Exchange Rates, Interest Rates, and the Risk Premium

Transcript

1 American Economic Review 2016, 106(2): 436–474 http://dx.doi.org/10.1257/aer.20121365 † , , and the Risk Premium Exchange Rates Interest Rates Charles Engel* By The uncovered interest parity puzzle concerns the empirical regularity that high interest rate countries tend to have high expected returns on short term deposits. A separate puzzle is that high real interest rate countries tend to have currencies that are stronger than can be accounted for by the path of expected real interest differentials under uncovered interest parity. These two findings have apparently contradictory implications for the relationship of the foreign-exchange risk premium and interest-rate differentials. We document these puzzles and show that existing models appear , unable to account for both. A model that might reconcile the findings is discussed. ( JEL E43, F31, G15 ) There are two well-known empirical relationships between interest rates and foreign exchange rates, one concerning the rate of change of the exchange rate and the other concerning the level of the exchange rate. Each of these empirical relation- ships presents challenges to traditional economic models in international finance, and each has spurred advances in the modeling of investor behavior and macro- economic relationships. Both are important for understanding the role of openness in financial markets and aggregate economic relationships. What has been here- tofore unnoticed is that the two relationships taken together constitute a paradox; the explanations advanced for one empirical finding are completely inadequate for explaining the other. ( ) puzzle in foreign exchange markets The interest parity or forward premium ( from a week to a quarter ) finds that over short time horizons when the interest rate ( one country relative to another ) is higher than average, the short-term deposits of the high-interest rate currency tend to earn an excess return. That is, the high interest rate country tends to have the higher expected return in the short run. The empirical literature on the forward premium anomaly is vast. Classic early references include * Department of Economics, University of Wisconsin, 1180 Observatory Drive, Madison, WI 53706 e-mail: ( [email protected] ) . I thank Bruce Hansen and Ken West for many useful conversations and Mian Zhu, Dohyeon Lee, and especially Cheng-Ying Yang for excellent research assistance. I thank David Backus, Gianluca Benigno, Martin Evans, Cosmin Ilut, Keyu Jin, Richard Meese, Michael Melvin, Anna Pavlova, John Prins, Alan Taylor, and Adrien Verdelhan for comments and helpful discussions. I have benefited from helpful comments at many semi- nars and from support from the following organizations at which I was a visiting scholar: Federal Reserve Bank of Dallas, Federal Reserve Bank of St. Louis, Federal Reserve Bank of San Francisco, Federal Reserve Board, European Central Bank, Hong Kong Institute for Monetary Research, Central Bank of Chile, and CREI. I acknowl- edge support from the National Science Foundation, award 0850429 and award 1226007. The author has no relevant or material financial interests that relate to the research described in this paper to disclose. † Go to http://dx.doi.org/10.1257/aer.20121365 to visit the article page for additional materials and author disclosure statement. 436

2 VOL. 106 NO. 2 437 ENGEL: EXCHANGE RATES, INTEREST RATES, AND THE RISK PREMIUM surveys the empirical work that 1981 ( 1984 ) . Engel ( 1996, 2014 ) and Fama ( Bilson ) establishes this puzzle and discusses the problems faced by the literature that tries to account for the regularity. A risk-based explanation of this anomaly requires that the short-term deposits in the high-interest rate country are relatively riskier ( the risk arising from exchange rate movements, since the deposit rates in their own currency , and therefore incorporate an expected excess return as a are taken to be riskless ) time-varying reward for risk-bearing. The ex ante risk premium must therefore be and covary with the interest differential. Standard exchange rate models, such as the textbook Mundell-Fleming model ( ) model, assume that interest parity holds: that 1976 or the well-known Dornbusch there are no ex ante excess returns from holding deposits in one country relative to another. These models have a prediction about the level of the exchange rate. The level of the exchange rate is important in international macroeconomics because it will help to determine demand for traded goods, especially when some nominal prices are sticky. These models predict that when a country has a higher than average relative interest rate, the price of foreign currency should be lower than average. This relationship is borne out in the data, but the strength of the home currency tends to be greater than is warranted by rational expectations of future short-term interest differentials as the models posit under interest parity; there is excess comovement or volatility. One way to rationalize this finding is to appeal to the influence of expected future risk premiums on the level of the exchange rate. That is, the country with the relatively high interest rate has the lower risk premium and hence the stronger currency. When a country’s interest rate is high, its currency is appreciated not only 1 because its deposits pay a higher interest rate but also because they are less risky. These two predictions about risk go in opposite directions: the high interest rate country has higher expected returns in the short run, but a stronger currency in levels. The former implies the high interest rate currency is riskier, the latter that it is less risky. That is the central puzzle of this paper. This study confirms these empir - ical regularities in a unified framework for the exchange rates of the G7 countries ) relative to the ( Canada, France, Germany, Italy, Japan, and the United Kingdom United States. be the difference ρ It is helpful to express this puzzle mathematically. Let 1 t + t and t + 1 on a foreign short-term deposit and between the return between period the home short-term deposit, inclusive of the return from currency appreciation. This * - − r be the differ r study always takes the United States to be the home country. Let t t ( inflation adjusted ) interest rate in the foreign country and the ence in the ex ante real United States. We use the * notation throughout to denote the foreign country. The literature on interest parity has struggled to account for the robust empiri- * 0 ) > r − ρ , . Here, “cov” refers to the unconditional r cov E ( cal finding that + t t t t 1 . The ex ante excess ρ ρ to the conditional expectation of covariance, and E 1 t 1 + t t + interest return on the foreign deposit is positively correlated with the foreign less US differential. This is a correlation between two variables known at time t : the risk pre- mium and the interest rate differential. It is not a correlation between two unexpected 1 ( ( 1989 ) and Obstfeld and Rogoff Hodrick 2003 ) incorporate risk into macroeconomic models of the level of the exchange rate. The latter includes a role for risk in a micro-founded model similar to a Dornbusch sticky-price model.

3 438 UAR Y 2016 FEBR THE AMERICAN ECONOMIC REVIEW returns, which may be the source of a risk premium. Instead it is an unconditional correlation between two ex ante returns, suggesting that the factor ) that drive time ( s ( s variation in the foreign exchange risk premium and the factor ) that drive time variation in the interest rate differential have a common component. An analogy would be a finding that the risk premium on stocks is positively correlated with the short-term interest rate. Models with standard preferences in a setting of undistorted financial markets are unable to account for this empirical finding by appealing to a risk premium arising from foreign exchange fluctuations. The consumption vari- ρ - in such models do not also lead to an inter E ances and covariances that drives 1 + t t 2 ρ . est differential that covaries positively with E t 1 + t Recent advances have found that the interest parity puzzle can be explained with the same formulations of nonstandard preferences that have been used to account for other asset-pricing anomalies. These studies model the ex ante excess return as a risk premium related to the variances of consumption in the home and foreign ( 2010 ) builds on the model of external habits of Campbell and country. Verdelhan ( 1999 ) ; and Bansal and Shaliastovich ( 2007, 2013 ) ; Colacito ( 2009 ) ; and Cochrane Colacito and Croce ( ) develop the model of preferences in Epstein and 2011, 2013 to account for this anomaly. Those studies show how ( ) and Weil ( Zin ) 1989 1990 the foreign exchange risk premium can be related to the difference in the conditional variance of consumption in the foreign country relative to the home country, in a setting of undistorted, complete financial markets. These papers are important not only to our understanding of the interest parity puzzle, but also to our understanding of asset pricing more generally because they show the power of a single model of preferences to account for a number of asset pricing regularities. A different approach to explaining the interest parity puzzle advances an expla- nation akin to the model of rational inattention of Mankiw and Reis ( ) and 2002 ( 2003 . This explanation builds on a standard model of exchange rates such Sims ) 1976 ) ( as Dornbusch . A monetary contraction increases the interest rate and leads to an appreciation of the currency. However, some investors are slow to adjust their portfolios, perhaps because it is costly to monitor and gather information constantly. As more investors learn of the monetary contraction, they purchase home assets, leading to a further home appreciation. So when the home interest rate increases, the return on the home asset increases both from the higher interest rate and the currency appreciation. This model of portfolio dynamics was proposed informally by Froot and Thaler ( ) and called “delayed overshooting.” Eichenbaum and 1990 ( provide empirical evidence that is consistent with this hypothesis, and ) Evans 1995 ( Bacchetta and van Wincoop ) develop a rigorous model. 2010 In the data for currencies of major economies relative to the United States, * ) relative to its mean ( is high − , the level of the foreign currency tends r r when t t ) ( ) to be stronger ( 1976 appreciated and Frankel ( 1979 ) are the original . Dornbusch papers to draw the link between real interest rates and the level of the exchange rate in the modern, asset-market approach to exchange rates. The connection has not gone unchallenged, principally because the persistence of exchange rates and interest differentials makes it difficult to establish their comovement with a high 2 On this point, see for example Bekaert, Hodrick, and Marshall ( 1997 ) and Backus, Foresi, and Telmer ( 2001 ) . Also see the surveys of Engel ( 1996, 2014 ) .

4 VOL. 106 NO. 2 439 ENGEL: EXCHANGE RATES, INTEREST RATES, AND THE RISK PREMIUM 1988 ( ) degree of certainty. For example, Meese and Rogoff and Edison and Pauls treat both series as nonstationary and conclude that evidence in favor of ) ( 1993 cointegration is weak. However, more recent work that examines the link between real interest rates and the exchange rate, such as Engel and West ( 2006 ) , Alquist and Chinn , and Mark ( 2009 ) , has tended to reestablish evidence of the empirical ( 2008 ) link. Another approach connects surprise changes in interest rates to unexpected changes in the exchange rate. There appears to be a strong link of the exchange see, for example, Faust et al. 2007; rate to news that alters the interest differential ( . Andersen et. al. 2007; and Clarida and Waldman 2008 ) It is widely recognized that exchange rates are excessively volatile relative to the predictions of monetary models that assume interest parity or no foreign exchange ) and Rogoff ( 1987 ) are prominent papers risk premium. Frankel and Meese ( 1996 that make this point. Evans ) refers to the “exchange-rate volatility puzzle” as ( 2011 one of six major empirical challenges in the study of exchange rates. Recent contri- butions that examine aspects of this excess volatility include Engel and West ( 2004 ) ; ( 2006 ) ; and Evans ( 2012 ) . Bacchetta and van Wincoop ( This excessive volatility in the level of the exchange rate arises ) by definition from the effect of deviations from uncovered interest parity on the level of the exchange rate. This effect is forward looking, and can be summarized in the vari- _ ∞ ∑ − , to denote the uncon- ρ ) . We use the overbar notation, as in x ( ρ E able t + j + 0 t 1 j = ̅ . When this sum of the ex ante risk premiums on foreign ditional mean of a variable x t deposits increases, the home currency appreciates. The second empirical finding we ∞ * E r ∑ − 0 . That means that ρ < r , cov focus on can be summarized as 0 t + t + t 1 t j ) ( * ) relative to its mean r − , the home currency is strong for two rea- ( is high r when t t sons: the influence of interest rates under uncovered interest parity ( as in Dornbusch and the influence of deviations from uncovered interest parity. and Frankel ) It is clear from examining the two covariances that are at the heart of the empiri- cal puzzle of this paper, it must be the case that while the interest parity puzzle has * ) 0 > ρ j that is, for some ( , r , for some period in the future , − r 0 ) > cov E ( t t t 1 + t * 0 ρ ) < , r , the reverse sign. − r E cov ( t + j + t t 1 t Neither modern models of the foreign exchange risk premium nor of delayed overshooting can account for the finding concerning the level of the exchange rate, ∞ * E ∑ r − ρ 0 . We explain why these models are not capable < , r that cov 1 + t 0 t t + t j ) ( of accounting for both puzzles. The very features that make them able to account for the interest parity puzzle work against explaining the level puzzle. As we show, both the models of the risk premium and of delayed overshooting imply a sort of muted adjustment in financial markets, which can account for the interest parity puzzle, but the excess comovement puzzle requires a sort of magnified adjustment. We describe the features of a model that can reconcile the empirical findings. We suggest that there may be multiple factors that drive the relationship between interest rates and exchange rates. We embed a simple model of liquidity risk based on Nagel ( 2014 ) within a standard open-economy macroeconomic model. In that framework, an asset may earn a liquidity premium that increases as nominal interest rates rise, or as there are shocks to the financial system. Both the macroeconomic shocks ( for example, to monetary policy ) that drive interest rates as well as financial shocks to liquidity play a role in the exchange rate-interest rate nexus, and could potentially account for both empirical findings.

5 440 Y 2016 FEBR UAR THE AMERICAN ECONOMIC REVIEW Section I develops the approach of this paper. Section II presents empirical results. Section III explains why the empirical findings constitute a puzzle. We discuss the difficulties encountered by asset pricing approaches such as representative agent 3 Then this sec- models of the risk premium, and models of “delayed overshooting.” tion proposes the model of the liquidity premium that can potentially encompass both empirical findings. The study of risk premiums in foreign exchange markets sheds light on important - questions in asset pricing that go beyond the narrow interest of specialists in inter national asset markets. The foreign exchange rate is one of the few, if not the only, aggregate asset for an economy whose price is readily measurable, so its pricing offers an opportunity to investigate some key predictions of asset pricing theories. For example, in the absence of arbitrage, the rate of real depreciation of the home SDF ) for foreign country’s currency equals the log of the stochastic discount factor ( returns relative to the log of the corresponding SDF for home returns, while the risk ( ) is proportional to the conditional variance of premium as conventionally measured 4 the log of the SDF for home relative to the variance of the SDF for foreign returns. Thus, the behavior of the foreign exchange rate may give direct evidence on the fundamental building blocks of equilibrium asset pricing models. I. Excess Returns and Real Exchange Rates We develop here a framework for examining behavior of ex ante excess returns and the level of the exchange rate. Our set-up will consider a home and a foreign country. In the empirical work of Section II, we always take the United States as ( as does the majority of the literature ) , and consider other major the home country be the home one-period nominal interest for economies as the foreign country. Let i t * is the corresponding foreign 1 and deposits in period + t that pay off in period t i t denotes the log of the foreign exchange rate, expressed as the US s interest rate. t dollar price of foreign currency. The excess return on the foreign deposit held from period t to period t + 1 , inclusive of currency return is given by * i − s − ≡ i . + s ρ ) 1 ( t t 1 1 t + t + t This definition of excess returns corresponds with the definition in the literature. We can interpret this as a first-order log approximation of the excess return in home currency terms for a foreign security. As Engel 1996 ) notes, the first-order log approx- ( imation may not really be adequate for appreciating the implications of economic expected excess return. For example, if the exchange rate is condi- theories of the 1 _ s E ( ( S var / S ) = E , s ) + − s ln tionally log normally distributed, then ) ( t 1 + t t t t t t + 1 t 1 + 2 ( s refers to the conditional variance of the log of the exchange rate ) var where 1 t t + points out that this sec- ) ( not log ) of the exchange rate. Engel ( 1996 is the level S and t premiums ond-order term is approximately the same order of magnitude as the risk 3 “Representative agent models” may be an inadequate label for models of the risk premium that are developed off of the Euler equation of a representative agent under complete markets, generally taking the consumption stream as exogenous. 4 The SDFs for home and foreign returns are unique when asset markets are complete. See Backus, Foresi, and Telmer ( 2001 ) ; Brandt, Cochrane, and Santa-Clara ( 2006 ) ; and Section IIIA below.

6 VOL. 106 NO. 2 441 ENGEL: EXCHANGE RATES, INTEREST RATES, AND THE RISK PREMIUM implied by some economic models. However, we proceed with analysis of excess returns defined according to equation ( 1 ) both because it is the object of almost all of the empirical analysis of excess returns in foreign exchange markets, and because the theoretical literature that we consider in Section III seeks to explain expected ρ . excess returns defined precisely as E 1 t + t The well known uncovered interest parity puzzle comes from the empiri- cal finding that the change in the log of the exchange rate is negatively cor - * − i . That is, estimates related with the home less foreign interest differential, i t t * * − i − s − , i tend to be negative. As i i ) ) = cov ( E s , − s s ( of cov t t t 1 + t t 1 + t t t t 1996, 2014 surveys, this finding is consistent over time among pairs ) Engel ( ( 1 ) , we note that the of high-income, low-inflation countries. From equation * * i − s i ( var − s < , i 0 ) i − is equivalent to ) < 0 E ( cov relationship t t t 1 t + t t t * i . That is, when the foreign interest rate is relatively high, ) , ρ − i cov < E ( t 1 + t t t * − i is above average, the excess return on foreign assets also tends to be above i so t t average. This is considered a puzzle because it has been very difficult to find plausi- ble economic models that can account for this relationship. While almost all of the empirical literature on the interest parity puzzle has doc- * , we recast the puzzle in terms of ) i , i ρ − cov ( E umented evidence concerning t 1 + t t t * − ex ante real interest rate, ) ρ i.e., United States ( is the home , r r r ) . ( cov E t t + t t t 1 denotes the log of the p = i − E and π , where π p − ≡ p r defined as 1 t + t t t t + t + t 1 t 1 * is defined analogously. This is an consumer price index in the home country. r t approximation of the real interest rate. Analogous to the discussion above of the exchange rate, a different approximation would include a term for the variance of inflation. In essence, that variance is treated as a constant here. We conduct empirical work using real interest rates for three reasons. First, the theoretical discussion of the interest parity puzzle usually builds models to explain * , r ρ − r ) , essentially assuming there is no inflation risk. Second, in high ( E cov 1 t + t t t * 0 ) > i − ρ , is less robust; see i inflation countries, the evidence that E ( cov t t t t 1 + 2000 2010 and Frankel and Poonawala ( ( ) . The puzzle arises Bansal and Dahlquist ) mostly among country pairs where inflation is low and stable. Third, the theoretical ) ) and Frankel ( 1979 models of the level of the exchange rate, such as Dornbusch , ( 1976 relate the stationary component of the exchange rate to real interest differentials. To measure the relation between the interest differential and the level of the , subtracting off unconditional means, and iter - exchange rate, begin by rearranging ating forward to get ∞ T IP E s = s − ρ − ∑ , ) ( ρ 2 ) ( t t + j + 1 t t ̅ j 0 = * T IP ∞ − ≡ s − lim ( E s − k ( s s )) and s ≡ E ∑ i − s where ( ̅ = 0 →∞ t t t + + 1 k t + j t t t k j * ̅ − ( − . In deriving this expression, we have assumed that the interest i i ) i ) j + t 5 differential and the ex ante excess return are stationary random variables. The lim term is the permanent component of the ( E )) s s − − k ( s ̅ k 1 k + t t →∞ + exchange rate according to the decomposition of Beveridge and Nelson ( 1981 ) . 5 Specifically, these variables are square summable, so that the sums on the right side of the equation exist.

7 442 Y 2016 FEBR UAR THE AMERICAN ECONOMIC REVIEW That is, assuming that the nominal exchange rate is stationary in first differences, the Beveridge-Nelson decomposition allows us to define a permanent component that follows a pure mean zero random walk, and a stationary or transitory com- T is the transitory component. When we talk about the effect s ponent. Therefore, t of risk on the level of the exchange rate, we refer to this component: the actual log of the exchange rate, normalized by its permanent component. If interest , the transitory component of the 0 ρ ≥ j 0 for all = parity held, so that E 1 + j + t t exchange rate is equal to the infinite sum of the expected nominal interest differen- IP ). ( the IP superscript referring to interest parity s tials, which we have denoted by t The effect of ex ante excess returns on the level of the exchange rate is given in the ∞ ) ∑ − ρ ( ρ . E term 1 = t j t + 0 j + ̅ The Dornbusch and Frankel models that assume interest parity imply IP ∞ * * , r < − r > 0 . We show empirically that cov r E s − ∑ ρ 0 . r , cov t j t + t 0 t 1 t t t + ) ( ) ( 2 ) , it follows that there is excess comovement in the level of the station- From ( T IP * * . That is, , r − r r > cov − s , r s ary component of the exchange rate: cov t t t t t t ( ( ) ) T IP ∞ * * * , r 0 . − r r − − cov s > , r s − r r = − cov E ∑ , ρ cov + 1 t 0 t + t t j t t t t t t ) ( ( ) ) ( ( can be gleaned by looking at the relationship ) Some more insight into equation 2 log ) real exchange rate is given of exchange rates to consumer price levels. The ( * ≡ + p . Assume for simplicity there is no drift in the real exchange − p s by q t t t t 2 ) by adding and subtracting prices appropriately, ( rate. We can rewrite ∞ ∞ * * ̅ ( ρ − lim . ( E q ∑ ) = E E ∑ − ) ) r r − − r r − ( − ρ ( 3 ) q ( ) t t k t + t t j + j t + t + t + j 1 ̅ →∞ k = j = 0 0 j When purchasing power parity holds in the long run, so the real exchange rate ) q ( E is simply the uncondi- lim is stationary as we will find for our data, t →∞ k t + k ( 3 ) says the real exchange tional mean of the real exchange rate. Then equation rate is above its mean either when the sum of current and future expected real interest differentials ) are above average, or when the sum of foreign less home ( expected current and future excess returns are above average. In the Dornbusch model, an increase in the current real interest rate influences the level of the real exchange rate through the term involving current and expected future real inter - ∞ * * ̅ ∑ r ) − r − ( r . Our empirical finding that − r E est rate differentials: ( ) + j t + j t j = 0 t ∞ * 0 implies that there is excessive volatility in the real ∑ < r ρ E r − , cov 0 t + j + t t t 1 ) ( exchange rate level. * ∞ * ̅ ∑ i − is not the interest differen- i i − i − ( ) E It is important to note that ) ( t 0 = j t t tial on long-term bonds, even hypothetical infinite-horizon bonds. It is the difference between the expected return from holding an infinite sequence of short-term foreign bonds and the expected return from the infinite sequence of short-term home bonds. An investment that involves rolling over short term assets has different risk charac- teristics than holding a long-term asset, which might include a holding-period risk 6 premium. 6 Chinn and Meredith ( 2004 ) find that uncovered interest parity holds relatively well for long-term bonds. Under some further assumptions ( e.g., that the expectations hypothesis of the term structure holds and the interest rate differential is a first-order Markov process ) , this would imply the reversal in the sign of the covariance we highlight.

8 VOL. 106 NO. 2 443 ENGEL: EXCHANGE RATES, INTEREST RATES, AND THE RISK PREMIUM * ( E r ρ 0 and ) > , In the next section, we present evidence that cov − r t t 1 + t t ∞ * E < ∑ 0 . The short-run ex ante excess return on the foreign ρ r − , r cov t j + t t 0 1 t + ) ( ρ , is negatively correlated with the real interest differential, consistent security, E 1 + t t with the many empirical papers on the uncovered interest parity puzzle. But the sum of current and expected future returns is positively correlated. The empirical approach of this paper can be described simply. We estimate vector * * . From the p i − i , , and p − s in the variables ) VECMs ( error-correction models t t t t t * * * r − i . − i − ( π r = − π ) E VECM estimates, we construct measures of ( ) 1 t t + t t 1 t + t t T IP s and . s Using standard projection formulas, we can also construct estimates of t t ∞ IP T ∑ . From these s ( ρ and − ρ ) , we take the difference of s To measure E t + 0 = j t t 1 + j t ̅ 7 Our ECM estimates, we calculate our estimates of the covariances just discussed. approach of estimating undiscounted expected present values of interest rates is pre- saged in Campbell and Clarida ( 1987 ) ; Clarida and Galí ( 1994 ) ; and more recently in Froot and Ramadorai ( ) ; Brunnermeier, Nagel, and Pedersen ( 2009 ) ; and 2005 , 8 9 . ) ( Mark 2009 Empirical Results II. We investigate the behavior of exchange rates and interest rates for the United States relative to the other six countries of the G7: Canada, France, Germany, Italy, Japan, and the United Kingdom. We also consider the behavior of US variables rela- tive to an aggregate weighted average of the variables from these six countries, with weights measured as the value of each country’s exports and imports as a fraction of - the average value of trade over the six countries. This set of seven countries are par ticularly interesting for examining these exchange rate puzzles because these coun- tries have had floating exchange rates among themselves since the early 1970s, little foreign exchange intervention in the market for each currency relative to the dollar, ( especially for each very low capital controls, very little default risk, low inflation country relative to the United States , and very deep markets in foreign exchange. ) These facts narrow the possible explanations for the puzzles. Our study uses monthly data. Foreign exchange rates are noon buying rates in New York, on the last trading day of each month, culled from the daily data reported in the Federal Reserve ( 2010 ) historical database. The price levels are consumer price indexes from the Main Economic Indicators on the OECD ( ) database. 2010 Nominal interest rates are taken from the last trading day of the month, and are the midpoint of bid and offer rates for one-month Eurorates, as reported on Intercapital ( 2010 ) . The interest rate data begin in June 1979, and the empirical from Datastream work uses the time period June 1979 to October 2009. The choice of an end date of October 2009 represents a compromise. On the one hand, it is important for our pur - poses to include these data well into 2009 because it has been noted in some recent 7 We also consider VECMs that are augmented with data on stock market returns, oil prices, gold prices, and long-term interest rates, which are included solely for the purpose of improving the forecasts of future interest rates and inflation rates. 8 This method does not require estimation of expected long-term real interest rates, about which there is some controversy. See Bansal, Kiku, and Yaron ( 2012 ) . 9 See the online Appendix for a detailed discussion of the relation of the empirical work in this paper to Froot and Ramadorai ( 2005 ) .

9 444 FEBR UAR Y 2016 THE AMERICAN ECONOMIC REVIEW papers that there was a crash in the “carry trade” in 2008, so it would perhaps bias 10 On the other hand, it seems our findings if our sample ended prior to this crash. plausible that there were some changes in the driving processes for interest rates and exchange rates during the turbulent period from late 2008 until early 2013 because of the global financial crisis and the European debt crisis. A. Fama Regressions see Fama 1984 ) is the basis for the forward premium puz- The Fama regression ( zle. It is usually reported as a regression of the change in the log of the exchange rate t t + 1 on the time and interest differential: between time t * ̃ u ) + = ζ − s . + β i ( i − s t 1 + t t s , t + 1 s s t ̃ = 0 and β 1 . We can rewrite this regression = ζ Under uncovered interest parity, s s as * u ) + i − = ζ , + β ( i ρ ) 4 ( t s 1 + s , t + 1 t t s ̃ ≡ 1 − β . The left-hand side of the regression is the ex post excess return β where s s 0 , there is a positive correlation between the excess > β on the foreign security. If s return on the foreign currency and the foreign-home interest differential. We estimate the Fama regression for our currencies as a preliminary exercise. Table 1 reports the 90 percent confidence interval for the regression coefficients from , based on Newey-West standard errors. For all of the currencies, the point lies above are positive. The 90 percent confidence interval for β estimates of β s s ( Italy and France being the exceptions, where the confidence interval zero for four for the latter barely includes zero ). For four of the six, zero is inside the 90 percent . In the case of the United Kingdom, the ( confidence interval for the intercept term, ζ s confidence interval barely excludes zero, while for Japan we find strong evidence is greater than zero. ) that ζ s The G6 exchange rate ( the weighted average exchange rate, defined in the data section ) appears to be less noisy than the individual exchange rates. In all of our tests, the standard errors of the coefficient estimates are smaller for the G6 exchange rate than for the individual country exchange rates, suggesting that some idiosyncratic movements in country exchange rates get smoothed out when we look at averages. The weights in the G6 exchange rate are constant. We can think of this exchange rate as the dollar price of a fixed basket of currencies, and can interpret our tests as examining the properties of expected returns on this asset. Our discussion focuses on the returns on this asset because its returns appear to be more predictable than for the individual currencies. Examining the behavior of the returns on the weighted portfolio is a more appealing way of aggregating the data and reducing the effects of the idiosyncratic noise in the country data than estimating the Fama regression as a panel using all six exchange rates. There is not a good theoretical reason to believe 10 See, for example, Brunnermeier, Nagel, and Pedersen ( 2009 . and Jordà and Taylor ( 2012 ) )

10 VOL. 106 NO. 2 445 ENGEL: EXCHANGE RATES, INTEREST RATES, AND THE RISK PREMIUM * * + β − s + i − i = ζ s ( i Table 1—Fama Regressions: − i ) + u t 1 s t t t t s , t + 1 t + s 1979:6–2009:10 ̂ ̂ ζ 90% interval β Country 90% interval s s Canada ) 1.186, 3.355 ( 2.271 ) 0.250, 0.160 (− 0.045 − France 0.028 (− 0.346, 0.290 ) 1.216 (− 0.171, 2.603 ) − Germany 0.192 (− 0.136, 0.520 ) 2.091 ( 0.599, 3.583 ) ) Italy 0.032 (− 0.325, 0.389 ) 0.339 (− 0.680, 1.359 Japan ( 0.504, 1.343 ) 3.713 ( 2.390, 5.036 ) 0.924 United Kingdom ) 0.410 (− 0.768, − 0.051 ) 3.198 ( 1.170, 5.225 − ) ( G6 0.054 (− 0.184, 0.292 ) 2.467 0.769, 4.164 Note: 90 percent confidence intervals in parentheses based on Newey-West standard errors. that the coefficients in the Fama regression are the same across currencies, and the gains from panel methods are likely to be small from a panel that would impose no - restrictions across the equations on the coefficients. Table 1 reports that the 90 per cent confidence interval for this exchange rate lies well above zero, with a point estimate of 2.467. The intercept coefficient, on the other hand is very near zero, and the 90 percent confidence interval easily contains zero. B. Vector Error Correction Model A building block in our estimates of the central statistics of this study, ∞ * * ρ , r − r , is a vector error correction ) r − ρ and r , cov E ∑ cov ( E + j + 0 1 t t t t t t t + 1 t ) ( in order to construct ( model from which we extract measures of expected inflation * − , and the ex ante excess returns ) r the short-term real interest rate differential, r t t ρ . ) for future periods E ( t + j t Let s t * − p p = ( 5 ) , x t t t [ ] * i − i t * is the log of the dollar price of foreign currency, is the log of the US p p − where s t t t * i − is the US interest i consumer price level relative to the foreign price level, and t t rate less the foreign interest rate expressed as monthly returns. We estimate ) x − x ( − x = C + G x + C ( x − x ) + C x ) 6 ( 2 0 − 1 2 1 t t t − 2 t − 1 t − 3 t − 1 t − . ( x − x u ) + C + t 4 3 3 t t − − While the C matrices are unrestricted, the G matrix takes the form g − g g 11 13 11 g − g g G = . 21 21 23 [ ] g g g − 33 31 31

11 446 UAR Y 2016 FEBR THE AMERICAN ECONOMIC REVIEW This form implies that if there is cointegration, the nominal exchange rate and rel- ative inflation rate adjust to misalignments in the real exchange rate. The estimated 6 ) are reported in online Appendix Table A.1. This system coefficients of equation ( can be estimated efficiently using equation-by-equation ordinary least squares. We take as uncontroversial the presumption that the nominal interest rate differential is is large in absolute ( g stationary. For all the currency pairs, the point estimate of 33 and negative, consistent with stationarity. value ) A large literature has been devoted to the question of whether the nominal exchange rate and the nominal price differential are cointegrated, or whether the real exchange rate is stationary. Three recent studies of uncovered interest parity, 2005 ) ; Brunnermeier, Nagel, and Pedersen Froot and Ramadorai 2009 ) ; and Mark ( ( 2009 ) , estimate statistical models that assume the real exchange rate is stationary, ( but do not test for a unit root. Jordà and Taylor ( ) demonstrate that there is 2012 a profitable carry-trade strategy that exploits the uncovered interest parity puzzle when the trading rule is enhanced by including a forecast that the real exchange rate will return to its long-run level when its deviations from the mean are large. We find fairly strong evidence of mean reversion in the real exchange rate in this sample. The strength of this evidence may arise from using the interest rate differ - ential in the VECM as a covariate, which previous studies have not included. We . Table g , g 2 , and g − g construct bootstrap distributions of the estimates of 11 11 21 21 reports these distributions and the estimated coefficients. Subtracting the second - measures the monthly mean rever − g g equation from the first, the estimate of 11 21 sion in the real exchange rate. We find the estimated difference is significant at the 5 percent level for three of the currencies, and at the 10 percent level for two more currency pairs. A sixth case, Italy, is nearly significant at the 10 percent level. Only g − g the Canadian dollar shows no clear signs of cointegration. The estimates of 11 21 − 0.02 ( for the Canadian dollar range from approximately to − 0.04 ( for the UK ) pound ) , implying a strong tendency for the real exchange rate to mean revert. The 0.03, and sig- − for the G6 average currency is approximately − g g estimate of 21 11 nificant at the 5 percent level. The general formulation of the VECM in 6 ) allows us to construct a measure of ( ( as required in the calcu- the permanent component of the nominal exchange rate lation given in equation 2 )) whether or not the nominal exchange rate and relative ( nominal prices are cointegrated. Given the evidence presented in Table 2, we pro- ceed under the assumption of cointegration so that there is a permanent common component to the nominal exchange rate and the price differential. Below, we primarily report results from a VECM with three lags. We have cal- culated the Bayes Information Criterion for the optimal lag length in the VECM, and found that for all currency pairs, the optimal lag length is 1. However, we stick with the longer VECM because it seems possible that there are important dynamic relationships that are not captured in a model with a single lag. In fact, below we also report results of a VECM with 12 lags. We also report results from VECMs that include the dollar price of oil and gold, relative ( foreign to US ) long-term interest - rates and relative stock returns. These variables are included because they are for . However, in practice, there is a ρ ward looking and may improve forecasts of j t + penalty for using larger VECMs ( i.e., more lags and more variables ) because many more parameters are estimated. We find that while the pattern of our estimates of

12 VOL. 106 NO. 2 447 ENGEL: EXCHANGE RATES, INTEREST RATES, AND THE RISK PREMIUM g Table 2—Bootstrapped Distribution of , g from VECM 21 11 Left tail Country Estimates g 10 percent 5 percent 1 percent 11 0.0187 0.0545 Canada 0.0311 − − − 0.0381 − ) 0.0117 ( 0.0283 − France 0.0276 − 0.0510 − 0.0340 − ) ( 0.0113 0.0335 Germany 0.0250 − − 0.0479 − 0.0324 − ) 0.0105 ( − 0.0221 0.0268 − 0.0336 − 0.0503 Italy − ) ( 0.0117 − Japan 0.0242 0.0203 − 0.0423 − 0.0281 − ) ( 0.0105 0.0400 − 0.0265 − United Kingdom 0.0486 − 0.0327 − ) 0.0159 ( − 0.0297 0.0234 − 0.0485 − G6 − 0.0295 ) ( 0.0105 Right tail Country Estimates g 1 percent 5 percent 10 percent 21 0.0022 0.0024 Canada 0.0032 0.0051 ) 0.0013 ( 0.0022 France 0.0022 0.0013 0.0009 ( ) 0.0009 Germany 0.0029 0.0013 0.0324 0.0029 ) ( 0.0012 0.0027 0.0018 0.0011 0.0036 Italy 0.0012 ( ) 0.0010 Japan 0.0031 0.0014 0.0014 ) ( 0.0007 United Kingdom 0.0007 0.0019 0.0040 0.0027 ) ( 0.0025 0.0022 0.0031 0.0010 G6 0.0014 ) 0.0010 ( Left tail g Country − g 10 percent Estimates 1 percent 5 percent 21 11 Canada 0.0209 − 0.0318 − − 0.0563 − 0.0382 ) ( 0.0116 France 0.0305 0.0279 − 0.0510 − 0.0352 − − ( 0.0112 ) Germany − 0.0364 0.0257 − − 0.0480 − 0.0328 ) ( 0.0105 Italy 0.0258 0.0266 − 0.0339 − 0.0497 − − ) 0.0117 ( Japan − 0.0250 0.0207 − 0.0289 − 0.0416 − ) ( 0.0104 United Kingdom − 0.0408 0.0272 − 0.0333 − 0.0482 − ) ( 0.0162 G6 − 0.0328 − 0.0235 0.0299 − 0.0492 − ) ( 0.0105

13 448 UAR Y 2016 FEBR THE AMERICAN ECONOMIC REVIEW * ) E ρ cov are the same across all of the VECMs, the standard errors of , r ( − r t j + t t t some of the estimates of that covariance increase as the number of coefficients esti- mated increase. C. Fama Regressions in Real Terms 4 ) in real terms can be written as A counterpart to equation ( * * ˆ ˆ ˆ ˆ − q r + r . − − r = ζ u + β + r q ) 7 ( ) ( t t q t t t 1 q , t + 1 + t q We use a ^ over the real interest rate variables to emphasize that these vari- ables are estimated from our VECM. The dependent variable in this regres- ρ is the relative inflation forecast + u u sion is equal to , where p 1 t , p , t + 1 + + t 1 * * * * ˆ p − ≡ p p − − p , where we − p p − p − − E p u error, ) ( ( ) + t t 1 t + t 1 t + 1 + t , t p t 1 1 + t t ) ( ˆ to designate our estimate of the expected inflation differential. use the notation E t Much of the theoretical literature on the foreign exchange risk premium builds mod- els that interpret the Fama regression as one in which the dependent and indepen- 7 ) dent variables are defined in real terms as in , and assume no inflation risk. ( * ˆ ˆ − r are estimates. The first is There are two senses in which our measures of r t t that the parameters of the VECM are estimated. But even if the parameters were * ˆ ˆ because we are − r r known with certainty, we would still only have estimates of t t * ˆ ˆ on linear projections. Agents certainly have more − r basing our measures of r t t sophisticated methods of calculating expectations, and use more information than is contained in our VECM. The findings for regression 7 ) in real terms are similar to those when the regres- ( , sion is estimated on nominal variables. For all currencies, the estimates of β q - reported in Table 3, are positive, which implies that the high real interest rate cur rency tends to have high real expected excess returns. The estimated coefficient for the G6 aggregate is close to 2. Table 3 and all of the subsequent tables report the Newey-West standard errors ( 7 ) , and also report two sets of confidence intervals for each param- from regression eter estimate based on bootstraps. The Newey-West standard errors ignore the fact * ˆ ˆ is a generated regressor. The first bootstrap uses percentile intervals and − r r that t t 11 t the second percentile- intervals. β The 95 percent confidence interval for lies above zero for Germany, Japan, and q the United Kingdom. The 90 percent confidence interval contains zero for Canada and Italy. The 95 percent confidence interval contains zero for France, but the 90 percent confidence interval lies above zero for the first bootstrap. These findings are similar to those from the Fama regressions in nominal returns ( reported in Table 1 ) , with the exception of the Canadian dollar which was significantly positive in the nominal regression. for the separate countries are positive The fact that all six of the estimates of β q conveys more information than the individual tests of significance. A test of the joint 11 See Hansen ( 2010 ) . The online Appendix describes the bootstraps in more detail.

14 VOL. 106 NO. 2 449 ENGEL: EXCHANGE RATES, INTEREST RATES, AND THE RISK PREMIUM * * ˆ ˆ ˆ ˆ + r Table 3—Fama Regression in Real Terms: − r q = ζ + β r u + − r − q ) ( q , t t q + q 1 t t + 1 t t t 1979:6–2009:10 Parameter Estimates and Confidence Intervals ̂ ̂ 95% interval 90% interval β 90% interval Country ζ 95% interval q q ) 0.182, 0.208 (− 0.030 Canada ) 0.670, 2.665 (− ) 1.103, 3.065 (− 0.722 ) 0.141, 0.171 (− ( ( ) (− 0.151, 0.200 ) (− 0.118, 0.162 ) 0.111 0.768 ) (− 1.004, 2.749 ) (− 0.673, 2.492 ) ) France − 0.071 (− 0.321, 0.124 ) (− 0.274, 0.072 ) 1.482 (− 0.237, 3.283 ( 0.076, 3.004 ) 0.266, 0.061 0.834, 3.881 (− ) (− ) 1.089 ( ) 0.353, 3.514 (− ) 0.312, 0.107 (− ) ) 0.186 ( ( 0.274, 0.099 ) 0.321, 4.896 ) 0.040 ) ( Germany (− (− − 0.643, 4.531 0.232, 0.065 ) 1.733 ( 0.229, 0.058 (− ( 1.112 ) 0.257, 0.087 (− ) ) 0.246, 4.740 ) 0.183 ) ( ( 0.546, 4.405 ) 0.153, 0.255 Italy (− 0.222, 0.278 ) ) 0.069 (− 0.431 (− 1.154, 2.542 ) (− 0.881, 2.227 ) ) 0.186 (− 0.182, 0.262 ) ) 1.125, 2.196 (− ( (− 0.122, 0.244 ) 1.478, 2.633 (− ) ) ( 0.971 ) 2.360 0.024, 0.332 ( ( 0.593, 4.595 ) ) ( 0.985, 4.320 0.110 ) (− 0.018, 0.367 Japan ) 0.195 ( ( 0.297, 4.958 ) ) (− ) ( 0.946 ( 0.815, 4.558 ( ) 0.007, 0.363 0.023, 0.331 ) 0.028 0.447, − (− 0.521, 0.029 ) (− United Kingdom − 0.165 ) 1.850 ( 0.288, 4.055 ) ( 0.654, 3.771 ) − 0.024 ) ) ( 0.886 ) ( (− 0.527, 0.016 ( 0.176, 4.144 ) ) (− 0.211 0.492, ( 0.465, 3.913 ) 1.983 G6 (− 0.238, 0.127 ) (− 0.194, 0.091 ) 0.050 − ( 0.394, 4.335 ) ( 0.644, 3.969 ) 0.190, 0.078 ) 0.143 (− 0.976 ) 0.218, 0.110 ) ) ( 0.091, 4.241 ) ( ( (− ( 0.570, 3.934 ) Notes: The Newey-West standard error is reported below the coefficient estimate in parentheses. The confidence intervals are bootstrapped. The first reports a percentile interval bootstrap and the second a percentile- t interval bootstrap. See the online Appendix for details. null that all are less than or equal to zero can be rejected at a confidence level β q greater than 99.9 percent, using a bootstrapped statistic based on the joint distribu- tion of the residuals from the regressions in Table 3. The findings are clear using the G6 average exchange rate: the coefficient esti- is very mate is 1.98 and both confidence intervals lie above zero. The estimate of ζ q close to zero, and the confidence intervals contain zero. In summary, the evidence on the interest parity puzzle is similar in real terms as are positive and tend to be β in nominal terms. The point estimates of the coefficient q * r − ρ . Even in real 0 , r ) > E ( cov statistically significantly greater than zero, t t + t 1 t terms, the country with the higher interest rate tends to have short-run excess returns i.e., excess returns and the interest rate differential are positively correlated ). ( D. The Real Exchange Rate Real Interest Rates , and the Risk Premium , Table 4 reports estimates from * ˆ ˆ = ζ + . r + β − r u ( 8 q ) ) ( Q t t Q , t Q t In all cases, the coefficient estimate is positive. In almost all cases, although the con- fidence intervals are wide, the coefficient is significantly positive. This regression confirms the well-known relationship that the US dollar tends to be stronger when the US real interest rate is relatively high. Our chief interest is not this relationship, but whether the real exchange rate responds more or less to the real interest differen- tial than predicted by uncovered interest parity. That is, we are interested in the sign ∞ * r . , r − ) ρ ∑ ( of cov E + t t j 0 t j + 1 t =

15 450 FEBR UAR Y 2016 THE AMERICAN ECONOMIC REVIEW * * ˆ ˆ ˆ ˆ on r r − r u : q + = ζ + β r Table 4—Regression of − q ( ) t Q t t t t q Q t + 1 t , 1979:6–2009:10 Parameter Estimates and Confidence Intervals ̂ 90% interval Country β 95% interval Q Canada 46.996 ( 25.157, 95.390 ) ( 31.793, 90.162 ) ) ( ( 21.633, 145.736 29.714, 128.420 8.688 ) ) ( 3.549, 42.051 ( France 20.372 ) (− 1.998, 46.182 ) ( ) 10.854 ) 5.951, 57.766 ) (− (− 11.709, 65.220 ) Germany 52.410 ( 0.616, 91.078 ) 28.470, 87.010 ( 12.415 0.073, 132.108 (− ( ( 13.828, 118.588 ) ) ) 38.359 Italy ( 10.971, 73.668 ) ( 15.560, 68.766 ) ( 8.031, 93.569 ) ( 8.042 ) ( 13.416, 84.040 ) ) Japan 19.650 (− 2.817, 47.262 ) ( 3.032, 42.822 ( ) ) 5.013, 45.018 ( 0.515, 50.100 ( 6.582 ) ) 15.744 (− 0.793, 36.283 United Kingdom ( 4.006, 32.573 ) 7.875 (− 8.155, 48.868 ) ) ) ( (− 3.824, 43.002 G6 ) 23.549, 75.480 43.702 ( ) 17.664, 80.124 ( ) ( 10.506, 98.543 ) ( 10.124 ( 19.449, 89.191 ) parentheses. Notes: The Newey-West standard error is reported below the coefficient estimate in The confidence intervals are bootstrapped. The first reports a percentile interval bootstrap and the second a percentile- t interval bootstrap. See the online Appendix for details. * ∞ ∞ * ˆ ˆ ˆ ˆ ˆ ˆ r on ) ∑ r − r + β ζ ) = − ρ ( ρ ∑ E : u r − Table 5—Regression of E + − ρ ( ρ ) ( 0 j + 1 j 0 t = t ρ t t + ρ ρ 1 + j t + t t = j t t ̅ ̅ 1979:6–2009:10 Parameter Estimates and Confidence Intervals ̂ β Country 90% interval 95% interval ρ − 15.414 24.762 (− − 60.281, ) 10.757 ) − (− 52.700, Canada 68.054, (− ) ( 5.523 ) (− 10.812 − 14.849 ) 98.321, − 39.998, 3.105 (− 13.983 − France ) 34.960, 0.200 (− ) (− (− 45.244, 8.814 ) 8.268 ( ) 40.468, 4.248 ) ) (− Germany − 33.895 (− 62.299, − 5.924 58.804, ) 10.621 − ( (− ) 87.170, 3.844 (− 10.365 73.809, − 4.432 ) ) 4.446 (− Italy − 26.556 − ) ) (− 49.863, − 10.649 54.355, 9.335 − ) 6.206 57.032, ) 4.848 − 64.174, (− (− ( ) 41.927, 2.218 2.177 37.617, (− ) − (− 15.225 − Japan ) 42.394, (− 0.325 (− 38.379, ) 6.487 ( − − 3.176 ) ) 27.130, 1.060 United Kingdom − 10.717 (− (− 31.865, 3.436 ) ) (− 8.565 ) 37.710, 9.599 ) 42.105, 13.602 ) (− ( − (− 30.890 59.899, (− 56.359, − 14.642 ) 9.893 ) G6 − 13.478 ( 8.352 ) (− − -60.065, ( ) 9.665 − 68.593, ) The Newey-West standard error is reported below the coefficient estimate in parentheses. The confidence Notes: t intervals are bootstrapped. The first reports a percentile interval bootstrap and the second a percentile- interval bootstrap. See the online Appendix for details. Our central empirical finding is reported in Table 5. This table reports the regression ∞ * ˆ ˆ ˆ E ) 9 ( r ζ r + β u . ) = ∑ − ρ − + ( ρ ) ( t t t + + j 1 t ρ ρ ρ t ̅ j = 0

16 VOL. 106 NO. 2 451 ENGEL: EXCHANGE RATES, INTEREST RATES, AND THE RISK PREMIUM _ ∞ ˆ ∑ ( ρ is calculated as the difference between our measure of the tran- ) ρ E − 1 = + t 0 t + j j T , and the estimated sum of current and s sitory component of the exchange rate, t IP ( , following equation 2 ) . expected future interest differentials, s t For all of the currency pairs, the estimated slope coefficient is negative, implying ∞ * ∑ E ρ , r − r 0 . The 90 percent confidence intervals are wide, but < cov 0 t t + t j t + 1 ( ) with only a few exceptions, lie below zero. We can reject the joint null that this covariance is greater than or equal to zero for all six currencies at a confidence level greater than 99.9 percent. The confidence interval for the G6 average strongly excludes zero. To get an idea of magnitudes, a 1 percentage point difference in annual rates between the foreign / 12th percentage point difference in monthly and home real interest rates equals a 1 rates. The coefficient of around − 31 reported for the regression when we take the US relative to the average of the other G7 countries then translates into around ∞ ˆ ∑ ) − ρ ( ρ of a 1 percentage point change in the E a 2.6 percent effect on + + 1 t j 0 = j t ̅ interest rate differential. From Table 4, we can see that if the US real rate increases 1 annualized percentage point above the real rate in the other countries, the dollar percent, 1.0 / 12 ) percent 43.7 percent ( is predicted to be 3.6 stronger. Of that 3.6 is the amount attributable to the higher real interest differential under uncovered as in the Dornbusch 1976 model ) , and the remaining 2.6 percent interest parity ( represents the effect of the cumulated expected excess return on dollar deposits on the exchange rate. ∞ * E ∑ r 0 is surprising in light of the ρ − < , r This finding that cov t + t 0 t t 1 j + ) ( * − r well-known uncovered interest parity puzzle. We have documented that when r t t is above average, foreign deposits tend to have expected excess returns relative to US deposits. That seems to imply that the high interest rate currency is the riskier currency. But the estimates from equation ( 9 ) deliver the opposite message—the high interest rate currency has the lower cumulative anticipated risk premium. Since ∞ * * r ρ − r , r 0 , we − r ) > 0 and cov E , ∑ < ρ we have found cov E ( + t t t t 1 + t t + t 1 j 0 t ) ( * E 0 ρ , r > − r . That is, we must have < 0 for at least some j cov must have j t t t t + ) ( * r − a reversal in the correlation of the expected one-period excess returns with r t t as the horizon extends. This is illustrated in Figure 1, which plots estimates of the slope coefficient in a * ˆ ˆ ˆ , ( ρ ) on r ... 1, 120 : − r for j = regression of E t + j t t t j * ˆ ˆ ˆ E r + β r − ζ ( ρ ) = + u . ) ( j j t j t t t t + For the first few , this coefficient is positive, but it eventually turns negative at lon- j ger horizons. To summarize, when the US real interest rate relative to the foreign real inter - est rate is higher than average, the transitory component of the dollar is stronger than average. Crucially, it is even stronger than would be predicted by a model of uncovered interest parity. Ex ante excess returns or the foreign exchange risk pre- mium contribute to this strength. This implies that the expected sum of future excess returns on the foreign asset must increase when the US real interest rate rises, which is a reversal of the correlation between the interest differential and expected returns in the short run.

17 452 Y 2016 UAR FEBR THE AMERICAN ECONOMIC REVIEW da Panel B. F ranc e ana Panel A. C 2.5 1.5 2 1 1.5 1 0.5 coef ficient 0.5 coef ficient 0 0 − 0.5 Slope − 0.5 Slope 1 − 1 121 49 61 37 109 25 37 121 25 13 13 49 73 61 85 73 97 85 1 97 109 43 103 43 79 91 67 79 55 67 31 55 31 103 115 115 19 19 7 7 91 hs Mont hs Mont Panel D. Italy Panel C. Germany 3 1.5 2.5 2 1 1.5 0.5 1 0 0.5 coef ficient 0 − 0.5 0.5 − Slope coefficient 1 − 1 − Slope − 1.5 1.5 − 71 81 101 111 121 61 51 1 11 21 31 41 91 85 1 73 121 37 109 49 25 61 97 13 106 36 116 76 26 66 86 46 16 56 96 6 79 91 67 55 103 43 115 31 19 7 Mont hs Mont hs Panel F. United Kingdom Panel E. Japan 3 3 2.5 2.5 2 2 1.5 1.5 1 1 coef ficient 0.5 0.5 0 0 0.5 − Slope − 0.5 1 − Slope coefficient − 1 1.5 − 109 49 85 37 1 25 73 121 13 97 61 121 1 109 97 85 73 61 49 37 25 13 7 19 31 43 55 67 79 91 115 103 103 43 79 31 115 67 19 91 7 55 Mont hs Mont hs Panel G. G6 3 2.5 2 1.5 1 0.5 0 0.5 − − 1 Slope coefficient 1.5 − 1 13 109 37 49 61 73 85 97 121 25 55 19 67 103 79 31 91 7 43 115 hs Mont Figure 1. Slope of Ex Ante Return Regression ˆ E ( ρ Notes: Figures plot the slope coefficients and 90 percent confidence interval of these regressions: ) j + t t j * . Monthly data, 1979:6–2009:10. Confidence intervals calculated from Newey-West stan- + β ( r = ζ − r ) + u t t t j j dard errors.

18 VOL. 106 NO. 2 453 ENGEL: EXCHANGE RATES, INTEREST RATES, AND THE RISK PREMIUM Slope of ex ante return regression: G6 3.5 3 2.5 2 1.5 coef ficient 1 0.5 Slope 0 .5 0 − 1 − 5 9 1 61 57 65 45 53 49 37 29 25 17 97 93 89 85 77 69 41 33 21 13 81 73 10 1 10 5 10 9 12 1 11 3 11 7 s Month Figure 2. Slope Coefficients and 90 Percent Confidence Interval of the Regression: j * ˆ − i ( + β u ) = ζ ) + ( ρ E i t t + j j j t t t Monthly data, 1979:6–2009:10. Confidence intervals calculated from Newey-West standard errors. Notes: Figure 2 shows that a similar pattern occurs when we estimate j * ˆ . u + ( ρ i − ) = ζ + β i E ( ) j + t t t j j t t The usual empirical work on the interest parity puzzle, as in Table 1, reports on the correlation of the excess return with the nominal interest differential. These * − i > 0 . In Figure 2, we see nonetheless that E ρ , i findings imply cov t + 1 t t t ( ) * < E ρ i − , i 0 as the horizon extends out to greater than 20 months. cov + t t t t j ) ( The figure plots results for the G6 average exchange rate, but the pattern is sim- ilar for the individual bilateral exchange rates. The evidence is less strong that ∞ * i i < , 0 than it is for the analogous relationship involving E ∑ ρ − cov + 0 t 1 t j t t + ) ( real interest rate differentials. In Figure 3, for the G6 currency, we plot the slope coefficient estimates from the regression j * ˆ ˆ + β u r . = ζ − r + ρ ) ( j t t t t j + j That is, the dependent variable in the regression is actual ex post excess returns on the foreign deposit, rather than the measure of ex ante excess returns calcu- lated on forecasts from the VECM. The same pattern emerges as in the previous * ˆ ˆ − r r , > 0 , but as the horizon ρ j , we find cov plots. For small values of t + j t t ) ( increases, the sign of the covariance reverses. The pattern of coefficient estimates is not as smooth because the regressions use ex post rather than ex ante returns, but the reversal of sign is unmistakable. It is not possible, of course, to calculate ∞ * r ρ ∑ r − , because there are only a finite number of realizations of , cov + t j t 1 0 + t ( ) ex post returns in our sample. Although we could calculate a truncated sum, the disadvantages of this procedure are well known from the literature. This problem

19 454 Y 2016 FEBR UAR THE AMERICAN ECONOMIC REVIEW Slope of ex post return regression: G6 4 3 2 1 0 coef ficient − 1 2 − Slope − 3 4 − 5 − 4 1 7 40 82 94 46 43 25 55 16 22 58 28 61 64 34 13 52 10 76 79 37 49 31 70 88 19 91 67 73 85 97 10 0 Mont hs Figure 3. Slope Coefficients and 90 Percent Confidence Interval of the Regression: j * ˆ ˆ r r − u ) + ( + β = ζ ρ t + j j j t t t Monthly data, 1979:6–2009:10. Confidence intervals calculated from Newey-West standard errors. Notes: tech- ) 1987, 1988 ( partly motivated the development of the Campbell and Shiller nique, which is closely related to our approach. Figure 4 is analogous to Figure 3, except that the regressor is the nominal interest rate differential. Figure 4 plots the slope coefficients from the regression j * . = ζ + + β u i − i ρ ) ( j t t t t j + j Again we can see the pattern of initially positive slope estimates, and then a rever - 1 ( j sal of sign. The initial slope coefficient estimated ) is exactly the estimate = reported for the G6 average exchange rate in Table 1. These regressions are notable because they do not rely on our VECM analysis at all. In particular, they do not rely 12 Valchev 2014 ) ( on imposing the constraint that the real exchange rate is stationary. finds very similar results for a panel regression of excess returns for 18 countries against the US dollar, using data spanning 1976–2013, imposing the same slope coefficient across currencies. We consider two extensions of the empirical analysis to see if augmenting the simple VECM estimated here can sharpen the forecasts of future short-term real interest rates. The results reported so far are from a VECM with three lags, using monthly data. We estimated the model using 12 lags, which might capture longer run dynamics in the monthly data. The second extension added four variables to the VECM for each country. We include a stock price index and a measure of long-term nominal government bond yields. The long-term bond yields are from the IMF’s International Financial Statistics ( 2011 ) , “interest rates, government securities and government bonds.” 12 The same is true for the results in Figure 3, except to the extent that the VECM is used to estimate the real interest differential.

20 VOL. 106 NO. 2 455 ENGEL: EXCHANGE RATES, INTEREST RATES, AND THE RISK PREMIUM Slope of ex post retu rn regressi on: G6 5 4 3 2 1 0 1 − 2 − Slope coefficient 3 − − 4 − 5 4 1 7 10 16 13 19 76 97 49 94 52 46 25 37 67 61 43 22 28 73 31 40 70 64 91 79 88 55 58 34 85 82 100 Months Figure 4. Slope Coefficients and 90 Percent Confidence Interval of the Regression: j * i − ρ i ( u + β ) + = ζ t + j j t t t j Monthly data, 1979:6–2009:10. Confidence intervals calculated from Newey-West standard errors. Notes: 13 The yields and The stock price indexes are monthly, from Datastream ( 2011 . ) stock prices are taken relative to the corresponding US variable. We also include 14 As we have noted above, data on the dollar price of oil and the dollar price of gold. our VECM estimates of expected inflation and expected future interest rates are estimates both because the coefficients of the VECM must be estimated, but also because the simple VECM does not include all of the information and news the mar - ket uses to forecast future inflation and interest rates. The point of including these four variables is that they are asset prices that respond quickly to news about the future state of the economy. For example, the gold price is believed to be sensitive to news about US monetary policy. Oil prices are thought to react to expectations of global economic growth, which in turn may influence expectations of inflation and interest rates. The stock prices and long-term bond rates from each economy may contain information about local monetary policy and economic growth prospects. We estimated the same parameters as reported in Tables 2, 3, and 4 for each of these models. The point estimates for the augmented models were very similar to those reported for the baseline model. The confidence intervals for the slope coeffi- cients for the regressions reported in the tables based on expectations generated from the VECM with 12 lags were wider than for the VECM with 3 lags. These wider confidence intervals might reflect the greater imprecision in coefficient estimates in the VECM with 12 lags. The more parsimonious specification has far fewer coeffi- cients to estimate. On the other hand, the findings when the additional informational variables are included in the VECM are not very different than those reported in the tables. That is, not only the signs but the statistical significance of the coefficient estimates ( based on Newey-West statistics and on both bootstraps ) are similar. 13 The Datastream codes are TOTMKCN ( PI ) , TOTMKFR ( PI ) , TOTMKIT ( PI ) , TOTMKUK ( PI ) , ( TOTMKBD ( PI ) PI ) , and TOTMKUS ( PI ). , TOTMKJP 14 These data are from the Federal Reserve Bank of St. Louis database. The oil price is the spot price of West Dow Jones 2014 Texas Intermediate crude oil ) ( , and the gold price is the Gold Fixing Price in the London Bullion Market ( 2014 ) .

21 456 UAR Y 2016 FEBR THE AMERICAN ECONOMIC REVIEW Slope of ex ante retu rn regressi on: G6 2.5 2 1.5 1 0.5 0 Slope coefficient 0 .5 − 1 − − 1.5 1 9 5 69 29 93 25 33 49 17 45 37 13 57 21 73 89 61 85 41 65 77 97 53 81 10 1 11 3 10 9 10 5 12 1 11 7 Months Figure 5. Slope Coefficients and 90 Percent Confidence Interval of the Regression: j * ˆ ˆ ˆ ( ρ ) = ζ + β u VECM ( r ) + − r E . 12-Lag t t j j t j + t t Notes: Monthly data, 1979:6–2009:10. Confidence intervals calculated from Newey-West standard errors. Similarly to Figure 1, Figure 5 plots the slope coefficient estimates for the regression j * ˆ ˆ ˆ ( ρ u + ) = ζ + β r − r E ( ) j t t t t j j t + for the 12-lag VECM. Figure 6 plots these slope coefficients for the VECM aug- mented with information variables. The overall message is the same as in the previous specifications. We turn now to the implications of these empirical findings for models of the foreign exchange risk premium. III. The Puzzle We have found that there is excess comovement of the level of the exchange rate and the interest differential, in the sense that the covariance of the stationary com- ponent of the exchange rate with the foreign less US interest rate is more negative ∞ * ∑ − ρ , r 0 . This finding < E r than would hold under interest parity: cov t t t 0 t + j + 1 ( ) of excess comovement is not surprising in itself, and corresponds to previous findings of excess volatility in the literature. The difficulty resides in reconciling this find- * r ) > , ρ 0 . r − cov E ( ing with the well-known interest parity “puzzle” that finds t + t t t 1 A complete theory of exchange rate and interest rate behavior needs to explain not * j for some 0 ) < ρ r − , r 1 . > only the interest parity puzzle, but also why E cov ( t t t + j t Recent work has made progress in developing economic models to account for the uncovered interest parity puzzle. The models we review below—one set based on risk averse behavior of investors, the other on rational inattention—rely on a cur - tailed adjustment by markets to a change in interest rates to explain the interest par - * . In one set of models, the shock also r − ity puzzle. Suppose some shock raises r t t riskiness increases the riskiness of foreign assets for home investors relative to the

22 VOL. 106 NO. 2 457 ENGEL: EXCHANGE RATES, INTEREST RATES, AND THE RISK PREMIUM Slope of ex an te retu rn regressio n: G6 3 2.5 2 1.5 1 0.5 0 Slope coefficient 0.5 − 1 − − 1.5 9 5 1 17 29 21 77 85 37 57 61 73 25 93 69 81 41 49 65 45 89 97 13 53 33 11 7 10 5 10 9 11 3 12 1 10 1 Months Figure 6. Slope Coefficients and 90 Percent Confidence Interval of the Regression: j * ˆ ˆ ˆ ) + ( ρ + β u ) = ζ E ( r . Stock Prices, Long-Term Bond Yields, Gold Price, − r t j t t t t + j j Oil Price Included in VECM Notes: Monthly data, 1979:6–2009:10. Confidence intervals calculated from Newey-West standard errors. of home assets for foreign investors. Investors’ desire to increase investment in the * r − is muted by the increase in for - foreign deposits because of the increase in r t t eign exchange risk, which implies an increase in the risk premium on the foreign deposits. In the other set of models, there is initially partial adjustment by inves- tors, not based on risk aversion, but by slow reaction to news of the increase in * . Some investors do not adjust their portfolios immediately, which then gen- r − r t t erates higher expected returns on the foreign deposits in the short run before all investors rebalance their portfolios. These models do not allow for a channel of amplified adjustment, by which the high interest rate currency is considered more desirable by investors, leading to a lower expected return on that currency. That is, there is no channel by which * , r ) < ρ − r 0 for any j . cov ( E j t t t + t Models of the Foreign Exchange Risk Premium under Complete Markets A. Almost since the initial discovery of the interest-parity puzzle, there have been - attempts to account for the behavior of expected returns in foreign exchange mar kets without relying on any market imperfections, such as market incompleteness 15 The literature has built models of risk or deviations from rational expectations. premiums based on risk aversion of a representative agent. Those models formulate preferences in order to generate volatile risk premiums which are important not only for understanding the uncovered interest parity puzzle, but also a number of other puzzles in asset pricing regarding returns on equities and the term structure. See, for ) ( example, Bansal and Yaron . 2004 15 See Engel ( 1996, 2014 ) for surveys of the theoretical literature.

23 458 UAR Y 2016 FEBR THE AMERICAN ECONOMIC REVIEW Here we briefly review the basic theory of foreign exchange risk premiums in complete-markets models and relate the factors driving the risk premium to the state variables driving stochastic discount factors. See, for example, Backus, Foresi, and ) or Brandt, Cochrane, and Santa-Clara ( 2006 ) . Telmer 2001 ( that M When markets are complete, there is a unique stochastic discount factor, + t 1 prices returns denominated in units of the home consumption basket. The returns on r j , t + 1 ) e M = E ( any asset j denominated in units of home consumption satisfy 1 1 + t t * that prices returns expressed in units M , . Likewise, there is a unique SDF j for all 1 t + ( 2001 show, of the foreign consumption basket. As Backus, Foresi, and Telmer ) or, as an approximation ) ( when the SDF and returns are log-normally distributed , we can write 1 * _ m ρ var = ) − ( var m ( 10 ) E t t t t + 1 + 1 t t 1 + 2 1 * * * _ 11 r ) ( r )) = E ( m . − m m ( ) + ( ( var var − m ) − + 1 t t t 1 + t t + 1 1 + t t t t 2 * * M and m and , respectively. are the logs of M The lower case variables, m + t 1 t t + 1 + t 1 1 + ( ) We focus attention on the “long-run risks” model of Bansal and Yaron , 2004 have recently ( ) preferences. Colacito and Croce ( based on Epstein-Zin ) 1989 2011 applied the model to understand several properties of equity returns, real exchange rates and consumption. Bansal and Shaliastovich ( 2007, 2013 ) ; Lustig and Verdelhan ( 2007 ) ; Colacito ( 2009 ) ; and Backus et al. ( 2010 ) demonstrate how the “long-run risks” model can account for the interest-parity anomaly. Colacito and Croce ( 2013 ) build a general equilibrium two-good, two-good endowment economy in which agents in both countries have Epstein-Zin preferences, under both complete markets and portfolio autarky, and are able to account for the interest-parity puzzle as well as other asset-pricing anomalies. These papers directly extend equilibrium closed-economy models to a two-country open-economy setting. The closed economy models assume an exogenous stream of endowments, with consumption equal to the endowment. The open-economy versions assume an exogenous stream of consumption in each country. These could be interpreted either as partial equilibrium models, with con- sumption given but the relation between consumption and world output unmodeled. Or they could be interpreted as general equilibrium models in which each country consumes an exogenous stream of its own endowment and there is no trade between 16 Under the latter interpretation, the real exchange rate is a shadow price, countries. since in the absence of any trade in goods, there can be no trade in assets that have any real payoff. In each country, households are assumed to have Epstein-Zin preferences. The home agent’s utility is defined by the recursive relationship, /ρ 1 ρ ρ/α α = ( 1 − β C ) + β E . U ( 12 U ) ( ) + 1 t t t t [ ] } { 16 Colacito and Croce’s ( 2013 ) two-good model does allow for trade in goods.

24 VOL. 106 NO. 2 459 ENGEL: EXCHANGE RATES, INTEREST RATES, AND THE RISK PREMIUM − α > 0 is the In this relationship, 1 β measures the patience of the consumer, − ρ / ( is the intertemporal elasticity 1 ) > degree of relative risk aversion, and 1 0 that , which corre- ) 2004 ( of substitution. Assume, as in Bansal and Yaron α < ρ sponds to the case in which agents prefer an early resolution of risk, and 0 < ρ < 1 , We will consider so the intertemporal elasticity of substitution is greater than one. a somewhat more general version of the long-run risks model than is present in the literature, but one that nests several models. We present only a version in which real interest rates are determined, but discuss extensions to the nominal interest rate. Assume an exogenous path for consumption in each country. In the home country ln ( ≡ ) ) , C c ( with t t _ x c = μ + l . − u ε + 13 ) c ( √ + t t t t 1 t + 1 . The com- l The conditional expectation of consumption growth is given by μ + t represents a persistent consumption growth component modeled as a l ponent t first-order autoregression: _ l ε . w = φ l + l ) 14 ( √ 1 t 1 + t t t + l x l The innovations, ε and ε are assumed to be uncorrelated within each coun- 1 t + t 1 + , but each shock may be correlated with its foreign N 0, i.i.d. 1 ) ( try, distributed l * x * ). , which are mutually uncorrelated and ε counterpart ε ( t 1 1 + + t In the foreign country, we have _ * * x * * * * √ c = μ + l + u − ε ) ( c 15 1 t t t + t 1 t + _ l * * * * * √ l = φ ( 16 ) + l w ε . t t 1 + t l 1 + t + 1 The conditional variances are written as the sum of two independent compo- h nents. The component with the superscript is idiosyncratic to the home country. f c superscript refers to the foreign idiosyncratic component. The one with the An h c = u u + , superscript is common to the home and foreign country. So, we write u t t t f f * c h c c * + u , w = w . Conditional variances are sto- + w = u , and w = w + w u t t t t t t t t t chastic and follow first-order autoregressive processes: i i i iu ) u 17 ( ( 1 − φ ) θ + φ + σ ε = , i = h , f , c u u + t + 1 1 u t u t u iw i i i w ) ( 18 i = ( 1 − φ , ) θ f . + φ w , c + σ h ε = . 1 1 w t w w w t + t + iu iw ε The innovations, ε and are assumed to be uncorrelated, distrib- , f c , i = h , 1 + t 1 + t uted i.i.d . with mean zero and unit variance.

25 460 FEBR Y 2016 UAR THE AMERICAN ECONOMIC REVIEW We can log linearize the first-order conditions as shown in the online Appendix. We ignore terms that are not time-varying or that do not affect both the conditional means and variances of the stochastic discount factors, lumping those variables into * and Ξ . Ξ the catchall terms t t The home discount factor is given by 2 α α α − ρ α − ρ ) ( ( ) β _ _ _ u = w + − ( 19 ) m t t t 1 + ( ) 2 2 1 φ − β l _ _ β l x _ + u + Ξ ε 1 − α + ρ − α ε . w ) ) ( ( √ √ t t + 1 t 1 + t t ) ( 1 φ − β l The foreign discount factor is analogous, but with foreign parameters and vari- ables replacing home parameters and variables. 2007, 2013 ) , Colacito ( 2009 ) , and Backus et al. ( Bansal and Shaliastovich ) ( 2010 ( ) and ( 11 10 , we find assume identical parameters in the two countries. Applying ) 2 β f f 1 h 2 h 2 _ _ ρ α − 1 w u w − − u + α − ρ = E 20 ( ) ( ) ( ) ) ( ) ( t t 1 + t t t t ) ( [ ] 2 1 φ − β l ρ − α + − ρ 1 − α 1 ( ( ) ) f h * _________________ ( r 21 ) r = − u − u ( ) t t t t 2 2 ρ − α ρ ( ) β f h _ _ . w − w + ) ( t t ( ) 2 − β φ 1 l ( 1 − α > 0 , 0 < ρ < 1 , α < ρ ) , we Under the given parameter restrictions * > E ρ r − , r 0 , providing a resolution of the interest parity cov see that t t t t 1 + ( ) puzzle. Intuitively, when the relative variance of the home consumption stream f f h h − u is high, there are two effects. First, as Bansal and Shaliastovich or w u − w t t t t ) ( 2013, p. 18 ) put it, there is a “flight to quality”; home investors shift their portfolios ( to less risky assets. The increase in volatility “increases the uncertainty about future growth, so the demand for risk-free assets increases, and in equilibrium, real yields * r rises. Second, foreign exchange risk for home investors rises − ” that is fall, r t t ρ . E more than for foreign investors, leading to an increase in t t 1 + ( 2011 ) consider a linear factor model in which Lustig, Roussanov, and Verdelhan there is a common factor affecting risk in both countries. Their approach can be translated into the economic model in which investors have Epstein-Zin prefer - ences, if we allow for attitudes toward risk that are different in the two countries. That study emphasizes the importance of different responses to common shocks, rather than focusing on idiosyncratic shocks to consumption volatility. Suppose f f h h 0 = u = = w , and u = w these country-specific shocks are set to zero t t t t ) (

26 VOL. 106 NO. 2 461 ENGEL: EXCHANGE RATES, INTEREST RATES, AND THE RISK PREMIUM * ( ) but other parameters of preferences there are differences in risk aversion α ≠ α ) are identical. Then, and β ( ρ 2 1 c 2 * _ E ( 22 ) = u α − 1 − α − 1 ρ ) ( ) ( ( ) t t 1 + t [ 2 2 2 β 2 * c _ + − ρ α − ρ − w α ( ) ) ( ] [ t ) ( ] φ − β 1 l 2 * * − α α ρ − α α 2 − ρ ( ) ) ( ( ) β c c * _ _ ____________ w . + u = 23 ) ( − r r t t t t ( ) 2 2 φ 1 − β l * > 0 , 0 , 1 − α ( 1 − α Under the same set of parameter assumptions as above > * * > 0 . As Lustig, ) , one finds again cov E ρ < ρ , r r − , α α 1 , < ρ < 0 ) ( t 1 t t + t Roussanov, and Verdelhan 2011, p. 3760 ( ) explain, “When precautionary saving ( and, demand is strong enough, an increase in the volatility of consumption growth * > α ( for lowers interest rates.” When α consequently, of marginal utility growth ) ) example , the home country is more risk averse, and the precautionary effect is c c * . Also, − r comoves positively with u and w r larger in the home country, so t t t t the foreign exchange risk for home residents exceeds that for foreign investors, c c ρ w comoves positively with the common shocks u and . E so t 1 t t t + Under either specification—idiosyncratic shocks and identical preferences, or different preferences and common shocks—the interest rate differential and the for - eign exchange risk premium are responding to changes in the variance of consump- tion growth. A precautionary motive that drives down the home interest rate also increases the foreign exchange risk premium. There is an under-adjustment of inves- tors to the lower home interest rate. They do not flock to foreign deposits to the same extent that risk neutral investors would because of aversion to foreign exchange risk. But this muted adjustment implies that there is no force to account for the negative ∞ * r with ρ . Under the first specification, ∑ − r correlation of E j t t t + 0 t + 1 2 ∞ f f h h w u − − u w β t t t t 1 2 2 _ _ _ _ E α − 1 ρ + α − ρ , = ∑ ) ( ) ( + j t t + 1 ) ( ] [ − φ 1 2 1 − φ 1 − β φ w u l 0 and under the second specification, ∞ c 2 u t 1 * 2 _ _ E = α − 1 ∑ − α ρ − 1 ) ( ) ( ) ( t 1 + j + t [ 1 − φ 2 u 0 2 c 2 w β t * 2 _ _ + − − ρ α α − ρ . ) ( ( ) [ ] ) ( ] φ 1 − φ 1 − β w l ∞ * E > ∑ 0 , contra- ρ , r − r It is clear that under either specification, cov + 1 t 0 t + t j t ) ( vening our empirical findings.

27 462 Y 2016 FEBR UAR THE AMERICAN ECONOMIC REVIEW - The interest parity puzzle is sometimes portrayed as a relation between cur rency depreciation and the interest differential. Using real exchange rates and * q 0 . This > − q , r r − real interest rates, the puzzle is expressed as cov 1 t t t + t ) ( * E ρ 0 . It can be expressed as r , r − > is a stronger relationship than cov t t t t 1 + ) ( * * ρ - . Because the condition is stronger, the empir , r E − r > var r r − cov t t t t 1 + t t ) ( ( ) ical relationship is not as robust, but it is found to hold for many currencies and ( see the surveys of Engel 1996, 2014 ) . The model with time periods nonetheless identical preferences and idiosyncratic shocks is able to account for this relation- ship if risk aversion is strong enough. It is straightforward to see that if the coef- ( α < 0 ) , the model implies ficient of relative risk aversion is greater than one * 0 . Similarly in the model with common shocks, but in > − q , r q − r cov t t t 1 + t ( ) * , if relative risk aversion is greater than one in both countries, we find α ≠ α which * 0 . > − q , r r q − cov t t t 1 + t ) ( * q − q , r r − cov It is notable that both formulations of the model that derive t t 1 + t t ) ( * q < − lim ( E r q − 0 , in contra- ) , r ( using equation ( 3 )) cov > 0 imply t k t t →∞ t k + t ( ) diction to the ample evidence reported in Table 4. That is, when the models are able to account for the stronger form of the interest parity puzzle, they imply that the country with the higher interest rate tends to have the weaker currency: the transi- tory component of the exchange rate is negatively correlated with the interest differ - ential. This prediction of the models is not noted in the literature, and is in striking contrast to the widely accepted empirical prediction of the Dornbusch-style models that a higher real interest rate is associated with a stronger currency. In the Dornbusch model, investors are risk neutral. A monetary contraction in the foreign country raises the relative foreign real interest rate because nominal prices * is accompa- − r r are sticky. Uncovered interest parity holds, so the increase in t t q − q . E nied by an expected real depreciation of the foreign currency—a fall in + t 1 t t rises initially In order to generate the expected fall in the real exchange rate, q t and then falls over time ) so there is a real appreciation of the foreign currency. ( The intuition for the predictions of the risk-premium models is similar, except that the exchange rate behavior is reversed. Because of risk aversion, the shock that * is accompanied by an expected real appreciation if investors are − r drives up r t t . To generate this expected appre- q q − E sufficiently risk averse—an increase in + t t t 1 ciation of the foreign currency, the real exchange rate initially falls relative to its permanent component. There is a real depreciation of the foreign currency when * r − is high. r t t * is positive, the for − r - In economic terms, in the Dornbusch model, when r t t eign deposit is attractive to investors, who buy the foreign currency which leads to * . In these models − r r a positive correlation between the real exchange rate and t t * is driven by shocks to the variance of consumption − r of risk-averse behavior, r t t * to − r growth. When investors are sufficiently risk averse, a shock that causes r t t rise also makes the foreign deposit so risky that investors are attracted to home deposits, leading to a stronger home currency and a negative correlation between the * r − . The models explain the strong version of the interest r real exchange rate and t t parity puzzle with strong risk aversion—but the prediction of the model for the level of the real exchange rate is the opposite of the data. Some papers extend the above model to allow for changes in inflation, and are able to generate predictions about the correlation of the nominal interest rate differential

28 VOL. 106 NO. 2 463 ENGEL: EXCHANGE RATES, INTEREST RATES, AND THE RISK PREMIUM 17 ρ E , for example, inflation processes . ) In Bansal and Shaliastovich ( 2013 with t 1 + t are exogenous, but higher inflation is assumed to lead to lower consumption growth. Hence, inflation volatility influences the risk premium and interest rates through its influence on real consumption. The analysis is similar to that for changes in the variance of real consumption shocks—an increase in home inflation variance lowers the real and nominal interest rate through a precautionary effect and increases the risk premium on foreign deposits. 2010 ) builds a two-country endowment model, with a representative Verdelhan ( agent in each country whose preferences are of the form first proposed by Campbell 18 In Verdelhan’s approach, the real interest rate differential and Cochrane . ( 1999 ) and the foreign exchange risk premium are driven by a factor that is related to the - consumption “habit” in each country. Each agent’s utility depends on the “sur plus”—his consumption relative to an aggregate habit level that is determined as a function of the aggregate consumption level. Similar to the model with Epstein-Zin preferences, the real interest differential and the foreign exchange risk premium are determined by the same driving factor, in this case the surplus. When the surplus is small in the home country, a precautionary effect leads to a lower home interest rate. But also, home investors find foreign deposits riskier relative to the riskiness of home deposits for foreign investors, so the foreign exchange risk premium is high. They underreact to a relatively high foreign interest rate because of foreign exchange risk. The online Appendix shows in more detail that this model cannot account for the main empirical findings of this paper because the single factor that drives the risk premium and the interest differential does not allow for any source of excess adjustment by investors. B. / Delayed Reaction Delayed Overshooting The behavior of exchange rates and interest rates described here seems related to the notion of “delayed overshooting.” The term was coined by Eichenbaum and ( 1995 ) , but is used to describe a hypothesis first put forward by Froot and Evans Thaler ( 1990 ) . Froot and Thaler’s explanation of the forward premium anomaly was that when, for example, the home interest rate rises, the currency appreciates as it would in a model of interest parity such as Dornbusch’s 1976 ) classic paper. They ( hypothesize that the full reaction of the market is delayed, perhaps because some investors are slow to react to changes in interest rates, so that the currency keeps on appreciating in the months immediately following the interest rate increase. ( 2010 ) Bacchetta and van Wincoop build a model based on this intuition. Much of the empirical literature that has documented the phenomenon of delayed overshoot- ing has focused on the impulse response of exchange rates to identified monetary policy shocks, though in the original context, the story was meant to apply to any 19 shock that leads to an increase in relative interest rates. 17 ( 2013 ) ; Backus et al. ( 2010 Bansal and Shaliastovich ; Lustig, Roussanov, and Verdelhan ( 2011 ) . ) 18 Moore and Roche ( 2010 ) also use Campbell-Cochrane preferences to provide a solution to the interest parity puzzle. Stathopolous 2012 ) examines other international asset pricing puzzles in a two-good equilibrium model ( that assumes these preferences. 19 See, for example, Eichenbaum and Evans ( 1995 ) ; Kim and Roubini ( 2000 ) ; Faust and Rogers ( 2003 ) ; Scholl and Uhlig ( 2008 ) ; and Bjornland ( 2009 ) .

29 464 FEBR Y 2016 UAR THE AMERICAN ECONOMIC REVIEW ( present a descriptive model of delayed overshooting 1990 Froot and Thaler ) that, they say, can explain the interest parity puzzle: the hypothesis that at least some investors are , Consider as an example slow in responding to changes in the interest differential. It may be that these investors need some time to think about trades before executing them , or that they simply cannot respond quickly to recent information. These investors might also be called “central banks , ” who seem to “lean against the wind” by trading in such a way as to attenuate the apprecia- tion of a currency as interest rates increase. Other investors in the model , albeit risk averse , and even may try to exploit the first are fully rational group’s slower movements. A simple story along these lines has the poten- it yields negative coefficient tial for reconciling the above facts. First , 3 − estimates of as long as some changes in nominal interest differen- tials also reflect changes in real interest differentials. While changes in nominal interest rates have different instantaneous effects on the exchange , rate across different exchange-rate models most of these models predict all else equal should ( that an increase in the dollar real interest rate ) lead to instantaneous dollar appreciation. If only part of this appreciation and the rest takes some time occurs immediately then we might expect the , , exchange rate to appreciate in the period subsequent to an increase in the interest differential ( Froot and Thaler 1990, p. 188 ) . * i Some intuition can be gained in the case in which − i follows a first-order t t ( 2010 ) , autoregression as does the interest differential in Bacchetta and van Wincoop * * , 0 + ε − − i i = θ 1 . i ≤ θ < ) i 24 ( ( ) t t t t − t 1 − 1 IP from equation ( 2 ) , we have Then using the definition of s t IP IP − θ 1 / + ε . = θ s ( s ) 25 ( ) t 1 t t − For simplicity, assume inflation is constant in both countries, so the distinction between real and nominal rates is not important. While the model of Bacchetta and van Wincoop is complex and requires numer - ical solution, the essence of it can be described as a model in which the exchange rate only gradually approaches the level that would hold under interest parity, as in the Froot and Thaler story. The gradual adjustment can be modeled as IP IP , 0 ε ) + α − s 1 . = δ( s ≤ δ < − s s ) 26 ( t 1 t t t t 1 − − 0 . We find Assume that there is initial underreaction so α < 2 − θ ) 1 ) ( − δ 1 α( * * _____________ i − ) = − var . i − i − 1 + s , i s ( cov t + t t t 1 t t ) ( ) ( − θδ 1 The model will deliver the well-known result from the Fama regression, 2 * < θδ − θ 1 − s , i − δ 1 . This condition can be − − i 1 ) > 0 if α cov ( s ( ) ( ) t t + 1 t t satisfied if α is large enough in absolute value, so the initial underreaction of the exchange rate is sufficiently large. In order for the home currency to appreciate

30 VOL. 106 NO. 2 465 ENGEL: EXCHANGE RATES, INTEREST RATES, AND THE RISK PREMIUM − θ initially when the home interest rate rises, we must have 1 1 . It is nec- − α < ( ) for both conditions to be satisfied. Under these conditions, we essary that θ > δ * ). in real terms ( , as we find strongly in the data , i 0 This − i ) > s ( also find cov t t t contrasts with the risk premium models in the previous section, which could account * * q r − q − , r 0 . − r < > 0 but imply cov q r − lim , ( E ) q for cov + t →∞ k t t t k t t t 1 + t t ) ) ( ( It is easy to see that the impulse response function of the exchange rate to shocks to the interest rate differential will take on the hump shape that is found in the ( 1995 ) and replicated in the model of empirical estimates of Eichenbaum and Evans * ) i − i ρ , ( E ( cov . Nonetheless, in this model, ) 2010 Bacchetta and van Wincoop t t t t j + j . is positive for all The model is built to explain the uncovered interest parity puzzle, and delivers 1 − j * * ρ i , i − − i ) = δ ) = − α 1 − δ / i 1 − θδ > 0 . But cov ( E , ρ E ( cov ) ( ) ( + t 1 + t t t j t t t t * , ρ 0 ) > , so the covariance does not switch signs. This model has i i − ( E cov t 1 t + t t a single shock, the monetary shock, that drives both the interest rate differential and the expected excess return. There is underreaction to the monetary shock, and that underreaction dissipates over time, but there is no source of magnified adjustment. Liquidity Return C. A model that can successfully account for the empirical findings of this paper may need to incorporate both a source of under-adjustment and a source of over-adjustment to changes in interest rates. One model with such properties includes short-term assets valued not only for their return but also for some role they play as liquid assets. Here we sketch the implications of considering the role of ( liquidity return. Our model is based on Nagel ) . 2014 Nagel’s model extends the standard Keynesian liquidity preference for money demand. In the traditional model, money is the only asset that provides liquidity. A contraction in the supply of money, ceteris paribus, increases the value of liquid- ity services of remaining money balances held by households. The rate of return on interest-bearing assets must rise to equilibrate to the liquidity return on money. Nagel ( 2014 ) , in a closed-economy setting, allows for some interest-bearing assets also to be valued for liquidity. They provide liquidity services, though their services are inferior to that of money. When there is a reduction in the money supply, returns rise across the spectrum of interest-bearing assets. In particular, they rise on the liquid interest-bearing asset. The value of liquidity services has risen for those assets as the central bank reduces the supply of money, but the value rises less than the increase for money itself, so the pecuniary return must rise. Nonetheless, it increases less than the interest rate on those assets that offer no liquidity services. The simple extension of Nagel here is to an international setting. Consider a “home” country investor. Following Nagel, we assume the investor has three assets to choose among—demand deposits which are considered to be money, a near- money asset such as interest bearing Eurocurrency deposits denominated in the home currency, and an asset that is not as liquid for the home investor—in this case, the foreign Eurocurrency deposit. The near-money asset is not as liquid as money, but the demand deposit pays a lower rate of interest, assumed to be zero. The online Appendix lays out the model in detail, but here we provide a few key equations. Infinitely lived households each period receive utility in period t

31 466 FEBR Y 2016 UAR THE AMERICAN ECONOMIC REVIEW C from consumption and from the holding of liquid assets: , Q u , where utility is ) ( t t increasing in both elements and concave, and additively separable in utility from consumption and liquidity. The household’s holdings of liquid assets is given by D B t t _ _ ε = . + κ Q ) ( t t P P t t are less-liquid domestic currency are liquid nominal deposits, and B D t t . D ε is less liquid than < 1 , so that B < κ 0 interest-bearing assets. We assume ( ) t t t 0 . is a shock to the liquidity of B , with κ > ε ′ t t Home monetary policy affects the amount of liquid assets. A monetary expansion . This would clearly be true if the increase occurred through a Q would increase t , perhaps financed with D “helicopter drop” ( transfer to households that increased ) t - lump-sum taxes. But even an open-market operation in which the central bank pur D , would increase from the public in exchange for an equal amount of chased B t t κ < 1 . liquidity because ( Linearizing the first-order conditions for the household, we can derive see the online Appendix ) _ κ 1 − β ) ( ̅ ′ 1 + i − κ 1 _ _ _______ ̅ _ = − . + X ε i t t t ( ) β + 1 i i 1 β + ̅ ( ) The overbars represent steady-state values. The household’s utility discount factor β , with 0 < β < 1 . If the liquidity of home interest-bearing assets increases is X ) , then the interest rate paid by those assets will fall. is the marginal a rise in ( ε t t rate of substitution between liquidity services and consumption. Suppose monetary . The marginal utility of liquidity increases Q policy causes a drop in liquid assets, t 1 κ < , the increase at the margin in the , but as described above, because ) rises X ( t D , so the interest rate rises. is less than for B liquidity value of t t The foreign bond, however, provides no liquidity services to home households, so Q when the ex ante return on the foreign bond rises even more than the increase in i t t ( see the online Appendix ) falls. We can then derive − β 1 κ κ ̅ ′ _ _______ _______ ̅ _ _ E = ε i + ρ , t t 1 t + t ) ( ( ) β + 1 − κ 1 i i − κ + ̅ ̅ so the ex ante excess return on the foreign bond rises when the home interest rate rises. The foreign country investors are symmetric to the home investors. They consider the foreign currency Eurocurrency deposit to be a liquid near-money. For simplicity, we assume parameters are the same in the home and foreign countries, so we can express the key relationships in relative home to foreign terms. We can write the equilibrium relationship as * , + η i ρ − 0 . = − α α > i ( 27 ) E ) ( 1 t t t t + t

32 VOL. 106 NO. 2 467 ENGEL: EXCHANGE RATES, INTEREST RATES, AND THE RISK PREMIUM The first component relates the pecuniary excess return to the foreign less home subsumes the shocks to the liquidity return of home rela- η interest differential. t as in Nagel tive to foreign assets that might arise from financial market shocks ( ). raises the expected return on the foreign short-term asset η A positive realization of t relative to the expected return on the home asset. This represents an increase in the 20 relative to the foreign liquidity value of the home interest-bearing asset ( ). Nagel presents evidence that there are volatile shocks to the liquidity value of near-money. These liquidity shocks are the source of the partial adjustment by inves- tors. A negative shock that reduces the liquidity of near-money in the home country weakens demand for home assets, leading to a depreciation of the home currency. In our model, the monetary policymaker reacts to the depreciation by raising the home interest rate: either because the policymaker wants to stabilize exchange rates, or else because it is concerned about the inflationary impact of the depreciation. So when there is a negative liquidity shock to home near-money, the home interest rate rises which increases demand for that asset but at the same time the home asset is less liquid, so the increase in demand is muted. Consumer price inflation can be summarized by the equation * * − π . − π = − δ q ( π − q ) E + β ( 28 ) π ) ( t 1 t + t t + 1 t t t ̅ The parameter governing the speed of adjustment of prices, δ , depends on two underlying parameters in the model: β , and the probability that a firm will not ( in a Calvo pricing framework ) , θ . Specifically, change its price in a given period is the equilibrium real 1 − θβ)/θ , so that δ is decreasing in θ − θ q 1 δ ≡ ( ) ( . t ̅ exchange rate—the value the real exchange rate would take on if nominal prices were flexible. It is driven by exogenous productivity shocks in the two countries, so that an increase in relative home productivity leads to a real home apprecia- tion. Engel ( 2012 ) shows how such an equation can be derived in a model with is q − producer-currency price stickiness and home bias in preferences. When q t t ̅ - positive, prices of goods favored by foreign consumers are relatively high, so for eign inflation will fall relative to home inflation over time. Because productivity follows a first-order autoregressive process with q shocks are persistent, we assume t ̅ a high degree of serial correlation. Monetary policy is specified as a simple Taylor rule, * * * . 1 . − i = σ( π < φ < 0 , 0 − π ) + φ i σ > − i i ) 29 ( ( ) 1 t t t t t − 1 t − Equation 29 ) assumes in each country the policymaker targets its own consumer ( price inflation, and that the instrument rules are identical. The policy rule includes interest-rate smoothing, embodied in the lagged interest rate term. Almost all esti- mated Taylor rules find significant evidence of interest rate smoothing. 20 Valchev ( 2014 ) presents a model to account for the puzzle of this paper, also based on liquidity returns, but the economic mechanisms in that paper are very different than those here. Gabaix and Maggiori ( 2015 ) is a recent study in which a liquidity premium is introduced into a model of exchange rates. However, the notion of liquidity in that paper refers to the cost of making international financial transactions, as opposed to the liquidity of assets as in this model and Valchev’s. Also, the model of Gabaix and Maggiori is unable to explain the puzzle of this paper.

33 468 UAR Y 2016 FEBR THE AMERICAN ECONOMIC REVIEW While a closed-form solution for this simple system is not possible, it is helpful to develop intuition by solving the model as if it were static—solving in terms of current shocks, expected future values, and the lagged interest rate. In particular, we 1 ) ( 27 ) – ( 29 ) to find ( and can use equations αφ 1 + σδ ασδ ασδ * _ _ _ _ q + ρ + E i η − − = − i q E ( ) t − t t t − 1 1 1 + t t t 1 t + ̅ D Δ Δ Δ ασ δ + β ( ) * _ − E π − π ) ( t + 1 1 + t t Δ σ δ + β ) ( φ σδ σδ * * * _ _ _ _ q − π + η , + π i E − − i i − E = q + i ) ( ( ) ( ) 1 t − t 1 − t t t t t + 1 t + t t 1 t + 1 ̅ Δ Δ Δ Δ Δ ≡ + σδ where 1 1 + α . ( ) First consider a positive realization of the shock to the equilibrium real exchange > 0 . This tends to push up foreign inflation, inducing a foreign monetary q rate, t ̅ * . From the model of liquidity returns, − i i tightening, which directly increases t t must fall. The liquidity return of the foreign bond ρ ( E equation ) 27 implies 1 + t t has increased, so its relative pecuniary return declines. Holding the expected future * ρ E − i work and the decline in i exchange rate constant, both the increase in + 1 t t t t rises. This increase in q lowers q to depreciate the home currency in real terms, so t t * * i − π . In response, monetary policymakers lower i , − π foreign relative inflation, t t t t which offsets some of the initial increase in the interest differential. Hence the effect * is less than one-for-one in equilibrium. shock on i − i q of a t t t ̅ is positive, the If there is a positive liquidity shock in the home country, so η t , rises. Holding the expectation ρ ex E ante excess return on the foreign deposits, t + 1 t of the future real exchange rate constant, and holding interest rates constant, this ). This in turn causes induces an appreciation of the home currency ( a drop in q t foreign inflation to rise relative to home inflation, inducing policymakers to increase * * relative to i , ceteris paribus, contributes to a drop in . The increase in i i − i t t t t ρ ρ rises less η , so E , partially offsetting the direct effect on an increase in E t t t t + 1 t + 1 than one-for-one in equilibrium. The shock to the equilibrium real exchange rate contributes toward a negative correlation between the short-run ex ante excess return on the foreign deposit and the foreign less home interest differential, while the liquidity shock contrib- utes to a positive correlation. Can this model reproduce the empirical findings that ∞ * * r ρ − ? It can because , r 0 − r ) > 0 and cov ( E r ∑ , ) < ρ E ( cov t j t t t t 1 + + t t t 1 0 = + j there is protracted adjustment of the nominal interest rate differential, arising from interest-rate smoothing and from persistence in the underlying shocks to inflation sourced from the equilibrium real exchange rate. , has a high enough variance relative to the mon- If the liquidity shock, η t etary shock, it will be the dominant factor in determining the covariance between the interest differential and the short-run ex ante excess return, and will * ) > r − ρ - . At longer horizons, the effect of the inter r 0 , E ( cov deliver t t + t 1 t est rate differential on the liquidity premium will dominate, and we will find * ) < r − ρ 0 . , r cov ( E t t + t j t

34 VOL. 106 NO. 2 469 ENGEL: EXCHANGE RATES, INTEREST RATES, AND THE RISK PREMIUM over-adjustment This model incorporates incentives for both under-adjustment and by investors. It is too simple to take directly to the data, but for plausible param- eter values, the model can roughly reproduce the regression results reported in 0.014 , Tables 3 and 5. As a baseline set of parameters, we take δ = β = 0.998 , φ = 0.915 , α = 0.15 σ = 0.1275 , . The serial correlation of the equilibrium real exchange rate is set to 0.99 in monthly data. We set the shocks to liquidity to have a variance equal to four percent of the variance of the equilibrium real exchange rate. The online Appendix elaborates on this calibration and examines the model under different parameter values. The parameter for time preference, β , is calibrated at a standard level for a monthly frequency. The price adjustment parameter δ implies an expected life of a nominal price in the Calvo model of nine months, which is a and σ standard calibration. The parameters for the policymaker’s interest rate rule, φ imply a long-run increase in the interest rate of 1.5 basis points for each basis point increase in inflation, which is consistent with US data. The smoothing parameter is based on an estimated Taylor rule for the United States relative to the G6 over our data sample. The serial correlation of the equilibrium real exchange rate is set to produce a half-life of five years, in line with the very persistent real exchange α , the response of the liquidity rates among high-income countries. The value of premium to changes in the interest rate, is based on estimates reported by Nagel for spreads of various less liquid assets over the T-bill rate. Nagel reports a value of around 0.10 , but it is adjusted upward slightly here to capture the idea that foreign deposits are somewhat more illiquid for domestic investors. The variance of the liquidity shocks is more difficult to calibrate, and is treated as a free parameter here, though Nagel ( 2014 ) notes these are relatively volatile. While it may seem that a variance of liquidity shocks that is 4 percent of the variance of the equilibrium real exchange rate is small, this refers to the unconditional variance of the latter. Given the persistence of the equilibrium real exchange rate, and that the liquidity shocks have no persistence, the variance of innovations to the liquidity shock equal twice the variance of innovations to the equilibrium real exchange rate. This parameter mainly affects the comovement of the short-run excess return with the real interest differential. For these parameter values, we find the model produces a coefficient of 1.81 for * ρ r − on r , corresponding to the regression results reported the regression of E t t 1 + t t in Table 3. That table reports a coefficient of 1.98 for the United States relative to ∞ * r ∑ 20.66 ρ on produces a coefficient of − − r the G6. The regression of E j + 1 0 t t t j = t + in the model compared to reported in Table 5 for the United States relative 30.89 − to the G6. The model generates reasonable looking behavior for interest rates and inflation. One of the important elements of a Keynesian model is its ability to repli- cate the high correlation of real and nominal interest rates, which is 0.79 in our data for the United States relative to the G6 ) and 0.77 in the model. ( As the online Appendix shows, a greater persistence in nominal interest rates, particularly arising from more persistent shocks to inflation, generates a larger ( in ∞ * r − ∑ . More volatile ρ r on E regression coefficient of ) absolute value = j t j 1 + t t 0 t + * r − ρ . on r E liquidity shocks lead to a larger regression coefficient of t t t + t 1 The model illustrates the forces that a model requires to resolve the puzzle. The shocks that lead to muted adjustment by investors must be volatile enough that * r − ρ . But their and r E they determine the short-run positive correlation of t 1 + t t t

35 470 FEBR Y 2016 UAR THE AMERICAN ECONOMIC REVIEW effect relative to the forces of amplified adjustment must weaken over time, so that * . j for a large enough is negatively correlated with ρ − r r E t j + t t t IV. Conclusions To summarize: A large literature has been devoted to explaining the uncovered * 0 ) > r − ρ . Another stylized fact , r E ( cov interest parity puzzle which implies t t + t 1 t that is generally accepted is that when a country’s real interest rate is relatively high, its currency is relatively strong. However, exchange rates appear to be more volatile than can be accounted for if uncovered interest parity holds, suggesting ∞ * E < ∑ ρ 0 . Our empirical findings confirm these relationships. r − , r cov 0 + j + t t 1 t t ( ) These findings pose a puzzle. Models that have been built to account for the uncov- ∞ * E ∑ r 0 . ρ − < , r ered interest parity puzzle cannot also account for cov 1 + t 0 t t + t j ) ( Neither models of delayed reaction to monetary shocks nor models of foreign exchange risk premiums work because they imply a dampened response, not an excessively volatile response to interest rates. We suggest a possible avenue to explain our findings by introducing a non-pecuniary liquidity return on assets. When a country’s assets become more valued for their liquidity, the country’s currency appreciates. This eases inflationary pressure, allow- under-adjustment part ing policymakers to lower interest rates. This provides the of the story: investors trade off the lower pecuniary return from the lower interest rate with the higher liquidity return. On the other hand, when interest rates rise, then liquid interest-bearing assets are more valued for their liquidity. This is the over-adjustment part of the story: higher interest rates are accompanied by higher liquidity return, giving investors extra incentive to buy the high-interest rate asset. There may be other possible resolutions to the empirical puzzles presented here. Several recent papers have explored the implications for rare, large currency depre- ( forthcoming ) ciations for the uncovered interest parity puzzle. Farhi and Gabaix present a full general equilibrium model of rare disasters and real exchange rates. - Their model implies that when the home real interest rate is high, the home cur rency is weak in real terms, and so cannot account for the levels puzzle presented 21 This correlation occurs during “normal times” in their model; the antici- here. pation of a future disaster increases interest rates and weakens the currency of the country that is expected to experience the future problems. Nonetheless, there are two caveats that must be considered in light of Farhi and Gabaix forthcoming ) ( and the related literature. The first is that if rare disasters are important, than the linear VAR technology used in this paper may not correctly capture the stochastic process for real exchange rates and real interest rates. Burnside, Eichenbaum, and ( 2011 ) Rebelo ( 2011 ) and Farhi , Burnside, Eichenbaum, Kleshchelski, and Rebelo et. al. ( 2013 ) extract information from options to infer expectations about rare large movements in exchange rates. Moreover, if these large rare events are important, then the lognormal approximations that lie behind our analysis of the risk premium in Section IIIA are not correct. Higher order cumulants matter for the risk premium 22 In fact, since our technique only takes a first-order approximation to in that case. 21 See also Gourio, Siemer, and Verdelhan ( 2013 ) . 22 See Martin ( 2013 ) .

36 VOL. 106 NO. 2 471 ENGEL: EXCHANGE RATES, INTEREST RATES, AND THE RISK PREMIUM the solution in terms of means and variances, we may be missing some higher-order effects coming from a more general solution, as in the general equilibrium model of 2013 ) . Colacito and Croce ( It may be that it is necessary to abandon the assumption that all agents have fully rational expectations. Some version of the model proposed by Hong and Stein ( ) may account for the empirical results uncovered here, which perhaps could 1999 be described as a combination of overreaction and momentum trading. That is, the short-term behavior of the real exchange rate under high interest rates incorporates overreaction in that the currency appreciates more than it would under interest parity. But perhaps momentum trading leads to expectations of further apprecia- 2004 ) tion in the short run when the interest rate is high. Gourinchas and Tornell ( ( ) are recent approaches that have and Burnside, Han, Hirshleifer, and Wang 2011 2012 ) adopts an opti- relaxed the assumption of full rationality in some way. Ilut ( mizing approach in which ambiguity averse agents may underreact to good news and overreact to bad news. This study brings two strands of the literature together. The uncovered interest parity puzzle has, in recent years, been primarily addressed as a “finance” puz- zle—it has been shown that models with exotic preferences account for the empir - ical regularity as reaction to foreign exchange risk. The second puzzle—the excess comovement of the exchange rate with interest rates—has been addressed more as an “economics” puzzle. The literature has noted, though not successfully explained, the high volatility of the level of the exchange rate. In bringing the two strands of the literature together, we uncover a striking conflict in the implications of the two puzzles that poses challenges to both lines of research. REFERENCES Alquist, Ron, and Menzie D. Chinn. 2008. “Conventional and Unconventional Approaches to Exchange International Journal of Finance and Economics 13 ( Rate Modelling and Assessment.” ) : 2–13. 1 Andersen, Torben G., Tim Bollerslev, Francis X. Diebold, and Clara Vega. 2007. “Real-Time Price Journal of International Eco- Discovery in Global Stock, Bond and Foreign Exchange Markets.” nomics 73 2 ) : 251–77. ( Bacchetta, Philippe, and Eric van Wincoop. 2006. “Can Information Heterogeneity Explain the Exchange Rate Determination Puzzle?” American Economic Review 96 ( 3 ) : 552–76. Bacchetta, Philippe, and Eric van Wincoop. 2010. “Infrequent Portfolio Decisions: A Solution to the American Economic Review ( 3 ) : 870–904. Forward Discount Puzzle.” 100 Backus, David K., Silverio Foresi, and Chris I. Telmer. 2001. “Affine Term Structure Models and the Journal of Finance 56 ( 1 ) : 279–304. Forward Premium Anomaly.” 2010. “Monetary Policy Backus, David K., Federico Gavazzoni, Chris Telmer, and Stanley E. Zin. and the Uncovered Interest Parity Puzzle.” National Bureau of Economic Research Working Paper 16218. 2000. “The Forward Premium Puzzle: Different Tales from Bansal, Ravi, and Magnus Dahlquist. Journal of International Economics 51 Developed and Emerging Economies.” 1 ) : 115–44. ( Bansal, Ravi, Dana Kiku, and Amir Yaron. 2012. “An Empirical Evaluation of the Long-Run Risks Model for Asset Prices.” Critical Finance Review 1 ( 1 ) : 183–221. Bansal, Ravi, and Ivan Shaliastovich. 2007. “Risk and Return in Bond, Currency and Equity Markets.” Unpublished. Bansal, Ravi, and Ivan Shaliastovich. 2013. “A Long-Run Risks Explanation of Predictability Puzzles Review of Financial Studies 26 ( in Bond and Currency Markets.” ) : 1–33. 1 Bansal, Ravi, and Amir Yaron. 2004. “Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles.” Journal of Finance 59 ( 4 ) : 1481–1509. Bekaert, Geert, Robert J. Hodrick, and David A. Marshall. 1997. “The Implications of First-Order Risk Aversion for Asset Market Risk Premiums.” Journal of Monetary Economics 40 ( 1 ) : 3–39.

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