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On the Driving Forces of Real Exchange Rates: Is the Japanese Yen Different?

We estimate variance decompositions of the real exchange rates for 19 individual currencies based on a present-value relation. We document substantial heterogeneity across forecasting horizons and currencies regarding the determinants of exchange rates. At short horizons, the dominant force is predictability of the future spot rates. At long horizons, return predictability drives most of the variation in exchange rates, with predictability of interest differentials playing a secondary role. However, for the Japanese Yen the main determinant is long-run interest spread predictability, a pattern that occurs in the most recent period. We develop a liquidity-based exchange rate model to explain such evidence. Agents value the liquidity benefit of near-money assets and have non-separable utility over consumption and liquidity services. Cross-country differentials in liquidity demand determine exchange rates and affect both interest differentials and risk premiums in a way consistent with the data. The quantitative simulation largely replicates our empirical findings.


Introduction
What moves asset prices and returns? This fundamental question has received substantial attention from academic research. The rational asset pricing framework unambiguously dictates that changes in asset prices or returns should reflect changes in expected future fundamentals and/or discount rates. Earlier attempts such as Campbell and Shiller (1988), Campbell (1991), and Cochrane (1992) test this powerful prediction using aggregate stock market data. Subsequent research extends the analysis to individual stocks or segments of the stock market (Vuolteenaho (2002), Maio (2014), and Maio and Santa-Clara (2015)), bonds (Campbell and Ammer (1993)), commodities (Pindyck (1993)), and international stock markets (Ammer and Mei (1996) and Rangvid, Schmeling, and Schrimpf (2014)). In the context of the foreign exchange market, the largest financial market in the world, Froot and Ramadorai (2005) decompose changes in currency returns into real interest rate differential (cash flow) shocks and expected return shocks. They find that these two forces are negatively correlated and such decomposition is useful to rationalize the long-observed weak connection between macroeconomic fundamentals and exchange rates, the so-called "Exchange Rate Disconnect Puzzle" (see e.g., Meese and Rogoff (1983)). Indeed, the cash-flow shocks are closely tied to the fundamentals, yet the literature typically documents a major role for the risk premium force in explaining currency movements.
Our paper contributes to the international finance literature by studying comprehensively the forces driving exchange rates. Specifically, we define a variance decomposition for the log real exchange rate (q) based on the dynamic present-value relation derived in Froot and Ramadorai (2005). This present-value relation is similar to the widely used decomposition of the dividend yield derived by Campbell and Shiller (1988) (although it is exact rather than approximate). According to this identity, the current log exchange rate is positively correlated with future multiperiod log interest rate spreads and the exchange rate at some terminal date, and negatively correlated with future multiperiod log currency returns. By using this present-value relation, we define a variance decomposition for the log exchange rate where the slopes represent the fraction of the variance of the current exchange rate attributed to interest spread, return, and future exchange rate predictability. The variance decomposition for the exchange rate is based on a restricted first-order VAR, in which the only predictor is the real exchange rate. Under this approach, the coefficients for future returns, interest rate spreads, and exchange rate at multiple horizons are mechanically related with the corresponding one-period slopes. 1 We estimate a term-structure of variance decompositions in order to account for the different predictability patterns at short, intermediate, and long forecasting horizons. In the empirical analysis, we use the traditional G10 currencies as well as nine currencies from other developed countries and emerging/developing economies. The sample period is from 2004:04 to 2015:06. Our results indicate that the major force driving the variation in the current real exchange rate of most currencies is return predictability, with interest spread predictability assuming a secondary role. This pattern is especially notorious at long forecasting horizons: More than 76% of the variation in q is attributable to forecasting long-run returns, implying a maximum share for interest spread predictability of 24%. The major outlier to this pattern is the Japanese Yen (JPY), which is one of the most actively traded currencies and the main funding currency in the international financial markets. For that currency, the dominant driving force of the real exchange rate is predictability of future interest differentials (rather than return predictability), and this pattern is especially notorious at long horizons: more than 100% of the variation in the real exchange rate is due to predictability of future long-run interest spreads, which means that return predictability goes with the wrong sign. Hence, this predictability mix is in sharp contrast to those estimated for the other currencies. We also observe that predictability of the future exchange rate is the dominant driving force of variation in the current exchange rate at short horizons for all currencies. However, this effect dies off at a relatively fast pace for most currencies.
Variance decompositions estimated under less restrictive VAR specifications (unrestricted first-order VAR and restricted second-order VAR) yield qualitatively similar results to those obtained under the benchmark decomposition (based on a restricted first-order VAR). We also conduct Monte-Carlo and bootstrap simulations in order to obtain more robust statistical inferences for the estimated shares in the variance decompositions. The results from both simulations indicate that the shares of interest spread predictability, although substantially smaller than the shares of return predictability, are statistically significant for most of the G10 currencies. However, such evidence is much weaker among the non-G10 currencies.
We conduct variance decompositions for portfolios of currencies. The results suggest that there is a larger and more statistically significant share of interest spread predictability among G10 currencies in comparison to non-G10 currencies, while in the case of return predictability we observe the opposite pattern. Specifically, the long-run return slopes are −0.83 and −1.02 for the portfolios of G10 and other currencies, respectively. This provides further evidence that the currency markets of the G10 and non-G10 currencies behave in a quite different way.
We also estimate a variance decomposition for each of the G10 currencies by using a longer sample that starts in 1990. The results suggest there is some degree of time-variation in the estimation of the decomposition for the real exchange rate. This pattern is especially notorious in the case of the JPY, in which the variance decomposition estimated for the long sample yields quite an opposite predictability mix to that estimated for the shorter period.
Hence, the dominant share of interest spread predictability for JPY seems to be concentrated in the recent period. On the other hand, there is a slightly larger degree of interest spread predictability among the other G10 currencies over the longer period.
The newly documented evidence in this paper raises a significant challenge to the existing international asset pricing models, such as the habit model of Verdelhan (2010) or the long-run risk model of Bansal and Shaliastovich (2013). Despite their success in accounting for the puzzling behavior of currency risk premium, they fail to depict correctly the forces governing exchange rate dynamics. As also pointed out by Balduzzi and Chiang (2018), these models imply that the risk premium shocks account for over 100% of exchange rate movements, which cannot be consistent with the sizable role of cash-flow channel we documented for e.g., JPY. Simultaneously, the overstatement of risk premium predictability implies that the real exchange rate forecasts negatively the future interest spread. Nevertheless, we find that such predictive relation is positive for almost all currencies analyzed.
With the discrepancy in hand, we propose and solve a new exchange rate model based on the model of liquidityà la Nagel (2016). Liquidity-based resolution of exchange rate puzzles has gained popularity in the recent literature (see Gabaix and Maggiori (2015), Mueller, Tahbaz-Salehi, and Vedolin (2017), Fang and Liu (2018), and Sandulescu, Trojani, and Vedolin (2019)), yet it has not studied the variance decomposition of real exchange rates.
As the main ingredients of our model, we assume that investors value the liquidity benefit provided by the domestic near-money assets such as the domestic Treasury-bills, similar to Nagel (2016). A key novelty of our framework is that investors have non-separable utility over consumption and liquidity services, which dictates that the stochastic discount factor (SDF) is directly linked to the marginal utility of liquidity services. When the consumption and liquidity are substitutes, higher foreign-minus-home marginal utility of liquidity increases the relative SDF, appreciates the foreign currency, and decreases the risk premium. Higher marginal utility of consumption lowers the saving motive of the agent and thus pushes up the equilibrium interest rate, which leads to a higher foreign-minus-home interest rate spread. This helps explain the positive (negative) interest spread (risk premium) predictability we found for many currencies. However, when the elasticity of substitution between consumption and liquidity is sufficiently low (so that consumption and liquidity services become complements), the risk premium predictability (from the real exchange rate) can even become positive, as is the case for JPY.
Our model is highly tractable and in contrast to the habit or the long-run risk model, it delivers correct implications on the predictability mix that we document for a large basket of currencies. After calibrating the model to match several key moments of exchange rates and macroeconomic quantities for the G10 currencies, we show that our model is able to quantitatively match the predictability mix found in the data. The long-run slopes of interest spread predictability are all positive, while the risk premium coefficients are mostly negative.
In particular, we find that the distribution of the risk premium long-run slopes associated with JPY has a nontrivial probability of taking positive values based on simulated data, consistent with our empirical results.
Our work is related to the long literature on the time-series predictability of currency returns, and more broadly, on what drives exchange rates at multiple forecasting horizons.
A large portion of this literature has focused on the relation between exchange rates and macro fundamentals, such as Meese and Rogoff (1983), Mark (1995), Mark and Sul (2001), Rapach and Wohar (2002), Engel and West (2005), Abhyankar, Sarno, and Valente (2005), Colacito, Riddiough, andSarno (2019). The evidence in these studies usually consists of a weak relation between currencies and macro fundamentals. This is often interpreted as a fundamental failure of standard models of exchange rates and it has even been described as the major weakness of international macroeconomics. Yet, Engel, Mark, and West (2007) show that theoretical models imply near-random walk behaviour for short-term exchange rates. Furthermore, they show that the forecasting power of these models improves considerably when using panel regressions and long-horizon forecasts. Subsequently, other successful predictors of currency returns have been proposed. Examples include past forward interest rate differentials (Boudoukh, Richardson, and Whitelaw (2016)), yield curve information (Chen and Tsang (2013)), external imbalances (Della Corte, Sarno, and Sestieri (2012)), commodity productions (Ready, Roussanov, and Ward (2017)), monetary fundamentals (Eichenbaum, Johannsen, and Rebelo (2017)), peso event risk (Burnside, Eichenbaum, Kleshchelski, and Rebelo (2011) and ), and return-based factors Verdelhan (2011, 2014)).
Our methodology is in line with the branch of the literature that applies the return decomposition of Campbell and Shiller (1988) in the context of the foreign exchange market. Froot and Ramadorai (2005) employ a decomposition of the real exchange rate into cumulated future real interest differentials (the permanent component) and future expected return innovations (the transitory component). By expressing unexpected currency returns as the difference between "cash-flow" news and "expected return" news, their goal is to study the interaction between exchange rates, investor flows, and fundamentals. They find that investor flows are important in understanding transitory elements of currency returns, but not the long-run currency value. More recently, Atanasov and Nitschka (2015) also use the Campbell and Shiller (1988) decomposition to understand currency returns. They first decompose the return on the stock market into cash-flow and discount-rate news and then calculate the sensitivities of currency returns to these two stock return components. They find that the sensitivities to aggregate cash-flow news are significantly related to average excess returns on foreign currency portfolios. Engel (2016) uses a similar present-value relation, but he evaluates predictability from interest rate differentials rather than from the real exchange rate.
More directly related to our paper is the contemporaneous and independent work of Balduzzi and Chiang (2018), who also extend the analysis of Cochrane (2008) to currency returns by conducting long-run variance decompositions for the real exchange rates. 2 However, there are several key differences in the empirical design of the two papers, which enable us to reach several major conclusions in our study, which are absent from the study of Balduzzi and Chiang (2018). 3 First, the JPY represents a major outlier relative to the other currencies by having a dominant role for predictability of interest rate differentials in the long-run. Second, there is significant interest predictability among G10 currencies (while the same does not occur for the other currencies) and there is also stronger evidence of return predictability among the non-G10 currencies. Third, the predictability mix changes substantially with the forecasting horizon. In particular, at short horizons, what drives variation in the exchange rate is its own predictability. Fourth, there is evidence of instability over time in the estimation of the variance decompositions for the G10 currencies, and this is specially notable in the case of the JPY. The two papers also differ on the theoretical framework used to explain the empirical findings. Balduzzi and Chiang (2018) rely on the habit and the long-run risk models to explain their results on risk premium predictability, while we propose a new model based on liquidity frictions. Critically, our theoretical model is able to replicate the documented predictability pattern for both the interest spread and risk premium across most currencies, both qualitatively and quantitatively, which is not the case in Balduzzi and Chiang (2018).
The rest of the paper is organized as follows. Section 2 describes the methodology and data.
Section 3 presents the main empirical results, while Section 4 discusses results for alternative VAR specifications. Section 5 provides additional results for variance decompositions. In Section 6, we present a theoretical model to explain our results and explore its quantitative 2 The first draft of Balduzzi and Chiang (2018) appears in SSRN at May 2014, while our first draft appears in September of that year.
3 Specifically, we compute a variance decomposition for the real exchange rate currency-by-currency for 19 currencies, including non-G10 currencies. Balduzzi and Chiang (2018) only provide long-run return coefficient estimates for a selected number of G10 currencies, as shown in their Table 1. Moreover, Balduzzi and Chiang (2018) do not estimate the implied long-run shares of interest spread predictability, and hence we cannot evaluate the statistical significance of the implied interest channel in their variance decompositions. Assessing the contribution of both the risk premium channel and the interest spread channel in driving exchange rates represents a key aspect of our study (and this is especially important in the case of JPY). Further, we compute a term-structure of decompositions across several forecasting horizons, and we also conduct variance decompositions for portfolios of G10 and non-G10 currencies, unlike their study.

Methodology
Consider the gross return associated with a zero-cost investment strategy in foreign currency, in which S t denotes the spot exchange rate at time t (units of domestic currency per unit of foreign currency); I t+1 is the domestic short-term interest rate between t and t + 1, which is known at the beginning of the period; and I * t+1 represents the foreign interest rate. By applying logs to the previous identity, we obtain, where r t+1 ≡ ln(1 + R t+1 ) is the log currency return; s t+1 ≡ ln(S t+1 ) denotes the log exchange rate at the end of t + 1; and dn t+1 = ln(1 + I * t+1 ) − ln(1 + I t+1 ) ≡ i * t+1 − i t+1 represents the differential of log nominal interest rates.
We can restate the excess currency return in terms of real variables, where q denotes the log real exchange rate and d t+1 ≡ dn t+1 − π * t+1 + π t+1 represents the differential of real interest rates, with π * t+1 and π t+1 denoting the foreign and domestic inflation rates (difference in the log of price index) between t and t + 1, respectively.
Following Campbell andClarida (1987), Froot andRamadorai (2005), and Engel (2016), we can obtain a present-value relation for the real exchange rate at a given forecasting horizon K: According to this dynamic present-value relation, it follows that variation in the current log real exchange rate stems from variation in future currency log returns, future log real interest rate differentials, and/or the log real exchange rate at some future date. Specifically, the current log exchange rate is positively correlated with both future multiperiod log interest rate spreads and the exchange rate at some terminal date, while being negatively correlated with future log currency returns. Hence, this relation represents a valid benchmark to analyse predictability in currency markets, especially at long forecasting horizons. This decomposition is also similar to the Campbell and Shiller (1988) decomposition associated with the log dividend-to-price ratio in equity markets, except the fact that it holds exactly rather than approximately.
By employing the present-value relation above, we can define a variance decomposition for the log real exchange rate at each forecasting horizon K: In this equation, b K d , b K r , and b K q represent the fraction of the variance of the current exchange rate attributed to interest spread, return, and future exchange rate predictability, respectively.
Estimating a term-structure of variance decompositions (that is, a decomposition for each forecasting horizon K) enables us to account for the different predictability patterns at short and long horizons. Under this variance decomposition, if the shares associated with the predictability of interest rate spreads and the future exchange rate are small or close to zero, this reinforces the evidence of return predictability from the real exchange rate since the three slopes have to sum to one. In other words, since the real exchange rate varies over time, there must be at least one among the three sources of predictability that drives the exchange rate. 4 To implement the variance decomposition, we follow Cochrane (2008) in employing the 4 Cochrane (20084 Cochrane ( , 2011 and Maio and Xu (2018) use similar variance decompositions for the market dividend yield and earnings yield, respectively, while Maio and Santa-Clara (2015) apply a variance decomposition for portfolio dividend yields. In contrast to these cases, this variance decomposition for the exchange rate holds exactly rather than as an approximation (first-order Taylor expansion).
following restricted first-order VAR, The full derivation of this variance decomposition is shown in the appendix. The presentvalue relation for the log real exchange rate motivates q t for being the only state variable (predictor) in the VAR. In Section 4 below, we explore less restrictive, and likely more robust, VAR specifications. The t-statistics for the predictive coefficients at horizon K, , are based on the t-statistics associated with the VAR slopes above by using the delta method. The standard error formulas are presented in the appendix. To assess the statistical significance of the VAR coefficient estimates, we use Newey and West (1987) t-ratios (computed with one lag). The VAR estimation is by multiple-equation OLS (see Hayashi (2000)).
We note that the first-order VAR represents only an approximation of the true long-run dynamics that govern the variables in the predictive system above, and thus, may produce wrong estimates of the implied predictive slopes, especially at long horizons. An alternative method to estimate the variance decomposition, which does not suffer from that shortcoming, is the direct approach in which one runs long-horizon regressions of future interest spreads and returns on the current real exchange rate: The minus side of this method is the sharp decrease in the number of usable observations at long horizons and the resulting low statistical power of the long-horizon regressions. In our case, given the small size of the sample, the issue of low statistical power in the long-horizon regressions is particularly severe, hence we opted to use the indirect approach. 5 By allowing K → ∞ and imposing the terminal condition lim K→∞ q t+K = 0 (that is, Purchased Power Parity (PPP) holds in the long-run), we obtain a long-run (infinite horizon) dynamic decomposition for q, which represents the present-value relation derived in Froot and Ramadorai (2005) and Engel (2016).
The corresponding long-run VAR-based variance decomposition for q is given by As in the case of the horizon-specific coefficients, the t-ratios for the long-run slopes are obtained by applying the delta method to the standard errors associated with the VAR slope estimates.

Data
We use monthly data on spot and one-month forward exchange rates (amounts of U.S. dollars per unit of foreign currency) associated with ten currencies from developed countries (denominated as G10 currencies)-Canadian Dollar ( The exchange data are originally from Datastream and represent a subset of the data employed in Barroso, Kho, Rouxelin, and Yang (2018). The price index data for U.S. and the 19 countries indicated above are obtained from the OECD/MEI database. We use the German price index to proxy for the price level in the Euro area.
The log real exchange rate is given by where P * and P denote the price indices for the foreign country and U.S., respectively.
The differential of log real interest rates is computed as where π * t+1 and π t+1 are the inflation rates for the foreign country and U.S., respectively, with π t+1 = ln(P t+1 /P t ). f t+1 is the log forward exchange rate for t + 1, which is known at time t.
Based on the data for q and d, the monthly currency log returns are computed from equation (3) above.
The descriptive statistics for the log real exchange rate (q), log return (r), and log interest rate differential (d) associated with the G10 currencies are shown in Table 1, while Table 2 presents the same statistics for the other nine currencies. We can see that currency returns are substantially more volatile than the corresponding interest rate differentials across all 19 currency pairs. On the other hand, the interest rate spreads show higher serial correlation (with autoregressive slopes above 0.30 in the cases of JPY, CZK, HUF, MXN, PLN, KRW, ISK, TRY, and CLP) than currency returns (with autocorrelation coefficients very close to zero in nearly all cases). Yet, the log real exchange rates are significantly more persistent than interest differentials, with autoregressive coefficients above 0.90 in all cases. We can also see that the interest spreads are more persistent among the non-G10 currencies than among the G10 currencies.
In terms of contemporaneous correlations, r and q are positively correlated across all currencies, with the largest correlations holding for CHF, JPY, SEK, DKK, NOK, NZD, EUR, HUF, MXN, and KRW (correlations above 20%). On the other hand, d and q also have positive, yet more modest, correlations across most currencies, with the largest degree of association occurring for CHF, GBP, JPY, KRW, and CLP. In comparison, the correlations between r and d are substantially smaller across most of the currencies, the exceptions being the cases of GBP and CLP in which these two variables are positively associated.

What drives exchange rates?
In this section, we estimate the benchmark variance decomposition for the log real exchange rate associated with each of the 19 foreign currencies.

Main results
The estimation results for the restricted first-order VAR applied to each of the 19 currencies are presented in Table 3. The one-year return and interest rate differential slopes are negative and positive, respectively, for most currencies, which is consistent with the present-value relation presented in Equation (4). The few exceptions are the positive estimate of λ r in the case of JPY and the negative estimates of λ d in the cases of TRY and ILS. We can see that the magnitudes of the return slopes are significantly above the magnitudes of the corresponding interest spread coefficients. The sole exception holds for the Japanese Yen. Despite the larger magnitudes, it turns out that for about half of the currencies the estimates of λ r are not statistically significant at the 10% level: CHF, GBP, JPY, SEK, DKK, NOK, AUD, EUR, KRW, and ISK. This stems from the relatively large noise associated with currency returns, as discussed in the previous section. In comparison, there are nine currencies in which there is no significance for the interest differential coefficients: SEK, NZD, CZK, MXN, PLN, KRW, ISK, TRY, and ILS. Excluding the unusual case of the JPY, the return slopes vary between −0.024 (ISK) and −0.093 (MXN). On the other hand, excluding the already mentioned cases of TRY and ILS, the range for the interest differential coefficients is between 0.003 (CZK) and 0.015 (CHF), with this last estimate being significant at the 1% level.
The evidence for the autoregressive coefficients indicate that all exchange rates are persistent variables: the estimates of φ are above 0.90 and strongly significant (1% level) for all currencies. Specifically, the range of these estimates is between 0.900 (MXN) and 0.993 (JPY). This suggests the possible existence of a unit-root in the case of the Japanese Yen.
Not surprisingly, the R 2 values in the equations for the real exchange rate are significantly above the counterpart estimates obtained in the return and interest spread equations. The range of the explanatory ratios is between 0.78 (MXN) and 0.95 (JPY). As discussed in the previous section, the magnitudes of the restricted VAR slopes represent the weights in the variance decomposition for q at the one-month forecasting horizon. Therefore, what drives variation in the current real exchange rate of all currencies is mainly its own predictability one-month ahead. The predictability of the future monthly return plays a secondary role, with shares that vary between 2.4% (ISK) and 9.3% (MXN). On the other hand, interest spread predictability assumes a rather marginal role in quantitative terms at the one-month horizon, despite the fact that several of these slope estimates are statistically significant: the largest share among all currencies is only 1.5% and holds for the CHF.
We are particularly interested in evaluating predictability at long horizons. The reason hinges on the present-value relation for q, according to which, the log exchange rate should be a stronger predictor of either returns or interest rate differentials at long horizons. The results associated with the long-run (infinite horizon) variance decomposition for q are displayed in Table 4. It turns out that long-run return predictability is the main driver of variation in the current log exchange rate of most currencies. Indeed, excluding the particular case of JPY, the long-run return slopes vary between −0.76 (GBP) and −1.15 (TRY), and these estimates are strongly significant (1% level) in all 18 cases. In comparison, and again excluding the case of JPY, the long-run interest differential slopes vary between −0.15 (TRY) and 0.24 (GBP), and none of these 18 estimates are significantly different from zero (at the 10% level). 7 Actually, in the cases of both TRY and ILS, the long-run return coefficients are above one (in magnitude) due to the incorrect sign (negative) of b lr d . Therefore, for those 18 currencies we have that the major driving force of the real exchange rate is return predictability: more than 76% of the variation in q is attributable to forecasting long-run returns. In fact, we cannot reject the null that the estimates of b lr r are equal to −1 at the 10% level, that is, we cannot reject the assertion that there is only return predictability at long horizons for all 18 currencies. Looking across currencies, we tend to observe larger magnitudes of the long-run return slopes (and hence, lower magnitudes for the interest spread coefficients) among non-G10 currencies in comparison to the G10 currencies: for five of the currencies in the first group the magnitudes of b lr r are above 0.90 (CZK, MXN, PLN, TRY, and ILS), compared to only one currency (CAD) in the second group.
The Japanese Yen represents the clear outlier to the pattern described above. Indeed, the estimates for b lr r and b lr d are 0.60 and 1.60, respectively, which suggests that more than 100% (160%) of the variation in q comes from predictability of future interest rate differentials (in result of the positive VAR return coefficient). However, both point estimates are largely insignificant, that is, we cannot reject the null that the long-run interest spread coefficient (of 1.60) is equal to zero in statistical terms (t-ratio of 0.37). We also cannot reject the null that the return slope is equal to −1. In other words, we cannot differentiate (in statistical terms) between a scenario of 100% long-run return predictability and an exactly opposite scenario of 100% long-run interest predictability in the case of JPY. This stems from the large standard errors associated with the long-run predictive slopes in the case of JPY, which in turn arise from the large standard errors of the one-period VAR return slope estimate, combined with a very large persistence in q. Therefore, the long-run variance decomposition for q is very imprecisely estimated (based on the asymptotic standard errors) in the case of the Japanese Yen, and is clearly at odds with the predictability mix obtained for the other 18 currencies.
Although the long-run statistics show the variance decomposition at a very-long horizon, we are also interested in the "term structure" of such variance decompositions. More specifically, it is important to assess the "speed" at which the predictability mix associated with each currency converges from the one estimated at the one-month horizon to that obtained in the long run. Tables 5 (for the G10 currencies) and 6 (for the other currencies) present the variance decompositions estimated at selected forecasting horizons: one, 12, 60, 120, and 240 months ahead. Excluding the already mentioned case of JPY, for 11 of the 18 currencies, the return slopes assume the larger magnitudes (among the three variables) already at the one-year horizon (K = 12). In other words, return predictability is the major source of variation in q at that horizon. For the remaining currencies (GBP, SEK, NOK, AUD, CZK, ISK, and ILS), the dominant role of return predictability is established at the 5-year horizon (K = 60). We can also see that the return and interest slopes at the 120-month horizon are nearly identical to the long-run coefficients presented in Table 4. This implies that the slopes for future q at long horizons are around zero for nearly all currencies. This means that PPP holds in the long-run.
In terms of statistical significance, the return slopes are significant at the 10% level (or better) at K = 12 for the majority of the currencies. The few exceptions hold for GBP, NOK, and ISK. At K = 60, the return coefficients are strongly significant (1% level) across most currencies, the sole exception being ISK (in which case there is significance at the 10% level). Therefore, the evidence is that, apart from very short forecasting horizons, the predictability mix for most currencies is qualitatively similar to that obtained in the long-run, that is, the driving force of variation in q is return predictability.
Again, the major outlier to the pattern just described is the JPY. The return coefficients are positive at all forecasting horizons, yet all of these estimates are largely insignificant. More importantly, the interest spread slopes are statistically significant for horizons until K = 60.
At such horizon, the share of interest predictability is only slightly lower than the share of exchange rate predictability (54% versus 66%). However, at longer horizons (K = 120, 240), the estimates for b K d are clearly insignificant despite assuming the largest magnitudes (around or above 0.90), which is in line with the long-run decomposition discussed above. Therefore, at short and middle forecasting horizons, the main driver of q is the predictability of the exchange rate itself, while predictability of future interest differentials also assumes an important role, especially at intermediate horizons (K = 60). However, at longer horizons such effect dies off, as none of the shares of variation in q is statistically significant. Overall, this suggests that the predictability of future d from the current exchange rate in the case of JPY appears to be a transitory phenomenon (at least in statistical terms).

Simulation
In the asset return predictability literature, substantial attention has been devoted to the poor small-sample properties of long-horizon predictability and whether the asymptotic inference is valid when assessing the statistical significance of the slopes in those regressions (see Valkanov (2003), Torous, Valkanov, andYan (2004), and Boudoukh, Richardson, and Whitelaw (2008), among others for a related discussion). This issue becomes particularly important when the predictor is a persistent variable as is the case with the log real exchange rate. Indeed, the results above suggest that q is highly persistent and this pattern is specially notorious for the case of JPY, with a first-order autoregressive coefficient very close to one.
The existence of a unit-root in the predictor makes invalid the asymptotic inference in order to assess the statistical significance of the VAR slope estimates (e.g., Hayashi (2000)). Moreover, as shown in the appendix, the standard errors for the implied K-period return and interest spread slope estimates will be quite large if the real exchange rate is very persistent. To address this issue, we conduct both a Monte-Carlo simulation and a bootstrap simulation of the restricted VAR model estimated above.
To assess the statistical significance of currency return predictability, we impose a null hypothesis in which the log real exchange rate does not forecast the future currency return.
That is, under this null, all the variation of the exchange rate comes from predicting the future interest rate differential. Thus, we simulate the first-order VAR by imposing the restrictions (in the predictive slopes and residuals) associated with this null hypothesis: where all the variables are demeaned.
To assess predictability of future interest rate differentials, we simulate an alternative VAR: In this second VAR specification, we impose a null hypothesis in which the log exchange rate does not forecast the future interest rate differential. In other words, all the variation of the log exchange rate comes from predicting the future currency return. In related work, Cochrane (2008) and Maio and Santa-Clara (2015) conduct similar VAR simulations to assess the predictability of the dividend yield for future stock returns and dividend growth.
We conduct both a Monte-Carlo and a bootstrap experiment associated with each of the null hypotheses (e.g. VAR) specified above, for a total of four simulations. In the case of the Monte-Carlo, in drawing the VAR residuals (10,000 times), we assume that they are jointly normally distributed and use the corresponding sample covariance matrix. We use the same sample size as in the original sample in constructing the artificial data. The log exchange rate for the base period is simulated as In the case of the bootstrap, we draw the VAR residuals with replacement from the original VAR estimates discussed above. The log real exchange rate for the base period is chosen randomly from the original time-series of q t . The key advantage of the bootstrap over the Monte-Carlo is in avoiding the restrictive assumption that the variables in the system are jointly normally distributed.
In both the Monte-Carlo and bootstrap experiments, we compute the pseudo p-values associated with the implied VAR return slopes for each horizon, which represent the fraction of simulated estimates for the return coefficients (from the simulations associated with the no-return predictability null) that are lower than the estimate found in the data. Likewise, the p-values for the interest spread slopes represent the fraction of pseudo estimates of the interest coefficient (obtained from the simulations under the no-interest spread predictability hypothesis) that are higher than the sample estimate. 8 The Monte-Carlo-based p-values at selected forecasting horizons (K = 1, 12, 60, 120, 240) are presented in Table 7. In Panel A, we can see that the return coefficients at forecasting horizons equal or longer than 60 months are statistically significant (at the 5% or 1% level) for most currencies. The exceptions occur for JPY and ISK, in which cases the return coefficients are not significant at the 10% level. In the case of GBP, there is significance at the 10% level at those horizons. Hence, the key difference relative to the inference based on the asymptotic t-ratios occurs for ISK since there is no significance for the corresponding return slopes at long horizons, in contrast to the asymptotic p-values below 1%. In the case of GBP, the significance is also much weaker than that implied by the asymptotic t-ratios (10% versus 1% level).
The p-values associated with the interest spread coefficients (shown in Panel B) indicate strong significance (at the 5% or 1% level) at K ≥ 60 for several currencies: CHF, GBP, JPY, DKK, NOK, EUR, HUF, KRW, and ISK. In the cases of CAD, SEK, and AUD, those slope estimates are significant at the 10% level. Hence, only for remaining seven currencies (NZD, CZK, MXN, PLN, TRY, CLP, and ILS) it turns out that the coefficients for interest differentials are not significant at the 10% level. These results are in sharp contrast with the results above showing that, according to the inference based on the asymptotic t-ratios, there are virtually no cases of statistical significance at the 10% level (the few exceptions occur for CHF, JPY, and AUD at K = 60). Therefore, it turns out that for several currencies the shares of interest predictability, although substantially smaller than the return counterparts, are statistically significant. This is especially notable in the cases of CHF, JPY, and KRW, in which the interest spread coefficients for horizons at and beyond 60 months are significant at the 1% level. These results also suggest that the shares of interest predictability tend to be more significant among the G10 currencies than among the other currencies.
The p-values associated with the bootstrap simulations are reported in Table 8, which  is similar to Table 7. Regarding the significance of the return slopes, we obtain similar conclusions to the Monte-Carlo-based inference: At intermediate and long horizons, the return slopes are significant (at the 5% level) for most currencies. The exceptions are ISK and JPY, while in the case of GBP there is significance at the 10% level. On the other hand, the bootstrap-based p-values point to stronger significance of the interest spread coefficients than in the case of the Monte-Carlo simulation for K ≥ 60: Only in five cases (CZK, MXN, PLN, TRY, and ILS), those estimates are not significant at the 10% level. In all other cases, we have significance either at the 5% level (which holds for most currencies) or at the 10% level (as in the cases of SEK and NZD). This further suggests that the evidence of interest predictability is substantially weaker among the non-G10 currencies.
In sum, the simulation results suggest that expectations of currency fundamentals (interest spreads) are likely a more important partial determinant of the exchange rate than the evidence provided from asymptotic statistical inference. In particular, the variance decomposition for the JPY is estimated with much more statistical precision than what the asymptotic statistical inference would imply.
All in all, the results of this section indicate that what drives the variation in the current real exchange rate of most currencies is return predictability, with interest spread predictability assuming a secondary role. This pattern is especially notorious at long forecasting horizons and seems to be stronger among non-G10 currencies (in comparison to developed currencies).
Indeed, the shares of return predictability are either very close to one, or above one, for several of the Non-G10 currencies, with the later case indicating that the interest spread coefficients have the wrong sign. The major outlier to this pattern is the JPY, in which case the dominant driving force of the real exchange rate is predictability of future interest differentials rather than return predictability, and this pattern is especially notorious at long horizons. We also observe that predictability of the future exchange rate is the dominant driving force of variation in the current exchange rate at short horizons for all currencies, yet this effect dies off at a relatively fast pace. Again, the major exception is the JPY due to the very large persistence of the real exchange rate.
Our results provide a justification for the failure of UIP in time-series tests as found in Hansen andHodrick (1980) or Fama (1984). Theoretically, the UIP holds in a world of risk-neutral rational investors and frictionless foreign exchange markets (see, for example, Sarno and Taylor (2003), Chapter 2). In a more general setting of risk-averse investors, rationality is also compatible with currencies having risk premiums (Grauer, Litzenberger, andStehle (1976), Stockman (1978)). If currency risk premiums are constant over time, forward premiums would still predict offsetting changes in exchange rates, as the premium is captured by the intercept of time-series regressions, so results as those of Fama (1984) should not occur. Our results show, though, that exchange rates are mainly driven by time-varying risk premiums. In this setting, UIP does not need to hold at all, even theoretically, and so empirical rejections of the hypotheses are expectable.

Alternative variance decompositions
In this section, we estimate alternative variance decompositions for q, which are based on less restrictive VARs in comparison to the baseline variance decomposition computed in the last section.

Unrestricted VAR
In the first case, we consider an unrestricted VAR(1): This specification accounts for possible relevant effects of lagged returns and interest spreads on all three variables in the VAR, something that the benchmark VAR ignores. The VAR above can be rewritten in matrix form as follows, or equivalently, where the last equation defines the variables of interest.
The benchmark VAR represents a special case of this specification, in which the matrix of transition coefficients is equal to Define the indicator vectors, e r ≡ (1, 0, 0) , e d ≡ (0, 1, 0) , and e q ≡ (0, 0, 1) , which define the position of each state variable in the VAR. The covariance matrix of the state variables is given by Given these definitions, we show in the appendix that the variance decomposition for q at each forecasting horizon K is given by In the above expressions, I represents a conformable identity matrix. As in the benchmark case, the VAR is estimated by applying multiple-equation OLS, with Newey-West t-ratios.
The t-ratios of the horizon-specific coefficients are obtained from the delta method.
At an infinite horizon, we have lim K→∞ A K approaching to a matrix of zeros, and thus, the corresponding long-run VAR-based variance decomposition for q is given by The estimation results of the unrestricted VAR are displayed in the appendix. The results indicate that lagged returns help to predict a significant increase in the interest spreads for most currencies, the few exceptions being GBP, JPY, and NZD. On the other hand, the interest differential is significantly autocorrelated in the case of JPY as well as in the case of all nine non-G10 currencies. In comparison, the lagged return and lagged interest spread do not provide information for future returns and exchange rates in the majority of the cases.
Adding the lagged return and the lagged interest spread as predictors actually increases the already large persistence of the real exchange rate of JPY (from 0.993 to 0.998), reinforcing the suspicion of non-stationarity.
The results associated with the long-run variance decomposition are displayed in Table 9.
The long-run return and interest differential slopes associated with JPY are −0.20 and 0.80, respectively. Hence, the long-run decomposition associated with the unrestricted VAR is less extreme than in the benchmark case, as both coefficients assume the correct sign. However, both of these estimates are largely insignificant and we also cannot reject the null that all the variation in the Japanese real exchange rate is due to return predictability (b lr r = −1). Hence, as in the benchmark case, the decomposition for JPY is very imprecisely estimated based on the asymptotic statistical inference.
Among the remaining 18 currencies the return slope estimates vary between −0.77 (CHF) and −1.23 (TRY), and all of these estimates are strongly significant (1% level). In comparison, the range for the interest spread coefficient estimates is between −0.23 (TRY) and 0.23 (CHF), with all estimates being insignificant, similarly to the benchmark decomposition. It turns out that the return coefficients are above one in magnitude for most emerging currencies (except the cases of HUF and KRW), as a result of the negative interest coefficients associated with those currencies. Therefore, there is an even larger share of return predictability at the infinite horizon than in the benchmark decomposition, especially among the non-G10 currencies. Again, this suggests that the return channel is relatively more important for the emerging currencies than for the G10 currencies.

Higher-order VAR
Next, we estimate a variance decomposition for q, based on the following restricted secondorder VAR: The motivation for this specification is that the second lag of the real exchange rate can provide useful information for predicting the three variables in the system. This is especially relevant given the evidence above that the real exchange rate is relatively persistent across most currencies, although with various degrees. The VAR (2) is estimated by multiple-equation

OLS.
To derive the decomposition for q, it is convenient to write the VAR(2) as a VAR (1) in the equivalent companion form, or equivalently, We define the indicator vectors as follows, and the covariance matrix of z t corresponds to Based on these definitions, we use the same formulas presented above for the case of the unrestricted VAR(1) in order to obtain the predictive slopes at each forecasting horizon K. This variance decomposition represents an innovation in the paper as the previous papers analyzing asset return predictability by conducting variance decompositions for key instruments (e.g., dividend yield, book-to-market ratio, earnings yield) rely on first-order VAR specifications.
The estimation results for the restricted VAR (2), which are presented in the appendix, indicate that the estimates for both λ d1 and λ d2 are statistically significant across most currencies. On the other hand, the estimates of both λ r1 and λ r2 are insignificant in nearly all cases. Thus, including one additional lag in q in the equation for r makes the first-order slopes insignificant. We can also see that the estimates for φ 2 are not significant for all 19 currencies. However, the inclusion of q t−1 in the VAR has a clear effect on the magnitudes of φ 1 . For example, the estimate of φ 1 is above one (1.13) in the case of JPY, as a result of a negative estimate of φ 2 (−0.15) .

Additional results
In this section, we provide additional empirical findings that complement the results from the previous sections. First, we conduct variance decompositions for q associated with currency portfolios. Second, we estimate a variance decomposition for each of the G10 currencies by using a longer sample.

Currency portfolios
The results in Sections 3 and 4 suggest that the share of return (interest spread) predictability is larger (smaller) among non-G10 currencies compared to the group of developed currencies.
In this subsection, we investigate this aspect in more detail. Specifically, we conduct variance decompositions at the portfolio level, by constructing an equal-weighted portfolio of the 10 developed currencies (G10) and a similar portfolio containing the remaining nine currencies (non-G10). All the three variables in the predictive system (r, d, and q) at the portfolio level represent equal-weighted averages of the corresponding currency-specific variables. To control for the effects of outliers in each group, we construct both an alternative G10 portfolio (exc. JPY) that excludes JPY and an alternative non-G10 portfolio (exc. TRY) that excludes TRY. The motivation hinges on the evidence from the previous sections that JPY (TRY) represents the currency with the largest share of interest spread (return) predictability at long horizons.
The descriptive statistics for the portfolio VAR variables, which are presented in the appendix, indicate that currency returns are more volatile and more persistent among the group of non-G10 currencies than among the group of G10 currencies. We also observe that the interest differential is more persistent for the non-G10 portfolio than for the G10 portfolio, as indicated by the autoregressive slopes of 0.49 and 0.31, respectively. Moreover, the real exchange rate is slightly more volatile, while being marginally less persistent, among the emerging currencies in comparison to the developed currencies. Excluding the JPY from the group of developed currencies implies that the portfolio return becomes more volatile and on par with the non-G10 portfolio.
The VAR estimation results for both currency portfolios are displayed in Table 11. The estimates for the return coefficients (λ r ) are −0.05 and −0.09 for the G10 and non-G10 baskets, respectively. These estimates are significant at the 10% and 5% levels, respectively.
This suggests a larger effect of return predictability among the emerging currencies, which can also be confirmed by the higher R 2 in those regressions (4.4% versus 1.8%). To a large extent, we observe qualitatively opposite results in the regressions associated with future interest spreads: In the case of the G10 group, the estimate of λ d is 0.011, which is marginally insignificant at the 5% level (t-ratio of 1.92), while in the case of non-G10 group the interest slope has the wrong sign (−0.001) and is largely insignificant. Moreover, the explanatory ratios are 4.1% and 0.1% in the interest spread regressions corresponding to the G10 and non-G10 portfolios, respectively. Hence, there is relevant short-run predictability of future d from the current real exchange rate among the developed currencies, but there is no such effect for the other currencies. The results for the AR(1) regressions for q indicate that the real exchange rate is slightly more persistent among the G10 currencies, but the difference is small (0.94 versus 0.92). Excluding both JPY from the G10 group and TRY from the non-G10 group has virtually no effect on the results for the regressions associated with future d. Yet, the estimate of λ r associated with the G10 portfolio increases in magnitude (from −0.05 to −0.07) and becomes significant at the 5% level. Nevertheless, we still observe a larger share of return predictability for the non-G10 group.
The results for the implied long-horizon predictive statistics are provided in Table 12.
The long-run return slope estimates are −0.83 and −1.02 for the G10 and non-G10 portfolios, respectively, and both estimates are strongly significant. This implies a share of 17% for interest predictability in the case of G10, which is significant at the 10% level if we rely on one-sided p-values (t-ratio of 1.39). In comparison, the corresponding estimate in the case of non-G10 has the wrong sign (−0.02) and is largely insignificant. Excluding the outliers in both portfolios has a minor effect on the long-run decompositions, as the shares change by at most 3 percentage points relative to the benchmark portfolios. In particular, by using one-sided p-values, we reject the null of a zero interest spread coefficient (t-ratio of 1.44) in the case of G10. This indicates that there is a non-negligible share of long-run predictability from interest differentials in the case of the G10 currencies, which is totally absent among the other currencies.
To gain further insight on the significance of the horizon-specific predictive slopes, we compute simulated-based p-values for the estimates in the decompositions for each portfolio.
The results presented in the appendix indicate a clear pattern for the interest spread coefficients across the two groups of currencies: these estimates are significant (at the 5% level) at all forecasting horizons in the case of G10, while there is no significance within the emergingcurrencies group. There is also stronger evidence of return predictability among the non-G10 portfolio (relative to the G10 portfolio), as indicated by the corresponding lower p-values.
These results hold under both the Monte-Carlo and bootstrap simulations, with the bootstrap pointing to marginally higher evidence of predictability. Moreover, excluding the outliers has no relevant effect on those findings.
Overall, the results of this subsection suggest that there is a higher share of interest spread predictability among G10 currencies in comparison to non-G10 currencies, while in the case of return predictability we observe the opposite pattern.

Longer sample
The variance decompositions estimated in the previous sections are based on a relatively short sample that starts in 2004:04. This is to ensure a proper comparison between G10 and non-G10 currencies. In this subsection, we estimate a variance decomposition for each of the G10 currencies by using a longer sample that starts in 1990:01. The objective is to assess if the qualitative results obtained above for the predictability mix among the G10 currencies are stable over time, hence holding for a longer period.
The results for the decomposition for q at an infinite horizon are displayed in Table 13.
The most striking difference relative to the benchmark decomposition holds for the JPY, with a long-run predictability mix that seems approximately opposite to that estimated for the post-2004 period: the estimate for the long-run return slope is −0.84, which is strongly significant (1% level). 9 This large return share stems from a negative one-period VAR return slope estimate (−0.01) combined with a lower persistence of q (relative to the shorter sample), as shown by the VAR estimation results presented in the appendix. In comparison, the long-run interest differential coefficient estimate is relatively modest (0.17) and highly insignificant, in result of a one-period VAR interest coefficient estimate around zero. Hence, 84% of the variation in the real exchange rate of JPY is due to long-run return predictability, which is clearly at odds with the dominant role of interest spread predictability in the decomposition for the shorter sample. Moreover, we cannot reject the null that all the variation in q is due to the return predictability channel. Indeed, the JPY is among the G10 currencies with the largest share of long-run return predictability over the longer sample.
Turning to the other nine currencies, the predictability mix is more similar to that estimated for the post-2004 period. The long-run return coefficient estimates vary between −0.59 (AUD) and −0.88 (NOK), with most of these estimates being significant at the 1% level (as in the benchmark case). The few exceptions are the cases of SEK and AUD in which the long-run return shares are significant at the 5% level. This suggests a higher dispersion in the return slopes within these nine currencies relative to the short-sample decomposition. The corresponding range for the interest slope coefficient estimates is between 0.12 (NOK) and 0.41 (AUD). Interestingly, the long-run interest rate coefficients are statistically significant at the 10% level in the cases of CAD and GBP. This is in contrast with the benchmark decomposition, in which case there is no significance for the interest slopes associated with each of these nine currencies (based on two-sided p-values).
Overall, the results of this subsection suggest some degree of time-variation in the estimation of the variance decomposition for q in the case of the G10 currencies. This pattern is especially notorious in the case of the JPY, in which the variance decomposition estimated for the long sample yields quite an opposite predictability mix to that estimated for the shorter period. Hence, the dominant share of interest spread predictability for JPY, estimated in the previous sections, seems to be concentrated in the recent period. On the other hand, there is a somewhat larger degree of interest spread predictability among the other G10 currencies over the longer sample in comparison to the recent period.

Theoretical framework and quantitative implications
In the previous sections, we have provided ample evidence that real exchange rates predict future interest rate spreads with a positive sign and excess currency returns with a negative sign. That is, both cash-flow and risk premium channels are important to account for the variations in real exchange rates, despite the fact that the discount rate channel is dominant across most currencies. However, the usual international asset pricing models such as the habit model of Verdelhan (2010) or the long-run risk model of Bansal and Shaliastovich (2013) are inconsistent with those stylized facts. Despite their implications that currency appreciation predicts a lower risk premium, those models substantially overstate the importance of the risk premium channel, and hence wrongly imply negative predictability of future cash-flows (interest spreads) by current real exchange rates. Such shortcoming becomes even more severe when confronted with the empirical predictability mix we found for JPY, where both future cash flows and risk premiums are positively correlated with current real exchange rate movements.
To reconcile such discrepancy and understand how the predictability mix varies across economies, in this section we develop and solve a parsimonious model of the real exchange rate by extending the liquidity model of Nagel (2016). As the main ingredients of our model, we assume that investors value the liquidity benefit provided by the near-money assets such as the domestic Treasury-bills. Furthermore, investors have non-separable utility over consumption and liquidity services. The model is highly tractable, and in contrast to the habit or the long-run risk model, it delivers correct implications on the predictability mix we documented in the previous sections for a large basket of currencies.
Introducing a non-separable utility over consumption and liquidity services is a novel aspect of our framework. The non-separation dictates that the marginal utility of consumption, or the stochastic discount factor (SDF), is directly linked to the marginal utility of liquidity services. On the one hand, when the consumption and liquidity are substitutes, higher marginal utility of liquidity increases the SDF. Hence, if the marginal utility of liquidity in the foreign country is comparably higher, the foreign currency appreciates and the representative investor requires a lower currency risk premium. Simultaneously, higher marginal utility of consumption lowers the saving motive of the agent and therefore pushes up the equilibrium interest rate, which leads to a higher foreign-minus-home interest rate spread. Therefore, the model implies positive (negative) predictability of interest spread (excess returns) by the real exchange rate, as documented in the results discussed in the previous sections. On the other hand, when the elasticity of substitution between consumption and liquidity is sufficiently low (so that consumption and liquidity services become complements), higher marginal utility of liquidity decreases the log SDF. When the foreign-minus-home marginal utility of liquidity is higher, the foreign currency in fact depreciates and the interest spread also declines. As the spot rate adjustment is weaker than the interest spread adjustment, this even produces a lower currency risk premium. Put differently, our model postulates that under a specific scenario, the real exchange rate even predicts concurrently the interest spread and the risk premium with a positive sign, as we found empirically for the case of JPY.
Compared with Engel (2016) who also extends the model of Nagel (2016) to the international setting, there are several important differences. First, his model still relies on the separable utility and requires full specification of the monetary policy to pin down exchange rate, whereas ours is a consumption-based asset pricing model that is more tractable.
Second, Engel (2016) tries to understand the reversal in the relation between interest spread and risk premium across different horizons, but our model explores the driving forces of real exchange rate across different economies. Third and most importantly, the model of Engel (2016) is not flexible enough to generate both the positive and negative predictability for the currency risk premium we documented for the JPY and other G10 currencies.

Model specification
The model abstracts from money and inflation. In the model, there are two countries: home (we call it U.S.) and a foreign country. The two countries are symmetric and share the same model parameters. For the interest of clarity, below we only discuss the problem faced by the representative agent in the home country, as the problem for the foreign agent is symmetric.
All foreign quantities will be labeled with superscript * .
The representative investor in the home country maximizes the lifetime utility, where Y t is the exogenous endowment and Q t is the real exchange rate expressed in domestic consumption goods per foreign goods. B t and B * t denote holdings of home and foreign risk-free assets that do not provide liquidity services. B T B t is the holding of domestic Treasury-bills, the near-money assets, as in Nagel (2016), which provide liquidity services. The aggregator over consumption and liquidity services, takes the functional form of constant elasticity of substitution (CES), where L t = B T B t . The introduction of the liquidity term L t follows Nagel (2016) to let the investor value the liquidity benefit of near-money assets such as the domestic Treasury-bills, yet we extend his setting of separable utility to the case of non-separable utility over consumption and liquidity. 10 Under this new specification, α denotes the share of liquidity services in utility and ν captures the intratemporal elasticity of substitution between consumption and liquidity.
When γ = 1 ν , our specification reduces to the usual separable utility function. Critically, the non-separation between consumption and liquidity generates dependence of the pricing kernel on the liquidity factor. The stochastic discount factor is given by where for convenience we denote θ = 1/ν−γ 1−1/ν . We also define the marginal benefit of liquidity relative to consumption as Λ t = (Xt/Ct) 1−1/ν 1−α . The definition of Λ t is straightforward since we have the identity: We denote the log yield of home country Treasury-bills as i B t , and the log yields of home (foreign) country risk-free instruments that do not produce liquidity services as i t (i * t ). Because investors can only use Treasury-bills as the liquidity provider, we would expect the existence of a spread between i t and i B t in equilibrium. The Euler equations related to the pricing of these three assets are given by where small letters denote the log of the corresponding upper-case variables. 11 While the first two equations are standard in the literature, the last equation suggests that investors demand a lower rate of return from T-bills as long as they value the liquidity benefit provided by those assets (i.e., U L,t /U C,t > 0). macroeconomics and asset pricing literatures (see Boyle and Young (1988), Finn, Hoffman, and Schlagenhauf (1990), Bufman and Leiderman (1993), Bakshi and Chen (1996), Holman (1998), Buraschi and Jiltsov (2007), Lioui and Maio (2014), among others).
11 Detailed derivations can be found in the appendix. Note that i t+1 , i * t+1 , and i B t+1 are all known at the beginning of t + 1.
Next, we specify the forcing processes for the exogenous variables. First, as the real exchange rate is non-stationary under a random walk process for log consumption, we assume cointegrated foreign and home log-consumption to reach a stationary real exchange rate, following the suggestion in Engel (2016): Then, we assume the following dynamics for the log of marginal utility of liquidity where u t+1 and t+1 are independent and standard normal shocks. Similar to Verdelhan (2010) and Bansal and Shaliastovich (2013), we adopt a square-root process since a heteroskedastic log SDF is necessary in order to obtain predictable currency excess returns.

Equilibrium
Solutions for the risk-free rates are standard, The model implied forward discount is then given by By combining the pricing equations for both i B and i, it follows that the equilibrium yields on the Treasury bills are as follows: We then turn to the equilibrium determination of the real exchange rate. Following Backus, Foresi, and Telmer (2001), with complete markets and in the absence of arbitrage, the percentage change in the real exchange rate equates the difference between the log of the stochastic discount factors of the two countries: The expected currency return is then leading to the equilibrium currency risk premium: Defining the long-run real exchange rate asq, we obtain the expression for the real exchange rate by solving forward Equation (55) The real exchange rate is determined by the cross-country differentials in liquidity demand, in addition to the consumption differentials. In the simplistic case when γ = 1/ν (so that θ = 0), the non-separable utility reduces to the separable utility. This implies that the liquidity factors do not affect the real exchange rate and there is no risk premium predictability. Under that setup, the model is not different from the conventional consumption-based international asset pricing models such as the one studied by Backus and Smith (1993).
Imposing the non-separation (γ = 1/ν) provides a new perspective towards understanding the predictability mix of real exchange rate on both future interest spreads and risk premiums.
We focus our analysis on the range, 0 < ν < 1, as the literature documents a small elasticity of substitution between consumption and liquidity (see e.g., Bufman and Leiderman (1993) and Piazzesi, Rogers, and Schneider (2019)). First, when ν > 1/γ (so that θ > 0), the consumption and liquidity services are substitutes for investors. Higher λ * t − λ t leads to higher m * t − m t and hence higher q t . Simultaneously, E t (r t+1 ) becomes lower and we obtain negative predictability for risk premium by q t . Second, as long as 1 2 σ 2 θ 2 < (1 − ψ)θ, it follows that B will be negative so we have positive predictability for interest spread. The model predictions are thus consistent with our evidence on predictability patterns found for most currencies.
Nonetheless, when the elasticity of substitution between liquidity and consumption ν is low enough (such that ν < 1/γ), the consumption and liquidity services become complements.
Higher λ * t − λ t lowers the foreign-minus-home log SDF, and hence leads to a lower q t and also a lower E t (r t+1 ). The interest spread (d t+1 ) also decreases since B > 0. In this case we even obtain positive predictability for both interest spread and risk premium, in line with our empirical evidence for JPY.

Quantitative results
In this subsection, we calibrate our model to G10 economies at the monthly frequency and investigate whether it correctly replicates the sign and magnitude we documented for the variance decompositions of real exchange rates. The risk-aversion coefficient (γ) is set to 2, the subjective discount factor (β) is fixed at 0.99, and the share of liquidity benefit in utility (α) is set to 0.2. All these choices are conventional in the literature. Then we select g and σ u to match the mean and volatility of U.S. consumption growth. For the cointegrating coefficient (κ) between each economy against U.S., we collect real private final consumption expenditures for G10 economies from the quarterly database of OECD. 12 We then estimate the cointegrating relations (47) and (48) from the consumption levels of U.S. and foreign economies. The persistence and volatility of the real exchange rate will be used to pin down ψ and σ . 13 We then describe how to calibrate the intratemporal elasticity of substitution between 12 We use German consumption data to proxy for the economic activity in the Euro area. 13λ is then set at σ 2 2(1−ψ) to ensure the positiveness of λ t in the square-root process (49). consumption and liquidity (ν). Our model implies a closed-form solution for the U.S. Treasury premium (USTP) studied by Du, Im, and Schreger (2018): which is linked to the cross-country differentials in the marginal utility of liquidity services.
So the sign of such correlation is informative on the sign of ν − 1/γ. Once the risk-aversion coefficient γ is fixed, we can select ν that is consistent with the data.
Empirically, we compute this correlation based on the data of the U.S. Treasury premium from Du, Im, and Schreger (2018), which covers all G10 currencies and spans seven horizons ranging from three months to ten years. 14 The result is reported in the last row of Panel B in Table 15. We find that for almost all G10 currencies, the correlation is negative. Interestingly, the JPY is an exception and it appreciates against the USD when the US Treasury premium is higher. Such finding is consistent with the previous literature showing that the JPY is a safe haven currency, we thus set ν < 1/γ for JPY and ν > 1/γ for other currencies.
All calibrated parameter values are listed in Panels A and B of Table 14. To evaluate the model's fit, Table 15 compares the empirical moments used to fix parameters and their modelimplied counterparts. We simulate 135 months of data from the model, which corresponds to the sample length in our empirical study in Section 3, and we compute the target moments.
Simulations are repeated for 1,000,000 times and we report the median of target moments (based on all simulations) as the model-implied moments. The results suggest that our model reasonably fits the stylized facts in the data such as the dynamics of consumption growth, as well as the persistent and volatile real exchange rates. We also match the positive correlation between the real exchange rate and the U.S. Treasury premium for the JPY, and the negative correlation for all other currencies.
After suitable parameters that align our model with the data, we then proceed to discuss the quantitative implications for the variance decompositions of real exchange rates. Table   16 reports the AR(1) coefficients of real exchange rates (φ), the long-run risk premium slope (b lr r ), and the long-run interest spread coefficient (b lr d ) based on our model simulations. These statistics are computed by applying the same methodology introduced in Section 2 on each of our simulated paths. To get a comprehensive picture, we report the 5% and 95% quantiles of the simulated quantities, in addition to the median. The model performance is satisfactory and reproduces the long-run shares attributed to the risk premium and cash-flow channels in the data. The risk premium slopes are negative, yet we also obtain positive and non-negligible interest spread coefficient estimates. This is more consistent with the data and strongly against the results of Balduzzi and Chiang (2018), as they obtain risk premium shares over 100% based on the standard long-run risk or habit models. Notably, we find that the distribution of the risk premium slopes for JPY admits a nontrivial probability of positive values, echoing with the positive predictability for risk premium uncovered in the data.
Another important fact we documented in the last section is the predictability reversal for JPY based on longer sample dating back to January of 1990, as shown in Table 13. We therefore explore whether our theoretical framework can mirror this pattern after re-calibrating the model on a longer sample. 15 Panel C of Table 14 and the appendix report the parameters and simulated moments from the new calibration exercises. The model still captures correctly the target moments. Interestingly, the correlation between the real exchange rate and the U.S. Treasury premium now switches to negative for the case of JPY. The switch reflects a change in the elasticity of substitution between liquidity and consumption (ν) in the longer sample. Consequently, the landscape of predictability mix will also change. Results tabulated in the appendix confirm such findings. While the interest spread predictability slopes are still positive and nontrivial, we find that now the 5% and 95% quantiles of risk premium coefficients for JPY are both negative. In other words, after considering the time-varying correlation between the real exchange rate and the U.S. Treasury premium, our model is able to reproduce the predictability reversal in the case of JPY.
15 As the U.S. Treasury premium data are only available after 1990, we use one-month Libor rates in replacement of government bond yields to recompute the liquidity premium difference defined in (58).

Conclusion
We specify a variance decomposition for the log real exchange rate based on a dynamic present-value relation. In such decomposition, the estimated coefficients represent the fraction of the variance of the current exchange rate attributed to interest spread, return, and future exchange rate predictability. The variance decomposition for the log exchange rate is based on a restricted first-order VAR, in which the only predictor is the real exchange rate. We estimate a term-structure of variance decompositions in order to account for the different predictability patterns at short, intermediate, and long forecasting horizons. We use the traditional G10 currencies in the analysis, as well as nine currencies from other developed countries and emerging/developing economies.
Our results indicate that the major force driving the variation in the current real exchange rate of most currencies is return predictability, with interest spread predictability assuming a secondary role. This pattern is especially notorious at long forecasting horizons: More than 76% of the variation in the exchange rate is attributable to forecasting long-run returns, implying a maximum share for interest spread predictability of 24%. The major outlier to this pattern is the Japanese Yen (JPY), in which case the dominant driving force of the real exchange rate is predictability of future interest differentials (rather than return predictability), and this pattern is especially notorious at long horizons: more than 100% of the variation in the real exchange rate is due to predictability of future long-run interest spreads, which means that return predictability goes with the wrong sign. We also observe that predictability of the future exchange rate is the dominant driving force of variation in the current exchange rate at short horizons for all currencies. However, this effect dies off at a relatively fast pace for most currencies.
Variance decompositions estimated under less restrictive VAR specifications (unrestricted first-order VAR and restricted second-order VAR) yield qualitatively similar results to those obtained under the benchmark decomposition (based on a restricted first-order VAR). We also conduct Monte-Carlo and bootstrap simulations in order to obtain more robust statistical inferences for the estimated shares in the variance decompositions. The results from both simulations indicate that the shares of interest spread predictability, although substantially smaller than the shares of return predictability, are statistically significant for most of the G10 currencies. We also conduct variance decompositions for portfolios of currencies. The results suggest that there is a higher and more statistically significant share of interest spread predictability among G10 currencies in comparison to non-G10 currencies, while in the case of return predictability we observe the opposite pattern. This provides further evidence that the currency markets of the G10 and non-G10 currencies behave in a different way.
Furthermore, we estimate a variance decomposition for each of the G10 currencies by using a longer sample that starts in 1990. The results suggest some degree of time-variation in the estimation of the decomposition for the real exchange rate. This pattern is especially notorious in the case of the JPY, in which the variance decomposition estimated for the long sample yields quite an opposite predictability mix to that estimated for the shorter period.
Hence, the dominant share of interest spread predictability in the case of JPY seems to be concentrated in the recent period.
In the last part of the paper, we develop and solve a parsimonious model of the real exchange rate by extending the liquidity model of Nagel (2016). As the main ingredients of our model, we assume that investors value the liquidity benefit provided by the near-money assets, such as the domestic Treasury-bills. Furthermore, investors have non-separable utility over consumption and liquidity services. The model is highly tractable, and in contrast to the habit or the long-run risk model, it delivers correct implications on the predictability mix that we document for a large basket of currencies. After calibrating the model to match several key moments of exchange rates and macroeconomic quantities, we find that our model is able to match relatively well the sign and magnitude of the share of risk premium and interest spread predictability for all G10 currencies, therefore providing a more reasonable justification on what drives real exchange rates in comparison to alternative theoretical frameworks.  This table reports descriptive statistics for the log currency return (r), log interest rate differential (d), and log real exchange rate (q). "AR(1)" designates the first-order autoregressive coefficient. The columns labeled r-q represent the correlation matrix of the three variables. The sample is 2004:04-2015:06. A description of the G10 foreign currencies is provided in Section 2.    (1). The variables in the VAR are the log currency return (r), log interest rate differential (d), and log real exchange rate (q). λ r , λ d , φ denote the VAR slopes associated with lagged q, while t denotes the respective Newey-West t-statistics (calculated with one lag). R 2 is the coefficient of determination for each equation in the VAR. The original sample is 2004:04-2015:06. Italic, underlined, and bold numbers denote statistical significance at the 10%, 5%, and 1% levels, respectively. A description of the 19 foreign currencies is provided in Section 2.  This table reports the variance decomposition for the real exchange rate (q) at an infinite horizon, based on a restricted VAR(1). The variables in the VAR are the log currency return (r), log interest rate differential (d), and log real exchange rate. b lr r , b lr d denote the long-run coefficients associated with future returns and interest rate differentials, respectively. t(b lr r = 0), t(b lr d = 1) and t(b lr r = −1), t(b lr d = 0) denote the t-statistics associated with the null hypotheses H 0 : b lr r = 0, b lr d = 1 and H 0 : b lr r = −1, b lr d = 0, respectively. The original sample is 2004:04-2015:06. Italic, underlined, and bold numbers denote statistical significance at the 10%, 5%, and 1% levels, respectively. A description of the 19 foreign currencies is provided in Section 2.  This table reports the variance decomposition for the real exchange rate (q) at several forecasting horizons. The implied predictive slopes are obtained from a restricted VAR(1). The variables in the VAR are the log currency return (r), log interest rate differential (d), and log real exchange rate. b K denotes the coefficient associated with future returns, interest rate differentials, or the real exchange rate, at horizon K. t denotes the respective Newey-West t-statistic. The original sample is 2004:04-2015:06. Italic, underlined, and bold numbers denote statistical significance at the 10%, 5%, and 1% levels, respectively. A description of the G10 foreign currencies is provided in Section 2.  This table reports the variance decomposition for the real exchange rate (q) at several forecasting horizons. The implied predictive slopes are obtained from a restricted VAR(1). The variables in the VAR are the log currency return (r), log interest rate differential (d), and log real exchange rate. b K denotes the coefficient associated with future returns, interest rate differentials, or the real exchange rate, at horizon K. t denotes the respective Newey-West tstatistic. The original sample is 2004:04-2015:06. Italic, underlined, and bold numbers denote statistical significance at the 10%, 5%, and 1% levels, respectively. A description of the nine non-G10 foreign currencies is provided in Section 2.        (1). The variables in the VAR are the log currency return (r), log interest rate differential (d), and log real exchange rate. b lr r , b lr d denote the long-run coefficients associated with future returns and interest rate differentials, respectively. t(b lr r = 0), t(b lr d = 1) and t(b lr r = −1), t(b lr d = 0) denote the t-statistics associated with the null hypotheses H 0 : b lr r = 0, b lr d = 1 and H 0 : b lr r = −1, b lr d = 0, respectively. The original sample is 2004:04-2015:06. Italic, underlined, and bold numbers denote statistical significance at the 10%, 5%, and 1% levels, respectively. A description of the 19 foreign currencies is provided in Section 2.    (2) This table reports the variance decomposition for the real exchange rate (q) at an infinite horizon, based on a restricted VAR(2). The variables in the VAR are the log currency return (r), log interest rate differential (d), and log real exchange rate. b lr r , b lr d denote the long-run coefficients associated with future returns and interest rate differentials, respectively. t(b lr r = 0), t(b lr d = 1) and t(b lr r = −1), t(b lr d = 0) denote the t-statistics associated with the null hypotheses H 0 : b lr r = 0, b lr d = 1 and H 0 : b lr r = −1, b lr d = 0, respectively. The original sample is 2004:04-2015:06. Italic, underlined, and bold numbers denote statistical significance at the 10%, 5%, and 1% levels, respectively. A description of the 19 foreign currencies is provided in Section 2.   (1). The variables in the VAR are the log currency return (r), log interest rate differential (d), and log real exchange rate (q). λ r , λ d , φ denote the VAR slopes associated with lagged q, while t denotes the respective Newey-West t-statistics (calculated with one lag). R 2 is the coefficient of determination for each equation in the VAR. The original sample is 2004:04-2015:06. Italic, underlined, and bold numbers denote statistical significance at the 10%, 5%, and 1% levels, respectively. The estimation is conducted for portfolios of G10 and non-G10 foreign currencies.   (1). The variables in the VAR are the log currency return (r), log interest rate differential (d), and log real exchange rate. b lr r , b lr d denote the long-run coefficients associated with future returns and interest rate differentials, respectively. t(b lr r = 0), t(b lr d = 1) and t(b lr r = −1), t(b lr d = 0) denote the t-statistics associated with the null hypotheses H 0 : b lr r = 0, b lr d = 1 and H 0 : b lr r = −1, b lr d = 0, respectively. The original sample is 2004:04-2015:06. Italic, underlined, and bold numbers denote statistical significance at the 10%, 5%, and 1% levels, respectively. The estimation is conducted for portfolios of G10 and non-G10 foreign currencies.  This table reports the variance decomposition for the real exchange rate (q) at an infinite horizon, based on a restricted VAR(1). The variables in the VAR are the log currency return (r), log interest rate differential (d), and log real exchange rate. b lr r , b lr d denote the long-run coefficients associated with future returns and interest rate differentials, respectively. t(b lr r = 0), t(b lr d = 1) and t(b lr r = −1), t(b lr d = 0) denote the t-statistics associated with the null hypotheses H 0 : b lr r = 0, b lr d = 1 and H 0 : b lr r = −1, b lr d = 0, respectively. The original sample is 1990:01-2015:06. Italic, underlined, and bold numbers denote statistical significance at the 10%, 5%, and 1% levels, respectively. A description of the G10 foreign currencies is provided in Section 2.

A Online appendix: Benchmark variance decomposition
By multiplying both sides of the present-value relation for the log real exchange rate by q t − E(q t ), and taking unconditional expectations, we obtain the following variance decomposition By dividing both sides by Var(q t ), we have, where β(y, x) denotes the slope from a regression of y on x.
By using the property of regression coefficients, β(y + z, x) = β(y, x) + β(z, x), we have: Under the restricted first-order VAR, we have, and by combining with the VAR equation for the currency return, implies the following equation for r t+j : Since Cov(ε q t+l , q t ) = 0, l > 0 and Cov(ε r t+j , q t ) = 0, by construction, it follows that Similarly, we have, On the other hand, given the expanded expression for q t+K , By simplifying the sums above, we obtain the VAR-based variance decomposition associated with q: To compute the t-statistics for the predictive coefficients, b , we use the Delta method. From the standard errors associated with the VAR slopes, b ≡ (λ r , λ d , φ) , we have: The matrix of derivatives is given by (A.14) We can see that the magnitudes of ∂b K d ∂φ and ∂b K r ∂φ will be large if φ is very close to one. In turn, this implies that the standard errors for the implied K-period return and interest spread slopes will be quite large if the real exchange rate is very persistent.

B Online appendix: Alternative variance decompositions
After recursive substitution, the vector of state variables at t + j can be written as, (B.15) or equivalently, This implies that the regression coefficient of r t+j on q t is given by β(r t+j , q t ) = Cov(e r z t+j , e q z t ) Var(e q z t ) = Cov(e r A j z t , e q z t ) Var(e q z t ) = e r A j Σe q e q Σe q , (B.17) where we use the fact that Cov(ε t+l , z t ) = 0 for l > 0.
By using the result above, it follows that the K-period return slope is given by The K-period interest spread slope is defined in an analogous way. The slope associated with future q at t + K is derived as follows: β(q t+K , q t ) = Cov(e q z t+K , e q z t ) Var(e q z t ) = Cov(e q A K z t , e q z t ) Var(e q z t ) = e q A K Σe q e q Σe q . (B.19) In the case of the unrestricted VAR(1), the t-ratios associated with the horizon-specific , are obtained by using the Delta method, In the case of the restricted VAR(2), we apply the same formula, with b ≡ (λ r1 , λ r2 , λ d1 , λ d2 , φ 1 , φ 2 ) .

C Online appendix: Model solution derivation
We first derive Equation (43). Write X t as (C.21) and dividing both sides by C From the definition of Λ t , the above equation leads to (C.25) Next, we turn to the derivation of the Euler equations. Recall the problem of the home investor, where Y t is the exogenous endowment, B t and B * t denote respectively the holdings of home and foreign risk-free assets that do not provide liquidity services, and B T B t is the holding of domestic Treasury-bills.
Solving the problem yields the following FOCs (denote ζ t as the Lagrangian multiplier on time-t budget constraint): (C.28) As the SDF is defined as M t+1 = U C,t+1 /U C,t , we can obtain the Euler equations presented in the main text.   (2) This table reports the estimation results associated with the restricted VAR(2). The variables in the VAR are the log currency return (r), log interest rate differential (d), and log real exchange rate (q). λ r1 , λ r2 , λ d1 , λ d2 , φ 1 , φ 2 denote the VAR slopes associated with lagged q, while t denotes the respective Newey-West t-statistics (calculated with one lag). Italic, underlined, and bold numbers denote statistical significance at the 10%, 5%, and 1% levels, respectively. A description of the 19 foreign currencies is provided in Section 2.   These are obtained from two alternative Monte-Carlo simulations (with 10,000 replications each) under the nulls of no return and no interest spread predictability in Panels A and B, respectively. The predictive variable is the real log exchange rate. The numbers indicate the fraction of pseudo samples under which the return (interest) coefficient is lower (higher) than the corresponding estimates from the original sample. K represents the forecasting horizon (in months). The original sample is 2004:04-2015:06. Italic, underlined, and bold numbers denote statistical significance at the 10%, 5%, and 1% levels, respectively. The estimation is conducted for portfolios of G10 and non-G10 foreign currencies.  This table reports the simulated p-values for the VAR-based return (b K r ) (Panel A) and interest spread (b K d ) (Panel B) slopes at several forecasting horizons. These are obtained from two alternative bootstrap simulations (with 10,000 replications each) under the nulls of no return and no interest spread predictability in Panels A and B, respectively. The predictive variable is the real log exchange rate. The numbers indicate the fraction of pseudo samples under which the return (interest) coefficient is lower (higher) than the corresponding estimates from the original sample. K represents the forecasting horizon (in months). The original sample is 2004:04-2015:06. Italic, underlined, and bold numbers denote statistical significance at the 10%, 5%, and 1% levels, respectively. The estimation is conducted for portfolios of G10 and non-G10 foreign currencies. This table reports the estimation results associated with the restricted VAR(1). The variables in the VAR are the log currency return (r), log interest rate differential (d), and log real exchange rate (q). λ r , λ d , φ denote the VAR slopes associated with lagged q, while t denotes the respective Newey-West t-statistics (calculated with one lag). R 2 is the coefficient of determination for each equation in the VAR. The original sample is 1990:01-2015:06. Italic, underlined, and bold numbers denote statistical significance at the 10%, 5%, and 1% levels, respectively. A description of the G10 foreign currencies is provided in Section 2.   (1) coefficients of real exchange rates (φ) in Equation (8), Panel B and C reports respectively the shares of risk premium (b lr r ) and interest spread predictability (b lr d ) in Equation (16). The model is simulated with 306 months that corresponds to our sample length from 1990:01 to 2015:06, and simulations are repeated 1,000,000 times for each economy. We report the median, and the 5% and 95% quantiles of relevant statistics based on all simulations.