Estimating the effect of exchange rate changes on total exports *

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1 Estmatng the effect of exchange rate changes on total exports * Therry Mayer, Walter Stengress PLEASE DO NOT CITE - PRELIMINARY Abstract Ths paper re-evaluates the theoretcal foundatons of Real Effectve Exchange Rate REER regressons, the standard approach to quantfy the response of aggregate exports to exchange rate changes. We frst derve the theory-consstent aggregaton of blateral trade flows mcro-founded by the gravty equaton. We then compare ths new deal-reer wth the typcal regressons found n the real-reer lterature. The man dfferences come from functonal form approxmatons, ms-specfcaton due to omtted varables and naccurate assumptons about the weghtng scheme. Our smulatons show that the magntude of the resultng bases n elastctes s small for the exchange rate and large for demand the two man varables that should be ncluded n those regressons. Usng annual blateral trade flow data, we fnd emprcal support for ths predcton. The dfference between the aggregate and the blateral elastctes reduces sgnfcantly when applyng an deal-reer regresson rather than the standard real-reer approach. JEL classfcatons: F11, F12, F31, F32 Keywords: nternatonal trade, trade elastcty, real effectve exchange rate, gravty equaton *We would lke to thank Raphael Auer, Arel Bursten, Gullaume Gauler, Phlppe Martn and Fernando Pérez- Cervantes for ther useful comments and suggestons. Ths research has receved fundng from the European Research Councl ERC under the Grant Agreement No The vews expressed n ths paper are those of the authors and do not necessarly reflect those of the Banque de France, the Eurosystem or the Bank of Canada. Scences Po, Banque de France; emal: therry.mayer@scencespo.fr Bank of Canada - Banque du Canada; emal: wstengress@bank-banque-canada.ca. 1

2 1 Introducton Regressons explanng total exports wth Real Effectve Exchange Rate REER are ubqutous n appled polcy work, 1 but are only loosely based on trade theory. Most mportant, those open-economy macro-level regressons rely on monadc export equatons,.e. they use changes n total exports of a country as the dependent varable, wth the REER as the man varable of nterest n addton wth a control for change n demand wth whch ths country s faced. Whle there has been several dmensons of debates and progress n the econometrc aspects of such work, n partcular regardng the choce of approprate measurement of the REER, the use of new tme-seres estmators or accountng for heterogenety n the estmated elastctes, 2 the fundamental specfcaton has been vrtually unchanged over the last decades. Most of those macro papers cte as a theoretcal motvaton the work of McGurk 1986, tself based on Armngton 1969, whch can be summarzed as a model where a representatve consumer exhbts CES demand over varetes that are dfferentated by country of orgn. Durng the same perod, consderable progress has been made n modelng the mcro-level foundatons of dyadc blateral trade equatons, called gravty equatons. A very mportant paper n that lterature s the one by Anderson and Van Wncoop 2003, whch turns out to be also based on Armngton The subsequent work n ths feld has however shown that the same estmatng equaton can be obtaned from very dfferent theoretcal setups, rangng from the Rcardan model to models featurng heterogeneous frms wth monopolstc competton see Head and Mayer 2014 for a summary of those foundatons. In the gravty lterature, very lttle room has been gven to estmatng the mpact of the exchange rate changes see Anderson et al for a notable excepton, and most of the nterest has been geared towards the measurement of the mpact of blateral trade costs, and n partcular on other polcy-relevant varables, such as the mpact of Regonal Trade Agreements RTAs, tarffs and membershp of WTO to name just a few. A common pont to both strands of the lterature has been the nterest for the trade elastcty, that s the reacton of trade flows to changes n prce shfters: exchange rate n one case, tarffs or tarff equvalent of other protectonst measures n the other case. In ths paper, we re-evaluate the REER macro lterature by assessng whether the standard approach of monadc macro regresson s compatble wth the mcro-founded gravty equaton that predcts a very precse functonal form for an estmaton explanng blateral flows. Smlar to Imbs and Méjean 2015, our paper s methodologcal n sprt, n that t seeks to derve the approprate monadc aggregaton of blateral flows, and compare our new deal-reer wth the typcal regresson 1 Global projecton models of central banks and nternatonal polcy nsttutons for example, Jakab et al. 2015, the lterature on global VAR models D Mauro and Pesaran 2013 as well as on global mbalances see, amongst others, Obstfeld and Rogoff 2005 andlane and Mles-Ferrett 2012 rely heavly on REER equatons and ther elastctes to summerze the trade lnkages between the dfferent regons n ther model. 2 see Chnn 2006 for a prmer on alternatve measures of REER. More recently, Bennett and Zarnc 2009 ntroduce product heterogenety and servces exports. Bayoum et al and Bems and Johnson 2017 contruct REER ndexes based on value added trade to account for Global Value Chans. Patel et al buld on Bayoum et al and Bems and Johnson 2017 to study how nput lnkages affect REER measurement n a mult-sector economy. 2

3 from ths real-reer lterature. Dfferences comes from the fact that the deal-reer regresson s based on proportonal changes often referred to as exact hat algebra, see Arkolaks et al. 2012, whle McGurk 1986 s REER s based on partal dervatves log changes and consttutes an approxmaton, whch holds for small changes. The key mplcaton of McGurk 1986 s approach s that t allows for the REER regresson, where total exports are a log lnear functon of log changes of the real effectve exchanges rate and foregn demand. On the hand, the aggregate export equaton followng the structural gravty approach mply a log lnear functon of log changes n the real exchange rate and changes n foregn demand deflated by changes n the destnaton specfc prce ndexes. To shed lght on the mportance of these dfferences, we resort to smulatons of the theoretcal model and quantfy how departures from the model n ts strct form generate bases n estmated trade elastctes. To that purpose, we use the Dekle et al verson of the structural gravty model compatble wth most of the commonly used theoretcal models of nternatonal trade. Monte Carlo smulatons show that whle there s dscordance between the deal-reer and the real- REER regressons, the magntude of the bases n elastctes wll depend on the precse regresson specfcaton as well as the underlyng assumptons on the exchange rates shock. Overall, the bas n the exchange rate elastcty s rather small close to 1 percent, whle the bas n the demand elastcty s more pronounced close to 10 percent and ncreases wth measurement errors. The second part of the paper compares the emprcal performance of the deal-reer and real-reer approach n the data. We use annual blateral trade data for the perod 1964 to 2011 and estmate the exchange rate and demand elastcty for each regresson specfcatons. The evdence based on the pooled sample across all countres as well as country-specfc regressons confrms the smulaton results and shows a sgnfcant reducton n the dfferences between the blateral and the aggregate elastctes when applyng the deal-reer rather than the real-reer approach. Our paper contrbutes to the macroeconomc lterature on exchange rate ndexes as a measure of prce compettveness. The REER s an mportant statstcal ndcator, produced by nternatonal agences such as the IMF see Bayoum et al. 2005, the BIS see Klau and Fung 2006, many central banks see Schmtz et al for the ECB and Barnett et al for the Bank of Canada reference and wdely used n applcatons. We see our analyss as complementary to the recent papers that study the theoretcal foundatons for these ndexes. Whle ther focus les more n ntroducng global lnkages nto the measurement of compettveness, see Bayoum et al. 2013, Patel et al and Bems and Johnson 2017, we quantfy bases that arse from functonal form assumptons n the aggregaton of blateral trade flows common n these papers and propose an alternatve estmaton equaton. Our analyss also speaks to the old macroeconomc lterature on estmatng trade elastctes at the macro-level started by Orcutt 1950, Houthakker and Magee 1969 and Goldsten et al and, more recently, Splmbergo and Vamvakds 2003, Freund and Perola 2012, Bussere et al and Ahmed et al All of these papers rely on effectve exchange rates to estmate demand and exchange rate elastctes. Our results warrant cauton n the use of these estmates for 3

4 counterfactual polces, partcularly n the case of the demand elastcty, where the bas can be sgnfcant. As an alternatve, we suggest our deal-reer regresson framework, whch can mtgate these concerns. These conclusons also apply to studes that use aggregate tme seres data to estmate exchange rate pass-through coeffcents such as Campa and Goldberg 2005, Vgfusson et al and Bussère et al The remander of the paper s structured as follows. In secton 2 we derve the gravty-compatble aggregate export equaton n the partal as well as n the general equlbrum. Secton 3 descrbes the standard macro approach of McGurk 1986 and dscusses the functonal form dfferences wth respect to the gravty approach. Secton 4 presents the dfferent emprcal approaches used n the appled lterature together wth the smulaton procedure used to quantfy potental bases. Secton 5 presents the smulaton results. Secton 6 ntroduces the data and provdes emprcal evdence on the key predctons of our theoretcal framework. Secton 7 concludes. 2 The Gravty approach We start our exposton wth the structural gravty equaton based on Head and Mayer 2014 derved from the Armngton model. The central aspect of the Armngton model s that t s not only at the heart of the gravty equaton lterature but also forms the bass of the Real Exchange Rate regressons as n Artus and McGurk 1981, McGurk 1986 and Splmbergo and Vamvakds Structural gravty Consder a world economy consstng of N countres. Each country s endowed wth Q n unts of a dstnct good. In each country, the representatve agent has Constant Elastcty of Substtuton CES preferences. Trade between countres s costly and takes the form of ceberg trade costs. In order to sell one unt of good n country n, exporters of country have to shp τ n 1 unts. Trade wthn the country s cost-less,.e. τ = 1. To prevent arbtrage opportuntes, we assume that tradng blaterally between and n s always cheaper than tradng va an ntermedate country k τ nk τ k τ n for, j, k. Note that gven country s endowment Q and total ncome Y, we can express supply S = Y /Q, the correspondng producer prce of good as P = S and the prce charged to consumers n country n as P n = τ n S. Gven our assumptons, we can wrte the gravty equaton as the product of the share of expendture that mporter n allocates on goods from exporter, π n, and the overall expendture, X n. Followng Head and Mayer 2014, π n can be expressed n the followng multplcatve separable form: π n = S φ n Φ n where Φ n = N S l φ nl and Ω = l=1 N φ l X l 1 Φ l=1 l where φ n captures the blateral accessblty of n to exporter and s a functon of ceberg trade costs. 4

5 Φ n represents the multlateral resstance term and Ω represents the weghted sum of the exporter capabltes. In order to quantfy potental estmaton bases as well as to produce counterfactual smulaton estmates of the structural gravty model, we need to specfy the underlyng supply structure. In ths paper, we opt for the Rcardan Comparatve Advantage specfcaton wth ntermedate nputs from Dekle et al based on Eaton and Kortum 2002 and Alvarez and Lucas In ths model, each country produces a large number of tradable ntermedate goods that are homogeneous across countres. Productvty z dffers across goods and s assumed to be dstrbuted Fréchet wth a cumulatve dstrbuton functon of exp T z θ, where T s a technology parameter and θ determnes the amount of heterogenety n the productvty dstrbuton. For each ntermedate good z, buyers choose the lowest cost suppler n the world. Aggregatng across all goods and assumng that factor prces are denomnated n local currency as well as complete pass-through, the expendture share that buyers n n spend on goods from can be wrtten as: π n = T r w β θ p1 β τ n r n P n θ where P θ n = N θ rl T l w β r l p1 β l τ nl. 2 l=1 n where w s the wage pad to workers n, p s the prce of tradable ntermedate goods, r s the nomnal exchange rate that converts local prces nto a common nternatonal currency USD for nstance and β s the share of ntermedate goods n the producton of manufacturng goods. Comparng equaton 2 to equaton 1 shows that the three structural gravty terms are gven by θ, S = T r w β p1 β φn = τ θ n wth θ as the trade elastcty and the multlateral resstance term n nternatonal currency Φ n = r n P n θ. At ths pont t s mportant to note that whle we chose the Eaton-Kortum varant of the gravty model as the underlyng theoretcal foundaton, all other theoretcal models that adhere to structural gravty are compatble wth the followng analyss and produce the same elastcty of exports wth respect to the real and nomnal exchange rate. The man dfference wll be underlyng mplcatons on the mcro-structure and the correspondng nterpretaton. For example, n other gravty models wthout ntermedate goods, the wage elastcty s 1, whle n ths model t s smaller snce β < Counterfactual Analyss To produce counterfactual analyss, we follow Dekle et al and use the exact hat notaton, see Arkolaks et al. 2012, to derve the aggregate export equaton. More precsely, hats denote the rato of post-shock to pre-shock values. We start by assessng the change n blateral exports and then aggregate up to obtan the change for total exports. For blateral exports of country to country n n nternatonal currency we have: X n X n X n, 5

6 where X n s blateral trade after the change and X n the value before any change. The resultng change n total exports after the polcy change s smply the sum over all mport countres wth the excepton of country s home market. Furthermore, we can decompose the changes n total exports Ê n nternatonal currency as the sum of changes n the expendture shares and the aggregate expendture n nternatonal currency across all mportng countres weghted by the ntal share of exports gong to country n ω n = X n /E : Ê = 1 E π n X n = ω n π n X n, 3 Ths equaton s extremely general and smply states that the change n total exports of a country s the weghted sum of all changes n blateral exports, weghts beng ntal share of each blateral flow n total exports. The change n each blateral flow can have two orgns: an ncrease n total expendture n destnaton n, a change n compettveness of country relatve to all other tradng partners of n. 2.3 Modular Trade Impact MTI verson As a frst polcy shock, we consder a change n the nomnal exchange rate of country, r, whch s ndependent to the exchange rate n all other countres. The shock to r acts lke a shfter of delvered prces to the consumer whch assumes complete pass-through at ths stage. 3 The envsoned change n the blateral trade shares therefore wrtes, assumng for the moment that local wages and therefore GDPs are unaffected by the shock, π n = ˆr ˆp 1 β θ ˆrn ˆP n θ where the change n the prce ndex n country n s equal to the change n country s exchange rate tmes the weght of country n country n s mport basket, P n θ N θ ˆrk = π nk ˆp 1 β ˆr k. 4 k=1 n As a result, we can rewrte the change n exports n nternatonal currency as: Ê = ω n π n X n = ω n ˆr ˆp 1 β θ ˆrn ˆP n θ X n or, takng logs, we get the log change n exports as a functon of the log change n the nomnal exchange rate and the foregn demand: 3 In ths model the exchange rate shock s a nomnal shock that changes the unt of account of prces. 6

7 ln Ê = θ ln ˆr ˆP + ln X n ω n θ. 5 ˆrn ˆP n where the change n the export prce of country n n local currency changes because of ntermedate nput prces,.e. ˆP = ˆp 1 β. Equaton 4 descrbes how prces change after an exchange rate shock. Consder an apprecaton of the exchange rate n country n ˆr n. Ths leads to an ncrease n the export prce of country n and, by consequence, to an ncrease n the ntermedate goods prces of all other countres that mport from country n. At the same tme, the apprecaton reduces unt costs of producton through lower ntermedate mport prces and dampens the export prce response. Gven that n the MTI counterfactual wages do not adjust, the pass-through nto export prces s gven by the change n the exchange rate mnus the change n the unt cost of producton mplyng ncomplete pass-through. Note that the co-varance between the two varables s negatve,.e. an exchange rate apprecaton leads to a reducton n the unt costs. 2.4 General Equlbrum Trade Impact GETI verson The analyss thus far has kept the level of wages and GDPs unchanged. Followng the analyss by Arkolaks et al. 2012, we use a second equaton to take those general equlbrum effects nto account. Ths equaton mposes market clearng such that total output, Y, corresponds to total spendng, X, mnus the changes n the trade defct. Wth labor as the sole factor, the value of producton n local currency n the country of orgn s gven by Y = w L. Followng Dekle et al and treatng fnal demand as an aggregate of manufactures and non-manufactures produced n the same proportons, the market clearng condtons n nternatonal currency become: r w L + D = π n r n w n L n + D n. n where D s the overall trade defct n local currency. 4 Wthout mgraton or other sources of changes n populatons, we have changes n output gven by changes n the nomnal exchange rate and wages, ˆr ŵ = Ŷ. Assume that the overall defct s exogenously gven, changes n π resultng from a change n the nomnal exchange rate take the followng form ˆπ n = π n l π nl ˆr ŵ β ˆp 1 β θ ˆr l ŵ β l ˆp 1 β l where we explctly ndcate that changes n supply depend on the changes n wages,.e. Ŝ ŵ = 4 For exposton purposes we do not present the dervatons of the market clearng condton and refer to Dekle et al for the detals. θ 6 7

8 ˆr ŵ β ˆp 1 β. Pluggng changes n trade shares back nto the market clearng condton, one can solve for the changes n producton of each country of orgn. The resultng changes n equlbrum wages pad n local currency are: ˆr ŵ Y + D = n π n l π nl ˆr ŵ β ˆp 1 β θ ˆr l ŵ β l ˆp 1 β l θ ˆr nŵ n Y n + D n. 7 Note that n ths model a nomnal exchange rate shock wll affect relatve prces and lead to changes n trade flows. The reason s that we fx the trade defct n nternatonal currency. Gven ths assumpton, the wage adjustment wll not be one-to-one wth exchange rate change and depend on the dollar value of the current account defct. 5 As such, ths scenaro wll be nstructve to the understandng of potental aggregaton bases when constructng an effectve exchange rate n general equlbrum. 6 To obtan the change n aggregate exports n the general equlbrum verson, we can smply plug equaton 6 nto equaton 3 and wrte: Ê = ω n π n l π nl ˆr ŵ β ˆp 1 β θ ˆr l ŵ β l ˆp 1 β l X n θ where X n = ˆr n ŵ n Y n + D n represents changes n absorpton. Usng the fact that Pn θ = l π ˆrl nl ˆr n ŵ β l ˆp 1 β θ l and takng logs, we get the log change n exports as a functon of the log change n the real exchange rate and foregn demand: ln Ê = θ ln RER + ln ω n X n θ 8 ˆr n P n where the real exchange rate RER s defned as the change n the nomnal exchange rate and the change n the producer prce n the exportng country n local currency: 5 Alternatvely, f one assumes that the rato of the currenct account defct to GDP s constant, then changes n the exchange rate wll be one-to-one off-set by changes n the local wage and have no real effects. In ths case the equlbrum condton looks as follow: ˆr ŵ Y + D = n π n l π nl ˆr ŵ β ˆp 1 β θ ˆr l ŵ β l ˆp 1 β l θ ˆr nŵ n Y n + D n. 6 Anderson et al consder an alternatve approach that leads to changes n relatve prces across countres after a nomnal exchange rate shock. They assume an exogenous degree of ncomplete exchange rate pass-through nto mport prces ˆr = ˆr ρ n where ρ n s the exogenous degree of mport pass-through. However, n ths case aggregaton of the blateral flows to an effectve exchange rate wthout bas wll not be possble. The change n the exporter s prce wll depend on the mportng country n and cannot be separated from the demand effect as n equaton 5. 8

9 ln RER = ln ˆr ˆP 9 The change n the producer prce s gven by the changes n the unt costs of producton,.e., changes n ntermedate nput prces and wages, ˆP = ŵ β ˆp 1 β. Overall, equatons 5 and 8 mply that, under the assumpton of a constant elastcty of substtuton and complete pass-through, 7 there s no aggregaton bas n aggregate macro exchange rate regresson. In order to obtan these unbased estmates, one needs to regress the log of the change n nomnal exports on the log of the change n the real exchange rate and the foregn demand deflated by changes n the mporter s prce ndex. However, to our knowledge, there s no paper runnng regressons based on equatons 5 and 8. Instead, the lterature focus on trade weghted real or nomnal effectve exchange rate regressons, whch are blateral exchange rates aggregated to a sngle country-specfc ndcator. Next, we dscuss the theoretcal underpnnngs of effectve exchange rate regressons used n appled work and relate them to the structural gravty framework. 3 The Real Effectve Exchange Rate approach McGurk, 1986 McGurk 1986 s the standard reference that descrbes the methods used by polcy nsttutons to construct a prce compettveness ndcator, or the real effectve exchange rate. These ndcators attempt to measure a country prce changes n the tradable sector relatve to those of other countres after convertng each of them nto a common currency. Insofar that the prce changes n the tradable sector extend to the non-tradable sector, these ndces may also serve as a measure of overall compettveness of the real sector. The purpose of the compettveness ndcator s to measure changes n nternatonal compettveness, n partcular relatve prce changes between any two supplers of tradable goods. These prce changes wll then lead to a shft n demand from the hgher prce suppler to the lower prce suppler. To model the demand for tradable goods, McGurk 1986 reles on the assumpton of mperfect sustanablty between goods and a constant elastcty of substtuton CES, as n Armngton These assumptons are consstent wth structural gravty. Hence, we use the assumptons of DEK and wrte the blateral mport expendture of country n for goods from country, X n, as a functon of total expendture X n and the blateral shares defned n equaton 2. Takng the log and consderng the change from the ntal equlbrum to the new equlbrum ln X n = ln X n ln X n, we get: r P ln X n = θ ln n + ln X n 10 r n P n where ln P n denotes the change n the prce that exportng country charges to consumers n 7 The model also permts aggregaton wth exporter-specfc pass-through ρ common to all mportng destnatons. Smply redefne the exchange rate shock as ˆr = ê ρ. In ths case, ncomplete pass-through reduces the magntude of the exchange rate shock. 9

10 country n and ln P n the change n the prce ndex of the mportng country n both denoted n local currency. 8 Concernng the changes n the mportng country s prce changes, McGurk 1986 expresses the log change of the mporter s prce as a weghted average of the log changes n the producton prces ln r n P n = K π nk ln r k P kk 11 k=1 where π nk represents the expendture shares of country n. Usng further smplfcatons outlned n the appendx, we can substtute the mport prce ndex n equaton 10 wth equaton 11 and aggregate across all mportng countres to obtan the change of total exports of country. To do so, we sum the blateral trade flows over all markets excludng the home market by weghtng the flows by the share of s output sold n each market. r P ln E = θ ω n π nk ln + r k = k P kk ω n ln X n where the weght ω n s defned as the share of export revenues of country that come from sales n destnaton n wth respect to total export sales. Next, we defne the double-weghted Real Effectve Exchange Rate REER as follows: ln REER = and wrte the standard REER regresson as k = r P ω n π nk ln r k P kk 12 ln E = θ ln REER + ω n ln X n. 13 Note that McGurk 1986 devates from the defntons n equaton 12 and 13 n two mportant ways. Frst, she consders total demand for goods from country ncludng the home market,.e. she sums over all markets and ncludes country n the second sum of equaton 12. Second, she assumes that the change n the producer prce s proportonal to the change n the prce ndex,.e. ln P ln P. Ths s not an nnocuous assumpton. The change n producer prce s gven by the change n unt costs of producton ln P = ln w β p1 β n the GETI and ln P = ln p 1 β n the MTI, whereas the change n the prce ndex equals the change n the ntermedate goods prce ln P = ln p. We wll analyze the precse mplcatons of these assumptons n specfcatons 4 and 5 n the smulaton part. 8 Based on the model n secton 2 and the assumpton that trade costs are fx, the change n the foregn prce equals the change n the producton prce ln P n = ln n the MTI and ln P n = ln n the GETI verson p 1 β w β p1 β and the change n the prce ndex s gven by the change n the ntermedate goods prce ln P n = ln p n. 10

11 3.1 Comparng Gravty and McGurk 1986 Let s consder the macro equaton from McGurk, equaton 13, and the one mpled by the gravty equaton 8. The man dfference s that McGurk 1986 reles on a weghted sum of partal dervatves of blateral exports, ln X n, whereas the gravty one s based on the exact hat algebra notaton consdered n the prevous secton,.e. X n. For small changes, these two equatons lead to the same aggregate changes. However, for large shocks ths s not true any more. To see the relatonshp between the two, we need to ntegrate the change of logs of blateral trade flows from the ntal polcy r to the new polcy state r. X n = ˆ r r ln X n dr where X n X n /X n denotes the proportonal change n blateral trade flows between the ntal and counterfactual equlbrum. Ths establshes that for any small change n the nomnal exchange rate from r to r, real exchange rate regressons can be used to estmate the trade elastcty, θ. An mplcaton of ths dfference s that the real exchange rate regresson of McGurk 1986, equaton 13, s smply a log lnear equaton of the real effectve exchange rate and foregn demand. Ths approach s easy to mplement because both varables can be readly calculated usng data on output and nternatonal trade flows. On the other hand, the real exchange rate regresson mpled by the exact hat algebra, equaton 8, s a lnear functon of the real exchange rate RER and foregn demand dvded by the multlateral resstance term. Ths approach s more demandng because the multlateral resstance term s a complex non-lnear term of the deal prce ndex and the trade elastcty. One can ask under whch condtons we can translate the REER of McGurk nto a RER based on the hat notaton and how does t relate to the Macro equaton 8? To start wth, we rewrte the REER n equaton 10 as a weghted average of the blateral real exchange rate drectly wthout substtutng for the mporter s prce ndex usng equaton 11. We then can wrte the REER as the log dfference between the changes n the RER of exporter and the weghted average of the prce changes n all tradng partners of country ln REER = ln P r ω n ln P n r n. 14 Substtutng the mpled REER n equaton 14 back nto the standard REER regresson defned n equaton 13, we can compare t drectly to the RER regresson n equaton 8. The dfference s that the approach of McGurk 1986 allows us to separate the foregn prce ndex from foregn demand. Substtutng the ntermedate goods prce for the prce ndex and usng the relatonshp between the hat notaton and the log change, we apply Jensen s nequalty and express the dfferences between the two exchange rate regressons more formally: 11

12 ln ω n X n θ ω n ln ˆX n ω n ln θ ˆP nˆr n. 15 ˆP nˆr n In essence equaton 15 states that the log of the weghted arthmetc mean the LHS s always larger than the weghted geometrc mean the RHS. To quantfy the magntude of the dfference n means, we use the fact that, on average, the geometrc mean equals the arthmetc mean mnus half the varance. 9 Ths mples that the nequalty n equaton 15 wll be larger for larger relatve prce changes, exchange rate shocks and larger values of the trade elastcty. To see the later pont, note that the log change n the prce ndex n McGurk s approach equaton 11 s tself an approxmaton. Accordng to the hat notaton, the actual log change n the prce ndex s gven by the log of the arthmetc mean of the exporters prce changes lnˆr n ˆP n θ = ln K k=1 π nkˆr k ˆP kk θ rather than the geometrc average of exporters prce changes defned n equaton 11. Gven that the export prce s a functon of the trade elastcty, hgher values of the elastcty mply larger prce changes and larger errors n the approxmaton. In the followng secton, we wll quantfy the estmaton bases due to the approxmaton of assumng small polcy shocks as well as other defntons of real effectve exchange rate regressons found n the lterature. 4 Smulaton Havng descrbed the underlyng mcro-structure, we quantfy potental estmaton bas n varous exchange rate regresson specfcatons usng the data set provded by Dekle et al These data comprse blateral trade flows n manufactures, GDP as well as balance of payments nformaton for 39 countres plus the Rest of the World n the year Baselne deal-reer The frst specfcaton baselne captures the Real Exchange Rate model mpled by the exact hat algebra n equaton 8. ln Ê = β 1 RER ln RER + β 1 X ln ω n X n + ɛ β 1 16 ˆrn ˆP RER n where β 1 RER s the estmated exchange rate elastcty of specfcaton 1 and β1 X the demand elastcty of specfcaton 1. ɛ represents the estmaton error. The baselne regresson s the benchmark specfcaton and wll produce unbased estmates of the trade elastcty β 1 RER = θ and the foregn 9 w More formally, x n n ω n x n 1 2 ω n x n x 2 where x represents the weghted arthmetc mean. 10 The blateral trade share matrx s a suffcent statstc for a set of parameters ncludng trade costs. For further detals, please refer drectly to Dekle et al

13 demand elastcty β 1 X = 1 wth no error. Note that the regresson equaton 16 s non-lnear because the trade elastcty s not known. We estmate the relevant elastctes usng SILS structurally terated least squares descrbed n Head and Mayer Gold medal mstake An mportant problem when estmatng REER regressons s the presence of the multlateral resstance term, Φ n ˆr n ˆP n θ, n equaton 8. Ths term captures the mpled changes n the rest of the world due to polcy shocks. To solve for Φ n, researchers rely on a gravty model and smulate the mpled equlbrum changes. Thus, we want to assess the estmaton bas n the absence of the multlateral terms ln Ê = β 2 RER ln RER + β 2 X ln ω n X n + ɛ. 17 In the MTI verson, the exchange rate elastcty β 2 RER and the demand elastcty β2 X are based upwards. The reason s that the multlateral resstance term the omtted varable s negatvely correlated wth changes n the real exchange rate and changes n mport demand. An ncrease n the exchange rate of country, r, reduces the multlateral resstance terms for all countres n, dependng on the weght of country n the mport basket of country n. Wth respect to the demand elastcty, n the MTI verson the demand for mports n country wll ncrease one-to-one wth the ncrease n the exchange rate complete pass-through. For countres that do not experence an exchange rate shock, mport demand does not change. The decrease n the multlateral resstance term n combnaton wth the ncrease n mport demand mply a negatve correlaton between the two varables. If mport demand s not deflated by the multlateral resstance term, the varaton n the second term of equaton 17 s dampened and the estmated demand elastcty wll be hgher than n the baselne specfcaton. 3. Approxmaton va log changes As we saw n secton 2, the mpled aggregate regresson s based on the real exchange rate. Instead, one mght be tempted to approxmate the baselne regresson wth log changes, see equaton 15. The baselne equvalent specfcaton wrtes as follows: ln Ê = β 3 RER ln RER ˆ + β 3 X ω n ln ˆX n ln ˆr β 3 + ɛ 18 n ˆP RER n where frst term on the RHS, ln RER ˆ, s dentcal to the defnton n equaton 9, whle the second term s log approxmaton of the second term on the RHS n the baselne specfcaton. Equaton 15 shows that accordng to Jensen s nequalty the weghted log changes of foregn demand and the 13

14 multlateral resstance term wll be smaller than the log change of the sum of the changes as defned by the hat notaton n the baselne specfcaton. On the other hand, for exports and for the RER, the change n logs equals the log of the change,.e. ln E = ln Ê and ln RER = ln RER. Compared to the baselne specfcaton, the bas of both the demand β 3 X and RER elastcty β3 RER wll ncrease n the sze of exchange rate shock. In general, we expect these bases to be small for the RER and large for the demand elastcty because the latter depends on the trade elastcty, whch magnfes the approxmaton error. Smlar to the baselne specfcaton, we estmate equaton 18 non-lnearly usng SILS. 4. Real Effectve Exchange Rate à la McGurk The fourth smulaton conssts of the double-weghted REER regresson derved by McGurk ln Ê = β 4 RER ln REER ˆ + β 4 X ω n ln ˆX n + ɛ 19 For small exchange rate changes, we expect that the double-weghted REER regresson produces very smlar results compared to specfcaton 3 log approxmaton. The key dfference wth respect the log approxmaton s that we deflate the real exchange rate by the multlateral resstance term rather than demand. We expect a small downward bas n the exchange rate elastcty β 4 RER because of the log approxmaton of the foregn prce ndex and the negatve correlaton between the exchange rate and the foregn prce ndex. The log approxmaton of foregn demand mples that foregn demand vares less than n the baselne specfcaton and, by consequence, ntroduces an upward bas. Overall, we expect that both bases to be smlar to specfcaton Real Effectve Exchange Rate wth IMF weghts The Effectve Exchange Rate calculated by polcy nsttutons such as the IMF Bayoum et al. 2005, the European Central Bank ECB Schmtz et al and the Bank of Internatonal Settlement BIS Klau and Fung 2006 also follow the theoretcal mpled weghtng scheme of Armngton 1969 and McGurk ln Ê = β 4 RER ln REER ˆ IMF + β 4 X ω n ln ˆX n + ɛ 20 However, ther double weghtng approach dffers slghtly from the mpled weghts n equaton 12. Ther calculatons ncludes the sales of the exporter country n ts domestc market and uses the change n the countres prce ndex rather than the changes n the producer prce. 11 The REER s defned as follows: 11 The startng pont s equaton 7 n McGurk 1986, whch corresponds to our blateral equaton 10. As detaled n the appendx of her paper, the blateral equaton s then aggregated over all markets ncludng the domestc market. 14

15 ln REER ˆ IMF ˆP = TW k ln ˆr k = ˆP kˆr k and the mpled weghts are gven by TW k = N ψ n π nk 21 n=1 where ψ k s the export weght ncludng the home market shares. ψ k relates to our export weghts n the followng way: ψ n = X n N n=1 X n X n = ω n n=1 N X. n However, as Bayoum et al notes, the weghts mpled by equaton 21 do not sum up to 1. As a result, polcy nsttutons normalze the weghts and defne ther real exchange rate as follows: TW k = N n=1 ψ nπ nk N n=1 ψ n 1 π n 22 The denomnator of the equaton 22 ensures that TW k sums to 1. The resultng estmaton equaton s then gven by: ln Ê = β 5 RER ln REER ˆ IMF + β 5 X ω n ln ˆX n + ɛ k = where the weghts for the foregn demand contnue to be the export shares ω n. We do not have any pror about the drecton of the bas n the demand β 5 X and the exchange rate elastcty β5 RER caused by the weghtng scheme but presume t wll be smlar to specfcaton 4. For each of the 5 specfcatons, we consder two types of polcy experments. The frst one s the partal equlbrum approach, where we consder an exogenous change n the nomnal exchange rate and assume that wages do not adjust however ntermedate nput prces do adjust. In the second experment, the general equlbrum approach, wages and prces wll fully adjust gven a shock to the nomnal exchange rate. 5 Results Ths secton contans the results from our partal and general equlbrum smulaton exercses. 15

16 Table 1: MTI smulaton elastctes of country-specfc random exchange rate shocks Baselne REER McGurk GM mstake d ln approx. IMF weghts "deal-reer" "real-reer" Exchange rate Mean Bas Std dev MSE 1/ Foregn demand Mean Bas Std dev MSE 1/ Notes: The sample conssts of 39 countres and each country receves 100 random exchange rate shocks. The total number of estmated exchange rate and demand elastctes s 3900 for each of the 5 dfferent specfcatons. The top cell labelled "Mean" reports the sample mean of the respectve estmates of each type of elastcty. The rows labeled Bas gve the estmated bas, where est. bas = sample mean - true value. Rows labelled Std Dev. gve the sample standard devaton of the 3900 pont estmates. Rows labelled MSE 1/2 gve the square root of the estmated mean squared error MSE, where estmated MSE = est. bas 2 + std dev Modular Trade Impact MTI counter factual The MTI smulatons conssts of a random exchange rate shock to each country n our sample. We consder a nomnal exchange rate shock ˆr to a specfc country at the tme for example France and assume the exchange rates of all other countres fxed. Each shock s drawn from a normal dstrbuton wth varance 10 equvalent to a 10 percent apprecaton or deprecaton. For every country, we run 100 dfferent exchange rate shocks to obtan a sample of 39x39x100 data ponts. Next, we run a cross-secton OLS regresson for each replcaton and record the demand and exchange rate coeffcents. Table 1 contans the results. The top value n Table 1 reports the average exchange rate and demand elastcty for the 3900 estmated coeffcents. The second row corresponds to the mpled bas of the estmate,.e. the dfference between the sample mean and the true value of the coeffcents. The thrd row shows the standard devaton of the estmates. Lastly, the fourth row gves the square root of the estmated mean squared error MSE, our selecton crtera for the specfcaton wth the lowest aggregaton bas. The frst column shows the results for the baselne specfcaton. The estmates are equal to the true values of the elastctes 4 and 1 respectvely and therefore unbased. 12 All other specfcatons lead to a bas. In general, the bas s more pronounced for the demand than for the exchange rate elastctes. The MSE for the demand elastctes s sgnfcantly larger to the MSE n the exchange 12 In the baselne smulaton we calbrate θ = 4, whch s a common value found n the lterature see Smonovska and Waugh However, the magntude of the bas does not depend on the level of ths elastcty. Table 6 n the appendx contans smulaton results when the elastcty s θ =

17 Table 2: GETI smulaton elastctes of country-specfc random exchange rate shocks Baselne REER McGurk GM mstake d ln approx. IMF weghts deal-reer real-reer Exchange rate Mean Bas Std dev MSE 1/ Foregn demand Mean Bas Std dev MSE 1/ Notes: The sample conssts of 39 countres and each country receves 100 random exchange rate shocks. The total number of estmated exchange rate and demand elastctes s 3900 for each of the 5 dfferent specfatons. The top cell labelled "Mean" reports the sample mean of the respectve estmates of each type of elastcty. The rows labeled Bas gve the estmated bas, where est. bas = sample mean - true value. Rows labelled Std Dev. gve the sample standard devaton of the 3900 pont estmates. Rows labelled MSE 1/2 gve the square root of the estmated mean squared error MSE, where estmated MSE = est. bas 2 + std dev 2. rate elastctes. Specfcaton 2,.e. the Gold Medal mstake wth omttng the multlateral resstance term shown n column 2, produces the largest bases for the exchange rate elastcty The specfcatons wth the smallest bas are the log approxmaton of the baselne specfcaton n column 3 and the double-weghted REER approach a la McGurk n column 4. The pont estmate of the exchange rate elastcty s -4 wth a standard devaton smaller than The coeffcent of the demand elastcty for the log approxmaton s slghtly larger than 1 wth Fnally, the last column 5 shows the bas n the case of IMF weghts, whch overestmates the exchange rate elastcty General Equlbrum Trade Integraton GETI counter factual The MTI smulaton dd not take nto account general equlbrum effects and focused only on prce changes by keepng wages fxed. In the GETI we relax ths assumpton and suppose that wages also have to adjust followng an exchange rate shock. In partcular, we mpose that wages have to adjust n order to keep the current account mbalanceconstant n nternatonal currency. To engneer an exchange rate shock n the general equlbrum, we use the same exchange rate shocks as n the MTI part. A country receves an dosyncratc exchange rate shocks whle for all other countres the exchange rate does not change. However, exports change for all countres because prces and wages adjust after the shock. For every country, we run 100 dfferent exchange rate shocks. Table 2 presents the general equlbrum results usng a trade elastcty of 4 for the smulatons. 17

18 The sgns and the magntude of the underlyng bases n the exchange rate elastctes are very smlar to the MTI verson. Wth respect to the demand elastctes, the bases are more pronounced. However, overall the general equlbrum results confrm the key message from the partal analyss: the bases on the exchange rate elastcty and on the demand elastcty are small. 5.3 What explans the dfference n the coeffcents for each specfcaton? Both the MTI and GETI results show bases n the demand and the exchange rate elastctes. In specfcaton 2 the reason s an omtted varable the mporter s prce ndex and n specfcaton 5 the weghtng scheme. In specfcatons 3 and 4 the sources of the bas are related to functonal form assumptons. In both cases we can reduce the bas and the predcton error wth small adaptons. In the log approxmaton specfcaton we consder the weghted geometrc mean rather than the weghted arthmetc mean of changes n foregn demand deflated by the prce ndex. The dfference between the two means depends on the varance of the changes n the deflated foregn demand. Ths varance ncreases n the sze of an exchange rate shock to a partcular country as well as n the crosscountry varance of these shocks. In addton, shocks to mportant countres n the tradng network for example the US and Germany matter more. Shocks n these countres nduce larger demand and prce changes n tradng partners because ther weght n the partner s prce ndex s larger. To see how much the varance matters for the bas we re-estmate specfcaton 3 wth the geometrc average of foregn demand plus the varance: ln Ê = β 3 RER ln where the varance s defned as: Var ωn RER ˆ + β 3 X ln ˆX n θ + 1 ˆrn ˆP n 2 Var ω n ˆX n ˆr n ˆp n θ = ˆX n ω n ˆr n ˆp n θ ω n ω n ˆX n θ + ɛ 23 ˆrn ˆP n ˆX n ˆr n ˆp n θ Smlarly, the defnton of the REER n specfcaton 4, equaton 12, s based on an approxmaton of the change n the prce ndex by a weghted average of the changes n the producton prce of exporters see equaton 11. Instead, one can use the actual change of the prce ndex wthout the double weghtng and run the followng specfcaton: ln Ê = β 4 RER r P ω n ln + β 4 X r k = n P n ω n ln ˆX n + ɛ 24 The RMSE error n specfcaton 3 for the exchange rate and the demand elastcty reduces n both by 80 percent, whereas n specfcaton 4 only by 9 percent, see column 2 versus column 3 as well as column 4 and column 5 n Table

19 Table 3: GETI smulaton elastctes of country-specfc random exchange rate shocks Baselne d ln approx. REER McGurk d ln approx. REER McGurk deal-reer less bas less bas Exchange rate Mean Bas Std dev MSE 1/ Foregn demand Mean Bas Std dev MSE 1/ Notes: The sample conssts of 39 countres and each country receves 100 random exchange rate shocks. The total number of estmated exchange rate and demand elastctes s 3900 for each of the 5 dfferent specfatons. The top cell labelled "Mean" reports the sample mean of the respectve estmates of each type of elastcty. The rows labeled Bas gve the estmated bas, where est. bas = sample mean - true value. Rows labelled Std Dev. gve the sample standard devaton of the 3900 pont estmates. Rows labelled MSE 1/2 gve the square root of the estmated mean squared error MSE, where estmated MSE = est. bas 2 + std dev Monte-Carlo wth stochastc trade costs Up untl now, all our smulatons are based on the assumpton that structural gravty equaton s the true data generatng process for blateral trade flows. To allow for the fact that the gravty equaton does not perfectly explan blateral trade flows, we run a Monte Carlo exercse and assume, as Head and Mayer 2014, that blateral trade flows change due to a stochastc term η n n the trade cost functon. φ n = exp ln Dst n η n 25 where Dst n s the dstance between mporter n and exporter. η n s a log-normal random term and the only stochastc term n the smulaton snce the GDPs and dstance are all set by actual data. We calbrate the varance of lnη n to replcate the root mean squared error RMSE of the least squares dummy varables LSDV regresson on real data. In the Monte Carlo smulaton of the general equlbrum, we use the method of Dekle et al and solve the model n changes. Based on actual data n trade flows and ncomes, we, frst, smulate changes n the trade costs wth a random shock ˆη n,.e. ˆτ n = ˆη n, and calculate then the changes n ncomes ŵ, prces ˆp and the aggregate prce ndexes ˆP n due to the random exchange rate shock. Table 4 reports the results. As expected, ntroducng stochastc trade costs ncreases the bas and the overall performance of the estmaton method. The RMSE ncreases sgnfcantly for both, the pont estmates of the exchange rate and the demand elastcty. Interestngly, the bas n the demand elastcty becomes more pronounced than n the exchnage rate elastcty. Across specfcatons, the 19

20 Table 4: GETI smulaton elastctes of country-specfc random exchange rate shocks Baselne REER McGurk GM mstake d ln approx. IMF weghts deal-reer real-reer Exchange rate Mean Bas Std dev MSE 1/ Foregn demand Mean Bas Std dev MSE 1/ Notes: The sample conssts of 39 countres and each country receves 100 random exchange rate and trade cost shocks. The total number of estmated exchange rate and demand elastctes s 3900 for each of the 5 dfferent specfcatons. The top cell labelled "Mean" reports the sample mean of the respectve estmates of each type of elastcty. The rows labeled Bas gve the estmated bas, where est. bas = sample mean - true value. Rows labelled Std Dev. gve the sample standard devaton of the 3900 pont estmates. Rows labelled MSE 1/2 gve the square root of the estmated mean squared error MSE, where estmated MSE = est. bas 2 + std dev 2. average magntude of the bas s close to 20 percent for the demand elastcty, whereas only close to 1 percent for the exchange rate elastcty. The specfcaton wth the lowest RMSE s the log approxmaton n column 3, suggestng that deflatng the changes n foregn demand by changes n the aggregate prce ndex s mportant to reduce the bas n the demand elastcty. Overall, our smulaton results suggest that the deal-reer approach allows us to estmate the response of aggregate exports to exchange rate changes wthout bas f the data generatng process follows structural gravty. Next, we test whether the deal-reer approach also reduces the estmaton bas when aggregatng blateral data emprcally. 6 Emprcal evdence A key mplcaton from the theoretcal analyss s that under the assumpton of a unque trade elastcty and an exporter-specfc pass-through coeffcent runnng exchange rate regressons on the blateral or on the aggregate level does not lead to a sgnfcant bas n the elastctes. Whle recognzng that n practce exchange rate elastctes dffer across countres, see for example Splmbergo and Vamvakds 2003 or Bussère et al. 2016, we test the theoretcal predcton of no-sgnfcant bas n exchange rate regressons when poolng across countres. Later we relax ths assumpton and test for sgnfcant dfferences n the blateral versus aggregate elastctes on a country-per-country bass. Our emprcal approach uses data for blateral exchange rates from the IMF fnancal statstcs. Data on Real Effectve Exchange Rates s comng from the Bank of Internatonal Settlement BIS. We 20

21 use the Narrow ndex as t has the advantage of coverng a longer sample perod. The Narrow ndex s a trade weghted effectve exchange rate over 25 economes excludng euro area and starts n Inflaton proxed by the GDP deflator and GDP data are comng from Natonal Accounts and are converted nto nternatonal currency USD usng the blateral exchange rate to the USD. We use GDP as a proxy for foregn demand. For changes n export prces, we use Producer Prce Indexes PPI from the OECD and the IMF dependng on data avalablty. Overall, our sample comprses yearly data from 1964 to 2011 for 25 countres. The blateral regressons contan all major countres and ther tradng partners ncluded n the defnton of the Effectve Exchange Rate. In the case of the Narrow ndex, the sample conssts of 25 countres and ther 24 partner countres over the perod 1964 to We run the followng blateral regresson pooled across all countres: lnx jt = β BIL lnrer jt + γ BIL lngdp jt + f + ɛ jt 26 where X jt s the level of nomnal exports n local currency, RER jt s the real blateral exchange rate between country and j, whch s defned as the log change n nomnal exchange rate multpled by the relatve prce changes between the PPI n country and nflaton n country j. GDP jt s country j s level of nomnal GDP n USD and f s an exporter fxed effect allowng us to abstract from tmeconstant exporter specfc effects. To compare the blateral estmates to the aggregate estmates, we run the emprcal equvalents of specfcaton 1 to 5 defned n equatons 16 to 20 and nclude an exporter specfc fxed effect f n those regressons as well. To estmate the emprcal equvalent aggregate regressons specfed n equatons 16 to 20, we use the followng data. The real exchange rate of country RER t s defned as the change n the nomnal exchange rate of country vs-à-vs the US dollar multpled by the change n the relatve prce between the PPI of country and the prce level n the US. We consder the US as the reference country. For the US, the reference country s the Unted Kngdom. Further, we use GDP denomnated n US dollars n country j as a proxy for demand n the mportng destnaton X jt. Inflaton proxes for changes n the prce ndex ˆP jt, whch s measured relatve to the reference country by multplyng nflaton wth the change n nomnal exchange rate of country j and the US dollar dvded by US nflaton. For the weghts ω nt, we use the share of exports of country gong to country n n year t. Ths defnton of the varables apples to all aggregate regressons wth the excepton of REER defned by the IMF weghtng scheme equaton 20, where we use the BIS Narrow ndex as the REER. The results are shown n Table 5. Note that there are no sgnfcant dfferences between the blateral elastctes and the aggregate baselne elastctes. However, the pont estmate of the exchange rate elastcty s slghtly lower 0.41 n column 2 compared to the blateral one 0.46 n column 1. On the other hand, the demand elastctes are almost dentcal wth pont estmates of 1.18 on the blateral and 1.19 on the aggregate level. Wth respect to all other aggregate regressons, the es- 21