Real Exchange Rates and Currency Risk Premiums

Balduzzi,, Pierluigi; Chiang, I-Hsuan, Ethan

doi:10.1093/rapstu/raz002

Real Exchange Rates and Currency Risk Premiums

Balduzzi,, Pierluigi;Chiang, I-Hsuan, Ethan 2020-02-01 00:00:00 Abstract Standard finite horizon tests uncover only weak evidence of the predictive power of the real exchange rate for excess currency returns. On the other hand, in long-horizon tests, the real exchange rate strongly and negatively predicts future excess currency returns. Conversely, we can attribute most of the variability in real exchange rates to changes in currency risk premiums. The “habit” and “long-run risks” models replicate the predictive power of the real exchange rate for excess currency returns, but substantially overstate the fraction of the volatility of the real exchange rate due to risk premiums. Received December 14, 2017; Editorial decision October 14, 2018 by Editor: Raman Uppal. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online. The main research question addressed in this paper is whether real exchange rates predict excess currency returns.1 The literature has mainly focused on other instruments—most notably, the nominal interest rate differential—to characterize the time variation in currency risk premiums. Moreover, the conventional view is that the predictive ability of real exchange rates for currency returns is weak (see, e.g., the review article by Rossi (2013). We address the question above by performing finite horizon (i.e., “short-horizon”) and infinite horizon (i.e., “long-horizon”) tests of the null hypothesis that excess (log) currency returns are unpredictable. Specifically, we test whether long-horizon cumulative excess currency returns are time varying as a function of the real exchange rate and other instruments. Moreover, we use a present value representation to decompose the variance of the real exchange rate into shares due to risk premiums and due to expected cash flows. Our empirical results, based on bootstrap inference, are easily summarized. Consider the case in which the real exchange rate is the only predictor, following a univariate AR(1) process.2 For individual currencies, we find little evidence of predictive power of the real exchange rate in finite horizon tests. For example, consider the six major currencies at the 3-month forecast horizon.3 The real exchange rate negatively predicts excess returns for all six currencies, with coefficients ranging from –0.16 (British pound) to –0.02 (Australian dollar), but none of the coefficients is significant at the 5% level (bootstrap p-value). On the other hand, the long-horizon regressions are more supportive of the predictive power of the real exchange rate: the long-horizon regression coefficient is significant for 3 of 6 major currencies. Moreover, the absolute magnitude of the coefficients is large. For example, in the case of the Japanese yen, the long-horizon regression coefficient equals –0.91, which implies that 91% of the variance of the real exchange rate can be attributed to the time variation in risk premiums. Results are similar, although statistically much more significant, in a panel regression setting.4 In particular, the panel regression estimates quantify at 86% the risk premium share of the variance of the real exchange rate. This is an important stylized fact that the literature has all but overlooked. For example, Cochrane (2011) does not recognize the predictive power of the real exchange rate for excess currency returns, and its implications for the source of real exchange rate variability, focusing on the nominal interest rate differential. A size and power study demonstrates that long-horizon tests are indeed substantially more powerful than the standard finite horizon tests. We also show that our results are robust to possible spurious-regression biases, alternative parameterizations of the null, the correction for small-sample biases in the parameterization of the null hypothesis, the inclusion of additional predictors, and the use of alternative reference currencies. Having established that (1) real exchange rates do predict excess currency returns and (2) the risk premium component dominates real exchange rate variation, in the second part of the paper, we relate our analysis to the implications of no-arbitrage models of exchange rate determination, specifically, the habit and long-run risks models. We amend the existing versions of the two models (see Verdelhan, 2010;Bansal and Shaliastovich, 2013) to ensure the stationarity of the real exchange rate, and we derive population values for regression and long-horizon statistics. We show that both models replicate the result that the real exchange rate (nominal interest rate differential) negatively (positively) predicts excess currency returns. Moreover, both models replicate the fact that long-horizon tests are more powerful than finite horizon tests. Both models also imply that most of the real exchange rate variability is due to risk premiums, rather than expected cash flows, although they overstate the magnitude of the effect. Indeed, in both models the real exchange rate and the nominal interest rate differential are negatively related, leading to a negative covariance between the real exchange rate and the differential between foreign and domestic expected real bond returns. This negative covariance translates into a negative expected cash-flow variance share, and a risk premium share in excess of 100%. The literature on exchange rate determination and predictability is vast. In the remainder of this section, we focus on the studies that are most relevant for our long-horizon tests and variance-decomposition analysis. Froot and Ramadorai (2005) use the real exchange rate as a predictor of currency returns, in the context of a vector autoregression (VAR) estimated with daily data in a panel setting, for the sample from June 20, 1994, to February 9, 2001. Following Campbell (1991), they decompose the variance of innovations in currency returns into shares attributable to risk premium news and fundamental news, finding that risk premium news dominates. However, the main focus of their analysis—the effect of net order flow on currency prices—is different from ours. In addition, they do not make inference on implied long-horizon regression coefficients. Engel and West (2005) study the relation between exchange rates and macro variables, using quarterly data for the January 1974–September 2001 sample. They investigate whether nominal currency returns Granger-cause macro variables (differentials in nominal money growth, inflation, nominal interest rates, and gross domestic product growth), finding some supportive evidence. Moreover, similarly to our variance decomposition exercise, they test whether nominal currency returns are correlated with changes in the present values of several macro variables, finding that in most cases the correlations are insignificant. We improve on their analysis by providing a framework—the present value representation in Equation (4)—that identifies the real bond return differential as the single macro variable being anticipated by exchange rates. This unifying framework allows us to relate the predictive power of real exchange rates for excess currency returns and real bond return differentials, respectively, to the roles of risk premiums and expected cash flows in determining (real) exchange rate variation. Engel (2016) studies the relation between the long-horizon cumulative UIP deviation and the current expected real bond return differential. He finds that the covariance between the two quantities is negative: high real-interest-rate currencies tend to experience low cumulative excess returns in the long run. This result stands in contrast with the well-documented deviations from UIP: high real-interest-rate currencies tend to experience high excess returns in the short run. Engel (2016) demonstrates that the two stylized facts cannot be accommodated by existing no-arbitrage models. Our analysis differs from that of Engel (2016), because we focus on the relation between the long-horizon excess currency returns and the real exchange rate, rather than the expected real bond return differential. Indeed, we show that the real exchange rate is the most robust predictor of cumulative excess currency returns. Moreover, we show that once we control for the real exchange rate, the expected real bond return differential predicts long-horizon excess currency returns with a positive sign.5 In a recent working paper Dahlquist and Penasse (2017) show how both the nominal interest rate differential and the real exchange rate are needed to predict excess currency returns. Indeed, they find that accounting for the real exchange rate raises the R-squared in predictive regressions for monthly returns by about 30% on a portfolio of currencies. Their empirical result is in line with the amended habit and long-run risks models in Section 5: expected currency returns are captured by a linear combination of both the nominal interest rate differential and the real exchange rate (in addition to the inflation differential).6,7 In summary, we make the following main contributions to the existing literature. First, we show that, relative to the finite horizon tests, the long-horizon tests uncover stronger evidence of the predictive power of the real exchange rate for excess currency returns. Second, by performing a variance-decomposition exercise, we show that the predictability patterns that we document translate into the economically important result that a dominant fraction of the variance of the real exchange rate can be attributed to changes in risk premiums. Third, we modify the existing habit and long-run risks models (Verdelhan, 2010; Bansal and Shaliastovich, 2013) to accommodate stationarity of the real exchange rate. We show how these models replicate several salient predictability patterns, for example, the negative correlation between real exchange rates and future excess currency returns. However, they also overstate the risk premium share of the variance of the real exchange rate. 1. Preliminaries Let rt and rtf denote the continuously compounded domestic and foreign nominal risk-free rates and let drt≡rtf−rt denote the interest rate differential. Throughout, the superscript f denotes a variable of the foreign country, and d denotes the differential between foreign and domestic variables: dX≡Xf−X ⁠. Δ denotes first differences: ΔXt≡Xt−Xt−1 ⁠. Finally, X˜ denotes a real quantity, and X^ denotes an estimate. Let st denote the log of the directly quoted nominal exchange rate and let ξt+1≡st+1−st+drt denote the time t + 1 excess log currency return, that is, the log currency return in excess of the cost of carry (⁠ −drt ⁠). Our empirical analysis focuses on testing the null hypothesis that expected excess currency returns are constant: Et(ξt+1)=E(ξt+1). (1) We denote this null hypothesis the no-predictability null. Let pt and ptf denote the log domestic and foreign price levels. We define the log real exchange rate s˜t as s˜t≡st+ptf−pt. (2) Note that if absolute PPP holds, the real exchange rate equals one, and the log real exchange rate equals zero. Taking first differences of (2), we have Δs˜t+1≡s˜t+1−s˜t=st+1−st+pt+1f−ptf−(pt+1−pt)≡st+1−st+πt+1f−πt+1=st+1−st+drt−drt+dπt+1≡ξt+1−dr˜t+1, (3) where πt+i and πt+if are the domestic and foreign continuously compounded inflation rates, respectively, dπt+1≡πt+1f−πt+1 is the inflation differential, and dr˜t+1≡drt−dπt+1 is the real bond return differential. The variables defined above appear in the following present value representation of the real exchange rate (it can be obtained by iterating (3) forward, taking conditional expectations, and assuming that relative PPP is expected to hold in the long run (see, e.g., Froot and Ramadorai, 2005; Engel, 2016): s˜t−E(s˜t)=−∑i=1∞Et(ξt+i)+∑i=1∞Et(dr˜t+i). (4) The two terms on the right-hand side of (4) capture long-horizon cumulative deviations from UIP and real rate equality (RRE), respectively. Note that while (4) implies that the real exchange rate is related to risk premiums and expected future real bond return differentials, it does not imply that the real exchange rate is the sole predictor of them. Indeed, in the analysis that follows, we consider predictive regressions both with the real exchange rate alone and with several additional variables. 2. Methodology We now turn to the econometric methodology. We test for violations of the no-predictability null, with a specific focus on the predictive power of the real exchange rate, as the ability of the real exchange rate to predict excess currency returns has direct implications for our understanding of the economic drivers of the real exchange rate itself. We start from the case in which the real exchange rate is the only predictor of excess currency returns and in which the finite horizon tests involve one-period returns. We then extend the analysis to multiple predictors and multiperiod returns. We conclude with a discussion of the long-horizon tests. 2.1 The single-predictor case 2.1.1 Single-period horizon The present value representation (4) motivates the real exchange rate as a predictor of excess currency returns. Hence, consider the “UIP,” “PPP,” and “RRE” regressions, with the real exchange rate as the predictor:8 ξt+1=αξ+βξs˜t+eξ,t+1 (5) Δs˜t+1=αΔs˜+(ρs˜−1)s˜t+eΔs˜,t+1 (6) dr˜t+1=αdr˜+βdr˜s˜t+edr˜,t+1, (7) where ρs˜ is the serial correlation coefficient of s˜t and, per Equation (3): βdr˜=βξ+1−ρs˜. (8) Hence, the restriction tested in the finite horizon tests of no-predictability of excess currency returns is βξ=βdr˜+(ρs˜−1)=0. (9) Hence, the predictability of excess currency returns is the result of the combined effect of the predictability of real bond returns and of the appreciation of the real exchange rate. Absent predictive power of the real exchange rate for the real bond return differential (⁠ βdr˜=0 ⁠), the mean reversion in the real exchange rate translates into predictive power of the real exchange rate for excess currency returns (⁠ βξ=(ρs˜−1)<0 ⁠). 2.1.2 Long-horizon tests Consider now forming expectations by projecting the long-horizon excess currency returns ∑i=1∞ξt+i and real bond return differentials ∑i=1∞dr˜t+i on the current real exchange rate and let:9 βξ,∞≡βξ1−ρs˜ (10) βdr˜,∞≡βdr˜1−ρs˜ (11) denote the long-horizon regression coefficients. We have10 s˜t=−βξ,∞s˜t+βdr˜,∞s˜t. (12) Hence, we can decompose the variance of the real exchange rate as var(s˜t)=cov(s˜t,−βξ,∞s˜t)+cov(s˜t,βdr˜,∞s˜t)=−βξ,∞var(s˜t)+βdr˜,∞var(s˜t). (13) Normalizing each term by var(s˜t) ⁠, we obtain 1=−βξ,∞+βdr˜,∞, (14) where the first term—the “risk premium variance share”—captures the share of the variance due to changes in expected future risk premiums, and the second term—the “cash-flow variance share”—captures the variance due to changes in expected future real bond return differentials. Equation (14) is also useful in interpreting the implications of no-arbitrage models of currency pricing. Both the habit and long-run risks models imply a risk premium share of the volatility of the real exchange rate in excess of 100%. Because the variance-decomposition results in the univariate and multivariate settings are quite similar, based on (14), we can interpret this implication of the two models as being due to the fact that they imply a negative covariance between s˜t and dr˜t+1 ⁠. 2.2 The multiple-predictor case In the multiple-predictor case, in addition to the real exchange rate, our baseline choice of instruments is as follows. To the extent that currency risk premiums exhibit serial correlation, the lagged excess currency return also should be used as a predictor. Moreover, given the evidence from existing studies, we include the interest rate differential as a predictor. Finally, the inflation differential is also a natural candidate as a predictor, as inflation is one of the macro variables typically considered in the international finance literature (see, e.g., Engel and West, 2005). Hence, we define zt≡[s˜t,ξt,drt,dπt,xt⊤]⊤ ⁠, where xt is a vector of K possible additional instruments. The multiple-predictor case is meant to capture patterns of predictability above and beyond the simple AR(1) process (Equation (6)) for the real exchange rate assumed in the previous section. This setup accounts, for example, for the well-documented predictive power of the nominal interest rate differential for excess currency returns. 2.2.1 Finite horizon tests We estimate the monthly predictive regressions: ξH,t+H=αξ,H+βξ,H⊤zt+eξ,t+H, (15) where ξH,t+H≡∑h=1Hξt+h is the roll-over H-month excess return. We test the individual null hypotheses: βξ,H,k=0, (16) where βξ,H,k is the k-th element of βξ,H ⁠, and k=1,…,K+4 ⁠. In addition, we test the joint null hypotheses: βξ,H=04+K. (17) 2.2.2 Long-horizon tests We assume that zt follows a stationary VAR(1):11 zt=A+Bzt−1+vt, (18) where the eigenvalues of B lie inside the unit circle.12 Based on the VAR in (18), we compute the theoretical regression coefficients from projecting long-horizon cumulative excess returns on the real exchange rate and other instruments: β^ξ,∞⊤=β^ξ,1⊤(I−B^)−1. (19) Hence, we can test the individual restrictions, βξ,∞,k=0 ⁠, and the joint restriction: βξ,∞=04+K. (20) Using the VAR in Equation (18), we can revisit the present value representation of the real exchange rate in Equation (4). We have s˜t−E(s˜t)=−∑i=1∞Et(ξt+i−dr˜t+i)=−∑i=1∞Et [ξt+i−E(ξt)]+∑i=1∞Et[dr˜t+i−E(dr˜t)]−∑i=1∞ [E(ξt)−E(dr˜t)]=−∑i=1∞Et[ξt+i−E(ξt)]+∑i=1∞Et[dr˜t+i−E(dr˜t)]=−βξ,∞⊤[zt−E(zt)]+βdr˜,∞⊤[zt−E(zt)], (21) where, given the assumed stationarity of s˜t, E(ξt)−E(dr˜t)≡E(Δs˜t)=0 ⁠, and β^dr˜,∞⊤=β^dr˜,1⊤(I−B^)−1 ⁠. Therefore, we have var(s˜t)=cov(s˜t,−βξ,∞⊤zt)+cov(s˜t,βdr˜,∞⊤zt); (22) and 1=cov(s˜t,−βξ,∞⊤zt)var(s˜t)+cov(s˜t,βdr˜,∞⊤zt)var(s˜t). (23) In the empirical analysis, we use the difference between the risk premium variance share in the single- and multiple-predictor cases as an indicator of the role of the real exchange rate in capturing the long-term dynamics of ξt. If the difference is small, as it is in our sample, we can then conclude that the real exchange rate alone captures most of the long-term dynamics of excess currency returns. 3. Empirical Analysis 3.1 Data and empirical strategy Our data set, available from Datastream, includes monthly observations, over the period December 1983–April 2012, of foreign exchange rates, interest rates, and (seasonally unadjusted) consumer price indexes (CPIs) of the following 34 countries:13 G10 countries: Australia, Canada, Germany, Japan, New Zealand, Norway, Sweden, Switzerland, and the United Kingdom. Non-G10-developed countries: Austria, Belgium, Denmark, Finland, France, Greece, Ireland, Italy, Netherlands, Portugal, Singapore, and Spain. Emerging countries: Czech Republic, Hungary, India, Indonesia, Kuwait, Malaysia, Mexico, Philippines, Poland, South Africa, South Korea, Taiwan, and Thailand. In most of our tests, we parameterize the restricted VAR based on a panel data estimator, which uses data on all currencies and controls for currency-specific fixed effects. The panel data approach has two, related, advantages relative to the estimation of separate VARs for each country. First, we employ more data, and, as a result, we obtain more precise estimates. Second, the panel data VAR estimates lead to stable dynamics of the state variables, whereas some country-specific VAR estimates lead to unstable dynamics. Indeed, it is well documented that panel data estimation leads to more reliable estimates of exchange rate dynamics than pure time-series approaches: Mark and Sul (2001), Rapach and Wohar (2002), and Jordà and Taylor (2012), all use panel data vector VECMs, similar to our VAR, to produce robust forecasts of exchange rates. We consider four finite investment horizons: 1, 3, 6, and 12 months. Wald statistics are Newey-West adjusted for heteroscedasticity and moving-average serial correlation of order equal to the investment horizon minus one. We report both asymptotic and bootstrap results, where the bootstrap inference is based on a bootstrap of the VAR residuals. We bootstrap the residuals from the VAR equations, separately for each currency, but jointly across all the predictors for the same currency, to preserve properties of the joint distribution, such as asymmetry, fat tails, and cross-sectional dependence. The number of bootstrap repetitions is 5,000. To minimize the possible bias introduced by an arbitrary choice of starting values, in each bootstrap repetition we employ a new starting value randomly drawn from the time series of zt, and we also use a warm-up period of 60 months. 3.2 The single-predictor case We start with a discussion of the single-predictor (the real exchange rate) results, which illustrate in the simplest way the advantages of our approach. 3.2.1 Six major currencies Table 1 presents results for the six major currencies—Australian dollar, Canadian dollar, Deutsche Mark, Japanese yen, Swiss franc, and British pound—in the single-predictor case, at the 3-month and long horizons. Panel A presents results for the finite horizon, 3-month tests. While the real exchange rate predicts excess currency returns in the “right” direction (all coefficients are negative), the evidence is weak: none of the bootstrap p-values is below 5%. In addition, R2s never exceed 7%. Table 1 Testing no-predictability null, single-predictor case, six major currencies A. 3-month finite horizon tests . Country . . β^ξ,3 . R2 . Australia Estimate −0.02 .00 Asy. p .46 Boot. p .62 Canada Estimate −0.01 .00 Asy. p .56 Boot. p .69 Germany Estimate −0.10 .05 Asy. p .01 Boot. p .08 Japan Estimate −0.08 .05 Asy. p .02 Boot. p .11 Switzerland Estimate −0.06 .02 Asy. p .11 Boot. p .29 United Kingdom Estimate −0.16 .07 Asy. p .01 Boot. p .10 A. 3-month finite horizon tests . Country . . β^ξ,3 . R2 . Australia Estimate −0.02 .00 Asy. p .46 Boot. p .62 Canada Estimate −0.01 .00 Asy. p .56 Boot. p .69 Germany Estimate −0.10 .05 Asy. p .01 Boot. p .08 Japan Estimate −0.08 .05 Asy. p .02 Boot. p .11 Switzerland Estimate −0.06 .02 Asy. p .11 Boot. p .29 United Kingdom Estimate −0.16 .07 Asy. p .01 Boot. p .10 B. Long-horizon regression statistics . Country . β^ξ,∞ . Asy. p . Boot. p . Australia −0.51 .27 .46 Canada −0.44 .39 .51 Germany −0.90 .00 .00 Japan −0.91 .00 .00 Switzerland −0.72 .00 .10 United Kingdom −0.82 .00 .01 B. Long-horizon regression statistics . Country . β^ξ,∞ . Asy. p . Boot. p . Australia −0.51 .27 .46 Canada −0.44 .39 .51 Germany −0.90 .00 .00 Japan −0.91 .00 .00 Switzerland −0.72 .00 .10 United Kingdom −0.82 .00 .01 This table reports the results of 3-month finite horizon tests (panel A) and long-horizon tests (panel B) of the no-predictability null for the six major currencies during the December 1983–April 2012 sample period. In the 3-month finite horizon tests, we run linear regressions of 3-month excess currency returns on the real exchange rate and obtain the predictive coefficients (⁠ β^ξ,3 ⁠) and R2. The long-horizon tests are based on the long-horizon predictive coefficients (⁠ β^ξ,∞ ⁠), implied from the monthly VAR estimates for the instruments. Asy. p and Boot. p denote the asymptotic and bootstrap p-values, respectively, where the bootstrap inferences are under the no-predictability null, characterized by panel regression estimates of a separate, restricted VAR system. Open in new tab Table 1 Testing no-predictability null, single-predictor case, six major currencies A. 3-month finite horizon tests . Country . . β^ξ,3 . R2 . Australia Estimate −0.02 .00 Asy. p .46 Boot. p .62 Canada Estimate −0.01 .00 Asy. p .56 Boot. p .69 Germany Estimate −0.10 .05 Asy. p .01 Boot. p .08 Japan Estimate −0.08 .05 Asy. p .02 Boot. p .11 Switzerland Estimate −0.06 .02 Asy. p .11 Boot. p .29 United Kingdom Estimate −0.16 .07 Asy. p .01 Boot. p .10 A. 3-month finite horizon tests . Country . . β^ξ,3 . R2 . Australia Estimate −0.02 .00 Asy. p .46 Boot. p .62 Canada Estimate −0.01 .00 Asy. p .56 Boot. p .69 Germany Estimate −0.10 .05 Asy. p .01 Boot. p .08 Japan Estimate −0.08 .05 Asy. p .02 Boot. p .11 Switzerland Estimate −0.06 .02 Asy. p .11 Boot. p .29 United Kingdom Estimate −0.16 .07 Asy. p .01 Boot. p .10 B. Long-horizon regression statistics . Country . β^ξ,∞ . Asy. p . Boot. p . Australia −0.51 .27 .46 Canada −0.44 .39 .51 Germany −0.90 .00 .00 Japan −0.91 .00 .00 Switzerland −0.72 .00 .10 United Kingdom −0.82 .00 .01 B. Long-horizon regression statistics . Country . β^ξ,∞ . Asy. p . Boot. p . Australia −0.51 .27 .46 Canada −0.44 .39 .51 Germany −0.90 .00 .00 Japan −0.91 .00 .00 Switzerland −0.72 .00 .10 United Kingdom −0.82 .00 .01 This table reports the results of 3-month finite horizon tests (panel A) and long-horizon tests (panel B) of the no-predictability null for the six major currencies during the December 1983–April 2012 sample period. In the 3-month finite horizon tests, we run linear regressions of 3-month excess currency returns on the real exchange rate and obtain the predictive coefficients (⁠ β^ξ,3 ⁠) and R2. The long-horizon tests are based on the long-horizon predictive coefficients (⁠ β^ξ,∞ ⁠), implied from the monthly VAR estimates for the instruments. Asy. p and Boot. p denote the asymptotic and bootstrap p-values, respectively, where the bootstrap inferences are under the no-predictability null, characterized by panel regression estimates of a separate, restricted VAR system. Open in new tab Panel B presents results for the long-horizon inference, where the long-horizon statistics are based on VARs estimated separately for each country, although the UIP null is still parameterized based on the panel data VAR. In this single-predictor case, the B matrix of the VAR has nonzero elements only in the first column. As discussed earlier, in this single-predictor setting, at the long horizon, the negative of the long-horizon regression coefficients can be interpreted as the shares of the real exchange rate variance explained by future risk premium variation. The long-horizon evidence is stronger than the finite horizon evidence: three of the six coefficients are significant at the 5% level or better (bootstrap p-values) and point estimates range between −0.91 (Japan) and −0.44 (Canada). 3.2.2 Individual currencies and currency baskets Table 2 summarizes the single-predictor evidence across all currencies and currency baskets. The overall picture that emerges is consistent with what discussed above for the six major currencies. Based on bootstrap inference, the 3-month finite horizon tests (panel A) uncover only weak evidence of predictability, with only 4 rejections of 39 tests. The long-horizon evidence (panel B) is much stronger, with 12 rejections. Table 2 Testing no-predictability null, single-predictor case, summary A. 3-month finite horizon tests . . β^ξ,3=0 . No. of asy. p <.05 17 Frac. of asy. p <.05 .44 No. of boot. p <.05 4 Frac. of boot. p <.05 .10 No. of assets: 39 Avg. R2:.04 A. 3-month finite horizon tests . . β^ξ,3=0 . No. of asy. p <.05 17 Frac. of asy. p <.05 .44 No. of boot. p <.05 4 Frac. of boot. p <.05 .10 No. of assets: 39 Avg. R2:.04 B. Long-horizon regression statistics . . β^ξ,∞=0 . . Asy. . Boot. . No. of p <.05 27 12 Frac. of p <.05 .79 .35 No. of assets: 34 B. Long-horizon regression statistics . . β^ξ,∞=0 . . Asy. . Boot. . No. of p <.05 27 12 Frac. of p <.05 .79 .35 No. of assets: 34 This table summarizes the results of 3-month finite horizon tests (panel A) and long-horizon tests (panel B) of the no-predictability null for 34 currencies and 5 currency baskets during the December 1983–April 2012 sample period. The 3-month finite horizon tests run linear regressions of 3-month currency excess returns on the real exchange rate and obtain predictive coefficients (⁠ β^ξ,3 ⁠). The long-horizon tests are based on the long-horizon predictive coefficients (⁠ β^ξ,∞ ⁠), implied from monthly VAR estimates for the instruments. Each panel reports the numbers and the fractions of asymptotic tests or of bootstrap tests yielding p-values smaller than.05. The number of test assets (No. of assets) that generate valid results and the average R2 are also reported. The bootstrap inferences are under the no-predictability null, characterized by panel regression estimates of a separate, restricted VAR system. Open in new tab Table 2 Testing no-predictability null, single-predictor case, summary A. 3-month finite horizon tests . . β^ξ,3=0 . No. of asy. p <.05 17 Frac. of asy. p <.05 .44 No. of boot. p <.05 4 Frac. of boot. p <.05 .10 No. of assets: 39 Avg. R2:.04 A. 3-month finite horizon tests . . β^ξ,3=0 . No. of asy. p <.05 17 Frac. of asy. p <.05 .44 No. of boot. p <.05 4 Frac. of boot. p <.05 .10 No. of assets: 39 Avg. R2:.04 B. Long-horizon regression statistics . . β^ξ,∞=0 . . Asy. . Boot. . No. of p <.05 27 12 Frac. of p <.05 .79 .35 No. of assets: 34 B. Long-horizon regression statistics . . β^ξ,∞=0 . . Asy. . Boot. . No. of p <.05 27 12 Frac. of p <.05 .79 .35 No. of assets: 34 This table summarizes the results of 3-month finite horizon tests (panel A) and long-horizon tests (panel B) of the no-predictability null for 34 currencies and 5 currency baskets during the December 1983–April 2012 sample period. The 3-month finite horizon tests run linear regressions of 3-month currency excess returns on the real exchange rate and obtain predictive coefficients (⁠ β^ξ,3 ⁠). The long-horizon tests are based on the long-horizon predictive coefficients (⁠ β^ξ,∞ ⁠), implied from monthly VAR estimates for the instruments. Each panel reports the numbers and the fractions of asymptotic tests or of bootstrap tests yielding p-values smaller than.05. The number of test assets (No. of assets) that generate valid results and the average R2 are also reported. The bootstrap inferences are under the no-predictability null, characterized by panel regression estimates of a separate, restricted VAR system. Open in new tab 3.2.3 Panel evidence Table 3 presents the results from panel regression tests at the four finite horizons and the long horizon. Panel A presents the results of the finite horizon tests. The real exchange rate is a significant predictor, at all four finite horizons. At the long horizon (panel B), the predictive coefficient is a significant –0.86: this implies that 86% of the variance of the real exchange rate can be attributed to changes in currency risk premiums.14 Table 3 Testing no-predictability null, single-predictor case, panel regressions A. Finite horizon tests . Horizon . β^ξ,H . β^ξ,H=0 Wald (Asy. p) . 1 −0.02 10.81 (.00) 3 −0.07 21.93 (.00) 6 −0.15 33.20 (.00) 12 −0.30 31.69 (.00) A. Finite horizon tests . Horizon . β^ξ,H . β^ξ,H=0 Wald (Asy. p) . 1 −0.02 10.81 (.00) 3 −0.07 21.93 (.00) 6 −0.15 33.20 (.00) 12 −0.30 31.69 (.00) B. Long-horizon regression statistics . β^ξ,∞ . −0.86 . Asy. p (.00) B. Long-horizon regression statistics . β^ξ,∞ . −0.86 . Asy. p (.00) This table reports the results of finite horizon tests (panel A) and long-horizon tests (panel B) of the no-predictability null, obtained from panel regressions using all 34 currencies and controlling for currency fixed effects during the December 1983–April 2012 sample period. The finite horizon tests run linear regressions of 1-, 3-, 6-, and 12-month currency excess returns on the real exchange rate and obtain predictive coefficients β^ξ,H ⁠, where H=1,3,6, or 12. The long-horizon tests are based on the long-horizon predictive coefficients (⁠ β^ξ,∞ ⁠), implied from monthly VAR estimates for the instruments. Asymptotic p-values, based on covariance matrix estimators clustered by both currency and time, are reported in parentheses under the estimates or the test statistics. Open in new tab Table 3 Testing no-predictability null, single-predictor case, panel regressions A. Finite horizon tests . Horizon . β^ξ,H . β^ξ,H=0 Wald (Asy. p) . 1 −0.02 10.81 (.00) 3 −0.07 21.93 (.00) 6 −0.15 33.20 (.00) 12 −0.30 31.69 (.00) A. Finite horizon tests . Horizon . β^ξ,H . β^ξ,H=0 Wald (Asy. p) . 1 −0.02 10.81 (.00) 3 −0.07 21.93 (.00) 6 −0.15 33.20 (.00) 12 −0.30 31.69 (.00) B. Long-horizon regression statistics . β^ξ,∞ . −0.86 . Asy. p (.00) B. Long-horizon regression statistics . β^ξ,∞ . −0.86 . Asy. p (.00) This table reports the results of finite horizon tests (panel A) and long-horizon tests (panel B) of the no-predictability null, obtained from panel regressions using all 34 currencies and controlling for currency fixed effects during the December 1983–April 2012 sample period. The finite horizon tests run linear regressions of 1-, 3-, 6-, and 12-month currency excess returns on the real exchange rate and obtain predictive coefficients β^ξ,H ⁠, where H=1,3,6, or 12. The long-horizon tests are based on the long-horizon predictive coefficients (⁠ β^ξ,∞ ⁠), implied from monthly VAR estimates for the instruments. Asymptotic p-values, based on covariance matrix estimators clustered by both currency and time, are reported in parentheses under the estimates or the test statistics. Open in new tab 3.3 The multiple-predictor (baseline) case We now turn to the discussion of the results for the baseline case, with the four instruments: real exchange rate, excess currency return, nominal interest rate differential, and inflation differential. 3.3.1 Individual currencies and currency baskets We start with the results for 39 currencies and currency baskets in total. Based on the bootstrap inference, the finite horizon tests uncover only weak evidence of predictability. For example, at the 3-month horizon (Table 4), in the finite horizon tests, the no-predictability null is rejected in seven, one, ten, and zero instances, for the real exchange rate, excess currency return, interest rate differential, and inflation differential, respectively.15 Table 4 Testing no-predictability null, baseline case, 3-month investment horizon, summary . Explanatory variables . . s˜ . ξ . dr . dπ . No. of asy. p <.05 22 3 15 0 Frac. of asy. p <.05 .56 .08 .38 .00 No. of boot. p <.05 7 1 10 0 Frac. of boot. p <.05 .18 .03 .26 .00 No. of assets: 39 Avg. adj R2:.05 . Explanatory variables . . s˜ . ξ . dr . dπ . No. of asy. p <.05 22 3 15 0 Frac. of asy. p <.05 .56 .08 .38 .00 No. of boot. p <.05 7 1 10 0 Frac. of boot. p <.05 .18 .03 .26 .00 No. of assets: 39 Avg. adj R2:.05 This table summarizes the results of the 3-month finite horizon tests of the no-predictability null for 34 currencies and 5 currency baskets during the December 1983–April 2012 sample period. The 3-month finite horizon tests run linear regressions of 3-month excess returns of currencies and currency baskets on four instrumental variables: the real exchange rate (⁠ s˜ ⁠), the 1-month excess return (ξ), the nominal interest rate differential (dr), and the inflation differential (⁠ dπ ⁠). Below each instrument, the table reports the numbers and the fractions of asymptotic tests or of bootstrap tests yielding p-values smaller than.05 for the associated coefficient estimate. The number of test assets that generate valid results and the average R2 are also reported. The bootstrap tests are under the no-predictability null, characterized by panel regression estimates of a separate, restricted VAR system. Open in new tab Table 4 Testing no-predictability null, baseline case, 3-month investment horizon, summary . Explanatory variables . . s˜ . ξ . dr . dπ . No. of asy. p <.05 22 3 15 0 Frac. of asy. p <.05 .56 .08 .38 .00 No. of boot. p <.05 7 1 10 0 Frac. of boot. p <.05 .18 .03 .26 .00 No. of assets: 39 Avg. adj R2:.05 . Explanatory variables . . s˜ . ξ . dr . dπ . No. of asy. p <.05 22 3 15 0 Frac. of asy. p <.05 .56 .08 .38 .00 No. of boot. p <.05 7 1 10 0 Frac. of boot. p <.05 .18 .03 .26 .00 No. of assets: 39 Avg. adj R2:.05 This table summarizes the results of the 3-month finite horizon tests of the no-predictability null for 34 currencies and 5 currency baskets during the December 1983–April 2012 sample period. The 3-month finite horizon tests run linear regressions of 3-month excess returns of currencies and currency baskets on four instrumental variables: the real exchange rate (⁠ s˜ ⁠), the 1-month excess return (ξ), the nominal interest rate differential (dr), and the inflation differential (⁠ dπ ⁠). Below each instrument, the table reports the numbers and the fractions of asymptotic tests or of bootstrap tests yielding p-values smaller than.05 for the associated coefficient estimate. The number of test assets that generate valid results and the average R2 are also reported. The bootstrap tests are under the no-predictability null, characterized by panel regression estimates of a separate, restricted VAR system. Open in new tab As the return horizon increases, the evidence from direct tests is similar, with an increase in the predictive power of the inflation differential. At the 12-month horizon, for example, the no-predictability null is rejected in 9, 0, 11, and 13 instances, for the real exchange rate, excess currency return, interest rate differential, and inflation differential, respectively. Table 5 reports results for the long-horizon inference. Across currencies, there are 12 rejections of 34 tests, for the real exchange rate, and 10 rejections for the nominal interest rate differential.16 Hence, for the real exchange rate, the rejection rate (35%, i.e., 12 of 34 test assets) is higher than the rejection rates from the finite horizon tests (at most 26%). The excess currency return and inflation differential, on the other hand, are significant in only one and three tests, respectively. The joint tests lead to 13 rejections, the risk premium variance share is significant in nine instances, and the increase in variance share due to the addition of instruments to the real exchange rate is significant in seven instances. Table 5 Testing no-predictability null, baseline case, long-horizon regressions, summary . Explanatory variables . β^ξ,∞=0 . Variance . Δ Variance . . s˜ . ξ . dr . dπ . Wald . share . share . No. of asy. p <.05 25 2 7 4 30 22 0 Frac. of asy. p <.05 .74 .06 .21 .12 .88 .65 .00 No. of boot. p <.05 12 1 10 3 13 9 7 Frac. of boot. p <.05 .35 .03 .29 .09 .38 .26 .21 No. of assets: 34 . Explanatory variables . β^ξ,∞=0 . Variance . Δ Variance . . s˜ . ξ . dr . dπ . Wald . share . share . No. of asy. p <.05 25 2 7 4 30 22 0 Frac. of asy. p <.05 .74 .06 .21 .12 .88 .65 .00 No. of boot. p <.05 12 1 10 3 13 9 7 Frac. of boot. p <.05 .35 .03 .29 .09 .38 .26 .21 No. of assets: 34 This table summarizes the results of long-horizon tests of the no-predictability null for 34 currencies and 5 currency baskets during the December 1983–April 2012 sample period. The tests are based on the long-horizon predictive coefficients, implied from monthly VAR estimates for the four instrumental variables: the real exchange rate (⁠ s˜ ⁠), the 1-month excess return (ξ), the nominal interest rate differential (dr), and the inflation differential (⁠ dπ ⁠). The table reports the numbers and the fractions of asymptotic tests or of bootstrap tests yielding p-values smaller than.05; below the instruments are the individual tests of the associated coefficient estimates; below “ β^∞=0,Wald” is the joint tests of all the coefficient estimates; below “Variance Share” is the tests of the risk premium share of the real exchange rate variance; and below “Δ Variance Share” is the tests of the incremental risk premium share, when compared with the single-predictor case. The bootstrap tests are under the no-predictability null, characterized by panel regression estimates of a separate, restricted VAR system. Open in new tab Table 5 Testing no-predictability null, baseline case, long-horizon regressions, summary . Explanatory variables . β^ξ,∞=0 . Variance . Δ Variance . . s˜ . ξ . dr . dπ . Wald . share . share . No. of asy. p <.05 25 2 7 4 30 22 0 Frac. of asy. p <.05 .74 .06 .21 .12 .88 .65 .00 No. of boot. p <.05 12 1 10 3 13 9 7 Frac. of boot. p <.05 .35 .03 .29 .09 .38 .26 .21 No. of assets: 34 . Explanatory variables . β^ξ,∞=0 . Variance . Δ Variance . . s˜ . ξ . dr . dπ . Wald . share . share . No. of asy. p <.05 25 2 7 4 30 22 0 Frac. of asy. p <.05 .74 .06 .21 .12 .88 .65 .00 No. of boot. p <.05 12 1 10 3 13 9 7 Frac. of boot. p <.05 .35 .03 .29 .09 .38 .26 .21 No. of assets: 34 This table summarizes the results of long-horizon tests of the no-predictability null for 34 currencies and 5 currency baskets during the December 1983–April 2012 sample period. The tests are based on the long-horizon predictive coefficients, implied from monthly VAR estimates for the four instrumental variables: the real exchange rate (⁠ s˜ ⁠), the 1-month excess return (ξ), the nominal interest rate differential (dr), and the inflation differential (⁠ dπ ⁠). The table reports the numbers and the fractions of asymptotic tests or of bootstrap tests yielding p-values smaller than.05; below the instruments are the individual tests of the associated coefficient estimates; below “ β^∞=0,Wald” is the joint tests of all the coefficient estimates; below “Variance Share” is the tests of the risk premium share of the real exchange rate variance; and below “Δ Variance Share” is the tests of the incremental risk premium share, when compared with the single-predictor case. The bootstrap tests are under the no-predictability null, characterized by panel regression estimates of a separate, restricted VAR system. Open in new tab In summary, the long-horizon tests uncover more predictive power for the real exchange rate than the finite horizon tests. The evidence on the predictive power of the nominal interest rate differential is consistent with the extensive existing evidence on deviations from UIP reviewed by, for example, Sarno (2005). On the other hand, the evidence on the predictive power of the real exchange rate is largely a novel contribution of this paper. 3.3.2 Panel evidence Table 6 presents evidence based on the panel regression estimation. Given the large total number of observations—7,826 currency-month observations—inference is based on asymptotics, where standard errors are clustered by currency and time and where we control for currency fixed effects. Table 6 Testing no-predictability null, baseline case, panel regressions A. Finite horizon tests . . Explanatory variables . . s˜ . ξ . dr . dπ . Horizon . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . 1 −0.02 0.10 0.57 −0.01 (.00) (.03) (.00) (.97) 3 −0.07 0.17 1.26 0.02 (.00) (.00) (.02) (.91) 6 −0.15 0.19 2.55 0.63 (.00) (.05) (.00) (.02) 12 −0.29 0.20 3.68 1.90 (.00) (.06) (.03) (.00) A. Finite horizon tests . . Explanatory variables . . s˜ . ξ . dr . dπ . Horizon . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . 1 −0.02 0.10 0.57 −0.01 (.00) (.03) (.00) (.97) 3 −0.07 0.17 1.26 0.02 (.00) (.00) (.02) (.91) 6 −0.15 0.19 2.55 0.63 (.00) (.05) (.00) (.02) 12 −0.29 0.20 3.68 1.90 (.00) (.06) (.03) (.00) B. Long-horizon regression statistics . Explanatory variables . β^ξ,∞=0 . Variance . Δ Variance . s˜ . ξ . dr . dπ . Wald . share . share . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . −0.93 0.04 1.57 −0.14 817.16 0.93 0.07 (.00) (.01) (.00) (.07) (.00) (.00) (.02) B. Long-horizon regression statistics . Explanatory variables . β^ξ,∞=0 . Variance . Δ Variance . s˜ . ξ . dr . dπ . Wald . share . share . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . −0.93 0.04 1.57 −0.14 817.16 0.93 0.07 (.00) (.01) (.00) (.07) (.00) (.00) (.02) This table reports the results of finite horizon tests (panel A) and long-horizon tests (panel B) of the no-predictability null, obtained from panel regressions using all 34 currencies and controlling for currency fixed effects during the December 1983–April 2012 sample period. In the finite horizon tests, we run linear regressions of 1-, 3-, 6-, and 12-month currency excess returns on the real exchange rate (⁠ s˜ ⁠), the 1-month excess return (ξ), the nominal interest rate differential (dr), and the inflation differential (⁠ dπ ⁠). The long-horizon tests are based on the long-horizon predictive coefficients, implied from monthly VAR estimates for the instruments. Below the instruments are the associated coefficient estimates. Below “ β^∞=0,Wald” is the joint test statistic on all the coefficient estimates. Below “Variance Share” is the risk premium share of the real exchange rate variance. Finally, below “Δ Variance Share” is the incremental risk premium share, when compared with the single-predictor case. Asymptotic p-values, based on covariance matrix estimators clustered by both currency and time, are reported in parentheses under the estimates or the test statistics. Open in new tab Table 6 Testing no-predictability null, baseline case, panel regressions A. Finite horizon tests . . Explanatory variables . . s˜ . ξ . dr . dπ . Horizon . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . 1 −0.02 0.10 0.57 −0.01 (.00) (.03) (.00) (.97) 3 −0.07 0.17 1.26 0.02 (.00) (.00) (.02) (.91) 6 −0.15 0.19 2.55 0.63 (.00) (.05) (.00) (.02) 12 −0.29 0.20 3.68 1.90 (.00) (.06) (.03) (.00) A. Finite horizon tests . . Explanatory variables . . s˜ . ξ . dr . dπ . Horizon . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . 1 −0.02 0.10 0.57 −0.01 (.00) (.03) (.00) (.97) 3 −0.07 0.17 1.26 0.02 (.00) (.00) (.02) (.91) 6 −0.15 0.19 2.55 0.63 (.00) (.05) (.00) (.02) 12 −0.29 0.20 3.68 1.90 (.00) (.06) (.03) (.00) B. Long-horizon regression statistics . Explanatory variables . β^ξ,∞=0 . Variance . Δ Variance . s˜ . ξ . dr . dπ . Wald . share . share . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . −0.93 0.04 1.57 −0.14 817.16 0.93 0.07 (.00) (.01) (.00) (.07) (.00) (.00) (.02) B. Long-horizon regression statistics . Explanatory variables . β^ξ,∞=0 . Variance . Δ Variance . s˜ . ξ . dr . dπ . Wald . share . share . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . −0.93 0.04 1.57 −0.14 817.16 0.93 0.07 (.00) (.01) (.00) (.07) (.00) (.00) (.02) This table reports the results of finite horizon tests (panel A) and long-horizon tests (panel B) of the no-predictability null, obtained from panel regressions using all 34 currencies and controlling for currency fixed effects during the December 1983–April 2012 sample period. In the finite horizon tests, we run linear regressions of 1-, 3-, 6-, and 12-month currency excess returns on the real exchange rate (⁠ s˜ ⁠), the 1-month excess return (ξ), the nominal interest rate differential (dr), and the inflation differential (⁠ dπ ⁠). The long-horizon tests are based on the long-horizon predictive coefficients, implied from monthly VAR estimates for the instruments. Below the instruments are the associated coefficient estimates. Below “ β^∞=0,Wald” is the joint test statistic on all the coefficient estimates. Below “Variance Share” is the risk premium share of the real exchange rate variance. Finally, below “Δ Variance Share” is the incremental risk premium share, when compared with the single-predictor case. Asymptotic p-values, based on covariance matrix estimators clustered by both currency and time, are reported in parentheses under the estimates or the test statistics. Open in new tab As one would expect, imposing a panel structure on the data leads to substantial additional power in all tests. Panel A shows evidence of predictability in the finite horizon tests, especially strong for the real exchange rate and the interest rate differential. To assess the economic magnitude of the effects, we also compute the scaled regression coefficients, which give us effects on the dependent variable in units of standard deviation, for a 1-standard-deviation change in the independent variable. While not tabulated here, the scaled regression coefficients associated with the real exchange rate range between –0.36 and –0.11, whereas the scaled coefficients associated with the interest rate differential range between 0.07 and 0.12, corroborating the notion of stronger predictive power of real exchange rates than interest rate differentials. As to the long-horizon evidence, we have very strong rejections of the no-predictability null for both the real exchange rate and the nominal interest rate differential. The risk premium variance share is a strongly significant 93%, and the addition of other instruments to the real exchange rate further increases the risk premium share only by 7%.17 Finally, we also performed the panel data analysis separately for the three groups of countries: G10, non-G10, and emerging. Results are qualitatively similar to those obtained for a single panel of countries: the long-horizon tests uncover much stronger evidence of predictability than the finite horizon tests, especially for the real exchange rate; the implied long-horizon regression coefficients are also especially significant for the real exchange rate; and a large fraction of the variance of the real exchange rate can be attributed to changes in risk premiums (between 82% and 100%), where the inclusion of instruments in addition to the real exchange rate makes little difference (at most 9%). 3.4 Relation to Engel (2016) At this point, it is interesting to reexamine the evidence of Engel (2016). Engel (2016) finds that the expected real bond return differential, Et(dr˜t+1) ⁠, correlates positively with short-horizon excess currency returns, Et(ξt+1) ⁠, but negatively with long-horizon excess currency returns, Et(∑i=1∞ξt+i) ⁠, for all G6 currencies. We replicate and extend the analysis of Engel (2016) to all 34 currencies in our data set, by running regressions of expected one-period and long-horizon excess currency returns on the expected real bond return differential, with or without the real exchange rate as an additional control (see the results in Table 7), where expectations are computed using both our VAR and Engel’s VECM.18 The table presents results for the panel data evidence (The Internet Appendix, Table IA.4, provides results for the individual currencies). When the dependent variable is the expected short-horizon excess currency return, the expected real bond return differential enters the regression with a positive coefficient, both when the real exchange rate is controlled for and when it is not. When the dependent variable is the expected long-horizon excess currency return and the real exchange rate is not controlled for, the expected real bond return enters with a negative coefficient, consistent with Engel (2016). However, when the real exchange rate is controlled for, the sign turns positive. Hence, it appears that once deviations from PPP are accounted for, a high expected real bond return differential positively predicts the appreciation of the currency at both short and long horizons. Similarly consistent across horizons is the sign of the coefficient associated with the real exchange rate, which is negative in both cases. Table 7 Expected real bond differential versus short-horizon and long-horizon risk premium, panel evidence . VECM . VAR . . Model (1) . Model (2) . Model (1) . Model (2) . Dependent variable . Et(dr˜t+1) . Et(dr˜t+1) . s˜t . Et(dr˜t+1) . Et(dr˜t+1) . s˜t . Et(ξt+1) 0.77 0.95 −0.03 0.74 0.93 −0.02 Et(∑i=1∞ξt+i) −3.01 3.70 −0.97 −5.33 1.84 −0.93 . VECM . VAR . . Model (1) . Model (2) . Model (1) . Model (2) . Dependent variable . Et(dr˜t+1) . Et(dr˜t+1) . s˜t . Et(dr˜t+1) . Et(dr˜t+1) . s˜t . Et(ξt+1) 0.77 0.95 −0.03 0.74 0.93 −0.02 Et(∑i=1∞ξt+i) −3.01 3.70 −0.97 −5.33 1.84 −0.93 This table reports estimates of regression coefficients associated with expected real bond return differential (⁠ Et(dr˜t+1) ⁠) below “Model (1),” or with real bond return differential and real exchange rate (⁠ s˜t ⁠) below “Model (2),” where the dependent variable is expected long-horizon excess return, Et(∑i=1∞ξt+i) ⁠. The estimates are obtained from panel regressions using all 34 currencies and controlling for currency fixed effects during the December 1983–April 2012 sample period. The conditional expectations are formed using the VECM by Engel (2016) or our baseline VAR model. Open in new tab Table 7 Expected real bond differential versus short-horizon and long-horizon risk premium, panel evidence . VECM . VAR . . Model (1) . Model (2) . Model (1) . Model (2) . Dependent variable . Et(dr˜t+1) . Et(dr˜t+1) . s˜t . Et(dr˜t+1) . Et(dr˜t+1) . s˜t . Et(ξt+1) 0.77 0.95 −0.03 0.74 0.93 −0.02 Et(∑i=1∞ξt+i) −3.01 3.70 −0.97 −5.33 1.84 −0.93 . VECM . VAR . . Model (1) . Model (2) . Model (1) . Model (2) . Dependent variable . Et(dr˜t+1) . Et(dr˜t+1) . s˜t . Et(dr˜t+1) . Et(dr˜t+1) . s˜t . Et(ξt+1) 0.77 0.95 −0.03 0.74 0.93 −0.02 Et(∑i=1∞ξt+i) −3.01 3.70 −0.97 −5.33 1.84 −0.93 This table reports estimates of regression coefficients associated with expected real bond return differential (⁠ Et(dr˜t+1) ⁠) below “Model (1),” or with real bond return differential and real exchange rate (⁠ s˜t ⁠) below “Model (2),” where the dependent variable is expected long-horizon excess return, Et(∑i=1∞ξt+i) ⁠. The estimates are obtained from panel regressions using all 34 currencies and controlling for currency fixed effects during the December 1983–April 2012 sample period. The conditional expectations are formed using the VECM by Engel (2016) or our baseline VAR model. Open in new tab 4. Further Analysis 4.1 Size and power Two natural questions arise at this stage: First, does the bootstrap adjustment lead to the correct inference? Second, are the long-horizon tests indeed more powerful than the standard finite horizon tests? To the best of our knowledge, we are the first to perform a size and power study of the long-horizon tests introduced by Cochrane (2008). Our simulation evidence (see Section IA.1 in the Internet Appendix for details) shows that (1) the bootstrap correction of the inference is needed and eliminates size distortions and (2) the power of the long-horizon test is substantially higher than that of the traditional finite horizon tests. The reason for the superior properties of the long-horizon regression coefficients can be easily seen in the single-predictor case and has to do with the correlation between the regression coefficients in the different one-period regressions. In a simulation calibrated to the U.S. dollar/Deutsche Mark currency pair, we find that corr(−β^ξ,1−ρ^s˜)=0.99 and corr(β^dr˜,1−ρ^s˜)=0.13 ⁠. Thus, the estimated numerator and denominator in −β^ξ,∞ and β^dr˜,∞ are positively correlated, increasing the precision of the estimates of the two ratios.19 A similar effect is likely to hold in the multivariate case, as the real exchange rate captures most of the long-term dynamics of excess currency returns. 4.2 Extensions and robustness checks We perform several extensions of our analysis and we implement several robustness checks of our baseline results:20 (1) nonstationarity of the real exchange rate; (2) correction of possible spurious-regression biases; (3) alternative choices of parameters for the process for the real exchange rate; (4) no-predictability null hypothesis parameterized according to currency-specific VARs; (5) correction of VAR parameter estimates for small-sample biases; (6) controlling for additional predictors; (7) use of alternative approaches to bootstrap; and (8) use of different reference currencies. In summary, we find that assuming a nonstationary real exchange rate leads to the implausible result that essentially 100% of the variance of the real exchange rate appreciation is due to revisions of long-horizon PPP deviations. We also find that the predictability patterns uncovered in this study are very robust, and that long-horizon tests are crucial to uncover such predictability. 5. Implications of the Theory In this section, we discuss the theory behind the predictability patterns in excess currency returns. First, we consider a general no-arbitrage model. Second, we specialize the analysis to two specific models: the habit model (Verdelhan, 2010) and the long-run risks model (Bansal and Shaliastovich, 2013).21 Both models are modified relative to their original formulations by assuming that domestic and foreign log consumption are cointegrated, rather than independent random walks. This assumption delivers a stationary real exchange rate. 5.1 A general framework Consider a general no-arbitrage model. The differential between the foreign and domestic real log pricing kernels is given by dm˜t ⁠. Absent arbitrage, we have Δs˜t=dm˜t ⁠. Hence, assuming stationarity of the log pricing kernel differential, we have dm˜t−E(dm˜t)=−∑i=1∞Et(dm˜t+i)=−∑i=1∞Et(Δs˜t+i)=s˜t−E(s˜t). (24) The (log) real exchange rate, in deviation from its unconditional mean, reflects transitory fluctuations in the log pricing kernel differential. As to the implications of the model for predictability patterns, if we assume lognormality, symmetric economies (same preferences and same stochastic processes for the relevant state variables), and homoskedastic and neutral inflation, we have Et(ξt+1)=−12dσm˜t2 and22 cov[s˜t,Et(ξt+1)]=−12cov(dm˜t,dσm˜t2), (25) where dσm˜t2 is the differential between the foreign and domestic volatilities of real log pricing kernels. Equation (25) highlights the stationarity of the real exchange rate alone does not mechanically translate in the real exchange rate negatively predicting excess currency returns, as it is prevalent in the data. What is needed is a positive covariance between the level and volatility differentials dm˜t and dσm˜t2 ⁠. In the following, we verify whether this condition is met by realistically parameterized versions of the habit and long-run risks models. 5.2 The habit model 5.2.1 Preferences and equilibrium quantities The symmetric habit model Verdelhan (2010) assumes that the real log pricing kernel differential equals dm˜t=−γ(Δdht+Δdc˜t), (26) where dht is the differential in log consumption surplus ratios, dht≡ln[(C˜tf−X˜tf)/C˜tf]−ln[(C˜t−X˜t)/C˜t] ⁠, where C˜tf and C˜t are foreign and domestic real per capita consumption, respectively; X˜tf and X˜t are the foreign and domestic external “habit” consumption, respectively; and dc˜t≡ln(C˜tf/C˜t) is the differential in real log per capita consumption.23 We have s˜t−E(s˜t)=−γ(dht+dc˜t)=−γ[dln(C˜t−X˜t)]. (27) The equation above highlights how the real exchange rate is driven by the differential in log excess consumption: what matters is not the differential in the absolute consumption level, but the differential in consumption relative to the external habit. Hence, two countries may have the same level of consumption, but experience different marginal utilities of consumption because of different external habit levels. In turn, the difference in marginal utilities leads to deviations from PPP. Like in Verdelhan (2010), we have24,25 Et(ξt+1)=γ2σϵc2H¯2dht, (28) where H¯ is a “tuning” parameter. Note that dht affects the real exchange rate and the currency risk premium in opposite directions. A higher differential in the habit level of consumption increases both the differential in marginal utility of consumption and the differential in the volatility of the marginal utility of consumption. The first effect drives the real exchange rate up, whereas the second effect drives the currency risk premium down. Indeed, the covariance between the real exchange rate and the currency risk premium is given by26 cov[s˜t,Et(ξt+1)]=−γ3σϵc˜2H¯2[var(dht)+cov(dht,dc˜t)]=−γ3σϵc2H¯2{var(dht)+2E[λ(ht)](1−ρϵc˜ϵc˜f)σϵc˜2}. (29) For our choice of parameters, this covariance is negative. Turning to the nominal interest rate differential, we have drt=−bdht+γ(φc−1)dc˜t+φπdπt, (30) where b≡γ(1−φh)−γ2σϵc2H¯2 ⁠. Note that the second and third term in the expression above are absent in the solutions of Verdelhan (2010), as he does not impose the stationarity of the real exchange rate and derives the real, rather than the nominal, interest rate differential.27 Following Verdelhan (2010), we choose b < 0. Hence, an increase in dht affects the nominal interest differential and the currency risk premium in the same direction. The covariance between the two quantities equals cov[drt,Et(ξt+1)]=γ2σϵc2H¯2[−bvar(dht)+γ(φc−1)cov(dht,dc˜t)]. (31) For our choice of parameter values, the covariance above is positive, implying that the nominal interest rate differential positively predicts the excess currency return. Finally, the expected real bond return differential equals Et(dr˜t+1)=drt−φπdπt≡−bdht+γ(φc−1)dc˜t. (32) Note that dht affects the real exchange rate and the expected real bond return differential in opposite directions and the covariance between the two quantities is given by cov[s˜t,Et(dr˜t+1)]=γbvar(dht)−γ2(φc−1)var(dc˜t). (33) For our choice of parameter values, the covariance above is negative and the real exchange rate predicts the real bond return differential in the “wrong” direction: as a result, the model implies a risk premium variance share of the real exchange rate in excess of 100%. 5.2.2 Calibration Like in Verdelhan (2010) and Bansal and Shaliastovich (2013), we focus on the implications of the model for the dollar/British pound exchange rate. Following Verdelhan (2010) we set as baseline values: γ = 2, φh=0.99 ⁠, and b=−0.01 ⁠. Based on our data for the United States and the United Kingdom, we set (quarterly frequency): φc=0.97 ⁠, φπ=−0.24 ⁠, σϵc˜=0.0066 ⁠, σϵπ=0.0078 ⁠, ρϵc˜fϵc˜=0.44 ⁠, and ρϵπfϵπ=0.24 ⁠. Table 8 reports the results from the calibration exercise. The table also considers the effect of deviations of γ and b from the baseline scenario: “high” denotes a value twice as large as the baseline, whereas “low” denotes a value equal to one-half of the baseline value. The population finite horizon regression statistics are based on regressions with three regressors: s˜t ⁠, drt, and dπt ⁠, because there are only three state variables in this habit model. Similarly, the long-horizon statistics are based on a VAR(1) modeling the joint dynamics of s˜t ⁠, drt, and dπt ⁠.28 We calculate population p-values associated with the regression coefficients, assuming that the analysis is performed with a sample of 96 quarterly observations, to match the number of quarterly observations in the sample. Table 8 Calibration A. Summary statistics . . σΔs˜ . ρs˜ . σdr . σξ . σdr˜ . Habit, baseline 0.3395 .9889 0.0081 0.3406 0.0203 Habit, high γ 0.4620 .9885 0.0051 0.4625 0.0193 Habit, low γ 0.1929 .9881 0.0079 0.1946 0.0202 LRR, baseline 0.1060 .9940 0.0082 0.1082 0.0009 LRR, high γ 0.4527 .9940 0.0278 0.4554 0.0006 LRR, low γ 0.0242 .9934 0.0050 0.0306 0.0009 LRR, high ψ 0.1083 .9940 0.0085 0.1105 0.0012 LRR, low ψ 0.1019 .9939 0.0077 0.1040 0.0002 Sample 0.1027 .8054 0.0051 0.1043 0.0881 A. Summary statistics . . σΔs˜ . ρs˜ . σdr . σξ . σdr˜ . Habit, baseline 0.3395 .9889 0.0081 0.3406 0.0203 Habit, high γ 0.4620 .9885 0.0051 0.4625 0.0193 Habit, low γ 0.1929 .9881 0.0079 0.1946 0.0202 LRR, baseline 0.1060 .9940 0.0082 0.1082 0.0009 LRR, high γ 0.4527 .9940 0.0278 0.4554 0.0006 LRR, low γ 0.0242 .9934 0.0050 0.0306 0.0009 LRR, high ψ 0.1083 .9940 0.0085 0.1105 0.0012 LRR, low ψ 0.1019 .9939 0.0077 0.1040 0.0002 Sample 0.1027 .8054 0.0051 0.1043 0.0881 B. UIP tests . . R2 . βξ,s˜ . p(βξ,s˜=0) . βξ,dr . p(βξ,dr=0) . Habit, baseline .0089 –0.0129 .7959 0.4286 .9803 Habit, high γ .0058 –0.0115 .4680 0.3846 .9886 Habit, low γ .0133 –0.0150 .5306 0.5000 .9155 LRR, baseline .0133 –0.0107 .8647 0.3252 .9710 LRR, high γ .0132 –0.0105 .9668 0.3207 .9933 LRR, low γ .0085 –0.0110 .6241 0.3353 .8944 LRR, high ψ .0140 –0.0108 .9342 0.3304 .9855 LRR, low ψ .0118 –0.0103 .7239 0.3144 .9428 Sample .0689 –0.2096 0.0022 1.6850 .1309 B. UIP tests . . R2 . βξ,s˜ . p(βξ,s˜=0) . βξ,dr . p(βξ,dr=0) . Habit, baseline .0089 –0.0129 .7959 0.4286 .9803 Habit, high γ .0058 –0.0115 .4680 0.3846 .9886 Habit, low γ .0133 –0.0150 .5306 0.5000 .9155 LRR, baseline .0133 –0.0107 .8647 0.3252 .9710 LRR, high γ .0132 –0.0105 .9668 0.3207 .9933 LRR, low γ .0085 –0.0110 .6241 0.3353 .8944 LRR, high ψ .0140 –0.0108 .9342 0.3304 .9855 LRR, low ψ .0118 –0.0103 .7239 0.3144 .9428 Sample .0689 –0.2096 0.0022 1.6850 .1309 C. Long-horizon statistics . . Variance share ξ . Δ Variance share ξ . Variance share dr˜ . Habit, baseline 1.4145 0.1448 –0.4145 Habit, high γ 1.1576 0.1491 –0.1576 Habit, low γ 1.8059 0.2936 –0.8059 LRR, baseline 2.1484 0.0101 –1.1484 LRR, high γ 2.1067 0.0005 –1.1067 LRR, low γ 2.2019 0.1975 –1.2019 LRR, high ψ 2.2021 0.0025 –1.2021 LRR, low ψ 2.0374 0.0414 –1.0374 Sample 0.6032 –0.2586 0.3968 C. Long-horizon statistics . . Variance share ξ . Δ Variance share ξ . Variance share dr˜ . Habit, baseline 1.4145 0.1448 –0.4145 Habit, high γ 1.1576 0.1491 –0.1576 Habit, low γ 1.8059 0.2936 –0.8059 LRR, baseline 2.1484 0.0101 –1.1484 LRR, high γ 2.1067 0.0005 –1.1067 LRR, low γ 2.2019 0.1975 –1.2019 LRR, high ψ 2.2021 0.0025 –1.2021 LRR, low ψ 2.0374 0.0414 –1.0374 Sample 0.6032 –0.2586 0.3968 This table reports the empirical implications of the habit and long-run risks (LRR) models, calibrated for a sample of 96 quarterly observations using U.S. and U.K. data. Panel A reports the following annualized summary statistics: volatilities (σ) of real currency appreciation (⁠ Δs˜ ⁠), nominal interest rate differential (dr), 1-month excess return (ξ), and real bond return differential (⁠ dr˜ ⁠), as well as autocorrelation in real exchange rate (⁠ ρs˜ ⁠). Panel B reports statistics of the UIP regression, including R2, the predictive coefficients associated with the real exchange rate and the interest rate differential, and their p-values. Panel C reports long-horizon statistics, including the risk premium share of the real exchange rate variance, the incremental variance share relative to one-predictor case, and the cash-flow share of real exchange rate variance. “High” and “low” denote scenarios in which a parameter—risk aversion (γ) for the habit model, or elasticity of intertemporal substitution (ψ) for the LLR model—is set to a value double or one-half of its baseline value, respectively. The sample counterparts are reported in the rows labeled “Sample.” Open in new tab Table 8 Calibration A. Summary statistics . . σΔs˜ . ρs˜ . σdr . σξ . σdr˜ . Habit, baseline 0.3395 .9889 0.0081 0.3406 0.0203 Habit, high γ 0.4620 .9885 0.0051 0.4625 0.0193 Habit, low γ 0.1929 .9881 0.0079 0.1946 0.0202 LRR, baseline 0.1060 .9940 0.0082 0.1082 0.0009 LRR, high γ 0.4527 .9940 0.0278 0.4554 0.0006 LRR, low γ 0.0242 .9934 0.0050 0.0306 0.0009 LRR, high ψ 0.1083 .9940 0.0085 0.1105 0.0012 LRR, low ψ 0.1019 .9939 0.0077 0.1040 0.0002 Sample 0.1027 .8054 0.0051 0.1043 0.0881 A. Summary statistics . . σΔs˜ . ρs˜ . σdr . σξ . σdr˜ . Habit, baseline 0.3395 .9889 0.0081 0.3406 0.0203 Habit, high γ 0.4620 .9885 0.0051 0.4625 0.0193 Habit, low γ 0.1929 .9881 0.0079 0.1946 0.0202 LRR, baseline 0.1060 .9940 0.0082 0.1082 0.0009 LRR, high γ 0.4527 .9940 0.0278 0.4554 0.0006 LRR, low γ 0.0242 .9934 0.0050 0.0306 0.0009 LRR, high ψ 0.1083 .9940 0.0085 0.1105 0.0012 LRR, low ψ 0.1019 .9939 0.0077 0.1040 0.0002 Sample 0.1027 .8054 0.0051 0.1043 0.0881 B. UIP tests . . R2 . βξ,s˜ . p(βξ,s˜=0) . βξ,dr . p(βξ,dr=0) . Habit, baseline .0089 –0.0129 .7959 0.4286 .9803 Habit, high γ .0058 –0.0115 .4680 0.3846 .9886 Habit, low γ .0133 –0.0150 .5306 0.5000 .9155 LRR, baseline .0133 –0.0107 .8647 0.3252 .9710 LRR, high γ .0132 –0.0105 .9668 0.3207 .9933 LRR, low γ .0085 –0.0110 .6241 0.3353 .8944 LRR, high ψ .0140 –0.0108 .9342 0.3304 .9855 LRR, low ψ .0118 –0.0103 .7239 0.3144 .9428 Sample .0689 –0.2096 0.0022 1.6850 .1309 B. UIP tests . . R2 . βξ,s˜ . p(βξ,s˜=0) . βξ,dr . p(βξ,dr=0) . Habit, baseline .0089 –0.0129 .7959 0.4286 .9803 Habit, high γ .0058 –0.0115 .4680 0.3846 .9886 Habit, low γ .0133 –0.0150 .5306 0.5000 .9155 LRR, baseline .0133 –0.0107 .8647 0.3252 .9710 LRR, high γ .0132 –0.0105 .9668 0.3207 .9933 LRR, low γ .0085 –0.0110 .6241 0.3353 .8944 LRR, high ψ .0140 –0.0108 .9342 0.3304 .9855 LRR, low ψ .0118 –0.0103 .7239 0.3144 .9428 Sample .0689 –0.2096 0.0022 1.6850 .1309 C. Long-horizon statistics . . Variance share ξ . Δ Variance share ξ . Variance share dr˜ . Habit, baseline 1.4145 0.1448 –0.4145 Habit, high γ 1.1576 0.1491 –0.1576 Habit, low γ 1.8059 0.2936 –0.8059 LRR, baseline 2.1484 0.0101 –1.1484 LRR, high γ 2.1067 0.0005 –1.1067 LRR, low γ 2.2019 0.1975 –1.2019 LRR, high ψ 2.2021 0.0025 –1.2021 LRR, low ψ 2.0374 0.0414 –1.0374 Sample 0.6032 –0.2586 0.3968 C. Long-horizon statistics . . Variance share ξ . Δ Variance share ξ . Variance share dr˜ . Habit, baseline 1.4145 0.1448 –0.4145 Habit, high γ 1.1576 0.1491 –0.1576 Habit, low γ 1.8059 0.2936 –0.8059 LRR, baseline 2.1484 0.0101 –1.1484 LRR, high γ 2.1067 0.0005 –1.1067 LRR, low γ 2.2019 0.1975 –1.2019 LRR, high ψ 2.2021 0.0025 –1.2021 LRR, low ψ 2.0374 0.0414 –1.0374 Sample 0.6032 –0.2586 0.3968 This table reports the empirical implications of the habit and long-run risks (LRR) models, calibrated for a sample of 96 quarterly observations using U.S. and U.K. data. Panel A reports the following annualized summary statistics: volatilities (σ) of real currency appreciation (⁠ Δs˜ ⁠), nominal interest rate differential (dr), 1-month excess return (ξ), and real bond return differential (⁠ dr˜ ⁠), as well as autocorrelation in real exchange rate (⁠ ρs˜ ⁠). Panel B reports statistics of the UIP regression, including R2, the predictive coefficients associated with the real exchange rate and the interest rate differential, and their p-values. Panel C reports long-horizon statistics, including the risk premium share of the real exchange rate variance, the incremental variance share relative to one-predictor case, and the cash-flow share of real exchange rate variance. “High” and “low” denote scenarios in which a parameter—risk aversion (γ) for the habit model, or elasticity of intertemporal substitution (ψ) for the LLR model—is set to a value double or one-half of its baseline value, respectively. The sample counterparts are reported in the rows labeled “Sample.” Open in new tab Across choices of parameter values, a few implications of the habit model stand out. First, similarly to Verdelhan (2010), the model overstates the volatility of changes in the (log) real exchange rate and the persistence of the level of the (log) real exchange rate. This is a result of the fact that the variable that dominates the variability of the log pricing kernel differential is dht, which is very volatile and persistent. Second, in the UIP regressions, the model underestimates the magnitudes of the coefficients associated with the real exchange rate and the nominal interest rate differential, although it reproduces their signs. Finally, the model overstates the risk premium variance share: for example, 141% for the baseline choice of parameters, as compared to a sample value of 60%. This result is due to the fact that the real exchange rate negatively predicts the real bond return differential. In summary, the habit model replicates a number of features of the data, including the fact that a dominant portion of the variance of real exchange rates can be attributed to risk premiums. On the other hand, the model overstates the volatility of real exchange rate appreciation, as well as the risk premium share of the real exchange rate variance. 5.3 The long-run risks model 5.3.1 Preferences and equilibrium quantities Following Bansal and Yaron (2004) and Bansal and Shaliastovich (2013), we assume m˜t=constant−θψΔc˜t+(θ−1)r˜ct, (34) where θ≡(1−γ)/(1−1/ψ) ⁠. ψ is the elasticity of intertemporal substitution; and r˜ct is the rate of return on the portfolio that delivers aggregate consumption, the “consumption asset.” Following the literature, we assume γ>1 and ψ>1 ⁠. As a result, we have θ<0 ⁠. Given the assumptions above, we have the log real pricing kernel differential:29 dm˜t=−γΔdc˜t+my⊤(κdyt−dyt−1); (35) where dyt≡[dc˜t dvt dπt]⊤ and my≡[mdc mdv 0]⊤ ⁠, where dvt denotes the differential in the conditional variance of log consumption in the two countries. We also have30 s˜t−E(s˜t)=−γdc˜t+my⊤dyt=(−γ+mdc)dc˜t+mdvdvt. (36) The differential in marginal utilities reflects not only the difference in current consumption but also the difference in equity valuations in the two countries. This difference, in turn, is driven by the log consumption differential as well as the volatility differential dvt. Hence, two countries may have the same level of current consumption, but different expected consumption growth and volatility of consumption growth, leading to deviations from PPP. Turning to the currency risk premium, it is driven by the differential consumption volatility process, dvt: Et(ξt+1)=−12mdc2dvt. (37) Given our choice of parameters, mdv > 0: an increase in consumption volatility reduces the valuation of the consumption asset, and an increase in the consumption volatility differential increases the foreign pricing kernel relative to the domestic pricing kernel. Hence, the covariance: cov[s˜t,Et(ξt+1)]=−12mdvmdc2var(dvt) (38) is negative. The intuition for the result is that when dvt increases, both the level and the volatility of the foreign pricing kernel increase, relative to the domestic pricing kernel. The relative increase in the foreign pricing kernel drives the real exchange rate higher, whereas the relative increase in the volatility of consumption growth drives the currency risk premium lower, leading to the negative covariance above. The nominal interest rate differential equals drt=(−γ+mdc)(1−φc)dc˜t+[mdv(1−φv)−12mdc2]dvt+φπdπt. (39) Note that the consumption volatility differential, dvt, affects the nominal interest rate differential through two channels. First, an increase in dvt reduces the valuation of the foreign consumption asset and increases its expected rate of return, relative to the domestic consumption asset. This effect drives down the expected pricing kernel differential—the two pricing kernels are inversely related to the rate of return on the corresponding consumption assets—and drives up the interest rate differential. Second, an increase in the consumption volatility differential increases the precautionary savings motive in the foreign country relative to the domestic country, driving down the interest rate differential. Indeed, the (conditional) covariance between the nominal interest rate differential and the currency risk premium is given by cov[drt,Et(ξt+1)]=[12mdc2−mdv(1−φv)]12mdc2var(dvt), (40) whose sign depends on the sign of the quantity inside the brackets. For our choice of parameter values, this quantity is positive. Finally, the expected real bond return differential equals Et(dr˜t+1)=[(−γ+mdc)(1−φc)]dc˜t+[mdv(1−φv)−12mdc2]dvt; (41) and cov[s˜t,Et(dr˜t+1)]=(−γ+mdc)2(1−φc)var(dc˜t)+mdv[mdv(1−φv)−12mdc2]var(dvt). (42) Given our choice of parameter values, the covariance above is negative. Hence, like in the habit model, a higher real exchange rate predicts a lower real bond return differential, and a variance decomposition exercise attributes more than 100% of the variance of the real exchange rate to risk premium variation. 5.3.2 Calibration Following Bansal and Shaliastovich (2013), we set γ = 12 and ψ=1.81 ⁠. The mean equations for consumption growth and inflation are parameterized similarly to the habit model. As to the process for the volatility of consumption innovations, we follow Bansal and Shaliastovich (2013) and assume φv=0.994 and ρϵvfϵv=0.94 ⁠.31 Like in the case of the habit model, we consider the effect of deviations of the preference parameters (γ and ψ) from the baseline scenario. We derive population statistics for the UIP regressions and for the long-horizon analysis, and we assume that the instruments in the predictive regressions and the VAR are s˜t ⁠, drt, and dπt ⁠. Similarly to the habit model, across parameter choices, the long-run risks model matches some features of the data, but misses other features (see Table 8). In particular, the model roughly matches the predictive power of the nominal interest rate differential for the excess currency return. On the other hand, the model grossly overstates the risk premium share of the variance of the real exchange: 215% for the baseline choice of parameters versus 60% in the data. 5.4 Interpretation The result that both the habit and long-run risks model imply a risk premium variance share of the real exchange above 100% can be understood as follows. The two models imply that cov[s˜t,−Et(∑i=1∞ξt+i)]>var(s˜t). (43) In turn, this means that cov[Et(∑i=1∞ξt+i),Et(∑i=1∞dr˜t+i)]>var[Et(∑i=1∞dr˜t+i)]. (44) In other words, both models imply a large positive covariance between the expected long-horizon excess currency return and the expected long-horizon real bond return differential. In the habit (long-run risks) model, this positive covariance is driven by the differential in log excess consumption (log excess consumption volatility). In the data (based on our panel data VAR estimates, for example), on the other hand, this covariance is negative, thus leading to positive variance shares for both the risk premium and the expected-cash flows components of the real exchange rate.32 6. Conclusions We provide new evidence that real exchange rates contain important information about—and they vary over time mainly because of—currency risk premiums, rather than expected cash flows. We perform multivariate finite- and long-horizon tests of the predictive power of the real exchange rate and a variance decomposition exercise. While the finite horizon tests uncover little predictive power, the implied long-horizon regression statistics reject the no-predictability null. The variance decomposition exercise shows that 93% of the variability of the real exchange rate can be attributed to changing expectations of future excess currency returns (panel regression evidence). Moreover, the real exchange rate alone captures most of the long-horizon predictability of excess currency returns. We relate our empirical findings to the implications of the habit and long-run risks models. Both models reproduce the predictive power of the real exchange rate and the nominal interest rate differential for excess currency returns. On the other hand, both models imply that the risk premium and expected cash-flow components of the real exchange rate are strongly negatively correlated. As a result, they substantially overstate the fraction of the volatility of the real exchange rate due to risk premiums. Acknowledgement We especially thank Rui Albuquerque for several useful conversations. We also thank Yi-Ting Chen (session chair), Pasquale Della Corte (discussant), Jin-Chuan Duan, Eyal Dvir, Chang-Tai Hsieh, Huichou Huang (session chair), Can Inci (discussant), Michael Kisser (discussant), Carolin Pflueger (discussant), Dmitriy Muravyev, Andreas Stathopoulos (discussant), Georg Strasser, Allan Timmermann, Ruey Tsay, Rosen Valchev, and Adrien Verdelhan and seminar participants at American University, Board of Governors of the Federal Reserve System, Boston College (Finance and Economics), Mississippi State University, UNC Charlotte, the 10th International Symposium on Econometric Theory and Applications (SETA 2014), the 2015 Eastern Finance Association Annual Meetings, the 2015 Financial Management Association Annual Meetings, the 2015 Taiwan Economics Research Workshop, the 2016 European Finance Association Meetings, the 2016 Midwestern Finance Association Annual Meetings, the 2017 American Finance Association Annual Meetings, and the 2017 Financial Management Association Meetings for useful comments. Ethan Chiang acknowledges the support of a Belk College Summer Research Grant. Supplementary data can be found on The Review of Asset Pricing Studies Web site. Footnotes 1 Although predictability in excess currency returns constitutes a violation of uncovered interested rate parity (UIP), even a constant, nonzero expected currency excess return violates UIP. Our analysis focuses on this specific type of violation of UIP, because it is related to the economically relevant question of what drives real exchange rate variation. 2 The evidence from the multiple-predictor case, where the instruments follow a VAR(1) process, is similar. Indeed, the difference between the risk premium share of the variance of the real exchange rate when the real exchange is the only predictor and when other predictors are accounted for is small, suggesting that the real exchange rate alone captures most of the long-term dynamics of the excess currency returns. 3 The six major currencies are the Australian dollar, Canadian dollar, Deutsche Mark/Euro, Japanese yen, Swiss franc, and British pound. 4 Note that the panel regression analysis also uncovers significant predictability patterns in the finite horizon tests. However, we believe that it is important to derive significant results at the individual currency level, as the panel data evidence may be driven by a few currencies with long time series. Moreover, the panel data analysis implicitly assumes homogeneity across currencies, which may not hold in the data. 5 Evans (2017) also starts from a present value representation of the log real exchange rate, but assumes that investors constantly update their expectation of long-horizon deviations from absolute purchasing power parity (PPP). His empirical analysis attributes most of the variance of changes in log real exchange rates to “dark matter,” which combines changes in risk premiums with revisions of expected long-horizon PPP deviations. When we apply the methodology of Evans (2017) to our setting, we find that virtually all of the variability of changes in real exchange rates is due to revisions of expected long-horizon PPP deviations. This result seems implausible, and we mainly attribute it to the misspecification of the joint dynamics of the real exchange rate, currency returns, and inflation differentials. 6 The recent working paper by Eichenbaum, Johannsen, and Rebelo (2017), on the other hand, compares the predictive power of the real exchange rate for the appreciation of the nominal exchange rate versus the inflation differential. Similar to an earlier version of the current paper, they find that the predictive ability of the real exchange rate is limited to the former. 7 Authors, such as Lustig and Verdelhan (2007), Farhi, Fraiberger, Gabaix, Ranciere, and Verdelhan (2015), Ang and Chen (2010), Burnside, Eichenbaum, and Rebelo (2011), Lustig, Roussanov, and Verdelhan (2011), Asness, Moskwitz, and Pedersen (2013), and Menkhoff, Sarno, Schmeling, and Schrimpf (2016), focus on cross-sectional predictability patterns. Menkhoff, Sarno, Schmeling, and Schrimpf (2016), in particular, show that adjusting real exchange rates for productivity and the quality of export goods generates a measure of currency value, which negatively predicts excess currency returns and is consistent with the spot exchange rate reverting toward fundamental value. 8 As noted earlier, we use the term “UIP” loosely to denote the null hypothesis of constant expected excess currency returns. In the international finance literature, UIP typically denotes the case of constant zero expected currency returns. 9 We are using the fact that Et(ξt+i)=Et[Et+i−1(ξt+i)]=constant+Et(βξs˜t+i−1)=constant+ρs˜i−1βξs˜. An analogous result holds for Et(dr˜t+i) ⁠. 10 See Cochrane (2008), Equation (10), for an analogous decomposition of the variance of the log dividend-price ratio. 11 Note that a restricted version of this VAR is used to generate bootstrap samples under the UIP null. Specifically, we estimate the restricted VAR at the monthly frequency and then we use powers of the restricted B matrix to parameterize the indirect null at different horizons. See the Internet Appendix (Section IA.2) for details. 12 If we replace zt with [s˜t,Δst,dπt]⊤ in the VAR above, we obtain a conventional vector error correction model (VECM). In the VECM, st and ptf−pt are cointegrated with cointegrating vector [1,1]⊤ ⁠, and, like in our VAR, s˜t is stationary (see Jordà and Taylor, 2012; Engel, 2016). Relative to the VECM, the VAR in (18) differs by including the dynamics of the interest rate differential, drt. 13 See the Internet Appendix (Section IA.3) for further details on the data set. Note that the G10 countries are selected based on trading volume in the currency market, not on measures of economic development. In addition, one of the G10 countries is the United States, and, hence, the list only contains nine countries. 14 For a comparison, see Cochrane (2008), who estimates at 109% the fraction of the variance of the aggregate dividend yield explained by expectations of future stock returns, using U.S. yearly data for the 1926–2004 sample. 15 In the interest of space, we relegate the results for the other horizons to the Internet Appendix (Tables IA.1–IA.3). 16 We exclude five currencies from the long-horizon analysis, because the corresponding VARs imply nonstationary dynamics. 17 The risk premium variance share in the single-predictor case is 86%, which, combined with the 7% change in share, leads to the 93% overall share. 18 Note that projecting Et(∑i=1∞ξt+i) onto Et(dr˜t+1) (and s˜t ⁠) is equivalent to projecting ∑i=1∞ξt+i onto Et(dr˜t+1) (and s˜t ⁠). In other words, our regression exercises document the predictive ability of the expected real bond return differential for excess currency returns in the infinite future (without or with the real exchange rate as a control). 19 Cochrane (2008) documents a similar effect: the estimates of the coefficients of regressions of stock returns and dividend growth on the log dividend price ratio are negatively correlated with the estimate of the serial correlation coefficient of the log dividend price ratio and the resultant long-horizon regression coefficients’ estimated numerator and denominator are again positively correlated. 20 In the interest of space, we relegate the details of this analysis to the Internet Appendix (Section IA.4). 21 We do not study the liquidity model of Engel (2016). His model has no direct channel connecting the real exchange rate to expected excess currency returns: the connection between the two quantities only takes place through the reaction function of monetary authorities in the two countries. See the Internet Appendix (Section IA.5) for further discussion. Additionally, we do not consider the “rare disasters” model of Farhi and Gabaix (2016). Their model implies that, as a country becomes more risky, the country’s currency depreciates and its interest rate increases; that is, it implies a strong negative correlation between the real exchange rate and the expected real bond return differential. This implication is at odds with the evidence from our sample. 22 In the formulation of the habit and long-run risks models, we also assume homoskedastic and neutral inflation, consistent with Verdelhan (2010). Bansal and Shaliastovich (2013) show that the nonneutrality of inflation is important to match the key features of the term structure in the data. However, we show that the long-run risks model does not need inflation nonneutrality to reproduce the predictability patterns in excess currency returns. 23 The domestic representative agent’s intertemporal marginal rate of substitution (i.e., the domestic economy’s pricing kernel) is given by exp (m˜t)=β(C˜t−X˜tC˜t−1−X˜t−1)−γ=β(HtHt−1)−γ(C˜tC˜t−1)−γ, where β is the time discount factor and Ht≡(C˜t−X˜t)/C˜t is the surplus consumption ratio. Taking logs, we have m˜t=ln(β)−γΔht−γΔc˜t ⁠, where ht≡ln(Ht) and c˜t≡lnC˜t ⁠. Analogous expressions apply to the foreign economy. 24 Here and in the following, μXt, σXt, σXYt ⁠, and ρXYt denote the mean, standard deviation, covariance, and correlation coefficient of the corresponding X and Y variables, conditional on time t available information. Below, we also introduce φX ⁠, denoting the serial correlation coefficient of the X variable, and ϵX, denoting the innovation in X. In the absence of a time subscript, the moments should be interpreted as constant over time. 25 See the Internet Appendix (Section IA.6.1) for details. 26 See the Internet Appendix (Section IA.6.2) for the derivation of the conditional moments of dht. 27 When the log per capita consumption differential dc˜t follows a random walk, φc=1 ⁠, and the second term on the right-hand side of (30) disappears. 28 The Internet Appendix (Section IA.7) shows that by employing as many instruments in the VAR as the number of underlying state variables—three, in this case—the VAR-implied variance decomposition coincides with the theoretical variance decomposition. 29 See the Internet Appendix (Sections IA.8.1 and IA.8.2) for details of the assumptions and the solution. 30 See the Internet Appendix (Section IA.8.3) for details. 31 See the Internet Appendix (Section IA.8.4) for details of the calibration of the volatility process. 32 Note that this feature of the data is akin to the finding in Engel (2016) that cov[Et(∑i=1∞ξt+i),Et(dr˜t+1)]<0 for the G6 countries (see Section 3.4). 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For permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Review of Asset Pricing Studies Oxford University Press http://www.deepdyve.com/lp/oxford-university-press/real-exchange-rates-and-currency-risk-premiums-4GlIk0BDvI

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Publisher: Oxford University Press
Copyright: © The Author(s) 2019. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For permissions, please email: journals.permissions@oup.com
ISSN: 2045-9920
eISSN: 2045-9939
DOI: 10.1093/rapstu/raz002
Publisher site: See Article on Publisher Site

Abstract

Abstract Standard finite horizon tests uncover only weak evidence of the predictive power of the real exchange rate for excess currency returns. On the other hand, in long-horizon tests, the real exchange rate strongly and negatively predicts future excess currency returns. Conversely, we can attribute most of the variability in real exchange rates to changes in currency risk premiums. The “habit” and “long-run risks” models replicate the predictive power of the real exchange rate for excess currency returns, but substantially overstate the fraction of the volatility of the real exchange rate due to risk premiums. Received December 14, 2017; Editorial decision October 14, 2018 by Editor: Raman Uppal. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online. The main research question addressed in this paper is whether real exchange rates predict excess currency returns.1 The literature has mainly focused on other instruments—most notably, the nominal interest rate differential—to characterize the time variation in currency risk premiums. Moreover, the conventional view is that the predictive ability of real exchange rates for currency returns is weak (see, e.g., the review article by Rossi (2013). We address the question above by performing finite horizon (i.e., “short-horizon”) and infinite horizon (i.e., “long-horizon”) tests of the null hypothesis that excess (log) currency returns are unpredictable. Specifically, we test whether long-horizon cumulative excess currency returns are time varying as a function of the real exchange rate and other instruments. Moreover, we use a present value representation to decompose the variance of the real exchange rate into shares due to risk premiums and due to expected cash flows. Our empirical results, based on bootstrap inference, are easily summarized. Consider the case in which the real exchange rate is the only predictor, following a univariate AR(1) process.2 For individual currencies, we find little evidence of predictive power of the real exchange rate in finite horizon tests. For example, consider the six major currencies at the 3-month forecast horizon.3 The real exchange rate negatively predicts excess returns for all six currencies, with coefficients ranging from –0.16 (British pound) to –0.02 (Australian dollar), but none of the coefficients is significant at the 5% level (bootstrap p-value). On the other hand, the long-horizon regressions are more supportive of the predictive power of the real exchange rate: the long-horizon regression coefficient is significant for 3 of 6 major currencies. Moreover, the absolute magnitude of the coefficients is large. For example, in the case of the Japanese yen, the long-horizon regression coefficient equals –0.91, which implies that 91% of the variance of the real exchange rate can be attributed to the time variation in risk premiums. Results are similar, although statistically much more significant, in a panel regression setting.4 In particular, the panel regression estimates quantify at 86% the risk premium share of the variance of the real exchange rate. This is an important stylized fact that the literature has all but overlooked. For example, Cochrane (2011) does not recognize the predictive power of the real exchange rate for excess currency returns, and its implications for the source of real exchange rate variability, focusing on the nominal interest rate differential. A size and power study demonstrates that long-horizon tests are indeed substantially more powerful than the standard finite horizon tests. We also show that our results are robust to possible spurious-regression biases, alternative parameterizations of the null, the correction for small-sample biases in the parameterization of the null hypothesis, the inclusion of additional predictors, and the use of alternative reference currencies. Having established that (1) real exchange rates do predict excess currency returns and (2) the risk premium component dominates real exchange rate variation, in the second part of the paper, we relate our analysis to the implications of no-arbitrage models of exchange rate determination, specifically, the habit and long-run risks models. We amend the existing versions of the two models (see Verdelhan, 2010;Bansal and Shaliastovich, 2013) to ensure the stationarity of the real exchange rate, and we derive population values for regression and long-horizon statistics. We show that both models replicate the result that the real exchange rate (nominal interest rate differential) negatively (positively) predicts excess currency returns. Moreover, both models replicate the fact that long-horizon tests are more powerful than finite horizon tests. Both models also imply that most of the real exchange rate variability is due to risk premiums, rather than expected cash flows, although they overstate the magnitude of the effect. Indeed, in both models the real exchange rate and the nominal interest rate differential are negatively related, leading to a negative covariance between the real exchange rate and the differential between foreign and domestic expected real bond returns. This negative covariance translates into a negative expected cash-flow variance share, and a risk premium share in excess of 100%. The literature on exchange rate determination and predictability is vast. In the remainder of this section, we focus on the studies that are most relevant for our long-horizon tests and variance-decomposition analysis. Froot and Ramadorai (2005) use the real exchange rate as a predictor of currency returns, in the context of a vector autoregression (VAR) estimated with daily data in a panel setting, for the sample from June 20, 1994, to February 9, 2001. Following Campbell (1991), they decompose the variance of innovations in currency returns into shares attributable to risk premium news and fundamental news, finding that risk premium news dominates. However, the main focus of their analysis—the effect of net order flow on currency prices—is different from ours. In addition, they do not make inference on implied long-horizon regression coefficients. Engel and West (2005) study the relation between exchange rates and macro variables, using quarterly data for the January 1974–September 2001 sample. They investigate whether nominal currency returns Granger-cause macro variables (differentials in nominal money growth, inflation, nominal interest rates, and gross domestic product growth), finding some supportive evidence. Moreover, similarly to our variance decomposition exercise, they test whether nominal currency returns are correlated with changes in the present values of several macro variables, finding that in most cases the correlations are insignificant. We improve on their analysis by providing a framework—the present value representation in Equation (4)—that identifies the real bond return differential as the single macro variable being anticipated by exchange rates. This unifying framework allows us to relate the predictive power of real exchange rates for excess currency returns and real bond return differentials, respectively, to the roles of risk premiums and expected cash flows in determining (real) exchange rate variation. Engel (2016) studies the relation between the long-horizon cumulative UIP deviation and the current expected real bond return differential. He finds that the covariance between the two quantities is negative: high real-interest-rate currencies tend to experience low cumulative excess returns in the long run. This result stands in contrast with the well-documented deviations from UIP: high real-interest-rate currencies tend to experience high excess returns in the short run. Engel (2016) demonstrates that the two stylized facts cannot be accommodated by existing no-arbitrage models. Our analysis differs from that of Engel (2016), because we focus on the relation between the long-horizon excess currency returns and the real exchange rate, rather than the expected real bond return differential. Indeed, we show that the real exchange rate is the most robust predictor of cumulative excess currency returns. Moreover, we show that once we control for the real exchange rate, the expected real bond return differential predicts long-horizon excess currency returns with a positive sign.5 In a recent working paper Dahlquist and Penasse (2017) show how both the nominal interest rate differential and the real exchange rate are needed to predict excess currency returns. Indeed, they find that accounting for the real exchange rate raises the R-squared in predictive regressions for monthly returns by about 30% on a portfolio of currencies. Their empirical result is in line with the amended habit and long-run risks models in Section 5: expected currency returns are captured by a linear combination of both the nominal interest rate differential and the real exchange rate (in addition to the inflation differential).6,7 In summary, we make the following main contributions to the existing literature. First, we show that, relative to the finite horizon tests, the long-horizon tests uncover stronger evidence of the predictive power of the real exchange rate for excess currency returns. Second, by performing a variance-decomposition exercise, we show that the predictability patterns that we document translate into the economically important result that a dominant fraction of the variance of the real exchange rate can be attributed to changes in risk premiums. Third, we modify the existing habit and long-run risks models (Verdelhan, 2010; Bansal and Shaliastovich, 2013) to accommodate stationarity of the real exchange rate. We show how these models replicate several salient predictability patterns, for example, the negative correlation between real exchange rates and future excess currency returns. However, they also overstate the risk premium share of the variance of the real exchange rate. 1. Preliminaries Let rt and rtf denote the continuously compounded domestic and foreign nominal risk-free rates and let drt≡rtf−rt denote the interest rate differential. Throughout, the superscript f denotes a variable of the foreign country, and d denotes the differential between foreign and domestic variables: dX≡Xf−X ⁠. Δ denotes first differences: ΔXt≡Xt−Xt−1 ⁠. Finally, X˜ denotes a real quantity, and X^ denotes an estimate. Let st denote the log of the directly quoted nominal exchange rate and let ξt+1≡st+1−st+drt denote the time t + 1 excess log currency return, that is, the log currency return in excess of the cost of carry (⁠ −drt ⁠). Our empirical analysis focuses on testing the null hypothesis that expected excess currency returns are constant: Et(ξt+1)=E(ξt+1). (1) We denote this null hypothesis the no-predictability null. Let pt and ptf denote the log domestic and foreign price levels. We define the log real exchange rate s˜t as s˜t≡st+ptf−pt. (2) Note that if absolute PPP holds, the real exchange rate equals one, and the log real exchange rate equals zero. Taking first differences of (2), we have Δs˜t+1≡s˜t+1−s˜t=st+1−st+pt+1f−ptf−(pt+1−pt)≡st+1−st+πt+1f−πt+1=st+1−st+drt−drt+dπt+1≡ξt+1−dr˜t+1, (3) where πt+i and πt+if are the domestic and foreign continuously compounded inflation rates, respectively, dπt+1≡πt+1f−πt+1 is the inflation differential, and dr˜t+1≡drt−dπt+1 is the real bond return differential. The variables defined above appear in the following present value representation of the real exchange rate (it can be obtained by iterating (3) forward, taking conditional expectations, and assuming that relative PPP is expected to hold in the long run (see, e.g., Froot and Ramadorai, 2005; Engel, 2016): s˜t−E(s˜t)=−∑i=1∞Et(ξt+i)+∑i=1∞Et(dr˜t+i). (4) The two terms on the right-hand side of (4) capture long-horizon cumulative deviations from UIP and real rate equality (RRE), respectively. Note that while (4) implies that the real exchange rate is related to risk premiums and expected future real bond return differentials, it does not imply that the real exchange rate is the sole predictor of them. Indeed, in the analysis that follows, we consider predictive regressions both with the real exchange rate alone and with several additional variables. 2. Methodology We now turn to the econometric methodology. We test for violations of the no-predictability null, with a specific focus on the predictive power of the real exchange rate, as the ability of the real exchange rate to predict excess currency returns has direct implications for our understanding of the economic drivers of the real exchange rate itself. We start from the case in which the real exchange rate is the only predictor of excess currency returns and in which the finite horizon tests involve one-period returns. We then extend the analysis to multiple predictors and multiperiod returns. We conclude with a discussion of the long-horizon tests. 2.1 The single-predictor case 2.1.1 Single-period horizon The present value representation (4) motivates the real exchange rate as a predictor of excess currency returns. Hence, consider the “UIP,” “PPP,” and “RRE” regressions, with the real exchange rate as the predictor:8 ξt+1=αξ+βξs˜t+eξ,t+1 (5) Δs˜t+1=αΔs˜+(ρs˜−1)s˜t+eΔs˜,t+1 (6) dr˜t+1=αdr˜+βdr˜s˜t+edr˜,t+1, (7) where ρs˜ is the serial correlation coefficient of s˜t and, per Equation (3): βdr˜=βξ+1−ρs˜. (8) Hence, the restriction tested in the finite horizon tests of no-predictability of excess currency returns is βξ=βdr˜+(ρs˜−1)=0. (9) Hence, the predictability of excess currency returns is the result of the combined effect of the predictability of real bond returns and of the appreciation of the real exchange rate. Absent predictive power of the real exchange rate for the real bond return differential (⁠ βdr˜=0 ⁠), the mean reversion in the real exchange rate translates into predictive power of the real exchange rate for excess currency returns (⁠ βξ=(ρs˜−1)<0 ⁠). 2.1.2 Long-horizon tests Consider now forming expectations by projecting the long-horizon excess currency returns ∑i=1∞ξt+i and real bond return differentials ∑i=1∞dr˜t+i on the current real exchange rate and let:9 βξ,∞≡βξ1−ρs˜ (10) βdr˜,∞≡βdr˜1−ρs˜ (11) denote the long-horizon regression coefficients. We have10 s˜t=−βξ,∞s˜t+βdr˜,∞s˜t. (12) Hence, we can decompose the variance of the real exchange rate as var(s˜t)=cov(s˜t,−βξ,∞s˜t)+cov(s˜t,βdr˜,∞s˜t)=−βξ,∞var(s˜t)+βdr˜,∞var(s˜t). (13) Normalizing each term by var(s˜t) ⁠, we obtain 1=−βξ,∞+βdr˜,∞, (14) where the first term—the “risk premium variance share”—captures the share of the variance due to changes in expected future risk premiums, and the second term—the “cash-flow variance share”—captures the variance due to changes in expected future real bond return differentials. Equation (14) is also useful in interpreting the implications of no-arbitrage models of currency pricing. Both the habit and long-run risks models imply a risk premium share of the volatility of the real exchange rate in excess of 100%. Because the variance-decomposition results in the univariate and multivariate settings are quite similar, based on (14), we can interpret this implication of the two models as being due to the fact that they imply a negative covariance between s˜t and dr˜t+1 ⁠. 2.2 The multiple-predictor case In the multiple-predictor case, in addition to the real exchange rate, our baseline choice of instruments is as follows. To the extent that currency risk premiums exhibit serial correlation, the lagged excess currency return also should be used as a predictor. Moreover, given the evidence from existing studies, we include the interest rate differential as a predictor. Finally, the inflation differential is also a natural candidate as a predictor, as inflation is one of the macro variables typically considered in the international finance literature (see, e.g., Engel and West, 2005). Hence, we define zt≡[s˜t,ξt,drt,dπt,xt⊤]⊤ ⁠, where xt is a vector of K possible additional instruments. The multiple-predictor case is meant to capture patterns of predictability above and beyond the simple AR(1) process (Equation (6)) for the real exchange rate assumed in the previous section. This setup accounts, for example, for the well-documented predictive power of the nominal interest rate differential for excess currency returns. 2.2.1 Finite horizon tests We estimate the monthly predictive regressions: ξH,t+H=αξ,H+βξ,H⊤zt+eξ,t+H, (15) where ξH,t+H≡∑h=1Hξt+h is the roll-over H-month excess return. We test the individual null hypotheses: βξ,H,k=0, (16) where βξ,H,k is the k-th element of βξ,H ⁠, and k=1,…,K+4 ⁠. In addition, we test the joint null hypotheses: βξ,H=04+K. (17) 2.2.2 Long-horizon tests We assume that zt follows a stationary VAR(1):11 zt=A+Bzt−1+vt, (18) where the eigenvalues of B lie inside the unit circle.12 Based on the VAR in (18), we compute the theoretical regression coefficients from projecting long-horizon cumulative excess returns on the real exchange rate and other instruments: β^ξ,∞⊤=β^ξ,1⊤(I−B^)−1. (19) Hence, we can test the individual restrictions, βξ,∞,k=0 ⁠, and the joint restriction: βξ,∞=04+K. (20) Using the VAR in Equation (18), we can revisit the present value representation of the real exchange rate in Equation (4). We have s˜t−E(s˜t)=−∑i=1∞Et(ξt+i−dr˜t+i)=−∑i=1∞Et [ξt+i−E(ξt)]+∑i=1∞Et[dr˜t+i−E(dr˜t)]−∑i=1∞ [E(ξt)−E(dr˜t)]=−∑i=1∞Et[ξt+i−E(ξt)]+∑i=1∞Et[dr˜t+i−E(dr˜t)]=−βξ,∞⊤[zt−E(zt)]+βdr˜,∞⊤[zt−E(zt)], (21) where, given the assumed stationarity of s˜t, E(ξt)−E(dr˜t)≡E(Δs˜t)=0 ⁠, and β^dr˜,∞⊤=β^dr˜,1⊤(I−B^)−1 ⁠. Therefore, we have var(s˜t)=cov(s˜t,−βξ,∞⊤zt)+cov(s˜t,βdr˜,∞⊤zt); (22) and 1=cov(s˜t,−βξ,∞⊤zt)var(s˜t)+cov(s˜t,βdr˜,∞⊤zt)var(s˜t). (23) In the empirical analysis, we use the difference between the risk premium variance share in the single- and multiple-predictor cases as an indicator of the role of the real exchange rate in capturing the long-term dynamics of ξt. If the difference is small, as it is in our sample, we can then conclude that the real exchange rate alone captures most of the long-term dynamics of excess currency returns. 3. Empirical Analysis 3.1 Data and empirical strategy Our data set, available from Datastream, includes monthly observations, over the period December 1983–April 2012, of foreign exchange rates, interest rates, and (seasonally unadjusted) consumer price indexes (CPIs) of the following 34 countries:13 G10 countries: Australia, Canada, Germany, Japan, New Zealand, Norway, Sweden, Switzerland, and the United Kingdom. Non-G10-developed countries: Austria, Belgium, Denmark, Finland, France, Greece, Ireland, Italy, Netherlands, Portugal, Singapore, and Spain. Emerging countries: Czech Republic, Hungary, India, Indonesia, Kuwait, Malaysia, Mexico, Philippines, Poland, South Africa, South Korea, Taiwan, and Thailand. In most of our tests, we parameterize the restricted VAR based on a panel data estimator, which uses data on all currencies and controls for currency-specific fixed effects. The panel data approach has two, related, advantages relative to the estimation of separate VARs for each country. First, we employ more data, and, as a result, we obtain more precise estimates. Second, the panel data VAR estimates lead to stable dynamics of the state variables, whereas some country-specific VAR estimates lead to unstable dynamics. Indeed, it is well documented that panel data estimation leads to more reliable estimates of exchange rate dynamics than pure time-series approaches: Mark and Sul (2001), Rapach and Wohar (2002), and Jordà and Taylor (2012), all use panel data vector VECMs, similar to our VAR, to produce robust forecasts of exchange rates. We consider four finite investment horizons: 1, 3, 6, and 12 months. Wald statistics are Newey-West adjusted for heteroscedasticity and moving-average serial correlation of order equal to the investment horizon minus one. We report both asymptotic and bootstrap results, where the bootstrap inference is based on a bootstrap of the VAR residuals. We bootstrap the residuals from the VAR equations, separately for each currency, but jointly across all the predictors for the same currency, to preserve properties of the joint distribution, such as asymmetry, fat tails, and cross-sectional dependence. The number of bootstrap repetitions is 5,000. To minimize the possible bias introduced by an arbitrary choice of starting values, in each bootstrap repetition we employ a new starting value randomly drawn from the time series of zt, and we also use a warm-up period of 60 months. 3.2 The single-predictor case We start with a discussion of the single-predictor (the real exchange rate) results, which illustrate in the simplest way the advantages of our approach. 3.2.1 Six major currencies Table 1 presents results for the six major currencies—Australian dollar, Canadian dollar, Deutsche Mark, Japanese yen, Swiss franc, and British pound—in the single-predictor case, at the 3-month and long horizons. Panel A presents results for the finite horizon, 3-month tests. While the real exchange rate predicts excess currency returns in the “right” direction (all coefficients are negative), the evidence is weak: none of the bootstrap p-values is below 5%. In addition, R2s never exceed 7%. Table 1 Testing no-predictability null, single-predictor case, six major currencies A. 3-month finite horizon tests . Country . . β^ξ,3 . R2 . Australia Estimate −0.02 .00 Asy. p .46 Boot. p .62 Canada Estimate −0.01 .00 Asy. p .56 Boot. p .69 Germany Estimate −0.10 .05 Asy. p .01 Boot. p .08 Japan Estimate −0.08 .05 Asy. p .02 Boot. p .11 Switzerland Estimate −0.06 .02 Asy. p .11 Boot. p .29 United Kingdom Estimate −0.16 .07 Asy. p .01 Boot. p .10 A. 3-month finite horizon tests . Country . . β^ξ,3 . R2 . Australia Estimate −0.02 .00 Asy. p .46 Boot. p .62 Canada Estimate −0.01 .00 Asy. p .56 Boot. p .69 Germany Estimate −0.10 .05 Asy. p .01 Boot. p .08 Japan Estimate −0.08 .05 Asy. p .02 Boot. p .11 Switzerland Estimate −0.06 .02 Asy. p .11 Boot. p .29 United Kingdom Estimate −0.16 .07 Asy. p .01 Boot. p .10 B. Long-horizon regression statistics . Country . β^ξ,∞ . Asy. p . Boot. p . Australia −0.51 .27 .46 Canada −0.44 .39 .51 Germany −0.90 .00 .00 Japan −0.91 .00 .00 Switzerland −0.72 .00 .10 United Kingdom −0.82 .00 .01 B. Long-horizon regression statistics . Country . β^ξ,∞ . Asy. p . Boot. p . Australia −0.51 .27 .46 Canada −0.44 .39 .51 Germany −0.90 .00 .00 Japan −0.91 .00 .00 Switzerland −0.72 .00 .10 United Kingdom −0.82 .00 .01 This table reports the results of 3-month finite horizon tests (panel A) and long-horizon tests (panel B) of the no-predictability null for the six major currencies during the December 1983–April 2012 sample period. In the 3-month finite horizon tests, we run linear regressions of 3-month excess currency returns on the real exchange rate and obtain the predictive coefficients (⁠ β^ξ,3 ⁠) and R2. The long-horizon tests are based on the long-horizon predictive coefficients (⁠ β^ξ,∞ ⁠), implied from the monthly VAR estimates for the instruments. Asy. p and Boot. p denote the asymptotic and bootstrap p-values, respectively, where the bootstrap inferences are under the no-predictability null, characterized by panel regression estimates of a separate, restricted VAR system. Open in new tab Table 1 Testing no-predictability null, single-predictor case, six major currencies A. 3-month finite horizon tests . Country . . β^ξ,3 . R2 . Australia Estimate −0.02 .00 Asy. p .46 Boot. p .62 Canada Estimate −0.01 .00 Asy. p .56 Boot. p .69 Germany Estimate −0.10 .05 Asy. p .01 Boot. p .08 Japan Estimate −0.08 .05 Asy. p .02 Boot. p .11 Switzerland Estimate −0.06 .02 Asy. p .11 Boot. p .29 United Kingdom Estimate −0.16 .07 Asy. p .01 Boot. p .10 A. 3-month finite horizon tests . Country . . β^ξ,3 . R2 . Australia Estimate −0.02 .00 Asy. p .46 Boot. p .62 Canada Estimate −0.01 .00 Asy. p .56 Boot. p .69 Germany Estimate −0.10 .05 Asy. p .01 Boot. p .08 Japan Estimate −0.08 .05 Asy. p .02 Boot. p .11 Switzerland Estimate −0.06 .02 Asy. p .11 Boot. p .29 United Kingdom Estimate −0.16 .07 Asy. p .01 Boot. p .10 B. Long-horizon regression statistics . Country . β^ξ,∞ . Asy. p . Boot. p . Australia −0.51 .27 .46 Canada −0.44 .39 .51 Germany −0.90 .00 .00 Japan −0.91 .00 .00 Switzerland −0.72 .00 .10 United Kingdom −0.82 .00 .01 B. Long-horizon regression statistics . Country . β^ξ,∞ . Asy. p . Boot. p . Australia −0.51 .27 .46 Canada −0.44 .39 .51 Germany −0.90 .00 .00 Japan −0.91 .00 .00 Switzerland −0.72 .00 .10 United Kingdom −0.82 .00 .01 This table reports the results of 3-month finite horizon tests (panel A) and long-horizon tests (panel B) of the no-predictability null for the six major currencies during the December 1983–April 2012 sample period. In the 3-month finite horizon tests, we run linear regressions of 3-month excess currency returns on the real exchange rate and obtain the predictive coefficients (⁠ β^ξ,3 ⁠) and R2. The long-horizon tests are based on the long-horizon predictive coefficients (⁠ β^ξ,∞ ⁠), implied from the monthly VAR estimates for the instruments. Asy. p and Boot. p denote the asymptotic and bootstrap p-values, respectively, where the bootstrap inferences are under the no-predictability null, characterized by panel regression estimates of a separate, restricted VAR system. Open in new tab Panel B presents results for the long-horizon inference, where the long-horizon statistics are based on VARs estimated separately for each country, although the UIP null is still parameterized based on the panel data VAR. In this single-predictor case, the B matrix of the VAR has nonzero elements only in the first column. As discussed earlier, in this single-predictor setting, at the long horizon, the negative of the long-horizon regression coefficients can be interpreted as the shares of the real exchange rate variance explained by future risk premium variation. The long-horizon evidence is stronger than the finite horizon evidence: three of the six coefficients are significant at the 5% level or better (bootstrap p-values) and point estimates range between −0.91 (Japan) and −0.44 (Canada). 3.2.2 Individual currencies and currency baskets Table 2 summarizes the single-predictor evidence across all currencies and currency baskets. The overall picture that emerges is consistent with what discussed above for the six major currencies. Based on bootstrap inference, the 3-month finite horizon tests (panel A) uncover only weak evidence of predictability, with only 4 rejections of 39 tests. The long-horizon evidence (panel B) is much stronger, with 12 rejections. Table 2 Testing no-predictability null, single-predictor case, summary A. 3-month finite horizon tests . . β^ξ,3=0 . No. of asy. p <.05 17 Frac. of asy. p <.05 .44 No. of boot. p <.05 4 Frac. of boot. p <.05 .10 No. of assets: 39 Avg. R2:.04 A. 3-month finite horizon tests . . β^ξ,3=0 . No. of asy. p <.05 17 Frac. of asy. p <.05 .44 No. of boot. p <.05 4 Frac. of boot. p <.05 .10 No. of assets: 39 Avg. R2:.04 B. Long-horizon regression statistics . . β^ξ,∞=0 . . Asy. . Boot. . No. of p <.05 27 12 Frac. of p <.05 .79 .35 No. of assets: 34 B. Long-horizon regression statistics . . β^ξ,∞=0 . . Asy. . Boot. . No. of p <.05 27 12 Frac. of p <.05 .79 .35 No. of assets: 34 This table summarizes the results of 3-month finite horizon tests (panel A) and long-horizon tests (panel B) of the no-predictability null for 34 currencies and 5 currency baskets during the December 1983–April 2012 sample period. The 3-month finite horizon tests run linear regressions of 3-month currency excess returns on the real exchange rate and obtain predictive coefficients (⁠ β^ξ,3 ⁠). The long-horizon tests are based on the long-horizon predictive coefficients (⁠ β^ξ,∞ ⁠), implied from monthly VAR estimates for the instruments. Each panel reports the numbers and the fractions of asymptotic tests or of bootstrap tests yielding p-values smaller than.05. The number of test assets (No. of assets) that generate valid results and the average R2 are also reported. The bootstrap inferences are under the no-predictability null, characterized by panel regression estimates of a separate, restricted VAR system. Open in new tab Table 2 Testing no-predictability null, single-predictor case, summary A. 3-month finite horizon tests . . β^ξ,3=0 . No. of asy. p <.05 17 Frac. of asy. p <.05 .44 No. of boot. p <.05 4 Frac. of boot. p <.05 .10 No. of assets: 39 Avg. R2:.04 A. 3-month finite horizon tests . . β^ξ,3=0 . No. of asy. p <.05 17 Frac. of asy. p <.05 .44 No. of boot. p <.05 4 Frac. of boot. p <.05 .10 No. of assets: 39 Avg. R2:.04 B. Long-horizon regression statistics . . β^ξ,∞=0 . . Asy. . Boot. . No. of p <.05 27 12 Frac. of p <.05 .79 .35 No. of assets: 34 B. Long-horizon regression statistics . . β^ξ,∞=0 . . Asy. . Boot. . No. of p <.05 27 12 Frac. of p <.05 .79 .35 No. of assets: 34 This table summarizes the results of 3-month finite horizon tests (panel A) and long-horizon tests (panel B) of the no-predictability null for 34 currencies and 5 currency baskets during the December 1983–April 2012 sample period. The 3-month finite horizon tests run linear regressions of 3-month currency excess returns on the real exchange rate and obtain predictive coefficients (⁠ β^ξ,3 ⁠). The long-horizon tests are based on the long-horizon predictive coefficients (⁠ β^ξ,∞ ⁠), implied from monthly VAR estimates for the instruments. Each panel reports the numbers and the fractions of asymptotic tests or of bootstrap tests yielding p-values smaller than.05. The number of test assets (No. of assets) that generate valid results and the average R2 are also reported. The bootstrap inferences are under the no-predictability null, characterized by panel regression estimates of a separate, restricted VAR system. Open in new tab 3.2.3 Panel evidence Table 3 presents the results from panel regression tests at the four finite horizons and the long horizon. Panel A presents the results of the finite horizon tests. The real exchange rate is a significant predictor, at all four finite horizons. At the long horizon (panel B), the predictive coefficient is a significant –0.86: this implies that 86% of the variance of the real exchange rate can be attributed to changes in currency risk premiums.14 Table 3 Testing no-predictability null, single-predictor case, panel regressions A. Finite horizon tests . Horizon . β^ξ,H . β^ξ,H=0 Wald (Asy. p) . 1 −0.02 10.81 (.00) 3 −0.07 21.93 (.00) 6 −0.15 33.20 (.00) 12 −0.30 31.69 (.00) A. Finite horizon tests . Horizon . β^ξ,H . β^ξ,H=0 Wald (Asy. p) . 1 −0.02 10.81 (.00) 3 −0.07 21.93 (.00) 6 −0.15 33.20 (.00) 12 −0.30 31.69 (.00) B. Long-horizon regression statistics . β^ξ,∞ . −0.86 . Asy. p (.00) B. Long-horizon regression statistics . β^ξ,∞ . −0.86 . Asy. p (.00) This table reports the results of finite horizon tests (panel A) and long-horizon tests (panel B) of the no-predictability null, obtained from panel regressions using all 34 currencies and controlling for currency fixed effects during the December 1983–April 2012 sample period. The finite horizon tests run linear regressions of 1-, 3-, 6-, and 12-month currency excess returns on the real exchange rate and obtain predictive coefficients β^ξ,H ⁠, where H=1,3,6, or 12. The long-horizon tests are based on the long-horizon predictive coefficients (⁠ β^ξ,∞ ⁠), implied from monthly VAR estimates for the instruments. Asymptotic p-values, based on covariance matrix estimators clustered by both currency and time, are reported in parentheses under the estimates or the test statistics. Open in new tab Table 3 Testing no-predictability null, single-predictor case, panel regressions A. Finite horizon tests . Horizon . β^ξ,H . β^ξ,H=0 Wald (Asy. p) . 1 −0.02 10.81 (.00) 3 −0.07 21.93 (.00) 6 −0.15 33.20 (.00) 12 −0.30 31.69 (.00) A. Finite horizon tests . Horizon . β^ξ,H . β^ξ,H=0 Wald (Asy. p) . 1 −0.02 10.81 (.00) 3 −0.07 21.93 (.00) 6 −0.15 33.20 (.00) 12 −0.30 31.69 (.00) B. Long-horizon regression statistics . β^ξ,∞ . −0.86 . Asy. p (.00) B. Long-horizon regression statistics . β^ξ,∞ . −0.86 . Asy. p (.00) This table reports the results of finite horizon tests (panel A) and long-horizon tests (panel B) of the no-predictability null, obtained from panel regressions using all 34 currencies and controlling for currency fixed effects during the December 1983–April 2012 sample period. The finite horizon tests run linear regressions of 1-, 3-, 6-, and 12-month currency excess returns on the real exchange rate and obtain predictive coefficients β^ξ,H ⁠, where H=1,3,6, or 12. The long-horizon tests are based on the long-horizon predictive coefficients (⁠ β^ξ,∞ ⁠), implied from monthly VAR estimates for the instruments. Asymptotic p-values, based on covariance matrix estimators clustered by both currency and time, are reported in parentheses under the estimates or the test statistics. Open in new tab 3.3 The multiple-predictor (baseline) case We now turn to the discussion of the results for the baseline case, with the four instruments: real exchange rate, excess currency return, nominal interest rate differential, and inflation differential. 3.3.1 Individual currencies and currency baskets We start with the results for 39 currencies and currency baskets in total. Based on the bootstrap inference, the finite horizon tests uncover only weak evidence of predictability. For example, at the 3-month horizon (Table 4), in the finite horizon tests, the no-predictability null is rejected in seven, one, ten, and zero instances, for the real exchange rate, excess currency return, interest rate differential, and inflation differential, respectively.15 Table 4 Testing no-predictability null, baseline case, 3-month investment horizon, summary . Explanatory variables . . s˜ . ξ . dr . dπ . No. of asy. p <.05 22 3 15 0 Frac. of asy. p <.05 .56 .08 .38 .00 No. of boot. p <.05 7 1 10 0 Frac. of boot. p <.05 .18 .03 .26 .00 No. of assets: 39 Avg. adj R2:.05 . Explanatory variables . . s˜ . ξ . dr . dπ . No. of asy. p <.05 22 3 15 0 Frac. of asy. p <.05 .56 .08 .38 .00 No. of boot. p <.05 7 1 10 0 Frac. of boot. p <.05 .18 .03 .26 .00 No. of assets: 39 Avg. adj R2:.05 This table summarizes the results of the 3-month finite horizon tests of the no-predictability null for 34 currencies and 5 currency baskets during the December 1983–April 2012 sample period. The 3-month finite horizon tests run linear regressions of 3-month excess returns of currencies and currency baskets on four instrumental variables: the real exchange rate (⁠ s˜ ⁠), the 1-month excess return (ξ), the nominal interest rate differential (dr), and the inflation differential (⁠ dπ ⁠). Below each instrument, the table reports the numbers and the fractions of asymptotic tests or of bootstrap tests yielding p-values smaller than.05 for the associated coefficient estimate. The number of test assets that generate valid results and the average R2 are also reported. The bootstrap tests are under the no-predictability null, characterized by panel regression estimates of a separate, restricted VAR system. Open in new tab Table 4 Testing no-predictability null, baseline case, 3-month investment horizon, summary . Explanatory variables . . s˜ . ξ . dr . dπ . No. of asy. p <.05 22 3 15 0 Frac. of asy. p <.05 .56 .08 .38 .00 No. of boot. p <.05 7 1 10 0 Frac. of boot. p <.05 .18 .03 .26 .00 No. of assets: 39 Avg. adj R2:.05 . Explanatory variables . . s˜ . ξ . dr . dπ . No. of asy. p <.05 22 3 15 0 Frac. of asy. p <.05 .56 .08 .38 .00 No. of boot. p <.05 7 1 10 0 Frac. of boot. p <.05 .18 .03 .26 .00 No. of assets: 39 Avg. adj R2:.05 This table summarizes the results of the 3-month finite horizon tests of the no-predictability null for 34 currencies and 5 currency baskets during the December 1983–April 2012 sample period. The 3-month finite horizon tests run linear regressions of 3-month excess returns of currencies and currency baskets on four instrumental variables: the real exchange rate (⁠ s˜ ⁠), the 1-month excess return (ξ), the nominal interest rate differential (dr), and the inflation differential (⁠ dπ ⁠). Below each instrument, the table reports the numbers and the fractions of asymptotic tests or of bootstrap tests yielding p-values smaller than.05 for the associated coefficient estimate. The number of test assets that generate valid results and the average R2 are also reported. The bootstrap tests are under the no-predictability null, characterized by panel regression estimates of a separate, restricted VAR system. Open in new tab As the return horizon increases, the evidence from direct tests is similar, with an increase in the predictive power of the inflation differential. At the 12-month horizon, for example, the no-predictability null is rejected in 9, 0, 11, and 13 instances, for the real exchange rate, excess currency return, interest rate differential, and inflation differential, respectively. Table 5 reports results for the long-horizon inference. Across currencies, there are 12 rejections of 34 tests, for the real exchange rate, and 10 rejections for the nominal interest rate differential.16 Hence, for the real exchange rate, the rejection rate (35%, i.e., 12 of 34 test assets) is higher than the rejection rates from the finite horizon tests (at most 26%). The excess currency return and inflation differential, on the other hand, are significant in only one and three tests, respectively. The joint tests lead to 13 rejections, the risk premium variance share is significant in nine instances, and the increase in variance share due to the addition of instruments to the real exchange rate is significant in seven instances. Table 5 Testing no-predictability null, baseline case, long-horizon regressions, summary . Explanatory variables . β^ξ,∞=0 . Variance . Δ Variance . . s˜ . ξ . dr . dπ . Wald . share . share . No. of asy. p <.05 25 2 7 4 30 22 0 Frac. of asy. p <.05 .74 .06 .21 .12 .88 .65 .00 No. of boot. p <.05 12 1 10 3 13 9 7 Frac. of boot. p <.05 .35 .03 .29 .09 .38 .26 .21 No. of assets: 34 . Explanatory variables . β^ξ,∞=0 . Variance . Δ Variance . . s˜ . ξ . dr . dπ . Wald . share . share . No. of asy. p <.05 25 2 7 4 30 22 0 Frac. of asy. p <.05 .74 .06 .21 .12 .88 .65 .00 No. of boot. p <.05 12 1 10 3 13 9 7 Frac. of boot. p <.05 .35 .03 .29 .09 .38 .26 .21 No. of assets: 34 This table summarizes the results of long-horizon tests of the no-predictability null for 34 currencies and 5 currency baskets during the December 1983–April 2012 sample period. The tests are based on the long-horizon predictive coefficients, implied from monthly VAR estimates for the four instrumental variables: the real exchange rate (⁠ s˜ ⁠), the 1-month excess return (ξ), the nominal interest rate differential (dr), and the inflation differential (⁠ dπ ⁠). The table reports the numbers and the fractions of asymptotic tests or of bootstrap tests yielding p-values smaller than.05; below the instruments are the individual tests of the associated coefficient estimates; below “ β^∞=0,Wald” is the joint tests of all the coefficient estimates; below “Variance Share” is the tests of the risk premium share of the real exchange rate variance; and below “Δ Variance Share” is the tests of the incremental risk premium share, when compared with the single-predictor case. The bootstrap tests are under the no-predictability null, characterized by panel regression estimates of a separate, restricted VAR system. Open in new tab Table 5 Testing no-predictability null, baseline case, long-horizon regressions, summary . Explanatory variables . β^ξ,∞=0 . Variance . Δ Variance . . s˜ . ξ . dr . dπ . Wald . share . share . No. of asy. p <.05 25 2 7 4 30 22 0 Frac. of asy. p <.05 .74 .06 .21 .12 .88 .65 .00 No. of boot. p <.05 12 1 10 3 13 9 7 Frac. of boot. p <.05 .35 .03 .29 .09 .38 .26 .21 No. of assets: 34 . Explanatory variables . β^ξ,∞=0 . Variance . Δ Variance . . s˜ . ξ . dr . dπ . Wald . share . share . No. of asy. p <.05 25 2 7 4 30 22 0 Frac. of asy. p <.05 .74 .06 .21 .12 .88 .65 .00 No. of boot. p <.05 12 1 10 3 13 9 7 Frac. of boot. p <.05 .35 .03 .29 .09 .38 .26 .21 No. of assets: 34 This table summarizes the results of long-horizon tests of the no-predictability null for 34 currencies and 5 currency baskets during the December 1983–April 2012 sample period. The tests are based on the long-horizon predictive coefficients, implied from monthly VAR estimates for the four instrumental variables: the real exchange rate (⁠ s˜ ⁠), the 1-month excess return (ξ), the nominal interest rate differential (dr), and the inflation differential (⁠ dπ ⁠). The table reports the numbers and the fractions of asymptotic tests or of bootstrap tests yielding p-values smaller than.05; below the instruments are the individual tests of the associated coefficient estimates; below “ β^∞=0,Wald” is the joint tests of all the coefficient estimates; below “Variance Share” is the tests of the risk premium share of the real exchange rate variance; and below “Δ Variance Share” is the tests of the incremental risk premium share, when compared with the single-predictor case. The bootstrap tests are under the no-predictability null, characterized by panel regression estimates of a separate, restricted VAR system. Open in new tab In summary, the long-horizon tests uncover more predictive power for the real exchange rate than the finite horizon tests. The evidence on the predictive power of the nominal interest rate differential is consistent with the extensive existing evidence on deviations from UIP reviewed by, for example, Sarno (2005). On the other hand, the evidence on the predictive power of the real exchange rate is largely a novel contribution of this paper. 3.3.2 Panel evidence Table 6 presents evidence based on the panel regression estimation. Given the large total number of observations—7,826 currency-month observations—inference is based on asymptotics, where standard errors are clustered by currency and time and where we control for currency fixed effects. Table 6 Testing no-predictability null, baseline case, panel regressions A. Finite horizon tests . . Explanatory variables . . s˜ . ξ . dr . dπ . Horizon . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . 1 −0.02 0.10 0.57 −0.01 (.00) (.03) (.00) (.97) 3 −0.07 0.17 1.26 0.02 (.00) (.00) (.02) (.91) 6 −0.15 0.19 2.55 0.63 (.00) (.05) (.00) (.02) 12 −0.29 0.20 3.68 1.90 (.00) (.06) (.03) (.00) A. Finite horizon tests . . Explanatory variables . . s˜ . ξ . dr . dπ . Horizon . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . 1 −0.02 0.10 0.57 −0.01 (.00) (.03) (.00) (.97) 3 −0.07 0.17 1.26 0.02 (.00) (.00) (.02) (.91) 6 −0.15 0.19 2.55 0.63 (.00) (.05) (.00) (.02) 12 −0.29 0.20 3.68 1.90 (.00) (.06) (.03) (.00) B. Long-horizon regression statistics . Explanatory variables . β^ξ,∞=0 . Variance . Δ Variance . s˜ . ξ . dr . dπ . Wald . share . share . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . −0.93 0.04 1.57 −0.14 817.16 0.93 0.07 (.00) (.01) (.00) (.07) (.00) (.00) (.02) B. Long-horizon regression statistics . Explanatory variables . β^ξ,∞=0 . Variance . Δ Variance . s˜ . ξ . dr . dπ . Wald . share . share . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . −0.93 0.04 1.57 −0.14 817.16 0.93 0.07 (.00) (.01) (.00) (.07) (.00) (.00) (.02) This table reports the results of finite horizon tests (panel A) and long-horizon tests (panel B) of the no-predictability null, obtained from panel regressions using all 34 currencies and controlling for currency fixed effects during the December 1983–April 2012 sample period. In the finite horizon tests, we run linear regressions of 1-, 3-, 6-, and 12-month currency excess returns on the real exchange rate (⁠ s˜ ⁠), the 1-month excess return (ξ), the nominal interest rate differential (dr), and the inflation differential (⁠ dπ ⁠). The long-horizon tests are based on the long-horizon predictive coefficients, implied from monthly VAR estimates for the instruments. Below the instruments are the associated coefficient estimates. Below “ β^∞=0,Wald” is the joint test statistic on all the coefficient estimates. Below “Variance Share” is the risk premium share of the real exchange rate variance. Finally, below “Δ Variance Share” is the incremental risk premium share, when compared with the single-predictor case. Asymptotic p-values, based on covariance matrix estimators clustered by both currency and time, are reported in parentheses under the estimates or the test statistics. Open in new tab Table 6 Testing no-predictability null, baseline case, panel regressions A. Finite horizon tests . . Explanatory variables . . s˜ . ξ . dr . dπ . Horizon . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . 1 −0.02 0.10 0.57 −0.01 (.00) (.03) (.00) (.97) 3 −0.07 0.17 1.26 0.02 (.00) (.00) (.02) (.91) 6 −0.15 0.19 2.55 0.63 (.00) (.05) (.00) (.02) 12 −0.29 0.20 3.68 1.90 (.00) (.06) (.03) (.00) A. Finite horizon tests . . Explanatory variables . . s˜ . ξ . dr . dπ . Horizon . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . 1 −0.02 0.10 0.57 −0.01 (.00) (.03) (.00) (.97) 3 −0.07 0.17 1.26 0.02 (.00) (.00) (.02) (.91) 6 −0.15 0.19 2.55 0.63 (.00) (.05) (.00) (.02) 12 −0.29 0.20 3.68 1.90 (.00) (.06) (.03) (.00) B. Long-horizon regression statistics . Explanatory variables . β^ξ,∞=0 . Variance . Δ Variance . s˜ . ξ . dr . dπ . Wald . share . share . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . −0.93 0.04 1.57 −0.14 817.16 0.93 0.07 (.00) (.01) (.00) (.07) (.00) (.00) (.02) B. Long-horizon regression statistics . Explanatory variables . β^ξ,∞=0 . Variance . Δ Variance . s˜ . ξ . dr . dπ . Wald . share . share . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . (Asy. p) . −0.93 0.04 1.57 −0.14 817.16 0.93 0.07 (.00) (.01) (.00) (.07) (.00) (.00) (.02) This table reports the results of finite horizon tests (panel A) and long-horizon tests (panel B) of the no-predictability null, obtained from panel regressions using all 34 currencies and controlling for currency fixed effects during the December 1983–April 2012 sample period. In the finite horizon tests, we run linear regressions of 1-, 3-, 6-, and 12-month currency excess returns on the real exchange rate (⁠ s˜ ⁠), the 1-month excess return (ξ), the nominal interest rate differential (dr), and the inflation differential (⁠ dπ ⁠). The long-horizon tests are based on the long-horizon predictive coefficients, implied from monthly VAR estimates for the instruments. Below the instruments are the associated coefficient estimates. Below “ β^∞=0,Wald” is the joint test statistic on all the coefficient estimates. Below “Variance Share” is the risk premium share of the real exchange rate variance. Finally, below “Δ Variance Share” is the incremental risk premium share, when compared with the single-predictor case. Asymptotic p-values, based on covariance matrix estimators clustered by both currency and time, are reported in parentheses under the estimates or the test statistics. Open in new tab As one would expect, imposing a panel structure on the data leads to substantial additional power in all tests. Panel A shows evidence of predictability in the finite horizon tests, especially strong for the real exchange rate and the interest rate differential. To assess the economic magnitude of the effects, we also compute the scaled regression coefficients, which give us effects on the dependent variable in units of standard deviation, for a 1-standard-deviation change in the independent variable. While not tabulated here, the scaled regression coefficients associated with the real exchange rate range between –0.36 and –0.11, whereas the scaled coefficients associated with the interest rate differential range between 0.07 and 0.12, corroborating the notion of stronger predictive power of real exchange rates than interest rate differentials. As to the long-horizon evidence, we have very strong rejections of the no-predictability null for both the real exchange rate and the nominal interest rate differential. The risk premium variance share is a strongly significant 93%, and the addition of other instruments to the real exchange rate further increases the risk premium share only by 7%.17 Finally, we also performed the panel data analysis separately for the three groups of countries: G10, non-G10, and emerging. Results are qualitatively similar to those obtained for a single panel of countries: the long-horizon tests uncover much stronger evidence of predictability than the finite horizon tests, especially for the real exchange rate; the implied long-horizon regression coefficients are also especially significant for the real exchange rate; and a large fraction of the variance of the real exchange rate can be attributed to changes in risk premiums (between 82% and 100%), where the inclusion of instruments in addition to the real exchange rate makes little difference (at most 9%). 3.4 Relation to Engel (2016) At this point, it is interesting to reexamine the evidence of Engel (2016). Engel (2016) finds that the expected real bond return differential, Et(dr˜t+1) ⁠, correlates positively with short-horizon excess currency returns, Et(ξt+1) ⁠, but negatively with long-horizon excess currency returns, Et(∑i=1∞ξt+i) ⁠, for all G6 currencies. We replicate and extend the analysis of Engel (2016) to all 34 currencies in our data set, by running regressions of expected one-period and long-horizon excess currency returns on the expected real bond return differential, with or without the real exchange rate as an additional control (see the results in Table 7), where expectations are computed using both our VAR and Engel’s VECM.18 The table presents results for the panel data evidence (The Internet Appendix, Table IA.4, provides results for the individual currencies). When the dependent variable is the expected short-horizon excess currency return, the expected real bond return differential enters the regression with a positive coefficient, both when the real exchange rate is controlled for and when it is not. When the dependent variable is the expected long-horizon excess currency return and the real exchange rate is not controlled for, the expected real bond return enters with a negative coefficient, consistent with Engel (2016). However, when the real exchange rate is controlled for, the sign turns positive. Hence, it appears that once deviations from PPP are accounted for, a high expected real bond return differential positively predicts the appreciation of the currency at both short and long horizons. Similarly consistent across horizons is the sign of the coefficient associated with the real exchange rate, which is negative in both cases. Table 7 Expected real bond differential versus short-horizon and long-horizon risk premium, panel evidence . VECM . VAR . . Model (1) . Model (2) . Model (1) . Model (2) . Dependent variable . Et(dr˜t+1) . Et(dr˜t+1) . s˜t . Et(dr˜t+1) . Et(dr˜t+1) . s˜t . Et(ξt+1) 0.77 0.95 −0.03 0.74 0.93 −0.02 Et(∑i=1∞ξt+i) −3.01 3.70 −0.97 −5.33 1.84 −0.93 . VECM . VAR . . Model (1) . Model (2) . Model (1) . Model (2) . Dependent variable . Et(dr˜t+1) . Et(dr˜t+1) . s˜t . Et(dr˜t+1) . Et(dr˜t+1) . s˜t . Et(ξt+1) 0.77 0.95 −0.03 0.74 0.93 −0.02 Et(∑i=1∞ξt+i) −3.01 3.70 −0.97 −5.33 1.84 −0.93 This table reports estimates of regression coefficients associated with expected real bond return differential (⁠ Et(dr˜t+1) ⁠) below “Model (1),” or with real bond return differential and real exchange rate (⁠ s˜t ⁠) below “Model (2),” where the dependent variable is expected long-horizon excess return, Et(∑i=1∞ξt+i) ⁠. The estimates are obtained from panel regressions using all 34 currencies and controlling for currency fixed effects during the December 1983–April 2012 sample period. The conditional expectations are formed using the VECM by Engel (2016) or our baseline VAR model. Open in new tab Table 7 Expected real bond differential versus short-horizon and long-horizon risk premium, panel evidence . VECM . VAR . . Model (1) . Model (2) . Model (1) . Model (2) . Dependent variable . Et(dr˜t+1) . Et(dr˜t+1) . s˜t . Et(dr˜t+1) . Et(dr˜t+1) . s˜t . Et(ξt+1) 0.77 0.95 −0.03 0.74 0.93 −0.02 Et(∑i=1∞ξt+i) −3.01 3.70 −0.97 −5.33 1.84 −0.93 . VECM . VAR . . Model (1) . Model (2) . Model (1) . Model (2) . Dependent variable . Et(dr˜t+1) . Et(dr˜t+1) . s˜t . Et(dr˜t+1) . Et(dr˜t+1) . s˜t . Et(ξt+1) 0.77 0.95 −0.03 0.74 0.93 −0.02 Et(∑i=1∞ξt+i) −3.01 3.70 −0.97 −5.33 1.84 −0.93 This table reports estimates of regression coefficients associated with expected real bond return differential (⁠ Et(dr˜t+1) ⁠) below “Model (1),” or with real bond return differential and real exchange rate (⁠ s˜t ⁠) below “Model (2),” where the dependent variable is expected long-horizon excess return, Et(∑i=1∞ξt+i) ⁠. The estimates are obtained from panel regressions using all 34 currencies and controlling for currency fixed effects during the December 1983–April 2012 sample period. The conditional expectations are formed using the VECM by Engel (2016) or our baseline VAR model. Open in new tab 4. Further Analysis 4.1 Size and power Two natural questions arise at this stage: First, does the bootstrap adjustment lead to the correct inference? Second, are the long-horizon tests indeed more powerful than the standard finite horizon tests? To the best of our knowledge, we are the first to perform a size and power study of the long-horizon tests introduced by Cochrane (2008). Our simulation evidence (see Section IA.1 in the Internet Appendix for details) shows that (1) the bootstrap correction of the inference is needed and eliminates size distortions and (2) the power of the long-horizon test is substantially higher than that of the traditional finite horizon tests. The reason for the superior properties of the long-horizon regression coefficients can be easily seen in the single-predictor case and has to do with the correlation between the regression coefficients in the different one-period regressions. In a simulation calibrated to the U.S. dollar/Deutsche Mark currency pair, we find that corr(−β^ξ,1−ρ^s˜)=0.99 and corr(β^dr˜,1−ρ^s˜)=0.13 ⁠. Thus, the estimated numerator and denominator in −β^ξ,∞ and β^dr˜,∞ are positively correlated, increasing the precision of the estimates of the two ratios.19 A similar effect is likely to hold in the multivariate case, as the real exchange rate captures most of the long-term dynamics of excess currency returns. 4.2 Extensions and robustness checks We perform several extensions of our analysis and we implement several robustness checks of our baseline results:20 (1) nonstationarity of the real exchange rate; (2) correction of possible spurious-regression biases; (3) alternative choices of parameters for the process for the real exchange rate; (4) no-predictability null hypothesis parameterized according to currency-specific VARs; (5) correction of VAR parameter estimates for small-sample biases; (6) controlling for additional predictors; (7) use of alternative approaches to bootstrap; and (8) use of different reference currencies. In summary, we find that assuming a nonstationary real exchange rate leads to the implausible result that essentially 100% of the variance of the real exchange rate appreciation is due to revisions of long-horizon PPP deviations. We also find that the predictability patterns uncovered in this study are very robust, and that long-horizon tests are crucial to uncover such predictability. 5. Implications of the Theory In this section, we discuss the theory behind the predictability patterns in excess currency returns. First, we consider a general no-arbitrage model. Second, we specialize the analysis to two specific models: the habit model (Verdelhan, 2010) and the long-run risks model (Bansal and Shaliastovich, 2013).21 Both models are modified relative to their original formulations by assuming that domestic and foreign log consumption are cointegrated, rather than independent random walks. This assumption delivers a stationary real exchange rate. 5.1 A general framework Consider a general no-arbitrage model. The differential between the foreign and domestic real log pricing kernels is given by dm˜t ⁠. Absent arbitrage, we have Δs˜t=dm˜t ⁠. Hence, assuming stationarity of the log pricing kernel differential, we have dm˜t−E(dm˜t)=−∑i=1∞Et(dm˜t+i)=−∑i=1∞Et(Δs˜t+i)=s˜t−E(s˜t). (24) The (log) real exchange rate, in deviation from its unconditional mean, reflects transitory fluctuations in the log pricing kernel differential. As to the implications of the model for predictability patterns, if we assume lognormality, symmetric economies (same preferences and same stochastic processes for the relevant state variables), and homoskedastic and neutral inflation, we have Et(ξt+1)=−12dσm˜t2 and22 cov[s˜t,Et(ξt+1)]=−12cov(dm˜t,dσm˜t2), (25) where dσm˜t2 is the differential between the foreign and domestic volatilities of real log pricing kernels. Equation (25) highlights the stationarity of the real exchange rate alone does not mechanically translate in the real exchange rate negatively predicting excess currency returns, as it is prevalent in the data. What is needed is a positive covariance between the level and volatility differentials dm˜t and dσm˜t2 ⁠. In the following, we verify whether this condition is met by realistically parameterized versions of the habit and long-run risks models. 5.2 The habit model 5.2.1 Preferences and equilibrium quantities The symmetric habit model Verdelhan (2010) assumes that the real log pricing kernel differential equals dm˜t=−γ(Δdht+Δdc˜t), (26) where dht is the differential in log consumption surplus ratios, dht≡ln[(C˜tf−X˜tf)/C˜tf]−ln[(C˜t−X˜t)/C˜t] ⁠, where C˜tf and C˜t are foreign and domestic real per capita consumption, respectively; X˜tf and X˜t are the foreign and domestic external “habit” consumption, respectively; and dc˜t≡ln(C˜tf/C˜t) is the differential in real log per capita consumption.23 We have s˜t−E(s˜t)=−γ(dht+dc˜t)=−γ[dln(C˜t−X˜t)]. (27) The equation above highlights how the real exchange rate is driven by the differential in log excess consumption: what matters is not the differential in the absolute consumption level, but the differential in consumption relative to the external habit. Hence, two countries may have the same level of consumption, but experience different marginal utilities of consumption because of different external habit levels. In turn, the difference in marginal utilities leads to deviations from PPP. Like in Verdelhan (2010), we have24,25 Et(ξt+1)=γ2σϵc2H¯2dht, (28) where H¯ is a “tuning” parameter. Note that dht affects the real exchange rate and the currency risk premium in opposite directions. A higher differential in the habit level of consumption increases both the differential in marginal utility of consumption and the differential in the volatility of the marginal utility of consumption. The first effect drives the real exchange rate up, whereas the second effect drives the currency risk premium down. Indeed, the covariance between the real exchange rate and the currency risk premium is given by26 cov[s˜t,Et(ξt+1)]=−γ3σϵc˜2H¯2[var(dht)+cov(dht,dc˜t)]=−γ3σϵc2H¯2{var(dht)+2E[λ(ht)](1−ρϵc˜ϵc˜f)σϵc˜2}. (29) For our choice of parameters, this covariance is negative. Turning to the nominal interest rate differential, we have drt=−bdht+γ(φc−1)dc˜t+φπdπt, (30) where b≡γ(1−φh)−γ2σϵc2H¯2 ⁠. Note that the second and third term in the expression above are absent in the solutions of Verdelhan (2010), as he does not impose the stationarity of the real exchange rate and derives the real, rather than the nominal, interest rate differential.27 Following Verdelhan (2010), we choose b < 0. Hence, an increase in dht affects the nominal interest differential and the currency risk premium in the same direction. The covariance between the two quantities equals cov[drt,Et(ξt+1)]=γ2σϵc2H¯2[−bvar(dht)+γ(φc−1)cov(dht,dc˜t)]. (31) For our choice of parameter values, the covariance above is positive, implying that the nominal interest rate differential positively predicts the excess currency return. Finally, the expected real bond return differential equals Et(dr˜t+1)=drt−φπdπt≡−bdht+γ(φc−1)dc˜t. (32) Note that dht affects the real exchange rate and the expected real bond return differential in opposite directions and the covariance between the two quantities is given by cov[s˜t,Et(dr˜t+1)]=γbvar(dht)−γ2(φc−1)var(dc˜t). (33) For our choice of parameter values, the covariance above is negative and the real exchange rate predicts the real bond return differential in the “wrong” direction: as a result, the model implies a risk premium variance share of the real exchange rate in excess of 100%. 5.2.2 Calibration Like in Verdelhan (2010) and Bansal and Shaliastovich (2013), we focus on the implications of the model for the dollar/British pound exchange rate. Following Verdelhan (2010) we set as baseline values: γ = 2, φh=0.99 ⁠, and b=−0.01 ⁠. Based on our data for the United States and the United Kingdom, we set (quarterly frequency): φc=0.97 ⁠, φπ=−0.24 ⁠, σϵc˜=0.0066 ⁠, σϵπ=0.0078 ⁠, ρϵc˜fϵc˜=0.44 ⁠, and ρϵπfϵπ=0.24 ⁠. Table 8 reports the results from the calibration exercise. The table also considers the effect of deviations of γ and b from the baseline scenario: “high” denotes a value twice as large as the baseline, whereas “low” denotes a value equal to one-half of the baseline value. The population finite horizon regression statistics are based on regressions with three regressors: s˜t ⁠, drt, and dπt ⁠, because there are only three state variables in this habit model. Similarly, the long-horizon statistics are based on a VAR(1) modeling the joint dynamics of s˜t ⁠, drt, and dπt ⁠.28 We calculate population p-values associated with the regression coefficients, assuming that the analysis is performed with a sample of 96 quarterly observations, to match the number of quarterly observations in the sample. Table 8 Calibration A. Summary statistics . . σΔs˜ . ρs˜ . σdr . σξ . σdr˜ . Habit, baseline 0.3395 .9889 0.0081 0.3406 0.0203 Habit, high γ 0.4620 .9885 0.0051 0.4625 0.0193 Habit, low γ 0.1929 .9881 0.0079 0.1946 0.0202 LRR, baseline 0.1060 .9940 0.0082 0.1082 0.0009 LRR, high γ 0.4527 .9940 0.0278 0.4554 0.0006 LRR, low γ 0.0242 .9934 0.0050 0.0306 0.0009 LRR, high ψ 0.1083 .9940 0.0085 0.1105 0.0012 LRR, low ψ 0.1019 .9939 0.0077 0.1040 0.0002 Sample 0.1027 .8054 0.0051 0.1043 0.0881 A. Summary statistics . . σΔs˜ . ρs˜ . σdr . σξ . σdr˜ . Habit, baseline 0.3395 .9889 0.0081 0.3406 0.0203 Habit, high γ 0.4620 .9885 0.0051 0.4625 0.0193 Habit, low γ 0.1929 .9881 0.0079 0.1946 0.0202 LRR, baseline 0.1060 .9940 0.0082 0.1082 0.0009 LRR, high γ 0.4527 .9940 0.0278 0.4554 0.0006 LRR, low γ 0.0242 .9934 0.0050 0.0306 0.0009 LRR, high ψ 0.1083 .9940 0.0085 0.1105 0.0012 LRR, low ψ 0.1019 .9939 0.0077 0.1040 0.0002 Sample 0.1027 .8054 0.0051 0.1043 0.0881 B. UIP tests . . R2 . βξ,s˜ . p(βξ,s˜=0) . βξ,dr . p(βξ,dr=0) . Habit, baseline .0089 –0.0129 .7959 0.4286 .9803 Habit, high γ .0058 –0.0115 .4680 0.3846 .9886 Habit, low γ .0133 –0.0150 .5306 0.5000 .9155 LRR, baseline .0133 –0.0107 .8647 0.3252 .9710 LRR, high γ .0132 –0.0105 .9668 0.3207 .9933 LRR, low γ .0085 –0.0110 .6241 0.3353 .8944 LRR, high ψ .0140 –0.0108 .9342 0.3304 .9855 LRR, low ψ .0118 –0.0103 .7239 0.3144 .9428 Sample .0689 –0.2096 0.0022 1.6850 .1309 B. UIP tests . . R2 . βξ,s˜ . p(βξ,s˜=0) . βξ,dr . p(βξ,dr=0) . Habit, baseline .0089 –0.0129 .7959 0.4286 .9803 Habit, high γ .0058 –0.0115 .4680 0.3846 .9886 Habit, low γ .0133 –0.0150 .5306 0.5000 .9155 LRR, baseline .0133 –0.0107 .8647 0.3252 .9710 LRR, high γ .0132 –0.0105 .9668 0.3207 .9933 LRR, low γ .0085 –0.0110 .6241 0.3353 .8944 LRR, high ψ .0140 –0.0108 .9342 0.3304 .9855 LRR, low ψ .0118 –0.0103 .7239 0.3144 .9428 Sample .0689 –0.2096 0.0022 1.6850 .1309 C. Long-horizon statistics . . Variance share ξ . Δ Variance share ξ . Variance share dr˜ . Habit, baseline 1.4145 0.1448 –0.4145 Habit, high γ 1.1576 0.1491 –0.1576 Habit, low γ 1.8059 0.2936 –0.8059 LRR, baseline 2.1484 0.0101 –1.1484 LRR, high γ 2.1067 0.0005 –1.1067 LRR, low γ 2.2019 0.1975 –1.2019 LRR, high ψ 2.2021 0.0025 –1.2021 LRR, low ψ 2.0374 0.0414 –1.0374 Sample 0.6032 –0.2586 0.3968 C. Long-horizon statistics . . Variance share ξ . Δ Variance share ξ . Variance share dr˜ . Habit, baseline 1.4145 0.1448 –0.4145 Habit, high γ 1.1576 0.1491 –0.1576 Habit, low γ 1.8059 0.2936 –0.8059 LRR, baseline 2.1484 0.0101 –1.1484 LRR, high γ 2.1067 0.0005 –1.1067 LRR, low γ 2.2019 0.1975 –1.2019 LRR, high ψ 2.2021 0.0025 –1.2021 LRR, low ψ 2.0374 0.0414 –1.0374 Sample 0.6032 –0.2586 0.3968 This table reports the empirical implications of the habit and long-run risks (LRR) models, calibrated for a sample of 96 quarterly observations using U.S. and U.K. data. Panel A reports the following annualized summary statistics: volatilities (σ) of real currency appreciation (⁠ Δs˜ ⁠), nominal interest rate differential (dr), 1-month excess return (ξ), and real bond return differential (⁠ dr˜ ⁠), as well as autocorrelation in real exchange rate (⁠ ρs˜ ⁠). Panel B reports statistics of the UIP regression, including R2, the predictive coefficients associated with the real exchange rate and the interest rate differential, and their p-values. Panel C reports long-horizon statistics, including the risk premium share of the real exchange rate variance, the incremental variance share relative to one-predictor case, and the cash-flow share of real exchange rate variance. “High” and “low” denote scenarios in which a parameter—risk aversion (γ) for the habit model, or elasticity of intertemporal substitution (ψ) for the LLR model—is set to a value double or one-half of its baseline value, respectively. The sample counterparts are reported in the rows labeled “Sample.” Open in new tab Table 8 Calibration A. Summary statistics . . σΔs˜ . ρs˜ . σdr . σξ . σdr˜ . Habit, baseline 0.3395 .9889 0.0081 0.3406 0.0203 Habit, high γ 0.4620 .9885 0.0051 0.4625 0.0193 Habit, low γ 0.1929 .9881 0.0079 0.1946 0.0202 LRR, baseline 0.1060 .9940 0.0082 0.1082 0.0009 LRR, high γ 0.4527 .9940 0.0278 0.4554 0.0006 LRR, low γ 0.0242 .9934 0.0050 0.0306 0.0009 LRR, high ψ 0.1083 .9940 0.0085 0.1105 0.0012 LRR, low ψ 0.1019 .9939 0.0077 0.1040 0.0002 Sample 0.1027 .8054 0.0051 0.1043 0.0881 A. Summary statistics . . σΔs˜ . ρs˜ . σdr . σξ . σdr˜ . Habit, baseline 0.3395 .9889 0.0081 0.3406 0.0203 Habit, high γ 0.4620 .9885 0.0051 0.4625 0.0193 Habit, low γ 0.1929 .9881 0.0079 0.1946 0.0202 LRR, baseline 0.1060 .9940 0.0082 0.1082 0.0009 LRR, high γ 0.4527 .9940 0.0278 0.4554 0.0006 LRR, low γ 0.0242 .9934 0.0050 0.0306 0.0009 LRR, high ψ 0.1083 .9940 0.0085 0.1105 0.0012 LRR, low ψ 0.1019 .9939 0.0077 0.1040 0.0002 Sample 0.1027 .8054 0.0051 0.1043 0.0881 B. UIP tests . . R2 . βξ,s˜ . p(βξ,s˜=0) . βξ,dr . p(βξ,dr=0) . Habit, baseline .0089 –0.0129 .7959 0.4286 .9803 Habit, high γ .0058 –0.0115 .4680 0.3846 .9886 Habit, low γ .0133 –0.0150 .5306 0.5000 .9155 LRR, baseline .0133 –0.0107 .8647 0.3252 .9710 LRR, high γ .0132 –0.0105 .9668 0.3207 .9933 LRR, low γ .0085 –0.0110 .6241 0.3353 .8944 LRR, high ψ .0140 –0.0108 .9342 0.3304 .9855 LRR, low ψ .0118 –0.0103 .7239 0.3144 .9428 Sample .0689 –0.2096 0.0022 1.6850 .1309 B. UIP tests . . R2 . βξ,s˜ . p(βξ,s˜=0) . βξ,dr . p(βξ,dr=0) . Habit, baseline .0089 –0.0129 .7959 0.4286 .9803 Habit, high γ .0058 –0.0115 .4680 0.3846 .9886 Habit, low γ .0133 –0.0150 .5306 0.5000 .9155 LRR, baseline .0133 –0.0107 .8647 0.3252 .9710 LRR, high γ .0132 –0.0105 .9668 0.3207 .9933 LRR, low γ .0085 –0.0110 .6241 0.3353 .8944 LRR, high ψ .0140 –0.0108 .9342 0.3304 .9855 LRR, low ψ .0118 –0.0103 .7239 0.3144 .9428 Sample .0689 –0.2096 0.0022 1.6850 .1309 C. Long-horizon statistics . . Variance share ξ . Δ Variance share ξ . Variance share dr˜ . Habit, baseline 1.4145 0.1448 –0.4145 Habit, high γ 1.1576 0.1491 –0.1576 Habit, low γ 1.8059 0.2936 –0.8059 LRR, baseline 2.1484 0.0101 –1.1484 LRR, high γ 2.1067 0.0005 –1.1067 LRR, low γ 2.2019 0.1975 –1.2019 LRR, high ψ 2.2021 0.0025 –1.2021 LRR, low ψ 2.0374 0.0414 –1.0374 Sample 0.6032 –0.2586 0.3968 C. Long-horizon statistics . . Variance share ξ . Δ Variance share ξ . Variance share dr˜ . Habit, baseline 1.4145 0.1448 –0.4145 Habit, high γ 1.1576 0.1491 –0.1576 Habit, low γ 1.8059 0.2936 –0.8059 LRR, baseline 2.1484 0.0101 –1.1484 LRR, high γ 2.1067 0.0005 –1.1067 LRR, low γ 2.2019 0.1975 –1.2019 LRR, high ψ 2.2021 0.0025 –1.2021 LRR, low ψ 2.0374 0.0414 –1.0374 Sample 0.6032 –0.2586 0.3968 This table reports the empirical implications of the habit and long-run risks (LRR) models, calibrated for a sample of 96 quarterly observations using U.S. and U.K. data. Panel A reports the following annualized summary statistics: volatilities (σ) of real currency appreciation (⁠ Δs˜ ⁠), nominal interest rate differential (dr), 1-month excess return (ξ), and real bond return differential (⁠ dr˜ ⁠), as well as autocorrelation in real exchange rate (⁠ ρs˜ ⁠). Panel B reports statistics of the UIP regression, including R2, the predictive coefficients associated with the real exchange rate and the interest rate differential, and their p-values. Panel C reports long-horizon statistics, including the risk premium share of the real exchange rate variance, the incremental variance share relative to one-predictor case, and the cash-flow share of real exchange rate variance. “High” and “low” denote scenarios in which a parameter—risk aversion (γ) for the habit model, or elasticity of intertemporal substitution (ψ) for the LLR model—is set to a value double or one-half of its baseline value, respectively. The sample counterparts are reported in the rows labeled “Sample.” Open in new tab Across choices of parameter values, a few implications of the habit model stand out. First, similarly to Verdelhan (2010), the model overstates the volatility of changes in the (log) real exchange rate and the persistence of the level of the (log) real exchange rate. This is a result of the fact that the variable that dominates the variability of the log pricing kernel differential is dht, which is very volatile and persistent. Second, in the UIP regressions, the model underestimates the magnitudes of the coefficients associated with the real exchange rate and the nominal interest rate differential, although it reproduces their signs. Finally, the model overstates the risk premium variance share: for example, 141% for the baseline choice of parameters, as compared to a sample value of 60%. This result is due to the fact that the real exchange rate negatively predicts the real bond return differential. In summary, the habit model replicates a number of features of the data, including the fact that a dominant portion of the variance of real exchange rates can be attributed to risk premiums. On the other hand, the model overstates the volatility of real exchange rate appreciation, as well as the risk premium share of the real exchange rate variance. 5.3 The long-run risks model 5.3.1 Preferences and equilibrium quantities Following Bansal and Yaron (2004) and Bansal and Shaliastovich (2013), we assume m˜t=constant−θψΔc˜t+(θ−1)r˜ct, (34) where θ≡(1−γ)/(1−1/ψ) ⁠. ψ is the elasticity of intertemporal substitution; and r˜ct is the rate of return on the portfolio that delivers aggregate consumption, the “consumption asset.” Following the literature, we assume γ>1 and ψ>1 ⁠. As a result, we have θ<0 ⁠. Given the assumptions above, we have the log real pricing kernel differential:29 dm˜t=−γΔdc˜t+my⊤(κdyt−dyt−1); (35) where dyt≡[dc˜t dvt dπt]⊤ and my≡[mdc mdv 0]⊤ ⁠, where dvt denotes the differential in the conditional variance of log consumption in the two countries. We also have30 s˜t−E(s˜t)=−γdc˜t+my⊤dyt=(−γ+mdc)dc˜t+mdvdvt. (36) The differential in marginal utilities reflects not only the difference in current consumption but also the difference in equity valuations in the two countries. This difference, in turn, is driven by the log consumption differential as well as the volatility differential dvt. Hence, two countries may have the same level of current consumption, but different expected consumption growth and volatility of consumption growth, leading to deviations from PPP. Turning to the currency risk premium, it is driven by the differential consumption volatility process, dvt: Et(ξt+1)=−12mdc2dvt. (37) Given our choice of parameters, mdv > 0: an increase in consumption volatility reduces the valuation of the consumption asset, and an increase in the consumption volatility differential increases the foreign pricing kernel relative to the domestic pricing kernel. Hence, the covariance: cov[s˜t,Et(ξt+1)]=−12mdvmdc2var(dvt) (38) is negative. The intuition for the result is that when dvt increases, both the level and the volatility of the foreign pricing kernel increase, relative to the domestic pricing kernel. The relative increase in the foreign pricing kernel drives the real exchange rate higher, whereas the relative increase in the volatility of consumption growth drives the currency risk premium lower, leading to the negative covariance above. The nominal interest rate differential equals drt=(−γ+mdc)(1−φc)dc˜t+[mdv(1−φv)−12mdc2]dvt+φπdπt. (39) Note that the consumption volatility differential, dvt, affects the nominal interest rate differential through two channels. First, an increase in dvt reduces the valuation of the foreign consumption asset and increases its expected rate of return, relative to the domestic consumption asset. This effect drives down the expected pricing kernel differential—the two pricing kernels are inversely related to the rate of return on the corresponding consumption assets—and drives up the interest rate differential. Second, an increase in the consumption volatility differential increases the precautionary savings motive in the foreign country relative to the domestic country, driving down the interest rate differential. Indeed, the (conditional) covariance between the nominal interest rate differential and the currency risk premium is given by cov[drt,Et(ξt+1)]=[12mdc2−mdv(1−φv)]12mdc2var(dvt), (40) whose sign depends on the sign of the quantity inside the brackets. For our choice of parameter values, this quantity is positive. Finally, the expected real bond return differential equals Et(dr˜t+1)=[(−γ+mdc)(1−φc)]dc˜t+[mdv(1−φv)−12mdc2]dvt; (41) and cov[s˜t,Et(dr˜t+1)]=(−γ+mdc)2(1−φc)var(dc˜t)+mdv[mdv(1−φv)−12mdc2]var(dvt). (42) Given our choice of parameter values, the covariance above is negative. Hence, like in the habit model, a higher real exchange rate predicts a lower real bond return differential, and a variance decomposition exercise attributes more than 100% of the variance of the real exchange rate to risk premium variation. 5.3.2 Calibration Following Bansal and Shaliastovich (2013), we set γ = 12 and ψ=1.81 ⁠. The mean equations for consumption growth and inflation are parameterized similarly to the habit model. As to the process for the volatility of consumption innovations, we follow Bansal and Shaliastovich (2013) and assume φv=0.994 and ρϵvfϵv=0.94 ⁠.31 Like in the case of the habit model, we consider the effect of deviations of the preference parameters (γ and ψ) from the baseline scenario. We derive population statistics for the UIP regressions and for the long-horizon analysis, and we assume that the instruments in the predictive regressions and the VAR are s˜t ⁠, drt, and dπt ⁠. Similarly to the habit model, across parameter choices, the long-run risks model matches some features of the data, but misses other features (see Table 8). In particular, the model roughly matches the predictive power of the nominal interest rate differential for the excess currency return. On the other hand, the model grossly overstates the risk premium share of the variance of the real exchange: 215% for the baseline choice of parameters versus 60% in the data. 5.4 Interpretation The result that both the habit and long-run risks model imply a risk premium variance share of the real exchange above 100% can be understood as follows. The two models imply that cov[s˜t,−Et(∑i=1∞ξt+i)]>var(s˜t). (43) In turn, this means that cov[Et(∑i=1∞ξt+i),Et(∑i=1∞dr˜t+i)]>var[Et(∑i=1∞dr˜t+i)]. (44) In other words, both models imply a large positive covariance between the expected long-horizon excess currency return and the expected long-horizon real bond return differential. In the habit (long-run risks) model, this positive covariance is driven by the differential in log excess consumption (log excess consumption volatility). In the data (based on our panel data VAR estimates, for example), on the other hand, this covariance is negative, thus leading to positive variance shares for both the risk premium and the expected-cash flows components of the real exchange rate.32 6. Conclusions We provide new evidence that real exchange rates contain important information about—and they vary over time mainly because of—currency risk premiums, rather than expected cash flows. We perform multivariate finite- and long-horizon tests of the predictive power of the real exchange rate and a variance decomposition exercise. While the finite horizon tests uncover little predictive power, the implied long-horizon regression statistics reject the no-predictability null. The variance decomposition exercise shows that 93% of the variability of the real exchange rate can be attributed to changing expectations of future excess currency returns (panel regression evidence). Moreover, the real exchange rate alone captures most of the long-horizon predictability of excess currency returns. We relate our empirical findings to the implications of the habit and long-run risks models. Both models reproduce the predictive power of the real exchange rate and the nominal interest rate differential for excess currency returns. On the other hand, both models imply that the risk premium and expected cash-flow components of the real exchange rate are strongly negatively correlated. As a result, they substantially overstate the fraction of the volatility of the real exchange rate due to risk premiums. Acknowledgement We especially thank Rui Albuquerque for several useful conversations. We also thank Yi-Ting Chen (session chair), Pasquale Della Corte (discussant), Jin-Chuan Duan, Eyal Dvir, Chang-Tai Hsieh, Huichou Huang (session chair), Can Inci (discussant), Michael Kisser (discussant), Carolin Pflueger (discussant), Dmitriy Muravyev, Andreas Stathopoulos (discussant), Georg Strasser, Allan Timmermann, Ruey Tsay, Rosen Valchev, and Adrien Verdelhan and seminar participants at American University, Board of Governors of the Federal Reserve System, Boston College (Finance and Economics), Mississippi State University, UNC Charlotte, the 10th International Symposium on Econometric Theory and Applications (SETA 2014), the 2015 Eastern Finance Association Annual Meetings, the 2015 Financial Management Association Annual Meetings, the 2015 Taiwan Economics Research Workshop, the 2016 European Finance Association Meetings, the 2016 Midwestern Finance Association Annual Meetings, the 2017 American Finance Association Annual Meetings, and the 2017 Financial Management Association Meetings for useful comments. Ethan Chiang acknowledges the support of a Belk College Summer Research Grant. Supplementary data can be found on The Review of Asset Pricing Studies Web site. Footnotes 1 Although predictability in excess currency returns constitutes a violation of uncovered interested rate parity (UIP), even a constant, nonzero expected currency excess return violates UIP. Our analysis focuses on this specific type of violation of UIP, because it is related to the economically relevant question of what drives real exchange rate variation. 2 The evidence from the multiple-predictor case, where the instruments follow a VAR(1) process, is similar. Indeed, the difference between the risk premium share of the variance of the real exchange rate when the real exchange is the only predictor and when other predictors are accounted for is small, suggesting that the real exchange rate alone captures most of the long-term dynamics of the excess currency returns. 3 The six major currencies are the Australian dollar, Canadian dollar, Deutsche Mark/Euro, Japanese yen, Swiss franc, and British pound. 4 Note that the panel regression analysis also uncovers significant predictability patterns in the finite horizon tests. However, we believe that it is important to derive significant results at the individual currency level, as the panel data evidence may be driven by a few currencies with long time series. Moreover, the panel data analysis implicitly assumes homogeneity across currencies, which may not hold in the data. 5 Evans (2017) also starts from a present value representation of the log real exchange rate, but assumes that investors constantly update their expectation of long-horizon deviations from absolute purchasing power parity (PPP). His empirical analysis attributes most of the variance of changes in log real exchange rates to “dark matter,” which combines changes in risk premiums with revisions of expected long-horizon PPP deviations. When we apply the methodology of Evans (2017) to our setting, we find that virtually all of the variability of changes in real exchange rates is due to revisions of expected long-horizon PPP deviations. This result seems implausible, and we mainly attribute it to the misspecification of the joint dynamics of the real exchange rate, currency returns, and inflation differentials. 6 The recent working paper by Eichenbaum, Johannsen, and Rebelo (2017), on the other hand, compares the predictive power of the real exchange rate for the appreciation of the nominal exchange rate versus the inflation differential. Similar to an earlier version of the current paper, they find that the predictive ability of the real exchange rate is limited to the former. 7 Authors, such as Lustig and Verdelhan (2007), Farhi, Fraiberger, Gabaix, Ranciere, and Verdelhan (2015), Ang and Chen (2010), Burnside, Eichenbaum, and Rebelo (2011), Lustig, Roussanov, and Verdelhan (2011), Asness, Moskwitz, and Pedersen (2013), and Menkhoff, Sarno, Schmeling, and Schrimpf (2016), focus on cross-sectional predictability patterns. Menkhoff, Sarno, Schmeling, and Schrimpf (2016), in particular, show that adjusting real exchange rates for productivity and the quality of export goods generates a measure of currency value, which negatively predicts excess currency returns and is consistent with the spot exchange rate reverting toward fundamental value. 8 As noted earlier, we use the term “UIP” loosely to denote the null hypothesis of constant expected excess currency returns. In the international finance literature, UIP typically denotes the case of constant zero expected currency returns. 9 We are using the fact that Et(ξt+i)=Et[Et+i−1(ξt+i)]=constant+Et(βξs˜t+i−1)=constant+ρs˜i−1βξs˜. An analogous result holds for Et(dr˜t+i) ⁠. 10 See Cochrane (2008), Equation (10), for an analogous decomposition of the variance of the log dividend-price ratio. 11 Note that a restricted version of this VAR is used to generate bootstrap samples under the UIP null. Specifically, we estimate the restricted VAR at the monthly frequency and then we use powers of the restricted B matrix to parameterize the indirect null at different horizons. See the Internet Appendix (Section IA.2) for details. 12 If we replace zt with [s˜t,Δst,dπt]⊤ in the VAR above, we obtain a conventional vector error correction model (VECM). In the VECM, st and ptf−pt are cointegrated with cointegrating vector [1,1]⊤ ⁠, and, like in our VAR, s˜t is stationary (see Jordà and Taylor, 2012; Engel, 2016). Relative to the VECM, the VAR in (18) differs by including the dynamics of the interest rate differential, drt. 13 See the Internet Appendix (Section IA.3) for further details on the data set. Note that the G10 countries are selected based on trading volume in the currency market, not on measures of economic development. In addition, one of the G10 countries is the United States, and, hence, the list only contains nine countries. 14 For a comparison, see Cochrane (2008), who estimates at 109% the fraction of the variance of the aggregate dividend yield explained by expectations of future stock returns, using U.S. yearly data for the 1926–2004 sample. 15 In the interest of space, we relegate the results for the other horizons to the Internet Appendix (Tables IA.1–IA.3). 16 We exclude five currencies from the long-horizon analysis, because the corresponding VARs imply nonstationary dynamics. 17 The risk premium variance share in the single-predictor case is 86%, which, combined with the 7% change in share, leads to the 93% overall share. 18 Note that projecting Et(∑i=1∞ξt+i) onto Et(dr˜t+1) (and s˜t ⁠) is equivalent to projecting ∑i=1∞ξt+i onto Et(dr˜t+1) (and s˜t ⁠). In other words, our regression exercises document the predictive ability of the expected real bond return differential for excess currency returns in the infinite future (without or with the real exchange rate as a control). 19 Cochrane (2008) documents a similar effect: the estimates of the coefficients of regressions of stock returns and dividend growth on the log dividend price ratio are negatively correlated with the estimate of the serial correlation coefficient of the log dividend price ratio and the resultant long-horizon regression coefficients’ estimated numerator and denominator are again positively correlated. 20 In the interest of space, we relegate the details of this analysis to the Internet Appendix (Section IA.4). 21 We do not study the liquidity model of Engel (2016). His model has no direct channel connecting the real exchange rate to expected excess currency returns: the connection between the two quantities only takes place through the reaction function of monetary authorities in the two countries. See the Internet Appendix (Section IA.5) for further discussion. Additionally, we do not consider the “rare disasters” model of Farhi and Gabaix (2016). Their model implies that, as a country becomes more risky, the country’s currency depreciates and its interest rate increases; that is, it implies a strong negative correlation between the real exchange rate and the expected real bond return differential. This implication is at odds with the evidence from our sample. 22 In the formulation of the habit and long-run risks models, we also assume homoskedastic and neutral inflation, consistent with Verdelhan (2010). Bansal and Shaliastovich (2013) show that the nonneutrality of inflation is important to match the key features of the term structure in the data. However, we show that the long-run risks model does not need inflation nonneutrality to reproduce the predictability patterns in excess currency returns. 23 The domestic representative agent’s intertemporal marginal rate of substitution (i.e., the domestic economy’s pricing kernel) is given by exp (m˜t)=β(C˜t−X˜tC˜t−1−X˜t−1)−γ=β(HtHt−1)−γ(C˜tC˜t−1)−γ, where β is the time discount factor and Ht≡(C˜t−X˜t)/C˜t is the surplus consumption ratio. Taking logs, we have m˜t=ln(β)−γΔht−γΔc˜t ⁠, where ht≡ln(Ht) and c˜t≡lnC˜t ⁠. Analogous expressions apply to the foreign economy. 24 Here and in the following, μXt, σXt, σXYt ⁠, and ρXYt denote the mean, standard deviation, covariance, and correlation coefficient of the corresponding X and Y variables, conditional on time t available information. Below, we also introduce φX ⁠, denoting the serial correlation coefficient of the X variable, and ϵX, denoting the innovation in X. In the absence of a time subscript, the moments should be interpreted as constant over time. 25 See the Internet Appendix (Section IA.6.1) for details. 26 See the Internet Appendix (Section IA.6.2) for the derivation of the conditional moments of dht. 27 When the log per capita consumption differential dc˜t follows a random walk, φc=1 ⁠, and the second term on the right-hand side of (30) disappears. 28 The Internet Appendix (Section IA.7) shows that by employing as many instruments in the VAR as the number of underlying state variables—three, in this case—the VAR-implied variance decomposition coincides with the theoretical variance decomposition. 29 See the Internet Appendix (Sections IA.8.1 and IA.8.2) for details of the assumptions and the solution. 30 See the Internet Appendix (Section IA.8.3) for details. 31 See the Internet Appendix (Section IA.8.4) for details of the calibration of the volatility process. 32 Note that this feature of the data is akin to the finding in Engel (2016) that cov[Et(∑i=1∞ξt+i),Et(dr˜t+1)]<0 for the G6 countries (see Section 3.4). 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Real Exchange Rates and Currency Risk Premiums

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