**BIS Working Papers **

**No 786**

**Estimating the effect of exchange rate changes on total exports**

by Thierry Mayer and Walter Steingress

**Monetary and Economic Department **

May 2019

JEL classification: D22, D84, E31

Keywords: inflation expectations, firms' survey, new information.

This publication is available on the BIS website (www.bis.org).

*© *Bank for International Settlements 2019. All rights reserved. Brief excerpts may be reproduced or translated provided the source is stated.

ISSN 1020-0959 (print)

ISSN 1682-7678 (online)

**Estimating the effect of exchange rate changes on total exports**

by Thierry Mayer and Walter Steingress

**Abstract **

This paper shows that real effective exchange rate (REER) regressions, the standard approach for estimating the response of aggregate exports to exchange rate changes, imply biased estimates of the underlying elasticities. We provide a new aggregate regression specification that is consistent with bilateral trade flows micro-founded by the gravity equation. This theory-consistent aggregation leads to unbiased estimates when prices are set in an international currency as postulated by the dominant currency paradigm. We use Monte-Carlo simulations to compare elasticity estimates based on this new “ideal-REER” regression against typical regression specifications found in the REER literature. The results show that the biases are small (around 1 percent) for the exchange rate and large (around 10 percent) for the demand elasticity. We find empirical support for this prediction from annual trade flow data. The difference between elasticities estimated on the bilateral and the aggregate levels reduce significantly when applying an ideal-REER regression rather than a standard REER approach

JEL classification: D22, D84, E31

Keywords: inflation expectations, firms' survey, new information.

Regressions that explain total exports with real effective exchange rates (REERs) are ubiquitous in applied policy work, but their analytical structures are only loosely based on trade theory. These open-economy "macro-level" regressions rely on monadic export equations where a country's change in total exports is the dependent variable with the REER as the main variable of interest and an additional control is considered for demand changes faced by the country. Despite considerable progress in the econometric aspects, the fundamental specification of McGuirk (1986), which derives its regression specification from Armington (1969), has remained virtually unchanged over the past few decades. During the same period, considerable progress has been made in modeling the "microlevel" foundations of dyadic (bilateral) trade equations, which are called gravity equations. A seminal paper in the gravity literature is Anderson and Van Wincoop (2003), which is also based on Arming- ton (1969). In the gravity literature, little attention has been given to estimate the impact of changes in the exchange rate (see Anderson et al. (2016) as a notable exception). In this paper, we reconnect the two strands of the literature.

We start by assessing whether the standard approach of monadic macro regression is compatible with the micro-founded gravity equation that predicts a very precise functional form for an estimation explaining bilateral flows. Similar to Imbs and Mejean (2015), our paper is methodological in that we seek to derive the appropriate monadic aggregation of bilateral flows and compare our new "ideal-REER" approach with typical regressions from the standard REER literature. Differences arise due to the fact that the ideal-REER regression is based on proportional changes, while McGuirk (1986)'s REER is based on partial derivatives (log changes). In the standard REER regression total exports are a log linear function of log changes of the real effective exchange rate and foreign demand. On the other hand, the aggregate export equation that follows a structural gravity approach implies a log-linear function of log changes in the real exchange rate (RER) and differences in foreign demand deflated by the destination-specific price index. The standard REER approach constitutes an approximation, which holds for small changes, and causes an aggregation bias.

The aggregation of bilateral trade flows inherently depends on the currency denomination. In our baseline gravity model, prices are set in a common international currency following the dominant currency paradigm (DCP) as in Casas et al. (2017). DCP implies that the exchange rate variation relative to the international (dominant) currency and not the bilateral exchange rate determines passthrough. As a result, we can split the relative price between the exporter and the importer into the sum of an exporter-specific (the RER) and an importer-specific component (foreign demand deflated by the importer's price index) each denominated in the international currency. This separation provides for theory-consistent aggregation and unbiased estimation of the trade and demand elasticities. If prices are set in the producer's currency (producer currency pricing (PCP)) or in the local currency (local currency pricing (LCP)), the pass-through depends on the bilateral exchange rate and estimation of the aggregate elasticities without bias is only possible if pass-through is complete. Extending the model to allow markups to vary across trading partners reduces the response of prices to the exchange rate change and lowers the bias in both the demand and the trade elasticities.

To reveal the importance of theory-consistent aggregation for parameter inference, we resort to theoretical model simulations and quantify how departures from the model's strict form can generate biases in trade and demand elasticity estimates. To that purpose, we use the Dekle et al. (2007) version of the structural gravity model (compatible with most of the commonly used theoretical models of international trade). Monte Carlo simulations show that while there is discordance between the ideal-REER and standard REER regressions, the magnitude of the biases in elasticities are dependent on the precise regression specification as well as the underlying assumptions on the exchange rate shock. Overall, the bias in the trade (exchange rate) elasticity is rather small (close to 1 percent), while the bias in the demand elasticity is more pronounced (close to 10 percent) and increases with measurement errors. If, instead, we re-estimate the same specifications and denominate trade in the producer currency, the aggregation bias increases significantly, the overall fit of the regression is lower, the point estimates of the elasticities are smaller and the standard errors are larger in all specifications.

Next, we build on the theoretical basis of our paper and compare the empirical performance of our ideal-REER with the standard REER approach. We use annual bilateral trade and price data for 25 countries that are part of the Bank for International Settlements (BIS) narrow effective exchange rate indexes (see Klau and Fung (2006)) for the period 1964 to 2014. As a first step, we follow Boz et al. (2017) and run exchange rate pass-through regressions to test (1) whether pass-through is complete, and (2) whether the pass-through depends on the currency denomination of export and import prices. Our results show that, overall, pass-through is incomplete (with an average coefficient of 0.16) and both exporter and importer prices vary mainly with the US dollar supporting the dominant currency paradigm. In the second step, we aggregate bilateral trade flows denominated in US dollars and estimate the trade and demand elasticities for both the ideal-REER and the standard REER regression specifications following the DCP. Evidence based on the pooled sample across all countries, in addition to country-specific regressions, corroborate the simulation results. The difference between the elasticities estimated on the bilateral and the aggregate levels reduces significantly when applying an ideal-REER regression rather than the standard REER approach.

Taken together, these results highlight two additional advantages of the dominant currency paradigm, which currently is the yardstick used in evaluating pricing and exchange rate pass-through (see Boz et al. (2017) and Casas et al. (2017)). First, DCP allows for consistent aggregation of microlevel founded bilateral trade equations with a minimum set of assumptions on the pricing behavior and, second, the estimation performance of aggregate regressions improves significantly when denominating the variables in US dollars.

Our paper contributes to the macroeconomic literature on exchange rate indexes as a measure of price competitiveness. The REER is an important statistical indicator, produced by international agencies such as the IMF (see Bayoumi et al. (2005)), the BIS (see Klau and Fung (2006)) and many central banks (see Schmitz et al. (2011) for the ECB and Barnett et al. (2016) for the Bank of Canada references). We see our analysis as complementary to the recent papers that study the theoretical foundations for these indexes. These papers' focus lies more in introducing global linkages into the measurement of competitiveness (see Bayoumi et al. (2013), Patel et al. (2017) and Bems and Johnson (2017)). We show that using the REER to estimate the aggregate response of exports leads to biased estimates of the underlying elasticities. The bias exists due to functional form assumptions, currency denomination and inconsistent weighting of trade flows in the aggregation of bilateral trade flows.

Our analysis also speaks to the old macroeconomic literature on estimating trade elasticities started by Orcutt (1950), Houthakker and Magee (1969) and Goldstein et al. (1985) and recent contributions by Spilimbergo and Vamvakidis (2003), Freund and Pierola (2012), Bussiere et al. (2014) and Ahmed et al. (2016). All of these papers rely on effective exchange rates to estimate demand and trade elasticities. Our results warrant caution in the use of these estimates for counterfactual policies, particularly in the case of the demand elasticity, where bias can be significant. As an alternative, we suggest our ideal-REER regression framework to mitigate these concerns. These conclusions also apply to studies that use aggregate time-series data to estimate exchange rate pass-through coefficients such as Campa and Goldberg (2005), Vigfusson et al. (2009) and Bussiere et al. (2014).

The remainder of the paper is structured as follows. Section 2 derives the gravity-compatible aggregate export equation in the partial as well as in the general equilibrium. Section 3 describes the standard empirical approaches used in the applied literature and discusses the functional form differences with respect to the gravity approach. Section 4 quantifies the biases of the different specifications using Monte-Carlo simulations of partial as well as general equilibrium counterfactuals of exchange rate shocks. Section 5 discusses the importance of the underlying pricing behavior and currency denomination. Section 6 introduces the data and provides empirical evidence on the key predictions of our theoretical framework. Section 7 concludes.

We start with a description of the structural gravity equation based on Head and Mayer (2014) derived from the Armington model. The central feature of the Armington model is that it is not only at the heart of the gravity equation literature but also forms the basis of the Real Exchange Rate regressions as in Artus and McGuirk (1981), McGuirk (1986) and Spilimbergo and Vamvakidis (2003).

**1.1. Structural gravity**

Consider a world economy consisting of N countries. Each country is endowed with Qn units of a distinct good. In each country, the representative agent has Constant Elasticity of Substitution (CES) preferences. Trade between countries is costly and takes the form of iceberg trade costs. In order to sell one unit of good i in country n, exporters from country i have to ship тп; > 1 units. Trade within the country is costless, i.e. тц = 1. To prevent arbitrage opportunities, we assume that trading bilaterally between i and n is always cheaper than trading via an intermediate country k ( TnkTki > Tni for Vi, j, k). Note that given country i's endowment Qi and total income Yi, we can express supply as Si = Yi/ Qi, the corresponding producer price of good i as Pii = Si and the price charged to consumers in country n to be Рп; = TniSi. Given our assumptions, the gravity equation is a product of the share of expenditure that importer n allocates on goods from exporter i, пп;, and the overall expenditure, Xn. Following Head and Mayer (2014), nni can be expressed in the following multiplicative separable form:

where Фп; captures the bilateral accessibility of n to exporter i and is a function of iceberg trade costs. Фп represents the multilateral resistance term and Qi represents the weighted sum of the exporter capabilities.

In order to quantify potential estimation biases and to produce counterfactual simulation estimates of the structural gravity model, we need to specify the underlying supply structure. In this paper, we opt for the Ricardian Comparative Advantage specification with intermediate inputs from Dekle et al. (2007) based on Eaton and Kortum (2002) and Alvarez and Lucas (2007). In this model, each country produces a large number of tradable intermediate goods that are homogeneous across countries. Productivity z differs across goods and is assumed to follow a Frechet distribution with a cumulative distribution function of exp(-Tiz-e), where Ti is a technology parameter and в determines the amount of heterogeneity in the productivity distribution. For each intermediate good z, buyers choose the lowest cost supplier in the world.

Aggregating across all goods, assuming that factor prices are denominated in the domestic currency as well as assuming complete pass-through, the expenditure share that buyers in n spend on goods from i can be written as where Wi is the wage paid to workers in i, p. is the price of tradable intermediate goods, r. is the nominal exchange rate that converts domestic prices into a common international currency (US dollar, for instance) and в is the share of intermediate goods in the production of manufacturing goods. Comparing equation (2) to equation (1) shows that the three structural gravity terms are given by Si = Ti {riwep1 ^ , фni = T-iв with в as the trade elasticity and the multilateral resistance term in international currency Фп = (rnPn)-в.

Asserting a market clearing condition implies that total output in i, Yi, corresponds to total spending, Xi. With labor as the sole factor, the value of production in local currency in the country of origin is given by Yi = wL. Following Dekle et al. (2007), the market clearing conditions for the manufacturing sector in international currency become

At this point, it is important to note that while we chose the Eaton-Kortum variant of the gravity model as the underlying theoretical foundation, all other theoretical models that adhere to structural gravity are compatible with the following analysis on the estimation bias. The main difference will be the underlying implications on the micro-structure and its corresponding interpretations.

**1.2. Counterfactual analysis**

To produce counterfactual analysis, we use the exact hat notation, see Dekle et al. (2007) and Arko- lakis et al. (20 2), to derive the aggregate export equation. More precisely, hats denote the ratio of post-shock to pre-shock values. We start by assessing the change in bilateral exports and then aggregate to obtain the change in total exports. For bilateral exports of country i to country n in international currency, we have

where X'n. is bilateral trade after the change and Xni is the value before any change. The resulting change in total exports after the policy change is simply the sum over all import countries with the exception of country i's home market. Furthermore, we can decompose the change in total exports (Ei) as the sum of changes in the expenditure shares and the aggregate expenditure in international currency across all importing countries weighted by the initial share of exports going to country n

This equation is extremely general and simply states that the change in a country's total exports corresponds to the weighted sum of all proportional changes in bilateral exports. Weights are the initial share of each bilateral flow in total exports. The change in each bilateral flow can have two origins: i) an increase in total expenditure in destination n, or ii) a change in competitiveness of country i relative to all other trading partners of n.

As a policy shock, we consider an exogenous change in the nominal exchange rate of country i, ti, which is independent of the exchange rate in all other countries. The shock to t{ acts like a shifter of delivered prices to the consumer (which assumes complete pass-through at this stage). Without migration or other affects to the population, we have changes in output given by changes in the nominal exchange rate and wages, Ггг&г- = Yi. Assuming that technology and trade costs are unaffected by the shock, the implied change in the bilateral trade shares resulting from a change in the nominal exchange rate takes the following form:

where the change in the price index of country n is equal to the change in country i's exchange rate times the weight of country i in country n's import basket,

Plugging the calculated trade shares back into the market clearing condition, one can solve for the changes in production of each country of origin. The resulting changes in equilibrium wages (wi) paid in local currency are

Note that, in this model, a nominal exchange rate shock will affect relative prices and lead to changes in trade flows. The reason is that we fix the trade deficit in international currency, i.e., in US dollars. Given this assumption, the wage adjustment will not be one-to-one with exchange rate change and depends on the dollar value of the current account deficit. As such, this scenario will be instructive to understand potential aggregation biases when constructing an effective exchange rate in general equilibrium.8

To obtain the export equation, we can simply apply equation 4 into equation 3. Taking logs and assuming that prices are set in the international currency,9 we can write the log change in exports as a function of the log change in the real exchange rate (RERi) and foreign demand as

Xn = fnwnYn + Dn represents changes in absorption and the real exchange rate is defined as the change in the nominal exchange rate and the change in the producer price in the exporting country in local currency, i.e., changes in intermediate input prices and wages, RERi = fiPu = fiW-pj в.

Equation 7 implies that, under the assumption of a constant elasticity of substitution, there is no aggregation bias in aggregate macro exchange rate regressions. In order to obtain these unbiased estimates, one needs to regress the log of the change in nominal exports on the log of the change in the real exchange rate and the foreign demand deflated by changes in the importer's price index:

в RER will be the estimated exchange rate elasticity, в x the corresponding demand elasticity and £i the error term. However, to our knowledge, there is no paper running regressions based on equation 8. Instead, the literature focuses on trade-weighted real (or nominal) effective exchange rate regressions, which aggregate bilateral exchange rates to a single country-specific indicator. Next, we discuss the theoretical underpinnings of the effective exchange rate regressions used in applied work and relate them to the structural gravity framework.

McGuirk (1986) is the standard reference that describes the methods used to construct an aggregate price competitiveness indicator, or the real effective exchange rate. This indicator attempts to measure a country's price change in the tradable sector relative to those of other countries after converting each of them into a common currency. In this section we show that aggregation based on the effective exchange rate relies on a functional form approximation and that using this indicator for parameter inference will lead to bias in the estimated coefficients.

The theoretical foundations of the McGuirk (1986) approach rests on the Armington (1969) assumption of imperfect substitutability between goods and a constant elasticity of substitution (CES), which is consistent with structural gravity. The starting point is writing the bilateral import demand function of country n for goods from country i, Xni, as a function of total expenditure Xn and the bilateral shares defined in equation 2. Taking the log and considering the change from the initial equilibrium to the new equilibrium ln Xni- = Д ln Xni = (ln xni - ln Xni), we get

where ln Pni- denotes the change in the price that exporting country i charges to consumers in country n and ln Pn is the change in the price index of the importing country n, both denoted in local currency. Concerning the changes in the importing country's price changes, McGuirk (1986) approximates the log change of the importer's price as a weighted average of the log changes in the production prices:

where Pkk represents the change in producer price index in country k and nnk is the expenditure shares of country n. Using further simplifications outlined in the appendix, we can substitute the import price index in equation 9 with equation 10 and aggregate across all importing countries. To do so, we sum the bilateral trade flows over all markets (excluding the home market) by weighting the flows by the share of i's output sold in each market:

where the weight wn{ is defined as the share of export revenues of country i that comes from sales in destination n with respect to total export sales. Next, we define the double-weighted Real Effective Exchange Rate (REER) as follows:

where в REER is the estimated exchange rate elasticity of specification and в х the demand elasticity. £i represents the estimation error. The estimation produces unbiased estimates if the exchange rate elasticity (вв.^в.) is equal to в and if the foreign demand elasticity (вх) is equal to one.

**1.1. Aggregation bias**

The essential difference between the REER regression in equation 12 and the ideal-REER regression implied by gravity in equation 7 is that the change in total exports is approximately equal to the weighted sum of the bilateral export changes, i.e., ln Ei & En=i wni ln Xni. To see the difference between the two approaches explicitly, we rewrite the REER in equation 9 as a weighted average of the bilateral real exchange rate without substituting for the importer's price index using equation 10. The resulting change in the REER is simply the log difference between the changes in the real exchange rate (RER) of exporter i and a weighted average of the price index changes of every trading partner:

Substituting the implied REER in equation 13 back into the standard REER regression defined in equation 12, we can compare the ideal REER regression in equation 7 with the standard REER regression:

Canceling the RERi on both sides and rewriting the destination-specific terms, we can apply Jensen's inequality and arrive at the following inequality:

Equation 14 states that the log of total exports (the LHS) is larger than the weighted average of the bilateral trade flows (the RHS). The difference between the two expressions is related to the Theil inequality index, which measures the heterogeneity in the destination-specific changes. Intuitively, the Theil index and the resulting bias will be larger the higher the variance is from destination- specific shocks. In the extreme case, if prices and demand change only in one destination, the Theil index is at a maximum value and the estimation biases in the demand and exchange rate elasticities of the REER approach will be at their largest. On the other hand, when the price and demand changes in all destinations are the same, the Theil index will be zero and both regression specifications will lead to the same elasticities. In the simulation section, we quantify the estimation biases as a function of the exchange rate shocks and the trade elasticity. Before, we discuss other real effective exchange rate regression specifications found in the literature starting with an alternative weighting scheme.

**1.2. Weighting scheme**

ln RE ERjMF and the implied weights are given by

The Effective Exchange Rate calculated by policy institutions such as the IMF (Bayoumi et al. (2005)), the European Central Bank (ECB) (Schmitz et al. (2011)) and the Bank for International Settlements (BIS) (Klau and Fung (2006)) follows the double weighting approach approach of McGuirk (1986) but uses slightly different weights. Their calculations include the sales of the exporter country in its domestic market. The REER is defined as follows:

where .ik is the export weight including the home market shares. .ik relates to our export weights in the following way:

However, as Bayoumi et al. (2005) note, the weights implied by equation 15 do not sum up to 1. As a result, the weights are normalized leading to the following definition:

The denominator of equation 16 ensures that TWki sums to 1. The resulting estimation equation is then given by

where the weights for the foreign demand continue to be the export shares wni. We do not have any prior on the direction of the bias in the demand (в x) and the exchange rate elasticity (в RER) caused by the weighting scheme and refer to section 4 for the quantitative simulation results.

**1.3. Multilateral resistance term**

The value of exports depends not only on the country's export price and foreign demand, but also on the willingness of foreigners to substitute domestic for foreign goods. The multilateral resistance term Фп defined in equation 1 picks this idea up. This subsection discusses the estimation bias in the demand and trade elasticities provoked by the absence of the multilateral terms in the ideal-REER and the standard REER regression approach.

The first point to note is that the change in the multilateral resistance term summarizes all the price changes due to variations in the competitiveness of exporter n as well as changes in country i's own competitiveness in its domestic market:

In our framework, the price change in exporter's competitiveness Pkk is simply given by the change in the exporter's per unit costs of production (wages and intermediate input prices) and the exchange rate. The parameter в describes consumers' willingness to substitute domestic for foreign goods following a price change. Omitting the multilateral resistance term in the export equation, we run the following regression:

Equation 19 implies that the multilateral resistance term enters the error мг- = — ln En=i шт (rnPn) - + ei. The omitted variable bias in the estimate of в depends on the co-variance between the change in the exporter's price and the multilateral resistance term:

In general, the multilateral resistance term will be positively correlated with changes in the real exchange rate. The following example with the US as the exporter and Canada as the importer illustrates this correlation. Suppose that the US dollar depreciates (fi t), which increases competitiveness of US exports in all destinations. Depending on the Canadian expenditure share on goods from the US, this change lowers the Canadian price index directly by making US goods cheaper and indirectly by reducing Canadian producer prices through lower costs for intermediate inputs. More generally, countries with the highest expenditure share on US goods (Canada and Mexico) will experience the largest decline in prices. Given that the multilateral resistance term in equation 18 depends negatively on the price index, the correlation with the exchange rate shock is positive and the trade elasticity will be biased towards zero.

Similarly, the bias in the demand elasticity depends on the co-variance between the change in the foreign demand and the multilateral resistance term in the error:

In this case we expect a downward bias in the demand elasticity due to a positive correlation between the multilateral resistance term and foreign demand. The reasoning parallels the correlation with the real exchange rate. An exchange rate depreciation of country i, fi, reduces the price index in destination n and increases the multilateral resistance term. The lower prices increase demand and imply a positive correlation between the multilateral resistance term and foreign demand.

So far, we have shown that if the assumptions of structural gravity hold, one can consistently aggregate bilateral exchange rate changes and estimate the trade and demand elasticities using the aggregate export equation. However, the empirical literature follows an alternative approach based on real effective exchange rates that essentially approximates the aggregate change of exports using a weighted average of changes in bilateral exports. As we have shown, this approximation will lead to an aggregation bias in the estimated elasticities if the underlying exogenous shock causes a heterogeneous response in bilateral trade. We have also discussed alternative approaches in the applied literature that are based on differences in the weighting scheme as well as highlighted the importance of accounting for the multilateral resistance term. In the following section, we simulate a standard structural gravity model in order to quantify the magnitude of the estimation biases when using real effective exchange rate regressions for parameter inference.

Having described the underlying micro-structure, we quantify potential estimation bias in various exchange rate regression specifications using the data set provided by Dekle et al. (2007). These data comprise bilateral trade flows in manufactures, GDP as well as balance of payments information for 39 countries (plus the Rest of the World) in the year 2004.

- Ideal-REER

The first specification captures the Real Exchange Rate model implied by the exact hat algebra in equation 8. This ideal-REER regression is the benchmark specification and will produce unbiased estimates of the trade elasticity (^rer = — $) and the foreign demand elasticity (вХ = 1) with no error. Note that the regression equation 8 is non-linear because the trade elasticity is unknown. We estimate the relevant elasticities using SILS (structurally iterated least squares) as described in Head and Mayer (2014).

- Multilateral resistance ("Gold medal mistake")

An important problem when estimating REER regressions is the presence of the multilateral resistance term. This term captures the implied price changes in the rest of the world due to policy shocks. We assess the importance of the omitted variable bias due to the absence of the multilateral terms by estimating equation 19.

- Real effective exchange rate a la McGuirk

The third specification quantifies the aggregation bias when estimating the double-weighted REER regression in equation 12.

- Approximation via log changes

One difference between the double-weighted REER regression a la McGuirk and the ideal-REER regression is that the multilateral resistance term deflates export prices rather than foreign demand. To assess the importance of this functional form assumption, we approximate the baseline regression with log changes and keep the non-linear form of equation 8:

Similar to the baseline specification, we estimate equation 22 non-linearly using SILS.

- Real effective exchange rate with different weights

The fifth specification quantifies the estimation bias when running a double-weighted REER regression and using an alternative weighting scheme, i.e., IMF weights, as defined in equation 17.

For each of the 5 specifications, we consider two types of policy experiments. The first one is the partial equilibrium approach (Modular Trade Impact (MTI) counterfactual), where we consider an exogenous change in the nominal exchange rate and assume that wages do not adjust (however, intermediate input prices do adjust). In the second experiment, the General Equilibrium Trade Integration (GETI) counterfactual, wages and prices will fully adjust following a shock to the nominal exchange rate.

**1.1. Modular Trade Impact (MTI) counterfactual**

The MTI counterfactual considers changes in prices and leaves wages constant by allowing the trade deficit to change. This case is identical to the one considered by McGuirk (1986). This approach allows us to abstract from income changes in the interpretation of the bias in the exchange rate elasticity. The counterfactual consists of a random nominal exchange rate shock (ri) specific to country i (France, for example), keeping the exchange rates of all other countries fixed. Each shock is drawn from a normal distribution with a variance of 0.1 (equivalent to a 10 percent appreciation or depreciation). For every country, we run 100 different exchange rate shocks to obtain a sample of 39x39x100 data points. Next, we run a cross-section OLS regression for each replication and record the demand and exchange rate coefficients. Table 1 contains the results.

The top value in Table 1 reports the average exchange rate and demand elasticity for the 3900 estimated coefficients. The second row corresponds to the implied bias of the estimate, i.e., the difference between the sample mean and the true value of the coefficients. The third row shows the standard deviation of the estimates. Lastly, the fourth row gives the square root of the estimated mean squared error (MSE), our selection criteria for the specification with the lowest aggregation bias. The first column shows the results for the baseline specification. The estimates are equal to the true values of the elasticities (-4 and 1, respectively) and therefore unbiased. All other specifications lead to a bias. In general, the bias is more pronounced for the demand than for the exchange rate elasticities. The MSE for the demand elasticities is significantly larger than the MSE for the exchange rate elasticities. The omission of the multilateral resistance term in Specification 2 (the "Gold medal mistake") shown in column (2) produces the largest biases for the exchange rate elasticity (-3.872). The specifications with the smallest bias are the double-weighted REER approach a la McGuirk in column (3) and the log approximation of the baseline specification in column (4). The point estimate

of the exchange rate elasticity is -4 with a standard deviation smaller than 0.001. The coefficient of the demand elasticity for the log approximation is slightly larger than 1 with 1.002. Finally, the last column (5) shows the bias in the case of alternative weights, which overestimates the exchange rate elasticity (-3.957).

In addition to the average elasticities reported in table 1, figure 1 plots the country-specific bias in the exchange rate elasticity as a function of the exchange rate shock. The magnitude of the bias increases in the size of the exchange rate shock in all specifications. We also observe that in the case of the "McGuirk" (sub-figure b) and the "log-approximation" (sub-figure c) that the direction of the bias is the opposite of the exchange rate shock (a depreciation leads to a downward bias while an appreciation leads to an upward bias). The other two scenarios show the importance of using the adequate functional form. Omitting the variation in the weighted average of the competitor's price index (scenario "Gold medal mistake") or having the wrong weighting scheme (scenario "Alternative weights"), leads to larger biases for countries that have more important implications in the international trading system. For example, the United States has the largest bias because it is, on average, the most important trading partner for the countries in our sample and, as a result, an exchange rate shock to the US dollar induces the largest variation in the partner countries' price indexes.

**1.2. General Equilibrium Trade Integration (GETI) counterfactual**

In the GETI counterfactual, we allow wages to adjust following an exchange rate shock. In particular, we impose that wages have to adjust in order to keep the current account imbalance constant in international currency. To engineer an exchange rate shock in the general equilibrium, we use the same exchange rate shocks as in the MTI part. When a country receives an idiosyncratic exchange rate shock, the exchange rate does not change for other countries. However, exports change for all countries because prices and wages adjust after the shock. Table 2 presents the general equilibrium results using a trade elasticity of 4.

The direction and the magnitude of the underlying biases in the exchange rate elasticities are similar to the MTI version. Wrong functional form assumptions and incorrect weighting schemes in the scenarios "Gold medal mistake" and "Alternative weights" bias the exchange rate elasticity towards zero. The key difference is the increase in the estimation error captured by the MSE. Figures 2 and 3 illustrate this change by plotting the bias in the exchange rate and the demand elasticity as a function of the exchange rate shock for the four different scenarios. In the GETI counterfactual an exchange rate shock leads to a much larger estimation bias than in the MTI. The reason why the average biases are small is that they move in the opposite direction of the exchange rate shock (appreciation leads to upward bias and depreciation leads to downward bias) and cancel each other out when averaging.

**1.3. Monte-Carlo with stochastic trade costs**

Up until now, our simulations are based on the assumption that the structural gravity equation is the true data-generating process for bilateral trade flows. To allow for the fact that the gravity equation does not perfectly explain bilateral trade flows, we run a Monte Carlo exercise and assume, as Head and Mayer (2014), that bilateral trade flows change due to a stochastic term qni in the trade cost function:

where Distni- is the distance between importer n and exporter i. qni is a log-normal random term and the only stochastic term in the simulation since GDP and distance are all set by actual data. We calibrate the variance of \n(qn{) to replicate the root mean squared error of the least squares dummy variables (LSDV) regression on real data. We use the method of Dekle et al. (2007) and solve the model in changes. Based on actual data in trade flows and incomes, we first simulate changes in the trade costs with a random shock qni, i.e. Tni = qni, and then calculate the changes in wages, the prices for intermediate goods and the aggregate price indexes resulting from the random exchange rate shock.

Table 3 reports the results. Introducing stochastic trade costs increases the bias and the estimation error. The MSE significantly increases for the point estimates of the exchange rate and the

demand elasticities. Interestingly, the bias in the demand elasticity becomes more pronounced than in the exchange rate elasticity. Across specifications, the average magnitude of the bias is close to 20 percent for the demand elasticity, whereas it is close to 1 percent for the exchange rate elasticity. The specification with the lowest MSE is the log approximation in column (4), suggesting that deflating the changes in foreign demand by changes in the aggregate price index is important to reduce the bias in the demand elasticity. These simulation results suggest that with the ideal-REER approach it is possible to estimate the response of aggregate exports to exchange rate changes without bias even if the data-generating process does not fully adhere to structural gravity.

We derived our ideal-REER regression specification under two important assumptions. First, markups do not vary with the exchange rate. Second, prices are set in an international currency. In this section, we relax these assumptions and discuss alternative estimation methods based on different pricing assumptions.

One way to change the exporter's price response to exchange rates is to follow Anderson et al. (2016) and assume an exogenous pass-through coefficient Qjn. The pass-through coefficient Q{n captures the exporter's price adjustment following a variation in the exchange rate, for example, due to changes in the markup or price rigidities in the invoicing currency. The coefficient is defined on the unit interval 0 < Q{n < 1. If the pass-through is complete, the exporter will pass on the entire bilateral exchange rate variation to the importer and Qin = 0. If the export price is fixed in the destination currency and the exporter absorbs all of the exchange rate variation by changing the mark-up, the pass-through coefficient equals one, Qin = 1. Given this definition, we can write the change in the export price denominated in importer n's currency as follows:

However, if the pass-through coefficient varies at the bilateral level, we will not be able to identify all parameters at the aggregate level:

because the number of parameters to estimate is larger than the number of export equations.

Instead, we rely on the dominant currency paradigm (DCP, see Casas et al. (2017)) and assume that exporter and importer prices are fixed in the dominant (international) currency. In this case, the pass-through coefficient depends only on the exchange rate changes vis-^-vis the international currency (Qin = pipn). As a result, we can write the bilateral exchange rate as a product of both the

exporter exchange rate and the importer exchange rate to the international currency and neglect the co-variance between the two. Reducing the dimensionality further by assuming a common passthrough coefficient, we can write the DCP export equation as follows

where the estimated exchange rate elasticity represents the product of the trade elasticity and the exchange rate pass-through coefficient (p), i.e., fiREER = в(1 — p), for both the exporter-specific as well as the importer-specific component. Introducing incomplete pass-through reduces the effect of the exchange rate but leaves the overall functional form unchanged compared with the baseline specification in equation 7. The lower exchange rate elasticity implies a reduction in the magnitude of the estimation biases in standard REER specifications discussed above. Both the qualitative results and the country-specific biases plotted in figures 1 to 3 remain the same.

An alternative pricing assumption is that exporters set their prices in the producer's currency (Producer Currency Pricing, or PCP) and absorb part of the changes in marginal costs by the common pass-through coefficient p. In this case, we cannot split the bilateral exchange rate into an exporter/importer component because the importer-specific component depends on the exporter's price reaction following an exchange rate shock. The resulting exporter equation is a function of the change in producer prices in the exporter's currency and a weighted average of foreign demand deflated by the price index converted into the exporter's currency:

The estimated exchange rate elasticity is the same for the exporter- and importer-specific components only if exchange rate pass-through is complete, i.e., q = 0. If exchange rate pass-through is incomplete (q > 0), the importer-specific component depends on the exporter's price reaction following an exchange rate shock and the estimated exchange rate elasticity fiREER will be biased.

Finally, the third common pricing assumption is that exporters set their prices in the destination's currency (Local Currency Pricing, or LCP). Under LCP, exporters absorb the bilateral exchange rate variation and the pass-through on import prices is zero. In this case, the exchange rate shock does not affect prices, demand and exports. As a result, there is no counterfactual variation.

Overall, consistent aggregation under incomplete pass-through is only possible when prices are set in an international currency as postulated by the dominant currency paradigm. If prices are set in the producer's currency (producer currency pricing (PCP)) or in the local currency (local currency pricing (LCP)), the pass-through depends on the bilateral exchange rate and estimation of the aggregate elasticities without bias is only possible if pass-through is complete. Next, we test whether the ideal-REER approach also reduces the estimation bias when aggregating bilateral data empirically.

A key implication from the theoretical analysis is that running exchange rate regressions on the bilateral or on the aggregate level does not lead to a significant bias in the elasticities if the trade elasticity is the same for all countries. While recognizing that in practice exchange rate elasticities differ across countries (see, for example, Spilimbergo and Vamvakidis (2003) or Bussiere et al. (2016)), we test the theoretical prediction of no-significant bias in exchange rate regressions when pooling across countries. Later we relax this assumption and test for significant differences between elasticities estimated on the bilateral and aggregate levels on a country-per-country basis.

Our empirical approach uses data for bilateral exchange rates from the IMF financial statistics database. Data on Real Effective Exchange Rates are coming from the Bank for International Settlements. We use the Narrow Index as it has the advantage of covering a longer sample period. The Narrow Index is a trade-weighted effective exchange rate over 25 economies (excluding the euro area). Inflation (proxied by the GDP deflator) and nominal GDP data are from national accounts and converted into international currency (USD) using the bilateral exchange rate with the USD. Bilateral trade data are retrieved from CEPII's historical trade database (see Fouquin and Hugot (2016)). We use absorption, defined as GDP minus net exports, to be a proxy for foreign demand. For changes in aggregate export prices, we use producer price indexes (PPI) from national accounts, the OECD and the IMF depending on data availability. Taken together, our sample comprises a panel of 25 countries with yearly observations from 1964 to 2014.

Given that the ideal-REER estimation equation depends on the currency denomination of trade flows, the first step is to run exchange rate pass-through regressions and shed light on the pricing behavior of exporters. To estimate the reaction of export prices to exchange rate changes, we follow Casas et al. (2017) and regress the export price of country i to country j (denominated in country i's currency,PiEj t) on country i's exchange rate with the USD (riiusD,t), the bilateral exchange rate between country i and country j (rj) and the producer price index of country i (PPIi,t), a proxy for changes in marginal costs:

The results in Table 4 show that the export price varies mainly with the USD exchange rate. The estimated coefficient of the USD exchange rate is -0.15 compared with a bilateral exchange rate coefficient of -0.05 (see column (4)).

We repeat the exercise for the import side and estimate the reaction of the import price denominated in importer's currency to exchange rate changes with the following specification:

where Plj t denotes the import price of country i from country j (denominated in country j's currency). Similar to the export price, the variation of the import price is predominantly driven by the USD exchange rate. Table 5 column (4) shows that the pass-through coefficient for the USD import price is -0.89 while the coefficient of the bilateral exchange rate is -0.21. These findings are similar to Casas et al. (2017) and support the dominant currency paradigm.

simulation section. We convert all variables into US dollars and use the following definitions for the real exchange rate and foreign demand. The real exchange rate of exporter i (RERit) is defined as the change in the nominal exchange rate of country i visA-vis the USD multiplied by the change in the PPI of country i. To calculate foreign demand, we use the current value of absorption in the importing country j denominated in the international currency (Xj). Inflation proxies for changes in the importer's price index (Pj). For the weights wnit, we use the share of exports of country i going to country n in year t. This definition of the variables applies to all aggregate regressions with the exception of the REER in the "alternative weights" specification defined in equation 17. In this case, we use the BIS Narrow index as the REER's empirical counterpart.

The results are shown in table 6. The first observation is that the elasticities estimated from bilateral trade flows and from aggregate trade flows using the ideal-REER approach are not significantly different from each other. The point estimate of the exchange rate elasticity using bilateral flows is -0.46 (column (1)) and the estimate using the ideal-REER is -0.41 (column (2)). The demand elasticities are almost identical with the point estimates of 1.18 and 1.19, respectively. For all other specifications (columns (3) to (6)), the aggregation method results in significant differences in the estimated elasticities compared with the bilateral level in column (1). The specification with the largest gap is the "Gold medal (GM) mistake" approach in column (3) with a point estimate of -0.38 for the exchange rate and 0.76 for the demand elasticity. The standard REER approach based on McGuirk (column (4)) fits the data best with the highest R2 but produces a higher exchange rate (-0.58) and a lower demand elasticity (1.09). The specification with the lowest R2 is the "Alternative weights" approach in column (6). Overall, these results highlight the trade-off one faces when

deciding which elasticities to use. If forecasting is the main objective, then the model with the best fit may be the preferred one. If the aim is to use the elasticities of the standard REER approach to calibrate a macro-model, which relies on aggregate export equations to summarize trade linkages between the different regions in the model, then the counterfactual predictions are inconsistent with the "micro-level" foundations implied by structural gravity. In this case, the estimates obtained from the ideal-REER regression may be the preferred ones.

The pooled estimates in table 6 hide potential heterogeneity in the exchange rate elasticities across countries. Spilimbergo and Vamvakidis (2003) (and many others later) documented this crosscountry heterogeneity. We investigate this point further by running the bilateral and the aggregate real exchange rate regressions on a country-per-country basis. We obtain country-specific exchange rate and demand elasticities to test whether they differ in a significant way.

Figure 4 plots the relationship between the elasticities obtained from bilateral and aggregate regressions based on the ideal-REER approach (panel a) as well as the standard REER approach a la

McGuirk (panel b). All four panels exhibit significant variation in elasticities across countries and these results are smaller for elasticities estimated on the bilateral than on the aggregate level. One possible explanation is the limited number of observations. The aggregate country-specific regressions consist of a time-series analysis with 50 years of observations. In contrast, the bilateral variation consists of up to 1200 observations (50 years multiplied by the number of trading partners). In the absence of an aggregation bias, we would expect that the difference in the elasticities does not move in a particular direction, i.e., the difference between the elasticities should be more or less proportional to the size of the country-specific elasticities. In order words, the best linear fit between the two sets of elasticities should have a slope close to 1. For this reason, we include the best linear fit (in blue) with the 95 percent confidence interval as well as the 45 degree line (in red) into figure 4. Note that in all cases the estimated slope is lower than 1, which confirms the initial observation that the cross-country variation in the elasticities estimated on the bilateral level is smaller than those obtained from the aggregate level.

With respect to a potential aggregation bias, we expect that the ideal-REER approach will result in a smaller differences between the elasticities than the standard REER approach. Panels (a) and (b) in figure 4 show that this is indeed the case: the absolute difference between the point estimates of the elasticities obtained from bilateral trade flows and the ones obtained from aggregate trade flows is smaller for the ideal-REER approach compared with the standard REER approach. Moreover, the difference is less biased in a particular direction, i.e., the best linear fit (in blue) of the ideal-REER estimates is closer to the 45 degree line (in red) than the best linear fit of the standard REER estimates. Taken together, the empirical results support our theoretical findings, namely that the difference in the elasticities is related to the functional form assumption in the aggregation of bilateral trade flows. The ideal-REER approach aggregates the bilateral variation in accordance with the gravity model and reduces bias from the aggregation.

Before concluding, we consider an alternative pricing paradigm and estimate our empirical specifications in accordance with the PCP shown in equation 25. Table 7 shows the estimated coefficients. The main difference with respect to the evidence based on DCP is that denominating the variables in the producer's currency worsens the estimation performance. Compared with table 6, the R2 and the point estimates of the elasticities are lower while the standard errors of the elasticities are higher in all specifications of table 7. With respect to the differences in elasticities across different levels of aggregation, the ideal-REER elasticities are closest to the ones estimated on the bilateral level. All other specifications lead to significant differences in elasticities with respect to the bilateral level. Overall, these results suggest that currency denomination matters for the estimation performance and the quantitative results but not for the qualitative results on the aggregation bias.

One reason why country-specific bilateral exchange rate and demand elasticities differ from aggregate elasticities is incorrect aggregation. The standard approach taken in the literature is based on functional form assumptions and an incorrect weighting scheme. A consequence of these assumptions is that both the trade (exchange rate) and the demand elasticity are biased. However, our simulations and the empirical evidence show that these biases are quantitatively small but statistically significant.

The fact that many macroeconomics models are calibrated with elasticities based on standard REER regressions has important policy implications. In particular, models calibrated with the elasticities estimated from aggregate data lead to inaccurate predictions by exaggerating the response of exports and, by extension, output following an exchange rate shock. Calibrating these models with elasticity estimates using our new ideal-REER regression specification with variables denominated in the dominant international currency (i.e. US dollar) improve the model's fit and results in predictions consistent with microeconomic behavior in bilateral trade equations.

We would like to thank Joshua Aizenman, Enrique Alberola, Raphael Auer, Ariel Burstein, Guillaume Gaulier, Philippe Martin and Fabrizio Zampolli for their useful comments and suggestions. Special thanks to our discussants Fernando Perez-Cervantes and Mike Waugh. This research has received funding from the European Research Council (ERC) under the Grant Agreement No. 313522. The views expressed in this paper are those of the authors and do not necessarily reflect those of the Bank of Canada.

Ahmed, S., M. Appendino, and M. Ruta (2016): "Global value chains and the exchange rate elasticity of exports," The BE Journal of Macroeconomics, 17.

Alvarez, F. and R. J. Lucas (2007): "General equilibrium analysis of the Eaton Kortum model of international trade," Journal of Monetary Economics, 54(6), 1726-1768.

Anderson, J. E. and E. van Wincoop (2003): "Gravity with Gravitas: A Solution to the Border Puzzle," American Economic Review, 93.

Anderson, J. E., M. Vesselovsky, and Y. V. Yotov (2016): "Gravity with scale effects," Journal of International Economics, 100,174-193.

Arkolakis, C., A. COSTINOT, and A. RODRIGUEZ-CLARE (2012): "New trade models, same old gains?" American Economic Review, 102,94-130.

Armington, P. S. (1969): "A Theory of Demand for Products Distinguished by Place of Production," IMF Staff Papers, 159-178.

Artus, J. R. and A. K. McGuirk (1981): "A Revised Version of the Multilateral Exchange Rate Model," IMF Staff Papers, 28, 275-309.

Barnett, R., K. B. Charbonneau, and G. Poulin-Bellisle (2016): "A New Measure of the Canadian Effective Exchange Rate," Discussion Papers 16-1, Bank of Canada.

BAYOUMI, T., J. Lee, and S. JAYANTHI (2005): "New rates from new weights," IMF Working Paper.

BAYOUMI, T., M. Saito, and J. TURUNEN (2013): "Measuring Competitiveness: Trade in Goods or Tasks?" IMF Working Papers.

Bems, R. AND R. C. JOHNSON (2017): "Demand for value added and value-added exchange rates," American Economic Journal: Macroeconomics, 9,45-90.

BENNETT, H. Z. and Z. ZARNIC (2009): "International Competitiveness of the Mediterranean Quartet: A Heterogeneous-Product Approach," IMF Staff Papers, 56,919-957.

BOZ, E., G. GOPINATH, and M. PLAGBORG-M0LLER (2017): "Global trade and the dollar," Tech. rep., National Bureau of Economic Research.

BUSSIERE, M., G. Callegari, F. Ghironi, G. Sestieri, and N. Yamano (2014): "Estimating trade elasticities: Demand composition and the trade collapse of 2008-09," American Economic Journal: Macroeconomics, 5,118-51.

BUSSIERE, M., S. Delle Chiaie, AND T. A. PELTONEN (2014): "Exchange Rate Pass-Through in the Global Economy: The Role of Emerging Market Economies," IMF Economic Review, 62,146-178.

BUSSIERE, M., G. Gaulier, AND W. STEINGRESS (2016): "Global Trade Flows: Revisiting the Exchange Rate Elasticities," Working papers 608, Banque de France.

CAMPA, J. M. AND L. S. Goldberg (2005): "Exchange rate pass-through into import prices," The Review of Economics and Statistics, 87, 679-690.

Casas, C., F. Diez, G. GOPINATH, and P.-O. GOURINCHAS (2017): "Dominant Currency Paradigm: A New Model for Small Open Economies," IMF Working Papers 17/264, International Monetary Fund.

Chinn, M. D. (2006): "A primer on real effective exchange rates: determinants, overvaluation, trade flows and competitive devaluation," Open Economies Review, 17,115-143.

COSTINOT, A. and A. RODRIGUEZ-CLARE (2014): "Trade theory with numbers: Quantifying the consequences of globalization," in Handbook of International Economics, Elsevier, vol. 4,197-261.

Dekle, R., J. EATON, AND S. KORTUM (2007): "Unbalanced Trade," American Economic Review, 97, 351-355.

Di Mauro, F. and M. H. Pesaran (2013): The GVAR handbook: Structure and applications of a macro model of the global economy for policy analysis, Oxford University Press.

EATON, J. AND S. KORTUM (2002): "Technology, Geography, and Trade," Econometrica, 70(5), 17411779.

FOUQUIN, M. and J. HUGOT (2016): "Two Centuries of Bilateral Trade and Gravity Data: 18272014," Working Papers 2016-14, CEPII.

Freund, C. and M. D. Pierola (2012): "Export surges," Journal of Development Economics, 97,387395.

GOLDSTEIN, M., M. S. KHAN, ET AL. (1985): "Income and price effects in foreign trade," Handbook of International Economics, 2,1041-1105.

HEAD, K. and T. MAYER (2014): "Gravity Equations: Workhorse, Toolkit, and Cookbook," Handbook of International Economics, 4.

HOUTHAKKER, H. S. and S. P. MAGEE (1969): "Income and price elasticities in world trade," The Review of Economics and Statistics, 111-125.

Imbs, J. M. and I. Mejean (2015): "Elasticity optimism," American Economic Journal: Macroeconomics, 5, 43-83.

JAKAB, Z., P. LUKYANTSAU, AND S. WANG (2015): "A Global Projection Model for Euro Area Large Economies," IMF Working Papers.

KLAU, M. AND S. S. FUNG (2006): "The new BIS effective exchange rate indices," BIS Quarterly Review, March.

LANE, P. R. and G. M. Milesi-Ferretti (2012): "External adjustment and the global crisis," Journal of International Economics, 88,252-265.

McGuirk, A. (1986): "Measuring price competitiveness for industrial country trade in manufactures," IMF Working Paper, 87.

OBSTFELD, M. AND K. S. ROGOFF (2005): "Global current account imbalances and exchange rate adjustments," Brookings Papers on Economic Activity, 2005, 67-123.

ORCUTT, G. H. (1950): "Measurement of price elasticities in international trade," The Review of Economics and Statistics, 117-132.

Patel, N., Z. Wang, and S.-J. Wei (2017): "Global value chains and effective exchange rates at the country-sector level," BIS Working Papers 637, Bank for International Settlements.

Schmitz, M., M. De Clercq, M. Fidora, B. Lauro, and C. Pinheiro (2011): "Revisting the Effective Exchange Rates of the Euro," ECB Occasional paper.

SIMONOVSKA, I. AND M. E. WAUGH (2014): "The elasticity of trade: Estimates and evidence," Journal of International Economics, 92,34-50.

SPILIMBERGO, A. AND A. VAMVAKIDIS (2003): "Real effective exchange rate and the constant elasticity of substitution assumption," Journal ofInternational Economics, 60, 337-354.

VIGFUSSON, R. J., N. SHEETS, AND J. GAGNON (2009): "Exchange Rate Passthrough to Export Prices: Assessing Cross-Country Evidence," Review of International Economics, 17,17-33