Monetary Policy and the Predictability of Nominal Exchange Rates

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Monetary Policy and the Predictability of Nominal Exchange Rates Martin Eichenbaum, Benjamin K. Johannsen, and Sergio Rebelo 2017-037 Please cite this paper as: Eichenbaum, Martin, Benjamin K. Johannsen, and Sergio Rebelo (2017). “Monetary Policy and the Predictability of Nominal Exchange Rates,” Finance and Economics Discussion Series 2017-037. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2017.037r1. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Monetary Policy and the Predictability of Nominal Exchange Rates Martin Eichenbaum Benjamin K. Johannsen∗ Sergio Rebelo October 2017 Abstract This paper documents two facts about countries with floating exchange rates where monetary policy controls inflation using a short-term interest rate. First, the current real exchange rate predicts future changes in the nominal exchange rate at horizons greater than two years both in sample and out of sample. This predictability improves with the length of the horizon. Second, the real exchange rate is virtually uncorrelated with future inflation rates both in the short run and in the long run. We show that a large class of open-economy models is consistent with these findings and that, empirically and theoretically, the ability of the real exchange rate to forecast changes in the nominal exchange rate depends critically on the nature of the monetary regime. ∗The views expressed here are those of the authors and do not necessarily reflect the views of the Board of Governors, the Federal Open Market Committee, or anyone else associated with the Federal Reserve System. We thank Adrien Auclert, Charles Engel, Gaetano Gaballo, Zvi Hercovitz, Oleg Itskhoki, Dmitry Mukhin, Paulo Rodrigues, Christopher Sims, and Oreste Tristani for their comments and Martin Bodenstein for helpful discussions. 1

1 Introduction This paper studies how the monetary policy regime affects the relative importance of nominal exchange rates (NERs) and inflation rates in shaping the response of real exchange rates (RERs) to shocks. To describe our findings, we define the RER as the price of the foreign-consumption basket in units of the home-consumption basket and the NER as the price of the foreign currency in units of the home currency. Webeginbydocumentingtwofactsaboutrealandnominalexchangeratesforasetofbenchmark countries. These countries have two characteristics in common over our sample period: they have flexible exchange rates and the central bank uses short-term interest rates to keep inflation near its target level. Our first fact, is that the current RER is highly negatively correlated with future changes in the NER at horizons greater than two years. This correlation is stronger the longer is the horizon. Our second fact, is that the RER is virtually uncorrelated with future inflation rates at all horizons. Taken together, these facts imply that the RER adjusts in the medium and long runs overwhelminglythroughchangesintheNER,notthroughdifferentialinflationrates. Whenacountry’s consumption basket is relatively expensive, its NER eventually depreciates by enough to move the RER back to its long-run level. These conclusions are consistent with those of Cheung, Lai, and Bergman (2004).1 Critically, we argue that these facts depend on the monetary policy regime in effect. To show this dependency, we re-do our analysis for China which is on a quasi-fixed exchange rate regime versus the U.S. dollar; for Hong Kong which has a fixed exchange rate versus the U.S. dollar; and for the euro-area countries, which have fixed exchange rates with each other. In all of these cases, the current RER is highly negatively correlated with future relative inflation rates. In contrast to the flexible exchange rate countries, the RER adjusts overwhelmingly through predictable inflation differentials. Additional evidence on the importance of the monetary policy regime comes from a set of countries that had crawling pegs or heavily managed floating exchange rates and then moved to floating exchange rates and inflation targeting. This set of countries consists of Brazil, Chile, Colombia, Indonesia, Israel, Mexico, South Korea, and Thailand. We show that when these countries adopted floating exchange rates and inflation targeting, the dynamic co-movements of the NER, the RER, and inflation became qualitatively similar to those in our benchmark countries. This type of sensitivity to the monetary policy regime is precisely what we would expect given the Lucas (1976) critique. Before discussing a class of models that accounts for our findings, we confront the concern that thesefindingsmightbespuriousinthesensethattheymightprimarilyreflectsmallsamplesizesand persistent RERs.2 We address these concerns in two ways. First, using a bootstrap methodology, 1These authors use an alternative statistical methodology to study the behavior of the exchange rates for four European countries and Japan. Their sample spans the period between the collapse of the Bretton Woods system and the establishment of the euro. 2Similarconcernslieattheheartofongoingdebatesaboutthepredictabilityoftheequitypremiumbasedonvariables 2

we find that it is implausible that our empirical findings could be produced by a data generating process (DGP) with a very persistent RER that is uncorrelated with future changes in the NER. Second, we show that out-of-sample forecasts of the NER based on the RER beat a random walk forecast at medium and long horizons. We argue that this finding is extremely unlikely if the NER is not predictable, regardless of whether the underlying DGP for the RER is stationary. Viewed overall, these results are strongly supportive of the view that our key empirical findings for the benchmark countries are not spurious. Having established our key facts, we turn to the underlying economics. We show that there is a wide class of models consistent with the fact that, for our benchmark countries, the current RER predicts future movements in the NER. This consistency holds in models both with and without nominal rigidities. The key elements of these models are that monetary policy is governed by a Taylor rule and there is home bias in consumption. We analyze versions of the same class of models in which the foreign central bank follows a managed float. We show that these models are consistent with the fact that, under a managed float, the RER is useful for predicting future movements in differential inflation rates. While the previous findings hold for all versions of the model that we consider, a dynamic stochastic general equilibrium (DSGE) model with nominal rigidities does the best job quantitatively. We begin our theoretical analysis with a simple flexible-price model where labor is the only factor in the production of intermediate goods. The intuition for why this simple model accounts for our empirical findings about Taylor-rule regimes is as follows. Consider a persistent fall in domestic productivity or an increase in domestic government spending. Both shocks lead to a rise intherealcostofproducinghomegoodsthatdissipatessmoothlyovertime. Homebiasmeansthat domestically produced goods have a high weight in the domestic consumer basket. So, after the shock, the price of the foreign consumption basket in units of the home consumption basket falls, i.e. the RER falls. The Taylor rule followed by both central banks keeps inflation relatively stable inthetwocountries. Asaconsequence, mostoftheadjustmentintheRERoccursthroughchanges in the NER. In our model, the NER behaves in a way that is reminiscent of the overshooting phenomenon emphasized by Dornbusch (1976). After a technology shock, the foreign currency depreciates on impact and then slowly appreciates to a level consistent with the return of the RER to its steady-state value. The longer the horizon, the higher is the cumulative appreciation of the foreign currency. So in this simple model, the current RER is highly negatively correlated with the value of the NER at future horizons, and this correlation is stronger the longer is the horizon. These predictable movements in the NER can occur in equilibrium because they are offset by the interest rate differential, i.e., uncovered interest parity (UIP) holds. An obvious shortcoming of the flexible-price model is that purchasing power parity (PPP) holds at every point in time. To remedy this shortcoming, we modify the model so that monopolist producers set the nominal prices of domestic and exported goods in the local currency where they are sold. They do so subject to Calvo-style pricing frictions. For simplicity, suppose for now that there is a complete set of domestic and international asset markets. Consider a persistent fall in like the price–dividend ratio (see Stambaugh (1999); Boudoukh, Richardson and Whitelaw (2006); and Cochrane (2008)). 3

domesticproductivityoranincreaseindomesticgovernmentspending. Bothshocksleadtoarisein domestic marginal cost. The domestic firms that can reoptimize their prices increase them at home and abroad, so inflation rises. Because of home bias, domestic inflation rises by more than foreign inflation. The Taylor principle implies that the domestic real interest rate rises by more than the foreign real interest rate. So, domestic consumption falls by more than foreign consumption. With complete asset markets, the RER is proportional to the ratio of foreign to domestic marginal utilities of consumption. So, the fall in the ratio of domestic to foreign consumption implies a fall in the RER. As in the flexible price model, the Taylor rule keeps inflation relatively low in both countries so that most of the adjustment in the RER is attributable to movements in theNER. Again, theimpliedpredictablemovementsintheNERcanoccurinequilibriumbecause they are offset by the interest rate differential, i.e. UIP holds. Whiletheintuitionislessstraightforward,ourresultsarenotsubstantivelyaffectedifwereplace complete markets with incomplete markets or assume producer-currency pricing instead of localcurrency pricing. Risk premiums aside, UIP holds conditional on the realization of many types of shocks to the model economy. We introduce shocks to the demand for bonds, for which UIP does not hold. So, when the variance of these shocks is sufficiently large, traditional tests of UIP applied to data from our model would reject that hypothesis. Finally,weassesswhetherempirically-plausibleversionsofourmodelcanquantitativelyaccount for the facts that we document by studying an open-economy medium-sized DSGE version of our model. Among other features, the model allows for Calvo-style nominal wage and price frictions and habit formation in consumption of the type considered in Christiano, Eichenbaum, and Evans (2005). A key question is whether the models we study are consistent with other features of the data stressed in the open-economy literature. It is well known that, under flexible exchange rates, real and nominal exchange rates co-move closely in the short run (Mussa (1986)). This property, and the fact that RERs are highly inertial (Rogoff (1996)) constitute bedrock observations that any plausible open-economy model must be consistent with. We show that our medium-sized DSGE model with nominal rigidities is, in fact, consistent with these observations. Finally,weshowthatourDSGEmodelcanquantitativelyaccountfortheextenttowhichRERbased medium- and long-run forecasts of the NER outperform random walk forecasts. Specifically, the model is consistent with the fact that the comparative advantage of RER-based forecasts increases with the forecasting horizon. In addition, the model accounts quantitatively for the average ratio of the root mean squared prediction error (RMSPE) of the RER-based and random walk-based forecasts at all horizons. Our work is related to four important strands of literature. The first strand demonstrates the existence of long-run predictability in NERs (e.g. Mark (1995) and Engel, Mark, and West (2007)). Our contribution here is to show that the ability of the RER to predict the NER at medium and long-run horizons depends critically on the monetary policy regime in effect. The second strand of the literature, which goes back to Meese and Rogoff (1983), studies the out-ofsample predictability of the NER. Authors like Engel and West (2004, 2005) and Molodtsova and 4

Papell (2009) have proposed using variables that might enter into a Taylor rule to improve outof-sample forecasting. Such variables includes output gaps, inflation, and possibly RERs. Rossi (2013) provides a thorough review of this literature. Recently, Cheung, Chinn, Pascual, and Zhang (2017) highlight the potential role of the RER in helping forecast the NER. Ca’Zorzi, Muck, and Rubaszek (2016) study the forecasting performance of the Justiniani and Preston (2010) DSGE model. Citing an earlier version of this paper, these authors note the potential usefulness of the RER in forecasting the NER. Our contribution relative to these two papers is to thoroughly document that role and show how it depends on the monetary policy regime. The third strand of the literature seeks to explain the persistence of RERs. See, for example, Rogoff (1996); Kollmann (2001); Benigno (2004); Engel, Mark, and West (2007); and Steinsson (2008). Our contribution relative to that literature is to show that we can account for the relationship between the RER and future changes in inflation and the NER in a way that is consistent with the observed inertia in the RER. The fourth strand of the literature emphasizes the importance of the monetary regime for the behavior of the RER. See, for example, Baxter and Stockman (1989); Henderson and McKibbin (1993);Engel,Mark,andWest(2007); andEngel(2012). Ourcontributionrelativetothisliterature is to document the importance of the monetary regime in determining the relative roles of inflation and the NER in the adjustment of the RER to its long-run levels. Ourpaperisorganizedasfollows. Section2containsourempiricalresults. Section3describesa sequence of models consistent with these results. We start with a model that has flexible prices and complete asset markets and where labor is the only factor in the production of intermediate goods. We then replace complete markets with a version of incomplete markets where only one-period bonds can be traded. Next, we introduce Calvo-style frictions in price setting. In Section 4, we consider an estimated medium-scale DSGE model. Section 5 concludes. 2 Some empirical properties of exchange rates In this section, we present our empirical results regarding NERs, RERs, and relative inflation rates. We use consumer price indexes for all items and average quarterly NERs versus the U.S. dollar. 2.1 Data We initially focus on a benchmark group of advanced economies—Australia, Canada, Norway, Sweden, and Switzerland—that had floating exchange rates in the period from 1973 to 2007.3 In choosing the sample period, we face the following trade off. On the one hand, we would like as long a time series as possible. On the other hand, we would like the monetary regime to be reasonably stable in our sample. To balance these considerations, we exclude from our sample data from 2008 to the present because short-term nominal interest rates in the United States were at or near 3Unless indicated otherwise, a year means that the entire year’s worth of data was used. 5

their effective lower bound. We include data since 2008 as a part of our robustness analysis. We exclude Japan from our set of benchmark countries because its short-term interest rates have been at or close to the effective lower bound since 1995. We exclude the United Kingdom, which left the European Exchange Rate Mechanism of the European Monetary System in 1992 after a large devaluation. WeincludedatafrombothJapanandtheUnitedKingdominourrobustnessanalysis, where we also consider countries that eventually adopted the euro.4 We compare results for the benchmark flexible exchange rate economies with those for China (from 1994 through 2007), which has been on a quasi-fixed exchange rate vis-`a-vis the U.S. dollar, and for Hong Kong (from 1985 through 2007), which has a fixed exchange rate vis-`a-vis the U.S. dollar. We also analyze data starting in 1999 for France, Ireland, Italy, Portugal, and Spain where the RER and relative inflation rates are defined relative to Germany. In addition, we consider a group of countries that had crawling pegs or heavily managed floating exchange rates and then moved to floating exchange rate regimes along with a form of inflation targeting. This set of countries consists of Brazil, Chile, Colombia, Indonesia, Israel, Mexico, South Korea, Thailand, and Turkey. 2.2 Results for flexible exchange rate countries We define the RER for country i relative to the United States as: NER P i,t i,t RER = , (1) i,t P t where NER is the nominal exchange rate, defined as U.S. dollars per unit of foreign currency. i,t The variables P and P denote the consumer price index in the U.S. and in country i, respectively. t i,t We assume that the RER is stationary and offer supporting evidence later in this section. Given this assumption, the RER must adjust back to its mean after a shock via changes in the NER or changes in relative prices. Figure 1 displays scatter plots for Canada of the log(RER ) against log(NER /NER ) at i,t i,t+h i,t different horizons, h. The analogue figures for the other benchmark flexible exchange rate countries are displayed in the appendix. Two properties of this figure are worth noting. First, consistent with the notion that exchange rates behave like random walks at high frequencies, there is no obvious relationship between the log(RER ) and log(NER /NER ) at a one-year horizon. i,t i,t+h i,t However, as the horizon expands, the correlation between log(RER ) and log(NER /NER ) i,t i,t+h i,t rises. The negative relation is very pronounced at longer horizons. This pattern holds for all of the benchmark flexible exchange rate countries included in the appendix. 4For bilateral exchange rate data between the United States and other countries, we use the H.10 exchange rate data published by the Federal Reserve, available at http://www.federalreserve.gov/releases/H10/Hist. We compute quarterly averagesofthedailydata. WhentheH.10datadonotincludeacountry,weuseexchangeratedatafromtheInternational Monetary Fund’s International Financial Statistics database. For price indexes, we also use the International Monetary Fund’s International Financial Statistics database. When consumer price indexes are not available from the International Financial Statistics database, we use data from the Organization for Economic Cooperation and Development (OECD), which were downloaded from FRED, a database maintained by the Federal Reserve Bank of St. Luois (“Main Economic Indicators - complete database,” Main Economic Indicators (database)). 6

2.2.1 Nominal exchange rate regressions We now discuss results based on the following NER regression: (cid:18) (cid:19) NER log i,t+h = αNER+βNERlog(RER )+εNER , (2) NER i,h i,h i,t i,t,t+h i,t for country i at horizon h = 1,2,...,H years. Panel (a) of Table 1 reports estimates of βNER, i,h along with standard errors, for the benchmark flexible exchange rate countries.5 A number of features are worth noting. First, for every country and every horizon, the estimated value of βNER i,h is negative. Second, for almost all countries, the estimated value of βNER is statistically significant i,h at three-year horizons or longer. Third, in most cases, the estimated value of βNER increases in i,h absolutevaluewiththehorizon, h. Moreover, βNER ismorepreciselyestimatedforlongerhorizons. i,h Panel (a) of Table 1 also reports the R2s of the fitted regressions. Consistent with the visual impression from the scatter-plots, the R2s are relatively low at short horizons but rise with the horizon. Strikingly, for the longest horizons, the R2 exceeds 50 percent for all of our benchmark countries and is 88 percent for Canada. Taken together, the results in Table 1 strongly support the conclusion that, for our benchmark countries, thecurrentRER isstronglycorrelatedwithchangesinfutureNERs, athorizonsgreater than roughly two years. 2.2.2 Relative price regressions We now consider results based on the following relative-price regression: (cid:18) (cid:19) P /P log i,t+h t+h = απ +βπ log(RER )+επ . (3) P /P i,h i,h i,t i,t,t+h i,t t This regression quantifies how much of the adjustment in the RER occurs via changes in relative ratesofinflationacrosscountries. Panel (a)ofTable 2reportsourestimatesandstandarderrorsfor the slope coefficient βπ . In most cases, the coefficient is statistically insignificant, though positive. i,h In some cases, it is negative instead of positive. Panel (a) of Table 2 also reports the R2s of the fitted regressions. These R2s are all much lower than those associated with regression (2). These results as a whole suggest that very little of the adjustment in the RER occurs via differential inflation rates. This conclusion is consistent with the results of Cheung, Lai, and Bergman (2004) based on an earlier sample period for Japan and four European countries. 2.2.3 Robustness: Other countries We now assess the robustness of the previous results by considering other advanced economies with flexible exchange rates—the euro area, Japan, and the United Kingdom. Because the samples for these countries are relatively short, we only estimate regressions (2) and (3) out to a five-year 5We compute standard errors using a Newey-West estimator with the number of lags equal to the forecasting horizon plus two quarters. 7

horizon.6 Our results are reported in panel (b) of Table 1 and panel (b) of Table 2. The estimated regression coefficients are similar to those obtained for the benchmark countries. The appendix reports results for both these countries and the benchmark countries when we extend the sample to end in 2016:Q4. This change in sample period has little effect on our results. 2.3 Sensitivity to monetary policy Our basic hypothesis is that the process by which the RER adjusts to shocks depends critically on the monetary policy regime. We provide two types of evidence in favor of this hypothesis. First, we redo our analysis for countries that are on fixed or quasi-fixed exchange regimes. Second, we consider countries that, initially, heavily managed their exchange rates but later allowed their exchange rates to float. 2.3.1 Fixed and quasi-fixed exchange rates In this subsection, we report the results of redoing our analysis for countries with fixed or quasifixed exchange rates. Results for China and Hong Kong, which have quasi-fixed and fixed exchange rates, respectively, are reported in panel (c) of Table 1 and panel (c) of Table 2. Several features of these results are worth noting. First, the estimated values of βNER are small relative to the i,h estimates for our benchmark countries. Second, values of βπ are large relative to the estimates for i,h our benchmark countries and statistically significant at every horizon. Third, the estimated value of βπ rises with the horizon, h. Fourth, the R2 values associated with regression (3), reported in i,h panel (c) of Table 2, are large and increase with the horizon. We also consider several euro-area countries—France, Ireland, Italy, Portugal, and Spain—vis- `a-vis Germany. For these countries, the NER is fixed. Results for regression (3) are reported in Table 3. As was the case for China and Hong Kong vis-`a-vis the United States, the estimated values of βπ are large, rise in magnitude with the horizon, and are statistically significant at long i,h horizons. In addition, the R2 values are large and increase with the horizon, with regression (3) explaining 94 percent of relative price movements between Germany and Portugal at a five-year horizon. Insum,foreconomieswithfixedorquasi-fixedexchangerates,theRERadjustsoverwhelmingly through predictable inflation differentials, not through changes in the NER. 2.3.2 Countries with changes in exchange rate policy In this subsection, we redo our analysis for a set of countries—Brazil, Chile, Colombia, Indonesia, Israel, Mexico, South Korea, and Thailand—that had crawling pegs or heavily managed floating exchange rates and then adopted floating exchange rates. 6We begin the sample for the euro in 1999, when it was created. We start the sample for Japan in 1973 and end the sample in 1994, because Japan has had nominal interest rates near their effective lower bound since the mid-1990s. We begin the sample for the United Kingdom in 1993 because the United Kingdom exited the European Exchange Rate Mechanism of the European Monetary System after a large devaluation in 1992. 8

Weconsidertwosampleperiods. Thefirstsampleisfrom1984:Q1to2016:Q4andcoversperiods in which all of the countries moved from a managed exchange rate to a floating exchange rate. The second sample spans the period from 1999:Q1 to 2016:Q4. We include the period in which the zero lower bound (ZLB) is binding in the United States and in some other countries in order to have enough observations to estimate our regressions at a five-year horizon. Our experience with the benchmark countries suggests that including the ZLB period has a mild effect on the coefficients and R2s of regressions (2) and (3). Tables(4)and(5)reportourestimatesofβNER andβπ ,aswellastheR2sfromtheregressions. i,h i,h In contrast to our benchmark countries, for the sample starting in 1984, the estimates of βNER and i,h βπ , and the R2 values, do not follow the consistent pattern observed for the benchmark flexible i,h exchangeratecountries. Inaddition, theestimatesdisplaynoapparentpatternacrossthecountries considered. Tables (4) and (5) also report results for the sample starting in 1999. Notice that for every country except Turkey and every horizon, the estimates of βNER are negative, grow in magnitude i,h with the horizon, and are statistically different from zero at longer horizons. In addition (again with the exception of Turkey), the R2 values for regression (2) using the sample starting in 1999 are much larger than the analogous R2s from the full sample. By contrast, the estimates of βπ are i,h relatively small in the sample starting in 1999, as are the R2 values for regression (3). Clearly, the post-1999 sample produces results that are more similar to those obtained with our benchmark flexible exchange rate countries. We view these results as being supportive of our hypothesis that the monetary policy regime is a central determinant of the way that the RER adjusts to shocks. 3 Are the empirical correlations spurious? In the previous section, we argue that for our benchmark countries, changes in the NER at long horizons display a strong negative correlation with the current level of the RER. A potential problem with this result is that if the RER is very persistent, we might statistically find in-sample predictability when none is actually present. Boudoukh et al. (2006) make this point in the context of the literature on equity returns predictability. In our context, their critique translates into the statementthattheasymptoticstandarderrorsfortheregressioncoefficientsreportedintheprevious section severely understate the importance of sampling uncertainty. In this section, we address these concerns in two steps. Following the approach proposed by Boudoukh et al. (2006), we examine the small-sample properties of the Wald statistic for the test that the slope coefficients in regression (2), βNER, are zero at all horizons. Under the null i,h hypothesis that the RER is a stationary process, we construct bootstrap p-values, which provide strong evidence against the hypothesis that βNER are all zero. i,h Analogue exercises conducted under the null hypothesis that the RER is a difference-stationary processturnouttohaveverylowpower,reflectingthediffusenatureofthesmall-sampledistribution oftheslopecoefficients. Fortunately,inourcase,testsbasedonout-of-sampleforecastsoftheNER 9

are more powerful. We show that over medium- and long-run horizons, our forecasts of the NER outperform random-walk forecasts. As discussed later, this finding is unlikely to reflect sampling uncertainty regardless of whether the RER has a unit root. 3.1 Testing whether slope coefficients are zero In this subsection, we test the joint null hypothesis that the slope coefficients in regression (2) are zero at all horizons up to 40 quarters (10 years), i.e., βNER = βNER = ··· = βNER = 0. (4) i,1 i,2 i,40 For each country i, we jointly estimate the slope coefficients βNER and compute the Wald statisi,h tic under the null hypothesis (4).7 We focus attention on our benchmark flexible exchange rate countries so that we have enough data to include regressions with a horizon of 10 years. Because the RER is highly persistent, we find in simulations that tests based on the asymptotic distribution of the Wald statistic have poor size. Accordingly, we test the null hypothesis (4) using thefollowingbootstrapprocedure. WeassumethatthestochasticprocessesforNER andRER i,t i,t are given by (cid:18) (cid:19) NER log i,t = εNER, (5) NER i,t i,t−1 A (L)log(RER ) = εRER. (6) i i,t i,t Here, A (L) is a polynomial in the lag operator with roots inside the unit circle so that the RER i is a stationary process. The random variables εNER and εRER are uncorrelated over time (though i,t i,t potentially correlated within a period). This DGP embeds the assumption that changes in the NER are unpredictable at all horizons.8 We consider up to 10 lags in A (L) and choose the i lag length separately for each country using the Akaike information criterion (AIC).9 Given the estimates of A (L), we back out a time series for εRER and εNER from the observed data. We i i,t i,t then jointly bootstrap εNER and εRER to compute 10,000 synthetic time series, each of length i,t i,t equal to our actual sample period.10 For each synthetic time series, we estimate regression (2) for h = 1,2,...,40 and compute the corresponding Wald statistics to produce a bootstrap distribution of that statistic. Table6reportsthefractionofthebootstrapWaldstatisticsthatarelargerthanthecorresponding Wald statistic that we computed in the data. With the exception of Norway, we can reject the nullhypothesis(4)atthe5percentsignificancelevel. ForNorway, wecanrejectitatthe10percent 7We compute standard errors using a Newey-West estimator with the number of lags equal to the forecasting horizon plus two quarters. 8Note that if log(NER /NER ) has a non-zero mean, that property is reflected in the fitted shocks from which i,t i,t−1 we construct the bootstrap samples. 9TheAICselectedfourlagsforAustralia, sevenlagsforCanada, eightlagsforNorway, fourlagsforSweden, andfour lags for Switzerland. 10We use 100 periods of initial burn in for our bootstraps. 10

significance level. Based on these tests, we infer that the negative correlations between the RER and the future changes of the NER that we documented are unlikely to be spurious. 3.2 Out-of-sample forecasts In this subsection, we use out-of-sample forecasting performance to test the null hypothesis that the NER is not predictable. In practice, quarterly consumer price indexes are available with one period lag. To avoid any look-ahead bias, we measure the RER for country i using lagged price indexes so that NER P i,t i,t−1 RER ≡ . (7) i,t P t−1 Our forecasting equation for the NER is (cid:18) (cid:19) NER log i,t+h = αNER+βNERlog(RER )+εNER . (8) NER i,h h i,t i,t,t+h i,t Notice that the parameter βNER is common across countries. This specification corresponds to a h balanced panel with country-specific intercepts (αNER) and common slopes.11 We set the training i,h period for the regression to the horizon of the forecast, h, plus 40 quarters. We assess our ability to forecast the NER relative to a forecast of no change. The latter is the benchmark in the literature and corresponds to the assumption that the NER is a random walk without drift. Define the RMSPE for country i associated with forecasts based on equation (8) as  1/2  1 (cid:88) T i,h(cid:20) (cid:18) NER i,t+h (cid:19)(cid:21)2 σ = f −log . (9) i,B,h i,t,t+h T NER  i,h i,t  t=0 Here, T denotes the number of forecasts for log(NER /NER ) in our sample, and f i,h i,t+h i,t i,t,t+h is the forecast of log(NER /NER ) based on equation (8). We denote by σ the correi,t+h i,t i,RW,h sponding RMSPE associated with the no-change forecast from a random walk model. For each country i, we report the ratio of the RMSPE associated with the benchmark and random walk specifications, σ /σ . We also compute a pooled RMSPE implied by our i,B,h i,RW,h forecasting equation for all of the countries in our sample, defined as  1/2  1 (cid:88)(cid:88) T i,h(cid:20) (cid:18) NER i,t+h (cid:19)(cid:21)2 σ = f −log . (10) B,h (cid:80) i,t,t+h T NER  i i,h i t=0 i,t  We denote by σ the pooled RMSPE implied by the random walk forecast and report the ratio RW,h of the pooled RMSPEs, σ /σ . B,h RW,h Weinitiallylimittheanalysistoourbenchmarkcountries. Panel (a)ofTable (7)reportsrelative RMSPEs for each country and for the pooled sample. Forecasts based on equation (8) outperform 11In adopting this approach, we follow Mark and Sul (2001), Groen (2005), and Engle, Mark, and West (2007), who use panel error-correction models to improve the forecasting power of exchange rate models. 11

the random walk model at horizons greater than two years. Remarkably, at the four- and sevenyear horizons, forecasting equation (8) outperforms the random walk by 23 percent and 45 percent, respectively.12 We now formally test the hypothesis that the relative RMSPEs reported in panel (a) of Table 7 were generated by a DGP in which the NER is a random walk. Under this hypothesis, changes in the NER should not be predictable. We test this hypothesis using a bootstrap procedure similar to the one described in the previous subsection. In particular, we assume that NER and RER i,t i,t are generated by equations (5) and (6), where we replace RER with RER . The lag length of i,t i,t A (L) is chosen using the AIC.13 We construct 10,000 synthetic time series, each of length equal i to the size of our sample, by randomly selecting a sequence of estimated disturbances. We jointly sample the disturbances so as to preserve contemporaneous correlations across the NER and RER and across countries.14 For each synthetic time series, we compute forecasts based on equation (8) and the random walk without drift. Using these forecasts, we compute RMSPEs for each country and for the pooled countries. Panel (b) of Table 7 shows the percentage of bootstrap simulations in which the value of the relative RMSPE is less than or equal to the analogue number reported in panel (a) at different horizons. The column labeled “Years 3–7” reports the percentage of bootstrap simulations where the relative RMSPEs are lower than in the data for all yearly horizons 3 through 7. For the horizon-specific tests using σ /σ , we can reject the random walk hypothesis at the 1 percent B,h RW,h significance level using the one-quarter forecasts and at the 5 percent significance level for all individual horizons of at least three years. At the five-, six-, and seven-year horizons, we can reject the null hypothesis at the 1 percent significance level. For the joint test of yearly horizons 3–7, we can also reject the random walk hypothesis at the 1 percent significance level. There is some variability in the results for different countries and horizons. But the joint-horizon test provides very strong evidence against the random walk hypothesis for all of our benchmark countries. Panel (c) of Table 7 reports robustness results for σ /σ . The first row repeats our bench- B,h RW,h mark results. The second row reports results for the case in which we use log(RER ) instead of i,t log(RER ) in forecasting equation (8). The results we obtain are very similar to the benchmark i,t case. The third row reports the results of extending the sample period until the end of 2016. There is a mild overall deterioration in forecasting performance at long horizons. The fourth row reports results obtained by adding Japan to our benchmark specification with the sample ending in December 2016. There is a further mild deterioration in forecasting performance at long horizons. The fifth row reports results based on an unbalanced panel that includes the euro area starting in 1999:Q1, the United Kingdom starting in 1993:Q1, and Japan starting in 1973:Q1 and ending 12Additional recent evidence against random-walk-based forecasts for the NER comes from Cheung, Chinn, Pascual, and Zhang (2017). These authors examine the ability of a host of economic models to forecast NERs. They find that, relative to random walk forecasts, relative-purchasing-power-parity-based forecasts outperform other economic models. 13TheAICselectedfourlagsforAustralia, sevenlagsforCanada, eightlagsforNorway, fourlagsforSweden, andfour lags for Switzerland. 14Weagainhaveaburn-inperiodof100quarterssothattheinitialvaluesoflog(RER )aredifferentacrossbootstrap i,t samples. 12

in 1994:Q4.15 The results are about the same as the benchmark results at short horizons and only somewhat worse at long horizons. Still, the model outperforms the random walk for all horizons. At the seven-year horizon, the RMSPE associated with our forecasting equation is 40 percent lower than that associated with a random walk. The panel structure of our benchmark specification assumes that the slope coefficients are the same across all countries. A natural question is, how sensitive are our results to this assumption? The sixth row of Table 7, labeled “Country-by-country regressions,” reports results obtained by estimating separate slope coefficients for each country. There is a slight deterioration in forecasting performance. But, even without imposing the panel structure, the model outperforms the random walk at long horizons (by 41 percent at the seven-year horizon). To this point, we have maintained the assumption that the RER is stationary. To assess the robustness of our results, we redo the out-of-sample bootstrap exercises assuming that log(RER ) i,t is difference stationary. In particular, we assume that B (L)(1−L)log (cid:0) RER (cid:1) = εRER. (11) i i,t i,t Here, B (L) is a polynomial in the lag operator with roots inside the unit circle. We maintain the i assumption that changes in the NER are given by (5). As previously, we choose the lag length by the AIC and compute the relative RMSPEs.16 The implied p-values are reported in panel (d) of Table 7. The critical point is that the results we obtain are very similar to those reported in panel (b) of that table. We infer that our results are not sensitive to whether we assume that the RER has a unit root.17 In summary, the results reported in this section strongly support the view that changes in the NER are predictable at medium- and long-run horizons. By implication, it is highly statistically unlikely that the correlations documented in the previous section are spurious. 4 Interpreting our empirical results: Economic models Inthissection,weuseasequenceofeconomicmodelstointerprettheempiricalfindingsdocumented earlier. We begin with a flexible-price, two-country, complete-markets model, allowing for different specifications of monetary policy, a Taylor rule, an exogenous money growth rule, and a regime where one country seeks to dampen fluctuations in the NER. Next, we consider a sticky-price model with an incomplete-markets setting in which the only assets traded internationally are bonds. It turns out that the complete- and incomplete-markets versions of our model have very similar implications. Finally, we consider a medium-sized DSGE model with Calvo-style nominal price and wage 15The euro did not exist until 1999. The United Kingdom left the European Exchange Rate Mechanism in 1992 after Black Wednesday. Japan’s short-term interest rate has been at or near the ZLB since 1995. 16The AIC selected one lag for Australia, three lags for Canada, one lag for Norway, three lags for Sweden, and one lag for Switzerland. 17The results are not sensitive to assuming that the RER is a random walk. 13

rigidities in which producers set prices in local currencies. We allow for technology shocks in each country and shocks to the demand for domestic bonds. The latter shocks imply that unconditional UIP does not hold in our model. 4.1 Flexible-price, complete-markets model The model consists of two completely symmetric countries. We first describe the households’ problems and then discuss the firms’ problems. 4.1.1 Households The domestic economy is populated by a representative household whose preferences are given by E (cid:88) ∞ βj (cid:34) log(C )− χ L1+φ+µ (M t+j /P t+j )1−σM (cid:35) . (12) t t+j 1+φ t+j 1−σ M j=0 Here, C denotes consumption, L hours worked, M end-of-period nominal money balances, P the t t t t price of consumption goods, and E the expectations operator conditional on time-t information. t We assume that 0 < β < 1, σ > 1, and χ and µ are positive scalars. M Households can trade in a complete set of domestic and international contingent claims. The domestic household’s flow budget constraint is given by B +NER B +P C +M = R B +NER R∗ B +W L +T +M . (13) H,t t F,t t t t t−1 H,t−1 t t−1 F,t−1 t t t t−1 Here,B andB arenominalbalancesofhomeandforeignbonds;NER isthenominalexchange H,t F,t t rate, defined as in our empirical section to be the price of the foreign currency unit (units of home currency per unit of foreign currency); R is the nominal interest rate on the home bond; R∗ is the t t nominal interest rate on the foreign bond; W is the nominal wage rate; and T denotes nominal t t lump-sumprofitsandtaxes. Fornotationalease, wehavesuppressedthehousehold’spurchasesand payoffs of contingent claims. With complete markets, the presence of one-period nominal bonds is redundant since these bonds can be synthesized using state-contingent claims. The first-order conditions with respect to labor supply, money balances, and consumption are W χLφC = t , (14) t t P t (cid:18) M (cid:19)−σM (cid:18) R −1 (cid:19) 1 t t µ = , (15) P R C t t t C t 1 = βR E , (16) t t C π t+1 t+1 where π ≡ P /P denotes the inflation rate. Equation (15) characterizes money demand by t t t−1 domestic agents. Since households derive utility only from their country’s money, domestic agents do not hold foreign money balances. 14

We use stars to denote the prices and quantities in the foreign country. The preferences of the foreign household are given by ∞  (cid:16) M∗ /P∗ (cid:17)1−σM  E (cid:88) βjlog (cid:0) C∗ (cid:1) − χ (cid:0) L∗ (cid:1)1+φ +µ t+j t+j . (17) t  t+j 1+φ t+j 1−σ  M j=0 The foreign household’s flow budget constraint is given by B∗ +NER−1B∗ +P∗C∗+M∗ = R∗ B +NER−1R B∗ +W∗L∗+T∗+M∗ . (18) F,t t H,t t t t t−1 F,t−1 t t−1 H,t−1 t t t t−1 The first-order conditions for the foreign household with respect to labor supply, money balances, and consumption are W∗ χ(L∗)φC∗ = t , (19) t t P∗ t (cid:18) M∗(cid:19)−σM (cid:18) R∗−1 (cid:19) 1 µ t = t , (20) P∗ R∗ C∗ t t t C∗ 1 = βR∗E t . (21) t t C∗ π∗ t+1 t+1 As in our empirical section, we define the real exchange rate, RER , as the price of the foreign t consumption good in units of the home consumption good: NER P∗ RER = t t . (22) t P t With this definition, an increase in RER corresponds to a rise in the relative price of the foreign t good. Complete markets and symmetry of initial conditions imply C t = RER . (23) C∗ t t Combining equations (21) and (23) we obtain C NER 1 = βR∗E t t+1 . (24) t t C π NER t+1 t+1 t Equations (16) and (23) imply C∗ NER 1 = βR E t t . (25) t t C∗ π∗ NER t+1 t+1 t+1 15

4.1.2 Firms The domestic final good, Y , is produced by combining domestic and foreign goods (Y and Y , t H,t F,t respectively) according to the technology (cid:104) (cid:105)1 Y = ω1−ρ(Y )ρ+(1−ω)1−ρ(Y )ρ ρ . (26) t H,t F,t Here, ω > 0 controls the importance of home bias in consumption. The parameter ρ ≤ 1 controls the elasticity of substitution between home and foreign goods. Similarly, the foreign final good, Y∗, is produced by combining domestic and foreign goods (Y∗ and Y∗ , respectively) according t H,t F,t to the technology (cid:104) (cid:105)1 Y∗ = ω1−ρ(cid:0) Y∗ (cid:1)ρ +(1−ω)1−ρ(cid:0) Y∗ (cid:1)ρ ρ . (27) t F,t H,t The domestic goods used in the production of the domestic final good (Y ) and in the pro- H,t duction of the foreign final good (Y∗ ) are produced according to the technologies H,t Y H,t = (cid:18)(cid:90) 1 X H,t (j) ν− ν 1 dj (cid:19) ν− ν 1 and Y H ∗ ,t = (cid:18)(cid:90) 1 X H ∗ ,t (j) ν− ν 1 dj (cid:19) ν− ν 1 . (28) 0 0 Here, X (j) and X∗ (j) are domestic intermediate goods produced by monopolist j using the H,t H,t linear technology X (j)+X∗ (j) = A L (j). (29) H,t H,t t t The variable L (j) denotes the quantity of labor employed by monopolist j, and A denotes the t t state of time-t technology, which evolves according to log(A ) = ρ log(A )+ε , (30) t A t−1 A,t where |ρ | < 1. The parameter ν > 1 controls the degree of substitutability between different A intermediate inputs. Theforeigngoodsusedintheproductionofthedomesticfinalgood(Y )andintheproduction F,t of the foreign final good (Y∗ ) are produced according to the technologies F,t Y F,t = (cid:18)(cid:90) 1 X F,t (j) ν− ν 1 dj (cid:19) ν− ν 1 and Y F ∗ ,t = (cid:18)(cid:90) 1 X F ∗ ,t (j) ν− ν 1 dj (cid:19) ν− ν 1 . (31) 0 0 Here,X (j)andX∗ (j)areforeignintermediategoodsproducedbymonopolistj usingthelinear F,t F,t technology X (j)+X∗ (j) = A∗L∗(j), (32) F,t F,t t t where L∗(j) is the labor employed by monopolist j in the foreign country and A∗ denotes the state t t of technology in the foreign country at time t, which evolves according to log(A∗) = ρ log(A∗ )+ε∗ . (33) t A t−1 A,t 16

Monopolists in the home country choose P˜ (j) and P˜∗ (j) to maximize per-period profits given H,t H,t by (cid:16) (cid:17) (cid:16) (cid:17) P˜ (j)−W /A X (j)+ NER P˜∗ (j)−W /A X∗ (j), (34) H,t t t H,t t H,t t t H,t subject to the demand curves of final good producers: (cid:32) P˜ (j) (cid:33)−ν (cid:32) P˜∗ (j) (cid:33)−ν X (j) = H,t Y and X∗ (j) = H,t Y∗ . (35) H,t P H,t H,t P∗ H,t H,t H,t The aggregate price indexes for X and X∗ , denoted by P and P∗ , can be expressed as H,t H,t H,t H,t (cid:18)(cid:90) 1(cid:104) (cid:105)1−ν (cid:19) 1− 1 ν (cid:18)(cid:90) 1(cid:104) (cid:105)1−ν (cid:19) 1− 1 ν P ≡ P˜ (j) dj and P∗ ≡ P˜∗ (j) dj . (36) H,t H,t H,t H,t 0 0 Monopolists in the foreign country choose P˜ (j) and P˜∗ (j) to maximize profits given by F,t F,t (cid:16) (cid:17) (cid:16) (cid:17) P˜∗ (j)−W∗/A∗ X∗ (j)+ NER−1P˜ (j)−W∗/A∗ X (j), (37) F,t t t F,t t F,t t t F,t subject to the demand curves of final good producers: (cid:32) P˜ (j) (cid:33)−ν (cid:32) P˜∗ (j) (cid:33)−ν X (j) = F,t Y and X∗ (j) = F,t Y∗ . (38) F,t P F,t F,t P∗ F,t F,t F,t Here, the aggregate price index for X and X∗ , denoted by P and P∗ , can be expressed as F,t F,t F,t F,t (cid:18)(cid:90) 1(cid:104) (cid:105)1−ν (cid:19) 1− 1 ν (cid:18)(cid:90) 1(cid:104) (cid:105)1−ν (cid:19) 1− 1 ν P ≡ P˜ (j) dj and P∗ ≡ P˜∗ (j) dj . (39) F,t F,t F,t F,t 0 0 The first-order conditions for the monopolists imply ν W P˜ (j) = NER P˜∗ (j) = t , (40) H,t t H,t ν −1 A t where P˜ (j) and P˜∗ (j) are prices that the home monopolist charges in the home and foreign H,t H,t markets, respectively. Similarly, ν W∗ NER−1P˜ (j) = P˜∗ (j) = t . (41) t F,t F,t ν −1 A∗ t Here, P˜ (j) and P˜∗ (j) are the prices that the foreign monopolist charges in the home and F,t F,t foreign markets, respectively. Equations (40) and (41) imply that the law of one price holds for intermediate goods. 17

4.1.3 Monetary policy, market clearing, and the aggregate resource constraint In our first specification of monetary policy, the domestic monetary authority sets the nominal interest rate according to the following Taylor rule: (cid:16) (cid:17)1−γ R = (R )γ Rπθπ exp(ε ). (42) t t−1 t R,t We assume that the Taylor principle holds, so that θ > 1. In addition, R = β−1, and εR is an π t independently and identically distributed (iid) shock to monetary policy. To simplify, we assume that the inflation target is zero in both countries. The foreign monetary authority follows a similar rule: R∗ = (cid:0) R∗ (cid:1)γ (cid:16) R(π∗)θπ (cid:17)1−γ exp (cid:0) ε∗ (cid:1) . (43) t t−1 t R,t We abstract from the output gap in the Taylor rule to ease the comparison between the flexibleprice version of the model, which has a zero output gap, and the sticky-price version of the model. In practice, the output gap coefficients in estimated versions of the Taylor rule are quite small (see, e.g. Clarida, Gali, and Gertler (1998)). Modifying the Taylor rule to include empirically plausible responses to the output gap has a negligible effect on our results.18 In our second specification of monetary policy, the domestic monetary authority sets the growth rate of the money supply according to: (cid:18) (cid:19) M log t = xM, where xM = ρ xM +εM. (44) M t t XM t−1 t t−1 Here, |ρ | < 1 and εM is an iid shock to monetary policy. For convenience, we assume that the XM t unconditionalmeangrowthrateofnominalmoneybalancesiszero. Theforeignmonetaryauthority follows a similar rule so that (cid:18) M∗ (cid:19) log t = xM∗, where xM∗ = ρ xM∗ +εM∗. (45) M∗ t t XM t−1 t t−1 In our third specification of monetary policy, the domestic monetary authority sets the nominal interest rate as in (42), but the foreign monetary authority uses an augmented Taylor rule that includes a term that targets the NER. (cid:16) (cid:17) R∗ = R NER−θNER exp (cid:0) ε∗ (cid:1) . (46) t t R,t We assume that the Taylor principle holds so that θ > 1, and that θ > 0 so that the nominal π NER interest rate in the foreign country rises whenever there is a depreciation of the foreign currency. We refer to the three specifications of monetary policy as the Taylor rule, the exogenous money growth rule, and the exchange rate targeting rule, respectively. 18Suppose we define the output gap as the percentage deviation of output from its steady-state value and include it in the Taylor rule with a coefficient equal to 0.5. The resulting impulse functions are very similar to those obtained for a version of the model with a Taylor rule that excludes the output gap. 18

We assume that government purchases, G , evolve according to: t (cid:18) (cid:19) (cid:18) (cid:19) G G log t = ρ log t−1 +εG, (47) G G G t and that, without loss of generality, the government budget is balanced each period using lumpsum taxes. Here, |ρ | < 1 and εG is an iid shock to government purchases. The composition of G t government expenditures in terms of domestic and foreign intermediate goods (Y and Y ) is the H,t F,t same as that of the domestic household’s final consumption good. Similarly, government purchases in the foreign country, G∗, evolve according to t (cid:18) G∗(cid:19) (cid:18)G∗ (cid:19) log t = ρ log t−1 +εG∗, (48) G G G t where εG∗ is an iid shock to government purchases and the government budget is balanced each t period using lump-sum taxes. The composition of government expenditures in terms of domestic and foreign intermediate goods (Y∗ and Y∗ ) is the same as that of the foreign household’s final F,t H,t consumption good. Since bonds are in zero net supply, bond-market clearing implies B +B∗ = 0 and B +B∗ = 0. (49) H,t H,t F,t F,t Labor market clearing requires that: (cid:90) 1 (cid:90) 1 L = L (j)dj and L∗ = L∗(j)dj. (50) t t t t 0 0 Market clearing in the intermediate input markets requires that X (j)+X∗ (j) = A L and X (j)+X∗ (j) = A∗L∗. (51) H,t H,t t t F,t F,t t t Finally, the aggregate resource constraints are given by Y = C +G and Y∗ = C∗+G∗. (52) t t t t t t 4.1.4 Impulse response functions Intheexamplesinthissectionweusethefollowingparametervalues. WeassumeaFrischelasticity of labor supply equal to 1 (φ = 1) and, as in Christiano, Eichenbaum, and Evans (2005), set σ = 10.62. We set the value of β so that the steady-state real interest rate is 3 percent. As M in Backus, Kehoe, and Kydland (1992), we assume that the elasticity of substitution between domestic and foreign goods in the consumption aggregator is 1.5 (ρ = 1/3) and the import share is 10 percent (ω = 0.9) so that there is home bias in consumption. We set ν = 6, which implies an average markup of 20 percent. This value falls well within the range considered by Altig et al. (2011). We normalize the value of χ, which affects the marginal disutility of labor, so that hours worked in the steady state is equal to 1. We assume that monetary policy is given by the Taylor 19

rules (42) and (43). We set θ to 1.5 so as to satisfy the Taylor principle. For ease of exposition, in π this section, we set γ = 0 so that there is no interest rate smoothing. We set ρ = 0.946, a value A that is very similar to those used in the literature (e.g., Hansen (1985)). In section 5.1.2, we discuss the estimation procedure underlying our choice of this value. We solve the model by log-linearizing the equilibrium conditions. Figure 2 displays the impulse response to a negative technology shock in the home country. Home bias in consumption plays a critical role in the impulse response function. First, the RER fallssincehomegoodsaremorecostlytoproduceandthehomeconsumptionbasketplacesahigher weight on these goods. Second, domestic consumption falls by more than foreign consumption because domestic agents consume more of the home good whose relative cost of production has risen. Third, the households’ Euler equations imply that the domestic real interest rate must rise by more than the foreign real interest rate. The Taylor rule and the Taylor principle imply that high real interest rates are associated with high nominal interest rates and high inflation rates. It follows that the domestic nominal interest rate and the domestic inflation rate rise by more than their foreign counterparts. This result is inconsistent with the naive intuition that inflation has to be lower in the home country in order for the RER to return to its pre-shock level. In fact, inflation is persistently higher in the home country. So the RER reverts to its steady-state value via changes in the NER. The NER has to change by enough to offset both the initial movement in the RER and the difference between the domestic and foreign inflation rates. From Figure 2, we see that the Taylor rule keeps prices relatively stable and the RER falls at time zero via an appreciation of the home currency. To understand the last result, it is useful to consider the model’s log-linearized equilibrium conditions. These conditions imply that the response of the RER, to a technology shock is given by R(cid:100)ER t = κAˆ t . (53) Here, xˆ denotes the log deviation of x from its steady state level and t (cid:26)(cid:20) 2ω 2(ω−1) 2ω−1 (cid:21) C 1 (cid:27)−1 κ = + φ+ (φ+1). (54) ρ−1 2ω−1 σ Y 2ω−1 Equation (53) implies that the RER inherits the first-order autoregressive (AR(1)) nature of the technology shock so that E t R(cid:100)ER t+1 = ρ A R(cid:100)ER t . (55) Combining the linearized home- and foreign-country intertemporal Euler equations (16) and (21), the relation between the two countries’ marginal utilities implied by complete markets (23), and the Taylor rules for the two countries (42) and (43), we obtain 1−ρ πˆ t −πˆ∗ t = − θ −ρ A R(cid:100)ER t . (56) π A (cid:12) (cid:12) Since the Taylor principle holds (θ > 1), we have (cid:12) 1−ρA (cid:12) < 1. Given that the RER is equal to π (cid:12)θπ−ρA (cid:12) 20

NER P∗/P , equation (56) implies that, on impact, the RER falls by more than P∗/P . It follows t t t t t t that NER must initially fall, i.e., the home currency appreciates on impact. t Recallthatinresponsetothetechnologyshock, boththerealandthenominalinterestratesrise more at home than abroad. The technology shock is persistent, so there is a persistent gap between the domestic and foreign nominal interest rates. Since UIP holds in the log-linear equilibrium, the domestic currency must depreciate over time to compensate for the nominal interest rate gap. So the home currency appreciates on impact and then depreciates. This pattern is reminiscent of the overshooting phenomenon emphasized by Dornbusch (1976).19 Domestic inflation is persistently higher than foreign inflation, so the domestic price level rises by more than the foreign price level. This result, along with the law of one price for intermediate goods, implies that the home currency depreciates over time to an asymptotically lower value (the figure displays the price of the foreign currency, which is rising to a higher value).20 As the previous discussion makes clear, home bias plays a critical role in our results. Absent that bias, the consumption basket would be the same in both countries, and the RER would be equal to 1 in each period after the technology shock. Equation (56) implies that if the RER is constant, so, too, is the relative inflation and the NER. 4.1.5 Implied regression coefficients We now assess the model’s ability to account for the basic regressions that motivate our analysis (equations(2)and(3)). InaversionofourmodeldrivenonlybyshockstoA andA∗,theprobability t t limits (plims) of the regression coefficients, βNER and βπ , are given by i,h i,h 1−ρh βNER = − A , (57) i,h 1−ρ /θ A π and 1−ρh βπ = A . (58) i,h θ /ρ −1 π A Equation (57) implies that βNER is negative for all h and increases in absolute value with h. The i,h intuition for these results is as follows. In the model, a low current value of the RER predicts a future appreciation of the foreign currency, so the slope of the regression is negative. The slope increases in absolute value with the horizon because the cumulative depreciation of the home currency increases over time. Notice that the more aggressive is monetary policy (i.e., the larger is θ ), the smaller is the π absolute value of βNER. The intuition for this result is as follows. After a domestic technology i,h shock, π is higher than π∗. Equation (55) implies that the RER must revert to its steady-state t t 19InDornbusch(1976),anunanticipatedpermanentchangeinthemoneysupplycausestheNERtoovershootrelative to its new long-run level. 20In this version of the model, temporary technology shocks generate permanent changes in the NER. This property doesnotgenerallyholdinversionsofthemodelwherebothcountriesadoptTaylorrulesoftheform(46). Inournumerical experiments, we find that when we place a small weight on the exchange rate (θ ), the NER becomes stationary. At NER the same time, the model’s quantitative properties as summarized by the implied values of βNER and βπ are virtually i,h i,h unchanged. 21

level at a rate ρ . The higher is θ , the lower is |π − π∗|, and the less the domestic currency A π t t needs to depreciate to bring about the required adjustment in the RER. So the absolute value of βNER is decreasing in θ . Equation (58) implies that βπ is positive for all h and converges to i,h π i,h ρ /(θ −ρ ). Consistent with the previous intuition, the higher is θ , the lower is βπ for all h. A π A π i,h The sum of the two slopes is given by: βNER+βπ = −(1−ρh). (59) i,h i,h A This sum converges to negative 1 as h goes to infinity, reflecting the fact that the RER must eventually converge to its pre-shock steady-state level through changes in relative prices or changes in the NER. Figure 4 displays the small-sample average estimates of βNER and βπ . These statistics are i,h i,h calculatedasfollows. Wesimulate10,000synthetictimeseriesusingthemodelwithonlytechnology shocksastheDGP.Eachsynthetictimeseriesisoflengthequaltooursamplesize. Foreachsample, we estimate βNER and βπ . We then compute the average values across the different samples. i,h i,h Consistent with our analytic expressions for the plims of these regressions, the absolute value of each coefficient grows with the horizon.21 The ability of the model to rationalize the regression coefficients does not depend on technology shocks per se. Consider a government spending shock that is mean reverting. An increase in government spending causes domestic consumption to fall by more than foreign consumption because the government consumes the domestic consumption bundle. As a result, the RER falls. Because the shock is mean reverting, domestic consumption and foreign consumption rise over time, but by more in the home country. As a result, real interest rates are higher at home than abroad. Under a Taylor rule, relatively high real interest rates accompany relatively high inflation rates, which, all else being equal, would cause the RER to decline further. As a result, the RER converges to its pre-shock level through the NER. The exact magnitude of the response of inflation and the NER depends on other model features, such as sticky prices and wages. 4.1.6 Economy with money growth rule Consistent with the intuition in Engel (2012), we now show that, when monetary policy follows a money growth rule (equation (44)), the flexible-price model is much less successful at accounting for our regression result. The impulse response functions to a technology shock are displayed in Figure 3. The following features are worth noting. First, prices in both countries move by much more than they did under the Taylor rule. So the movements in the NER required to validate the given equilibrium path of the RER are much smaller than under a Taylor rule. Second, since the growth rate of money does not increase after the shock, the price level eventually reverts to its pre-shock steady-state level. As a result, the NER also reverts to its steady state. Third, not all of the adjustment in 21In practice, we do not find a large difference between the plims of βNER and βπ and the small-sample average i,h i,h estimates. 22

the RER occurs via the price level, so there are still predictable movements in the NER. But these movements are much smaller than under a Taylor rule. This property is reflected in the model-implied plim of the small-sample regression slopes for our NER regression. These slopes are much smaller than under a Taylor rule (see Figure 4). The reason that movements in the NER are smaller than under a Taylor rule is that relative inflation rates help move the RER back to the steady state. Under a Taylor rule, prices move the RER away from the steady state. 4.1.7 Economy with NER targeting rule We now discuss the response of the economy to a productivity shock when monetary policy is given by equation (46) in the foreign country and by equation (42) in the domestic country. We set θ = 1, which reduces the volatility of the per-period change in the NER by 75 percent relative NER to our benchmark calibration. This setting corresponds to a situation in which foreign monetary policy places a high priority on exchange rate stabilization. Figure 5 shows the response of this economy to a technology shock. Since prices are flexible, the behavior of the real variables is the same as when both countries follow a Taylor rule. The NER is much more stable in this version of the model. Since UIP holds, this stability requires the nominal interest rates to be similar in the two countries. After the shock, consumption growth is higher in the domestic economy. As a consequence, the domestic real interest rate exceeds the foreign real interest rate. Since the Fisher equation holds, to make the two nominal interest rates similar, foreign monetary policy must ensure that foreign expected inflation is higher than domestic inflation. As a result, P∗ rises faster than P , so t t the behavior of prices drives the RER back to its steady state. This property is reflected in the regression coefficients displayed in panel (b) of Figure 4. We see that the bulk of the adjustment of the RER toward the steady state is accomplished by prices, not by the NER. 4.2 Sticky-price, incomplete-markets model As it turns out, the implications of the flexible price model discussed previously and those of the sticky price model discussed in this section are very similar when markets are complete or when the only assets that can be traded internationally are one-period nominal bonds. The basic structure is as in the previous subsection, with addition features similar to models used in Kollmann (2001) and in Gali and Monacelli (2005). 4.2.1 Incomplete asset markets Itiswellknownthatwithincompleteassetmarkets, theequilibriumprocessfortheRER inmodels like ours has a unit root. To avoid this implication, authors like Schmitt-Grohe and Uribe (2003) assume that there is a small quadratic cost to holding bonds. We make a similar assumption in our 23

model. The domestic household’s budget constraint is given by φ (cid:18) NER B (cid:19)2 B t F,t B +NER B +P C +M + P = H,t t F,t t t t t 2 P t R B +NER R∗ B +W L + T +M . (60) t−1 H,t−1 t t−1 F,t−1 t t t t−1 We assume that the quadratic cost of holding bonds applies to bonds from the other country. The foreign household’s budget constraint is given by φ (cid:32) NER−1B∗ (cid:33)2 B∗ +NER−1B∗ +P∗C∗+M∗+ B t H,t P∗ = F,t t H,t t t t 2 P∗ t t R∗ B∗ +NER−1R B∗ +W∗L∗ + T∗+M∗ . (61) t−1 F,t−1 t t−1 H,t−1 t t t t−1 Equation (23) no longer holds, and the home household’s budget constraint is used to close the model. 4.2.2 Sticky prices Monopolist producers set nominal prices in currency units that are local to where the good is sold. The technology for producing final goods is still given by equation (26). Intermediate-goodsproducing firms set prices according to a variant of the mechanism spelled out in Calvo (1983). In each period, a firm faces a constant probability, 1−ξ, of being able to reoptimize its nominal price. The ability to reoptimize prices is independent across firms and time. Domestic intermediate goods firms choose P˜ (i) and P˜∗ (i) to maximize the objective function H,t H,t E (cid:88) ∞ βjΛ (cid:40)(cid:32) P˜ H,t (i) −MC (cid:33) X (i)+ (cid:32) NER P˜ H ∗ ,t (i) −MC (cid:33) X∗ (i) (cid:41) , t t+j P t+j H,t+j t+j P t+j H,t+j t+j t+j j=0 (62) subject to the demand equations (35). Here, MC denotes the real marginal cost in period t+j, t+j and βjΛ is the utility value of profits in period t+j to the household in period t. t+j Foreign intermediate goods firms choose P˜ (i) and P˜∗ (i) to maximize the objective function H,t H,t E (cid:88) ∞ βjΛ∗ (cid:40)(cid:32) P˜ F ∗ ,t (i) −MC∗ (cid:33) X∗ (i)+ (cid:32) NER−1 P˜ F,t (i) −MC∗ (cid:33) X (i) (cid:41) , (63) t t+j P∗ t+j F,t+j t+j P∗ t+j F,t+j j=0 t+j t+j subject to equations (38). In all other respects, the model is the same as in the previous subsection. 4.2.3 Impulse response to a technology shock Figure 6 displays the response of the economy to a negative technology shock in the home country. These effects are similar to those in the flexible-price model. The key difference is that in the sticky-price model, the response of π , π , π∗ , π∗ is attenuated relative to the flexible-price H,t F,t H,t F,t model. Interestingly, the effect of sticky prices on overall inflation is ambiguous. When prices are 24

flexible, producers of the foreign good initially reduce the price they charge in the home market. Thiseffecthelpsreducethedomesticrateofinflationintheflexible-pricemodel. Withstickyprices, this effect is attenuated relative to the flexible-price model. So, depending on parameter values, domestic inflation can be higher or lower in the sticky price model than in the flexible-price model. Because the negative technology shock leads to a decline in RER followed by a persistent t depreciation of the home currency, the model-implied values for βNER in the economy with only i,h technology shocks are negative and grow in absolute value with the horizon. As in the flexible-price model, the basic intuition is that a negative technology shock drives down the RER. Over time, the NER rises to its new steady-state value. So a low value of the RER is associated with future increases in the NER. 4.3 Medium-scale model with nominal rigidities Inthissubsection, weinvestigatewhetheranempiricallyplausibleversionofourmodelcanaccount for the facts that we document. By “empirically plausible,” we mean that the model is consistent with the persistence of RERs, the short-run failure of UIP and PPP, and the high correlation between the RER and the NER. In addition, the model should be able to account for our out-of sample forecasting results for the NER. 4.3.1 Monopolists The production of domestic and foreign final goods (Y and Y∗) and domestic (Y and Y∗ ) and t t H,t H,t foreign (Y and Y∗ ) intermediate goods is the same as in the benchmark models. The final good F,t F,t is used for consumption (C and C∗), investment (I and I∗), and government purchases (G and t t t t t G∗) so that t Y = C +I +G and Y∗ = C∗+I∗+G∗. (64) t t t t t t t t Differentiated intermediate goods supplied by monopolist i (X (i) and X∗ (i)) are produced H,t H,t with capital (K (i)) and labor (L (i)): t t X (i)+X∗ (i) = A K (i)αL (i)1−α. (65) H,t H,t t t t The variable A denotes the time-t level of technology, which again evolves according to equation t (30). Marginal cost in the home country is given by (R )α(W /P )1−α K,t t t MC = . (66) t (1−α)1−αααA t As before, domestic monopolist i sets prices in local currency subject to Calvo-style frictions. The monopolist maximizes the objective function given by equation (62) subject to the demand curvesforitsgoods. Toconserveonspace,wedonotdescribetheproblemoftheforeignmonopolist i. That problem is entirely symmetric to that of the domestic monopolist i. 25

4.3.2 Households Each household has a continuum of members indexed j ∈ (0,1). Each member of the household belongs to a union that monopolistically supplies labor of type j. The union sets the wage W j,t subject to constraint (68) and Calvo-style wage frictions, modeled as in Erceg, Henderson, and Levin (2000). The wage for labor of type j remains constant with probability ξ and is updated w with probability 1−ξ . w Intermediate producers purchase a homogeneous labor input from a representative labor contractor. Thelatterproducesthehomogeneouslaborinputbycombiningdifferentiatedlaborinputs, l , j ∈ (0,1), using the technology j,t (cid:34) (cid:35) νL (cid:90) 1 νL−1 νL−1 L = l νL dj . (67) t j,t 0 Let W denote the cost of hiring a unit of L . Labor contractors are perfectly competitive and take t t the nominal wage rates, W , and W , j ∈ (0,1), as given. Profit maximization on the part of t j,t contractors implies (cid:20) W (cid:21)−νL j,t l = L . (68) j,t t W t Perfect competition and equation (67) imply W = (cid:20)(cid:90) 1 W 1−νLdj (cid:21) 1− 1 νL . (69) t j,t 0 The preferences of the household are given by E (cid:88) ∞ βi  log (cid:0) C −hC¯ (cid:1) − χ (cid:90) 1 L1+φdj +µ (cid:16) M Pt t + + i i (cid:17)1−σM +log(η )V (cid:18) B H,t+i (cid:19)  . t  t+i t+i−1 1+φ j,t+i 1−σ t+i P  0 M t i=0 (70) Here, C is consumption, C¯ is average aggregate consumption, and L is hours worked. t t t Recall that the only assets that can be traded internationally are one-period nominal bonds. As in McCallum (1994), we allow for shocks that break UIP in log-linearized versions of the model. Instead of introducing a shock directly into the UIP condition, we assume that households derive utility from domestic bond holdings and that this utility flow varies over time. The function V that governs the utility flow from the stock of domestic bonds is increasing, is strictly concave, and has both positive and negative support. For convenience, we assume that η t is 1 in the steady state, so the flow utility from bonds is zero in the steady state. Outside of the steady state, there may be shocks that put a premium on one bond or the other—those arising from flights to safety or liquidity, for example. 26

The household budget constraint is B +NER B +P C +P I = R B +NER R∗ B +P R K (71) H,t t F,t t t I,t t t−1 H,t−1 t t−1 F,t−1 t K,t t φ (cid:18) NER B (cid:19)2 (cid:90) 1 B t F,t − P + W L dj +T , t j,t j,t t 2 P t 0 where B and B are nominal balances of home and foreign bonds; P is the price of final goods H,t F,t t in the home country; R is the nominal interest rate on the home bond and R∗ is the nominal t t interest rate on the foreign bond; W is the wage rate; R is the rental rate on capital; K , I are t K,t t t investment goods; and T are lump-sum profits and taxes. For notational ease, we have suppressed t the household’s purchases and payoffs of domestic contingent claims. The capital accumulation equation is (cid:34) (cid:35) φ (cid:18) I (cid:19)2 K t K = I 1− −1 +(1−δ)K . (72) t+1 t t 2 I t−1 The sequence of events within each period is as follows. First, the technology shocks and spread shocks are realized. Second, the household makes its consumption and asset decisions. Third, wage rates are updated. The problem of the foreign household is entirely symmetric with one exception. We assume that foreign households derive utility from their holdings of the domestic country’s bonds, which we conceptualize as bonds that are internationally desired for special reasons. 4.3.3 Breaking UIP In a log-linearized version of the model without shocks to the utility flow from real bond holdings, UIP holds. To show this result, consider the log-linearized first-order conditions of the home household with respect to B and B : H,t F,t (cid:16) (cid:17) Cˆ = CV(cid:48)(0)ηˆ +Rˆ +E −Cˆ −πˆ , (73) t t t t t+1 t+1 (cid:16) (cid:17) Cˆ t +φ B b F,t = Rˆ t ∗+E t −Cˆ t+1 −πˆ t+1 +∆N(cid:100)ER t+1 . (74) Here, ∆N(cid:100)ER t+1 ≡ log(NER t+1 /NER t ), and C is the steady-state level of C t . It is convenient to normalize V(cid:48)(0) to be equal to 1/C. Combining equations (73) and (74) and ignoring the small term associated with φ , we obtain B (cid:16) (cid:17) Rˆ t −Rˆ t ∗ = E t ∆N(cid:100)ER t+1 −ηˆ t . (75) This equation is identical to the reduced-form equation assumed by McCallum (1994).22 Absent the spread shocks ηˆ , equation (75) corresponds to the classic UIP condition t (cid:16) (cid:17) Rˆ t −Rˆ t ∗ = E t ∆N(cid:100)ER t+1 . (76) (cid:16) (cid:17) 22If we do not ignore φ B , equation (75) is replaced by Rˆ t −Rˆ t ∗ =E t ∆N(cid:100)ER t+1 −ηˆ t −φ B b F,t . 27

All of the other shocks in our model induce movements in nominal interest rates and exchange rates that are consistent with equation (76). Conditional on these shocks occurring, UIP holds. However, UIP does not hold unconditionally in the presence of spread shocks, and traditional tests would reject the hypothesis of UIP. For example, the classic Fama (1984) test involves running the regression (cid:16) (cid:17) ∆N(cid:100)ER t+1 = α 0 +α 1 Rˆ t −Rˆ t ∗ +ε t , (77) andtestingthenullhypothesisthatα = 0andα = 1. Ourmodelimpliesthatthisnullhypothesis 0 1 should be rejected because of a negative covariance between the error term and the interest rate differential. To see this result, consider a positive iid shock to ηˆ . A rise in ηˆ is equivalent to a rise t t in ε . Since domestic bonds are in zero net supply, the yield on domestic bonds must fall, leading to t a decline in Rˆ −Rˆ∗ . So ε covaries negatively with Rˆ −Rˆ∗ which causes the plim of an ordinary t t t t t least squares estimate of α to be negative in an economy driven only by spread shocks. 1 4.3.4 Calibration For the purposes of calibration and estimation, we assume that the domestic and foreign monetary authorities follow Taylor rules (42) and (43), respectively. We divide the parameters into two categories: those that we calibrate and those that we estimate. The calibrated parameters are listed in Table 8. We maintain the parameter values used in the previous sections and set the habit persistenceparameter, h; theprobabilitythatfirmscannotadjusttheirprice, ξ; andtheprobability that labor suppliers cannot readjust their nominal wage, ξ , to the point estimates reported in W Christiano, Eichenbaum, and Evans (2005). We set the value of ν so as to imply a 5 percent L steady state markup. We calibrate the parameters ρ = 0.85. This value is equal to the persistence η of the spread shock in the closed-economy version of the new-Keynesian model estimated by Gust et al. (2016). We estimate the remaining parameters ρ , σ , and σ so that the model is consistent with the A A η following moments of the data. Technology shocks are assumed to be uncorrelated across countries. We require that the first-order autocorrelation of HP-filtered model output and the standard deviation of the innovation to a fitted AR(1) coincide with the analogous statistics estimated using quarterly U.S. data for the period from 1973:Q1 through 2007:Q4.23 In addition, we require that the model be consistent with the slope coefficient of the Fama regression—defined by equation (77)—being equal to 0.5.24 We find that the parameters that give us the best fit are ρ = 0.946, A σ = 0.010, and σ = 0.001. A η 4.3.5 Model results Here, we discuss the model’s implications for the statistics that we emphasized in our empirical analysis. Table 9 reports the model’s implications for the coefficients in regressions (2) and (3). We 23We measure output using per capita real gross domestic product. We calculate model moments as small-sample average estimates using 140 periods—the same length as in our data. 24This value is well within standard errors of the slope coefficients reported in the literature. 28

report results for two versions of the model (with flexible prices and wages and with sticky prices and wages) and three monetary regimes (Taylor rule, money growth rule, and NER targeting). Two results are worth noting. First, the model with a Taylor rule does a good job of accounting fortheestimatedvaluesofβNERandtheirriseinabsolutevaluewiththeregressionhorizon. Second, i,h if sampling uncertainty is taken into account, the model with a Taylor rule is also consistent with the positive values of βπ and the fact that they rise with the horizon. With a money growth rule i,h and an NER targeting rule, the model predicts negative values of βπ and values of βNER that are i,h i,h small in magnitude relative to the data. Table 11 reports the standard deviations of ∆RER and ∆NER for the countries in our sample and the Taylor-rule version of our model. In addition, we report estimates for the autocorrelation of the RER. Four features of Table 11 are worth noting. First, our model is consistent with the well-known fact that real and nominal exchange rates are equally volatile (Mussa (1986), Rogoff (1996), and Burstein and Gopinath (2015)). Second, the model understates the volatility of ∆RER and∆NER. Thisunderstatementreflectsinpartthesmallnumberofshocksincludedinthemodel (two technology shocks and a spread shock). Third, the model produces persistent estimates of the cyclical component of HP-filtered RERs that are within sampling uncertainty of the data. Fourth, the model with nominal rigidities accounts for the classic Mussa observations that changes in real and nominal exchange rates are highly correlated. Taken as a whole, these results indicate that our model is broadly consistent with the properties of the data stressed by Mussa (1986) and Rogoff (1996). Table 11 also reports the key properties of a version of our model without nominal rigidities. This version of the model captures many qualitative features of the data. However, the model does not account for the high correlation between the NER and RER. For every country in our sample, that correlation is above 0.95. In the model without nominal rigidities, this correlation is 0.65. 4.3.6 Forecasting implications Here, we assess whether the model can account for the key characteristics of our out-of-sample forecasting results. The results from our panel regressions in the data are repeated in the first row of Table 10. We use our economic model to generate 10,000 synthetic samples, each of length equal to the number of quarters in our sample, for the same number of countries as our benchmark flexible-exchange rate countries. We redo our panel forecasting exercise on each of the synthetic data sets and compute the expected value of the small-sample RMSPEs, i.e., we compute the average of the forecasts at different horizons across the 10,000 synthetic samples. In addition, we compute the standard deviation of the small-sample RMSPEs across the synthetic data sets. The second row of Table 10 reports the average value of the ratio of the RMSPE of our forecasts to the RMSPEoftherandomwalkforecastsacrossthesyntheticdatasetswhenweassumethatmonetary policy follows a Taylor rule. Standard deviations are reported in brackets. Two key results emerge. First, the model reproduces the fact that the relative performance of our benchmark specification improveswiththehorizon. Second, theRMSPEsarewellwithinstandarderrorsatlongerhorizons. To provide intuition for the performance of the model, it is useful to consider a simplified 29

version of the model in which prices and wages are flexible, there are complete markets across the two countries, and there is no capital or habit formation in utility. The only shocks in this simple modelare country-specific technologyshocks that followan AR(1) process with autocorrelationρ . A The standard deviation of the innovation to the shock is equal to σ . Using equations (53) and A (56), along with the definition of the RER, we have that log(NER )−log(NER ) = log(NER )−E log(NER ) (78) t+h t t+h t t+h h + (cid:16) ρh −1 (cid:17) κ (cid:16) Aˆ −Aˆ∗ (cid:17) + (cid:88) ρ A −1 ρkκ (cid:16) Aˆ −Aˆ∗ (cid:17) . A t t θ −ρ A t t π A k=1 The left-hand side of equation (78) is the forecast error associated with a random walk forecast. The first term on the right-hand side is the forecast error associated with the rational expectations forecast assuming that the structural model is the DGP. Squaring both sides of equation (78) using the fact that the rational expectations forecast error is orthogonal to variables in the time-t information set, we obtain E(log(NER )−log(NER ))2 = E(log(NER )−E log(NER ))2 (79) t+h t t+h t t+h +κ2 (cid:34) (cid:16) 1−ρh (cid:17) + (cid:88) h 1−ρ A ρk (cid:35)2 σ A 2 . A θ −ρ A 1−ρ2 k=1 π A A Two important conclusions follow from equation (79). First, as ρ approaches 1, the difference A between the two mean-squared forecast errors converges to zero. This convergence is due to the fact that when ρ approaches 1, the RER and the NER become increasingly like random walks. A The closer ρ is to 1, the more difficult it is in small samples, to reject the hypothesis that the A NER is a random walk. This observation is reminiscent of a key result in Engle and West (2005). That paper works with a reduced form class of models in which the NER depends on current and future expected ‘fundamentals’ like relative money and output growth rates. Engle and West show that as the rate of time discounting (β) approaches 1, the NER becomes increasingly difficult to distinguish from a random walk. Note that β plays no role in our analytic results. Second, in equation (79), the difference between the two mean-squared forecast errors increases with the horizon, h, similar to what we find in the data. Monetary policy plays a key role in the model’s implications for the average squared predictions of a rational forecast relative to a forecast of no change in the NER. For example, equation (79) implies that a larger value of θ reduces the difference between the average squared prediction π error of the rational expectations forecast and that of the random walk forecast. For intuition, consider the extreme case in which θ is equal to infinity so that domestic and foreign prices are π fixed. Then movements in the NER and RER are equal to each other. Since the RER is close to a random walk, it is difficult to distinguish both the NER and the RER from a random walk in small samples. When θ is very close to 1, prices respond by a relatively large amount to shocks, π which causes the movements in the NER to be much larger than movements in the RER. The magnitude of the movements in the NER after a shock can then be exploited in small samples to 30

predict future values of the NER. MonetarypoliciesotherthanTaylorrulesalsoaffectaveragesquaredpredictionerrors. Suppose thatmonetary policy follows aconstant moneygrowth ruleoranNER targetingrule. Wegenerate 10,000synthetictimeseriesfromversionsofthemodelunderthesepoliciesandgeneratethemodel’s implicationsforourforecastingexercises. Table10reportstheaverageRMSPEfromourforecasting equation relative to the average RMSPE from a random walk forecast. Notice that our forecasting specification no longer outperforms the random walk in the simulated data. The reason that forecasts based on the RER no longer outperform the random walk forecast is that the money growth rule and the NER targeting rule do not produce as strong of a correlation between the current level of the RER and future changes in the NER as is produced under Taylor rules, as shown in Table 9. The weaker correlations are more difficult to estimate and exploit for forecasting in small samples. The basic lesson from these results is that one should not expect forecasting performance to be robust across different monetary policy regimes. 5 Conclusion Thispapershowsthatincountrieswithfloatingexchangerateswheremonetarypolicyusesashortterm interest rate to control inflation, RERs adjust toward parity in the medium and long runs throughchangesinNERsnotviadifferencesininflationrates. Webasethisconclusionontwofacts. First,thecurrentvalueoftheRERishighlycorrelatedwithchangesintheNERathorizonslonger than two years. Second, the current value of the RER is uncorrelated with differential inflation rates at all horizons. Inourtheoreticalanalysis,weshowthatthereisalargeclassofopen-economymodelsconsistent with these facts: models with and without nominal rigidities as well as complete- and incompletemarkets models. But to account for our empirical findings, models must feature home bias in consumption and monetary policy guided by Taylor rules. 31

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6 Tables and Figures Figure 1: Canada: NER and RER data 0.25 0.00 ●● ● ●● ●●● ● ● ●● ● ●●●●●●●●● ●● ●● ● ●●● ●●● ●●●●●● ●● ● ● ●●● ● ● ● ● ●● ●●●● ●● ●● ●● ●●●● ● ● ● ● ● ●●●●●● ●● ●●●●●●●●●● ● ● ● ● ●●● ● ● ● ● ●● ●● ●● ● ● ●●●●● ● ● ● ● ● ●● ● ●● ● ● ●● ●●●●●●●●● −0.25 −0.2 −0.1 0.0 0.1 0.2 log(RERt ) ) tREN j+tREN(gol 1 year horizon −0 0 0 . . . 2 0 2 5 0 5 ●●● ●● ●●● ● ● ●● ● ●● ●●●●●●● ●● ●● ● ● ●●●● ● ●●●● ● ● ● ● ● ●●●●● ● ● ● ● ●●● ● ● ● ● ● ● ●●●●● ● ● ● ●●● ● ●●●●●● ●●●●●● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ●●●●●●●● ●● ● ●● ●● ●●●● ●● ● ● ●●● −0.2 −0.1 0.0 0.1 0.2 log(RERt ) ) tREN j+tREN(gol 3 year horizon ● ● 0.25 ●●● ●●●● ● −0 0 . . 2 0 5 0 ●●● ●● ● ●● ● ● ● ● ●●● ● ●● ●●●● ● ● ● ● ●● ● ●● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●●● ● ● ● ● ● ● ● ● ●● ●● ●●● ● ●●●●●●●●●● ●● ● ●● ●● ●●●●●●●●●●● −0.2 −0.1 0.0 0.1 0.2 log(RERt ) ) tREN j+tREN(gol 7 year horizon ● −0 0 0 . . . 2 0 2 5 0 5 ●● ● ● ●●● ● ● ● ● ● ● ●● ●●●● ● ●●● ●● ● ● ● ●● ●●● ● ●● ● ● ● ● ●●●● ●● ●●● ● ●● ●●●● ● ● ● ● ●●● ● ● ●● ●●● ● ●●●●●● ● ●●● ●● ● ●● ●●●●●●●● ● ●●●● −0.2 −0.1 0.0 0.1 0.2 log(RERt ) ) tREN j+tREN(gol 10 year horizon Sources: International Monetary Fund, International Financial Statistics; Federal Reserve Board, H.10 Foreign Exchange Rates; authors’ calculations. 35

Table 1: NER regression results βNER R2 i,h Horizon (in years) Horizon (in years) 1 3 5 7 10 1 3 5 7 10 (a) Flexible, benchmark Australia -0.20 -0.70 -1.06 -1.13 -1.59 0.10 0.39 0.59 0.60 0.75 (0.10) (0.19) (0.21) (0.22) (0.14) Canada -0.12 -0.55 -0.94 -1.16 -1.66 0.08 0.35 0.59 0.69 0.88 (0.07) (0.18) (0.18) (0.14) (0.12) Norway -0.21 -0.76 -1.29 -1.47 -1.25 0.07 0.29 0.55 0.65 0.51 (0.12) (0.15) (0.25) (0.29) (0.05) Sweden -0.20 -0.75 -1.14 -1.37 -1.28 0.11 0.41 0.65 0.76 0.67 (0.10) (0.16) (0.19) (0.13) (0.21) Switzerland -0.31 -0.91 -1.37 -1.30 -1.13 0.15 0.45 0.71 0.79 0.71 (0.12) (0.14) (0.19) (0.12) (0.13) (b) Flexible, other Euro Area -0.13 -0.86 -0.89 0.03 0.46 0.67 (0.17) (0.29) (0.13) Japan -0.06 -0.45 -0.68 0.01 0.16 0.33 (0.11) (0.25) (0.29) United Kingdom -0.29 -1.31 -1.66 0.10 0.58 0.65 (0.16) (0.35) (0.17) (c) Fixed Hong Kong 0.00 -0.01 -0.02 -0.03 -0.03 0.04 0.32 0.62 0.76 0.77 (0.00) (0.01) (0.01) (0.00) (0.00) China -0.12 -0.21 -0.26 0.26 0.29 0.45 (0.03) (0.06) (0.10) Sources: International Monetary Fund, International Financial Statistics; Federal Reserve Board, H.10 Foreign Exchange Rates; OECD Main Economic Indicators; authors’ calculations. 36

Table 2: Relative price regression results βπ R2 i,h Horizon (in years) Horizon (in years) 1 3 5 7 10 1 3 5 7 10 (a) Flexible, benchmark Australia 0.01 0.05 0.10 0.20 0.48 0.00 0.01 0.04 0.09 0.24 (0.04) (0.09) (0.08) (0.08) (0.18) Canada 0.01 0.03 0.04 0.08 0.26 0.01 0.02 0.01 0.02 0.10 (0.01) (0.04) (0.06) (0.11) (0.18) Norway -0.07 -0.15 -0.11 -0.06 -0.06 0.11 0.11 0.04 0.01 0.00 (0.02) (0.11) (0.17) (0.19) (0.20) Sweden 0.01 0.08 0.11 0.05 -0.02 0.01 0.06 0.06 0.01 0.00 (0.02) (0.06) (0.10) (0.19) (0.21) Switzerland -0.02 0.00 0.08 0.10 0.01 0.03 0.00 0.02 0.03 0.00 (0.02) (0.06) (0.09) (0.16) (0.18) (b) Flexible, other Euro Area -0.04 -0.08 0.03 0.51 0.63 0.07 (0.01) (0.01) (0.01) Japan -0.08 -0.15 -0.11 0.22 0.15 0.06 (0.04) (0.11) (0.12) United Kingdom -0.02 -0.03 -0.04 0.02 0.03 0.03 (0.01) (0.05) (0.03) (c) Fixed Hong Kong -0.09 -0.45 -0.93 -1.32 -1.63 0.13 0.37 0.66 0.88 0.99 (0.05) (0.14) (0.16) (0.14) (0.03) China -0.43 -0.93 -1.05 0.37 0.67 0.91 (0.19) (0.20) (0.07) Sources: International Monetary Fund, International Financial Statistics; Federal Reserve Board, H.10 Foreign Exchange Rates; OECD Main Economic Indicators; authors’ calculations. 37

Table 3: Euro area relative price regression results βπ R2 i,h Horizon (in years) Horizon (in years) 1 3 5 1 3 5 France -0.22 -1.00 -1.31 0.12 0.59 0.78 (0.12) (0.17) (0.19) Italy -0.17 -0.47 -0.61 0.40 0.71 0.80 (0.05) (0.08) (0.06) Ireland -0.30 -0.83 -1.10 0.40 0.73 0.83 (0.09) (0.08) (0.09) Portugal -0.22 -0.66 -0.83 0.48 0.85 0.94 (0.060) (0.06) (0.04) Spain -0.16 -0.44 -0.65 0.49 0.75 0.88 (0.03) (0.08) (0.07) Sources: International Monetary Fund, International Financial Statistics; OECD Main Economic Indicators; authors’ calculations. 38

Table 4: NER regression results for other countries Sample Starting in 1984 Sample Starting in 1999 βNER R2 βNER R2 i,h i,h Horizon (in years) Horizon (in years) Horizon (in years) Horizon (in years) 1 3 5 1 3 5 1 3 5 1 3 5 Brazil -0.31 0.38 1.75 0.02 0.01 0.05 -0.17 -0.59 -1.11 0.07 0.25 0.59 (0.39) (0.57) (0.71) (0.12) (0.24) (0.13) Chile -0.22 -0.59 -0.89 0.05 0.12 0.14 -0.32 -0.98 -1.24 0.16 0.50 0.74 (0.15) (0.39) (0.57) (0.14) (0.17) (0.16) Colombia -0.07 -0.29 -0.76 0.01 0.03 0.07 -0.15 -0.50 -0.93 0.05 0.19 0.43 (0.14) (0.42) (0.69) (0.12) (0.27) (0.16) Indonesia -0.25 -0.59 -0.89 0.08 0.20 0.29 -0.14 -0.46 -0.55 0.06 0.36 0.43 (0.09) (0.16) (0.29) (0.12) (0.14) (0.16) Israel 0.70 0.57 0.45 0.07 0.03 0.01 -0.39 -0.64 -1.13 0.20 0.37 0.63 (0.71) (1.00) (1.33) (0.14) (0.21) (0.19) Mexico 0.78 1.33 1.48 0.23 0.13 0.12 -0.32 -0.96 -0.54 0.06 0.23 0.10 (0.32) (0.75) (0.90) (0.38) (0.13) (0.34) South Korea -0.42 -1.04 -1.35 0.21 0.55 0.70 -0.49 -1.07 -1.35 0.22 0.54 0.78 (0.12) (0.22) (0.15) (0.23) (0.22) (0.11) Thailand -0.26 -0.59 -1.02 0.16 0.33 0.52 -0.13 -0.42 -0.73 0.09 0.34 0.56 (0.12) (0.20) (0.25) (0.06) (0.13) (0.10) Turkey 0.40 1.22 1.97 0.15 0.22 0.20 0.16 0.13 -0.21 0.03 0.01 0.02 (0.11) (0.38) (0.59) (0.20) (0.42) (0.30) Sources: International Monetary Fund, International Financial Statistics; OECD Main Economic Indicators; authors’ calculations. 39

Table 5: Relative price regression results for other countries Sample Starting in 1984 Sample Starting in 1999 βπ R2 βπ R2 i,h i,h Horizon (in years) Horizon (in years) Horizon (in years) Horizon (in years) 1 3 5 1 3 5 1 3 5 1 3 5 Brazil -0.28 -1.24 -2.80 0.02 0.06 0.12 -0.02 0.00 0.07 0.05 0.00 0.10 (0.22) (0.56) (0.74) (0.02) (0.05) (0.03) Chile -0.05 -0.23 -0.46 0.01 0.03 0.06 0.06 0.16 0.14 0.17 0.42 0.24 (0.10) (0.35) (0.56) (0.03) (0.03) (0.04) Colombia -0.14 -0.39 -0.52 0.12 0.11 0.07 -0.04 -0.06 -0.08 0.12 0.10 0.11 (0.08) (0.31) (0.57) (0.02) (0.05) (0.03) Indonesia -0.07 -0.02 0.01 0.06 0.00 0.00 -0.06 -0.12 -0.23 0.12 0.33 0.61 (0.03) (0.07) (0.14) (0.03) (0.03) (0.03) Israel -0.99 -1.32 -1.56 0.16 0.13 0.13 0.07 0.09 0.00 0.08 0.05 0.00 (0.67) (1.07) (1.35) (0.04) (0.11) (0.09) Mexico -1.10 -2.15 -2.55 0.62 0.36 0.32 0.04 0.07 -0.06 0.03 0.02 0.02 (0.24) (0.61) (0.83) (0.03) (0.06) (0.13) South Korea 0.06 0.17 0.21 0.19 0.42 0.34 -0.02 0.02 0.12 0.02 0.01 0.34 (0.02) (0.03) (0.05) (0.02) (0.06) (0.03) Thailand 0.04 0.11 0.14 0.12 0.21 0.18 0.01 0.01 -0.01 0.02 0.01 0.00 (0.02) (0.06) (0.07) (0.01) (0.03) (0.02) Turkey -0.57 -1.58 -2.50 0.46 0.43 0.37 -0.39 -0.77 -0.81 0.49 0.37 0.32 (0.09) (0.32) (0.50) (0.10) (0.28) (0.24) Sources: International Monetary Fund, International Financial Statistics; OECD Main Economic Indicators; authors’ calculations. Table 6: Test of null-hypothesis of no predictability with Wald statistic Australia Canada Norway Sweden Switzerland Bootstrap p-value 0.02 0.01 0.07 0.04 0.00 Sources: International Monetary Fund, International Financial Statistics; Federal Reserve Board, H.10 Foreign Exchange Rates;authors’calculations. Notes: Weused10,000bootstrapsamples. Thetabledisplaysthepercentageofthosesamples that produced Wald statistics larger than the Wald statistic calculated from the data. 40

Table 7: Out-of-sample forecasting for the NER (a) RMSPE from panel regression relative to a random walk Forecast horizon 1Q 2Q 1Y 2Y 3Y 4Y 5Y 6Y 7Y All Countries 1.01 1.01 1.02 0.98 0.87 0.77 0.65 0.60 0.55 Australia 1.00 1.00 0.99 1.02 0.89 0.73 0.63 0.58 0.60 Canada 1.01 1.02 1.06 1.10 1.02 0.88 0.77 0.74 0.64 Norway 1.01 1.03 1.05 1.04 0.96 0.90 0.77 0.73 0.64 Sweden 1.01 1.03 1.05 0.99 0.81 0.72 0.57 0.52 0.46 Switzerland 1.00 0.99 0.97 0.86 0.73 0.66 0.53 0.42 0.37 Note: The left-hand-side variable is the change in the quarterly average nominal exchange rate at the indicated horizons. The right-hand-side variable is the quarterly real exchange rate (calculated using the quarterly average of the nominal exchange rate and lagged price levels). The sample periods is from 1973Q1 through 2007Q4. We use a training sample of 10 years plus the horizon of the forecast. Sources: International Monetary Fund, International Financial Statistics; Federal Reserve Board, H.10 Foreign Exchange Rates; authors’ calculations. (b) Bootstrap p-values, stationary RER Forecast horizon 1Q 2Q 1Y 2Y 3Y 4Y 5Y 6Y 7Y Years 3-7 All Countries 0.71 0.75 0.58 0.20 0.05 0.03 0.01 0.01 0.01 0.00 Australia 0.40 0.35 0.20 0.36 0.11 0.04 0.03 0.03 0.07 0.03 Canada 0.79 0.71 0.83 0.74 0.23 0.10 0.06 0.07 0.05 0.04 Norway 0.83 0.90 0.83 0.39 0.11 0.09 0.05 0.06 0.04 0.03 Sweden 0.83 0.84 0.80 0.22 0.03 0.03 0.01 0.01 0.01 0.01 Switzerland 0.38 0.43 0.33 0.14 0.09 0.08 0.04 0.02 0.02 0.01 41

(c) RMSPE relative to a random walk, robustness Forecast horizon 1Q 2Q 1Y 2Y 3Y 4Y 5Y 6Y 7Y Benchmark 1.01 1.01 1.02 0.98 0.87 0.77 0.65 0.60 0.55 Time t price levels 1.01 1.01 1.02 0.98 0.88 0.78 0.66 0.61 0.56 Sample ends in 2016Q4 1.00 1.00 1.00 0.97 0.92 0.86 0.81 0.77 0.72 Sample ends in 2016Q4 (with Japan) 1.00 1.00 0.99 0.95 0.92 0.89 0.87 0.83 0.77 Unbalanced panel 1.01 1.01 1.02 0.97 0.89 0.83 0.79 0.67 0.60 Country-by-country regressions 1.02 1.03 1.05 1.07 1.02 0.89 0.72 0.66 0.59 Note: The row labeled “Benchmark” is the row labeled “All countries” in panel (a). The other rows are variants of the panel regression considered and described in panel (a) and show the averge mean squared prediction error for all countries. The row labeled “Sample ends in 2016Q4” extends the sample to 2016Q4. The row labeled “Sample ends in 2016Q4(withJapan)”addsjapantothelistofcountriesinthepanel. Therowlabeled“Unbalancedpanel”addstheeuro tothepanelstarting1999Q1, addstheUKstartingin1993Q1, andaddsJapanstartingin1973Q1andendingin1994Q4. The row labeled “Country-by-country regressions” does not impose the panel structure. Sources: International Monetary Fund, International Financial Statistics; Federal Reserve Board, H.10 Foreign Exchange Rates; OECD Main Economic Indicators; authors’ calculations. (d) Bootstrap p-values, non-stationary RER Forecast horizon 1Q 2Q 1Y 2Y 3Y 4Y 5Y 6Y 7Y Years 3-7 All Countries 0.49 0.53 0.40 0.15 0.04 0.02 0.00 0.01 0.01 0.00 Australia 0.33 0.30 0.18 0.33 0.11 0.04 0.03 0.04 0.06 0.02 Canada 0.68 0.58 0.71 0.61 0.24 0.09 0.05 0.06 0.05 0.03 Norway 0.66 0.75 0.65 0.31 0.12 0.10 0.06 0.06 0.05 0.03 Sweden 0.68 0.68 0.64 0.18 0.03 0.02 0.01 0.01 0.01 0.00 Switzerland 0.34 0.38 0.31 0.14 0.08 0.08 0.04 0.03 0.03 0.01 42

Table 8: Calibrated parameters Parameter Value Model counterpart σ 10.62 Elasticity of money demand M µ 1 Steady state money stock β 1.03−0.25 Steady state interest rate h 0.65 Habit persistence σ 1 log utility φ 1 Disutility of labor γ 0.75 Policy rate smoothing θ 1.5 Taylor principle π ν 6 Intermediate goods firm’s markups ρ 0.85 Persistence of interest rate differential η ρ 1 Substitutability of home and foreign goods 3 ξ 0.6 Frequency of price adjustment φ 0.001 Cost of foreign bond holdings B ν 21 Differentiated wage markup L ξ 0.65 Frequency of wage adjustment W ω 0.90 Home bias in consumption Table 9: Model-implied regression results βNER βπ i,h i,h Horizon (in years) Horizon (in years) 1 3 5 7 10 1 3 5 7 10 Taylor rule: Flexible -0.31 -0.66 -1.03 -1.32 -1.54 -0.03 0.12 0.30 0.41 0.43 Sticky -0.35 -0.77 -1.19 -1.52 -1.72 0.02 0.22 0.43 0.56 0.59 Money growth rule: Flexible -0.12 -0.31 -0.47 -0.59 -0.67 -0.21 -0.23 -0.27 -0.33 -0.42 Sticky -0.22 -0.47 -0.64 -0.76 -0.85 -0.09 -0.10 -0.13 -0.19 -0.28 NER targeting rule: Flexible -0.03 -0.03 -0.03 -0.04 -0.05 -0.32 -0.53 -0.72 -0.90 -1.08 Sticky -0.04 -0.03 -0.04 -0.05 -0.07 -0.17 -0.46 -0.69 -0.88 -1.07 Sources: authors’ calculations. 43

Table 10: Model-implied RMSPE relative to a random walk Forecast horizon 1Q 2Q 1Y 2Y 3Y 4Y 5Y 6Y 7Y All Countries 1.01 1.01 1.02 0.98 0.87 0.77 0.65 0.60 0.55 Taylor rule 0.97 0.95 0.92 0.89 0.87 0.85 0.82 0.80 0.80 (0.01) (0.02) (0.04) (0.07) (0.10) (0.12) (0.14) (0.16) (0.18) Money growth rule 1.00 1.00 1.00 1.01 1.02 1.02 1.01 1.00 0.99 (0.01) (0.01) (0.02) (0.05) (0.07) (0.10) (0.12) (0.15) (0.17) NER targeting rule 1.01 1.01 1.02 1.03 1.04 1.06 1.07 1.09 1.10 (0.01) (0.01) (0.01) (0.01) (0.02) (0.03) (0.04) (0.05) (0.08) Note: The left-hand-side variable is the change in the quarterly average nominal exchange rate at the indicated horizons. The right-hand-side variable is the quarterly real exchange rate (calculated using the quarterly average of the nominal exchangerateandlaggedpricelevels). Thelinemarked“AllCountries”isthesameasinTable 7. Formodelsimulations, wesimulated5countriesworthofdatafor140periodseach. Sources: InternationlMonetaryFund,InternationalFinancial Statistics; FederalReserveBoard, H.10ForeignExchangeRates; OECDMainEconomicIndicators; authors’calculations. 44

Figure 2: Response to technology shock under Taylor rule 0 10 20 30 40 50 2.0− 6.0− 0.1− At tnecreP 0 10 20 30 40 50 0.0 4.0− 8.0− RERt 0 10 20 30 40 50 2.0− 6.0− 0.1− Ct and C t * 0 10 20 30 40 50 0.1 5.0 0.0 0.1− p H,t and p H * ,t 0 10 20 30 40 50 0.1 5.0 0.0 0.1− p F,t and p F * ,t 0 10 20 30 40 50 0.1 5.0 0.0 0.1− p t and p t * Quarters tnecreP 0 10 20 30 40 50 01.0 00.0 01.0− mct and mc t * 0 10 20 30 40 50 01.0 50.0 00.0 Rt and R t * 0 10 20 30 40 50 0.1 5.0 5.0− NERt Note: The vertical axis is expressed in percent. The horizontal axis shows quarters after the shock. Red-dashed lines indicate the variables with a ∗. Figure 3: Response to technology shock under money growth rule 0 10 20 30 40 50 2.0− 6.0− 0.1− At tnecreP 0 10 20 30 40 50 0.0 4.0− 8.0− RERt 0 10 20 30 40 50 2.0− 6.0− 0.1− Ct and C t * 0 10 20 30 40 50 0.1 5.0 0.0 0.1− p H,t and p H * ,t 0 10 20 30 40 50 0.1 5.0 0.0 0.1− p F,t and p F * ,t 0 10 20 30 40 50 0.1 5.0 0.0 0.1− p t and p t * Quarters tnecreP 0 10 20 30 40 50 01.0 00.0 01.0− mct and mc t * 0 10 20 30 40 50 10.0 10.0− Rt and R t * 0 10 20 30 40 50 0.0 2.0− 4.0− NERt Note: The vertical axis is expressed in percent. The horizontal axis shows quarters after the shock. Red-dashed lines indicate the variables with a ∗. 45

Figure 4: Implied small-sample values of βNER and βπ from small-scale models i,h i,h (a) Taylor and money growth rules 0 10 20 30 40 2 1 0 1− 2− Horizon (in quarters) eulav tneiciffeoC NER regression (Taylor) Relative price regression (Taylor) NER regression (money) Relative price regression (money) (b) Taylor and NER targeting rules 0 10 20 30 40 2 1 0 1− 2− Horizon (in quarters) eulav tneiciffeoC NER regression (Taylor) Relative price regression (Taylor) NER regression (NER target) Relative price regression (NER target) Note: The model-implied values come from our model with no nominal rigidities and only technology shocks. 46

Figure 5: Response to technology shock under NER targeting rule 0 10 20 30 40 50 2.0− 6.0− 0.1− At tnecreP 0 10 20 30 40 50 0.0 4.0− 8.0− RERt 0 10 20 30 40 50 2.0− 6.0− 0.1− Ct and C t * 0 10 20 30 40 50 0.1 5.0 0.0 0.1− p H,t and p H * ,t 0 10 20 30 40 50 0.1 5.0 0.0 0.1− p F,t and p F * ,t 0 10 20 30 40 50 0.1 5.0 0.0 0.1− p t and p t * Quarters tnecreP 0 10 20 30 40 50 01.0 00.0 01.0− mct and mc t * 0 10 20 30 40 50 01.0 50.0 00.0 Rt and R t * 0 10 20 30 40 50 00.0 60.0− 21.0− NERt Note: The vertical axis is expressed in percent. The horizontal axis shows quarters after the shock. Red-dashed lines indicate the variables with a ∗. Figure 6: Response to technology shock under Taylor rule with incomplete markets and sticky prices 0 10 20 30 40 50 2.0− 6.0− 0.1− At tnecreP 0 10 20 30 40 50 0.0 4.0− 8.0− RERt 0 10 20 30 40 50 2.0− 6.0− 0.1− Ct and C t * 0 10 20 30 40 50 0.1 5.0 0.0 0.1− p H,t and p H * ,t 0 10 20 30 40 50 0.1 5.0 0.0 0.1− p F,t and p F * ,t 0 10 20 30 40 50 0.1 5.0 0.0 0.1− p t and p t * Quarters tnecreP 0 10 20 30 40 50 51.0 00.0 51.0− mct and mc t * 0 10 20 30 40 50 01.0 00.0 Rt and R t * 0 10 20 30 40 50 5.1 5.0 5.0− NERt Note: The vertical axis is expressed in percent. The horizontal axis shows quarters after the shock. Red-dashed lines indicate the variables with a ∗. 47

Figure7: Response totechnologyshockunderTaylor rulewithincompletemarkets, medium-scale model 0 10 20 30 40 50 2.0− 6.0− 0.1− At tnecreP 0 10 20 30 40 50 0.0 4.0− 8.0− RERt 0 10 20 30 40 50 0.0 4.0− 8.0− Ct and C t * 0 10 20 30 40 50 0.1 5.0 0.0 5.0− p H,t and p H * ,t 0 10 20 30 40 50 0.1 5.0 0.0 5.0− p F,t and p F * ,t 0 10 20 30 40 50 0.1 5.0 0.0 5.0− p t and p t * Quarters tnecreP 0 10 20 30 40 50 8.0 4.0 0.0 mct and mc t * 0 10 20 30 40 50 21.0 60.0 00.0 Rt and R t * 0 10 20 30 40 50 0.1 5.0 0.0 5.0− NERt Note: The vertical axis is expressed in percent. The horizontal axis shows quarters after the shock. Red-dashed lines indicate the variables with a ∗. Table 11: Empirical facts about exchange rates ρ σ σ cor(∆RER,∆NER) RER ∆RER ∆NER Australia 0.971 0.040 0.040 0.968 (0.848,0.986) (0.003) (0.003) (0.007) Canada 0.986 0.022 0.022 0.969 (0.872,0.997) (0.002) (0.002) (0.007) Norway 0.948 0.043 0.042 0.975 (0.824,0.972) (0.002) (0.002) (0.005) Sweden 0.970 0.047 0.048 0.978 (0.849,0.986) (0.004) (0.004) (0.004) Switzerland 0.934 0.052 0.052 0.989 (0.828,0.963) (0.003) (0.003) (0.002) Nominal rigidities 0.900 0.014 0.012 0.934 Without nominal rigidities 0.859 0.013 0.010 0.693 Note: confidence intervals for ρ are constructed from a parametric bootstrap for an AR(1) model of log(RER ). We RER t used 10,000 bootstrap draws and report the 0.025% and 0.975% quantiles of the bootstrap distribution of the statistic of interest. Standard errors for σ and σ are GMM standard errors. Source: International Monetary Fund, ∆RER ∆NER InternationalFinancialStatistics;FederalReserveBoard,H.10ForeignExchangeRates;OECDMainEconomicIndicators; authors’ calculations. 48

A Data analysis appendix A.1 Scatter plots Figure 8: Australia: NER and RER data 0.4 0.0 llll ll ll l l l l ll l ll l l ll l l l lll ll ll lll l l ll ll l l l l l ll l lllll ll llll lll l l l l ll l lll lll ll l l l lllll l lll ll l l l ll l ll l l ll l l ll l ll lllllll l ll lllllll l lllllll l −0.4 −0.4 −0.2 0.0 0.2 ( ) log RER t ) ( t REN j+t REN gol 1 year horizon −0 0 0 . . . 4 0 4 llllll lll l lll l lll l l l l ll l ll l lll l lll l llllll l l l l l l l l l l ll l l ll lll l l l l ll l l ll l ll ll l ll l l l llll lll l l ll ll ll l ll l l l ll l l ll l l ll ll lll llll l l llll l −0.4 −0.2 0.0 0.2 ( ) log RER t ) ( t REN j+t REN gol 3 year horizon l 0.4 ll −0 0 . . 4 0 ll l ll lll l ll l ll ll l l l l llll l l ll l l l l l l l l ll ll l l l l l ll l l l l lll ll ll l ll l l l l l l l ll l l l l l l l l l l l l l l ll l l l l l l l lll l ll ll ll l lllll −0.4 −0.2 0.0 0.2 ( ) log RER t ) ( t REN j+t REN gol 7 year horizon 0.4 −0 0 . . 4 0 l l ll l l l lll l l l ll ll l l l lll l l l l ll lll l l l l ll l l l llll ll l l l l l l l lll l lllllll ll lll l llllll lll l l l l l ll l ll ll l l llll ll −0.4 −0.2 0.0 0.2 ( ) log RER t ) ( t REN j+t REN gol 10 year horizon Sources: International Monetary Fund, International Financial Statistics; Federal Reserve Board, H.10 Foreign Exchange Rates; authors’ calculations. 49

Figure 9: Norway: NER and RER data 0.4 0.0 l ll l l l l l ll l ll lll l l l llll llll llll ll lllll ll ll l ll ll ll l lll l lllllllll ll ll ll lll ll l l l l l ll l ll l llll l l l l l l llllll l ll l llll l l l l l l llllll ll l ll l l l lll lll l −0.4 −0.50 −0.25 0.00 0.25 0.50 ( ) log RER t ) ( t REN j+t REN gol 1 year horizon −0 0 0 . . . 4 0 4 l llll lll ll lll ll l l l l l lll ll l l ll ll l l llll l l llllll ll ll l l l l l lllllll l llll l l lll l l l l l l l lllll l l l l l l l l l l ll l l l l l l ll l l l l l l ll lll l ll l l l l l l l lll l −0.50 −0.25 0.00 0.25 0.50 ( ) log RER t ) ( t REN j+t REN gol 3 year horizon l −0 0 0 . . . 4 0 4 l ll l l ll l llll l lll lll l l l lll ll ll l ll ll ll l ll ll l l l lllll l l lllll l l l l l ll ll lll l lll l l l l l l l l l l l l l l l l ll l l ll l ll l l ll ll l l l l ll l l −0.50 −0.25 0.00 0.25 0.50 ( ) log RER t ) ( t REN j+t REN gol 7 year horizon 0.4 l l −0 0 . . 4 0 l l l lll l l lll l l ll l l ll ll l l l ll ll l l l l l l l l l l l ll l ll l l l l l lll lll l l l ll l l l l l l l l l l l ll l l l l l lllll lllllll l l l ll l ll −0.50 −0.25 0.00 0.25 0.50 ( ) log RER t ) ( t REN j+t REN gol 10 year horizon Sources: International Monetary Fund, International Financial Statistics; Federal Reserve Board, H.10 Foreign Exchange Rates; authors’ calculations. Figure 10: Sweden: NER and RER data 0.5 0.0 ll l l l l l l l ll l l ll ll llllll l l lll l l l l l lll l l l lllll ll l l lll lllll lll l l l l l ll l lll llllll l l ll l l l l ll l l l lll l l ll lllll ll llll l l l l ll l ll l l ll ll l ll l ll l lllll l −0.5 l −0.3 0.0 0.3 ( ) log RER t ) ( t REN j+t REN gol 1 year horizon 0.5 −0 0 . . 5 0 ll l l lll lll ll l l l lll ll l ll l l l l l ll l l ll ll l l l ll ll l ll l ll l ll l l l l ll ll ll ll l ll l l l l ll l ll ll l l lll l l l l llll l l l l llll l l l l l l l l l l ll l lll ll ll l l l l ll l l l −0.3 0.0 0.3 ( ) log RER t ) ( t REN j+t REN gol 3 year horizon −0 0 0 . . . 5 0 5 llll ll l lllll lllll lll llll ll ll l ll lll l ll l l l llll ll ll l l ll l l l llll l ll l lll l l llll ll l ll l l l lll lll l ll l l l ll ll l l l ll ll ll ll ll l l l −0.3 0.0 0.3 ( ) log RER t ) ( t REN j+t REN gol 7 year horizon 0.5 −0 0 . . 5 0 lll ll llll l ll l llllll lll l lllllllll ll l l l lllll l ll l l l ll l l l l l l ll l llll l l l l l llll l l l l l l l llll l l lll ll l l lll l l l ll −0.3 0.0 0.3 ( ) log RER t ) ( t REN j+t REN gol 10 year horizon Sources: International Monetary Fund, International Financial Statistics; Federal Reserve Board, H.10 Foreign Exchange Rates; authors’ calculations. 50

Figure 11: Switzerland: NER and RER data 0.5 0.0 l l l l l l l l l l l l lll ll ll l lll l l l lllll l l l l l l l l ll l l l l l l ll l l l l l ll l l l l l ll l l l l lllllll l ll l ll l ll l l l llll llll l l llllll l ll ll l l l l ll l lllll l l l l l l ll l l l l l l lll −0.5 −0.6 −0.3 0.0 0.3 ( ) log RER t ) ( t REN j+t REN gol 1 year horizon l 0 0 . . 0 5 ll ll l lll llll l l ll l l l ll ll ll l l l l l l ll l l l l lll l l ll l l l l l ll ll l l l l l l ll l l l l l l ll l l ll ll l ll l l l lll l ll l l l l l ll l ll ll l l lll l l l l l l lllllll l ll l lllll l −0.5 −0.6 −0.3 0.0 0.3 ( ) log RER t ) ( t REN j+t REN gol 3 year horizon l 0 0 . . 0 5 l ll l l l l l lll l ll l l lllll l l l l l l l ll l l lll l l ll ll l ll lllll l l l l l l l llll llll l l llll ll llll l ll l l ll l l l l l l l l l l l ll l l l l ll l l l l l l ll l −0.5 −0.6 −0.3 0.0 0.3 ( ) log RER t ) ( t REN j+t REN gol 7 year horizon l l lll l 0 0 . . 0 5 lllllllll lll l l l llll l l l l l l l l l l l l l l l l ll l l l l l l l l l ll l ll l lll l l ll llll l ll l l l l l llll l lllll l l l l ll l ll llll −0.5 −0.6 −0.3 0.0 0.3 ( ) log RER t ) ( t REN j+t REN gol 10 year horizon Sources: International Monetary Fund, International Financial Statistics; Federal Reserve Board, H.10 Foreign Exchange Rates; authors’ calculations. 51

A.2 Additional regression results Table 12: NER regression results, sample ending in 2016Q4 βNER R2 i,h Horizon (in years) Horizon (in years) 1 3 5 7 10 1 3 5 7 10 (a) Flexible, benchmark Australia -0.19 -0.57 -0.94 -1.21 -1.95 0.09 0.28 0.42 0.52 0.83 (0.09) (0.15) (0.15) (0.18) (0.14) Canada -0.15 -0.57 -0.98 -1.29 -1.77 0.08 0.32 0.53 0.69 0.93 (0.07) (0.18) (0.19) (0.11) (0.08) Norway -0.24 -0.76 -1.26 -1.53 -1.58 0.08 0.27 0.50 0.63 0.62 (0.11) (0.14) (0.19) (0.25) (0.23) Sweden -0.20 -0.72 -1.12 -1.29 -1.33 0.09 0.38 0.64 0.77 0.76 (0.09) (0.16) (0.19) (0.12) (0.14) Switzerland -0.26 -0.73 -1.16 -1.19 -1.17 0.13 0.36 0.59 0.67 0.64 (0.10) (0.16) (0.19) (0.16) (0.13) (b) Flexible, other Euro Area -0.22 -0.80 -1.13 0.11 0.54 0.78 (0.15) (0.12) (0.12) Japan -0.15 -0.63 -0.82 0.06 0.31 0.48 (0.09) (0.18) (0.16) United Kingdom -0.46 -1.19 -1.61 0.16 0.47 0.61 (0.20) (0.25) (0.27) (c) Fixed Hong Kong 0.00 -0.01 -0.02 -0.03 -0.02 0.04 0.30 0.57 0.66 0.50 (0.00) (0.01) (0.01) (0.00) (0.00) China -0.07 -0.16 -0.31 0.10 0.11 0.14 (0.04) (0.12) (0.15) Sources: International Monetary Fund, International Financial Statistics; Federal Reserve Board, H.10 Foreign Exchange Rates; OECD Main Economic Indicators; authors’ calculations. Samples extended to 2016Q4 relative to Table 1. 52

Table 13: Relative price regression results, sample ending in 2016Q4 βπ R2 i,h Horizon (in years) Horizon (in years) 1 3 5 7 10 1 3 5 7 10 (a) Flexible, benchmark Australia 0.00 0.02 0.06 0.17 0.34 0.00 0.00 0.02 0.10 0.24 (0.02) (0.06) (0.07) (0.07) (0.15) Canada 0.01 0.03 0.05 0.10 0.22 0.01 0.01 0.02 0.06 0.17 (0.01) (0.03) (0.05) (0.08) (0.15) Norway -0.06 -0.13 -0.10 0.01 0.13 0.09 0.09 0.03 0.00 0.02 (0.03) (0.10) (0.16) (0.18) (0.23) Sweden 0.02 0.09 0.15 0.17 0.24 0.03 0.09 0.11 0.09 0.09 (0.02) (0.09) (0.15) (0.17) (0.24) Switzerland -0.01 0.01 0.07 0.09 0.01 0.01 0.00 0.02 0.02 0.00 (0.01) (0.04) (0.09) (0.15) (0.15) (b) Flexible, other Euro Area 0.00 0.00 0.05 0.00 0.00 0.27 (0.01) (0.02) (0.02) Japan -0.06 -0.07 -0.03 0.18 0.06 0.01 (0.03) (0.07) (0.06) United Kingdom -0.01 0.04 0.18 0.00 0.01 0.09 (0.03) (0.09) (0.11) (c) Fixed Hong Kong -0.09 -0.43 -0.88 -1.30 -1.62 0.12 0.35 0.61 0.82 0.98 (0.05) (0.15) (0.14) (0.08) (0.05) China -0.04 -0.05 -0.10 0.02 0.01 0.02 (0.07) (0.16) (0.26) Sources: International Monetary Fund, International Financial Statistics; Federal Reserve Board, H.10 Foreign Exchange Rates; OECD Main Economic Indicators; authors’ calculations. Samples extended to 2016Q4 relative to Table 2. 53

B Model Appendix B.1 Household Thehouseholdproblemfortherepresentativehouseholdinthehomecountryis maxEt j (cid:88) ∞ =0 βj      (cid:0)Ct+j−h 1 C − ¯ t+ σ j−1 (cid:1)1−σ − 1+ χ φ (cid:90) 0 1 Lt+j(i)1+φdi+µ (cid:18)M Pt t 1 + + − j j (cid:19) σM 1−σM +log(cid:0)ηt+j (cid:1)V (cid:32)B P H t , + t+ j j (cid:33) +log (cid:16) ηt ∗ +j (cid:17) V (cid:32)BF,t+ P j t N + E j Rt+j (cid:33)      (B.1) whereCtisconsumption,C¯ tisaggregateconsumption,Lt(i)ishoursworkedbymemberi, M Pt t arerealmoneybalances. Thebudgetconstraint is BH,t+NERtBF,t+PtCt+PI,tIt+Mt=Rt−1BH,t−1+NERtRt ∗ −1BF,t−1− φ 2 B (cid:18)NER P t t BF,t (cid:19)2 Pt+PtRK,tKt+(1+τW) (cid:90) 0 1 Wt(i)Lt(i)di+Tt+Mt−1 (B.2) whereandBH,tandBF,tarenominalbalancesofhomeandforeignbonds,NERtisthenominalexchangeratequotedasthepriceoftheforeign currencyunit,Ptisthepriceoffinalgoodsinthehomecountry,RtisthenominalinterestrateonthehomebondandRt ∗isthenominalinterest rateontheforeignbond,Wtisthewagerate,Rt K istherentalrateoncapital,Kt,Itareinvestmentgoods,PI,tisthepriceofinvestmentgoods, andTtarelump-sumprofitsandtaxes. Thecapitalaccumulationequationis Kt+1=It (cid:32) 1− φK (cid:32) It −1 (cid:33)2(cid:33) +(1−δ)Kt (B.3) 2 It−1 Thehousehold-widefirst-orderconditionsare (cid:0)Ct−hCt−1 (cid:1)−σ=Λt (B.4) Λt=log(ηt)V(cid:48) (cid:18)BH,t (cid:19) +βRtEt Λt+1 (B.5) Pt πt+1 Λt+φB (cid:18)NER P t t BF,t (cid:19) =log(cid:0)ηt ∗(cid:1)V(cid:48) (cid:18)BF,t P N t ERt(cid:19) +βRt ∗Et Λ πt t + + 1 1N N E E R R t+ t 1 (B.6) µ (cid:18)Mt(cid:19)−σM = log(ηt) V(cid:48) (cid:18)BH,t (cid:19) + (cid:18)Rt−1(cid:19) Λt (B.7) Pt Rt Pt Rt P P I t ,tΛt=Qt (cid:34)(cid:32) 1− φ 2 K (cid:32) It I − t 1 −1 (cid:33)2(cid:33) − It I − t 1 φK (cid:32) It I − t 1 −1 (cid:33)(cid:35) +βEtQt+1φK (cid:18)It I + t 1 −1 (cid:19)It I 2 + t 2 1 (B.8) Qt=βEt (cid:2)Qt+1(1−δ)+Λt+1RK,t+1 (cid:3) (B.9) Thehouseholdproblemfortherepresentativehouseholdintheforeigncountryis maxEt j (cid:88) ∞ =0 βj       (cid:16) Ct ∗ +j−h 1 C − ¯ t ∗ + σ j−1 (cid:17)1−σ − 1+ χ φ (cid:90) 0 1 L∗ t+j(i)1+φdi+µ (cid:32) M Pt ∗ t ∗ 1 + + − j j (cid:33) σM 1−σM +log(cid:0)ηt+j (cid:1)V   NE B R H ∗ t+ ,t j + P j t ∗ +j  +log (cid:16) ηt ∗ +j (cid:17) V   B P F ∗ t ∗ , + t+ j j         (B.10) Thebudgetconstraintis BF ∗ ,t+NERt −1BH ∗ ,t+Pt ∗Ct ∗+PI ∗ ,tIt ∗+Mt ∗=Rt−1BH ∗ ,t−1NERt −1+Rt ∗ −1BF ∗ ,t−1− φ 2 B (cid:32) NE B R H ∗ t ,t Pt ∗ (cid:33)2 Pt+Pt ∗RK ∗ ,tKt ∗+(1+τW) (cid:90) 0 1 Wt ∗(i)L∗ t(i)di+Tt ∗+Mt ∗ −1 (B.11) Thecapitalaccumulationequationis Kt ∗ +1=It ∗ (cid:32) 1− φ 2 K (cid:32) I I ∗ t ∗ −1 (cid:33)2(cid:33) +(1−δ)Kt ∗ (B.12) t−1 Thehousehold-widefirst-orderconditionsare (cid:16) Ct ∗−hCt ∗ −1 (cid:17)−σ =Λ∗ t (B.13) Λ∗ t +φB (cid:32) NE B R H ∗ t ,t Pt ∗ (cid:33) =log(ηt)V(cid:48) (cid:32) NE B R H ∗ t ,t Pt ∗ (cid:33) +βRtEt Λ π t ∗ ∗ t + + 1 1 N N E E R R t+ t 1 (B.14) Λ∗ t =log(cid:0)ηt ∗(cid:1)V(cid:48) (cid:32)B P F ∗ t ∗ ,t (cid:33) +βRt ∗Et Λ π t ∗ ∗ t + + 1 1 (B.15) µ (cid:32)M Pt ∗ t ∗(cid:33)−σM = log R (cid:0) t ∗ ηt ∗(cid:1) V(cid:48) (cid:32)B P F ∗ t ∗ ,t (cid:33) + (cid:32)Rt ∗ R − t ∗ 1(cid:33) Λ∗ t (B.16) P P I ∗ t ∗ ,tΛ∗ t =Q∗ t (cid:34)(cid:32) 1− φ 2 K (cid:32) I t I ∗ − t ∗ 1 −1 (cid:33)2(cid:33) − I t I ∗ − t ∗ 1 φK (cid:32) I t I ∗ − t ∗ 1 −1 (cid:33)(cid:35) +βEtQt+1φK (cid:32)It I ∗ + t ∗ 1 −1 (cid:33)(cid:32)It I ∗ + t ∗ 1 (cid:33)2 (B.17) 54

Q∗ t =βEt (cid:104) Q∗ t+1(1−δ)+Λ∗ t+1RK ∗ ,t+1 (cid:105) (B.18) Therearesimilarfirst-orderconditionsfortheforeignhousehold. Notethatwedefine RERt= NERtPt ∗ (B.19) Pt B.2 The labor market Weassumethatallofthehouseholdmembersconsumerthesameamount(perfectconsumptioninsurance). Eachhouseholdmemberofisamember ofaunionthatsuppliesitstypeoflabor,i. Laboriscombinedvia  (cid:90)1 νL−1  νL νL −1 Lt= Lt(i) νL di 0 toproducelaborservices,whichgototheproductionsector. Theaggregatorthatminimizesthecostofproducinglaborservicesis Wt= (cid:18)(cid:90)1 Wt(i)1−νLdi (cid:19)1− 1 νL 0 Thedemandforagivenlabortypeis (cid:18)Wt(i)(cid:19)−νL Lt(i)= Lt Wt Unionsnegotiatetheirwagewithprobability1−ξW. Whentheydo, theymaximizehouseholdutilitytakingdemandcurvesfortheirlaboras given. Thefirst-orderconditionwithrespecttothewageis Et j (cid:88) ∞ =0 (βξW)jLt+j  Λt+j Pt 1 +j (1−νL) (cid:34) W W˜ t+ t j (cid:35)−νL (1+τW)+χνL (cid:32)(cid:34) W W˜ t+ t j (cid:35)−νL Lt+j (cid:33)φ(cid:34) W W˜ t+ t j (cid:35)−νL W˜ 1 t  =0 whereW˜ tisthechosenwagebyaunionthatcanupdatesitswage. Thisissimplifiedtobe Et (cid:88) ∞ (βξW)jLt+j (cid:34) Wt (cid:35)−νL  Λt+j 1 νL−1 (1+τW)W˜ t−χ (cid:32)(cid:34) W˜ t (cid:35)−νL Lt+j (cid:33)φ =0 j=0 Wt+j Pt+j νL Wt+j Et j (cid:88) ∞ =0 (βξW)jLt+j (cid:34) W W t+ t j (cid:35)−νL  Λt+j Pt 1 P + t+ νL j φνL νL −1 (1+τW)w˜t 1+νLφ−χ (cid:32)(cid:34) wt 1 +j Pt 1 +j (cid:35)−νL Lt+j (cid:33)φ =0 Et j (cid:88) ∞ =0 (βξW)jLt+j (cid:34) W W t+ t j (cid:35)−νL  Λt+j P P t+ t j νL νL −1 (1+τW)w˜t 1+νLφ−χ (cid:32)(cid:34) wt 1 +j P P t+ t j (cid:35)−νL Lt+j (cid:33)φ =0 wherew˜tistherealwagethatissetbyunionsthatoptimize. Then,wecanwrite FW,tw˜t 1+νLφ=KW,t (B.20) where FW,t=LtΛt νL νL −1 (1+τW)+βξWEtπ t − + 1 1 (cid:18)w w t+ t 1πt+1 (cid:19)νL FW,t+1 (B.21) and KW,t=χL1 t +φwt φνL+βξWEt (cid:18)w w t+ t 1 (cid:19)νL(cid:0)πt+1 (cid:1)νL(1+φ)KW,t+1. (B.22) Thenwagesevolvesothat Wt= (cid:16) (1−ξW)W˜ t 1−νL+ξWW t 1 − − 1 νL(cid:17)1− 1 νL whichyields wt= (cid:32) (1−ξW)w˜t 1−νL+ξW (cid:18)w π t− t 1 (cid:19)1−νL (cid:33) 1− 1 νL (B.23) NotethatinthecasethatξW =0,wehavew˜t=wtand Λt νL νL −1 (1+τW)wt=χLt(i)φ=χLφ t sothatif νL νL −1(1+τW)=1wehavetheusualintratemporalEulerequation. Intheforeigneconomy,wehave FW ∗ ,t (cid:0)w˜t ∗(cid:1)1+νLφ=KW ∗ ,t (B.24) 55

FW ∗ ,t=L∗ tΛ∗ t νL νL −1 (1+τW)+βξWEt (cid:16) πt ∗ +1 (cid:17)−1 (cid:32)w w t ∗ + t ∗ 1πt ∗ +1 (cid:33)νL FW,t+1 (B.25) KW ∗ ,t=χ(cid:0)L∗ t (cid:1)1+φ(cid:0)wt ∗(cid:1)φνL+βξWEt (cid:32)w w t ∗ + t ∗ 1 (cid:33)νL(cid:16) πt ∗ +1 (cid:17)νL(1+φ) KW ∗ ,t+1 (B.26) wt ∗= (cid:32) (1−ξW)(cid:0)w˜t ∗(cid:1)1−νL+ξW (cid:32)w π t ∗ − t ∗ 1 (cid:33)1−νL(cid:33)1− 1 νL . (B.27) B.3 Goods aggregators Ineachcountry,perfectlycompetitivefirmsaggregatecountry-specificintermeidateinputsintoYH,t,YF,t,YH ∗ ,t,andYF ∗ ,t. Theseintermediate inputsareusedeitherforthecreationofconsumption,governmentpurchases,orinvestmentgoodssothat CH,t+GH,t+IH,t=YH,t (B.28) CF,t+GF,t+IF,t=YF,t (B.29) CH ∗ ,t+G∗ H,t+IH ∗ ,t=XH ∗ ,t (B.30) CF ∗ ,t+G∗ F,t+IF ∗ ,t=XF ∗ ,t (B.31) ThevaluesYH,tandYF,tarearecompositesofgoodspurchasedfrommonopolistsbyperfectlycompetitivefirmswhoproduceusing (cid:18)(cid:90)1 ν−1 (cid:19)ν− ν 1 YH,t= XH,t(i) ν di 0 (cid:18)(cid:90)1 ν−1 (cid:19)ν− ν 1 YF,t= XF,t(i) ν di 0 Demandcurvesarethenoftheform XH,t(i)= (cid:32)PH,t(i)(cid:33)−ν YH,t (B.32) PH,t XF,t(i)= (cid:32)PF,t(i)(cid:33)−ν YF,t. (B.33) PF,t Thezeroprofitcondition,alongwiththesedemandcurves,implies PH,t= (cid:18)(cid:90)1 PH,t(i)1−νdi (cid:19)1− 1 ν (B.34) 0 Similarly, PF,t= (cid:18)(cid:90)1 PF,t(i)1−νdi (cid:19)1− 1 ν (B.35) 0 Theforeigncountryissymmetric. Demandcurvesarethenoftheform XH ∗ ,t(i)=   PH P ∗ , ∗ t(i)   −ν XH ∗ ,t (B.36) H,t XF ∗ ,t(i)=   PF P ∗ , ∗ t(i)   −ν XF ∗ ,t. (B.37) F,t Thezeroprofitconditions,alongwiththesedemandcurves,imply PH ∗ ,t= (cid:18)(cid:90)1 PH ∗ ,t(i)1−νdi (cid:19)1− 1 ν (B.38) 0 and PF ∗ ,t= (cid:18)(cid:90)1 PF ∗ ,t(i)1−νdi (cid:19)1− 1 ν . (B.39) 0 B.4 Retailers Finalconsumptiongoods,Ct,arecreatedbycombininggoodsfromcountriesHandF(CH,tandCF,t)using Ct= (cid:16) ω1−ρ(cid:0)CH,t (cid:1)ρ+(1−ω)1−ρ(cid:0)CF,t (cid:1)ρ(cid:17) ρ 1 (B.40) 56

Profitsaregivenby Pt (cid:16) ω1−ρ(cid:0)CH,t (cid:1)ρ+(1−ω)1−ρ(cid:0)CF,t (cid:1)ρ(cid:17) ρ 1 −PH,tCH,t−PF,tCF,t. (B.41) wherePH,tisthenominalpriceofCH,t,PF,tisthenominalpriceofCF,t. Demandcurvesarethen CH,t= (cid:18)PH,t (cid:19)ρ− 1 1 ωCt (B.42) Pt CF,t= (cid:18)PF,t (cid:19)ρ− 1 1 (1−ω)Ct. (B.43) Pt Thereisfreeentryforretailers,soprofitsarezero. Substitutingdemandcurvesintotheprofitsexpressionyields Pt= (cid:32) ωP H ρ , − ρ t 1 +(1−ω)(cid:0)PF,t (cid:1)ρ− ρ 1 (cid:33)ρ− ρ 1 (B.44) Governmentpurchasesareproducedusingthesametechnologysothat Gt= (cid:16) ω1−ρ(cid:0)GH,t (cid:1)ρ+(1−ω)1−ρ(cid:0)GF,t (cid:1)ρ(cid:17) ρ 1 (B.45) Thisimpliesdemandcurvesoftheform GH,t= (cid:18)PH,t (cid:19)ρ− 1 1 ωGt (B.46) Pt and GF,t= (cid:18)PF,t (cid:19)ρ− 1 1 (1−ω)Gt. (B.47) Pt Inaddition,thepriceofgovernmentpurchasesisthesameasthepriceoftheconsumptiongood. Investmentgoodsareproducedusingthesame technologyso Finalconsumptiongoodsintheforeigncountry,Ct ∗,arecreatedbycombininggoodsforcountriesHandF(CH ∗ ,tandCF ∗ ,t)using Ct ∗= (cid:16) ω1−ρ(cid:16) CF ∗ ,t (cid:17)ρ +(1−ω)1−ρ(cid:16) CH ∗ ,t (cid:17)ρ(cid:17) ρ 1 (B.48) Profitsaregivenby Pt ∗(cid:16) ω1−ρ(cid:16) CF ∗ ,t (cid:17)ρ +(1−ω)1−ρ(cid:16) CH ∗ ,t (cid:17)ρ(cid:17) ρ 1 −PF ∗ ,tCF ∗ ,t−PH ∗ ,tCH ∗ ,t. (B.49) wherePH ∗ ,tisthenominalpriceofCH ∗ ,t,PF ∗ ,tisthenominalpriceofCF ∗ ,t. Demandcurvesaregivenby CH ∗ ,t= (cid:32)P P H ∗ t ∗ ,t (cid:33) ρ− 1 1 (1−ω)Ct ∗ (B.50) CF ∗ ,t= (cid:32)P P F ∗ t ∗ ,t (cid:33) ρ− 1 1 ωCt ∗ (B.51) Theconsumerpriceindexesaregivenby Pt ∗= (cid:18) ω (cid:16) PF ∗ ,t (cid:17) ρ− ρ 1 +(1−ω) (cid:16) PH ∗ ,t (cid:17) ρ− ρ 1 (cid:19)ρ− ρ 1 . (B.52) Governmentpurchasesareproducedusingthesametechnologysothat G∗ t = (cid:16) ω1−ρ(cid:16) G∗ F,t (cid:17)ρ +(1−ω)1−ρ(cid:16) G∗ H,t (cid:17)ρ(cid:17) ρ 1 (B.53) Thisimpliesdemandcurvesoftheform G∗ H,t= (cid:32)P P H ∗ t ∗ ,t (cid:33) ρ− 1 1 ωG∗ t (B.54) and G∗ F,t= (cid:32)P P F ∗ t ∗ ,t (cid:33) ρ− 1 1 (1−ω)G∗ t. (B.55) Inaddition,thepriceofgovernmentpurchasesisthesameasthepriceoftheconsumptiongood. 57

B.5 Investment goods Investmentgoods,It,arecreatedbycombininggoodsfromcountriesHandF(IH,tandIF,t)using It= (cid:16) ω I 1−ρ(cid:0)IH,t (cid:1)ρ+(1−ωI)1−ρ(cid:0)IF,t (cid:1)ρ(cid:17) ρ 1 (B.56) Profitsaregivenby PI,t (cid:16) ω I 1−ρ(cid:0)IH,t (cid:1)ρ+(1−ωI)1−ρ(cid:0)IF,t (cid:1)ρ(cid:17) ρ 1 −PH,tIH,t−PF,tIF,t. (B.57) wherePH,tisthenominalpriceofIH,t,PF,tisthenominalpriceofIF,t. NotethatweareimposingthatthepriceofIH,tandCH,tbethesame becausetheyarethesameinput. SimilarlyforIF,tandCF,t. First-orderconditionsare PI,t (cid:16) ω I 1−ρ(cid:0)IH,t (cid:1)ρ+(1−ωI)1−ρ(cid:0)IF,t (cid:1)ρ(cid:17)1− ρ ρ ω I 1−ρ(cid:0)IH,t (cid:1)ρ−1=PH,t PI,t (cid:16) ω I 1−ρ(cid:0)IH,t (cid:1)ρ+(1−ωI)1−ρ(cid:0)IF,t (cid:1)ρ(cid:17)1− ρ ρ (1−ωI)1−ρ(cid:0)IF,t (cid:1)ρ−1=PF,t Demandcurvesarethen IH,t= (cid:32)PH,t (cid:33) ρ− 1 1 ωIIt (B.58) PI,t IF,t= (cid:32)PF,t (cid:33) ρ− 1 1 (1−ωI)It. (B.59) PI,t Thereisfreeentryforretailers,soprofitsarezero. Substitutingdemandcurvesintotheprofitsexpressionyields PI,t= (cid:32) ωIP H ρ , − ρ t 1 +(1−ωI)(cid:0)PF,t (cid:1)ρ− ρ 1 (cid:33)ρ− ρ 1 (B.60) Becausewesetω=ωI,wehavethatPI,t=Pt. Investmentgoodsintheforeigncountry,It ∗,arecreatedbycombininggoodsforcountriesHand F(IH ∗ ,tandIF ∗ ,t)using It ∗= (cid:16) ω I 1−ρ(cid:16) IF ∗ ,t (cid:17)ρ +(1−ωI)1−ρ(cid:16) IH ∗ ,t (cid:17)ρ(cid:17) ρ 1 (B.61) Profitsaregivenby PI ∗ ,t (cid:16) ω I 1−ρ(cid:16) IF ∗ ,t (cid:17)ρ +(1−ωI)1−ρ(cid:16) IH ∗ ,t (cid:17)ρ(cid:17) ρ 1 −PF ∗ ,tIF ∗ ,t−PH ∗ ,tIH ∗ ,t. (B.62) wherePH ∗ ,tisthenominalpriceofYH ∗ ,t,PF ∗ ,tisthenominalpriceofYF ∗ ,t. Again,weareimposingthatthepriceofYH ∗ ,tandIH ∗ ,tarethesame. Similarly,thepriceofYF ∗ ,tandIF I ,tarethesame. Demandcurvesaregivenby IH ∗ ,t=   P P H ∗ ∗ ,t   ρ− 1 1 (1−ωI)It ∗ (B.63) I,t IF ∗ ,t=   P P F ∗ ∗ ,t   ρ− 1 1 ωIIt ∗ (B.64) I,t Thepriceindexeisgivenby PI ∗ ,t= (cid:18) ωI (cid:16) PF ∗ ,t (cid:17) ρ− ρ 1 +(1−ωI) (cid:16) PH ∗ ,t (cid:17) ρ− ρ 1 (cid:19)ρ− ρ 1 . (B.65) Becausewesetω=ωI,wehavethatPI,t=Pt. B.6 Bond market clearing Bondsareinzeronetsupplyso bH,t+b∗ H,t=0 (B.66) bF,t+b∗ F,t=0 (B.67) wherebH,t≡BH,t/Pt,b∗ H,t≡BH ∗ ,t/Pt,bF,t≡BF,t/Pt ∗,b∗ F,t≡BF ∗ ,t/Pt ∗. B.7 Monopolists WeintroducepricestickinessasaCalvo-styleprice-settingfriction. Monopolistsareonlyabletoupdatetheirpricewithprobabilityξ ineach period. Iftheyarenotabletoupdatetheirprice,itremainsthesameastheperiodbefore. IfmonopolistiinthecountryHcanupdateitsprice, 58

itchoosesP˜ H,t(i)andP˜ H ∗ ,t(i)tomaximize Et j (cid:88) ∞ =0 Λt+j    (cid:32)P˜ P H t , + t( j i) (1+τX)−MCt+j (cid:33)(cid:32)P P ˜ H H , , t t+ (i j )(cid:33)−ν YH,t+j+ (cid:32)NERt P + t j + P˜ j H ∗ ,t(i) (1+τX)−MCt+j (cid:33)  P P ˜ H H ∗ ∗ , , t t+ (i j )   −ν YH ∗ ,t+j    TheFOCwithrespecttoP˜ H,t(i)is Et (cid:88) ∞ (βξ)jΛt+j (cid:34)P˜ H,t Pt (cid:32) PH,t (cid:33)−ν YH,t+j− 1 ν MCt+j (cid:32) PH,t (cid:33)−ν YH,t+j (cid:35) =0 j=0 Pt Pt+j PH,t+j 1+τX ν−1 PH,t+j Thenwehave FH,tp˜H,t=KH,t (B.68) wherewedefineFH,tandKH,tasrecursivesumssothat FH,t≡Et (cid:88) ∞ (βξ)jΛt+j Pt (cid:32) PH,t (cid:33)−ν YH,t+j j=0 Pt+j PH,t+j KH,t≡Et (cid:88) ∞ (βξ)jΛt+j 1 ν MCt+j (cid:32) PH,t (cid:33)−ν YH,t+j. j=0 1+τX ν−1 PH,t+j Thesecanbewrittenas FH,t=ΛtYH,t+βξEtπ t − + 1 1 πH ν ,t+1FH,t+1 (B.69) KH,t=Λt 1+ 1 τX ν− ν 1 MCtYH,t+βξEtπH ν ,t+1KH,t+1 (B.70) where πH,t≡PH,t/PH,t−1 (B.71) Thepriceindexforhomegoodsinthehomemarketisgivenby PH,t= (cid:16) (1−ξ)P˜ H 1− ,t ν+ξP H 1− ,t ν −1 (cid:17) 1− 1 ν sothat  p1−ν  1− 1 ν pH,t=(1−ξ)p˜1 H − ,t ν+ξ π H t 1 , − t− ν 1  (B.72) TheFOCwithrespecttoP˜ H ∗ ,t(i)is Et j (cid:88) ∞ =0 (βξ)jΛt+j   N N E E R R t+ t j NE P R t tPt ∗ p˜∗ H,tP P t+ t j   P P H ∗ H ∗ ,t , + t j   −ν YH ∗ ,t+j− 1+ 1 τX ν− ν 1 MCt+j   P P H ∗ H ∗ ,t , + t j   −ν YH ∗ ,t+j  =0 Thenwehave FH ∗ ,tRERtp˜∗ H,t=KH ∗ ,t (B.73) where FH ∗ ,t≡Et j (cid:88) ∞ =0 (βξ)jΛt+j N N E E R R t+ t j P P t+ t j   P P H ∗ H ∗ ,t , + t j   −ν YH ∗ ,t+j KH ∗ ,t≡Et j (cid:88) ∞ =0 (βξ)jΛt+j 1+ 1 τX ν− ν 1 MCt+j   P P H ∗ H ∗ ,t , + t j   −ν YH ∗ ,t+j. Thesecanbewrittenas FH ∗ ,t=ΛtYH ∗ ,t+βξEt N N E E R R t+ t 1π t − + 1 1 (cid:16) πH ∗ ,t+1 (cid:17)ν FH ∗ ,t+1 (B.74) KH ∗ ,t=Λt 1+ 1 τX ν− ν 1 MCtYH ∗ ,t+βξEt (cid:16) πH ∗ ,t+1 (cid:17)ν KH ∗ ,t+1 (B.75) where πH ∗ ,t≡PH ∗ ,t/PH ∗ ,t−1 (B.76) . Thepriceindexforhomegoodsintheforeignmarketisgivenby PH ∗ ,t= (cid:18) (1−ξ) (cid:16) P˜ H ∗ ,t (cid:17)1−ν +ξ (cid:16) PH ∗ ,t−1 (cid:17)1−ν(cid:19) 1− 1 ν 59

sothat p∗ H,t=    (1−ξ) (cid:16) p˜∗ H,t (cid:17)1−ν +ξ (cid:16) p∗ H (cid:0)π ,t t ∗ − (cid:1)1 1 − (cid:17)1 ν −ν   1− 1 ν (B.77) Theforeignfirmsaresymmetricsymmetric. Ifmonopolisticanupdateitsprice,itchoosesP˜ F ∗ ,t(i)andP˜ F,t(i)tomaximize Et j (cid:88) ∞ =0 Λ∗ t+j      P˜ P F ∗ t , ∗ t + ( j i) (1+τX)−MCt ∗ +j     P P ˜ F F ∗ ∗ , , t t+ (i j )   −ν YF ∗ ,t+j+   NE P R ˜ F t , + t j ( P i) t ∗ +j (1+τX)−MCt ∗ +j   (cid:32)P P ˜ F F , , t t+ (i j )(cid:33)−ν YF,t+j    TheFOCwithrespecttoP˜ F ∗ ,t(i)is Et j (cid:88) ∞ =0 (βξ)jΛ∗ t+j   P˜ P F ∗ t ∗ ,t P P t ∗ + t ∗ j   P P F ∗ F , ∗ t , + t j   −ν YF ∗ ,t+j− 1+ 1 τX ν− ν 1 MCt ∗ +j   P P F ∗ F , ∗ t , + t j   −ν YF ∗ ,t+j  =0. Wewritethisas FF ∗ ,tp˜∗ F,t=KF ∗ ,t (B.78) where FF ∗ ,t=Λ∗ tYF ∗ ,t+βξEt (cid:16) πt ∗ +1 (cid:17)−1(cid:16) πF ∗ ,t+1 (cid:17)ν FF ∗ ,t+1 (B.79) and KF ∗ ,t=Λ∗ t1+ 1 τX ν− ν 1 MCt ∗YF ∗ ,t+βξEt (cid:16) πF ∗ ,t+1 (cid:17)ν KF ∗ ,t+1. (B.80) where πF ∗ ,t≡PF ∗ ,t/PF ∗ ,t−1 (B.81) . Thepriceindeximplies p∗ F,t=    (1−ξ) (cid:16) p˜∗ F,t (cid:17)1−ν +ξ (cid:16) p (cid:0) ∗ F π ,t t ∗ − (cid:1) 1 1− (cid:17)1 ν −ν   1− 1 ν (B.82) TheFOCwithrespecttoP˜ F,t(i)is Et j (cid:88) ∞ =0 (βξ)jΛ∗ t+j   N N E E R R t+ t j NE P R t tPt ∗ p˜F,t P P t ∗ + t ∗ j (cid:32) P P F F ,t , + t j (cid:33)−ν YF,t+j− 1+ 1 τX ν− ν 1 MCt ∗ +j (cid:32) P P F F ,t , + t j (cid:33)−ν YF,t+j  =0 Wecanwritethisas FF,t p˜F,t =KF,t (B.83) RERt where FF,t=Λ∗ tYF,t+βξEt N N E E R R t+ t 1 (cid:16) πt ∗ +1 (cid:17)−1(cid:0)πF,t+1 (cid:1)νFF,t+1 (B.84) and KF,t=Λ∗ t1+ 1 τX ν− ν 1 MCt ∗YF,t+βξEt (cid:0)πF,t+1 (cid:1)νKF,t+1. (B.85) where πF,t≡PF,t/PF,t−1 (B.86) Thepriceindeximpliesthat pF,t= (cid:32) (1−ξ)(cid:0)p˜F,t (cid:1)1−ν+ξ (cid:0)pF (π ,t t − )1 1 − (cid:1)1 ν −ν(cid:33)1− 1 ν (B.87) B.8 Marginal cost Monopolistsproducewithtechnologysothat XH,t(i)+XH ∗ ,t(i)=AtKt(i)αLt(i)1−α XF,t(i)+XF ∗ ,t(i)=A∗ tKt ∗(i)αL∗ t(i)1−α whereAt andA∗ t arestochasticprocessesand,inaslightabuseofnotation,the(i)meanstheamounthiredbyaparticularmonopolist. Cost minimizationimplies RK,t=αMCtAt(Kt)α−1(Lt)1−α (B.88) W Pt t =(1−α)MCtAtKt α(Lt)−α (B.89) and RK ∗ ,t=αMCt ∗A∗ t (cid:0)Kt ∗(cid:1)α−1(cid:0)L∗ t (cid:1)1−α (B.90) 60

W Pt ∗ t ∗ =(1−α)MCt ∗A∗ t (cid:0)Kt ∗(cid:1)α(cid:0)L∗ t (cid:1)−α. (B.91) B.9 Aggregation Aggregatingaccrossfirmsyields (cid:90) 0 1(cid:32)PH PH ,t , ( t i)(cid:33)−ν YH,tdi+ (cid:90) 0 1   PH P ∗ H , ∗ t , ( t i)   −ν YH ∗ ,tdi=At (cid:90) 0 1 Kt(i)α(Lt(i))1−αdi sothat dH,tYH,t+d∗ H,tYH ∗ ,t=AtKt αL1 t −α (B.92) wherethelastequationfollowswithoutthe(i)’sbecauseallfirmschoosethesamecapital-to-laborratiofromtheconstant-returns-to-scaleproductiontechnologyandthatwehaveameasure1offirms. Similarly dF,tYF,t+d∗ F,tYF ∗ ,t=At (cid:0)Kt ∗(cid:1)α(cid:0)L∗ t (cid:1)1−α (B.93) Herethedispersiontermscanbewrittenrecursivelyas dH,t=(1−ξ)pν H,t (cid:0)p˜H,t (cid:1)−ν+ξπH ν ,tdH,t−1 (B.94) d∗ H,t=(1−ξ) (cid:16) p∗ H,t (cid:17)ν(cid:16) p˜∗ H,t (cid:17)−ν +ξ (cid:16) πH ∗ ,t (cid:17)ν d∗ H,t−1, (B.95) d∗ F,t=(1−ξ) (cid:16) p∗ F,t (cid:17)ν(cid:16) p˜∗ F,t (cid:17)−ν +ξ (cid:16) πF ∗ ,t (cid:17)ν d∗ F,t−1, (B.96) dF,t=(1−ξ)(cid:0)pF,t (cid:1)ν(cid:0)p˜F,t (cid:1)−ν+ξ(cid:0)πF,t (cid:1)νdF,t−1. (B.97) B.10 Government ThemonetaryauthorityfollowsaTaylorrule Rt=(cid:0)Rt−1 (cid:1)γ(cid:16) Rπt θπ(cid:17)1−γ exp(cid:0)(cid:15)R,t (cid:1) whereθπ>1 (B.98) oralternativelyfollowsamoneygrowthrule log (cid:32) Mt (cid:33) =log(cid:0)xM,t (cid:1) (B.99) Mt−1 where log(cid:0)xM,t (cid:1)=ρxM log(cid:0)xM,t−1 (cid:1)+σxM (cid:15)xM,t (B.100) Thefiscalauthoritybalancesitsbudgetwithlumpsumtaxes. TheforeignmonetaryauthorityfollowsaTaylorrule Rt ∗= (cid:16) Rt ∗ −1 (cid:17)γ(cid:16) R(cid:0)πt ∗(cid:1)θπ(cid:17)1−γ exp (cid:16) (cid:15)∗ R,t (cid:17) whereθπ>1 (B.101) oralternativelyfollowsamoneygrowthrule log (cid:32) M M ∗ t ∗ (cid:33) =log (cid:16) x∗ M,t (cid:17) (B.102) t−1 where log (cid:16) x∗ M,t (cid:17) =ρxM log (cid:16) x∗ M,t−1 (cid:17) +σxM (cid:15)∗ xM,t (B.103) Thefiscalauthoritybalancesitsbudgetwithlumpsumtaxes. WealsoconsideranNERtargetingrulewhere Rt ∗=R (cid:16) NERt −θNER(cid:17) exp (cid:16) ε∗ R,t (cid:17) (B.104) B.11 Asset-market completeness Whenwehaveassetmarketcompleteness,itmustbethatφB =0. WeassumethatthereisacompletesetofArrowsecurities. Letst denote thestateoftheworldintimetandst=(cid:8)st,st−1,...(cid:9). ThehouseholdincountryHpricestheArrowsecuritiesthatpayoffoneunitoftheH currencyinstatest+1sothat Q s t t+1 Λt=β Λt+1Pr (cid:16) st+1|st(cid:17) Pt Pt+1 61

whereQ s t t+1 isthepriceattimetofthesecuritythatpaysoffattimet+1instatest+1. ThehouseholdincountryFpricestheArrowsecurities thatpayoffoneunitoftheHcurrencyinstatest+1sothat N Q E s t R t+ tP 1 t ∗ Λ∗ t =β NER Λ t ∗ t + + 1 1 P t ∗ +1 Pr (cid:16) st+1|st(cid:17) . Notethatitisaglobalmarket,sothepricesarethesame. Thendivide NERtPt ∗ Λt = NERt+1Pt ∗ +1Λt+1 Pt Λ∗ t Pt+1 Λ∗ t+1 RERt Λ Λ ∗ t t =RERt+1 Λ Λ ∗ t t + + 1 1 Giventhatthismusthappenforeverydateandstate, Λt RERt Λ∗ t =κ (B.105) forallt. B.12 Equilibrium Whenassetmarketsareincomplete,anequilibriumdeterminesthefollowing37endogenousobjects:Ct,CH,t,CF,t,GH,t,GF,t,Λt,Lt,wt≡ W Pt t, YH,t,YF,t,Rt,MCt,πt,mt≡ M Pt t,bF,t,bH,t,Kt,It,IH,t,IF,t,Qt,RK,t,pF,t≡ P P F t ,t,pH,t≡ P P H t ,t,p˜H,t,FH,t,KH,t,dH,t,πH,t,p˜F,t, FF,t,KF,t,dF,t,πF,t,w˜t,FW,t,KW,t,the37starversions,aswellas∆NERt ≡ N N E E R R t− t 1 andRERt. Todeterminethese76variables,we requirethatthefollowing76equationshold: (B.3),(B.4),(B.5),(B.6),(B.7),(B.8),(B.9),(B.12),(B.13),(B.14),(B.15),(B.16),(B.17),(B.18), (B.19),(B.20),(B.21),(B.22),(B.23),(B.24),(B.25),(B.26),(B.27),(B.28),(B.29),(B.30),(B.31),(B.42),(B.43),(B.44),(B.46),(B.47),(B.50), (B.51),(B.52),(B.54),(B.55),(B.58),(B.59),(B.63),(B.64),(B.66),(B.67),(B.68),(B.69),(B.70),(B.71),(B.72),(B.73),(B.74),(B.75),(B.76), (B.77),(B.78),(B.79),(B.80),(B.81),(B.82),(B.83),(B.84),(B.85),(B.86),(B.87),(B.88),(B.89),(B.90),(B.91),(B.92),(B.93),(B.94),(B.95), (B.96), (B.97)alongwitheither(B.98)and(B.101)or(B.99)and(B.102)or(B.98)and(B.104). Finally, weusethehomehouseholdbudget constraint(B.2). TheforeignhouseholdbudgetconstraintcanbeignoredbecauseofWalras’law. Ifwewantcompletemarkets,weuse(B.105) insteadof(B.2)andsetφB =0. Inaddition, wereplaceequations(B.6)and(B.14)withtheconditionsthatbH,t =bF,t =0. Ifwewantto excludecapitalaccumulation,wesetα=0andreplaceequations(B.3),(B.8),(B.9),and(B.88),bytheconditionsthatIt=Kt=Qt=RK,t=0. Similarly,wereplaceequations(B.12),(B.17),(B.18),and(B.90)bytheconditionsthatIt ∗=Kt ∗=Q∗ t =RK ∗ ,t=0. B.13 Steady State Todeterminesteadystate,weassumethattargetinflationinbothcountriesis1. So,π=π∗=1. TheintertemporalEulerequationsdetermine R=R∗ =β−1. WenormalizedL=L∗ =1. Fromthedefinitionofsteadystate,∆NER=1. WedefineinitialconditionssothatRER=1. Firmoptimalyandsymmetryoftheequilibrium,pH=p∗ H=pF =p∗ F =1. Asaresult,pI =p∗ I =1andQ=Λ. Marginalcostisgivenby ν−1 MC= (1+τX) ν Therentalrateofcapitalis 1−β(1−δ) RK= β Sothat K βαMC = XH+X H ∗ 1−β(1−δ) Then I K =δ XH+X H ∗ XH+X H ∗ Since XH+XH ∗ =Kα (cid:0)XH+XH ∗(cid:1)1−α= (cid:32) XH K +X H ∗ (cid:33)α Sothat XH+XH ∗ = (cid:32)βα 1 ν − − ν β 1 ( ( 1 1 − + δ τ ) X)(cid:33)1− α α =XF +XF ∗ wherethelastequalityfollowsbysymmetry. Withthis,wealsohaveK. Demandcurvesimply ω YH=YF 1−ω and YF ∗=YH ∗ 1− ω ω 62

Similarly, IH=IF ωI 1−ωI and IF ∗ =IH ∗ 1− ωI ωI Then YH+IH+YH ∗+IH ∗ =XH+XH ∗ Symmetryandthedemandcurvesimply YH+IH+YH 1− ω ω +IH 1− ωI ωI =XH+XH ∗ YH ω 1 +IH ω 1 I =XH+XH ∗ Y +I=XH+XH ∗ So Y = XH+XH ∗ −δ K K whichgivesusY. Then, YF = (1−ω)Y YH = ωY YF ∗ = ωY YH ∗ = (1−ω)Y and IF = (1−ωI)I IH = ωII IF ∗ = ωII IH ∗ = (1−ωI)I GivenGandG∗ thisgivesusC andC∗,whichdetermineΛandΛ∗. Thevaluesofmandm∗ aredeterminedbythemoneydemandequations. Thevaluesofwandw∗aredeterminedby w=(1−α)MCKα. Finally,wegetχfrom χ=Λ νL−1 (1+τW)w. νL B.14 Equilibrium with no capital, flexible prices/wages, complete markets, technology shocks, and a Taylor rule Inaddition,weassumethath=0andγ=0sothattherearenostatevariables. CollecttherelevantequationstodetermineCˆ t,Λˆ t,Rˆ t,πˆt,wˆt, Lˆ t,Cˆ H,t,Cˆ F,t,pˆH,t,pˆF,t,M(cid:100)Ct,Yˆ H,t,Yˆ F,t,thestarversions,and∆N(cid:100)ERt,R(cid:100)ERt. Herexˆt≡log(xt/x)isthelogdeviationfromsteadystate. −σCˆ t=Λˆ t −σCˆ t ∗=Λˆ∗ t Λˆ t=Rˆ t+Et (cid:16) Λˆ t+1−πˆt+1 (cid:17) Λˆ∗ t =Rˆ t ∗+Et (cid:16) Λˆ∗ t+1−πˆ∗ t+1 (cid:17) Λˆ t+wˆt=φLˆ t Λˆ∗ t +wˆt ∗=φLˆ∗ t CHCˆ H,t=YHYˆ H,t CFCˆ F,t=YFYˆ F,t CH ∗Cˆ H ∗ ,t=YH ∗Yˆ H ∗ ,t CF ∗Cˆ F ∗ ,t=YF ∗Yˆ F ∗ ,t 63

1 Cˆ H,t= pˆH,t+Cˆ t ρ−1 Cˆ H ∗ ,t= ρ− 1 1 pˆ∗ H,t+Cˆ t ∗ 1 Cˆ F,t= pˆF,t+Cˆ t ρ−1 Cˆ F ∗ ,t= ρ− 1 1 pˆ∗ F,t+Cˆ t ∗ 0=ωpˆH,t+(1−ω)pˆF,t 0=ωpˆ∗ F,t+(1−ω)pˆ∗ H,t. pˆH,t=M(cid:100)Ct pˆ∗ H,t+R(cid:100)ERt=M(cid:100)Ct pˆF,t−R(cid:100)ERt=M(cid:100)C∗ t pˆ∗ F,t=M(cid:100)C∗ t wˆt=M(cid:100)Ct+Aˆ t wˆt ∗=M(cid:100)C∗ t +Aˆ∗ t YHYˆ H,t+YH ∗Yˆ H ∗ ,t=Aˆ t+Lˆ t YFYˆ F,t+YF ∗Yˆ F ∗ ,t=Aˆ∗ t +Lˆ∗ t Rˆ t=θππˆt Rˆ t ∗=θππˆ∗ t Λˆ∗ t −Λˆ t=R(cid:100)ERt R(cid:100)ERt−R(cid:100)ERt−1=∆N(cid:100)ERt+πˆ∗ t −πˆt FirstsuboutYˆ H,t,Yˆ H ∗ ,t,Yˆ F,t,andYˆ F ∗ ,talongwithΛˆ tandΛˆ∗ t. NotethatthechangeinNER,inflationandthenominalinterestrateonlyenter 5equations(theyareblockrecursive). Deletethem. −σCˆ t+wˆt=φLˆ t −σCˆ t ∗+wˆt ∗=φLˆ∗ t 1 Cˆ H,t= pˆH,t+Cˆ t ρ−1 Cˆ H ∗ ,t= ρ− 1 1 pˆ∗ H,t+Cˆ t ∗ 1 Cˆ F,t= pˆF,t+Cˆ t ρ−1 Cˆ F ∗ ,t= ρ− 1 1 pˆ∗ F,t+Cˆ t ∗ 0=ωpˆH,t+(1−ω)pˆF,t 0=ωpˆ∗ F,t+(1−ω)pˆ∗ H,t. pˆH,t=M(cid:100)Ct pˆ∗ H,t+R(cid:100)ERt=M(cid:100)Ct pˆF,t−R(cid:100)ERt=M(cid:100)C∗ t pˆ∗ F,t=M(cid:100)C∗ t wˆt=M(cid:100)Ct+Aˆ t wˆt ∗=M(cid:100)C∗ t +Aˆ∗ t CHCˆ H,t+CH ∗Cˆ H ∗ ,t=Aˆ t+Lˆ t CFCˆ F,t+CF ∗Cˆ F ∗ ,t=Aˆ∗ t +Lˆ∗ t −σCˆ t ∗+σCˆ t=R(cid:100)ERt 64

Suboutwt. wt ∗. CH,t,CF,t,CH ∗ ,t,andCF ∗ ,t. Also,usethatCH= Y Cω,CF = Y C(1−ω),CF ∗ = Y Cω,CH ∗ = Y C(1−ω). −σCˆ t+M(cid:100)Ct+Aˆ t=φLˆ t −σCˆ t ∗+M(cid:100)C∗ t +Aˆ∗ t =φLˆ∗ t 0=ωpˆH,t+(1−ω)pˆF,t 0=ωpˆ∗ F,t+(1−ω)pˆ∗ H,t. pˆH,t=M(cid:100)Ct pˆ∗ H,t+R(cid:100)ERt=M(cid:100)Ct pˆF,t−R(cid:100)ERt=M(cid:100)C∗ t pˆ∗ F,t=M(cid:100)C∗ t Y C ω (cid:18) ρ− 1 1 pˆH,t+Cˆ t (cid:19) + Y C (1−ω) (cid:18) ρ− 1 1 pˆ∗ H,t+Cˆ t ∗ (cid:19) =Aˆ t+Lˆ t Y C (1−ω) (cid:18) ρ− 1 1 pˆF,t+Cˆ t (cid:19) + Y C ω (cid:18) ρ− 1 1 pˆ∗ F,t+Cˆ t ∗ (cid:19) =Aˆ∗ t +Lˆ∗ t −σCˆ t ∗+σCˆ t=R(cid:100)ERt SuboutpH,t,p∗ H,t,pF,t,p∗ F,t. −σCˆ t+M(cid:100)Ct+Aˆ t=φLˆ t −σCˆ t ∗+M(cid:100)C∗ t +Aˆ∗ t =φLˆ∗ t 0=ωM(cid:100)Ct+(1−ω) (cid:16) M(cid:100)C∗ t +R(cid:100)ERt (cid:17) 0=ωM(cid:100)C∗ t +(1−ω) (cid:16) M(cid:100)Ct−R(cid:100)ERt (cid:17) Y C ω (cid:18) ρ− 1 1 M(cid:100)Ct+Cˆ t (cid:19) + Y C (1−ω) (cid:18) ρ− 1 1 (cid:16) M(cid:100)Ct−R(cid:100)ERt (cid:17) +Cˆ t ∗ (cid:19) =Aˆ t+Lˆ t Y C (1−ω) (cid:18) ρ− 1 1 (cid:16) M(cid:100)C∗ t +R(cid:100)ERt (cid:17) +Cˆ t (cid:19) + Y C ω (cid:18) ρ− 1 1 M(cid:100)C∗ t +Cˆ t ∗ (cid:19) =Aˆ∗ t +Lˆ∗ t −σCˆ t ∗+σCˆ t=R(cid:100)ERt Nowstrategicallysubtractequationssoastoidentifydifferences,notlevels. −σ (cid:16) Cˆ t−Cˆ t ∗(cid:17) + (cid:16) M(cid:100)Ct−M(cid:100)C∗ t (cid:17) + (cid:16) Aˆ t−Aˆ∗ t (cid:17) =φ (cid:16) Lˆ t−Lˆ∗ t (cid:17) 0=(2ω−1) (cid:16) M(cid:100)Ct−M(cid:100)C∗ t (cid:17) +2(1−ω)R(cid:100)ERt Y C ρ− 1 1 (cid:16) M(cid:100)Ct−M(cid:100)C∗ t (cid:17) + Y C (2ω−1) (cid:16) Cˆ t−Cˆ t ∗(cid:17) − Y C 2(1−ω) ρ− 1 1 R(cid:100)ERt= (cid:16) Aˆ t−Aˆ∗ t (cid:17) + (cid:16) Lˆ t−Lˆ∗ t (cid:17) σ (cid:16) Cˆ t−Cˆ t ∗(cid:17) =R(cid:100)ERt NowsuboutCˆ t−Cˆ t ∗. −R(cid:100)ERt+ (cid:16) M(cid:100)Ct−M(cid:100)C∗ t (cid:17) + (cid:16) Aˆ t−Aˆ∗ t (cid:17) =φ (cid:16) Lˆ t−Lˆ∗ t (cid:17) 0=(2ω−1) (cid:16) M(cid:100)Ct−M(cid:100)C∗ t (cid:17) +2(1−ω)R(cid:100)ERt Y C ρ− 1 1 (cid:16) M(cid:100)Ct−M(cid:100)C∗ t (cid:17) + Y C (2ω−1) σ 1 R(cid:100)ERt− Y C 2(1−ω) ρ− 1 1 R(cid:100)ERt= (cid:16) Aˆ t−Aˆ∗ t (cid:17) + (cid:16) Lˆ t−Lˆ∗ t (cid:17) NowsuboutM(cid:100)Ct−M(cid:100)C∗ t. − 2ω 1 −1 R(cid:100)ERt+ (cid:16) Aˆ t−Aˆ∗ t (cid:17) =φ (cid:16) Lˆ t−Lˆ∗ t (cid:17) Y C(cid:20) (2ω−1) σ 1 − 2( ρ 1 − − 1 ω) 2ω 2ω −1 (cid:21) R(cid:100)ERt= (cid:16) Aˆ t−Aˆ∗ t (cid:17) + (cid:16) Lˆ t−Lˆ∗ t (cid:17) Finally,suboutLˆ t−Lˆ∗ t (1+φ) (cid:18) 2ω 1 −1 +φ Y C(cid:20)2ω σ −1 − 2( ρ 1 − − 1 ω) 2ω 2ω −1 (cid:21)(cid:19)−1(cid:16) Aˆ t−Aˆ∗ t (cid:17) =R(cid:100)ERt 65

B.14.1 Overshooting Wejustshowedthat R(cid:100)ERt=κ (cid:16) Aˆ t−Aˆ∗ t (cid:17) . TheintertemporalEulerequations,theTaylorrule,andcompleteassetmarketsimply −R(cid:100)ERt=θπ (cid:0)πˆt−πˆ∗ t (cid:1)+Et (cid:16) −R(cid:100)ERt+1− (cid:16) πˆt+1−πˆ∗ t+1 (cid:17)(cid:17) . AssumingthatthethattheAtandA∗ t areindependentAR(1)processeswithautocorrelationρAwehave: (ρA−1)R(cid:100)ERt=θπ (cid:0)πˆt−πˆ∗ t (cid:1)−Et (cid:16) πˆt+1−πˆ∗ t+1 (cid:17) . Solvethisforward (cid:0)πˆt−πˆ∗ t (cid:1)=(ρA−1)Et j (cid:88) ∞ =0θπ 1 1 +j R(cid:100)ERt+j= θ ρ π A − − ρ 1 A R(cid:100)ERt. Noticethat (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)θ ρ π A − − ρ 1 A (cid:12) (cid:12) (cid:12) (cid:12) <1 iftheTaylor-principleholds. SotheinflationdifferentialislessthantheRERt deviationfromsteadystate. Thismeansthat,inresponsetoa shocktotheRERt,therelativepricelevelofthetwocountriesmovesbylessthanthemovementintheRER. Thatis,thenominalexchange ratehastomoveinthesame directionastheRER. BecausetheRERismeanrevertingbuttheinflationdifferentialremainstheopositsignas thedeviationoftheRERtfromitssteadystatevalue,thenominalexchangeratemovesbacktowarditinitialvalue,andwegettheovershooting pattern. B.14.2 Regression coefficients Wecanwritethemulti-periodchangeintheRERas log(cid:0)RERt+h (cid:1)−log(RERt)=log(cid:0)NERt+h (cid:1)−log(NERt)+ (cid:88) h (cid:16) log (cid:16) πt ∗ +k (cid:17) −log(cid:0)πt+k (cid:1)(cid:17) k=1 Takeexpectationsanduseourdefinitionsoflog-deviationsfromsteadystatetoget EtR(cid:100)ERt+h−R(cid:100)ERt=Etlog(cid:0)NERt+h (cid:1)−log(NERt)+ (cid:88) h Et (cid:16) πˆ∗ t+k−πˆt+k (cid:17) k=1 EtR(cid:100)ERt+h−R(cid:100)ERt=Etlog(cid:0)NERt+h (cid:1)−log(NERt)− k (cid:88) = h 1 Et θ ρ π A − − ρ 1 A R(cid:100)ERt+k (cid:16) ρh A−1 (cid:17) R(cid:100)ERt=Etlog(cid:0)NERt+h (cid:1)−log(NERt)− θ ρ π A − − ρ 1 A R(cid:100)ERt k (cid:88) = h 1 ρk A (cid:16) ρh A−1 (cid:17) R(cid:100)ERt=Etlog(cid:0)NERt+h (cid:1)−log(NERt)− θ ρ π A − − ρ 1 A R(cid:100)ERt k (cid:88) = h 1 ρk A (cid:16) ρh A−1 (cid:17) R(cid:100)ERt=Etlog(cid:0)NERt+h (cid:1)−log(NERt)+ ρA θπ − − ρ ρ h A A +1 R(cid:100)ERt − 1− 1− ρA ρ / h A θπ R(cid:100)ERt=Etlog(cid:0)NERt+h (cid:1)−log(NERt) Define βh NER≡− 1− 1− ρA ρ / h A θπ ThiscorrespondstoourNERregression. Notethatβh NER<0,βh N + E 1 R<βh NER,and h l → im ∞ βh NER=− 1−ρ 1 A/θπ >1 Nowlet’sthinkabouttherelative-priceregression. Notethat k (cid:88) = h 1 Et (cid:16) πˆt+k−πˆ∗ t+k (cid:17) = θ ρ π A − − ρ 1 Ak (cid:88) = h 1 EtR(cid:100)ERt+k 66

− k (cid:88) = h 1 Et (cid:16) πˆ∗ t+k−πˆt+k (cid:17) = θ ρ π A − − ρ 1 A R(cid:100)ERt k (cid:88) = h 1 ρk A Etlog (cid:32)Pt ∗ +h/Pt ∗(cid:33) = 1−ρA ρA−ρh A +1 R(cid:100)ERt Pt+h/Pt θπ−ρA 1−ρA Etlog (cid:32)Pt ∗ +h/Pt ∗(cid:33) = 1−ρh A REˆRt Pt+h/Pt θπ/ρA−1 Define βh π≡ θπ 1 / − ρA ρh A −1 Thiscorrespondstoourrelative-priceregression. Notethatβh π>0,βh π +1>βh π,and h l → im ∞ βh π= θπ/ρ 1 A−1 . Nowitisapparentthat h l → im ∞ (cid:16) βh NER+βh π(cid:17) =−1 andthat βh NER=−βh π− (cid:16) 1−ρh A (cid:17) βh NER+βh π=− (cid:16) 1−ρh A (cid:17) 67