AN EMPIRICAL INVESTIGATION

This paper is an empirical investigation of the predictability and comovement of risk premia in the term structure of Euromarket interest rates. We show that variables which have been used as proxies for risk premia on uncovered foreign asset positions also predict excess returns in Euroniarket term structures, while variables which have been used as proxies for risk premia in the term structure also predict excess returns on taking uncovered foreign asset positions. These findings suggests that risk premia in the Euromarket term structures and on uncovered foreign asset positions move together. We test formally the hypothesis that risk premia on uncovered 3-month EuroDM and Eurosterling deposits move in proportion to a single latent variable. We are unable to reject this hypothesis. We are also unable to reject the hypothesis that the risk premia on these three strategies and those on rolling over 1-month Eurosterling (EuroDM) deposits versus holding a 3-month Eurosterlirig (EuroDN) deposit move in proportion to a single latent variable. The single latent variable model can be interpreted atheoretically, as a way of characterizing the extent to which predictable asset returns "move together"; or it can be interpreted as in Hansen and Hodrick (1983) and Hodrick and Srivastava (1983) as a specialization of the ICAPM in which assets have constant betas on a single, unobservable benchmark portfolio. John V. Campbell Richard H. Clarida Department of Economics The Cowles Foundation Dickinson Hall Yale University Princeton University Box 2125, Yale Station Princeton, NJ 08544 New Haven, CT 06520 THE TERN STRUCTURE OF EUROMARKET INTEREST RATES: AN EMPIRICAL INVESTIGATION John Y. Campbell and Richard H. Clarida


I. Introduction
There is by now a substantial body of empirical work on the predictability of excess returns across assets of the same maturity denominated in different currencies. Recent studies by Geweke and Feige (1979), Hansen andHodrick (1980, 1983), Cumby andObstfeld (1981, 1982), Hakkio (1981a), and Hodrick and Srivastava (1983) have rejected the "Fisher open" hypothesis that interest rate differentials on assets of the same maturity denominated in different currencies are offset by an expected exchange rate depreciation and the hypothesis, equivalent given covered interest arbitrage, that forward exchange rates are efficient predictors of future spot rates. The source of these rejections is that information variables such as the percentage forward premium [Hodrick and Srivastava (1983)) or lagged forward rate forecast errors [Hansen and Hodrick (1980)] help to predict realized excess returns on uncovered foreign asset positions. Hansen and Hodrick (1983) and Hodrick and Srivastava (1983) have tested whether or not the time varying risk premia which separate the dollar prices of 1-month forward contracts in several currencies from expected future spot exchange rates are driven by a single latent variable. This latent variable 1 is interpreted in the context of the intertemporal capital asset pricing model (ICAPM) as the expected dollar return on some benchmark portfolio in excess of the nominally riskiess 1-month dollar rate) Hansen and Hodrick (1983) were unable to reject the hypothesis that risk premia on 1-month yen, mark, pound, Swiss franc, and French franc forward contracts all move in proportion to a single latent variable. By contrast, using a somewhat longer data series which included an additional twenty-one months of data ending in September 1982, Hodrick and Srivastava (1983) rejected the single latent variable model for the same five contracts.2 The expectations theory of the term structure of 'on shore' U.S. interest rates has also been rejected in recent empirical work. Shiller, Campbell, and Schoenholtz (1983) and Mankiw and Summers (1984) have shown that the spread between 6-and 3-month treasury bill rates has predictive power for excess returns in the U.S. terni structure. Campbell (1985) has decisively rejected the hypothesis that the time varying risk premia on 2-month treasury bills, bonds, and stocks over 1-month bills all move in proportion to a single latent variable. This paper is an empirical investigation of the predictability and comovement of risk premia in the term structure of Euromarket interest rates. Our approach may be described as follows. We choose the dollar as our numeraire currency and focus on four alternative strategies for obtaining dollar payoffs three months hence. The first is to purchase a 3-month Eurodollar deposit, a nominally riskless investment under our assumptions.
The second is to roll over 1-month Eurodollar deposits. The third is to purchase a 3-month foreign currency Eurodeposit and to sell the proceeds received three months hence at the then prevailing spot exchange rate. The 2 fourth is to purchase a 1-month foreign currency Eurodeposit, reinvest the proceeds in 1-month Eurodeposits in the same currency, and to sell the proceeds accumulated after three months at the then prevailing spot exchange rate. The difference in Post excess returns on the latter two strategies is equal to the realized foreign currency return on holding a 3-month foreign currency Eurodeposit versus rolling over three 1-month deposits denominated in the same currency. Each of these latter three strategies generates a risky dollar payoff three months hence. We establish that the excess returns on these risky strategies relative to the 3-month Eurodollar deposit rate are predictable given information such as the spreads between 3-and 1-month Eurodeposit rates and the differentials between 3-month foreign currency Eurodeposit rates and the 3-month Eurodollar rate. The main task of the paper is to test whether or not the time varying risk premia on each of these risky strategies move in proportion to a single latent variable. This hypothesis can be interpreted atheoretically, as a way of characterizing the extent to which predictable asset returns "move together"; or it can be interpreted as in Hansen and Hodrick (1983) and Hodrick and Srivastava (1983) as a specialization of the ICAPM in which assets have constant betas on a single, unobservable benchmark portfolio.
The data are restricted in this framework with as few as two asset returns and two information variables.
The first contribution of this paper is to show that variables which have been used as proxies for risk premia on uncovered foreign asset positions also predict excess returns in at least two Euromarket term structures. In addition, varia'bles which have been used as proxies for risk premia in the term structure also predict excess returns on taking uncovered foreign asset positions. These findings suggest that risk premia in the Euromarket term structures an on uncovered foreign asset positions move together. We first test the hypothesis that risk premia on uncovered 3-month EuroDM and Eurosterling deposits and on rolling over 1-month versus holding 3-month Eurodollar deposits move in proportion to a single latent variable. We are unable to reject this hypothesis at standard levels of significance. We next test whether or not the risk premia on these three strategies and those on rolling over 1-month Eurosterling (EuroDM) deposits versus holding a 3-month Eurosterling (EuroDM) deposits move in proportion to a single latent variable. Again, we are unable to reject this hypothesis at any reasonable level of significance.
The plan of the paper is a.s follows. Section II discusses the data series used and the construction and interpretation of the relevant excess return variables. This section also shows how we construct proxies for risk premia, and summarizes the behavior of these information variables over the sample period. Section III provides an overview of the estimation and testing strategies employed in the analysis. Section IV investigates the existence and nature of predictable excess returns in the Eurodollar term structure and on uncovered 3-month EuroDM and Eurosterling deposits. A formal test of the proportionality of these predictable excess returns with a single latent variable is presented. The section concludes with a similar investigation of the predictability and coniovement of excess returns on these three strategies and those on rolling over 1-month Eurosterling (EuroDM) deposits versus holding a 3-month Eurosterling (EuroDM) deposit.
Section V provides some concluding remarks.

II. Construction and Interpretation of Variables
The data set used in this paper consists of weekly observations on DM and sterling spot exchange rates relative to the dollar, and on 1-and 3-month Eurodollar, EuroDM, and Eurosterling deposit rates.3 From these data, we construct excess returns on three alternative investment strategies relative to a baseline investment in a 3-month Eurodollar deposit. Each alternative requires a dollar investment and yields a risky dollar payoff after three months.
The log of the 3-month gross return on the baseline investment, undertaken at time t , is log(l + R3(us)/400) where R3(us) is the 3-month Eurodollar deposit rate quoted in percentage points per year at time t .
In forming excess returns, we subtract log gross returns on alternative investments from the baseline and simplify by using the approximation 4 . log(l+x)x . Finally we multiply by 400 to express excess returns in units of percentage points per year.
The first alternative investment strategy we consider is that of rolling over 3 successive 1-month Eurodollar deposits. The log gross return on this strategy is log[(1 + R1 t(us)/l200)(l + Ri+1(us)/1200) •(l + Rit+2(us)/1200)] . The approximate excess return on the baseline over the dollar rollover strategy, in percentage points per year, is The rational expectation at time t of this excess return is called the "rolling premium" on a 3-month Eurodollar deposit over 1-month Eurodollar deposits [Campbell and Shiller (1984)]. According to the "pure expec-tations theory" of the terni structure, the rolling premium is zero. A slightly weaker hypothesis, which we call the "expectations theory" and test in this paper, is that the rolling premium is constant through time at some The "Fisher open" hypothesis (uncovered interest parity) states that the rational expectation of (2) at time t is zero, or at least constant through time.
Finally, we consider mixed strategies which involve rolling over 1-month EuroDM or Eurosterling deposits. Returns on these strategies are uncertain both because future 1-month Eurodeposit rates are uncertain, and because future spot exchange rates are uncertain.7 The log gross return on a mixed DM strategy is The approximate excess return on the baseline over the mixed DM strategy, in percentage points per year, is 3 2 )4 '(us,g) - The difference between (2') and (2) is just the excess return in the DM term structure, the German equivalent of (1). In general we have, Recent theoretical work on asset pricing restricts the behavior of the excess returns defined in equations (1), (2) arid (3). Formulated in discrete time with a representative agent with time separable utility u defined over consumption c , the intertemporal capital asset pricing models of Merton (1973), Lucas (1978Lucas ( , 1982, Breeden (1979), and Hansen, Richard, and Singleton (1981) imply that for any asset i , the gross dollar return from time t to time t+n , 1 + h'(i) , must satisfy the first-order condition Here denotes expectation conditional on the representative agent's information set at time t and is the marginal rate of substitu-tion of dollars between t+n and t . In a one good world and in the absence of cash-in-advance constraints where S is a discount factor and is the dollar price of the consumption good at date t (Hansen and Hodrick (1983) and Mark (1985)). In a more complex model, with multiple goods or cash-in-advance constraints, equation (5) would have to be generalized but a relation of the form (4) would remain valid.
In our context, (4) implies that, for example, which is obtained by subtracting relation (4) for the DM rollover strategy from relation (4) for the baseline Eurodollar investment strategy.
Rearranging the expectation of the product into the sum of the covariance and the product of expectations, (6) implies Thus the risk premium is proportional to the conditional covariance of the interteniporal marginal rate of substitution of dollars and the excess dollar return on buying DM, rolling over 1-month EuroDM deposits, and selling the accumulated proceeds back for dollars at the exchange rate prevailing three months hence. The other risky returns studied in this paper are priced analogously in the ICAPM framework. Now consider a "benchmark" portfolio defined to have nominal return Clearly, for any asset i The nominally riskiess rate must obey (1 + R3 (us)) This can be rewritten as This relationship must hold for h3(i) -h3(b) , so we can substitute out E{l + h3(b)] to obtain R3(us) -E[h3(i)) 1ER3(us) -Eh3(b)I (12) or in more compact notation, Thus, the expected dollar excess return on asset i is proportional to the expected excess return on a benchmark portfolio whose payoff is perfectly conditionally correlated with the intertemporal 9 marginal rate of substitution of dollars. Although the betas can in general vary over time, our econometric work will test restrictions implied by constant betas. As pointed out by Hansen and Hodrick (1983), such tests are not tests of a fully specified general equilibriun model, but are tests of proportional co-movement of expected excess returns which are motivated by the ICAPM.
In Section IV, we test the hypotheses that the rational expectations of excess returns are constant through time by regressing excess returns on a constant and "information variables" which are known at time t , and testing the joint significance of the information variable coefficients.
If such tests are to be powerful against a broad range of alternatives, the information variables used should be ones which plausibly predict excess returns when the expectations and Fisher open hypotheses are false.
Some authors have used lagged excess returns [Hansen and Hodrick (1980)], thereby conducting a "weak-form" test. We adopt a different approach, using as information variables elements of excess returns themselves, as defined in equations (1), (2) and (3), which are known at time t The element of (1) which we use for our tests is R3(us) -R1(us) the spread between 3-month and 1-month Eurodollar interest rates. Using the linear approximation for the term structure, the Eurodollar spread can be decomposed into When the expectations theory of the term structure holds, the first term on the right hand side of (13) is a constant; unusually high longer term interest rates are then due entirely to expectations of short rate increases. However if there is any variation in the rolling premium, it will be reflected in the spread so that this variable should be a good instrument for our empirical work.
The element of (2) we use is R3(us) -R3 the interest differential between 3-month Eurodollar and EuroDM and Eurosterling rates.8 Following Fama (1984), we can decompose this variable into the sum of the cross-currency risk premium, and the expected rate of depreciation of the dollar relative to currency j When uncovered interest parity holds, the first term on the right hand side of (14) is constant, and unusually high differentials are then due entirely to expected dollar depreciation. However in general variation in the crosscurrency risk premium will be reflected in the differential, so this is also a good instrument for our purposes.
One final set of information variables corresponds to equation (3). We include foreign currency term structure spreads, R3(i) -R1 which can be decomposed in the manner of equation (l3).
If our tests are to be powerful, it is also desirable that our information variables should not be too highly correlated. The inclusion of variables which are highly collinear with other instruments is likely to use up degrees of freedom in estimation and testing, without contributing to the predictability of returns. To check for this, and to review the behavior of interest rates over our sample, we present summary statistics for information variables in Table 1.
The first row of Table 1 reports the mean Eurodollar spread, the mean differentials between 3-month Eurodollar and foreign currency rates, and the mean foreign currency spreads. Mean spreads in all currencies are positive but small, ranging from just over 5 basis points in the UK to almost 13 basis points in Germany.
The mean Eurodollar rate is more than 420 basis points higher than the mean EuroDM rate, but almost 175 basis points lower than the mean Eurosterling rate. The second row of Table 1 reports standard deviations, which are all quite large relative to the nieans)°T he bottom panel of Table 1 shows the correlations of the various instruments. The highest correlation, 0.646, is between the two differentials; evidently movements in US interest rates relative to the other two currencies account for a large part of the variation in the differentials.
There are positive but fairly small correlations among the spreads in all three currencies. None of the correlations are high enough to suggest that we should drop any of our information variables.

III. EcrQmetric Issues
Most of the empirical work of this paper is conducted within a standard regression framework. The major econometric difficulty in our application is one which is by now familiar from the work of Hansen and Hodrick (1980) and others; we have weekly observations but a 3-month (13-week) holding period, so the error terms in our regressions follow an MA-12 process under 11 the null hypothesis.
Serial correlation in the equation error does not affect the consistency of OLS coefficient estimates, T but it does require an adjustment in the estimated variance-covariance matrix of these estimates. Hansen and Hodrick (1980), following Hansen (1979), propose the following consistent estimator of the variance-covariance matrix of Here is the matrix of independent variables at sample size T , and the ()th element of T WT(1,j) , is defined by where k -i-j and n -12 (the order of the MA process).
The above estimator does not allow for conditional heteroskedasticity in the error terni, but it is straightforward to do so, using the method of White (1984) and following Hsieh (1984) and Cuniby and Obstfeld (1984), by redefining w(i.j)uu for k n and -0 otherwise.
One problem with both Hansen and Hodrick's estimator and its heteroskedasticity-consistent variant, is that need not be positive definite in finite samples. Thus use of 8 in practice may result in negative "CM-squared" test statistics or even negative standard errors; problems may also arise with nonlinear iterative estimation methods of the type we use to estimate the latent variable models. Newey and West (1985) propose a simple modification of which is consistent and constructed to be positive definite.
Allowing for condi-tional heteroskedasticity, this is cz)T(i,j) y(k,m)uu. for k in ; and = 0 otherwise, where -y(k,m) 1 -(k/(m+1)) and inas T -, , but no faster than the fourth root of T However, as shown in the second row of Table 5, the spreads between 3-and 1-month EuroDM and Eurosterling deposit rates both have significant explanatory power for excess returns on uncovered 3-month EuroDM deposits. To summarize, variables which have been used as proxies for risk premia on uncovered foreign asset positions also predict excess returns in the Eurodollar term structure, while variables which have been used as proxies for risk premia in the term structure also predict excess returns on taking uncovered foreign asset positions.
These findings suggest that risk premia in the Eurodollar term structure and on uncovered foreign asset positions move together. We now test formally the hypothesis that risk premia on uncovered 3-month EuroDM and Eurosterling deposits and on rolling over 1-month versus holding 3-month Eurodollar deposits move in proportion to a single latent variable.
As discussed in Section II, the ICAPM places the following restrictions on the excess returnof a particular asset i This equation is derived by projecting equation (12) onto a k-vector x of information variables, a subset of the market's full information set. The first element of x is just a constant.
is the "beta" of excess return i with the benchmark portfolio, defined as in equation (12) but now assumed to be constant through time. Since the expected excess return on the benchmark portfolio is unobservable, we substitute the best linear projection of this excess return on the variables in x , i.e., Consider now the implications of (16) and (17) for the system of p regression equations A-x+v (18) where A is a p-vector of excess returns, ' is a p by k matrix of regression coefficients with typical element , and V iS a p-vector of forecast errors which is orthogonal to The model (16) and (17)   (31) (33) The key excess returns are A (us,us) and A (us,uk) which are predictable in Table 3. The a coefficients in Table 4 are quite close to the unconstrained regression coefficients of A3' (us,us) in the first row of Table 3.
The constrained system must then fit A3'3(us,uk) Table 4. The estimated latent variable is most strongly correlated with the Eurodollar spread, at 0.697, but is also quite highly correlated with the two differentials.
In Tables 5 to 7 we repeat the above analysis for a large system of excess returns and information variables. The vector of excess returns is expanded to include the foreign term structure excess returns and ).3' (uk,uk) . We work with these excess returns, rather than the excess returns on ttmixed investment strategies defined in equation (2' Corresponding to the foreign term structure excess returns, we now include foreign terni structure spreads in our vector of information variables along with the dollar spread and the cross-currency interest differentials.
Regression results for the larger system are presented in Table 5. The two new information variables have considerable explanatory power for cross-currency excess returns on 3-month Eurodeposits, so that the excess return on the 3-month EuroDM investment strategy is now predictable along with the other excess returns. The excess return in the EuroDM term structure is highly predictable; by contrast the expectations theory of the term structure cannot be rejected in the Eurosterling term structure. In general there appear to be important cross-effects which suggest that once again a single latent variable model might fit the data well.
In Table 6 we test and are unable to reject the hypothesis that a single latent variable explains the results of Finally, in Table 7 we shrink the model to the first three rows of

VI. Conclusion
In this paper we document the existence of predictable time-varying excess returns on a variety of investment strategies, relative to a 3-month Eurodollar investment. The strategies we consider involve three currencies --the dollar, the DM and the pound sterling--and two maturities--3-and 1-month Eurodeposits.
We find not only that excess returns are predictable, but that there are important cross-effects. Variables which traditionally are used to predict term structure excess returns also have explanatory power for cross-currency excess returns, and vice versa. In fact a tightly specified model, constraining all expected excess returns to move in proportion with a single latent variable, cannot be rejected at any reasonable significance level in a variety of tests. The estimated latent variable is quite highly correlated with the spread between 3-and 1-month Eurodollar interest rates.
Our favorable results for the single latent variable model contrast with the rejections reported by Campbell (1985) for US domestic assets and by Hodrick and Srivastava (1984) for Eurocurrency assets. However Gibbons and Fersori (1985) and Hansen and Hodrick (1983), wrking with domestic and Eurocurrency data respectively, also failed to reject the model. An important topic for future research is to sort out these apparently conflicting results. It does seem to be the case that rejections occur when more heterogeneous assets are studied, and when a longer sample period is used.
Slow changes through time in relative betas of assets with widely divergent characteristics might explain this pattern of results. Our work should not be taken to imply that betas are constant for all assets in all sample periods, but merely as evidence that a simple latent variable structure describes our data.
These results are preliminary to a more structural investigation of the sources of predictable excess returns across currencies and maturities. If indeed predictable excess returns are driven by a single underlying latent variable, it seems plausible that conditional second moments of returns also move with this variable. We hope to explore the interrelationship between conditional first and second moments of returns in future work.        .195. Initial values are as in Table 6. See note to Table 4. are careful to point out, their approach to modelling is best construed as an interpretation of a parsimonious statistical representation of risk premia using intertemporal capital asset pricing theory.
In particular, their statistical tests are not tests of a fully specified equilibrium model.
However their rejection of the single latent variable model is not due to their different sample procedure.

3.
We are grateful to Richard Levich for supplying us with these data, derived originally from Harris Bank sources.
The data set also includes forward exchange rates and 6 and 12 month Eurodeposit interest rates, which we do not use here.

4.
This approximation is commonly used in empirical work on the term structure (Shiller, Campbell and Schoenholtz (1983), Mankiw and Summers (1984)).
For the short 3-month holding period considered in this paper, the approximation is extremely accurate since gross returns are very close to one when measured in natural units.

5.
Campbell and Shiller also discuss the "holding premium" on a long-term investment, sold before maturity, relative to a short-term investment. We do not consider holding premia in this paper since Eurodeposits are generally non-negotiable and must be held to maturity.

6.
It would also be possible to arrange at time t to convert the DM back to dollars at a known forward exchange rate. This would give a riskiess 3-month dollar return which by arbitrage ("covered interest parity") is always very close to the 3-month Eurodollar rate. Small deviations from covered interest parity in the Euromarkets appear to arise from transactions costs.

7.
Note that it would not be possible to fully hedge the exchange risk of a mixed strategy by engaging in a forward exchange transaction. This is because the number of DM which must be converted back to dollars at time t+3 depends on future 1-month DM rates, and thus is not known at time t. 8.
By covered interest parity, the interest differential is also the forward premium (400 times the log of the ratio of the 3-month ahead forward rate quoted at time t , to the time t spot rate). 9.
The decompositions of (17) and (18)  10. All the information variables have a fairly high degree of serial correlation, so the standard deviations should not be interpreted as if they come from independently and identically distributed variables.
11. This statement is true for regressions with cross-currency excess returns (2) as the dependent variable. With term structure excess returns (1) or (3) as the dependent variable, uncertainty is resolved after 2 months when the 2-month ahead 1-month Eurodollar rate becoses known, so the error process is only MA-8. We ignore this in our empirical work, thus obtaining conservative estimates of standard errors.
12. Alternative methods for constructing a positive definite T matrix have been developed by Hansen (1982) and Cumby, Huizinga and Obstfeld (1983).
However we found Newey and West's approach to be easier to apply.
13. However, for all three excess return, we can rejects at the 5% level the hypothesis that the first six autocovariances are jointly zero.
14. Point estimates of some of the coefficients in Table 3 are sensitive to the starting values used in the estimation, because these starting values help to determine the first-round weighting matrix in the GMM procedure. However, the Chi-squared test statistic is only minimally affected by starting values.