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A Sentiment-based Explanation of the Forward Premium Puzzle* Jianfeng Yu† March 2013 Abstract A sentiment-based model of the exchange rate is proposed to understand the forward premium puzzle. Agents over- or underestimate the growth rate of the economy. All else equal, when perceived domestic growth is higher than perceived foreign growth, the domestic interest rate is higher than the foreign interest rate. At the same time, an econometrician would expect an increase in the home currency value. Together, the model with investor misperception can account for the forward premium puzzle. In addition, it helps lower the correlation between consumption growth differentials and exchange rate growth. Finally, this paper provides empirical evidence supporting the mechanism in the sentiment-based explanation. JEL Classification: G12, F31, G14, G15 Keywords: investor sentiment, forward premium puzzle, misperception, long-run risk, Backus-Smith Puzzle *I would like to thank Raj Aggarwal, Ravi Bansal, Frederico Belo, John Boyd, Zhi Da, Paul Gao, Bob Goldstein, Jeremy Graveline, David Hirshleifer, Philippe Jorion, Hanno Lustig, Terry Odean, Stavros Panageas, Tracy Wang, Jeffrey Wurgler, Wei Xiong, Xiaoyun Yu, and seminar participants at the University of Minnesota, University of Notre Dame, UC-Irvine, the Federal Reserve Bank of Dallas, Peking University, Tsinghua University, 2010 Econometric Society World Congress, and 2011 WFA for comments. I especially thank Riccardo Colacito (WFA discussant), Urban Jermann (the editor), an anonymous referee, and Jian Wang for extremely insightful and detailed comments. I also thank Malcolm Baker, Jeffrey Wurgler, and Yu Yuan for making their investor sentiment data available. I gratefully acknowledge financial support from Dean’s small research grant at the University of Minnesota. All errors are my own. † Corresponding Author: Jianfeng Yu, Department of Finance, Carlson School of Management, University of Minnesota. Address: 321 19th Avenue South, Suite 3-122, Minneapolis, MN 55455, Tel: 612-625-5498, Email: [email protected]. 1. Introduction Uncovered interest rate parity (UIP) implies that the expected change in exchange rates (the foreign price of domestic currency) should equal the interest rate differential between the foreign and domestic countries (i.e., the forward premium). Therefore, the regression coefficient of future changes in exchange rates on interest rate differentials should equal one. However, Fama (1984) and subsequent studies consistently find a negative coefficient in such regressions. That is, an increase in the foreign interest rate forecasts an appreciation of the foreign currency. The violation of UIP is often referred as the ”forward premium puzzle” in the literature. This evidence also implies that there are predictable returns from investing in currency markets. The forward premium puzzle has inspired a vast theoretical work that arises to shed light on this puzzle. This paper offers a sentiment-based explanation of the forward premium puzzle. In particular, a representative agent endowment economy for each country with complete home bias is considered. Agents in different countries share a common but subjective belief about future fundamentals, and agents can be optimistic (i.e., high sentiment) or pessimistic (i.e., low sentiment) about future endowment growth. When domestic sentiment is high, it means that both the home and foreign agents are optimistic about future U.S. endowment growth.1 When agents are optimistic about the domestic economy, and all else is equal, the interest rate at home is higher than the foreign interest rate due to a strong desire to borrow to increase current consumption. Thus, the key for our model to reproduce the failure of UIP is that high sentiment in one country can forecast an appreciation of this country’s currency. In a complete market, the (log) change in exchange rates, measured in units of the foreign goods per domestic goods, equals the difference between the (log) stochastic discount factors of the domestic and foreign countries. Hence, with recursive preferences, the exchange rate has to adjust to reflect differences in both current and future relative consumption across countries. When domestic sentiment is high today, high future growth at home is anticipated. Hence, from the economic agents’ point of view, the home currency is expected to depreciate to reflect the anticipated difference in consumption across two countries in the future. The magnitude of this anticipated depreciation depends on the intertemporal elasticity of substitution (IES). However, an econometrician anticipates that the economic agent at home will receive a lower-than-expected endowment tomorrow, raising the marginal 1 It is certainly interesting to consider heterogeneous beliefs as in Scheinkman and Xiong (2003), Dumas, Kurshev, and Uppal (2009), and Xiong and Yan (2010). However, the purpose of the current paper is to illustrate that investor misperception can lead to the failure of UIP in the simplest possible model. Thus, a simpler modeling approach is chosen. 1 utility of consumption next period, and hence the home currency appreciates. The magnitude of this appreciation depends on the home agent’s risk aversion since risk aversion determines the reduction in the marginal utility. Therefore, the econometrician is likely to observe a smaller depreciation or even an appreciation of the home currency in the data. In particular, when risk aversion is greater than the inverse of IES, the second effect dominates and high domestic sentiment predicts an appreciation of the home currency from the econometrician’s point of view. All together, when domestic sentiment is relatively high, the home interest rate is pushed up, and at the same time an appreciation of the domestic currency is expected. This can qualitatively account for the violation of the UIP condition in the data. For reasonable parameterizations, the model can indeed produce a negative UIP coefficient. Meanwhile, since our model can be viewed as a misperception-augmented version of Colacito and Croce (2011), it can simultaneously match many asset pricing moments as well as several other stylized facts in the international markets. For example, the model can account for the high correlation of pricing kernels and the low correlation of fundamentals (see, e.g., Brandt, Cochrane, and Santa-Clara (2006)). Interestingly, the model can also help produce a relatively low correlation between consumption growth differentials and foreign exchange growth (i.e., the Backus and Smith (1993) puzzle). Exchange rates in our model are driven by both fundamentals and investor sentiment. As a result, investor sentiment breaks the link between changes in foreign exchange and consumption growth differentials, and hence reduces their correlation. In sum, our misperception-augmented model improves upon Colacito and Croce (2011) by producing a negative UIP coefficient and by lowering the correlation between consumption growth differentials and foreign exchange growth.2 The key refutable predictions of the model are tested using proxies for investor sentiment. The evidence in the data suggests that investor sentiment has significant power to forecast changes in exchange rates and returns on foreign exchange, with the same sign as predicted by the model. The predictive power of the ”true” expected growth rate is also explored. Compared with sentiment, the ”true” expected growth rate has opposite power in predicting changes in exchange rates. Whereas a high domestic-foreign sentiment differential predicts a higher future spot rate, a high true expected growth differential between the domestic and foreign countries predicts a lower spot rate in the future. This evidence highlights the distinct role of sentiment and its special ability to account for a negative UIP coefficient. In a standard 2 As shown in Section 3.4. below, the model also helps produce a large equity premium with lower risk aversion and IES. This is because sentiment can serve as additional ”long-run risk”. Since sentiment is misperception, the realized consumption growth is still fairly unpredictable. As a result, the model can potentially feature a large amount of long-run risk without increasing the predictability of actual consumption growth. 2 rational model, a high true expected growth differential between the domestic and foreign countries predicts depreciation of the home-country currency. Thus, our evidence supports this prediction. However, without the sentiment effect, this would lead to a positive UIP coefficient. On the other hand, a high domestic-foreign sentiment differential is associated with a low forward premium, but predicts an appreciation of the home currency. Therefore, misperception is crucial in generating a negative UIP coefficient. Considerable prior work on the forward premium puzzle has been done. A number of earlier studies have provided insightful analysis on the role of investor irrationality in foreign exchange markets.3 Frankel and Froot (1987, 1990a) provide an early application of irrationality to currency markets. Mark and Wu (1998) develop a model in the noise trading setting of De Long, Shleifer, Summers, and Waldmann (1990), in which traders overweigh the forward premium when predicting future changes in the exchange rate. Gourinchas and Tornell (2004) provide an explanation based upon a distortion in investors’ beliefs about the interest rate process. Beber, Breedon, and Buraschi (2010) find differences in beliefs impact implied volatility of currency options and currency returns. Finally, in a closely related study, Burnside, Han, Hirshleifer, and Wang (2011) offer an insightful explanation based on investor overconfidence on inflation. This paper adds to the previous literature by showing that investors’ misperception on the growth rate can potentially induce a negative UIP coefficient.4 Our modeling approach also allows us to utilize investor sentiment data to test the mechanisms inside our model. Indeed, our empirical evidence lends support for a sentiment-based explanation. In addition, our model highlights the importance of recursive preferences in generating a negative UIP coefficient. More important, the same mechanism appears to help account for many salient features in international markets simultaneously. In particular, investor sentiment can potentially generate substantial long-run risk under recursive preferences, 3 There are numerous other studies that attempt to explain this puzzle under rational expectations. Notable recent papers include Alvarez, Atkeson, and Kehoe (2009) who examine a segmented market, Bacchetta and Van Wincoop (2009) who use a rational inattention framework, Burnside, Eichenbaum, and Rebelo (2007) who include microstructure frictions, Farhi and Gabaix (2008) who employ a Rietz (1988) rare disaster framework, Verdelhan (2010) who uses habit formtion, and Lustig and Verdelhan (2007) and Lustig, Roussanov, and Verdelhan (2011, 2012) who perform a cross-sectional analysis of foreign exchange portfolios. Earlier equilibrium models that attempt to explain the forward premium puzzle include Backus, Gregory, and Telmer (1993) and Bekaert (1996). 4 Overconfidence a la Burnside et al (2011) is a natural ingredient to produce investor misperception. However, as shown in the example in the next section, the difference between investor perception and the objective belief does not have to be derived from investor overconfidence. It could be due to other effects such as extrapolation, conservatism, or even rational learning. More important, our asset pricing approach to exchange rates allows us to impose many asset pricing moments to discipline the model parameters. These asset pricing moments include the first two moments of stock returns, riskfree rate, and their correlations across countries. 3 which helps account for several asset pricing moments with lower risk aversion and less predictable actual consumption growth. Section 2 provides an illustrative example to convey the main idea. Section 3 presents the model, and Section 4 presents empirical evidence. Section 5 concludes. 2. An Illustrative Example This section presents a simple example to convey the main idea of the paper. To make the presentation transparent, complete home biases and complete markets are imposed. The two countries are symmetric and representative investors in both countries have power preferences 1−γ P t Ct and maximize their life-time utility: ∞ t=1 δ 1−γ . Below only domestic variables are considered, and the corresponding foreign variables are denoted with a superscript ”*”. Assume that the data generating process for consumption growth is i.i.d. normal: gt+1 ≡ log (Ct+1 ) − log (Ct ) = µg + σg g,t+1 . However, all investors (homogeneously) believe that they can forecast consumption growth according to gt+1 = µg + st + σg ˆg,t+1 , where ˆt is i.i.d. standard normal under investors’ perception. Here, st can be interpreted as investor sentiment. It is worth emphasizing that the term “sentiment” is used in a very broad sense. It simply represents the difference between economic agents’ (i.e., investors) belief and econometricians’ belief. This difference in beliefs could be due to either rational or irrational learning. For example, st could be a result of agents’ rational learning on news shocks to technology or production, which is not observed by the econometrician. Moreover, no further restriction is imposed on the dynamics of the sentiment process in this example. According to investors’ belief, risk-free rates are: rf,t = − log δ + γµg + γst − γ2 2 σ . 2 g (1) Thus, the risk-free rate differentials are: ∗ rf,t − rf,t = γ (s∗t − st ) . (2) Under a frictionless complete market, the (log) exchange rate et is linked to domestic and 4 foreign discount factors mt and m∗t by the following equation:5 ∗ et+1 − et = mt+1 − m∗t+1 = −γ gt+1 − gt+1 , (3) where the (log) exchange rate is defined as the foreign price per home consumption unit. Under investors’ perception Êt (et+1 − et ) = γ (s∗t − st ) , (4) where Êt (·) is the expectation under economic agents’ subjective belief conditional on information available up to time t. It follows from at equations (2) and (4) that UIP holds ex ante under investors’ perception. However, ex-post exchange rates evolve according to the actual data generating process, which is i.i.d. according to equation (3). Hence, an econometrician regressing equation (3) on equation (2) would estimate a coefficient zero. Thus, the UIP condition fails in this simple example. However, in this simple example, the correlation between consumption growth differentials and exchange rate growth is one (see equation (3)), whereas this correlation is very low in the data. The next section shows that under recursive preferences, the UIP coefficient can be negative and the correlation between consumption growth differentials and exchange rate growth can also be substantially reduced. 3. An Exchange Rate Model with Investor Sentiment This section presents a simple model of exchange rates with investor sentiment. To highlight the effect of investor sentiment on the forward premium puzzle, the model abstracts from money, and irrational belief is the only deviation from a standard two-country model. Hollifield and Yaron (2003) find that inflation risk is virtually unrelated to currency returns, and they conclude that currency models should focus on real risk. Therefore, inflation is abstracted from our analysis and the model focuses on the real side of the economy. A few recent studies on currency markets (e.g., Verdelhan (2010), Colacito and Croce (2011), and Engel (2011)) also shift the focus to real variables. In particular, Engel (2011) shows that Fama’s (1984) results also hold in real terms. 5 See Backus, Foresi, and Telmer (2001) for a rigorous argument. In particular, the argument in Backus, Foresi, and Telmer (2001) still follows even if the agents do not have correct beliefs. Among others, Backus, Foresi, and Telmer (2001), Brandt, Cochrane and Santa-Clara (2006), and Colacito and Croce (2011) exploit this equation to relate the dynamics of the exchange rate to the dynamics of the domestic and foreign discount factors. 5 3.1. Consumption Dynamics and Preferences We analyze an economy with two countries that are denoted as domestic and foreign. To simplify the setup, a separate pure exchange economy for each country is specified by following Colacito and Croce (2011). Consumption in the two countries is exogenously given and there is a complete home bias, meaning that the representative agent in each country only consumes the good with which he is endowed. Asset markets are complete. This setup allows us to analyze the asset prices in each country separately. For convenience, endowments, preferences, and prices are characterized only for the domestic country. Identical expressions indexed by a ”*” apply to the foreign country. A discrete-time economy with infinite horizons is considered. Following the long-run risk literature, under the econometrician’s view (i.e, the objective belief or the data-generating process), the dynamics for consumption growth gt are modeled as gt+1 = µg + xt + σg g,t+1 (5) xt+1 = ρx xt + σx x,t+1 , (6) where (g,t , x,t ) is i.i.d. joint standard normal under the data-generating process. The representative agent in the domestic economy (and the foreign economy) is sentimental and has a subjective belief that is different from the data-generating process observed by an econometrician. Under economic agents’ subjective belief, the dynamics of consumption growth are as follows: gt+1 = µg + xt + st + σg ˆg,t+1 (7) st+1 = ρs st + σs s,t+1 (8) xt+1 = ρx xt + σx x,t+1 , (9) where (ˆg,t , x,t , s,t ) is i.i.d. joint standard joint normal under economic agents’ belief. It is further assumed that the sentiment dynamics are the same with equation (8) under the datagenerating process.6 To simplify the setting, all the shocks are joint normal and independent of each other throughout the paper. Since the true expected growth is xt , one can interpret st as investor sentiment. When st is positive (negative), the agent is optimistic (pessimistic). 6 Strictly speaking, one needs to show that there exist objective and subjective probability beliefs such that the dynamics of consumption follow those given here under these two probability measures. These are purely technical requirements, and the proof is just a simple application of the Girsanov theorem. The learning model in the Internet Appendix provides such an example. 6 The above specification on consumption and sentiment represents the main departure from rational expectation models and hence deserves further discussion. There are potentially many ways to obtain a time-varying misperception on the growth rate (i.e., sentiment) endogenously. One natural approach is a learning model with overconfidence. Scheinkman and Xiong (2003) and Dumas, Kurshev, and Uppal (2009) elegantly provide such models. Nonetheless, an exogenous sentiment process is specified because modeling the origin of sentiment does not add economic insights for the purpose of this study, while it does complicate the analysis. Our overall approach is to keep the model as simple as possible while retaining the key ingredients needed to generate the failure of the UIP condition.7 Moreover, the Internet Appendix shows that all the results remain when the sentiment process is modeled endogenously. To complete the model, the investors’ preferences over uncertain consumption stream Ct is specified as the recursive utility function (e.g., Epstein and Zin (1989)) 1−γ θ Ut = (1 − δ) Ct +δ 1−γ Êt Ut+1 θ θ1 1−γ , (10) where Êt (·) is the expectation under economic agents’ subjective belief conditional on information available up to time t, the parameter 0 < δ < 1 is the time discount factor, γ ≥ 0 is the risk-aversion parameter, ψ ≥ 0 is the intertemporal elasticity of substitution 1−γ (IES) parameter, and θ = 1− 1 . When risk aversion is larger than the reciprocal of IES, ψ agents prefer early resolution of uncertainty of the consumption path (e.g., Kocherlakota (1990)). 3.2. The Stochastic Discount Factor As shown in Epstein and Zin (1989), the logarithm of the intertemporal marginal rate of substitution (IMRS) is given by mt+1 ≡ log (Mt+1 ) = θ log δ − θ gt+1 + (θ − 1) ra,t+1 , ψ (11) where ra,t+1 is the logarithm of the gross return on an asset that delivers aggregate consumption as its dividends each period. For any continuous return rt+1 = log (Rt+1 ) , 7 The Internet Appendix provides an exchange rate model based on learning with overconfidence. In such a model, a mean-reverting sentiment process can be generated endogenously. Empirically, the sentiment data from Baker and Wurgler (2006) also suggest that the sentiment process is mean-reverting. Thus, the sentiment process is specified as an AR(1) in equation (8). 7 including the one on the consumption claim, Êt [exp (mt+1 + rt+1 )] = 1. (12) With log-linear approximation, the return on wealth can be solved explicitly, and hence the IMRS can be obtained explicitly as well. 3.3. The Foreign Exchange Market A foreign country is introduced into the model. The setup is similar to that used in Colacito and Croce (2011). For tractability, complete symmetry is imposed, and all the model parameters are identical across countries. Furthermore, it is assumed that the agents at home and abroad share the same belief, although their beliefs could be wrong. Thus, when domestic sentiment is high, it means that all the representative agents are homogeneously optimistic about the U.S. economy. 3.3.1. The Exchange Rate and the Forward Premium Again, under a frictionless complete market, the (log) exchange rate et is linked to domestic and foreign discount factors mt and m∗t by the following equation: et+1 − et = mt+1 − m∗t+1 . (13) To solve for the exchange rate, the standard log-linearization argument in the long-run risk literature is used (e.g, Bansal and Yaron (2004) and Colacito and Croce (2011)). First compute the return on wealth, ra,t+1 , in the pricing kernel equation (11). It follows from the Campbell-Shiller log-linear approximation that ra,t+1 ≈ κ0 + κ1 zt+1 − zt + gt+1 . (14) The price-consumption ratio can be further approximated as a linear function of the state variables st and xt . That is, zt ≈ A0 + A1 st + A2 xt , (15) where A0 , A1 , and A2 are constants. Substituting equations (14) and (15) back into the Euler equation (12) yields A1 = 1 1− ψ 1−κ1 ρs and A2 = 1 1− ψ 1−κ1 ρx . Furthermore, substituting equations (14) and (15) back into equation (11) gives the 8 equilibrium solution to the IMRS, mt+1 1 = −γgt+1 + γ − (xt + st )+κ1 (θ − 1) A1 σs s,t+1 +κ1 (θ − 1) A2 σx x,t+1 +C0 , (16) ψ where C0 is a constant. Thus, the risk-free rate, which is determined by the agent’s subjective belief, is given by 1 rf,t = − log Êt exp (mt+1 ) = (xt + st ) + C1 , ψ (17) where C1 is a constant. As a result, the forward premium (f pt ), which equals the foreigndomestic interest rate differential, is given by ∗ f pt ≡ rf,t − rf,t = − 1 1 (st − s∗t ) − (xt − x∗t ) . ψ ψ (18) Thus, the first testable prediction is obtained as follows. Prediction 1 The forward premium (i.e., the foreign-domestic interest rate differential) is negatively associated with the sentiment differential (st − s∗t ) and the true expected growth diferential (xt − x∗t ). In addition, the interest rate is positively associated with both sentiment and true expected growth. Furthermore, it follows from equations (13) and (16) that the solution to changes in exchange rates is given by et+1 − et 1 (st − s∗t + xt − x∗t ) + κ1 (θ − 1) A1 σs s,t+1 − ∗s,t+1 = − γgt+1 + γ − ψ +κ1 (θ − 1) A2 σx x,t+1 − ∗x,t+1 . (19) ∗ γgt+1 Notice that under the data-generating process, all the shocks s,t+1 , x,t+1 , ∗s,t+1 , and ∗x,t+1 ∗ have mean zero, Et (gt+1 ) = xt , and Et gt+1 = x∗t . Here, Et (·) is the expectation under the econometrician’s view (i.e., the objective belief). Therefore, taking the conditional expectation under the data-generating process yields 1 1 · (st − s∗t ) − (xt − x∗t ) . Et (et+1 − et ) = γ − ψ ψ (20) When γ > 1/ψ, the true expected consumption growth and the misperceived growth have opposite predictive power for the change in the exchange rate. This highlights the distinct 9 role of the misperception in consumption growth and the true expected consumption growth. To emphasize the unique prediction of the model on the relation between foreign exchange growth and sentiment differentials, the second testable prediction is summarized as follows. Prediction 2 If γ > 1/ψ, the change in the exchange rate, et+1 − et , is positively associated with the sentiment differential (st − s∗t ), but negatively associated with the true expected growth differential (xt − x∗t ). 3.3.2. The UIP Coefficient First, it is worthy of note that under economic agents’ belief, the following equalities obtain Êt (et+1 − et ) = − 1 ∗ · (st − s∗t + xt − x∗t ) = rf,t − rf,t . ψ (21) Hence, when the sentiment about the domestic growth rate is high and all else equal, economic agents expect an appreciation of the foreign currency. Further, economic agents expect a UIP coefficient of one. This result is the same with traditional asset pricing models. The exchange rate has to reflect the differences in growth rates between the two countries. If agents in the model expect a high growth rate in the future, they also anticipate the home currency to depreciate to reflect the differences in growth rates between the two countries. More important, equation (21) is consistent with the findings in Frankel and Froot (1987, 1990b) that when ex-ante measures of expected exchange rate changes (based on survey data) instead of the ex post realizations are used as the dependent variable in UIP regression, the UIP coefficient is in the vicinity of one. To examine the traditional UIP regression of foreign exchange growth on the forward premium, one need to combine equations (18) and (20). The regression coefficients of nperiod changes in the exchange rate on the n-period forward premium (i.e., the long-horizon UIP coefficients) are given by ψγ βU IP,n = 1 − 1−ρn x 1−ρx 2 1−ρn s 1−ρs var(xt −x∗t ) var(st −s∗t ) 2 + 1−ρn s 1−ρs 2 (22) With straightforward algebra, the following proposition is obtained. Proposition 1 The UIP coefficient βU IP,n is increasing in horizon n, as long as ρx > ρs > 0. var(x −x∗ ) Further, if the variance ratio var(stt −s∗t ) < γψ − 1, the one-period UIP coefficient βU IP,1 is t 10 negative. var(x −x∗ ) If xt = x∗t as in Colacito and Croce (2011), then var(stt −s∗t ) < γψ − 1 always holds, as t long as investors prefer early resolution of uncertainty. Thus, the UIP coefficient is negative. The Internet Appendix shows that through learning with overconfidence, ρx > ρs > 0 holds for the endogenous sentiment process.8 Hence, Proposition 1 implies that the UIP condition holds better at longer horizons, which is consistent with the empirical findings in Gourinchas and Tornell (2004) and Meredith and Chinn (2004). In particular, at longer horizons of five and ten years, Chinn and Meredith (2004) find much less negative forward premium bias. The intuition on the failure of the UIP condition is elaborated below. As discussed earlier, from economic agents’ point of view, when domestic sentiment is higher than foreign sentiment today, the home currency is expected to depreciate in the future to reflect the difference in consumption across two countries. Moreover, equation (21) implies that the magnitude of this anticipated depreciation depends on the IES. The easier it is to substitute consumption intertemporally, the less the anticipated depreciation. However, to an econometrician who analyzes the historical data, there is another counter effect. The econometrician anticipates that the economic agent at home will receive a lowerthan-expected endowment tomorrow, thus raising the marginal utility of consumption next period. Through this effect, the econometrician expects the home currency to appreciate. The magnitude of this appreciation depends on the agent’s risk aversion since risk aversion determines the reduction in the marginal utility due to the lower-than-expected realized consumption endowment (e.g., equation (16)). Consequently, the econometrician is likely to observe a smaller depreciation or even an appreciation of the home currency. In particular, when risk aversion is greater than the inverse of IES, the second effect dominates and high domestic sentiment predicts an appreciation of home currency. Meanwhile, high domestic sentiment implies high domestic interest rates. Taken together, the high interest rate currency is expected to appreciate in the econometrician’s view, and hence a negative UIP coefficient obtains. 3.3.3. The Return on Foreign Exchange Finally, the failure of the UIP condition also implies predictable returns in currency markets. The return on foreign exchange is defined as the excess return of a domestic investor who 8 This result is intuitive. Misperception is the difference between the true expected growth rate and the perceived expected growth rate. Hence, the persistence of the difference process should be smaller than that of the original processes. 11 borrows at the domestic risk-free rate (in logs), rf,t , converts the funds into foreign currency, ∗ , and then converts his earnings back to lends at the foreign risk-free rate (in logs), rf,t domestic currency. That is, the return on foreign exchange is just the return on a carry trade strategy. Hence, in logs, the return on foreign exchange, rtF X , is given by ∗ FX − rf,t + et − et+1 . = rf,t rt+1 (23) It follows from equations (18), (20), and (23) that under the econometrician’s view, the expected returns on foreign exchange are given by9 FX Et rt+1 = −γ · (st − s∗t ) . (24) Note that the true expected growth has no effect on returns on foreign exchange. The above argument leads to the third testable prediction from our model. Prediction 3 Returns on foreign exchange are negatively associated with the sentiment differential (st − s∗t ). Moreover, combining equations (18) and (24), one can show that the regression coefficient of returns on foreign exchange on the forward premium is given by: βRF X = ψγ var(xt −x∗t ) var(st −s∗t ) . (25) +1 Under the special case where the true expected growth is a constant or xt = x∗t , equations (22) and (25) imply that βU IP,1 = (1 − ψγ) and βRF X = ψγ. The above result is stated as another proposition below, which is consistent with the empirical evidence in Fama (1984). var(x −x∗ ) Proposition 2 As long as var(stt −s∗t ) < γψ−1, the regression coefficient of returns on foreign t exchanges on the forward premium is greater than one. Under the special condition that the true expected growth is constant or that xt and x∗t are equal, the regression coefficient of returns on foreign exchanges on the forward premium is given by γψ and the UIP coefficient is given by 1 − ψγ. 9 Notice that from the economic agents’ perspective, the expected return on foreign exchange is a constant (zero) and independent of st . This follows from equation (21) and the definition of the return of foreign exchange in equation (23). 12 In sum, our simple model can reproduce the forward premium puzzle. However, to further test whether the proposed mechanism is in the data, one needs to test the refutable Predictions 1-3. Section 4 provides evidence in support of Predictions 1-3. 3.4. Additional Implications for Correlations and Volatilities Although the main purpose of this paper is to provide a simple sentiment-based explanation for the forward premium puzzle, we also explore the model’s ability to account for other major puzzles in international markets. More important, these additional moment conditions impose restrictions on the structural parameters of our model. Apart from the first two moments of the stock market, three additional stylized facts are considered here. First, it is well documented that consumption growth is smooth and poorly correlated across countries, whereas asset market returns are volatile and highly correlated across countries. Second, it is also well known that consumption growth differentials and foreign exchange growth are poorly correlated (e.g., the Backus-Smith (1993) puzzle). Third, more recently, Brandt et al. (2006) argue that high equity premia imply highly volatile pricing kernels. In addition, consumption growth is poorly correlated across countries. Within a power utility setting, this suggests a low correlation between pricing kernels across countries. Together, it implies that the volatility of exchange rates should be much larger than the 10 ∼ 15% value observed in the data (see equation (13)). In other words, pricing kernels should be much more correlated than is suggested by consumption data, and hence quantities and prices have dramatically different implications for the correlations between the pricing kernels.10 To solve the exchange volatility puzzle a la Brandt et al (2006), Colacito and Croce (2011) cleverly break the link between the correlation of consumption growth and the correlation of the pricing kernel by introducing highly correlated long-run consumption. In their model, pricing kernels are mostly driven by the small but persistent long-run consumption components, which are perfectly correlated across countries. Thus, a high correlation between pricing kernels can be reconciled with a low consumption correlation. Their model also reproduces many other stylized facts in both macroeconomic and financial variables. By incorporating investor misperception into Colacito and Croce (2011), our model can help partially account for several additional stylized facts. These include the forward premium 10 For our model with investor misperception, the exact Hansen-Jagannathan bound actually does not apply. Hence, in theory the pricing kernels in our model do not need to be as volatile as the maximum observed Sharpe ratio. Thus, the price-implied international correlation between the pricing kernels does not need to be as high as in Brandt et al. (2006). This already makes the issue less puzzling. 13 puzzle and the low correlation between consumption growth differentials and exchange rate changes. In particular, equation (19) implies that exchange rate growth and consumption growth differentials are not perfectly correlated in this model. This imperfect correlation is mainly due to fluctuation in sentiment and can help partially resolve the Backus-Smith (1993) puzzle. 3.5. Calibration This section provides quantitatively calibration of the model to account for several stylized facts simultaneously. 3.5.1. The Parameter Choice Table 1 lists the main parameter values for our calibration. The model is simulated at a monthly frequency. Most of the parameter values are borrowed directly from the long-run risk literature. Following Bansal and Yaron (2004), the mean consumption growth µg is set to be 0.0015, and the IES ψ is set to be 1.5. The risk aversion γ is set to be 3.5, much smaller than that in Bansal and Yaron (2004), but closer to the standard value in the macro literature. The stock market’s dividend is modeled as a levered claim on consumption.11 The leverage parameter φ is set to be 3, the same as that in Bansal and Yaron (2004). Following Colacito and Croce (2011), we set the correlation between short-run consumption innovation ρ g , ∗g = 0.3, the correlation between short-run dividend innovation ρ (d , ∗d ) = −0.1, the persistence ρx = 0.987, and the average dividend growth µd = 0.0007. Finally, the volatility of short-run consumption growth innovation σg = 0.006, and the volatility of short-run dividend growth innovation σd = 0.027. These values are chosen to match the consumption and dividend dynamics. Long-run consumption volatility σx is set to be 0.00042 , such that the volatility of the true expected growth (xt ) has roughly the same value as that in Bansal and Yaron (2004). Finally, the time discount δ is set to be 0.999, so that the model produces a low riskfree rate. There are a few less standard parameters. The first one is the volatility of investor sentiment. Most measures of investor sentiment are unit free, and the absolute level of sentiment has no particular meaning. Thus, it is hard to gauge the magnitude of sentiment 11 That is, the dividend growth is specified as gd,t+1 = µd + φ (xt + st ) + σd ˆd,t+1 under the agent’s perception and ˆd,t+1 is i.i.d. normal under agents’ perception. However, under the data-generating process, gd,t+1 = µd + φxt + σd d,t+1 , where the data-generating process for d,t is i.i.d. normal. As mentioned earlier, all the shocks are independent of each other. 14 volatility relative to consumption volatility.12 In the main calibration, the relative volatility of sentiment to consumption growth is set to be less than 5%, so that the model is very close to its rational benchmark. The second key parameter is the correlation of sentiment indices across countries. This parameter is especially hard to estimate due to the unobservability of investor sentiment. Using the proxies in Baker, Wurgler, and Yuan (2012), the average correlation across countries is 35%. Using additional survey data on international sentiment indices, this correlation is found to be even lower, around 25%. Given potentially large amount of noises in the measured sentiment indices, the correlation calculated based on sentiment proxies can significantly underestimate the true correlation. For example, even if the true sentiment correlation is perfect and the variance ratio of noise to signal is one, the measured sentiment correlation would be only 50%. Thus, in the calibration, three different values, 25%, 50%, and 90%, are chosen to provide a sensitivity analysis. Finally, the sentiment persistence parameter ρs is set to be 0.965, which is calculated based on Baker and Wurgler’s sentiment index. 3.5.2. The Main Calibration Results Table 2 reports the calibration analysis based on 1, 200, 000 months of artificial data. Column 1 reports the results for a model with i.i.d. consumption growth and without investor misperception. As expected, the model performance is poor. Column 2 introduces a perfectly correlated long-run risk component as in Colacito and Croce (2011). The model can successfully reproduce many stylized facts. For example, the equity premium is large despite low exchange rate volatility, and the correlation of stock market returns is also large despite a low fundamental correlation. However, with perfect correlation between long-run consumption components across countries and without investor sentiment, all of the exchange rate movements come from the consumption growth differential (see equation (19)). Thus, the correlation between consumption growth differentials and exchange rate growth is perfect in this setting.13 In addition, in this model the forward premium is a constant. Thus, the model cannot address the UIP puzzle. Column 3 relaxes this perfect correlation by setting ρx ,∗x = 0.98. In this case, the model still implies a UIP coefficient of exactly one. However, with imperfectly 12 One could potentially use professional forecasts on GDP growth to obtain a rough estimation of misperceived growth. However, one would need to estimate the true expected GDP growth before obtaining the misperceived growth. There is no consensus on how to estimate true expected GDP growth. 13 Colacito and Croce (2012) show that by relaxing the assumption of complete home bias, this correlation can be reduced. Instead, our study shows that introducing investor sentiment is an alternative mechanism for reducing this correlation. 15 correlated expected growth, the correlation between consumption growth differentials and exchange rate growth is reduced to 82%, still large but closer to the data. In column 4, which is our benchmark calibration, investor misperception is introduced. The international correlation of sentiment is set to be 50%. Compared with column 3, a few important changes are noticed. First, due to this additional long-run risk component, the equity premium is now larger and stock returns are also more volatile. Second, the correlation between consumption growth differentials and exchange rate growth is reduced even further to 60%, closer to the counterpart in the data. Third, the model produces a larger volatility in foreign exchange growth, but it is still within the conventional range of 10˜15 percent per year. Fourth, the UIP coefficient is negative (−3.55). This coefficient is a bit larger in absolute value than its empirical counterpart, which is typically in the range of −2 to 0. However, the 95% confidence interval for the UIP coefficient is (−11.51, −0.61) , obtained from 1, 000 repeated small sample simulations, each with 30 years of simulated data. Finally, the UIP regression has an R2 of 1%, which is largely consistent with the data. Columns 5 and 6 provide a sensitivity analysis by choosing different values for the crosscountry sentiment correlation (i.e., 25% and 90%). The UIP coefficient remains negative. Meanwhile, the model can simultaneously keep other asset pricing moments close to the data. However, there are a few changes. On the one hand, when the cross-country sentiment correlation is large, the magnitude of the UIP coefficient is smaller and foreign exchange growth is smoother, but the correlation between consumption growth differentials and exchange rate growth is too high. On the other hand, when sentiment correlation is low, the correlation between consumption growth differentials and exchange rate growth is reduced, but the exchange rate is a bit too volatile and the UIP coefficient is too negative. In addition, the misperception-augmented model cannot fully account for the Backus-Smith puzzle since the model-implied correlation is still larger than the empirical counterpart in the data, which is typically around 30% or smaller. Finally, column 7 reports the calibration for a pure misperception-based model with i.i.d. consumption growth. This calibration helps us understand how far the sentiment channel can go without the standard long-run risk channel. The variation of sentiment is set to be about 30% of the variation of the realized consumption growth. Compared with column 1, the model performance improves in many dimensions. For example, the equity premium increases from a negative value to 2.52% per annum despite low risk aversion. The mechanism of this model is similar in spirit to Colacito and Croce (2011) since the highly correlated sentiment serves as highly correlated long-run risk. Moreover, due to the misperception effect, the UIP coefficient is negative and the correlation between the observed 16 consumption growth differentials and the exchange rate growth is not perfect anymore. In addition, differing with Colacito and Croce (2011), even if the sentiment process is volatile and persistent, the observed consumption growth still behaves as an i.i.d. process, consistent with the consumption data. Thus, sentiment can potentially produce large long-run risk in the model while keeping realized consumption growth close to an i.i.d. process.14 Obviously, sentiment alone cannot quantitatively account for all of the patterns observed in the data. Nonetheless, it is comforting to know that the same mechanism can potentially help reproduce many stylized facts in international markets simultaneously. 4. Empirical Tests In this section, we first introduce notation for the predictive variables, then discuss the empirical design and data used in the analysis and present summary statistics. The empirical tests of the key model predictions (Predictions 1-3) follow. 4.1. Definition of Variables and Empirical Design denote the log nominal exchange rate (expressed as the foreign price of domestic Let enom t currency) and pt denote the log consumer price index at home. Following Engel (2011), the real exchange rate is calculated as et = enom − p∗t + pt . Further define changes in real t exchange rates as ∆et = et − et−1 . (26) In the empirical test that follows, 3-month forward rates are used as in Engel and West (2005). It is one of the most popular and easily available forward rates for all of the industrial countries. The 3-month forward exchange rate Ft is observed at the end of month t for a delivery at the end of month t + 3. Let ft = log(Ft ). The 3-month forward premium is defined as f pt = ft − enom . (27) t 14 One challenge faced by the traditional long-run risks model is that the long-run risk component is hard to identify using consumption data alone since actual consumption growth behaves like an i.i.d. process. With high-quality survey data available in the future, one could potentially use survey data to identify the perceived long-run risk, which is also highly priced under recursive preferences. Alternatively, recent studies (e.g., Bansal et al. (2009) and Colacito and Croce (2011)) have started to use financial variables to estimate the long-run consumption component. Asset prices can reflect the misperceived consumption growth, whereas the realized consumption growth cannot. Thus, our model provides another justification for looking at the financial data, rather than actual consumption data, to identify the long-run risk. 17 By covered interest parity, f pt equals the difference between the 3-month risk-free rate (from the end of month t to the end of month t + 3) in the foreign country at the end of month t and the 3-month domestic risk-free rate the end of month t. The forward premium in real term is defined as the ex-post real interest rate differential. The return on foreign exchange is therefore given by FX (28) = ft − enom rt+3 t+3 . Finally, the true expected growth rate, xt , is measured as the trailing average of the past 3-year real consumption growth rates as in Bansal, Dittmar, and Lundblad (2005). The empirical specifications are standard long-horizon predictive regressions (e.g., Fama and French (1988) and Hodrick (1992)). All variables are observed monthly. The future changes in spot rates and returns on foreign exchange rates are regressed onto investor sentiment, st . Therefore, the dependent variable is either the log change in foreign exchange FX . rate, et+h − et , with different horizons or the 3-month log return on foreign exchange, rt+3 The independent variables are investor sentiment, st , or other predictive variables. For example, to examine the ability of investor sentiment to predict changes in spot rates, the main regression is et+h − et = αh + βs Xt + t,t+h , (29) where the predictor Xt is either U.S. sentiment st or the sentiment differential, st − s∗t . The error term t,t+h is an element of the time t + h, and is autocorrelated due to overlapping observations when the forecast horizon, h, is greater than one. Newey-West (1987) standard errors are used to adjust for this autocorrelation and potential heteroscedasticity. The tstatistics resulting from the adjustment test the null hypothesis that a given slope coefficient equals zero. 4.2. Data The data set on foreign exchange includes monthly spot and 3-month forward exchange rates between the U.S. dollar and currencies in 19 industrial countries: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Ireland, Italy, Japan, the Netherlands, New Zealand, Norway, Portugal, Spain, Sweden, Switzerland, and the U.K. The data sample is between 1973m1 and 2007m12 for most countries and may vary in some countries due to data availability. Legacy exchange rates derived by Haver Analytics are used for currencies of Euro-area member countries after they switched to the euro. Monthly CPI data for these 19 countries and the United States over 1973m1 to 2007m12 18 are obtained from the IMF. Annual real consumption per capita from 1970 to 2007 is obtained from the OECD. Quarterly real U.S. consumption per capita comes from the BEA. Spot exchange rates come from the Federal Reserve, and 3-month forward exchange rates come from the International Financial Statistics (IFS). The spot and forward exchange rates are used to calculate interest rate differentials between the U.S. dollar and other currencies. For several countries, some of the forward exchange rates are missing in the IFS. In these cases, the forward rate is obtained from Bloomberg or is calculated from the 3-month interest rate obtained from the G-10 data set. The choice of data source is dictated by data availability. The U.S. investor sentiment is obtained from Baker and Wurgler (2006). Their index serves as a good proxy for misperception since it is orthogonalized against a set of fundamental variables. Moreover, it is a measure based on market participants rather than the general population. The monthly sentiment index spans over 42 years, from July 1965 to December 2007. A detailed data description is provided in the Internet Appendix. As to international sentiment, annual sentiment data for Canada, France, Germany, Japan, and the U.K. are obtained from Baker, Wurgler, and Yuan (2012). These data span from 1980 to 2005. Following Baker and Wurgler (2006), the annual sentiment indices are transferred into a monthly frequency by assigning the same value for the corresponding subsequent 12 months. In addition, sentiment data for Australia, Belgium, Denmark, Italy, the Netherlands, New Zealand, and Switzerland are collected. These additional seven countries are selected purely for data availability reasons. The sentiment data for the rest of countries are either too short or unavailable. A more detailed data description is provided in the Internet Appendix. In particular, these sentiment indices are based on surveys and are related to expected business conditions. Thus, these indices are orthogonalized against expected consumption growth before they are used in our regression analysis. Ideally, sentiment indices should be constructed using the same approach as in Baker and Wurgler (2006). However, data limitations make this approach hard to pursue. On the other hand, it is plausible that the survey-based sentiment should be correlated with market participants’ perceptions. Thus, these survey-based measures could serve as imperfect proxies for investor sentiment in our model. 4.3. Summary Statistics Because the absolute magnitude of sentiment indices typically has no meaning, all the sentiment indices are standardized to mean zero and unit standard deviation. All 13 sentiment indices are plotted in Figure 1. As expected, all of indices are quite persistent 19 except Australia. They also tend to move together with a positive average correlation. For example, most indices are at their highs during late 1990s. To facilitate comparison with prior literature, we report summary statistics for exchange rates in nominal terms. Table 3 reports summary statistics for the monthly change in spot rate (∆enom ). The means and standard deviations of variables are given in percentages. The t volatility of the monthly change in spot rate is about 3% for most of the sample countries, which is consistent with previous studies. The autocorrelation of the change in spot rate is very low, which suggests that spot rates behave like a random walk. Table 3 shows that the currencies of Austria, Denmark, Germany, Japan, the Netherlands and Switzerland appreciated the most against the U.S. dollar from 1973m1 to 2007m12. Precisely these countries (except Denmark) had lower interest rates than the United States during this period (or equivalently, a negative forward premium). This implies that the expectation hypothesis in currency markets seems to hold on average or in the long run. On the other hand, Table 3 also indicates that the countries with a high average interest rate also yield a slightly higher return on foreign exchange. This suggests that sorting on the interest rate differential could yield a significant spread (e.g., Lustig and Verdelhan (2007)). The volatility of returns on foreign exchange is much more volatile than the forward premium, consistent with the finding in Fama (1984). 4.4. Main Empirical Evidence In the model, all the results pertain to real exchange rates and real interest rates. Thus, the majority of our empirical analysis is in real terms. However, all of our empirical results are quantitatively similar if nominal exchange rates are used. Those results are omitted for brevity, and available upon request. In addition, because a majority of foreign exchange transactions involve U.S. dollars, the U.S. dollar is used as the domestic currency throughout the paper. Sentiment is elusive to measure, and international data usually have poor quality. Thus, the state variables of our model are measured with inevitable noise. Due to this issue, two sets of empirical tests are presented. One is based on the domestic-foreign sentiment differential. Since the model has direct implications on the relation between observables (such as returns on foreign exchange) and sentiment differentials, this is a natural test. However, sentiment indices are not available for all the 19 industrial foreign countries, and the international data are not of the best quality. Thus, another set of tests using U.S. sentiment only are also 20 presented since U.S. sentiment is arguably better measured than its foreign counterparts. With U.S. sentiment only, the second test can be applied to all 19 countries. However, the second test is based on indirect predictions of the model regarding the relations between foreign exchange growth (or returns on foreign exchange) and domestic sentiment. Thus, it is intended to be used as suggestive and corroborating evidence only. 4.4.1. Evidence Based on U.S. and Foreign Sentiment First consider Prediction 1, which predicts that the forward premium in real terms (i.e., the real interest rate differential) is negatively related to domestic-foreign sentiment differentials and expected growth differentials. Panel A of Table 4 shows some weak support for the sentiment channel. In addition, our model and standard rational models share a common prediction regarding the negative relation between the true expected growth differential and the forward premium. Panel A of Table 4 shows very weak evidence, probably due to the intrinsic difficulty of measuring the true expected growth from international consumption data. Nonetheless, the evidence on the model’s new and unique prediction related to sentiment is more supportive. Panel B tests the sharper prediction of the model on the relation of country-specific real interest rates with sentiment and expected growth. The evidence is not supportive of either our model or the traditional rational models. This weak result is actually in line with the existing literature. The property of real interest rates is extremely hard to establish.15 There are substantial measurement errors on both sides of the regression equation in Panel B. On the other hand, the stronger result based on interest rate differentials in Panel A appears to be consistent with the notion that measurement errors might be correlated across countries. For example, if inflation is overstated due to quality adjustment bias or substitution bias (see, e.g., Lebow and Rudd (2003)), then these measurement errors could vary over time and be correlated across countries. Thus, focusing on differentials could potentially alleviate the measurement error problem. Next consider Prediction 2, which predicts that changes in real exchange rates are positively related to sentiment differentials between domestic and foreign countries. The left part of Panel A of Table 5 reports results from the monthly overlapping predictive 15 Dotsey, Lantz, and School (2003), for example, show the behavior of the real interest rate in US is exactly the opposite of the stylized facts presented in Stock and Watson (1999). The result is typically sensitive to the sample period and the choice of inflation indices. More recently, in the context of habit models, Wachter (2006) argues for countercyclical real interest rates, whereas Verdelhan (2010) argues for the opposite. Thus, this empirical failure might be due to the intrinsic difficulty in measuring real interest rates, sentiment, and expected growth, especially in the international context. 21 regression of 3-month changes in real exchange rates on investor sentiment differentials. The evidence is supportive. In particular, 8 out of 12 countries are statistically significant, and one more is marginally significant. In addition, our model and rational models share the same prediction that changes in real exchange rates are negatively related to expected growth differentials between domestic and foreign countries. The right part of Panel A tests this common prediction by including both investor sentiment differentials and expected consumption growth differentials as regressors. Consistent with Prediction 2, the evidence in the right part suggests that sentiment is positively related to changes in exchange rates, whereas the true expected growth rate is slightly negatively related to changes in exchange rates. However, the evidence does not provide strong support on the negative relation between foreign exchange growth and true expected growth differentials. Nonetheless, it does appear to support the model’s distinct prediction on the relation between sentiment and foreign exchange. After all, to solve the UIP puzzle, one could completely shut down the standard long-run risk channel as in column 7 of Table 2. Panel B of Table 5 reports the results by regressing exchange rate changes on individual sentiment indices and true expected growth, rather than differentials. The evidence is not as strong as that in Panel A. However, it generally is supportive. For example, the coefficients for domestic sentiment are significant for 9 out of 12 cases, and 2 of them are marginally significant. The sign for foreign sentiment is generally consistent with the model prediction, albeit less significant. The sign for domestic expected growth is also largely consistent with the model prediction. Lastly, the evidence for foreign expected growth is quite weak. Finally, consider Prediction 3, which predicts that returns on foreign exchange are negatively related to sentiment differentials between domestic and foreign countries. Panel A of Table 6 provides empirical support. In particular, the regression coefficients for all 12 countries have the correct sign, and more than half of them are statistically significant. Although neither our model nor the standard rational model has implications on the relation between returns on foreign exchange and true expected growth, these results are reported in the right panel for completeness. There is some weak evidence that true expected growth differentials positively predict returns on foreign exchange. This might be due to other forces beyond misperception and the standard long-run risk channel. Lastly, Panel B of Table 6 reports the results of regressing returns on foreign exchange on individual sentiment indices and true expected growth, rather than differentials. These results are again less supportive than those based on differentials. 22 4.4.2. Evidence Based on U.S. Sentiment Only The evidence in Tables 5 and 6 provides preliminary support for the key channel in the sentiment-based explanation. As mentioned earlier, because sentiment data are not available for all 19 industrial countries, the analysis so far is based on 12 countries only. The second set of tests based on U.S. sentiment alone are now presented as corroborating evidence. These tests use the better-measured monthly sentiment data built by Baker and Wurgler (2006). Table 7 reports the results. An indirect implication of Prediction 2 is that all else equal, changes in real exchange rates are positively related to domestic sentiment and negatively related to true expected growth. The first two panels of Table 7 provide support for this indirect implication. All 19 countries show correct signs. About two thirds of the countries are statistically significant. Similarly, an indirect implication of Prediction 3 is that all else equal, returns on foreign exchange are negatively related to domestic sentiment. The last panel of Table 7 provides support for this indirect implication as well. All 19 countries show correct signs and most of them are statistically significant. Previous studies have proposed alternative models to reproduce the failure of UIP. Typically, these explanations work in the following manner. The forward premium is approximately a linear function of some state variable (e.g., investor sentiment in our model or surplus ratios in habit models), and the expected change in exchange rates and the expected return on foreign exchange are also approximately linear functions of the same state variable. If the coefficients on the forward premium and on the expected change in exchange rates are opposite, then the model yields a negative coefficient in the UIP regression. However, these models operate through different mechanisms and constitute fundamentally different views of the sources and the pricing of risk. Thus, it is particularly important to test the underlying model-implied relations between the state variable in the model and the exchange rate growth or returns on foreign exchange. However, it is usually difficult to implement such tests due to the unobservable state variable. Our evidence suggests that misperception is an empirically relevant channel in explaining the forward premium puzzle. Of course, other forces such as habit-formation (e.g., Verdelhan (2010)) and long-run risk (e.g., Bansal and Shaliastovich (2011)) should also play significant roles in the foreign exchange market. Our study simply provides preliminary evidence that investor sentiment is a relevant force in the determination of exchange rates.16 16 It is well known that the forward premium strongly predicts returns on foreign exchange. The Internet Appendix further documents that investor sentiment has significant additional power in forecasting returns on foreign exchange. For many currencies, the R2 of a regression in which both the forward premium and U.S. investor sentiment are included is more than double the R2 of a regression in which the forward premium 23 5. Conclusion A sentiment-based model of foreign exchange is proposed to account for the forward premium puzzle. In addition, the model also helps reproduce many observed patterns in international markets, such as the low correlation between consumption growth differentials and exchange rate changes. This paper shows that sentiment induced long-run risk is potentially useful in reproducing many key asset pricing moments. This observation could be applied to resolve other puzzles in asset markets in future research. Moreover, this paper also provides evidence in support of the key mechanism in the model. Our empirical evidence suggests that investor sentiment has substantial predictive power for the currency market. Thus, this study provides “out-of-sample” support for the important role of investor sentiment in the stock market noted by previous studies (see, e.g., Baker and Wurgler (2006) and Stambaugh, Yu, and Yuan (2012)). The evidence of expectation-driven fluctuations in the currency market has policy implications. 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Xiong, W., Yan, H., 2010. Heterogeneous expectations and bond markets. Review of Financial Studies 23, 1433–1466. 27 Figure 1: Investor Sentiment Indices This figure plots investor sentiment indices for the U.S. and 12 international countries. The monthly US investor sentiment index is from 1965m07 to 2007m12, obtained from Baker and Wurgler (2006). The annual sentiment indices from 1980 to 2005 for Canada, France, Germany, Japan, and U.K. are obtained from Baker, Wurgler, and Yuan (2012). The Australia sentiment is from 1974m10 to 2007m12, obtained from Westpac Banking Corporation. Belgium sentiment is from 1988m6-2007m12, obtained from National Bank of Belgium. Denmark sentiment is from 1974m10-2007m12, obtained from Statistics Denmark. Italy sentiment is from 1982m1-2007m12, obtained from ISAE/ISTAT. New Zealand sentiment is from 1988m62007m12, obtained from WESTPAC Banking Corporation. Netherlands sentiment is from 1987m1-2007m12, obtained from Dutch Statistics Office. Switzerland sentiment is from 1972m10-2007m12, obtained from Swiss National Bank. US Australia Belgium 4 4 4 2 2 2 0 0 0 -2 -2 -2 -4 -4 1970 1980 1990 2000 1980 Canada 4 1990 2000 1990 1995 2000 2005 Denmark France 4 2 2 2 -4 0 0 0 -2 -2 -2 1980 1990 2000 -4 Germany 4 1980 1990 2000 1990 Italy 4 2000 Japan 4 2 2 -4 1980 2 0 0 0 -2 -2 1980 1990 2000 -4 Netherlands 1985 1990 1995 2000 2005 -2 1980 2 2 0 0 0 -2 -2 -2 1990 1995 2000 2005 -4 1990 1995 2000 2005 UK 4 2 0 -2 1980 1990 2000 28 2000 Switzerland 2 -4 1990 New Zealand -4 1980 1990 2000 Table 1: Parameter Choices This table reports the parameter choices for the sentiment-augmented long-run risk models. Most of the parameter values are directly taken from Bansal and Yaron (2004) and Colacito and Croce (2011). As in Bansal and Yaron (2004), these parameter values are at a monthly frequency. Statistics Variable Risk aversion coefficient γ IES coefficient ψ Subjective discount factor δ Mean consumption growth (%) µg Mean dividend growth (%) µd Standard deviation of short-run consumption innovations (%) σg Standard deviation of long-run consumption innovations (%) σx Persistence coefficient in true expected growth ρx Standard deviation of dividend innovations σd Dividend exposure to long-run consumption (leverage) φ Cross-country correlation of consumption innovations ρg ,∗g Cross-country correlation of dividend innovations ρd ,∗d Persistence coefficient in the sentiment process ρs 29 Value 3.5 1.5 0.999 0.15 0.07 0.6 0.042 0.987 0.027 3 0.3 -0.1 0.965 Table 2: Calibration This table reports the key moments (in annual terms) of international markets for our symmetric calibration. The model is calibrated at a monthly frequency. The dividend growth is specified as gd,t+1 = µd + φ(xt + st ) + σd ˆd,t+1 under agents’ perception and ˆd,t is i.i.d. normal under agents’ perception. However, under the data-generating process, gd,t+1 = µd + φxt + σd d,t+1 , where the data-generating process for d,t is i.i.d. normal. As shown in Table 1, the following parameters are fixed through all calibrations: µg = 0.0015, µd = 0.0007, σg = 0.006, σd = 0.027, ρx = 0.987, ρs = 0.965, δ = 0.999, ρ(g , ∗g ) = 0.3, ρ(d , ∗d ) = −0.1, and all the unspecified correlations are set to zero. variable ρ(s , ∗s ) ρ(x , ∗x ) σs (basis point) σx (basis point) (1) 0.90 1.00 0 0 (2) 0.90 1.00 0 4.2 Corr. b/t log pricing kernel: ρ(m, m∗ ) 0.30 0.96 Volatility of FX growth σ(∆et+1 ) 0.09 0.09 AC(1) of FX growth ρ(∆et+1 , ∆et ) 0.00 0.00 Mean excess stock return (%) -0.46 3.89 Volatility of excess stock return (%) 9.36 22.76 Average interest rate E(rf ) (%) 2.29 1.80 Volatility of interest rate σ(rf ) (%) 0.00 0.60 Interest rate correlation ρ(rf , rf∗ ) 1.00 1.00 Corr. b/t excess market returns -0.10 0.81 ∗ Corr. b/t consumption growth ρ(g, g ) 0.30 0.41 Volatility of consumption growth 2.08 2.27 ∗ Corr. b/t dividend growth ρ(gd , gd ) -0.10 -0.01 Volatility of dividend growth 9.36 9.74 Volatility of pricing kernel 0.07 0.31 Ratio of var(s)/var(g) 0.00 0.00 Ratio of var(x)/var(g) 0.00 0.16 ∗ ρ(gt+1 − gt+1 , ∆et+1 ) UIP coefficient R2 1.00 NA NA 30 1.00 NA NA (3) 0.90 0.98 0 4.2 (4) 0.50 0.98 3.6 4.2 (5) 0.25 0.98 3.6 4.2 (6) 0.90 0.98 3.6 4.2 (7) 0.90 1.00 9.0 0 0.94 0.90 0.88 0.94 0.85 0.10 0.14 0.16 0.11 0.14 0.00 -0.01 -0.01 0.00 -0.01 3.89 4.32 4.32 4.32 2.52 22.76 23.69 23.69 23.69 21.62 1.80 1.75 1.75 1.75 1.93 0.60 0.68 0.68 0.68 0.79 0.98 0.88 0.82 0.96 0.90 0.80 0.76 0.74 0.80 0.71 0.41 0.41 0.41 0.41 0.30 2.27 2.27 2.27 2.27 2.08 -0.02 -0.02 -0.02 -0.02 -0.10 9.74 9.74 9.74 9.74 9.36 0.31 0.32 0.32 0.32 0.26 0.00 0.04 0.04 0.04 0.33 0.16 0.16 0.16 0.16 0.00 0.82 1.00 0.00 0.60 -3.55 0.01 0.54 -3.76 0.01 0.76 -1.98 0.00 0.61 -4.20 0.01 31 Country Australia Austria Belgium Canada Denmark Finland France Germany Ireland Italy Japan Netherlands New Zealand Norway Portugal Spain Sweden Switzerland U.K. E(∆enom ) σ(∆enom ) t t 0.09 2.95 -0.21 3.19 -0.11 3.19 -0.00 1.56 -0.07 3.12 -0.01 2.95 -0.03 3.10 -0.20 3.23 0.06 2.99 0.20 3.00 -0.24 3.31 -0.18 3.20 0.11 3.23 -0.04 2.91 0.39 3.15 0.14 3.03 0.08 3.07 -0.28 3.52 0.04 2.95 ρ(∆enom ) E(f pt ) σ(f pt ) ρ(f pt ) E(rtF X ) σ(rtF X ) ρ(rtF X ) t -0.00 0.90 1.04 0.69 0.52 5.29 0.69 0.04 -0.26 0.84 0.85 0.32 6.17 0.71 0.05 0.19 0.74 0.80 0.36 6.17 0.71 0.02 0.19 1.50 0.91 0.20 3.13 0.73 0.03 0.44 0.90 0.79 0.61 5.72 0.71 0.05 0.90 1.34 0.89 0.67 5.57 0.71 0.03 0.24 0.69 0.80 0.07 5.86 0.71 0.05 -0.49 0.76 0.93 0.06 6.20 0.70 0.04 0.41 0.93 0.82 0.92 5.49 0.71 0.09 1.47 1.36 0.51 0.74 5.81 0.74 0.04 -0.72 1.06 0.80 -0.07 6.17 0.72 0.03 -0.25 0.96 0.49 0.09 6.11 0.69 0.05 1.03 0.99 0.96 0.58 5.99 0.75 0.02 0.60 1.48 0.34 0.69 5.34 0.67 0.08 1.67 1.02 0.99 1.50 5.76 0.72 0.10 1.29 1.92 0.25 0.42 6.17 0.79 0.09 0.44 1.15 0.64 0.20 5.69 0.72 0.05 -0.82 0.87 0.81 -0.05 6.54 0.71 0.07 0.55 0.78 0.75 0.42 5.50 0.70 ), the 3-month This table reports mean (E), standard deviation (σ), and autocorrelation (ρ) for monthly changes in nominal exchange rates (∆enom t forward premium (f pt ) (e.g., the difference between the foreign interest rate and the U.S. rate), and 3-month (overlapping) returns on foreign exchange (rtF X ) (e.g., borrow from the U.S. dollar and invest in the foreign bond). All variables are observed at a monthly frequency. The sample periods vary across countries based on data availability. The Internet Appendix provides a detailed description on the data. The means and standard deviations are on a percentage basis. Table 3: Summary Statistics Table 4: The Relation Between Interest Rates and Sentiment Panel A reports results from the monthly overlapping regression of 3-month (ex-post) real interest rate ∗ differentials (rf,t − rf,t ) on sentiment differentials (st − s∗T ) and true expected growth differentials (xt − x∗t ). Panel B reports results from the monthly overlapping regression of 3-month (ex-post) real interest rate on sentiment (st ) and true expected growth (xt ). A constant is always included in the regressions, but not reported. The sample periods vary across countries based on data availability. The Internet Appendix provides a detailed description of the data. β is the coefficient of the corresponding regression. Similarly, t is the t-statistic, and R2 is the adjusted r-squared for the predictive regression. Newey-West standard errors are used to adjust for heteroscedasticity and autocorrelation in error terms. The coefficient β for the sentiment is in percentage. Australia Belgium Canada Denmark France Germany Italy Japan Netherlands New Zealand Switzerland U.K. Australia Belgium Canada Denmark France Germany Italy Japan Netherlands New Zealand Switzerland U.K. USA ∗ Panel A: Regress rf,t − rf,t on st − s∗t and xt − x∗t β(st − s∗t ) t(st − s∗t ) β(xt − x∗t ) t(xt − x∗t ) R2 -0.16 -1.92 -0.06 -0.77 0.04 0.03 0.49 -0.39 -5.82 0.38 0.05 0.32 0.40 1.89 0.06 -0.04 -0.45 -0.01 -0.08 -0.00 -0.14 -0.60 0.15 2.31 0.11 -0.25 -2.04 0.00 0.52 0.03 -0.11 -0.81 0.00 0.06 0.01 -0.15 -1.62 0.01 0.12 0.02 0.03 0.24 -0.16 -1.78 0.05 -0.10 -2.13 -0.24 -6.77 0.20 -0.12 -1.11 -0.04 -0.65 0.03 -0.15 -1.66 -0.08 -0.68 0.03 Panel β(st ) -0.17 0.22 -0.11 -1.33 -0.19 -0.12 1.21 0.04 -0.32 -0.61 -0.20 -1.14 1.71 B: Regress rf,t on st and xt t(st ) β(xt ) t(xt ) -1.00 -0.02 -0.03 1.37 1.67 12.89 -0.36 -1.50 -3.91 -4.05 -0.65 -2.63 -0.31 -0.14 -0.71 -0.56 -0.50 -1.79 2.79 -0.63 -7.43 0.17 -0.33 -1.34 -0.88 0.88 3.65 -2.26 1.41 6.66 -0.65 -0.19 -1.21 -3.81 0.91 2.02 5.63 0.19 0.53 32 R2 0.00 0.47 0.16 0.18 0.02 0.02 0.35 0.01 0.11 0.46 0.01 0.08 0.16 33 Australia Belgium Canada Denmark France Germany Italy Japan Netherlands New Zealand Switzerland UK Expected Sign 0.40 0.89 0.36 0.84 2.06 1.01 0.86 0.31 -0.13 0.73 1.17 0.85 + 2.05 2.01 1.67 3.40 3.28 0.81 2.26 0.69 -0.30 2.32 2.76 2.18 + 0.01 0.03 0.01 0.05 0.10 0.01 0.04 0.00 0.00 0.05 0.04 0.03 0.38 1.02 0.43 0.88 3.29 1.27 0.94 0.35 -0.14 0.96 1.24 1.01 + 1.97 2.41 1.78 3.29 4.59 1.07 2.39 0.76 -0.32 3.21 2.84 2.29 + 17.64 -50.71 29.61 -10.38 -73.13 -110.06 -15.42 29.21 -30.25 -88.47 -20.24 -85.20 - 0.62 -1.10 1.43 -0.39 -4.31 -2.09 -0.80 0.69 -0.89 -3.16 -0.76 -1.62 - Panel A: Regress Changes in Exchange Rates on on (st − s∗t ) and (xt − x∗t ) et+3 − et on st − s∗t and xt − x∗t et+3 − et on st − s∗t 2 ∗ ∗ ∗ R β(st − st ) t(st − s∗t ) β(xt − x∗t ) t(xt − x∗t ) β(st − st ) t(st − st ) 0.01 0.04 0.02 0.05 0.17 0.04 0.04 0.00 0.00 0.10 0.04 0.05 R2 The left part of Panel A reports results from the monthly overlapping predictive regression of 3-month changes in real exchange rates on sentiment differentials between domestic and foreign countries. The right part of Panel A reports results from the monthly overlapping predictive regression of 3-month changes in real exchange rates on sentiment differentials and true expected growth differentials. Panel B reports results from the monthly overlapping predictive regression of 3-month changes in real exchange rates on individual sentiment and true expected growth. A constant is always included in the regressions, but not reported. The sample periods vary across countries based on data availability. The Internet Appendix provides a detailed description of the data. β is the coefficient of the corresponding regression. Similarly, t is the t-statistic, and R2 is the adjusted r-squared for the predictive regression. Newey-West standard errors are used to adjust for heteroscedasticity and autocorrelation in error terms. The coefficient β for the sentiment differential is in percentage. Table 5: Forecasting Exchange Rate Changes with both US and Foreign Sentiment 34 Australia Belgium Canada Denmark France Germany Italy Japan Netherlands New Zealand Switzerland U.K. Expected Sign Panel B: Regress Exchange Rate Changes β(xt ) t(xt ) β(st ) t(st ) β(s∗t ) t(s∗t ) 0.74 2.30 -0.07 -0.27 -29.85 -0.65 1.22 2.29 -0.95 -1.99 -109.79 -1.74 0.77 2.72 -0.16 -0.71 2.36 0.08 0.73 1.72 -0.90 -2.33 -114.28 -2.48 2.33 4.23 -2.11 -2.77 -78.40 -1.35 1.14 1.54 0.37 0.55 -85.48 -1.52 0.99 2.20 -0.95 -1.74 -113.28 -1.85 0.96 2.04 0.23 0.41 24.66 0.42 0.36 0.66 0.55 1.10 -109.15 -1.85 1.27 2.12 -0.68 -1.56 -126.28 -1.76 1.49 3.08 -0.79 -1.49 -144.14 -2.61 1.39 2.76 -0.55 -1.07 -88.84 -1.48 + + - on st , s∗t , β(x∗t ) -40.91 10.69 -61.01 -30.45 49.45 58.01 4.85 -42.63 3.67 92.10 -3.23 45.62 + xt , and x∗t t(x∗t ) R2 -1.17 0.02 0.18 0.04 -2.43 0.06 -0.90 0.07 2.33 0.12 0.81 0.06 0.26 0.05 -0.88 0.02 0.09 0.02 1.89 0.10 -0.11 0.07 0.54 0.05 + 35 Australia Belgium Canada Denmark France Germany Italy Japan Netherlands New Zealand Switzerland UK Expected Sign -0.57 -0.89 -0.40 -0.88 -1.95 -1.25 -1.13 -0.46 -0.38 -0.87 -1.29 -1.02 - -2.42 0.02 -1.55 0.05 -1.57 0.01 -3.29 0.05 -2.80 0.08 -0.98 0.01 -2.51 0.07 -0.97 0.00 -0.52 -0.00 -2.78 0.07 -2.82 0.05 -2.45 0.04 - -0.55 -0.92 -0.38 -0.92 -3.43 -1.52 -1.65 -0.50 -0.39 -1.05 -1.33 -1.16 - -2.35 -1.66 -1.38 -3.22 -4.35 -1.25 -2.93 -1.03 -0.54 -3.41 -2.85 -2.50 - -21.84 8.12 10.51 10.15 88.06 114.18 54.64 -28.45 3.00 67.95 13.12 77.15 NA R2 -0.70 0.02 0.16 0.04 0.49 0.01 0.37 0.05 4.72 0.18 2.12 0.05 1.70 0.10 -0.64 0.00 0.06 -0.01 2.46 0.10 0.46 0.04 1.44 0.05 NA Panel A: Regress Returns on Foreign Exchange on (st − s∗t ) and (xt − x∗t ) FX FX on st − s∗t and xt − x∗t rt+3 on st − s∗t rt+3 2 ∗ ∗ ∗ R β(st − st ) t(st − s∗t ) β(xt − x∗t ) t(xt − x∗t ) β(st − st ) t(st − st ) The left part of Panel A reports results from the monthly overlapping predictive regression of 3-month returns on foreign exchange on sentiment differentials between domestic and foreign countries. The right part of Panel A reports results from the monthly overlapping predictive regression of 3-month returns on foreign exchange on sentiment differentials and true expected growth differentials. Panel B reports results from the monthly overlapping predictive regression of 3-month returns on foreign exchange on individual sentiment and true expected growth. A constant is always included in the regressions, but not reported. The sample periods vary across countries based on data availability. The Internet Appendix provides a detailed description of the data. β is the coefficient of the corresponding regression. Similarly, t is the t-statistic, and R2 is the adjusted r-squared for the predictive regression. Newey-West standard errors are used to adjust for heteroscedasticity and autocorrelation in error terms. The coefficient β for the sentiment differential is in percentage. Table 6: Forecasting Returns on Foreign Exchange with both US and Foreign Sentiment 36 Australia Belgium Canada Denmark France Germany Italy Japan Netherlands New Zealand Switzerland U.K. Expected Sign Panel β(st ) -1.02 0.13 -0.88 -1.30 -4.28 -1.97 -1.50 -1.20 -0.44 -1.43 -1.84 -1.71 - B: Regress Returns on Foreign t(st ) β(s∗t ) t(s∗t ) β(xt ) -2.44 0.14 0.52 40.62 0.09 1.37 1.89 104.15 -3.15 -0.13 -0.48 -22.17 -3.03 0.52 1.31 118.14 -4.08 3.09 3.66 75.61 -1.64 -0.45 -0.33 86.87 -1.54 1.69 2.49 135.76 -2.52 -0.22 -0.38 -46.37 -0.37 -0.20 -0.32 125.55 -2.40 0.72 1.74 109.80 -3.57 0.73 1.20 143.76 -3.24 0.52 1.01 77.36 + + NA Exchange on st , t(xt ) β(x∗t ) 0.79 52.73 0.96 99.93 -0.65 -29.03 2.48 11.81 0.95 -76.21 1.11 -135.59 1.54 -39.26 -0.76 33.39 1.28 28.86 1.51 -74.36 2.50 -14.90 1.29 -28.38 NA NA s∗t , xt , and x∗t t(x∗t ) R2 1.40 0.04 1.17 0.06 -0.94 0.09 0.34 0.08 -3.16 0.19 -1.66 0.08 -1.03 0.10 0.65 0.03 0.61 0.01 -1.55 0.10 -0.41 0.08 -0.34 0.07 NA 37 Australia Austria Belgium Canada Denmark Finland France Germany Ireland Italy Japan Netherlands New Zealand Norway Portugal Spain Sweden Switzerland U.K. Expected Sign (et+3 − et ) on st β(st ) t(st ) R2 0.75 2.07 0.02 0.97 2.31 0.02 1.16 2.62 0.03 0.08 0.37 0.00 0.98 2.31 0.02 0.64 1.78 0.01 0.92 2.07 0.02 1.01 2.33 0.02 1.13 2.00 0.02 0.60 1.30 0.01 1.18 2.76 0.03 1.02 2.34 0.02 0.82 1.49 0.01 0.51 1.31 0.01 0.49 1.03 0.00 1.07 2.36 0.03 1.09 2.62 0.03 0.93 1.79 0.02 0.88 1.68 0.02 + + (et+3 − et ) on st β(st ) t(st ) β(xt ) 0.77 2.16 -0.89 1.09 2.53 -3.92 1.29 2.89 -4.46 0.05 0.24 0.96 1.10 2.52 -4.23 0.75 1.96 -3.88 1.07 2.62 -4.91 1.14 2.59 -4.30 1.35 2.47 -4.99 0.75 1.75 -4.82 1.22 2.80 -1.21 1.16 2.62 -4.67 0.98 1.83 -5.32 0.60 1.51 -2.92 0.56 1.20 -2.46 1.19 2.53 -3.81 1.17 2.65 -2.78 1.11 2.14 -5.84 1.00 2.12 -4.02 + + and xt t(xt ) -0.47 -2.04 -2.26 0.95 -2.15 -2.05 -2.49 -2.17 -2.33 -2.36 -0.58 -2.38 -2.48 -1.63 -1.06 -1.70 -1.40 -2.64 -1.89 R 0.01 0.04 0.05 0.00 0.04 0.03 0.05 0.04 0.05 0.03 0.03 0.05 0.04 0.02 0.01 0.04 0.04 0.05 0.04 2 FX ) on (rt+3 β(st ) t(st ) -1.93 -3.98 -1.47 -2.75 -1.42 -2.64 -0.42 -1.72 -1.39 -3.14 -1.28 -2.96 -1.15 -1.95 -1.31 -2.51 -1.09 -1.27 -1.55 -1.80 -1.39 -3.14 -1.48 -2.75 -1.38 -2.27 -0.98 -2.32 -2.71 -0.91 -1.16 -1.57 -1.40 -2.99 -1.38 -2.45 -0.91 -1.51 - R2 0.07 0.05 0.05 0.01 0.05 0.04 0.03 0.04 0.01 0.03 0.04 0.05 0.04 0.03 0.02 0.03 0.05 0.04 0.02 st This table reports results from the monthly overlapping predictive regression of 3-month changes in real exchange rates and 3-month returns on foreign exchange on investor sentiment, st , respectively. The sample is monthly from 1973m1 to 2007m12 during which investor sentiment data are available. β is the coefficient of the corresponding regression. Similarly, t is the t-statistic, and R2 is the adjusted r-squared for the predictive regression. Newey-West standard errors are used to adjust for heteroscedasticity and autocorrelation in error terms. The coefficient β for sentiment is in percentage. Table 7: Forecasting Exchange Rate Changes and Returns on Foreign Exchange by Sentiment