Economic Modelling xxx (xxxx) xxx Contents lists available at ScienceDirect Economic Modelling journal homepage: www.journals.elsevier.com/economic-modelling The changing nature of the real exchange rate: The role of central bank preferences☆ Rodrigo Caputo a, ∗ , Michael Pedersen b a b CESS, Oxford and Universidad de Santiago, Chile Central Bank of Chile, Chile A R T I C L E I N F O JEL classiﬁcation: C32 E42 F31 F33 Keywords: Real exchange rate DSGE models Central bank preferences Structural VAR Sign restrictions A B S T R A C T We investigate the sources of real exchange rate ﬂuctuations. We do so, ﬁrst, in the context of a DSGE model that explicitly considers the central bank’s preferences. Then we estimate SVAR models, where shocks are identiﬁed by sign restrictions derived from the DSGE model. We perform this exercise for twelve countries, nine of which have adopted inﬂation targeting during the period analyzed. In sharp contrast to the previous evidence in the literature, we ﬁnd that exchange rate (country risk premium) shocks have become the main drivers of real exchange rate dynamics, while real shocks play a less important role. Evidence from the DSGE model reveals that, as the central bank becomes more averse to inﬂation movements, and cares less about nominal exchange rate ﬂuctuations, the impact of nominal shocks on the real exchange rate tends to increase, while the impact of real shocks decreases. Our results suggest that the adoption of inﬂation targeting, along with a ﬂoating exchange rate, contributes to a shift in the relative importance of demand and country risk premium shocks in determining the RER. 1. Introduction In open economies the real exchange rate (RER) is a key relative price and its changes have implications for the external equilibrium as well as for the internal resource allocation. Understanding the nature of shocks that drive the RER is a challenging issue in international economics. The quest to explain RER movements has been pursued by several scholars over the years. Initially, Structural Vector Autoregression (SVAR) methods and variance decomposition techniques were used to determine the relative importance of real and nominal shocks for the RER dynamics. Early studies include those of Lastrapes (1992), Clarida and Galí (1994), Enders and Lee (1997), and Rogers (1999). The identiﬁcation strategy proposed in these papers is based on the long run restrictions suggested by Blanchard and Quah (1989).1 A system- atic result is that in the post-Bretton Woods era nominal shocks played a minor role in the RER dynamics in the UK, Canada, Germany, Italy and Japan. More recently, Farrant and Peersman (2006), Juvenal (2011) and Craighead and Tien (2015), have employed SVAR models where shocks are identiﬁed with the sign restriction approach suggested by Faust (1998), Canova and De Nicolò (2002) and Uhlig (2005). This method imposes theoretically based sign restrictions on the dynamic responses of a vector of variables to determine the relative contribution of structural shocks to the RER dynamics. These papers conclude that demand shocks are still the main driver of the RER dynamics at diﬀerent horizons. Overall, the empirical evidence supports the notion that the RER is a shock-absorber, rather than a source of ﬂuctuations. ☆ The views expressed are those of the authors and do not necessarily represent the opinions of the Central Bank of Chile or its board members. We are grateful to an anonymous referee for constructive criticism and to Christiane Baumeister, Fabio Canova, Jordi Galí, Gustavo Leyva, James Stock, and Harald Uhlig for useful discussions and suggestions. We also thank the participants at the XVIII World Congress of the International Economic Association, the 4th Conference of the International Association for Applied Econometrics, the 2017 Annual Congress of the European Economic Association, the 49th Money, Macro and Finance Research Group Annual Conference, the 22nd Annual LACEA Meeting, seminars held in Danmarks Nationalbank and the Central Bank of Chile for their comments as well as Camila Figueroa for outstanding research assistance. ∗ Corresponding author. E-mail addresses: [email protected] (R. Caputo), [email protected] (M. Pedersen). 1 This methodology is consistent with the notion of long-run money neutrality, allowing nominal shocks to have a temporary eﬀect on the RER, but not a permanent one. https://doi.org/10.1016/j.econmod.2019.11.029 Received 24 July 2018; Received in revised form 7 September 2019; Accepted 26 November 2019 Available online XXX 0264-9993/© 2019 Elsevier B.V. All rights reserved. Please cite this article as: Caputo, R., Pedersen, M., The changing nature of the real exchange rate: The role of central bank preferences, Economic Modelling, https://doi.org/10.1016/j.econmod.2019.11.029 R. Caputo, M. Pedersen Economic Modelling xxx (xxxx) xxx There are, however, several unsettled issues regarding the RER’s behavior. In particular, with the adoption of inﬂation targeting (IT), several countries have experienced a sharp decline in inﬂation volatility,2 while the volatility of the RER has increased substantially. The present paper studies whether the relative contribution of real and nominal shocks to the RER has changed as countries have adopted IT. In doing so we perform two complementary exercises. First, in a Dynamic Stochastic General Equilibrium (DSGE) model, we analyze the determinants of the RER and the extent to which changes in the monetary policy regime could aﬀect the way in which structural shocks are transmitted to the RER. Second, using Bayesian techniques we estimate SVAR models, where shocks are identiﬁed by sign restrictions. We perform rolling window estimations to determine the extent to which the contributions of real and nominal shocks may have changed with the aand Gambettidoption of alternative monetary policy regimes. Unlike Farrant and Peersman (2006), Juvenal (2011) and Craighead and Tien (2015), we derive the sign restrictions from a DSGE model, which explicitly considers the preferences of the central bank. This model is a slightly modiﬁed version of that of Kam et al. (2009) and it is used to study how diﬀerent shocks aﬀect the main macroeconomic variables, particularly the RER, and the results are used to impose sign restrictions on SVAR models, which are estimated for diﬀerent subsamples during the period 1986–2014 for twelve countries. Nine of these adopted IT during the period analyzed, while the other three have conducted a ﬁxed exchange rate policy, though only one vis-à-vis the benchmark country. We ﬁnd that the relative contribution of shocks driving the RER has changed over time, particularly, in IT countries. In line with the existing literature, demand shocks were the main source of RER ﬂuctuations in the early years, whereas nominal shocks, particularly country-risk premium (CRP) shocks, are more important in the more recent period. The results for the only non-inﬂation-targeting (NIT) countries with a ﬁxed exchange rate vis-à-vis the benchmark country do not show the same changes. Evidence from the DSGE model reveals that as the central bank becomes more averse to inﬂation and policy rate movements, and cares less about nominal exchange rate ﬂuctuations, the impact of nominal shocks on the RER tends to increase, while the impact of real shocks decreases. The reason is that, under a ﬁxed exchange rate regime, the impact of nominal shocks on the RER is almost completely muted by the policy rate reaction. In contrast, under an IT regime with a fully ﬂoating exchange rate, the RER absorbs an important proportion of nominal shocks as the policy rate reacts much less to these innovations. For IT countries, our results suggest that the adoption of this regime along with a ﬂoating exchange rate, has contributed to the shift in the relative importance of demand and CRP shocks in determining the RER. The rest of the paper is organized as follows. The second section reviews the main structure of the theoretical DSGE model and discusses the sign restrictions that can be derived from it. Section 3 presents the empirical methodology and reports the results of the estimation. Section 4 explores the impact of alternative central bank preferences on both the relative importance of nominal and real shocks for the RER dynamics and for the overall volatility of the main macroeconomic variables. Section 5 concludes. endogenous persistence in both aggregate demand and supply equations, which is crucial for bringing the model closer to the data, as shown by Fukač and Pagan (2010). Importantly for the present analysis, the model explicitly considers the preferences of the monetary authority with respect to inﬂation, output, interest rate and exchange rate volatility.3 This type of models are generally used to assess the transmissions of diﬀerent shocks in small open economies (see, for instance, Medina and Soto (2016) and Bhattarai and Trzeciakiewicz (2017)). The model is useful for the present study for three reasons. Firstly, it clearly illustrates, from a theoretical point of view, the sources of RER volatility such that it is possible to investigate which shocks drive the RER dynamics. Secondly, it allows for an analysis of the signs of the responses to diﬀerent shocks. Thirdly, it makes it possible to easily assess how changes in the central bank’s preferences aﬀect the way diﬀerent structural shocks are transmitted to the RER. 2.1. The DSGE model In this subsection we brieﬂy summarize the main features of the Kam et al. (2009) DSGE model in order to introduce the shocks that will be analyzed. The main equations in this model are derived under the assumptions that households, ﬁrms and the central bank optimize their relevant loss functions. Appendix A contains a detailed description of the model. From the ﬁrst-order condition of households’ optimization problem we obtain the Euler equation for consumption. In particular, it is obtained by log-linearizing the inter-temporal optimality condition for a representative household that maximizes lifetime utility: ct − hct −1 = Et (ct +1 − hct ) − 1−h 𝜎 (rt − Et 𝜋t+1 ) + 𝜀d,t . (2.1) Variables are expressed as deviations from steady-state values. Consumption, ct , is expressed in logarithm, whereas the nominal interest rate, rt , and CPI inﬂation, 𝜋 t , are in percentage levels. The degree of habit formation is represented by h, and 𝜎 is the coeﬃcient of relative risk aversion of the households. There is an exogenous demand shock, 𝜀d,t .4 Households obtain utility from consuming a basket of domestic and foreign goods. The consumption basket is relative to an external habit stock, such that consumption has a degree of habit persistence. Firms producing domestic goods have a linear production technology hiring labor, which is oﬀered by the households as the only input. Domestic output evolves according to: YH,t = 𝜀s,t Nt , where 𝜀s,t is an exogenous productivity shock and Nt is the level of employment. The log-linear approximation of the optimal price decision rule can be expressed as the following hybrid Phillips curve for domestic goods inﬂation: 𝜋H,t = 𝛽 Et (𝜋H,t+1 − 𝛿H 𝜋H,t ) + 𝛿H 𝜋H,t−1 +𝜆H (mct ) − 𝜆H (1 + 𝜙)𝜀s,t , (2.2) where 𝛽 is the subjective discount factor, 𝜋 H,t is domestic inﬂation, 𝛿 H measures the degree of inﬂation indexation, and mct represent the marginal costs associated to the production of domestic goods. The 𝜆H coeﬃcient is the slope of this Phillips curve, which is inversely related to the degree of domestic price stickiness. The 𝜙 coeﬃcient 2. An open economy DSGE model To evaluate how nominal and real shocks are transmitted to the RER, we employ the small open economy DSGE model of Kam et al. (2009), which assumes, as does Monacelli (2005), that the pass-through of exchange rate movements to prices is incomplete. It also introduces 3 We consider concerns about nominal exchange rate ﬂuctuations, rather than RER ﬂuctuations as in Kam et al. (2009), since many countries had explicit targets for the nominal exchange rate before adopting IT. 4 This shock is not included in the original speciﬁcation of Kam et al. (2009) but, as shown by Galí (2015), it can be derived from an exogenous preference shifter that enters the households’ utility function. This shock can be interpreted as a shock to the eﬀective discount factor. We calibrate the size of this shock so that the relative contribution of demand innovations are in line with the empirical evidence for developed countries reported by Clarida and Galí (1994). 2 See the Great Moderation literature in Galí and Gambetti (2009) and Canova and Gambetti (2010). 2 R. Caputo, M. Pedersen Economic Modelling xxx (xxxx) xxx captures the disutility of labor. The domestic productivity shock, 𝜀s,t , reduces inﬂation and increases output. Marginal costs are increasing in terms of trade, st , consumption, ct , and output, yt . In particular, 𝜎 (c − hct−1 ), implying that demand shocks will mct = 𝜙yt + 𝛼 st + 1− h t eventually impact domestic inﬂation. As is standard in the literature, ﬁrms face an independent signal that allows a fraction of them to set prices in order to maximize their present value of the stochastic stream of proﬁts. Firms, that do not re-optimize partially, index their prices to past inﬂation. To be more speciﬁc, the slope of the New Keynesian Phillips curve, 𝜆H , depends on the probability that ﬁrms adjust prices optimally each period, 1 − 𝜃 H . In particular, 𝜆H = (1 − 𝛽𝜃H )(1 − 𝜃H )𝜃H−1 . If prices are more ﬂexible, i.e. if 𝜃 H tends to zero, the slope of (2.2) increases. Imported inﬂation, 𝜋 F,t , is obtained by log-linearizing the optimality conditions of importing retailers. They purchase imported goods at competitive world prices, but act as monopolistic competitive redistributors of these goods. This creates a gap between the price of imported goods, in domestic currency, and the domestic retail price of imported goods. Hence, if ﬁrms adjust prices infrequently, pass-through of the exchange rate to prices is incomplete. The New Keynesian Phillips curve for imported inﬂation is expressed as follows: 𝜋F ,t = 𝛽 Et (𝜋F ,t+1 − 𝛿F 𝜋F ,t ) + 𝛿F 𝜋F ,t−1 + 𝜆F (𝜓F ), that the 𝜀q,t innovation may be capturing anticipated shocks (news) as well as unanticipated ones.6 As noted by Chen and Zhang (2015), economic news may be capturing shocks to the RER that are expected by markets participants. The relative importance of anticipated and unanticipated shocks is certainly a relevant question, but in the present context 𝜀q,t is reﬂecting both types of shocks. We follow Neumeyer and Perri (2005) and Kam et al. (2009), among many others, and assume that the relevant foreign rate, rt∗ , is denominated in U.S. dollars.7 This assumes that the relevant foreign asset market for the economies under analysis is the U.S. For most countries, this is a reasonable assumption which is also supported by the empirical evidence on the UIP (see Engel (2016) and Kiley (2016), among others). As a consequence, the country risk premium is measured against a relatively stable country, the U.S., which also seems to be a sensible metric for measuring this premium. The real exchange rate is expressed as qt = et + p∗t − pt , where et is the nominal exchange rate and pt and p∗t are the CPI price levels in the domestic and the foreign economy, respectively. Using the deﬁnition of qt and the UIP condition in (2.5), it is possible to derive a condition that describes the evolution of qt : Et (qt +1 − qt ) = (rt − rt∗ ) − Et (𝜋t +1 − 𝜋t∗+1 ) + 𝜀q,t , (2.3) where the real exchange rate and the expected inﬂation diﬀerential are deﬁned relative to the U.S. because the interest rate diﬀerential, rt − rt∗ , is also relative to the U.S. where 𝛿 F measures the degree of imported inﬂation persistence. The variable ( )𝜓 F represents the law of one price gap, deﬁned as 𝜓F = et + p∗F ,t − pF ,t , where et is the nominal exchange rate, p∗F ,t is the price, 2.1.2. Monetary policy The central bank sets the nominal interest rate, rt , in order to minimize a quadratic loss function: [ ] 1 𝜇𝜋 𝜋̃2t + 𝜇y yt2 + 𝜇e Δe2t + 𝜇r (Δrt )2 , (2.7) L= 2 in foreign currency, of an imported consumption basket and pF,t represents the price, in domestic currency, of an imported consumer basket. The slope of (2.3) is 𝜆F = (1 − 𝛽𝜃F )(1 − 𝜃F )𝜃F−1 and increases if import retailers set prices more frequently (i.e. a lower 𝜃 F ). Imperfect passthrough is a direct consequence of import retailers that do not adjust prices instantly when the RER changes. If import retailers can set prices at any moment, then 𝜃 F = 0. In this case, the pass-through is immediate implying that 𝜓 F = 0. As a consequence, the pass-through is complete and we have that: pF ,t = et + p∗F ,t . where 𝜇y , 𝜇 e and 𝜇 r express the central bank’s concern with respect to output, the nominal exchange rate and policy rate stabilization, respectively. These objectives are expressed relative to a concern for annual inﬂation, 𝜋̃t , which is normalized to 𝜇 𝜋 = 1. A monetary policy shock, 𝜀m,t , is appended to the optimal interest rate path that minimizes (2.7). As noted by Kam et al. (2009), this speciﬁcation of macroeconomic objectives encompasses the expressed objectives of the so-called ﬂexible inﬂation targeting central banks by allowing for positive weights of all arguments in the loss function. In the Kam et al. (2009) speciﬁcation, foreign variables are assumed to follow uncorrelated AR(1) processes. In our empirical exercise, output, prices, and interest rates are measured relative to the same variable in the U.S., in line with Farrant and Peersman (2006) and Juvenal (2011). In the next subsection this approach is explained in the context of the model speciﬁcation. (2.4) 2.1.1. Country risk premium and the exchange rate We use the uncovered interest parity (UIP) to model the behavior of the nominal exchange rate, et . This is a no-arbitrage condition between investing in a domestic currency denominated asset and a foreign currency denominated asset such that: Et (et +1 − et ) = rt − rt∗ + 𝜀q,t . (2.6) (2.5) The expression in (2.5) links expected nominal devaluations and the nominal interest rate spread, rt − rt∗ . In particular, an increase in the foreign currency denominated interest rate, rt∗ , is associated with an expected nominal appreciation. The stochastic term in (2.5), 𝜀q,t , reﬂects the country risk premium. As shown by Neumeyer and Perri (2005), this premium may be aﬀected by domestic fundamentals (expected productivity) and, through the presence of working capital, may exacerbate the eﬀect of this shock on real activity. In addition, the shock could also be interpreted as changes in capital ﬂows or, indeed, capital controls imposed by the domestic economy, that have an impact on interest rate diﬀerentials.5 In Kam et al. (2009) this innovation is interpreted as an exogenous CRP shock, reﬂecting foreign and domestic elements not explicitly modeled. In this context, it is important to note 2.1.3. Equilibrium conditions: aggregate inﬂation and output Households consume domestic and foreign goods, and the relevant consumer price index is given by8 : [ ] 1 1−𝜂 1−𝜂 Pt = (1 − 𝛼)PH,t + 𝛼 PF ,t , (2.8) where 𝜂 denotes the elasticity of substitution between domestic and foreign goods and 𝛼 represents the proportion of imported goods in the household’s consumption basket. Log-linearizing (2.8) and ﬁrst diﬀerentiating gives rise to the CPI inﬂation equation: 6 See Nam and Wang (2015). For Argentina Neumeyer and Perri (2005) notice that in the early 1990s, the foreign rate faced by Argentina was very close to the U.S. dollar rate and during crisis times (the hyperinﬂation of 1989) the country risk increased signiﬁcantly. 8 Details are included in Appendix A. 7 5 As shown by Herrera and Valdés (2001), in emerging economies the eﬀect of capital controls on interest rate diﬀerentials, and on the RER, is considerably smaller than what static calculations suggest. 3 R. Caputo, M. Pedersen 𝜋t = (1 − 𝛼)𝜋H,t + 𝛼𝜋F ,t . Economic Modelling xxx (xxxx) xxx empirical model by including fewer observable variables.11 (2.9) Hence, CPI inﬂation is a weighted average of domestic and imported inﬂation, where the relative importance of imported goods inﬂation is given by 𝛼 . The evolution of domestic and foreign inﬂation is determined by equations (2.2) and (2.3), respectively. Thus, CPI inﬂation will be inﬂuenced by domestic productivity shocks aﬀecting 𝜋 H,t and shocks aﬀecting 𝜋 F,t . In the small open economy, domestic output equals total domestic and foreign demand for goods produced at home. As a result, domestic output will depend on total consumption, ct , foreign output, yt∗ , the RER, qt , and the terms of trade, st . In particular, the log-linear approximation of domestic output is given by9 : yt = (1 − 𝛼)ct + 𝛼 yF∗,t + 𝛼𝜂 qt + 𝛼𝜂 st , 2.2. Impulse-responses Kam et al. (2009) make inferences regarding structural coeﬃcients and shocks using Bayesian posterior distributions of the model parameters for three IT countries: Australia, Canada and New Zealand.12 To identify the impact that diﬀerent shocks have on the RER and the rest of the relevant variables, we use the coeﬃcients estimated by Kam et al. (2009) for Australia.13 Applying a given, and common, calibration enables us to derive the response of the main variables to any given structural shock. This calibration process is just the ﬁrst step to knowing how each of the economies we consider reacts to each shock. To be more speciﬁc, we will obtain diﬀerent posteriors for each country reﬂecting the individual nature of the structural coeﬃcients across countries which are not assumed to be equal ex-post. Hence the common calibration does not preclude us from obtaining diﬀerent responses across countries. Having said that, using Australia, or any other country, does not seem to change the results much. For instance, the slope of the IS curve, equah tion (2.10), is given by this expression: 1− . If we use the calibration 𝜎 for Australia, where h = 0.925 and 𝜎 = 1.029, the calibrated slope is 0.0729. If we use, instead, the estimated coeﬃcients for Canada in Kam et al. (2009), the slope is almost identical: 0.0732. If we used the estimated values for Chile, from Gomez et al. (2019), the slope would be 0.0607. Thus, it appears that the main features of the model are well reﬂected by our initial calibration, and the speciﬁc nature of each country will be reﬂected in the posterior results obtained from the country-speciﬁc SVAR estimation. Once the model is solved, it is possible to compute the responses of the main macroeconomic variables to the four structural innovations included in the model: 𝜀s , 𝜀d , 𝜀m , and 𝜀q .14 As shown in Fig. 1 (ﬁrst row), a supply shock (i.e. a positive technology shock relative to the U.S.) increases the relative output and, at the same time, reduces marginal costs and relative prices. Via the monetary policy, the interest rate declines importantly, although in the ﬁrst quarter it increases marginally. Given the policy reaction and the path of the relative prices, the RER depreciates almost immediately by 4%. A demand shock increases consumption and, as a consequence, increases output. In this context marginal costs are higher, determining an increase in the prices of domestic goods. The increase in the RER has a positive impact on imported inﬂation too. Hence, this shock induces an increase in relative CPI inﬂation and output. The optimal policy response implies that the interest rate increases. The RER appreciates nearly 4% as a consequence of the actual and expected increase (2.10) where 𝜂 > 0 is the elasticity of substitution between home and foreign goods and the variable st represent the terms of trade, which evolve according to the diﬀerence between imported and domestic inﬂation: st − st −1 = 𝜋F ,t − 𝜋H,t . (2.11) Several of the countries in the analysis (Australia, Canada, Chile and Norway) have an important commodity sector. In general, this sector is not explicitly modeled, neither in the seminal contribution of Farrant and Peersman (2006) nor in the more recent works of Kam et al. (2009) and Gomez et al. (2019). This sector is, however, implicitly considered in the open economy IS curve, equation (2.10). In particular, changes in the terms of trade or world output could be related to an increase in the demand for commodities (which is part of the domestic output) and, hence, the estimations of 𝛼 and 𝜂 are going to reﬂect the extent to which foreign variables aﬀect the demand for commodities in these countries. This in turn, will change domestic output and domestic marginal cost, and will determine, eventually, changes in the domestic policy rate, which have an impact on the RER. Hence, the commodity sector, and its implications for the RER and output, is implicitly considered in the model. The model contains ﬁve relevant relationships. The ﬁrst one reﬂects the determination of nominal exchange rate and is given by equation (2.5). The second describes the evolution of the real exchange rate, qt , which is described by equation (2.6). Monetary policy is described by the loss function criterion, equation (2.7). The fourth relevant relationship is CPI inﬂation, equation (2.9), which incorporates shocks aﬀecting domestic inﬂation and the variables that impact foreign inﬂation. Finally, the evolution of aggregate output is determined by equation (2.10). Two crucial relationships, the nominal exchange rate and the real exchange rate - equations (2.5) and (2.6) -, are measured relative to the U.S., since this is the relevant foreign rate for most countries. Accordingly, the relevant inﬂation diﬀerential is also measured relative to the U.S. To keep consistency with the equations for the nominal and real exchange rates, we also measure CPI inﬂation, the policy interest rate,10 and the output gap relative to the U.S. This is an approximation of the original model, but it does allow us to compare our results with the existing literature and, in particular, with the paper of Farrant and Peersman (2006). In addition, we can reduce the dimensionality of the 11 Another reason to measure the variables relative to the U.S. is that the structural model in the domestic economy and in the U.S. are similar. This is, perhaps, a strong assumption, so in the context of the small open economy model, measuring the variables relative to the U.S. is an approximation. In any case, and in order to analyze the extent to which our results may be driven by developments in the U.S., we estimate stochastic volatility models (Chan and Hsiao (2014)) for the raw data in each country: inﬂation, GDP growth and the real interest rate. Then we estimate the models for the same series but relative to the U.S.. It turns out that the relative series, which are the ones we use in our empirical analysis, mimic quite well the path of the country-level series and, hence, the behavior of the relative series does not seem to be driven by developments in the U.S., but rather do they reﬂect idiosyncratic changes in each country. 12 The Bayesian methodology closely follows related papers such as those of Smets and Wouters (2003) and Rabanal and Rubio-Ramírez (2005). 13 The numbers are from the baseline estimation, where 𝜇 = 0. The 𝜎 coefe d ﬁcient is calibrated such that demand shocks account for 50% of the RER’s variance, which is pretty much in line with the relative importance of demand shocks in previous empirical studies. 14 Shocks are independent and identically distributed and the size of them is one standard deviation, as reported by Kam et al. (2009) for Australia. 9 See Appendix A for a derivation. In terms of monetary policy identiﬁcation, we can say that since the early 1980s inﬂation has been very stable in the U.S. According to Ilbas (2012), inﬂation stabilization owes mainly to the change in monetary policy that took place at the beginning of Volcker’s mandate in 1979. In addition, as shown by Bauducco and Caputo (2020) inﬂation in the U.S. has been systematically around 2% since the early 1990. 10 4 R. Caputo, M. Pedersen Economic Modelling xxx (xxxx) xxx Fig. 1. Theoretical Accumulated IRF under Baseline Calibration (quarters after the shock). in the policy rate (Fig. 1, second row). A positive monetary policy shock (i.e. a decline in the policy rate) results in increases in consumption and output. A higher demand for domestic output increases marginal costs and prices. The consequence of this shock is that the RER initially increases and then declines in the second quarter (Fig. 1, third row). In this model, a CRP shock generates an increase in the RER. This is equivalent to a decline in the relative price of domestic goods. As a result, the demand for and production of home goods increase, generating higher marginal costs and prices. The optimal monetary policy response is to raise the interest rate, even though the RER is not, per se, a policy target (Fig. 1, fourth row). The sign restrictions imposed in the SVAR exercises are based on the impulse-responses derived from the DSGE model. Table 1 summarizes the restrictions that will be used to identify the structural shocks.15 Restrictions are binding for four quarters, with the exception of the interest rate, for which the restriction is binding one quarter. The response of the policy rate to a supply shock is left unrestricted because during the ﬁrst two quarters, the policy rate moves in the opposite direction in response to this shock (Fig. 1, ﬁrst row). NIT countries in a framework of vector autoregressive (VAR) models, where the shocks are identiﬁed by imposing the sign restrictions derived from the DSGE model. This approach has gained momentum in empirical analyses conducted since the beginning of the present millennium. Several early studies report the results by means of quantiles of the impulse-responses acknowledging that a drawback is that the responses shown are not necessarily from the same model and, hence, the shocks may not be orthogonal (see Fry and Pagan (2011)). In the present analysis we follow the approach presented by Inoue and Kilian (2013), who suggest a solution to this orthogonality problem by reporting the results of the most likely model chosen among those that satisfy the sign restrictions imposed. With this approach, however, the credibility intervals are often quite wide and, to assess uncertainty, we evaluate the histograms of all possible outcomes. After brieﬂy presenting the econometric model and the data utilized, we report, as a preliminary assessment, the full sample variance decompositions for individual IT countries and compare them to previous ﬁndings in the literature. The evidence indicates that changes have occurred over time. To gain further insight into these changes we present results, ﬁrstly, from panel VAR (PVAR) models and, secondly, for individual countries, IT as well as NIT, where the sample utilized for the estimations is changed over time.16 3. Empirical estimations This section presents exercises where we let the data “speak” in a less structured environment. We analyze data from nine IT and three 15 Our sign restrictions are comparable to the ones in Juvenal (2011), while those used to identify the supply shock are more restrictive than in Farrant and Peersman (2006). 16 Since the dates of the IT adoptions are known with certainty, we opted for a rolling approach where the time of IT increases markedly with each new subsample estimation. 5 R. Caputo, M. Pedersen Economic Modelling xxx (xxxx) xxx Table 1 Sign restrictions from the DSGE model. Supply shock Demand shock Monetary shock CRP shock 𝜕(y ∕y ∗ ) 𝜕𝜀s 𝜕(y ∕y ∗ ) 𝜕𝜀d 𝜕(y ∕y ∗ ) 𝜕𝜀m 𝜕(y ∕y ∗ ) 𝜕𝜀q 𝜕(p∕p∗ ) 𝜕𝜀s 𝜕(p∕p∗ ) 𝜕𝜀d 𝜕(p∕p∗ ) 𝜕𝜀m 𝜕(p∕p∗ ) 𝜕𝜀q ≥0 ≥0 ≥0 ≥0 3.1. Econometric model and data ≥0 ≥0 ≥0 𝜕(i−i∗ ) 𝜕𝜀s 𝜕(i−i∗ ) 𝜕𝜀d 𝜕(i−i∗ ) 𝜕𝜀m 𝜕(i−i∗ ) 𝜕𝜀q ≷0 ≥0 ≤0 ≥0 𝜕q 𝜕𝜀s 𝜕q 𝜕𝜀d 𝜕q 𝜕𝜀m 𝜕q 𝜕𝜀q ≥0 ≤0 ≥0 ≥0 Because of this, the NITs should not be regarded as a control group, i.e. a group for which one might expect results opposite of those of the ITs, maybe with the exception of Hong Kong. The group does, however, represent countries that have not experienced signiﬁcant changes in the monetary policy during the period investigated. The series applied start in 1986Q1 (Hong Kong: 1990Q4) and end in 2014Q2. In interpreting the results, it should be noted that only Australia and Canada have had, de jure, a freely ﬂoating exchange rate for the entire sample, while the Philippines has had a regime of managed ﬂoating exchange rate. In the UK and Sweden, freely ﬂoating currency started in 1992, in Israel in 1998, in Chile in 1999, and in Norway and South Africa in 2001. Before these years, some of the countries had a regime that can be characterized as ﬁxed exchange rate, while others had managed ﬂoating rates. The central banks have, however, to some extent been buying and selling foreign currencies as shown in Fig. 2, which illustrates average changes, in absolute values, in the foreign reserves before and after the IT implementation. Generally, the central banks have been less active in trading foreign currency after the IT implementation, but in Norway the reduction is quite limited and in Australia there is practically no change. For the empirical analysis we consider a four-dimensional reducedform VAR model with p lags: Y = X 𝛽 + 𝜀, ≤0 (3.1) where Y = [Y1 , Y2 · · · YT ]′ , X = [X1 , X2 · · · XT ]′ , Xt = [1, Yt′−1 , Yt′−2 · · · Yt′−p ]′ for t = 1, 2, … , T, 𝛽 = [c, 𝛽1 , 𝛽2 · · · 𝛽p ]′ , and 𝜀 = [𝜀1 , 𝜀2 · · · 𝜀T ]′ . It is assumed that 𝜀t is independent and identically distributed with zero mean and covariance matrix Ω. The endogenous variables are Yt = [Δ(yt ∕yt∗ ) Δqt Δ(pt ∕p∗t ) it − i∗t ]′ , where the variable yt ∕yt∗ is the logarithm of the ratio between real gross domestic product (GDP) in the domestic country and the real GDP in the foreign (∗) country (the U.S.), qt is the logarithm of the real exchange rate, pt ∕p∗t is the logarithm of the price ratio, and it − i∗t is the diﬀerence between the real interest rates.17 The measurement of the real exchange rate implies that an increase is a depreciation of the real domestic currency.18 A brief summary of the methodology is presented in Appendix B. To focus the analysis on how diﬀerent sources of RER volatility may have changed over time, the estimations are performed with rolling windows, each including a period of 15 years of data.19 The results are presented as variance decompositions of the most likely model as well as a ﬁtted trend to evaluate changes in the contributions over time. Furthermore, histograms of the variance decomposition contributions of all the models, that fulﬁll the imposed sign restrictions, are employed to assess uncertainty and to evaluate whether the most likely model is representative.20 The primary source of the data is the IMF’s International Financial Statistics (IFS) and observations from nine IT and three NIT countries are utilized as well as U.S. data for the benchmark economy.21 The economies classiﬁed as IT are: Australia (IT since 1993), Canada (1991), Chile (1991), Israel (1992), Norway (2001), the Philippines (2002), South Africa (2000), Sweden (1993), and the UK (1992), while the NITs are Denmark (ﬁxed exchange rate against a single currency, from 1982 the German mark and since 1999 the euro), Hong Kong (Currency Board, 1983), and Singapore (ﬁxed exchange rate against a revised basket of currencies, 1975). Hence, the monetary policy of the three NIT countries has not been changed during the period analyzed. Only Hong Kong, however, has a ﬁxed exchange rate vis-à-vis the USD, while the Danmarks Nationalbank states that the main objective of the monetary policy is to ensure low inﬂation and the Monetary Authority of Singapore (MAS) that the primary objective is to promote price stability.22 3.2. Full sample variance decomposition In order to make comparisons with previous literature, we estimate full-sample VARs for each IT country in our data set and compute the variance decomposition of RER ﬂuctuations.23 As shown in Table 2, demand and CRP shocks contribute to most of the RER volatility in the period 1986–2014. In terms of relative contribution, demand shocks explain between 5% (Australia) and 50% (UK) of RER volatility, whereas CRP shocks (or shocks to the RER) explain between 30% (UK) and 90% (Philippines) of RER volatility. The importance of demand shocks we ﬁnd for Australia are in sharp contrast to the ﬁndings of Farrant and Peersman (2006), who conclude that around 70% of RER volatility is explained by demand shocks in small open economies. The reason for this is that whereas Farrant and Peersman (2006) focus on a period that includes, mostly, observations under a non-IT regime (1974–2002), our sample contains many more observations under the IT regime, ranging from 1986 to 2014. Once we compare the results from 1986 to 2000, we ﬁnd that demand shocks tend to explain a substantial fraction of the RER volatility in Australia, pretty much in line with the ﬁndings of Farrant and Peersman (2006). The fact that, in general, CRP shocks explain a larger fraction of RER volatility is in sharp contrast with previous empirical evidence, as several studies have concluded that demand shocks are one of the main drivers of the RER dynamics. For Canada we ﬁnd that demand shocks explain between 39% and 42% of RER volatility at horizons of one quarter to four years (Table 2). In contrast, Farrant and Peersman (2006) ﬁnd that, between 1974 and 2002, demand shocks contributed 17 In terms of the model presented previously, the real rates diﬀerential is equivalent to (rt − Et 𝜋t +1 ) − (rt∗ − Et 𝜋t∗+1 ). 18 As a robustness check, the estimations were also made with level data ﬁltered by the Hodrick and Prescott (1997) ﬁlter. These estimations show results similar to the ones reported. 19 To speed up the process, the windows are shifted four quarters at a time. Hence, estimations are made for 15 overlapping periods. 20 ̃ are quite close among the models with the highest values, The values of f (Θ) while their variance decompositions may be very diﬀerent. Hence, a histogram comparison shows whether the changes are a more general feature of the data. 21 A detailed description of the data is included in Appendix C. 22 See Danmarks Nationalbank (2009) and the MAS article “Singapore’s Exchange Rate-Based Monetary Policy” available at http://www.mas.gov.sg/. 23 All VAR models include a constant term and, as suggested by the Schwarz information criterion, the lag order is set to one for all countries. The results presented are, however, generally robust when including up to four lags. According to the DSGE model, all variables should be stationary and when applying a 5% conﬁdence level, the Trace test of Johansen (1991) suggests that indeed all the VARs estimated are stationary. Chile, however, is a borderline case with a p-value of 0.0546. 6 R. Caputo, M. Pedersen Table 2 RER variance decomposition: Full sample 1986–2014. Horizon (Q) 7 Lower Demand Shock Modal Upper Lower Monetary Shock Modal Upper Lower CRP Shock Modal Upper Lower Australia 1 4 16 0.034 0.066 0.065 0.781 0.724 0.709 0.000 0.000 0.000 0.048 0.083 0.082 0.942 0.916 0.886 0.000 0.000 0.001 0.229 0.214 0.237 0.765 0.744 0.742 0.000 0.002 0.003 0.689 0.637 0.617 0.978 0.961 0.960 0.000 0.002 0.003 Canada 1 4 16 0.053 0.173 0.173 0.689 0.641 0.628 0.000 0.001 0.001 0.420 0.398 0.397 0.974 0.920 0.914 0.000 0.000 0.000 0.026 0.025 0.026 0.564 0.478 0.478 0.000 0.000 0.001 0.502 0.405 0.404 0.995 0.959 0.952 0.005 0.017 0.017 Chile 1 4 16 0.006 0.044 0.044 0.655 0.630 0.630 0.000 0.001 0.003 0.246 0.263 0.263 0.975 0.931 0.930 0.000 0.000 0.000 0.087 0.084 0.084 0.975 0.934 0.930 0.000 0.001 0.001 0.661 0.608 0.608 0.995 0.977 0.976 0.000 0.003 0.003 Israel 1 4 16 0.003 0.025 0.028 0.665 0.629 0.629 0.000 0.000 0.000 0.420 0.444 0.442 0.954 0.930 0.922 0.001 0.005 0.005 0.110 0.119 0.119 0.775 0.719 0.716 0.000 0.000 0.000 0.468 0.412 0.411 0.984 0.980 0.979 0.001 0.002 0.003 Norway 1 4 16 0.009 0.047 0.048 0.817 0.803 0.787 0.000 0.000 0.000 0.442 0.455 0.454 0.972 0.936 0.929 0.000 0.004 0.004 0.105 0.095 0.097 0.644 0.626 0.637 0.000 0.000 0.001 0.444 0.403 0.402 0.979 0.964 0.961 0.001 0.004 0.004 Philippines 1 4 16 0.001 0.031 0.031 0.270 0.300 0.301 0.000 0.001 0.002 0.092 0.093 0.094 0.817 0.801 0.799 0.092 0.083 0.082 0.000 0.005 0.006 0.091 0.139 0.138 0.000 0.001 0.001 0.906 0.871 0.869 0.906 0.871 0.869 0.145 0.156 0.156 South Africa 1 4 16 0.000 0.038 0.041 0.181 0.252 0.247 0.000 0.000 0.001 0.191 0.208 0.204 0.810 0.797 0.794 0.000 0.002 0.002 0.030 0.036 0.047 0.434 0.400 0.408 0.000 0.001 0.002 0.779 0.719 0.708 0.997 0.965 0.939 0.183 0.183 0.184 United Kingdom 1 4 16 0.054 0.056 0.057 0.559 0.541 0.540 0.000 0.001 0.001 0.519 0.577 0.573 0.995 0.977 0.974 0.018 0.033 0.035 0.052 0.051 0.057 0.571 0.537 0.537 0.000 0.001 0.001 0.375 0.316 0.313 0.940 0.907 0.901 0.000 0.004 0.004 Sweden 1 4 16 0.031 0.028 0.029 0.452 0.429 0.418 0.000 0.000 0.000 0.383 0.363 0.341 0.968 0.926 0.920 0.030 0.052 0.051 0.047 0.106 0.158 0.465 0.448 0.466 0.000 0.006 0.008 0.538 0.503 0.473 0.899 0.853 0.839 0.001 0.005 0.005 Note: Modal refers to the most likely model, while Upper (Lower) indicates the maximum (minimum) among the 68% most likely models. Economic Modelling xxx (xxxx) xxx Supply Shock Modal Upper R. Caputo, M. Pedersen Economic Modelling xxx (xxxx) xxx Fig. 2. Annual average of absolute changes in the foreign reserves. Notes: Calculated with observations from 1986 to 2014. Pre IT period includes the year of the IT adoption. Total reserves including gold measured in current USD. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the Web version of this article.) Source: Own calculations based on data from IMF (series FI.RES.TOTL.CD). between 70% and 80% of the RER’s volatility at similar horizons. In the same way, Clarida and Galí (1994) ﬁnd that for Canada demand shocks account for 94%–97% of the RER volatility between 1974 and 1994. Our estimates suggest that in this country CRP shocks explain between 40% and 50% of RER volatility at horizons of one quarter to four years, whereas Farrant and Peersman (2006) ﬁnd that this shock contributes much less, explaining between 5% and 20% of RER volatility at similar horizons. In the case of the UK, we ﬁnd that the relative importance of nominal shocks in the RER dynamics has increased. In particular, our results suggest that CRP shocks explain between 31% and 37% of RER volatility at horizons of one quarter to four years, whereas Farrant and Peersman (2006) ﬁnd, for similar horizons, that the weight of this shock is between 8% and 40%. For Chile, our results are also in sharp contrast with previous evidence documented by Soto (2003) for the 1990–1999 period. On the one hand, we ﬁnd that real shocks (supply and demand) contribute 25%–27% of RER volatility at horizons of one quarter to four years, whereas at similar horizons Soto (2003) ﬁnds that these shocks account for 70% and 90% of the RER variance. On the other hand, we ﬁnd that the contribution of nominal shocks to RER variance is between 61% and 67%, whereas Soto (2003) ﬁnds, at similar horizons, that nominal shocks contribute between 10% and 30% of RER volatility. To explore in greater detail the diﬀerences with results from studies employing earlier samples, we estimate the models across diﬀerent periods of time. As mentioned earlier, the IT countries included in our analysis experienced important policy changes during in the mid 1990s and early 2000s. In particular, they moved, with diﬀerent speed, towards a fully ﬂedged IT regime.24 As a ﬁrst step to detect possible structural changes, we estimate two PVAR(1) models, one for the IT economies and another for the NITs.25 The rolling variance decompositions are presented in Figs. 3 and 4, while Fig. 5 shows changes in the distribution of the weights in the variance decompositions of the RERs for the ﬁrst and last subsamples. There are several diﬀerences between the changes in the variance decompositions of IT and NIT countries, where most notable are the following: (1) The impact of supply shocks on inﬂation volatility has increased markedly in NIT countries,26 while the impact of monetary shocks has decreased. In IT countries, the impact of monetary and CRP shocks has increased, while that of demand shocks has decreased. This, as we will explain later, can probably be attributed to the shift to inﬂation targeting. (2) Across the whole period, monetary shocks explain the main part of the interest rate volatility in NITs, while there appears to have been a change in ITs, such that CRP shocks explained an important part in the early periods, while monetary shocks explain the main part in the later subsamples. For economies that have adopted IT policies, these results are expected: If there is some form of exchange rate control, then the policy rate should move in the face of CRP shocks to avoid exchange rate ﬂuctuations. As economies move towards an IT regime, monetary policy shocks become more important for interest rate ﬂuctuations. For the same reason, one could expect, a priori, the CRP to explain a larger part of interest rate volatility in the NIT countries. (3) Generally, demand shocks explain a smaller part of the RER’s volatility, while CRP shocks explain a larger part. This is more pronounced in the ITs. When looking at the distributions of the contributions to the RER variance decompositions (Fig. 5) it is notable how it, particularly in the ITs, moves to the left in the case of a demand shock and to the right for a CRP shock.27 To sum up the results with respect to RER volatility, in both NITs and ITs it seems that demand shocks (real shocks) explain a smaller part, while CRP shocks explain a larger part. This is particularly true for the economies that have adopted IT, which supports the view that when the central bank becomes more averse to inﬂation, the relative contribution of the CRP shocks to the volatility of the RER increases, while the importance of demand shocks is reduced. The similar results for the NITs may be because two of them, Denmark and Singapore, 24 The UK, for example, abandoned the exchange-rate-based nominal stabilization programs, in place since 1987, and left the Exchange Rate Mechanism of the European Monetary System and established an inﬂation target of 1%–4% in 1992 (see Levant and Ma (2016)). Another example is Chile that in the second half of 1999 implemented a number of changes in the macroeconomic policy framework, including the adoption of a fully-ﬂedged IT regime. In September 1999 it adopted a free-ﬂoating exchange rate regime (see Valdés (2007)). 25 The Abrigo and Love (2016) GMM estimates are utilized as input in the Inoue and Kilian (2013) routine. 26 With only three NIT countries in our sample, however, one should be careful not to interpret these results as general for NIT countries. 27 Even though there are apparent diﬀerences between the two groups of countries, there are also important similarities and one might suspect that the results could be driven by U.S.-speciﬁc factors. If this is the case, the variability of the variables included in the analysis would be explained by one factor and, given the results, the loadings would be higher for the IT countries. Estimations of simple factors models, for the relative as well as individual country variables, suggest that this is not the case. 3.3. PVAR estimations 8 R. Caputo, M. Pedersen Economic Modelling xxx (xxxx) xxx Fig. 3. Rolling Variance Decompositions for IT Countries. Horizon: Four Quarters. Notes: Dotted lines show the contribution to the variance decomposition of the most likely model for the 15-year period ending the year indicated on the ﬁrst axis. The solid line is a ﬁtted trend. CRP: Country Risk Premium. the CRP shock shifts to the right.28 Hence, also in this case the evidence suggests that CRP shocks have become more important as an underlying source of RER ﬂuctuations. Comparing the results of Canada and the UK from the ﬁrst subsample with those of Clarida and Galí (1994) and Farrant and Peersman (2006), our evidence suggests that demand shocks explain less and CRP shocks a greater part of the RER volatility. Both countries implemented IT in the early 1990s, and we will show that changes in the monetary policy regime could explain this pattern. Turning to the developing IT countries (Figs. 8 and 9), the results for Chile and Israel are similar to those of the developed economies. In particular, we ﬁnd for both countries that the relative importance of demand shocks has declined over time, whereas the importance of CRP shocks has increased. As mentioned earlier, Chile implemented IT in 1991 and adopted a regime of freely ﬂoating exchange rate in 1999. In the case of Israel, IT was adopted in 1992 although there was an exchange rate band in place until mid 2005. The changes in the distributions are also statistically signiﬁcant for both countries. The evidence for the Philippines and South Africa is less clear, even though for South Africa the histogram of the demand shocks does shift slightly to the left, and for the Philippines there appears to be a movement to the right of the histogram of the CRP shock. Both of these movements are statistically signiﬁcant. On the other hand, for South Africa the CRP histogram also seemed to have moved to the left, while it is not evident that the explicitly have stated that an objective of the central bank is related to price stability. Hence, only Hong Kong has a ﬁxed exchange rate policy with the USD as nominal anchor. The next subsection presents individual country results. 3.4. The changing nature of RER ﬂuctuations: evidence from individual countries In this subsection we assess the impact across time in individual countries. We perform the same rolling window estimations as in the previous subsection and report the results for three groups of countries: Developed ITs, developing ITs, and NITs, with an emphasis on the contributions of demand and CRP shocks to the overall RER volatility. For most developed IT countries (Australia, Canada, Sweden and the UK) the message is clear: The contribution of demand shocks to the RER has declined steadily since 2000 (Fig. 6). In contrast, the contribution of CRP has increased substantially and, as a consequence, it has become the main driver of the RER in the latter part of the sample (Fig. 7). The histograms show a similar pattern: the distribution of demand shock contributions shifted to the left in the latter sample, whereas the distribution of CRP shocks shifted to the right. For Norway the outcomes of the most likely models are less clear, but when looking at the histograms of all the admissible models, it appears that the most likely models are not representative. Also for this country, the distribution of the contributions of the demand shock shifts to the left, whereas that of 28 According to the Kolmogorov (1933) and Smirnov (1939) test - see e.g. Conover (1999) for a description - the distributions have in all cases moved statistically signiﬁcantly to the left for the demand shocks and to the right for the CRP shocks. 9 R. Caputo, M. Pedersen Economic Modelling xxx (xxxx) xxx Fig. 4. Rolling Variance Decompositions for NIT Countries. Horizon: Four Quarters. Notes: Dotted lines show the contribution to the variance decomposition of the most likely model for the 15-year period ending the year indicated on the ﬁrst axis. The solid line is a ﬁtted trend. CRP: Country Risk Premium. demand histogram has moved across time. These results may have to do with the fact that they adopted IT later, in 2002 and 2000, respectively. Finally, to evaluate the impact of the three countries that have had a ﬁxed exchange rate policy during the entire period analyzed, Figs. 10 and 11 show the results of the three NITs. For Denmark and Singapore, the tests suggest that the importance of demand and CRP shocks has moved in the same directions as in the IT countries. This may be, as mentioned earlier, because the exchange rate policy in the these countries is implemented with the objective of maintaining prices stable. On the other hand, for Hong Kong, the only country in the sample with a ﬁxed exchange rate vis-à-vis the U.S., there is no statistical evidence of movements in these two distributions. This may be due to a smallsample problem, but the same test indicates that the distribution of the contributions of supply shocks has moved to the left (are less important), while that of monetary shocks has moved to the right. Overall, our results show that the relative importance of real and nominal shocks has changed importantly in IT countries during the last couple of decades. In particular, during the ﬁrst 15 years of the sample, demand shocks accounted for a signiﬁcant fraction of RER variance, whereas CRP shocks were, in general, less important. As the estimation window moves forward to include only periods of IT policies, the results are reversed: demand shocks account for a smaller fraction of the RER volatility, whereas CRP shocks are the main driver. Similar results, although seemingly less pronounced compared to the IT countries, were obtained for Denmark and Singapore, while in the currency board economy, Hong Kong, there appeared to be no change. As argued in the previous subsection, it does not seem likely that the results are due to speciﬁc factors in the benchmark economy. In this context, it should also be mentioned that our results are, to some extent, coherent with Engel and West (2010), who ﬁnd that for the U.S. few of the movements in the dollar, during the recent ﬁnancial crisis of 2008–2009, were directly attributable to the real interest rate component, suggesting that most of the movements were due to the residual risk premium component. The present analysis indicates, however, that the importance of CRP shocks increased before the recent ﬁnancial crisis of 2008–2009 and was particularly evident in countries that followed an IT monetary regime. In addition, given the fact that the importance of the CRP shock has increased in recent years, and that the variance decomposition suggests that the importance of this shock is quite persistent, it could be argued that this is a real shock rather than a nominal shock. In this context, and in line with Nam and Wang (2015), one could claim that this is a misspeciﬁed news shock or real shock driving the U.S real exchange rate behavior, which in turn has an impact on the RER of the countries under analysis. However, since changes in the RER behavior appear to be more pronounced in IT countries, it seems less likely that common news shock aﬀecting the U.S. exchange rate is driving our results. 4. Central bank preferences, RER volatility and variance decomposition We ﬁnd that the relative importance of demand and CRP shocks changed gradually as countries moved towards full-ﬂedged IT regimes. In this context a relevant question is whether shifts in central bank objectives, namely an increase in policymakers’ preferences for inﬂation stability, can explain the relative contribution of structural shocks to 10 R. Caputo, M. Pedersen Economic Modelling xxx (xxxx) xxx Fig. 5. Histograms of Variance Decompositions. Horizon: Four Quarters. Notes: Histograms of contributions to the RER variance decomposition made with the models that fulﬁll the imposed sign restriction. The two periods shown are the ﬁrst and the last in the rolling sample. The vertical lines show the contributions of most likely models. Fig. 6. Developed IT countries. Variance Decompositions. Horizon: Four Quarters. Notes: In the ﬁrst row, dotted lines show the contribution to the variance decomposition of the most likely model for the 15-year period ending the year indicated on the horizontal axis. The solid line is a ﬁtted trend. Histograms of contributions to the RER variance decomposition, in the second and third row, are obtained from models that fulﬁll the imposed sign restrictions. The two periods shown are the ﬁrst and the last in the rolling sample. The vertical lines show the contributions of the most likely models. the overall RER volatility. In this section we show that central bank preferences play a crucial role in the way that shocks are transmitted to the RER. We interpret shifts in central bank preferences as changes in the monetary policy regime. Monetary policy objectives and targets are not necessarily constant over time. Furthermore, there is extensive evidence that central bank preferences have changed as monetary authorities have moved towards full-ﬂedged IT regimes. For instance, Clarida et al. (1998) conclude that, 11 R. Caputo, M. Pedersen Economic Modelling xxx (xxxx) xxx Fig. 7. Developed IT countries. Variance Decompositions. Horizon: Four Quarters. Note: In the ﬁrst row, dotted lines show the contribution to the variance decomposition of the most likely model for the 15-year period ending the year indicated on the horizontal axis. The solid line is a ﬁtted trend. Histograms of contributions to the RER variance decomposition, in the second and third row, are obtained from models that fulﬁll the imposed sign restrictions. The two periods shown are the ﬁrst and the last in the rolling sample. The vertical lines show the contributions of the most likely models. Fig. 8. Developing IT countries. Variance Decompositions. Horizon: Four Quarters. Notes: In the ﬁrst row, dotted lines show the contribution to the variance decomposition of the most likely model for the 15-year period ending the year indicated on the horizontal axis. The solid line is a ﬁtted trend. Histograms of contributions to the RER variance decomposition, in the second and third row, are obtained from models that fulﬁll the imposed sign restrictions. The two periods shown are the ﬁrst and the last in the rolling sample. The vertical lines show the contributions of the most likely models. from 1979 to 1990, the Bank of England reacted mainly to the German interest rate (DTD) and to a lesser extent to inﬂation and output. During this period, the reaction to inﬂation was quite mild and, because the UK was part of the Exchange Rate Mechanism of the European Monetary System, there were greater concerns about exchange rate ﬂuctuations. After the introduction of the IT regime in the UK, in October 1992, there has been a substantial increase in central bank preferences for inﬂation stability, as shown by Arestis et al. (2016). For Sweden, Adolfson et al. (2008) ﬁnd that, after the adoption of IT in 1993, the policy response to inﬂation increased, whereas the response to the RER and output declined. Similar evidence is found for Chile by Caputo et al. (2007). Findings for the U.S. are similar to those for IT countries. In particular, Clarida et al. (2000), Fernández-Villaverde and Rubio-Ramírez (2008), Dennis (2006) and Ilbas (2012) show that the Fed, in the post 12 R. Caputo, M. Pedersen Economic Modelling xxx (xxxx) xxx Fig. 9. Developing IT countries. Variance Decompositions. Horizon: Four Quarters. Notes: In the ﬁrst row, dotted lines show the contribution to the variance decomposition of the most likely model for the 15-year period ending the year indicated on the horizontal axis. The solid line is a ﬁtted trend. Histograms of contributions to the RER variance decomposition, in the second and third row, are obtained from models that fulﬁll the imposed sign restrictions. The two periods shown are the ﬁrst and the last in the rolling sample. The vertical lines show the contributions of the most likely models. Fig. 10. NIT countries. Variance Decompositions. Horizon: Four Quarters. Notes: In the ﬁrst row, dotted lines show the contribution to the variance decomposition of the most likely model for the 15-year period ending the year indicated on the horizontal axis. The solid line is a ﬁtted trend. Histograms of contributions to the RER variance decomposition, in the second and third row, made with the models that fulﬁll the imposed sign restriction. The two periods shown are the ﬁrst and the last in the rolling sample. The vertical line shows the contributions of most likely models. Volcker era, has assigned more importance to inﬂation and interest rate volatility.29 Our conjecture is that, during the inﬂation targeting period, the relevance of the exchange rate in the central bank loss function has declined importantly. There is evidence that, indeed, in inﬂation targeting countries this is the case. Based on panel data techniques, Aizenman et al. (2011) investigate IT in emerging markets, focusing on the role of the real exchange rate in the Taylor rule. The main ﬁndings of this contribution is that IT emerging markets appear to follow a strategy that puts substantially more weight on inﬂation and much less on real exchange rate ﬂuctuations once countries adopt an IT regime. To be more speciﬁc, once countries move from a NIT regime to an IT one, the response to inﬂation increases by a factor of two, whereas the response to exchange 29 Uribe and Yue (2006) and Caputo and Herrera (2017) assess the importance, in diﬀerent contexts, of the foreign rates (or the Fed funds rate) for emerging economies. 13 R. Caputo, M. Pedersen Economic Modelling xxx (xxxx) xxx Fig. 11. NIT countries. Variance Decompositions. Horizon: Four Quarters. Notes: In the ﬁrst row, dotted lines show the contribution to the variance decomposition of the most likely model for the 15-year period ending the year indicated on the horizontal axis. The solid line is a ﬁtted trend. Histograms of contributions to the RER variance decomposition, in the second and third row, are obtained from models that fulﬁll the imposed sign restrictions. The two periods shown are the ﬁrst and the last in the rolling sample. The vertical lines show the contributions of the most likely models. Table 3 Policy preferences and sources of RER volatility. Conﬁguration A B C D E F G Variance Decomposition Δ q Preferences 𝜇r 𝜇𝜋 𝜇y 𝜇e 𝜀q 𝜀d 𝜀d ∕𝜀q 0.000 0.517 0.517 6.067 10.660 23.727 40.442 1.000 1.000 1.000 1.205 1.182 1.342 1.352 0.000 0.404 0.404 0.404 0.404 0.404 0.404 1.000 0.700 0.500 0.000 0.000 0.000 0.000 1.6% 7.6% 8.4% 15.0% 17.0% 18.6% 20.0% 40.8% 42.0% 43.4% 40.0% 38.0% 35.9% 35.0% 24.8 5.5 5.2 2.7 2.2 1.9 1.8 once countries implemented IT. In this context, Gomez et al. (2019) ﬁnd support for the type of policies that these central banks implemented de jure. Given the evidence from these studies it appears that the preferences of several of the central banks in our sample did actually change with the adoption of IT. In particular, the exchange rate was given less, or zero, weight in the loss function once the new regime was implemented. To understand how changes in central bank preferences aﬀect the relative contribution of shocks to the RER variance, we simulate the DSGE model presented in Section 2 under alternative policy preferences.30 In particular, we assess the consequences of increasing, gradually, the central bank’s preference for inﬂation stabilization and reducing the importance it gives to exchange rate ﬂuctuations. In the ﬁrst scenario, conﬁguration A in Table 3, the central bank cares only about stabilizing inﬂation and the nominal exchange rate: 𝜇 𝜋 = 𝜇 e = 1 and 𝜇y = 𝜇r = 0. In this case, demand shocks contribute 41% of RER volatility whereas CRP shocks contribute only 2%. Hence, in this conﬁguration, the relative contribution of demand to CRP shocks is 24.8. If the central bank moves towards a more ﬂexible exchange rate regime (conﬁgurations B and C), the contribution of demand shocks declines importantly. In particular, if preferences for exchange rate sta- rate declines by 50%. As a result, the ratio between the inﬂation and exchange rate response increases by a factor of four. In terms of the welfare loss weights, Kam et al. (2009) provide robust evidence for three developed countries, which is coherent with the evidence discussed previously, on the importance of inﬂation objectives vis-à-vis exchange rate stabilization or other concerns. In particular, Kam et al. (2009) estimate underlying structural macroeconomic policy objectives for Australia, Canada and New Zealand from 1990 to 2005, concluding that none of the central banks shows a concern for stabilizing the real exchange rate. All three central banks share a concern for minimizing the volatility of the change in the nominal interest rate. For developing countries under IT, Gomez et al. (2019) ﬁnd similar results: Inﬂation stabilization in these countries has a higher priority than other objectives such as interest smoothing, GDP and exchange rate stabilization. In particular, they ﬁnd signiﬁcant evidence against a systematic objective to stabilize the exchange rate when the central banks set the interest rate. This is true when Gomez et al. (2019) consider the period of explicit IT in Chile, Colombia, Brazil and Peru. This is perhaps not surprising since many developing countries used the nominal exchange rate as a nominal anchor during the inﬂation stabilization periods. Examples from Latin America are Mexico, Peru, Chile, Brazil, and Argentina that used ﬁxed or managed exchange rate regimes in the 1980s and 1990s. In general, exchange rate targeting was abandoned 30 We modify only the central bank preferences in equation (2.4). The other coeﬃcients in the model are maintained. 14 R. Caputo, M. Pedersen Economic Modelling xxx (xxxx) xxx Table 4 Policy preferences and macroeconomic volatility. Conﬁguration A B C D E F G Preferences Macro Volatility (%) 𝜇r 𝜇𝜋 𝜇y 𝜇e 𝜎𝜋2̃ 𝜎Δ2 q 𝜎Δ2 y 𝜎Δ2 r 0.000 0.517 0.517 6.067 10.660 23.727 40.442 1.000 1.000 1.000 1.205 1.182 1.342 1.352 0.000 0.404 0.404 0.404 0.404 0.404 0.404 1.000 0.700 0.500 0.000 0.000 0.000 0.000 8.12 7.64 6.93 1.31 1.63 2.04 2.46 2.47 3.37 4.00 27.21 25.38 24.22 23.23 1.32 1.27 1.25 1.15 1.18 1.20 1.21 1.14 0.35 0.33 0.14 0.09 0.05 0.03 Fig. 12. Volatility of inﬂation, output growth, interest rate, and RER depreciation: 1986–2010 (in %). bility decline to 𝜇 e = 0.7 and 𝜇 e = 0.5 and the importance given to output and the interest rate increases to 𝜇y = 0.404 and 𝜇r = 0.517, respectively, then the relative contribution of demand to CRP shocks declines to 5.5 and 5.2. Clearly, as the central bank moves towards a more ﬂexible exchange rate regime (from conﬁguration A to C), the importance of CRP shocks increases. The contribution of a CRP shock is, however, still below what we ﬁnd empirically for the latter period and, to investigate possible explanations, we consider four additional policy conﬁgurations where the central bank is embracing a fully ﬂexible exchange rate regime, 𝜇e = 0, while 𝜇 𝜋 and 𝜇 r are calibrated to match speciﬁc values for the contribution of CRP and demand shocks (conﬁgurations D to G). In order to increase the contribution of CRP to 15% and reduce the contribution of demand shocks to 40%, the central bank should become more hawkish, increasing preferences for inﬂation stabilization to 𝜇 𝜋 = 1.205 and for interest rate stabilization to 𝜇r = 6.057. If the contribution of CRP shocks is increased further, up to 20%, and the contribution of demand shocks is reduced, down to 35%, the central bank should further increase preferences for inﬂation and interest rate stabilization to 𝜇 𝜋 = 1.352 and 𝜇 r = 40.442. In this last conﬁguration the relative contribution of demand to CRP shocks is 1.8. From these exercises we draw two important conclusions. First, the relative importance of demand and CRP shocks depends, crucially, on the preferences of the central bank. Second, as the central bank moves towards a ﬂexible exchange rate regime, in which the main objectives are inﬂation and interest rate stability, the importance of demand shocks, relative to CRP innovations, declines importantly. Our results suggest that the gradual adoption of full-ﬂedged IT regimes, in which the importance of exchange rate ﬂuctuations is reduced, can explain the changing nature of RER ﬂuctuations we observe. To lend additional support to our exercises, we need to check if the gradual convergence towards IT in the model is able to match additional features of the data. 15 R. Caputo, M. Pedersen Economic Modelling xxx (xxxx) xxx Fig. 13. Accumulated IRF under Exchange Rate Targeting (red) and IT (blue) (quarters after the shock). (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the Web version of this article.) As shown in Table 4, the volatility of inﬂation, interest rate and output declines as the central bank moves from managed exchange rate regimes (conﬁgurations A to C) to fully ﬂexible exchange rate regimes (conﬁgurations D to G). As expected, the volatility of the RER increases substantially between regimes. These results are in line with the empirical evidence for countries that have moved towards full-ﬂedged IT regimes over time, as shown in Fig. 12, where stochastic volatility models (Chan and Hsiao (2014)), are employed for estimations.31 shock, so the exchange rate, output and inﬂation are fully stabilized. For demand shocks, the results are completely diﬀerent under exchange rate targeting. In particular, if the central bank cares about the nominal exchange rate, the policy rate does not increase signiﬁcantly, so inﬂation and output absorb an important part of this shock. Even if the nominal rate does not react signiﬁcantly, the RER will appreciate as a consequence of increasing inﬂation (Fig. 13, second row). Similarly, in the case of a supply shock the inﬂation rate declines signiﬁcantly, inducing a RER appreciation. Under conﬁguration A, the impact of nominal shocks on the RER is almost fully oﬀset by an aggressive policy response. For nominal shocks the policy trade-oﬀ between stabilizing RER, output and inﬂation is almost absent as long as the policy rate can react. Real shocks, on the other hand, are absorbed by output, inﬂation and the RER (mainly through changes in prices). As a consequence, under conﬁguration A, the contribution of nominal shocks to the RER is almost completely muted given the fact that the policy rate can be adjusted. At the same time real shocks are the main drivers of RER dynamics. In the case of an inﬂation targeting regime, with fully ﬂoating exchange rate and concerns about inﬂation and interest rate stabilization (conﬁguration G), the relative importance of nominal shocks increases substantially. In particular, in the face of a CRP shock, the policy rate does not move signiﬁcantly and the RER absorbs an important proportion of the CRP shock. In the case of demand shocks, the policy rate increases so as to stabilize inﬂation generating also a decline in the nominal exchange rate. Consequently, the RER absorbs part of the 4.1. Responses to demand and CRP shocks under alternative preferences To understand the mechanism behind the declining importance of demand shocks and the increasing relevance of CRP innovations, we compare the dynamic response of the main macro variables under exchange rate targeting (conﬁguration A) and inﬂation targeting (conﬁguration G). As shown in Fig. 13, under exchange rate targeting, the impact of CRP shocks on the RER is almost fully oﬀset by an aggressive increase in the policy rate. The impact of this shock on output growth is also oﬀset to a great extent.32 In the case of a monetary shock, the aggressive increase in the policy rate fully oﬀsets the impact of this 31 The decline in inﬂation and output growth volatility has been documented for the U.S., by Galí and Gambetti (2009) and Canova and Gambetti (2010), and has been referred to as the Great Moderation episode. 32 There is still a small RER appreciation, which given the nature of the model has an expansive impact on output. The increase in the policy rate, however, determines minor contractions in output and inﬂation. 16 R. Caputo, M. Pedersen Economic Modelling xxx (xxxx) xxx shock through the nominal appreciation and increase in prices. Under a ﬂexible exchange rate regime (conﬁguration G), the RER is much more volatile for any given shock. When compared to conﬁguration A, however, the increase in volatility due to nominal shocks is substantially larger. The reason is that the policy rate cannot move freely, so both the nominal and the real exchange rates absorb an important proportion of monetary policy and CRP shocks. Hence, the relative contribution of CRP and demand shocks crucially depends on the preferences of the central bank. shocks tended to explain a relatively small proportion of RER volatility in most IT countries. In that period, nominal shocks were relatively more important in explaining RER ﬂuctuations. When we perform a sub-sample analysis, however, we can practically reconcile the empirical ﬁndings in the literature with our results for the early subsamples. We conclude that the relative importance of demand shocks has declined importantly over time. In contrast, the relative importance of nominal shocks and, in particular of CRP shocks, has increased in recent years. We show that our empirical results could be explained by gradual shifts in monetary policy regimes towards full-ﬂedged IT regimes. In particular, if exchange rate targets are abandoned, and inﬂation and policy rate stability become the main concerns of the central bank, the policy rate will not oﬀset the impact that nominal shocks have on the RER. Accordingly, nominal shocks, and in particular CRP shocks, will have a greater impact on the RER. Overall, we conclude that central bank preferences play a crucial role in the way that shocks are transmitted to the RER. In a world with reduced inﬂation rates, the variation in the nominal exchange rate dominates in the RER volatility. In this context, one might argue that a low U.S. inﬂation rate has contributed to the decline in the rates in other countries. In this sense, the U.S. may be the driving force behind the results presented in this study. However, existing evidence suggests that falling inﬂation rates, inﬂation volatilities and the sacriﬁce ratios in IT countries are due to the adoption of IT. One of the ﬁrst scholars to notice this was Rogoﬀ (2003) and subsequent research tends to conﬁrm his claim (see Gonçalves and Salles (2008), Svensson (2010) and, more recently, Huang et al. (2019)). In this setting, the results obtained in the present paper are in line with previous evidence on the reasons for lower inﬂation rates in IT countries. In this environment, the role of ﬁnancial integration is certainly an issue that deserves more analysis. This is, however, beyond the scope of the present paper and is left to future research. 5. Conclusions Understanding the sources of real exchange rate (RER) variability is one of the challenging issues in international economics. Based on earlier empirical evidence, there is general consensus that real shocks (i.e. demand and supply shocks) play the most important role in determining RER ﬂuctuations. On the contrary, nominal shocks (monetary policy and country risk premium (CRP) shocks) contribute very little to the RER dynamics. In this context, the objective of this paper is twofold. First, we examine a dynamic stochastic general equilibrium (DSGE) model in order to understand how diﬀerent shocks aﬀect the RER dynamics in a small open economy. Second, we estimate structural vector autoregressive (SVAR) models for several inﬂation targeting (IT) and non inﬂation targeting (NIT) countries using the sign restriction approach suggested by Inoue and Kilian (2013). These restrictions are derived from the open economy DSGE model of Kam et al. (2009). Estimations of the SVAR models are performed for diﬀerent sub-samples in order to detect changes in the relative importance of the contributions of real and nominal shocks to the RER dynamics. The sample employed is large enough to take into account possible changes of the monetary policy objectives. We perform the empirical exercise for the set of IT and NIT countries from 1986 to 2014. Our main ﬁndings are as follows. In sharp contrast with previous empirical evidence, we ﬁnd that for the period 1986–2014, demand Appendix A. The DSGE model Households Our model is based on Kam et al. (2009). Households have a period utility function of the form: 1+𝜙 U (Ct , Ht , Nt ) = N (Ct − Ht )1−𝜎 − t , 1−𝜎 1+𝜙 (5.1) where Ct represents the consumption basket, Ht = hCt −1 is an external habit stock with h between 0 and 1 and Nt is labor hours. Househols maximize the utility function U(.) subject to the sequential budget constraint: Bt ≥ PH,t CH,t + PF ,t CF ,t + Bt +1 − Wt Nt , Rt (5.2) where Bt is an Arrow security that pays out contingent on the state of the economy, Rt is the gross return on a nominal riskless one-period bond, and Wt Nt is the total wage income. The expenditure in home goods is given by PH,t CH,t , whereas the expenditure in foreign consumption goods is given by PF,t CF,t . The consumption index, Ct , is linked to domestic, CH,t , and foreign goods, CF,t , such that: [ 1 𝜂−1 𝜂 1 𝜂−1 𝜂 ] 𝜂−1 Ct = (1 − 𝛼) CH,t + 𝛼 CF ,t 𝜂 𝜂 𝜂 , (5.3) where the elasticity of substitution between home and foreign goods is given by 𝜂 > 0. It can be shown that the optimal allocation of expenditures across each good type gives rise to the demand functions: ( )−𝜂 P Ct , (5.4) CH,t = (1 − 𝛼) H,t Pt ( CF ,t = 𝛼 ) PF ,t −𝜂 Ct . Pt (5.5) Substitution of these demand functions into (5.3) yield the consumer price index as: [ ] 1 1−𝜂 1−𝜂 1−𝜂 Pt = (1 − 𝛼)PH,t + 𝛼 PF ,t . (5.6) 17 R. Caputo, M. Pedersen Economic Modelling xxx (xxxx) xxx The intertemporal optimality condition for the households yields the familiar stochastic Euler equation: ( )−𝜎 ( ) Ct +1 − Ht +1 Pt 𝛽 Rt Et = 1. Ct − Ht P t +1 (5.7) Log-linearizing this expression gives rise to the standard stochastic linear Euler expression, which is equation (2.1) in the main text. International risk sharing and the uncovered interest rate parity conditions Prices in the rest of the world are given by: 1 [ ( )1−𝜂 ( )1−𝜂 ] 1−𝜂 + 𝛼 ∗ P∗F ,t , P∗t = (1 − 𝛼 ∗ ) P∗H,t (5.8) where ∗ denotes foreign variables. Because the domestic economy is a small one, for the rest of the world 𝛼 ∗ is equal to zero. As a consequence, international prices are given by: P∗t = P∗H,t . (5.9) For the rest of the world, the ﬁrst-order condition for optimal consumption, i.e. the Euler equation, is analogous to the one derived for the domestic economy. In short, given complete international markets, we obtain the perfect risk sharing condition: ( )−𝜎 ( )( ( )−𝜎 ( ) ) Ct∗+1 − Ht +1 P∗t C − Ht+1 ERt Pt 𝛽 t +1 = Qt,t+1 = 𝛽 , (5.10) ∗ ∗ Ct − Ht P t +1 ERt +1 Ct − Ht P t +1 where Qt ,t +1 = 1 Rt is the discount factor, which is inversely related to the gross return on a nominal riskless one-period bond. The expression in (5.10) holds for all dates and states, where ERt represents the nominal exchange rate. From (5.10) it is possible to derive the non-arbitrage conditions for the exchange rates, or the uncovered interest rate parity condition in levels: Rt − R∗t ERt = 0. ERt +1 (5.11) Log-linearizing (5.11) we obtain the UIP condition: Et (et +1 ) − et = rt − rt∗ , (5.12) where et = ln(ERt ∕ERss ), the percentage deviation of the nominal exchange rate from its steady-state (ss) value, and the domestic and foreign rates of net return are rt = Rt − 1 and rt∗ = R∗t − 1. The expression in (5.12) gives rise to the stochastic UIP condition, equation (2.5) in the main text. Domestic ﬁrms optimal pricing Domestic goods ﬁrms operate a linear production technology. These ﬁrms face an independent signal that allows them to set prices optimally each period with probability 1 − 𝜃 H . In each period t, the remaining fraction 𝜃 H ∈ (0, 1) of ﬁrms partially index their prices to take into account past aggregate inﬂation. Given the Calvo price setting, it is possible to deﬁne the aggregate price level of domestic goods: 1 1−𝜖 ( ⎧ ( ) )1−𝜖 ⎫ ( )1−𝜖 PH,t −1 𝛿H ⎪ ⎪ new PH,t = ⎨(1 − 𝜃H ) PH,t + 𝜃H PH,t−1 ⎬ , PH,t −2 ⎪ ⎪ ⎩ ⎭ (5.13) is the optimal price set by ﬁrms that are able to optimally set prices, 𝛿 H ∈ (0, 1) measures the degree of inﬂation indexation, and 𝜖 where Pnew H ,t represents the elasticity of substitution between home diﬀerentiated goods. Consider a ﬁrm i that has set its price optimally in time t as PH,t (i). The ﬁrm faces the following demand for its product: [ ( ) ]−𝜖 P(i)H,t PH,t +s−1 𝛿H Y (i)H,t +s = (CH,t+s + CH∗ ,t+s ), (5.14) PH,t +s PH,t −1 The ﬁrst-order necessary condition characterizing domestic ﬁrms’ optimal pricing function in a symmetric equilibrium is: ( ) ( ) ∞ s ∑ 𝜃H Y (i)H,t+s PH,t +s−1 𝛿H ̃ Et PH,t − 𝜁 P(i)H,t+s MCH,t+s = 0, Rt +s−1 PH,t −1 s=0 (5.15) PH,t is the price set (optimally) today by ﬁrms, which has a probability 𝜃 H of remaining ﬁxed in the next period. The variable MCH,t +s is the where ̃ marginal cost of producing one unit of domestic goods. Let the home goods inﬂation rate be 𝜋 H,t = ln(Ph,t ∕PH,t −1 ). Then log-linearizing (5.15) it is possible to get a New Keynesian Phillips curve for home goods, which is given by: 𝜋H,t = 𝛽 Et (𝜋H,t+1 − 𝛿H 𝜋H,t ) + 𝛿H 𝜋H,t−1 + 𝜆H (mct + (1 + 𝜙)𝜖s,t ), (5.16) 𝜎 The above Phillips curve corresponds to equation (2.2) in the main text, where 𝜆H = (1 − 𝛽𝜃H )(1 − 𝜃H )𝜃H−1 and mct = 𝜙yt + 𝛼 st + 1− (c − hct−1 ). h t Import ﬁrms optimal pricing Import retailers are assumed to purchase imported goods at competitive world prices. These ﬁrms, however, act as monopolistically competitive redistributors of these goods creating a gap between the price of imported goods in domestic currency, et + p∗t , and the domestic retail price of imported goods, pF,t . This gap is reﬂected in the variable 𝜓 F,t as follows: 𝜓F ,t = et + p∗t − pF ,t . (5.17) 18 R. Caputo, M. Pedersen Economic Modelling xxx (xxxx) xxx Import ﬁrms face an independent signal that allows them to set prices optimally each period with probability 1 − 𝜃 F . In each period t, the remaining fraction 𝜃 F ∈ (0, 1) of ﬁrms partially index their prices to take into account past aggregate inﬂation. Given the Calvo price setting, it is possible to deﬁne the aggregate price level of foreign goods: 1 1−𝜖 ( ⎧ ( ) )1−𝜖 ⎫ ( )1−𝜖 PF ,t −1 𝛿F ⎪ ⎪ P + 𝜃 PF ,t = ⎨(1 − 𝜃F ) Pnew ⎬ , F F ,t −1 F ,t PF ,t −2 ⎪ ⎪ ⎩ ⎭ (5.18) where Pnew is the optimal price set by ﬁrms who are able to optimally set prices, 𝛿 F ∈ (0, 1) measures the degree of inﬂation indexation, and 𝜖 F ,t represents the elasticity of substitution between home diﬀerentiated goods. Consider a ﬁrm j that has set its price optimally at time t as PF,t (i). Then the ﬁrm faces the following demand for its product: [ ( ) ]−𝜖 PF ,t (j) PF ,t +s−1 𝛿F CF ,t +s . (5.19) Y (j)F ,t +s = PF ,t +s PF ,t −1 The ﬁrst-order necessary condition characterizing domestic ﬁrms’ optimal pricing function in a symmetric equilibrium is: [ ] ( ) ∞ s ∑ 𝜃F Y (j)F ,t+s PF ,t +s−1 𝛿F ∗ ̃ Et PF ,t − 𝜁 ERt+s Pt+s = 0, Rt +s−1 PF ,t −1 s=0 (5.20) PF ,t is the price set (optimally) today by ﬁrms that have a probability 𝜃 F of remaining ﬁxed in the next period. The variable ERt +s P∗t +s is the where ̃ cost, in domestic currency, of importing one unit of the foreign good. If equation (5.20) is log-linearized around the non-stochastic steady-state we get: 𝜋F ,t = 𝛽 Et (𝜋F ,t+1 − 𝛿F 𝜋F ,t ) + 𝛿F 𝜋F ,t−1 + 𝜆F (𝜓F ). (5.21) The above expression is the New Keynesian Phillips curve for imported goods, which corresponds to equation (2.3) in the main text. Aggregate output In log-linear terms, aggregated output is the sum of domestic goods consumed by nationals and domestic goods consumed by foreigners: yt = cH,t + c∗H,t . (5.22) From equations (5.4) and (5.5) we obtain the log-linear expressions for home, CH,t , and foreign, CF,t , consumption of domestic goods as cH,t = [ ( ) ] (1 − 𝛼)[𝛼𝜂 st + ct ] and c∗H,t = 𝛼 𝜂 st + 𝜓F ,t + yt∗ , respectively. Using the previous expressions and the deﬁnition of 𝜓 F,t , we can express yt as: yt = (1 − 𝛼)ct + 𝛼 yF∗,t + 𝛼𝜂 qt + 𝛼𝜂 st , (5.23) which is equation (2.10) in the main text. Appendix B. Econometric details The VAR models are estimated with a Bayesian method where the prior and posterior distributions belong to the normal-inverse Wishart distributions.33 Hence, the prior distribution for the parameters of (3.1) is vec(𝛽) ∣ Ω ∼ N (vec(𝛽 0 ), Ω ⊗ K0−1 ), Ω ∼ IWn (v0 S0 , v0 ), where K0 is a positive deﬁnite matrix, S0 a covariance matrix, and v0 > 0. The posterior is vec(𝛽) ∣ Ω ∼ N (vec(𝛽 T ), Ω ⊗ KT−1 ), Ω ∼ IWn (vT ST , vT ), where 𝛽 T = NT−1 (N0 𝛽 0 + X ′ X 𝛽̂ols ), NT = N0 + X′ X, vT = T + v0 , ST = (5.24) v0 vT S0 + T vT ̂+ Ω 1 vT ̂ ols are the (𝛽̂ols − 𝛽 0 )′ N0 N0−1 X ′ X (𝛽̂ols − 𝛽 0 ), and 𝛽̂ols and Ω ordinary least squares (OLS) estimates of (3.1). To obtain the random draws needed for the estimation, the covariance matrix is decomposed as Ω = AUU′ A′ , where A is the lower triangular Cholesky decomposition of Ω and U is an orthogonal matrix such that UU′ = I. As mentioned earlier, we follow the approach of Inoue and Kilian (2013) for a fully identiﬁed model such that U is obtained as in Uhlig (2005), i.e. the prior distribution is a uniform distribution, which is deﬁned ̃ denotes the structural impulse-responses, Inoue and Kilian (2013) show that the posterior density of on the space of orthonormal matrices U. If Θ ̃ is Θ ( ) | | −1 | 𝜕Ω | ̃ 𝜕 vec(Θ) | | ̃ | | f (Θ) ∝ | (5.25) | | 𝜕 A | f (𝛽 |𝛺)f (𝛺), | 𝜕[vec(𝛽)′ vech(A)′ veck(U )′ ] | | | | | where vech(A) is the vector that contains the elements of A that are on and below the diagonal, while veck(U) includes the elements of U that are above the diagonal. The following process is applied to obtain the results: (1) Take 20,000 random draws from (5.24). (2) For each (𝛽, Ω) take 5,000 random draws ̃ and ̃ for each (𝛽, Ω, U). (4) Keep the Θ ̃ s that satisfy the sign restrictions and delete the rest. (5) Calculate f (Θ) of the rotation U. (3) Compute Θ ̃ f (Θ)} ̃ in descending order by the value of f (Θ) ̃ . The so-called most likely model is the one with maximum value of the posterior sort the pairs {Θ, density (5.25). 33 See for example Gelman et al. (2014). 19 R. Caputo, M. Pedersen Economic Modelling xxx (xxxx) xxx Appendix C. Data description The IFS codes of the series applied are as follows: NGDP_R for the real gross domestic product (index, 2010 = 100), PCPI for the all items in the consumer price index (index, 2010 = 100), FILR for the lending interest rates (percent per annum), and ENDA for the exchange rates measured as units of national currency per US dollar, period average. Where the series were not available, the following were utilized: Chilean consumer prices are extended backwards from 2008Q4 with the annual changes of all CPI items, capital city (IFS: PCPI_A1). From 2003 onwards the Danish interest rate is the weighted average of all lending rates supplied by the Danmarks Nationalbank. The Norwegian interest rate is the rate of bank loans, which was extracted from Bloomberg. For the United Kingdom the price index utilized is the retail price index (IFS: CPRPTT01). 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