as the minimal expenditure in terms of country i's domestic currency needed to purchase one unit of consumption index.

6 We follow Erceg et al. (1999), Smets and wouters (2003), and Adolfson et al. (2007) when taking CPI in ‡ation as wage indexation. Some open DSGE model such as the SIGMA model by Erceg et al. (2006) and that of Jacquinot et al. (2006) instead use wage in ‡ation as indexation.

[START_REF] Betts | Exchange rate dynamics in a model of pricing-tomarket[END_REF] Notice that P d it = Piit, thus aggregate producer price index will be the same as[START_REF] Erceg | Sigma: A new open economy model for policy analysis[END_REF] for j = i .

[START_REF] Adjemian | Dynare: Reference manual, version 4[END_REF] One variable is quali…ed to be "long run forcing" when it is weakly exogenous. In the context of cointegrating models, it implies that there is no long-run feedback from domestic or endogeneous variables to the exogeneous variable. 2 2 See Dees et al. (2009) for derivation.