Trend Fundamentals and Exchange Rate Dynamics

We estimate a multivariate unobserved components stochastic volatility model to explain the dynamics of a panel of six exchange rates against the US dollar. The empirical model is based on the assumption that two countries’ monetary policy strategies may be well described by Taylor rules with a time‐varying inflation target, a time‐varying natural rate of unemployment, and interest rate smoothing. Compared to the existing literature, our model simultaneously provides estimates of the latent components included in a typical Taylor rule specification and the model‐based real exchange rate. Our estimates closely track major movements along with important time series properties of real and nominal exchange rates across all currencies considered, outperforming a benchmark model that does not account for changes in trend inflation and trend unemployment. More precisely, the proposed approach improves on competing models in tracking the actual evolution of the real exchange rate in terms of simple correlations while it appreciably improves on simpler competitors in terms of matching the persistence of the real exchange rate.


INTRODUCTION
To what extent do economic fundamentals explain exchange rate movements? Following the seminal work by Meese and Rogoff (1983), a wealth of studies have aimed to answer this question by comparing the out-of-sample predictive ability of economic exchange rate models to random walk forecasts, with mixed success (for an overview, see Rossi (2013)). Engel and West (2005) show, however, that exchange rates may be unpredictable if the fundamentals follow a random walk. Engel and West (2006) and Engel et al. (2019) therefore examine the in-sample explanatory power of fundamentals using variables typically included in Taylor-type policy rules. In this paper, we examine whether taking into account slow-moving trends-such as changes in the inflation target or the natural rate of unemployment-improves the in-sample explanatory power of exchange rate models.
To motivate our empirical framework, we derive a partial equilibrium expression for the bilateral real exchange rate, assuming that each country's central bank targets shortterm interest rates according to a Taylor rule with a time-varying inflation target and a time-varying natural rate of unemployment. Combining these Taylor rules with a noarbitrage condition reveals that the current real exchange rate depends on future expected trend inflation, inflation gaps, unemployment gaps and short-term interest rates. The model implies that the trend and gap components may have different impacts on the exchange rate. A decline in inflation is associated with an appreciation of the currency if the central bank lowers its inflation target. By contrast, a decline in inflation below target is associated with a depreciation, if the central bank cuts its policy interest rate more than one-for-one.
Such considerations imply that the researcher needs to estimate latent variables such as trend inflation and the inflation gap. In this paper, we propose estimating these quantities with an unobserved components stochastic volatility (UC-SV) model, similar to that of Stock and Watson (2007). Trend inflation and trend unemployment follow strong performance can be explained through our model selection prior, with posterior restriction probabilities pointing towards the necessity to decompose US inflation in its trend and cyclical component, while for the home country including the level of inflation appears to be sufficient. In terms of describing salient features of the involved exchange rates, the UC-SV model captures the major upturns and downturns of bilateral real exchange rates against the US dollar for a panel comprising six economies during the post-Bretton-Woods era. In fact, the correlations between the model-based predictions and the actual real exchange rates are as high as 0.58. A benchmark model, which is estimated on the same information set but does not discriminate between trend and gap components, yields lower correlations comparable to existing studies (see Engel and West 2006;Mark 2009). In terms of reproducing long-run trends of nominal exchange rates, the UC-SV model is capable of capturing all major low-frequency movements over the last 40 years. In particular, we show that the model works especially well before 1990, when trend inflation was volatile in many countries in our sample. Finally, the model successfully mimics the actual exchange rates with respect to several key time series properties. More specifically, we accurately reproduce the persistence of the real exchange rates and the correlations with other macroeconomic variables. Especially in terms of replicating the persistence of the real exchange rate, our approach sharply improves on simpler benchmark specifications.
In what follows, Section I motivates the UC-SV model by deriving a partial equilibrium expression for the real exchange rate in terms of future expected fundamentals. Then Section II outlines the empirical strategy adopted along with the corresponding prior specification. Finally, Section III presents the empirical results, while Section IV summarizes and concludes the paper.

I. THEORETICAL FRAMEWORK
Following Engel and West (2006), we derive an expression for the real exchange rate in terms of future expected fundamentals if monetary policy in two countries is characterized by Taylor rules. All equations are shown in log-linearized terms. Let the short-term policy interest rate i t in the home economy be determined as The central bank in the home economy targets the short-term interest rate as a function of deviations of expected inflation from the target (E t b p tþ1 ), of the expected unemployment rate from its natural level (E t b u tþ1 ), and of the lagged interest rate, whereas ɛ t is a monetary policy innovation. 1 The inflation and unemployment gaps are defined as b p t ¼ p t À p t and b u t ¼ u t À u t , respectively. Therefore the inflation target (p t ) as well as the natural rate of unemployment (u t ) change over time. As is standard in the literature, c p > 0 and c u < 0 such that the central bank increases its policy interest rate in response to a higher inflation gap or a lower unemployment gap.
We follow Engel and West (2006) and assume that the home central bank responds to the real exchange rate defined as q t ¼ e t À p t þ p Ã t . The nominal exchange rate (e t ) is expressed as the price of one unit of the foreign currency in terms of domestic currency such that a rise in the exchange rate implies a depreciation of the home currency. Furthermore, p t and p Ã t denote the domestic and foreign price levels, respectively. We assume that c q > 0, implying that the central bank lowers the interest rate when the exchange rate appreciates in real terms. We depart from Engel and West (2006) in three ways. First, we assume that the central bank responds to an unemployment gap rather than an output gap. This is mainly because our empirical analysis uses monthly data. Second, we assume that the central bank responds to the future expected unemployment gap rather than the current one. Therefore we take into account that many central banks publish forecasts and justify their interest rate decisions based on future expected developments. 2 Third, we assume that the central bank follows a Taylor rule with interest rate smoothing (see, for example, Coibion and Gorodnichenko 2012). According to Giannoni and Woodford (2003), interest rate smoothing is a feature of optimal simple interest rate rules.
The central bank in the foreign economy targets the short-term interest rate using an analogous rule, except that it does not respond to the real exchange rate: 3 where all foreign variables are labelled by an asterisk.
Furthermore, we assume that an uncovered interest parity relationship holds periodby-period: 4 Replacing the interest rate differential by the two policy rules and rearranging terms, we obtain with q = 1/(1 + c q ). Solving the equation forward allows us to express the real exchange rate in terms of future expected fundamentals. This expression is the present-value solution of Engel and West (2005): Despite the partial equilibrium nature of the analysis, some interesting insights emerge. Reflecting the findings by Engel and West (2005), this equation suggests that changes in trend inflation, if they occur, will dominate fluctuations of the real exchange rate. To see this, assume that the home inflation gap follows a stationary AR(1) process with autoregressive parameter φ, and home trend inflation follows a random walk. For J ? ∞, it is straightforward to show that the partial equilibrium effect of a change in the inflation gap and trend inflation amount to Analogous expressions with the opposite sign hold for changes in the foreign inflation gap and foreign trend inflation. For reasonable parameter values, changes in domestic or foreign trend inflation, if they occur, will have a larger effect on the exchange rate in absolute value. 5 In addition, changes in trend inflation and the inflation gap affect the exchange rate differently. If the Taylor principle holds, that is 1Àc p < 0, a decline in the inflation gap causes a depreciation because the central bank lowers the policy rate by more than one for one. 6 By contrast, a decline in trend inflation causes an appreciation because the central bank does not change the policy rate. Similar to Engel et al. (2019), the source of the shock that causes movements in inflation determines whether the exchange rate appreciates or depreciates.
To map the theoretical equation to empirical data, we need to form expectations about nominal short-term rates, and the inflation and unemployment gaps, as well as future trend inflation. In what follows we outline the empirical strategy to model the decomposition and the future expected evolution of these measures.

II. EMPIRICAL FRAMEWORK
We propose a multivariate unobserved components stochastic volatility (UC-SV) model to describe the dynamics of the fundamentals. The model may be viewed as an open economy variant of earlier UC-SV specifications that aim to model inflation and unemployment dynamics by decomposing the respective variables in non-stationary trend and stationary gap components. (For similar modelling approaches, see, for example, Gordon 1998;Stock and Watson 2007;Stella and Stock 2015;Chan et al. 2016Chan et al. , 2018Hwu and Kim 2019).

The unobserved components stochastic volatility model
Let us store the observed inflation and unemployment series measured at time t = 1,. . .,T in a 4 9 1 vector x t ¼ ðp t ; p Ã t ; u t ; u Ã t Þ 0 . We assume that x t may be decomposed as with f t ¼ ½p t ; p Ã t ; u t ; u Ã t 0 being a 4 9 1 vector of latent trend components of inflation and unemployment at home and abroad. Similarly, b f t ¼ ½b p t ; b p Ã t ; b u t ; b u Ã t 0 denotes a 4 9 1 vector of (stationary) latent gap components of inflation and unemployment. We assume that U ¼ diagð/ p ; / Ã p ; / u ; / Ã u Þ is a 4 9 4 matrix of autoregressive coefficients with absolute value below unity. This ensures that b f t is mean reverting and thus permits us to interpret b f t as a vector containing the inflation and unemployment gaps, respectively. Finally, e t and f t ¼ ½g 0 t ; b g 0 t 0 are normally distributed vector white noise errors with time-varying variance covariance matrices R t and V t . We assume that R t is a diagonal matrix with typical element r 2 jt (j = 1,. . . ,4), and V t is a full matrix that can be decomposed as where A is an 8 9 8 lower triangular matrix with unit diagonal and typical non-zero offdiagonal element a j , and S t ¼ diagðs 1t ; . . .; s 8t Þ contains the stochastic volatilities of the latent factors on its main diagonal.
We complete the description of our empirical model by stacking the logarithm of the volatilities in R t and S t in a generic vector h t , with typical element denoted by h it . Following Kastner and Fr€ uhwirth-Schnatter (2014), we assume that each h it evolves according to where l i is the level of the log-volatility, q i 2 ðÀ1; 1Þ denotes the autoregressive parameter, and ϑ i denotes the variance of the log-volatility. This choice ensures that the volatility is bounded in the limit and rules out odd behaviour related to random walk state equations for log-volatilities.
The UC-SV model explicitly discriminates between components that are nonstationary, capturing trends in the respective macroeconomic variable, and stationary processes that capture the high-frequency behaviour. To improve the fit of the model, we moreover assume that all components are allowed to follow distinct stochastic volatility processes.
The specification described by equations (7) and (8) is loosely related to the model put forward by Stella and Stock (2015). The key difference is that the multivariate unobserved component model of Stella and Stock (2015) assumes that while both unemployment and inflation feature a non-stationary trend term, there exists only a single common stationary factor that can be interpreted as the unemployment gap. This common factor enters the inflation equation as an additional covariate, implying that the stationary part of inflation is driven by the unemployment gap plus a white noise shock. Our approach is slightly more flexible. This increased flexibility is necessary since we found that for non-US data, using a single gap component is not sufficient to explain medium-frequency movements in inflation and the corresponding factor loading close to zero. Moreover, another distinction between our proposed model and this specification is that we model unemployment and inflation across two countries whereas the specification of Stella and Stock (2015) is estimated on US data only.

Relation to the real exchange rate
We can derive an approximation that maps the empirical model described in the previous subsection to the theoretical exchange rate model. If we assume that the discount factor q is close to unity and assume that the expectations hypothesis holds, then E t P JÀ1 j¼0 ði tþj À i Ã tþj Þ is approximately J times the interest rate differential for J-period bonds, which we denote as Jðb J;t À b Ã J;t Þ. 8 Furthermore, for a discount factor close to but below unity we have for large J that E t q J+1 q t+J+1 % 0. Finally, under the structure of the Since the trend components follow a random walk process, the expectations are given by For large J and a discount factor close to unity, we can approximate the exchange rate relationship in equation (5) as The terms involving the gap components are exact for J ? ∞ and q ? 1. However, it is worth noting that these approximating assumptions are accurate even for finite J and relatively persistent processes. 9 In the empirical specification, we relax the assumption of parameter homogeneity across both countries' Taylor rules. The empirical model that relates the system described in the previous subsection to equation (11) is therefore given by with and m t $ N ð0; r 2 m Þ being a homoscedastic white noise error term. While it would be straightforward to allow for stochastic volatility in equation (12), we leave this possibility aside because we are mainly interested in capturing the dynamics of the exchange rate related to the first moment of the corresponding predictive density.

A model specification prior
To assess whether or not we have to discriminate between gap and trend components in equation (12), we follow a Bayesian approach and develop a model selection prior that allows for testing restrictions such as Notice that this is only an approximation since we introduce measurement errors to the observation equation. However, as mentioned in note 7, the corresponding error variances are close to zero, implying that the state innovations are also small in magnitude. From a practical perspective, these restrictions effectively imply that we do [OCTOBER not discriminate between gap and trend components of inflation in the home country. In the following, we test these restrictions not only for inflation in the home country but also for the foreign economy.
In what follows, we discuss the specific prior setup for the home country exclusively. The corresponding prior distributions on foreign quantities are similar and indicated by an asterisk. The prior distribution on b b p reads as where d p is an indicator with Prob(d p = 1) = 1/2, and s p0 ( s p1 denotes prior scaling parameters with s p0 set close to zero. This prior essentially implies that if d p = 1, then we strongly push the corresponding posterior estimate towards parameter homogeneity, implying that we do not discriminate between trend inflation and the inflation gap. This is commonly referred to as the 'spike' component. By contrast, if d p = 0, then we introduce only little prior information and allow for differences across parameters. This is the 'slab' component of the Gaussian prior. Equation (14) can be rewritten as This hierarchical prior structure is closely related to stochastic search variable selection (SSVS) priors (George and McCulloch 1993) that allow for flexible testing of whether it is necessary to discriminate between gap and trend components in our regression model. The main difference between this and standard SSVS priors is that the spike component is centred on b e p ð1 À / p Þ. A similar specification has been used by Koop and Korobilis (2016) in panel VAR models to select cross-country restrictions within a Bayesian framework. The state space model described in the previous subsection and equation (12) are estimated using Bayesian methods. The prior setup related to the remaining quantities of the model and the Markov chain Monte Carlo (MCMC) algorithm is described in Online Appendix A.

III. EMPIRICAL FINDINGS
We estimate the model for the US dollar (USD) against the currencies of a panel of six economies: Germany, the UK, Japan, Canada, Sweden and Switzerland (see Online Appendix B for a detailed description of the data). For the DEM/USD exchange rate, the series is linked with the EUR/USD exchange rate after the introduction of the euro. The real exchange rate is calculated using the same consumer price indices that are used in the estimation for the trend inflation rate. We use 10-year government bond yields to approximate the sum of future expected short-term interest rates, and thus set J = 120 months. As short-term interest rates we use 3-month interbank or T-Bill rates. Finally, we use civilian unemployment rates to estimate the unemployment gaps.

Model selection criteria and posterior restriction probabilities
Before discussing selected in-sample features of our proposed modelling approach, it is worth illustrating that our model receives substantial data support. To this end, we  Spiegelhalter et al. 2002), that is, a Bayesian information criterion similar to the Akaike information criterion. The DIC is computed as where q ¼ ðq 1 ; . . .; q T Þ 0 , Ξ denotes generic notation that stacks all unknowns of the model, and Ξ denotes the posterior mean of Ξ.
The first term in equation (15) rewards in-sample fit while the second term penalizes model complexity and serves to measure the effective number of parameters. Since the set of models that we propose allows for discriminating between gap and cycle components, considering only the model fit would ignore increased model complexity and favour more complex specifications. This issue is circumvented by using the DIC. Notice that the DIC is a negatively oriented criterion and the model with the lowest DIC value is preferred. The reason why we adopt this measure as opposed to more traditional model selection criteria such as the Bayesian information criterion or the Akaike information criterion is that our model is a hierarchical latent variable model and computing standard information criteria is not straightforward. Specifically, it is not clear how to count the number of parameters and, in addition, what point estimators for the latent quantities in the model should be used to compute the likelihood.
Our benchmark model is a simple exchange rate regression that does not discriminate between gap and trend components. More specifically, this implies that X t is replaced by e X t given by The corresponding set of regression coefficients features a SSVS prior in the spirit of George and McCulloch (1993). The prior on the error variances is of inverted Gamma form with loosely informative hyperparameters.
To decide on how relevant different components of our model specification are, we not only compare the benchmark and UC-SV models, but also assess whether the introduction of stochastic volatility pays off. Table 1 shows the DIC values across different models and exchange rates, with lower values of the DIC signalling a better model performance. Two major findings emerge. First, comparing the homoscedastic to the heteroscedastic specifications indicates that allowing for stochastic volatility generally yields lower DIC values. This finding is closely related to recent findings in the literature on macroeconomic forecasting that highlight the relevance of controlling for heteroscedasticity for forecasting macroeconomic quantities (see, for instance, Clark 2011; Clark and Ravazzolo 2015). Second, by considering the first two columns of Table 1, we observe that applying a gap-cycle decomposition translates into lower DIC values across all currency pairs considered. Thus the increased complexity is outweighed by a better model fit.
After providing some evidence that our proposed model fits the data well, we now turn to analysing whether our stochastic model selection prior effectively pushes the model towards the benchmark specification. The corresponding probabilities that the parametric restrictions hold are shown in Table 2. The first column refers to the posterior probability that d p = 1, while the second column shows the posterior probability that Economica © 2020 The Authors. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science d Ã p ¼ 1. In this discussion, this refers to the probability that the restrictions are introduced for US-based quantities. Table 2 reveals that for most exchange rate pairs, the posterior probabilities of parameter homogeneity associated with domestic quantities exceed 60%. Only for the DEM/USD exchange rate do we find a restriction probability that is smaller than 50%. This finding, however, does not carry over to US-based gap and cycle components. For US fundamentals, we observe that posterior restriction probabilities are smaller than 50% in all cases except for the USD/JPY exchange rate. The main thing to take away from this discussion is that for domestic quantities, it is sufficient to include the inflation rate without discriminating between the inflation gap and trend. For US-based quantities, it seemingly pays off to estimate separate coefficients associated with gap and trend components. At this point, it is worth stressing that estimating a model that does not discriminate between domestic trend inflation and the inflation gap, while allowing for this feature for US measures, differs from our approach. Our Bayesian model selection prior implies that in a certain fraction of cases we differentiate between the inflation trend and gap, whereas in other cases we do not. This implies that the resulting posterior mean estimates can be viewed as a weighted average between different model  specifications. This shows some resemblance to Bayesian model averaging strategies that aim to explore a vast model space to identify promising models.

In-sample correlation
We next discuss the in-sample fit of the model in more detail. Figures 1 and 2 show the actual real and nominal exchange rates along with the mean and the 5th and 95th percentiles of the posterior distribution from the UC-SV model. The posterior distribution reflects uncertainty associated with the estimation of the UC-SV model as well as due to the estimation of the linear relationship of the exchange rate equation. For all countries, the posterior mean tracks major exchange rate movements well. The majority of turning points of the real exchange rate are captured by our model. Moreover, we match the appreciation trends of the nominal exchange rate well, in particular for Japan and Switzerland.
In what follows, we discuss the episodes when the actual exchange rate moves outside of the 5th and 95th percentiles. In the mid-1980s, the real exchange rate leaves the credible bands for all countries except Canada. Similar problems of matching the strong US dollar during this period are reported by Engel and West (2006), who note that this period has been frequently labelled a US dollar 'bubble'. This is in line with the idea that the fundamentals included in the extended model do not explain the strong US dollar.
Starting in 1998, the US dollar appreciated and rose outside of the 95th percentile for most currencies under consideration. We conjecture that this is closely related with several major economic crises that forced investors to reduce their non-USD exposure ('flight to safety'). More specifically, the Asian financial crisis that hit the region between 1997 and 1998 was closely followed by the sovereign default of Russia and the unwind of long-term capital management. Besides these developments in Asia, increased uncertainty surrounding the Argentinian crisis between 1998 and 2002 presumably contributed to the upward pressure on the US dollar. Such safe-haven considerations are probably not well captured in the factors affecting short-term interest rates via the Taylor rule.
Generally speaking, significant deviations from the model predictions occur when the Taylor rule is a poor approximation to monetary policy, for example, at the effective lower bound on short-term interest rates and during unconventional monetary policy actions. In 1978 and 2011, the real exchange rate leaves the credible bands for Switzerland when the short-term interest rate was constrained by the effective lower bound. B€ aurle and Kaufmann (2018) argue that a currency is likely to appreciate strongly at the effective lower bound in response to modestly deflationary risk premium shocks because of increasing instead of declining real interest rates. In the late 1970s as well as in 2011, the Swiss National Bank counteracted the appreciation by introducing a minimum exchange rate against the German mark and the euro, respectively. Also, for Japan we observe a substantial deviation from the prediction in 1995 when short-term interest rates fell to very low levels (to 0.4% in September 1995). These results are in line with Huber (2017) and Hauzenberger and Huber (2018) providing evidence for switching policy rules in exchange rate models when interest rates are constrained by the effective lower bound.
Another reason why our model fails may be that we ignore a risk premium that explains short-term fluctuations of exchange rates. To investigate this possibility, we compute the correlation between the residual of the exchange rate models with various measures of uncertainty (stock market volatility and economic policy uncertainty). For brevity, we show these correlations in the Online Appendix. For most exchange rates, we find significantly positive correlations between the residuals and the US VIX. The correlations range from 0.2 for the CHF/USD exchange rate to 0.4 for the CAD/USD exchange rate. This suggests that the exchange rates depreciate more than expected if stock market volatility in the USA is high. We also find some significant correlations with non-US measures of uncertainty; but the correlations are smaller and less precisely estimated.
Using the posterior distribution of the exchange rate prediction, we may investigate the model fit more formally by calculating the posterior distribution of the correlation with the actual exchange rate. The model predictions match the dynamics of the level of the exchange rate well; however, they do not explain exchange rate changes. Table 3 shows the posterior mean and percentiles for the correlation with the actual real and nominal exchange rates for each country. For each pair, the first line is our benchmark model that does not control for the fact that trend inflation and trend unemployment may change over time. The second line gives the UC-SV model specification with the decomposition. Using the benchmark model, we obtain correlations between 0.28 for the UK and 0.56 for Canada. The correlation for Germany at 0.42 is close to existing estimates by Engel and West (2006) and Mark (2009).
If we include trend inflation rates, the inflation gaps and the unemployment gaps separately, then the correlation rises to 0.34 for the UK and 0.58 for Canada. For Germany, the posterior mean correlation amounts to 0.48. The statistics broadly confirm our conclusions using model selection criteria. Quantitatively, however, the results are relatively modest. For the nominal exchange rate, the correlation is generally higher, reflecting that we match the trends for Japan and Switzerland particularly well. But the correlation is also substantial for Canada, where the nominal exchange rate does not exhibit a strong secular trend.
For changes in exchange rates, the model does not outperform the benchmark, while the posterior mean correlation is usually higher for the UC-SV model when compared with the benchmark. In fact, the percentiles always include zero for both specifications. This suggests that we mainly capture the major exchange rate movements while monthto-month movements are not very well captured. This also supports Chinn and Meredith (2004) and Chinn and Quayyum (2012), who find that UIP holds better in the long run rather than in the short run.
If our model works well because of changes in trend inflation, then we expect that the correlations differ mainly in the period before 1990. 10 Table 4 reports the correlations with the actual real exchange rates separately for the two time periods. Except for Canada, the correlation of the UC-SV prediction is higher before 1990 than afterward. In addition, except for Japan, the increase relative to the benchmark is higher before 1990. This suggests that our model works well mainly because we capture changes in trend inflation. An important aspect for an exchange rate model to match is the high persistence or near random walk properties of the real exchange rate. Table 5 shows the sample autocorrelations up to third order for the actual real exchange rate along with the autocorrelation of the posterior means of the predictions. The benchmark model already implies a highly persistent real exchange rate. Nevertheless, the persistence of the exchange rate based on the benchmark model is lower than that of the actual real exchange rate for all countries and all lags. The UC-SV model is capable of explaining the higher persistence of the real exchange rate and matches the actual persistence more closely. The only place where the model does not substantially improve over the benchmark is Canada.
Similar to the previous results, we do not make much progress matching the persistence of exchange rate changes. In the actual data, the first-order autocorrelation is larger than zero for all countries. By contrast, the benchmark model implies a negative first-order autocorrelation for exchange rate changes. Although the UC-SV model often yields a first-order autocorrelation closer to zero, the improvement over the benchmark is modest. As a further check of whether the exchange rate model predicts the real exchange rate with reasonable properties, we compare correlations of the real exchange rate with the fundamentals. Table 6 shows that the model closely matches the correlation between the actual exchange rate and the fundamentals. The posterior mean is close to the actual correlation, and the 5th and 95th percentiles mostly include the actual value. The actual value of the correlation lies outside the percentiles for inflation in Japan, Canada and Switzerland, and long-term bonds in Germany and Switzerland. However, even for these correlations the sign of the posterior mean is consistent with the sign of the actual correlation. For the benchmark model, the correlations also tend to track the actual relationship between the exchange rate and its fundamentals rather well.
Finally, we performed robustness checks with respect to the model specification (the results are available in the Online Appendix). Replacing the stochastic trends with an estimate from the Hodrick and Prescott (1997) filter yields similar correlations between the actual and predicted real exchange rate as in the UC-SV model. 11 Then we limited the sample to the pre-euro era for Germany. The main results are robust. We even find a somewhat higher correlation of our prediction and the actual real exchange rate of 0.7. For a similar sample, Engel and West (2006) find a correlation between their prediction and the real DEM/USD exchange rate of 0.3. Moreover, we also assess the impact of not only allowing for SV in terms of measures of model adequacy like the DIC but also how results change if we turn off heteroscedasticity. In that case, results change only quantitatively, with the vast majority of findings remaining valid.

IV. CLOSING REMARKS
Recent research has documented that trend inflation changes over time. We add an international dimension to this line of research and highlight that changes in trend inflation explain important aspects of exchange rate dynamics. We develop a multivariate UC-SV model that is theoretically motivated by assuming that central banks follow Taylor rules, but the inflation target as well as the natural rate of unemployment may change over time.
The UC-SV model succeeds in capturing major upturns and downturns of the real US dollar exchange rate against the currencies of six economies. In fact, the correlations of the model predictions with the actual real exchange rates are higher than in existing studies. While a benchmark model performs comparatively well, the improvements obtained by explicitly discriminating between non-stationary trend and stationary gap components are significant for all currencies under consideration. Looking at nominal exchange rates reveals that we are able to accurately reproduce major exchange rate trends observed over the last 40 years. Finally, the model successfully captures several key time series characteristics commonly found for real exchange rates. More specifically, we accurately reproduce the persistence of the real exchange rate and its correlation with other macroeconomic variables. As one would expect, however, accounting for changes in trend inflation has a more important effect for the pre-1990 period. Our discussion shows that although the model explains a larger share of exchange rate fluctuations than previous studies, it fails during episodes when the Taylor rule is unlikely to be an accurate description of the central banks' conduct of monetary policy. Improving the model predictions by accounting for unconventional monetary policy actions and constraints on the operational targets of central banks might be a promising avenue for future research.