Exchange rates and interest rates: can term structure models explain currency movements?

Journal of Economic Dynamics & Control 28 (2004) 1595 – 1624 www.elsevier.com/locate/econbase

Exchange rates and interest rates: can term structure models explain currency movements? Ahmet Can Incia;∗ , Biao Lub; c a College

of Business 424 RBA, Florida State University, Tallahassee, FL 32306, USA b University of Michigan, USA c Quantitative Financial Strategies, Inc., Grossman Asset Management, Four Stamford Plaza—Suite 500B, Stanford, CT 06902, USA Received 21 March 2002; accepted 11 March 2003

Abstract We construct an international term structure model that has excellent empirical performance in tracking movements of exchange rates and currency returns. The forward premium puzzle is accounted for, yet the model does not have the undesirable properties found in Backus et al. (J. Finance 56 (2001) 279). Examination of the estimated factor structure indicates that local factors are not important in the US, UK, and German markets. Moreover, we >nd that term structure factors alone cannot satisfactorily explain exchange rate movements. In other words, exchange rates are also a?ected by other factors that are not in the interest rate dynamics. ? 2003 Elsevier B.V. All rights reserved. JEL classi.cation: E43; F31; G12 Keywords: Exchange rates; Term structure of interest rates; Model selection

1. Introduction Many theoretical models suggest that exchange rates should be jointly determined with macroeconomic variables such as foreign and domestic money supplies, real growth rates, interest rates, price levels, and balance of international payments. However, the empirical performance of these models has been very poor. In fact, Meese (1990) concludes that “the proportions of (monthly or quarterly) exchange rate changes that current models can explain is essentially zero”. In this paper, our main purpose ∗

Corresponding author. Tel.: +1-850-645-1169; fax: +1-850-644-4225. E-mail addresses: [email protected] (A.C. Inci), [email protected] (B. Lu).

0165-1889/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0165-1889(03)00081-2

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is to construct an international term structure model that can capture the dynamics of exchange rates and currency returns well. There are economic rationales to highlight the importance of the term structures of interest rates in modeling exchange rates. First, one potential reason for the failure of current macroeconomic exchange rate models is that they may have incorrectly modeled the market’s expectations on future changes of macroeconomic fundamentals. So even if these models had included the right set of fundamentals, they still could not explain changes of exchange rates or currency returns well. The dynamics of the term structure reKect the market’s expectations of the future evolution of a country’s inKation rates and economic growth rates. Thus, information contained in both the domestic and foreign term structures should be useful in accounting for exchange rate movements. Second, the traditional theory of uncovered interest parity that links exchange rate changes to interest rate di?erentials is certainly too simplistic. It fails to capture the complementary and substitution e?ects between short- and long-term interest rates. It also fails to model time-varying foreign exchange risk premia. More sophisticated models like ours overcome these shortcomings. Our paper follows pioneering work in this area by Amin and Jarrow (1991), Ahn (1997), Backus et al. (1995, 2001), Bakshi and Chen (1997), Basal (1997), Saa’Requejo (1993) and Nielsen and Saa’-Requejo (1993). We provide a more comprehensive empirical study than the existing literature. In addition to studying the model’s ability to match the forward premium puzzle, we examine in detail the model’s ability to capture actual movements of exchange rates and currency returns. Our basic model is an international extension of a quadratic term structure model. It has a factor structure in which unobserved global state variables have asymmetric e?ects on pricing kernels in di?erent countries. This is a similar factor structure to that in Backus et al. (2001), but we are able to allow for a more Kexible conditional correlations structure among state variables 1 in our framework, and nominal interest rates are guaranteed to be positive. We estimate three- and >ve-factor models on the monthly US–German exchange rate and term structure data over a sample period of January 1974 through December 1998 and on the monthly US–UK exchange rate and term structure data for a sample period over February 1980 to December 1998. We compare the >tted values with actual exchange rate data. In the US–UK case, for example, the sample average and standard deviation of actual dollar–pound exchange rate are $1:6436=$ and 0.3560, while the contemporaneous values of the three-factor model are $1:6405=$ and 0.5315. The model’s >tted values track the actual movements very closely month by month, and always correctly predict directions of changes in the dollar–pound exchange rate. As a result, the three-factor model captures movements of pound returns well. Over the sample period, the average and standard deviation of annualized 1-month pound returns are −1:71% and 40.07%, while those implied from the three-factor model is −1:74% and 29.45%. The signs of model >tted pound returns are always in line with those of the actual returns. The >ve-factor model further improves over the >tting of 1 Dai and Singleton (2000) show that a Kexible conditional correlation structure is very important in >tting the US term structure dynamics.

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pound return volatility, giving a standard deviation of 36.77%. Our models show a similar performance in explaining the dollar–mark exchange rate movements over a longer sample period. Since returns on major currencies are highly volatile and largely unpredictable, the empirical performance of our models is fairly good. The model also succeeds in accounting for the uncovered interest parity (or forward premium) puzzle. Moreover, it does not have the shortcomings in Backus et al. (2001), who report that their model can imply a long-term yield of as high as 80% per annum as the result of an unreasonably high market price of risk. Finally, the model captures both the shape and volatility of domestic and foreign term structures well. Intuitively, the term structure of interest rates helps to explain currency movements because it contains information on some macroeconomic variables that are important determinants of exchange rates. However, an interesting question is whether term structure factors alone can explain currency movements. We tackle this issue in the following way. We re-estimate the models using only interest rate data for those currencies. The exchange rates are excluded from the estimation. The recovered state variables in these estimations are the domestic and foreign term structure factors only. Next, we construct model >tted values of exchange rates based on these state variables. The performance of this model has not been satisfactory. For instance, in the US–German case, the model has consistently underestimated the dollar value of mark. Finally, based on the Schwarz Information Criterion, we show that our model outperforms the aOne currency model in Backus et al. (2001) and the uncovered interest rate parity model. However, none of the models under consideration can beat the random walk forecast of the exchange rate. The rest of the paper is organized as follows. Section 2 presents the theoretical model of exchange rates and international interest rates and provides a detailed comparison with the literature. Section 3 describes the data and our empirical method. Section 4 discusses the results of the empirical analysis of three- and >ve-factor models on the US–German and US–UK exchange rate and interest rate data. Section 5 concludes. 2. Determination of exchange rates and interest rates 2.1. The model In our model, we assume the existence of a unique positive state price density process or pricing kernel, Mk (t), in each country k, where k = {d; f} for the domestic and foreign country. Exchange rates are traded asset prices and are determined by the pricing kernels together with prices of domestic and foreign bonds. For a zero-coupon bond that has maturity and pays one unit of currency in country k, its time-t price is given by Pk (t; ) = Et [Mk (t + )]=Mk (t);

(1)

where Et is the expectation conditional on information at time t. The yield of this zero-coupon bond is given by yk (t; ) = −ln Pk (t; )= . So the term structure of interest rates in each economy is uniquely determined by its pricing kernel. We de>ne the

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exchange rate as the number of units of domestic currency per unit of foreign currency. In this paper, the domestic currency will always be the US dollar. Under standard asset pricing theory, the spot exchange rate, S(t), is uniquely determined by 
Mf (t)=Mf (0) : (2) S(t) = S(0) Md (t)=Md (0) Thus, the spot exchange rate is the ratio of the pricing kernels of the two countries. 2 Alternatively, the log exchange rate, s(t) = ln S(t), is given by s(t) − s(0) = [ln Mf (t) − ln Mf (0)] − [ln Md (t) − ln Md (0)]: The dynamics of the pricing kernels in the two economies are driven by N unobserved global state variables, X (t), as Md (t) = exp[ − gd t + X
(t)d X (t)]

(3)

Mf (t) = exp[ − gf t + X
(t)f X (t)];

(4)

and where the subscripts (d,f) refer to domestic and foreign economies, respectively. We allow the state variables to be correlated and to follow multivariate Gaussian processes: dX (t) = K( − X (t)) dt +  dWt ;

(5)

where K is diagonal,  is a N × 1 vector,  is a lower triangular matrix, and Wt are N independent standard Weiner processes. The ith diagonal element of K is the mean-reverting parameter and the ith element of  is the long-run mean of the ith state variable. The lower-triangular structure of  captures conditional correlations among state variables. 3 The global factors have asymmetric e?ects on the two pricing kernels. We set d for the domestic country to be an identity matrix, IN , and set f for the foreign country to be diagonal with elements (1 ; : : : ; N ). These i coeOcients are restricted to be non-negative. Their magnitudes relative to one measure the relative importance of the ith state variable xi in a?ecting domestic and foreign pricing kernels and term structures of interest rates. If i far exceeds one, then xi is a factor with a much greater impact on the foreign than domestic pricing kernel and bond market. If i is a small fraction, then xi is a factor with a much greater impact on the domestic than foreign pricing kernel and bond market. If i is close to one, then xi has a similar impact on both pricing kernels and bond markets. Note that it is not a generalization of the model to also set d for the domestic country to be diagonal. Doing so would cause an identi>cation problem since one could simultaneously scale up or down d , f , and the parameters in the multivariate Gaussian processes in (5). Finally, the parameters gd and gf in the pricing kernels have the interpretation of being yields of long-maturity discount bonds in the two countries. 2

The derivation follows from Ahn (1997). Any arbitrary foreign asset, whose price is Pf (t), satis>es (1). The domestic price of this asset Pfd (t) (which is equal to Pf (t)S(t)) satis>es the domestic valuation equation of (1). From the uniqueness of the pricing kernels we get (2) at time 0. 3 An alternative and equivalent approach for modeling the correlations is to let K to be lower-triangular and  to be diagonal.

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2.2. Links to the literature Our model is a direct extension of the one-country quadratic term structure model in Constantinides (1992). Aside from working in a multi-country setting, we also allow the state variables to be correlated. This extension is necessary in light of the empirical >ndings in Dai and Singleton (2000) (for aOne models including Vasicek, 1977, among others) and Ahn et al. (2001) (for quadratic models), who show that orthogonality of state variables imposes signi>cant restrictions on >tting the US term structure. Many studies have explored the use of aOne models for explaining outstanding issues in international interest rates and exchange rates. As an important recent contribution, Backus et al. (2001) demonstrate that an aOne model, in which global state variables have asymmetric e?ects on di?erent pricing kernels, can account for the forward premium puzzle. However, they also >nd that the model has important shortcomings in >tting the term structure dynamics: when >tted to the data, the model implies an unreasonably high market price of risk, which causes the long-term yield to be as high as 80% per annum. Our model has a similar factor structure, and can account for the forward premium puzzle, but does not have the undesirable properties. We also go one step further by examining the model’s performance in explaining the actual time-series movements of exchange rate levels and currency returns. After all, accounting for the forward premium puzzle is only a minimum requirement for the model, whose usefulness ultimately hinges on its ability to capture actual dynamics of exchange rates and interest rates. Bakshi and Chen (1997) adopt a di?erent factor structure in which each country’s bond market is a?ected by independent local factors. In order to capture co-movements between foreign and domestic interest rates, Ahn (1997) further introduces common factors, which are uncorrelated with local factors but have impacts on both countries. In our framework, a local-common factor structure could be imposed by putting restrictions on d and f in the pricing kernels and  in the state variable dynamics. We do not do so because we feel that there is no theoretical guidance on how many of the term structure factors in the US or a foreign economy should be local or common. This is largely an interesting empirical issue, and in fact, our current model structure can shed some light on it. For instance, if the estimation results show that the 1 coeOcient far exceeds one and the 1st state variable x1 is uncorrelated with the others, then x1 can be regarded as a foreign local factor. On the other hand, if 1 is far less than one and the 1st state variable x1 is uncorrelated with the others, then x1 can be regarded as a domestic local factor; and in other situations, it can be regarded as a common factor. In the term structure literature, Ahn et al. (2001) has recently generalized various models, including Constantinides (1992), into a quadratic class. In their maximally Kexible version, the stochastic process of the pricing kernel is given as dM (t) (6) = −r(t) dt + (0 + 1 X (t))
dWt ; M (t) where 1 is lower triangular, and the short rate r(t) is given as r(t) =  + 
X (t) + X (t)
X (t);

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where  = 0 and is a symmetric matrix with diagonal elements equal to one. Our model amounts to imposing the following restrictions for each country k: 0 = 0N ; 1 = 2k ;  = gk − trace(k 
);  = −2k K;  = 2k (K − 
k ):

(7)

To further generalize our model towards (6) in a multi-country setting would require very careful consideration on identi>cation. For example, we have seen that in our model, even making d to be a non-identity diagonal matrix would cause identi>cation problems. Furthermore, identi>cation aside, the more important reason for using our current model structure, as an extension of Constantinides (1992), is that pricing kernels are explicitly expressed as functions of state variables as in (3) and (4), not in terms of stochastic di?erential equations. As shown below, this allows us to explicitly derive exchange rates as functions of state variables, just as we derive bond pricing formulas. After all, currencies should be treated as traded assets in the model, just like bonds. The explicit exchange rate formula further facilitates derivation of foreign exchange risk premium and incorporation of exchange rate data into empirical estimation of the model. 4 2.3. Exchange rates and interest rates Given the pricing kernels in (3) and (4), the (log) exchange rate can be explicitly derived as s(t) − s(0) = (gd − gf )t + X
(t)(f − d ) X (t) − X
(0)(f − d ) X (0):

(8)

Accordingly, the (annualized) log currency return from t to t + is given by s(t + ) − s(t) = (gd − gf ) + [X
(t + )(f − d )X (t + ) − X
(t)(f − d )X (t)]:

(9)

This log return has two components. The >rst term, gd − gf , is the di?erence of very long-term interest rates in domestic and foreign countries. The second term depends on temporal changes of state variables over the period and their asymmetric e?ects on the two pricing kernels. The bond prices and yields in each country can be derived as in Ahn et al. (2001). So we simply write down their general pricing formula with additional parameter 4

Models that specify pricing kernels in stochastic di?erential equations usually give rise to the dynamics of exchange rates in terms of di?usion processes, whose transitional densities are mostly unknown. Since the data is always discrete, it is diOcult to incorporate exchange rates into estimation of such models.

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restrictions. Their most general formula for the price and yield of a zero-coupon bond in each country k are as follows: Pk (t; ) = exp[Ak ( ) + Bk ( )
X (t) + X (t)
Ck ( )X (t)]

(10)

1 yk (t; ) = − [Ak ( ) + Bk ( )
X (t) + X (t)
Ck ( )X (t)];

(11)

and

where Ak , Bk , and Ck satisfy the following ordinary di?erential equations with the subscript k suppressed: dC( ) = 2C( )
C( ) + [C( )(−K − #1 ) + (−K − #1 )
C( )] − ; d dB( ) = 2C( )
B( ) + (−K − #1 )
B( ) + 2C( )(K − #0 ) − ; d dA( ) 1 = trace[
C( )] + B( )
B( ) + B( )
(K − #0 ) −  d 2 with initial conditions Ak (0)=0, Bk (0)=0N , Ck (0)=0N ×N . Our model puts in additional parameter restrictions in (7). 2.4. The uncovered interest rate parity The uncovered interest rate parity predicts that the currency of the country with the higher interest rate tends to depreciate. This theory implies that investors would essentially achieve the same return from holding the high-interest-rate currency as from holding the low-interest-rate currency. The uncovered interest rate parity has been extensively tested in the literature mainly via the following regression: s(t + ) − s(t) = 0 + 1 (yd (t; ) − yf (t; )) + $t ;

(12)

where s(t + ) − s(t) is the log return of the foreign currency and yd (t; ) − yf (t; ) is the di?erential of the domestic and foreign interest rates. The null hypothesis under the uncovered interest rate parity is that the slope coeOcient 1 is one. The empirical results depend on the forecasting horizons and terms of interest rates. Most of the studies have focused on forecasting horizons less than one year using short-term interest rates. These studies have found that not only the slope coeOcient 1 statistically di?erent from one, but also it is negative for all the major currencies, thus rejecting the uncovered interest rate parity. Some recent papers 5 have focused on longer-term forecasting horizons and/or longerterm interest rates. They have found results more in line with the uncovered interest rate parity. For example, Meredith and Chinn (1998) and Alexius (2000) reported evidence of positive slope coeOcient 1 for long-term interest rates over long horizons. 5

We thank an anonymous referee for directing us to these studies.

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One way to examine whether our model is consistent with these various empirical >ndings is to check the two conditions in Fama (1984), which attributes the negative slope coeOcients in Eq. (12) to certain dynamics of a time-varying foreign currency risk premium. Let f(t; ) be the -period forward exchange rate. We de>ne the foreign currency risk premium as p(t; ) = f(t; ) − Et s(t + ): This is what investors demand to bear the exchange rate risk. We further de>ne the (annualized) expected rate of depreciation of foreign currency as q(t; ) = Et s(t + ) − s(t): By virtue of the covered interest rate parity, which holds under the no-arbitrage condition, the interest rate di?erential yd (t; ) − yf (t; ) always equals the di?erential of the -period forward and current spot exchange rates, f(t; ) − s(t). Then interest rate differentials can be expressed as the sum of foreign currency risk premium and expected rate of depreciation: yd (t; ) − yf (t; ) = f(t; ) − s(t) = p(t; ) + q(t; ): Standard regression results show that the slope coeOcient in Eq. (12) is given by 1 =

Cov(q(t; ); p(t; ) + q(t; )) Cov(q(t; ); p(t; )) + var(q(t; )) = : var(p(t; ) + q(t; )) var(p(t; ) + q(t; ))

(13)

Thus, generating a negative 1 requires two conditions: (i) The currency risk premium is negatively correlated with expected rate of depreciation (Corr(q; p) ¡ 0) and (ii) the currency risk premium is more volatile than the expected rate of depreciation (STD(p) ¿ STD(q)). Under our model, the foreign currency risk premium and expected rate of depreciation can be recovered as follows. First, the interest rates di?erential yd (t; ) − yf (t; ) can be derived from yield functions in (11). Next, we derive the expected rate of depreciation using (9) as follows: q(t; ) = (gd − gf ) + Et [X
(t + )(f − d )X (t + )] − X
(t)(f − d )X (t);

(14)

where the expectation term is given by Et [X
(t + )(f − d )X (t + )] = trace[(f − d )Et X (t + )X
(t + )] = trace[(f − d )(Var t X (t + ) + Et X (t + )Et X (t + )
)]: The key pieces of information in the above derivation are the conditional means and variance–covariances of the state variables, which we recover by using a >ltering procedure in the empirical estimation of the model. Finally, the foreign currency risk

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premium can be derived as p(t; ) = [yd (t; ) − yf (t; )] − q(t; ):

(15)

We then check whether our model passes Fama’s conditions for both the short- and long-term interest rates.

3. Data and empirical method 3.1. The data The empirical analysis is based on monthly observations from January 1974 through December 1998 for the United States and Germany, and from February 1980 through December 1998 for the United States and United Kingdom. The monthly exchange rate data is obtained from Data Resources Incorporated (DRI) and Datastream. The month of January 1974 is chosen as the earliest starting point so that the sample would cover the Koating exchange rate period. More precisely, the international exchange rates became Koating in August of 1973; however, we have chosen to start the investigation from the beginning of 1974 so that the international markets absorb the changes and adjust to the new system. The data for the UK becomes available only since February 1980. The United States is the domestic country in all of our analysis. Again, the exchange rate is de>ned as the number of US dollars per German mark or British pound. Thus, an increase of exchange rate means depreciation of the US dollar or appreciation of German mark or British pound. For the interest rate data, we use eurocurrency deposit rates obtained from DRI under the Webstract database environment. The data consists of end-of-month ask quotes of 1-, 3-, 6-, 12-month, 2-, 3-, 4-, and 5-year Eurocurrency deposit rates for the US dollar and German mark, and ask quotes of 1-, 3-, 6-, and 12-month rates for the British pound. These ask quotes are reported at the close of the London market. Eurocurrency market is an enormous and very liquid interbank market, which is free from additional risks originating from investors. Access to this liquid market is reserved to institutions of top quality. The market has a single location, London. This single market ensures comparability across di?erent currency denominations. The use of eurocurrency rates minimizes possible distortions caused by di?erences in tax rules, capital controls, and other government regulations which vary signi>cantly from country to country. The 1-, 3-, 6-, and 12-month Eurocurrency deposits are essentially par zero-coupon bonds whose payo?s at maturity are the principal plus an interest payment. We convert the quoted rates on these deposits into continuously compounded yields on the same-maturity zero-coupon bonds whose only payo?s are the principal at maturity. The 2-, 3-, 4-, and 5-year Eurocurrency deposits are essentially equivalent to par bonds paying annual coupons at the quoted rates. We recover the 2-, 3-, 4-, and 5-year zero-coupon bond yields or spot rates from the quoted Eurocurrency rates by forward substituting the shorter spot rates to account for the intermediate coupon payments.

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3.2. The empirical method We estimate the model using the extended Kalman >lter. In addition to providing maximum-likelihood estimates of model parameters, the extended Kalman >lter generates conditional densities of unobserved state variables over time. These densities can be used to construct estimates of unobserved state variables and foreign currency risk premia in (14). A standard textbook treatment of the extended Kalman >lter can be found in Anderson and Moore (1979). We express the models in state-space forms, which consist of observation equations and state equations. The observation equations include the exchange rate equation in (8) and yield equations in (11), augmented by error terms. Thus, we estimate the models jointly using the exchange rate data and term structure data in the two countries. The observation errors are assumed to be normally distributed with zero means but di?erent variances. They are further assumed to be serially uncorrelated, cross-sectionally uncorrelated, and independent of the state variables, as in Chen and Scott (1993), DuOe and Singleton (1997), DuOe and Kan (1996) and Du?ee (2002), among others in the term structure literature. For the US–UK pair, the yield equations span 1-, 3-, 6-, and 12-month maturity for both countries; for the US–Germany pair, they span 1-, 3-, 6-, 12-month, 2-, 3-, 4-, and 5-year maturity. The state equations are based on transitional densities of the state variables implied by their stochastic processes in (5). The transitional densities are normally distributed with the mean vector and variance–covariance matrix given as Et X (t + ) =  + exp(− K)[X (t) − ]; 
ij (exp((−-i − -j ) ) − 1) Var t X (t + ) = ; −-i − -j N ×N

(16)

where ij is the (i; j)th element of  and -i is the ith diagonal element of K. We estimate a variety of models using interest rate and exchange rate data. Since it has been shown that three factors are needed to >t the US term structure movements (see Knez et al., 1991), we focus on models with at least three global state variables for any two countries. In all cases, we allow the state variables to be conditionally correlated with each other. 4. Empirical analysis 4.1. The US–German case 4.1.1. Dynamics of interest rates and exchange rate We >rst assess the models’ performance in accounting for the US and German term structure movements. Table 1 presents means and standard deviations of the Eurodollar and Euromark zero-coupon yields for maturities up to 5 years (at the >rst row of Panels A and B) during the period of 1/1974 –12/1998. These moments reKect the average shapes of the term structures and volatility structures of interest rates. Both

Eurodollar term structure Maturity A. Mean Actual 3-Factor model w/ exchange rate 5-Factor model w/ exchange rate 3-Factor model w/o B. Volatility (STD) Actual 3-Factor model w/ exchange rate 5-Factor model w/ exchange rate 3-Factor model w/o exchange rate

1M

3M

6M

Euromark term structure 1Y

2Y

3Y

4Y

5Y

1M

3M

6M

1Y

2Y

3Y

4Y

5Y

0.0823 0.0833 0.0840 0.0833 0.0870 0.0892 0.0910 0.0926 0.0626 0.0630 0.0633 0.0632 0.0672 0.0700 0.0724 0.0738 0.0818 0.0828 0.0840 0.0849 0.0872 0.0890 0.0904 0.0923 0.0638 0.0636 0.0636 0.0651 0.0674 0.0701 0.0727 0.0746 0.0844 0.0849 0.0854 0.0855 0.0872 0.0894 0.0916 0.0945 0.0655 0.0652 0.0651 0.0663 0.0684 0.0708 0.0731 0.0746 0.0839 0.0846 0.0854 0.0860 0.0881 0.0903 0.0923 0.0949 0.0633 0.0632 0.0633 0.0650 0.0676 0.0703 0.0729 0.0745 0.0357 0.0350 0.0338 0.0300 0.0281 0.0270 0.0263 0.0257 0.0257 0.0253 0.0244 0.0226 0.0200 0.0181 0.0165 0.0155 0.0248 0.0240 0.0227 0.0232 0.0204 0.0184 0.0167 0.0154 0.0353 0.0343 0.0332 0.0296 0.0274 0.0260 0.0252 0.0245 0.0259 0.0250 0.0236 0.0238 0.0205 0.0184 0.0166 0.0155 0.0377 0.0362 0.0344 0.0303 0.0281 0.0271 0.0265 0.0258 0.0149 0.0165 0.0180 0.0221 0.0206 0.0180 0.0149 0.0120 0.0377 0.0363 0.0346 0.0301 0.0277 0.0267 0.0264 0.0261

Means and standard deviations (volatilities) of the actual and model-implied Eurodollar and Euromark zero-coupon yields are presented. The data on Eurocurrency zero-coupon yields are constructed from end-of-month ask quotes of Eurocurrency deposit rates, which are taken from Data Resources, Inc. The model-implied yields are one-month-ahead forecasting values based on the prediction densities of unobserved state variables recovered in estimation. Reported are results from three models: 3-factor model and 5-factor model estimated using both interest rates and dollar–mark exchange rate data, and 3-factor model estimated using interest rate data only. The sample is monthly over the period from January 1974 through December 1998.

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Table 1 Actual and model implied term structures of Eurodollar and Euromark interest rates

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the Eurodollar and Euromark term structures are on average upward sloping during this sample period. The volatility structures are downward sloping so the long-term interest rates are generally less volatile than the short-term rates at the Eurocurrency markets. The Eurodollar yields are about 2% higher than the Euromark yields. The next two rows in each panel report the means and standard deviations of implied yields in 3- and 5-factor models. These model-implied yields are one-month-ahead in-sample forecasts based on the prediction values of unobserved state variables recovered in estimation. These two models are estimated using both interest rate and exchange rate data. The last row in each panel lists the results from a 3-factor model that is estimated using interest rate data only, and will be discussed in Section 4.3. The 3-factor model implies an upward-sloping term structure for both currencies, consistent with the data. Average pricing errors for the Eurodollar yields range from less than 1 basis point for 6-month maturity to 12 basis points for 1-year maturity, while the average pricing errors for the Euromark yields range from 1 basis point for 3-year maturity to 12 basis points for 1-month maturity. The model also produces a downward-sloping volatility term structure for both currencies, again consistent with the data. The model’s estimates of yield volatilities are reasonably close to the actual data, but it seems to slightly under-estimate volatilities of the Eurodollar yields and over-estimate volatilities of the Euromark yields. The 5-factor model has a similar performance in accounting for the term structure dynamics. Overall, our model has not produced any economically unreasonable results. We regard this as an improvement over Backus et al. (2001), who report that their model can imply a long-term yield as high as 80%. Next, we assess the models’ performance in accounting for the dollar–mark exchange rate dynamics. Table 2, left panel, presents means and standard deviations of exchange rates and continuously compounded monthly return of mark in the actual data and >tted values obtained from 3- and 5-factor models. The model >tted exchange rates are >ltered values and the model->tted 1-month returns are computed using these >lter values. All of the monthly returns are annualized. The data shows that the dollar–mark exchange rate is highly volatile. The actual average of monthly mark returns is only 1.83%, while the return volatility (standard deviation) is 39.05%. This explains why it is very diOcult for asset pricing models to forecast or predict exchange rate dynamics. The 3-factor model performs well, producing an average monthly return of 3.14% and volatility of 36.38%. The 5-factor model performs even better. It implies an average monthly return of 1.85% and volatility of 37.08%. Both model >ts are very close to the actual data. Fig. 1 provides further evidence on how close the model >ts are to actual dollar– mark exchange rate and mark return movements over time. The upper panel of Fig. 1 plots the time series of dollar–mark exchange rate (in dotted line) versus that of >tted values of the 3-factor model (in solid line). The dollar value of German mark has experienced large swings during the sample period. It rose from about $0.4 to $0.6 from the mid 1970s to early 80s, plunged to around $0.3 in the mid-80s, then rose back to around $0.6 in the late 1990s. Throughout these ups and downs, the model values track the actual exchange rates very well and are virtually always in the right direction with the changes of the exchange rate. The lower panel of Fig. 1 plots the actual mark

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Table 2 Actual and model->tted exchange rates dynamics Dollar–mark exchange rate A. Mean in % Actual 3-Factor model w/ exchange rate 5-Factor model w/ exchange rate 3-Factor model w/o exchange rate B. Volatility (STD in %) Actual 3-Factor model w/ exchange rate 5-Factor model w/ exchange rate 3-Factor model w/o exchange rate

Monthly mark return

Dollar–pound exchange rate

Monthly pound return

$0.5055 0.4587

0.0183 0.0314

$1.6436 1.6405

−0:0171 −0:0174

0.5053

0.0185

1.6390

−0:0140

0.2959

−0:0040

1.9253

−0:0204

0.1095 0.1486

0.3905 0.3638

0.3560 0.5315

0.4007 0.2945

0.1072

0.3708

0.3473

0.3677

0.0542

0.5502

0.3527

0.3162

Means and standard deviations (volatilities) of actual and model->tted exchange rates and monthly currency returns are presented. The exchange rate is de>ned as number of dollars per mark or per pound. The actual exchange rate data is taken from Datastream. The model exchange rates are contemporaneous values based on the >ltering densities of unobserved state variables recovered in estimation. All of the currency returns are continuously compounded and annualized. Reported are results from three models: 3- and 5-factor models estimated using both the interest rates and exchange rate data, and 3-factor model estimated using the interest rate data only. The sample period for the dollar–mark exchange rate is over January 1974 through December 1998. The sample period for the dollar–pound exchange rate is over February 1980 through December 1998.

monthly returns (in dotted line) versus the contemporaneous >lter values of the 3-factor model (in solid line). The actual return series is very volatile and noisy, and does not exhibit clear serial correlations or any other patterns. The model->tted returns track the actual ones very well. Their movements are always in the same direction and at similar magnitudes. However, the contemporaneous model >ts seem to slightly underestimate changes of the mark exchange rate so the actual mark returns are more volatile than the model-generated ones. Fig. 2 provides the same two plots for the 5-factor model. With >ve factors, the model >tted values show better performance in accounting for the exchange rate dynamics. In particular, the contemporaneous model monthly mark returns (solid line in the bottom panel) track the actual ones more closely and do not under-estimate the volatility of the exchange rate. Finally, Table 3 lists Fama’s conditions for both short-term interest rates (1-month yields) with short forecasting horizon (1-month) and long-term interest rates (5-year yields) with long forecasting horizon (5-year). Fama’s conditions are met for short-term interest rates with short horizon, indicating that uncovered interest rate parity is rejected. The conditions for the long-term interest rates with long horizon are not met, consistent with the results in Meredith and Chinn (1998) and Alexius (2000). Thus, our model is

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A.C. Inci, B. Lu / Journal of Economic Dynamics & Control 28 (2004) 1595 – 1624 Dollar-Deutsche Mark Exchange Rate

0.7

actual model-implied

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3 Feb74

Apr80

Jun86

Sept92

Dec98

Annualized 1-Month DM Return actual model-implied

0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

Apr80

Jun86

Sept92

Dec98

Fig. 1. Dollar–mark exchange rate and monthly mark return: actual versus >ltered values from three-factor model.

A.C. Inci, B. Lu / Journal of Economic Dynamics & Control 28 (2004) 1595 – 1624

1609

Dollar-Deutsche Mark Exchange Rate 0.75

actual model-implied

0.7

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3 Feb74

Apr80

Jun86

Sept92

Dec98

Annualized 1-DM Return actual model-implied

0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

Apr80

Jun86

Sept92

Dec98

Fig. 2. Dollar–mark exchange rate and monthly mark return: actual versus >ltered values from >ve-factor model.

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Table 3 Analysis of Fama’s conditions on forward premium puzzle Model

Condition I: Corr(q; p) ¡ 0

US–German 3-factor model w/ exchange rate US–German 5-factor model w/ exchange rate US–UK 3-factor model w/ exchange rate US–UK 5-factor model w/ exchange rate

−0:7422 ¡ 0

0:0469 ¿ 0:0316

−0:7668 ¡ 0

0:0490 ¿ 0:0346

−0:7701 ¡ 0

0:0424 ¿ 0:0245

−0:7049 ¡ 0

0:0419 ¿ 0:0237

Long-horizon results US–German 5-factor model w/ exchange rate 5-year horizon US–UK 5-Factor model w/ exchange rate 5-year horizon

Condition I: Corr(q; p) ¡ 0 −0:9972 ¡ 0 −0:9967 ¡ 0

Condition II: STD(p) ¿ STD(q)

Condition II: STD(p) ¿ STD(q) 0:2764 ¡ 0:2780 0:21749 ¡ 0:22846

This table presents evidence on whether the models satisfy the two conditions for accounting for the forward premium puzzle, as shown in Fama (1984). The foreign currency risk premium is denoted by p and the expected rate of depreciation is denoted by q. The two conditions are: (1) the currency risk premium (p) and expected rate of depreciation (q) are negatively correlated and (2) the currency risk premium is more volatile than the expected rate of depreciation.

capable of explaining various empirical >ndings regarding the uncovered interest rate parity. 4.1.2. Parameter estimates and factor structure Table 4 presents parameter estimates with standard errors for the 3- and 5-factor models. The parameters from the stochastic processes of state variables are listed in the >rst group. The nature and degree of conditional correlations among the state variables are reKected in the correlation coeOcients .ij . For the 3-factor model, the >rst and third state variables are negatively correlated with a correlation coeOcient of −0:2319, the second and third state variables are also negatively correlated with a correlation coeOcient of −0:2815, and the >rst and second state variables are positively correlated. For the 5-factor model, the correlation structure is more complex. Speci>cally, the second state variable is either positively or negatively correlated with the other four state variables, the >rst state variable is positively correlated with both the second and fourth state variables, but the correlations among the third, fourth, and >fth state variables are not statistically signi>cant. These results show that it is important to allow for a Kexible correlation structure among the factors. The parameters governing the foreign and domestic pricing kernels are listed in the next two groups. The >rst one, gf , has the interpretation of being a very long-term German government bond yield and is estimated at 10.01% in the 3-factor model and 17.47% in the 5-factor model. Although the sample spans the late 1970s and early 1980s, when interest rates were very high, the 10.01% makes more economic sense based on the macroeconomic fundamentals in Germany, while the 17.47% seems a

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Table 4 Parameter estimates of the US–German models 3-Factor model

5-Factor model

State variable dynamics -1 1 1 -2 2 2 -3 3 3 -4 4 4 -5 5 5 .12 .13 .14 .15 .23 .24 .25 .34 .35 .45

0.4492 0.1068 0.0117 0.1001 1.1761 0.1008 0.8281 0.2758 0.0590 — — — — — — −0:2319 0.1347 — — −0:2815 — — — — —

0.0752 0.8827 0.1029 0.0075 1.7571 0.0608 0.7633 0.0487 0.0101 0.5446 0.1350 0.1364 0.2735 0.2949 0.0612 0.7523 0.0090 0.2080 −0:1365 −0:1903 0.1992 0.2007 −0:0833 −0:0753 0.0197

German pricing kernel parameters gf 1 2 3 4 5

0.1001 (0.0021) 16.2242 (13.2698) 0.5824 (0.0104) 0.1444 (0.0211) — —

0.1747 (0.0028) 0.1842 (0.0387) 2.5257 (0.3407) 35.7762 (13.6289) 0.1325 (0.0366) 3.6677 (0.3205)

0.1264 (0.0034)

0.1334 (0.0068)

US pricing kernel parameters gd

(0.0102) (0.0454) (0.0050) (0.0058) (0.0517) (0.0043) (0.0222) (0.0107) (0.0038)

(0.0739) (0.0844) (0.0718)

(0.0028) (0.0664) (0.0053) (0.0025) (0.3827) (0.0086) (0.0246) (0.0097) (0.0020) (0.0092) (0.0124) (0.0015) (0.0078) (0.0218) (0.0041) (0.0474) (0.0891) (0.0939) (0.0827) (0.1041) (0.0821) (0.0949) (0.1135) (0.0982) (0.0896)

Parameter estimates of three- and >ve-factor models on the term structures of Eurodollar and Euromark deposit rates and dollar–mark exchange rate are presented. Standard errors of the estimates are reported in parentheses.

bit too high. The last one, gd , has the interpretation of being a very long-term US government bond yield and is estimated at 12.64% in the 3-factor model and 13.34% in the 5-factor model. The i coeOcients control the relative e?ects of the state variables on the German and US pricing kernels. Alternatively, these coeOcients determine the relative market prices of global factor risks in the two currencies. The corresponding coeOcients in

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A.C. Inci, B. Lu / Journal of Economic Dynamics & Control 28 (2004) 1595 – 1624

the US pricing kernel are all set equal to one. So the magnitude of each i relative to one indicates the importance the ith state variable on the two economies. For the 3-factor model, the >rst state variable (1 = 16:2242) is a more important risk factor in the German than in the US market; the third state variable (with 3 = 0:1444) is a more important risk factor in the US market; and the second state variable (with 2 = 0:5824) has relatively similar impacts on both countries. For the 5-factor model, the >rst and fourth state variables have larger impacts on the US markets, while the second, third, and >fth ones have larger impacts on the German market. The evidence so far does not support existence of ‘local’ factors in the US–German setting. A local factor, by de>nition, impacts only one economy, has negligible e?ects on the other, and is uncorrelated with other factors. The state variables in our model are shown to have di?erent impacts on the US and Germany economies, but none of them are negligible. Moreover, these state variables are correlated with each other in various ways. This evidence supports the approach that emphasizes the importance of common factors in modeling the interest rates and exchange rate dynamics in these countries. The issue of local versus common factors has interesting implications for portfolio diversi>cation across international bond markets. Consider a US investor who holds both US and German bonds but hedges currency risk. The return uncertainties of this currency-hedged bond portfolio only come from the risk factors in both bond markets. So there would be signi>cant diversi>cation bene>ts from holding hedged bond portfolios internationally if there would exist prominent local factors in the German bond market that are not present in the US market. Since we do not >nd strong evidence in support of local factors, diversi>cation bene>ts from holding a currency-hedged bond portfolio in these markets are likely to be small. 4.2. The US–UK case 4.2.1. Dynamics of interest rates and exchange rate We conduct the same analysis on the Europound yields and dollar–pound exchange rate in Tables 5 and 2 (the right panel). Due to data limitations, the sample period for the US–UK case is over February 1980 through December 1998 and the maturity of the yields is only up to 1 year. Table 5 shows that during the sample period, the short end of the Europound term structure is actually downward-sloping, while the short end of the Eurodollar term structure is relatively Kat. The yield volatilities decrease with maturity in both currencies. The Europound yields are on average higher than the Eurodollar yields. The 3-factor model produces correct shapes of the term structure and volatility structure of yields in both currencies. The average pricing errors on the Europound yields are between 1 and 6 basis points and the average pricing errors on the Eurodollar yields are between 1 and 10 basis points. This is similar to the performance of the 3-factor model on the US–German case. In terms of yield volatilities, the model seems to underestimate volatilities of the Eurodollar yields but overestimates volatilities of the Europound yields. The 5-factor model performs worse than the 3-factor model in simultaneously >tting the Europound and Eurodollar term structures. Note that the

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Table 5 Actual and model-implied term structures of Eurodollar and Europound interest rates Eurodollar term structure

A. Mean Actual 3-Factor model w/ exchange rate 5-Factor model w/ exchange rate 3-Factor model w/o exchange rate B. Volatility (STD) Actual 3-Factor model w/ exchange rate 5-Factor model w/ exchange rate 3-Factor model w/o exchange rate

Europound term structure

1M

3M

6M

1Y

1M

3M

6M

1Y

0.0814 0.0809

0.0817 0.0807

0.0819 0.0809

0.0816 0.0816

0.1037 0.1042

0.1030 0.1032

0.1012 0.1013

0.0972 0.0978

0.0827

0.0823

0.0822

0.0829

0.1080

0.1062

0.1037

0.0998

0.0815

0.0811

0.0811

0.0822

0.1045

0.1035

0.1017

0.0982

0.0376 0.0339

0.0368 0.0323

0.0357 0.0298

0.0323 0.0268

0.0342 0.0370

0.0329 0.0359

0.0309 0.0347

0.0272 0.0314

0.0368

0.0348

0.0320

0.0288

0.0395

0.0380

0.0364

0.0328

0.0337

0.0322

0.0297

0.0264

0.0368

0.0357

0.0348

0.0324

Means and standard deviations (volatilities) of the actual and model-implied Eurodollar and Europound zero-coupon yields are presented. The data on Eurocurrency zero-coupon yields are constructed from end-of-month ask quotes of Eurocurrency deposit rates, which are taken from Data Resources, Inc. The model-implied yields are 1-month-ahead forecasting values based on the prediction densities of unobserved state variables recovered in estimation. Reported are results from three models: 3-factor model and 5-factor model estimated using both interest rates and dollar–pound exchange rate data, and 3-factor model estimated using interest rates data only. The sample is monthly over the period from February 1980 through December 1998.

5-factor model has a lot more parameters to be estimated. Its inferior performance is likely due to over-parameterization. 6 The models’ performance in accounting for the dollar–pound exchange rate dynamics is reported in Table 2, right panel. Again, the data show that exchange rates are highly volatile. The actual average monthly pound returns is only −1:71% during the sample period, while the return volatility (standard deviation) is 40.07%. The 3-factor model performs really well, producing an average monthly pound return of −1:74% and return volatility of 29.45%. The 5-factor model produces an average monthly return of −1:4% and return volatility of 36.77%. So the 5-factor model does help improve the >tting of return volatility. Overall, these model-generated moments of pound returns are very close to the actual data. The upper panel of Fig. 3 plots the time series of dollar–pound exchange rate (in dotted line) versus that of contemporaneous >ltered values of the 3-factor model (in solid line). During the >rst half of the 1980s when Britain engaged in the Falkland Islands War, the pound lost half of its value, with the exchange rate diving from 6 The 5-factor model does not seem to have an over-parameterization problem in the US–German case. It is perhaps because we have more Euromark yield data obtained from a larger time horizon.

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A.C. Inci, B. Lu / Journal of Economic Dynamics & Control 28 (2004) 1595 – 1624 Dollar-Pound Exchange Rate 2.4

actual model-implied

2.2

2

1.8

1.6

1.4

1.2

Feb80

Sep84

Jun89

Mar95

Dec98

Annualized 1-Month Pound Return 1.5

actual model-implied

1

0.5

0

-0.5

-1

-1.5 Sep84

Jun89

Mar95

Dec98

Fig. 3. Dollar–pound exchange rate and monthly pound return: actual versus >ltered values from three-factor model.

A.C. Inci, B. Lu / Journal of Economic Dynamics & Control 28 (2004) 1595 – 1624

1615

around $2.4 to $1.1 per pound. It regained some of its value in the second half of the 1980s. The pound depreciated in 1992, during which Britain left the EMS. The pound remained weak against the dollar throughout the rest of 1990s in our sample period. The 3-factor model explains the dollar–pound exchange rate dynamics well. The model >tted values track the actual values very closely. The lower panel Fig. 3 plots the actual pound monthly returns (in dotted line) versus those of the 3-factor model (in solid line). It clearly shows that the model >tted values are always correct in the direction and magnitude of the exchange rate changes. Fig. 4 provides the same two plots for the 5-factor model, which tracts the actual dollar–pound exchange rate even more closely. Finally, Table 3 lists Fama’s conditions for both short-term interest rates (1-month yields) with short forecasting horizon (1-month) and long-term interest rates (5-year yields) with long forecasting horizon (5-year). Again, the Fama’s conditions are met for the short-term interest rates with short horizon but not met for the long-term interest rates with long horizon. This is again consistent with the results in Meredith and Chinn (1998) and Alexius (2000). 4.2.2. Parameter estimates and factor structure Table 6 presents parameter estimates with standard errors for the 3- and 5-factor models in the US–UK case. The correlation structure of state variables is reKected in the estimates of correlation coeOcients .ij , i; j = 1; : : : ; 5. For the 3-factor model, the >rst and third state variables are negatively correlated with a correlation coeOcient of −0:2819, the second and third state variables are positively correlated with a correlation coeOcient of 0.2290. Both correlations are statistically signi>cant at 5% level. The >rst and second state variables are negatively correlated with a coeOcient of −0:2041, but that is not statistically signi>cant. For the 5-factor model, the fourth state variable is positively correlated with the >rst, second, and >fth state variables, and the second and third state variables are positively correlated. All of these correlation coeOcients are highly statistically signi>cant. In both 3- and 5-factor models, no state variable is found to be independent of the others. Thus, again, it is important to allow for a Kexible correlation structure among the factors in these models. The state variables have di?erent impacts on the pricing kernels of the two currencies. This is reKected in the magnitudes of i coeOcients in the UK pricing kernel. Recall that the corresponding coeOcients in the US pricing kernel are set equal to ones. For the 3-factor model, the >rst two state variables (with 1 =4:7057 and 2 =3:9432) have larger impacts in the UK than in the US market, while the third state variable (with 3 = 0:1554) is a more important risk factor in the US market. For the 5-factor model, the third and fourth state variables have larger impacts on the UK market, while the >rst, second, and >fth ones have larger impacts on the US market. Overall, we have found similar results in the US–UK case to those in the US– Germany case. The model performs very well in accounting for the dynamics of the dollar–mark and dollar–pound exchange rates. Moreover, the results suggest that movements in those exchange rates and Eurocurrency interest rates are mostly driven by global common factors that a?ect all those countries. Local factors do not seem to be important.

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actual model-implied

2.2

2

1.8

1.6

1.4

1.2

Feb80

Sep84

Jun89

Mar95

Dec98

Annualized 1-Month Pound Return 1.5

actual model-implied

1

0.5

0

-0.5

-1

-1.5 Sep84

Jun89

Mar95

Dec98

Fig. 4. Dollar–pound exchange rate and monthly pound return: actual versus >ltered values from >ve-factor model.

A.C. Inci, B. Lu / Journal of Economic Dynamics & Control 28 (2004) 1595 – 1624

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Table 6 Parameter estimates of the US–UK models 3-Factor model

5-Factor model

State variable dynamics -1 1 1 -2 2 2 -3 3 3 -4 4 4 -5 5 5 .12 .13 .14 .15 .23 .24 .25 .34 .35 .45

0.0267 0.5125 0.0401 0.4848 0.2138 0.0240 0.3627 0.6016 0.0566 — — — — — — −0:2819 −0:2041 — — 0.2290 — — — — —

0.0628 1.0819 0.0734 0.3071 0.4713 0.0814 0.9359 0.1055 0.0956 0.0541 0.4091 0.0244 0.6848 0.1091 0.0171 0.2568 −0:0360 0.7297 0.0993 0.4246 0.4360 0.1780 0.0319 −0:0991 0.4427

UK pricing kernel parameters gf 1 2 3 4 5

0.0938 4.7057 3.9432 0.1554 — —

US pricing kernel parameters gd

(0.0088) (0.7288) (0.0515) (0.0214) (0.0410) (0.0044) (0.0251) (0.0422) (0.0043)

(0.1405) (0.1855) (0.1146)

(0.0072) (10.2240) (1.1924) (0.0895)

0.0916 (0093)

(0.0090) (0.4033) (0.0192) (0.0277) (0.0729) (0.0057) (0.0874) (0.0195) (0.0061) (0.0089) (0.2939) (0.0186) (0.0400) (0.0183) (0.0029) (0.2311) (0.1423) (0.1273) (0.1553) (0.1177) (0.1489) (0.1613) (0.1189) (0.0967) (0.1129)

0.1147 (0.0149) 0.0139 (0.1274) 0.0133 (0.0622) 12.5531 (17.7566) 8.6265 (2.3646) 0.0034 (0.0001) 0.1045 (0.0166)

Parameter estimates of three- and >ve-factor models on the term structures of Eurodollar and Europound deposit rates and dollar–pound exchange rate are given. Standard errors of the estimates are reported in parentheses.

4.3. Can term structure factors alone explain currency movements? Many economic factors a?ect exchange rate movements. These include money supplies, inKation rates, economic growth rates, and trade variables in the domestic and foreign economies. The term structure of interest rates has been shown to predict the future inKation rate and real GDP growth rate. Intuitively, this is why the term structure

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Table 7 Parameter estimates of three-factor models without exchange rates data US–Germany

US–UK

State variable dynamics -1 1 1 -2 2 2 -3 3 3 .12 .13 .23

0.0838 0.3637 0.0877 0.4612 0.1735 0.0527 0.1341 1.1541 0.0668 0.3932 0.2719 0.2788

(0.0042) (0.0477) (0.0053) (0.0101) (0.0108) (0.0035) (0.0057) (0.0706) (0.0037) (0.0551) (0.0651) (0.0647)

0.3636 0.4930 0.0573 0.8369 0.0300 0.0057 0.2225 0.4575 0.0585 −0:0753 0.0400 0.4375

Foreign pricing kernel parameters gf 1 2 3

0.1196 4.1938 4.6350 0.1312

(0.0023) (0.4673) (0.5040) (0.0176)

0.1023 (0.0131) 0.0522 (0.1920) 70.0570 (245.7675) 2.4988 (0.5148)

U.S. pricing kernel parameter gd

0.1230 (0.0091)

(0.0057) (0.0989) (0.0247) (0.0047) (0.6664) (0.0481) (0.0099) (0.2586) (0.1364) (0.0181) (0.1706) (0.1736)

0.0891 (0.0059)

The table presents parameter estimates of three-factor models on the joint term structures of Eurodollar– Euromark and Eurodollar–Europound interest rates, respectively. The dollar–mark and dollar–pound exchange rates are not used in estimation. Standard errors of the estimates are reported in parentheses.

of interest rates should contain information that helps account for exchange rate movements. It also raises an interesting empirical question: can term structure factors alone explain exchange rate movements? To address this issue, we re-estimate the 3-factor US–German and US–UK models using interest rate data only. Exchange rate data is excluded from the observation equation of the state-space representation. The state variables recovered in this estimation are term structure factors only. We then use these recovered state variables to construct exchange rates. Comparing the model performance in this case with the previous estimation results using both interest rate and exchange rate data gives us a clear answer to the question. The parameter estimates of the models are provided in Table 7. The upper panels of Figs. 5 and 6 plot the actual dollar–mark exchange rates and monthly mark returns (in dotted line) and the model generated values from the 3-factor model (in solid line). Based on the term structure factors alone, the contemporaneous model values have consistently underestimated the dollar value of mark throughout most of the sample period. The lower panels of Figs. 5 and 6 plot the actual dollar–pound exchange rates and pound returns (in dotted line) and the one-month-ahead predictions from the 3-factor

A.C. Inci, B. Lu / Journal of Economic Dynamics & Control 28 (2004) 1595 – 1624

1619

Dollar-Deutsche Mark Exchange Rate 0.7

actual model-implied

0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 Feb74

Apr80

Jun86

Sept92

Dec98

Dollar-Pound Exchange Rate 2.4

actual model-implied

2.2

2

1.8

1.6

1.4

1.2

Feb80

Sep84

Jun89

Mar95

Dec98

Fig. 5. Dollar–mark and dollar–pound exchange rates: actual versus >ltered values from three-factor models estimated without exchange rates

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A.C. Inci, B. Lu / Journal of Economic Dynamics & Control 28 (2004) 1595 – 1624 Annualized 1-Month DM Return 2.5

actual model-implied

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2 Apr80

Jun86

Sept92

Dec98

Annualized 1-Month Pound Return 1.5

actual model-implied

1

0.5

0

-0.5

-1

-1.5 Sep84

Jun89

Mar95

Dec98

Fig. 6. Monthly mark and pound returns: actual versus >ltered values from three-factor models estimated without exchange rates.

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model (in solid line). Contrary to the US–German case, the model generated >lter values have over-estimated the dollar value of pound throughout most of the sample period. The worst performance is seen in early 1980s, when the pound plunged by more than 50% while the term structure factors predicted a much more stable value of the pound. The plunge was likely caused by the Falkland Islands War, which obviously did not have as large an e?ect on the US–UK interest rate di?erentials. In summary, the term structure of interest rates contains useful information for exchange rate movements, but the term structure factors alone cannot explain exchange rate dynamics. Other economic factors such as trade variables are also very important for accounting for exchange rate dynamics. Moreover, large one-time shocks to an economy also have much larger impacts on the value of currency than the interest rates in the country.

4.4. Forecast accuracy comparison with other models We further compare the empirical performance of our model against the aOne models in Backus et al. (2001), the uncovered interest rate parity model, and the random walk model of exchange rates. We follow Inoue and Killian (2003) in evaluating the forecast performance of the term-structure models based on the Schwarz Information Criterion. Among the models we examine, our model and Backus et al. (2001) are models of both term structure of interest rates and exchange rates, while the uncovered interest parity and random walk models are on exchange rates only. We conduct two comparisons using the Schwarz Information Criterion, which penalizes the use of more parameters. First, we compare the performance of our model with that of Backus et al. (2001) in >tting both the interest rates and exchange rates. To do this, we estimate the best-performing model in Backus et al. (2001), the 2-factor aOne model with interdependent factors, using our interest rate and exchange rate data and the extended Kalman >lter, which generates a full log-likelihood value. We then compute the Schwarz Information Criteria for our model and Backus et al. (2001). The criteria scores are listed in Panel A of Table 8. Both of our 3- and 5-factor models beat that of Backus et al. (2001). Second, we compare our model with all three alternatives: Backus et al. (2001), the uncovered interest rate parity model, and the random-walk model. Although our model and that of Backus et al. (2001) use more parameters in the estimations and therefore have greater Kexibility, they estimate not only the exchange rate but also domestic and foreign bond yields with a number of maturities. To provide a fair comparison, we regressed actual exchange rate on each model’s predicted exchanged rate and determined Schwarz Information Criterion scores from the errors of these regressions. The results are reported in the second part of Table 8. The Schwarz Information Criterion values of our 3-factor and 5-factor models are lower than those of uncovered interest parity and Backus et al. (2001), indicating better empirical performance in >tting the exchange rates. However, they are still higher than those of the random walk model.

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Table 8 Comparison of model predictive power with Schwarz information criterion US–Germany Rank

Model

US–UK SIC values

Rank

Model

SIC values

Panel A. Interest rate and exchange rate estimation: full-likelihood values 1 5-Factor model −75:2 1 5-Factor model 2 3-Factor model −68:4 2 3-Factor model 3 BFT −47:3 3 BFT

−37:6 −35:1 −25:2

Panel B. Exchange rate estimation 1 RW −6:8303 2 3-Factor model −6:7881 3 5-Factor model −6:7871 4 UIP −6:1049 5 BFT −5:7413

−6:7775 −6:6999 −6:4160 −6:1228 −5:3728

1 2 3 4 5

RW 5-Factor model 3-Factor model UIP BFT

Panel A presents Schwarz Information Criteria (SIC) based on the full likelihood models (3-, 5-factor models and interdependent 2-factor model of Backus et al.) where not only the exchange rates but also domestic and foreign yields are estimated. Panel B compares SIC of 3-factor and 5-factor models of the study with the benchmark models of random walk and uncovered interest parity and Backus et al. SIC calculation is based only on exchange rate forecasting. Models are ranked based on their SIC, the lower the SIC the better the model. RW: Random walk, UIP: Uncovered interest parity, BFT: Interdependent 2-factor ∗ aOne model of Backus et al. (2001). The SIC values are computed using log(uˆ u=T ˆ ) + k ∗ ln(T )=T , where uˆ is the di?erence between actual and model estimated exchange rates, T is sample size, and k is the number of parameters estimated in the model. The lower the SIC, the better the overall forecasting performance of the model. The sample is monthly over the period from January 1974 through December 1998 for US–Germany and from February 1980 through December 1998 for US–UK.

5. Conclusion In this paper, we aim to build an international term structure model that can capture the exchange rate dynamics well. The economic justi>cation for the simultaneous modeling of international interest rates and exchange rates is that the term structure of interest rates contains information on future inKation rates and economic growth rates, both of which are important determinants of currency values. Our empirical studies show that the model with either three or >ve global factors can very well explain the contemporaneous movements of dollar–mark and dollar–pound exchange rates, as well as the mark and pound returns. We have shown that our model has better empirical performance compared to some existing models such as the uncovered interest rate parity model and the aOne models in Backus et al. (2001). However, our model still fails to outperform the random-walk model of exchange rates by the Schwarz Information Criterion. Thus, the current model is unlikely to add value to exchange rate forecasting. Improved predictive accuracy may require a multi-country model. Hodrick and Vassalou (2002) follow such an approach with an aOne model and >nd mixed results. Perhaps, the quadratic model here can be extended following their approach.

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Finally, the introduction and active trading of the euro neither changes nor diminishes the results presented in the study. For practical purposes, the pound is not part of the euro; the mark is the only currency, which is directly a?ected with euro. Given the economic power and >nancial inKuence of Germany in the European Union, one can reasonably assume a strong tie between the euro and the German mark. Therefore many of the results and conclusions of this paper may apply to the euro. A more systematic analysis of the euro is warranted for future research when more data becomes available. Acknowledgements We would like to thank Gautam Kaul, Nejat Seyhun, two anonymous referees, the editor Wouter den Haan, and seminar participants at the University of Michigan Business School for helpful comments. References Ahn, D.H., 1997. Common factors and local factors: implications for term structures and exchange rates. Working paper, Kenan-Flagler Business School, University of North Carolina, Chapel Hill, NC. Ahn, D.H., Dittmar, R.F., Gallant, A.R., 2001. Quadratic term structure models: theory and evidence. Review of Financial Studies 15, 243–288. Alexius, A., 2000. UIP for short investments in long-term bonds. Sveriges Riksbank Working Paper Series, 115. Amin, K., Jarrow, R.A., 1991. Pricing foreign currency options under stochastic interest rates. Journal of International Money and Finance 10, 310–329. Anderson, B.D., Moore, J.B., 1979. Optimal Filtering. Prentice-Hall, Englewood Cli?s, NJ. Backus, D., Foresi, S., Telmer, C., 1995. Interpreting the forward premium anomaly. Canadian Journal of Economics 28, S108–S119. Backus, D., Foresi, S., Telmer, C., 2001. AOne term structure models and the forward premium anomaly. Journal of Finance 56, 279–304. Bakshi, G.S., Chen, Z., 1997. Equilibrium valuation of foreign exchange claims. Journal of Finance 52, 799–822. Basal, R., 1997. An exploration of the forward premium puzzle in the currency markets. Review of Financial Studies 10, 369–403. Chen, R., Scott, L., 1993. Maximum likelihood estimation for a multifactor equilibrium model of the term structure of interest rates. The Journal of Fixed Income 3, 14–31. Constantinides, G.M., 1992. A theory of the nominal term structure of interest rates. Review of Financial Studies 5, 531–552. Dai, Q., Singleton, K.J., 2000. Speci>cation analysis of aOne term structure models. Journal of Finance 55, 1943–1978. Du?ee, G., 2002. Term premia and interest rate forecasts in aOne models. Journal of Finance 57, 405–444. DuOe, D., Kan, R., 1996. A yield-factor model of interest rates. Mathematical Finance 4, 379–406. DuOe, D., Singleton, K.J., 1997. An econometric model of the term structure of interest rate swap yields. Journal of Finance 52, 1287–1323. Fama, E., 1984. Forward and spot exchange rates. Journal of Monetary Economics 14, 319–338. Hodrick, R., Vassalou, M., 2002. Do we need multi-country models to explain exchange rate and interest rate and bond return dynamics? Journal of Economic Dynamics and Control 26, 1275–1299. Inoue, A., Killian, L., 2003. On the selection of forecasting models. European Central Bank Working Paper Series, 214.

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