The Term Structure of Interest Rates in the Onshore Markets of the United States, Germany, and Japan

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 382

June 1990

The Term Structure of Interest Rates in the Onshore Markets of the United States, Germany, and Japan

Helen Popper

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.

ABSTRACT

This paper investigates term premia behavior in U.S., German, and Japanese markets. Onshore returns are evaluated in order to focus on the co-movement of the term premia across a set of potentially heterogeneous markets. The paper extends the work of Campbell and Clarida [1987], who find that the term premia within the Euromarket appear to move together. In keeping with their approach, Hansen and Hodrick’s [1983] latent variable model is ised. The model constrains expected returns, conditional on an information set, to be proportional to one another. These restrictions are not rejected for the markets examined here,

implying that the term premia behave as if in a single market.

The Term Structure of Interest Rates in the Onshore Markets of the United States, Germany, and Japan

Helen Popper! 1. Introduction

This paper investigates term premia behavior in U.S., German, and Japanese markets.” Onshore returns are evaluated in order to focus on the co-movement of the term premia across a set of potentially heterogeneous markets. The paper extends the work of Campbell and Clarida [1987], who find that the term premia within the Euromarket appear to move together. In keeping with their approach, the latent variable model of Hansen and Hodrick [1983] is used. > The model constrains expected excess returns, conditional on an information set, to be proportional to one another. These restrictions are not rejected for the markets examined here, implying that the term premia

behave as if in a single market.

1. The author is a staff economist in the International Finance Division. This paper represents the views of the author and should not be interpreted as reflecting the views of the Board of Governors o: the Federal Reserve System or other members of its staff. This research is based on a dissertation chapter, done at the University of California at Berkeley, and I am grateful to my thesis advisors, Roger Craine, Jeffrey Frankel, and Greg Connor for their comments. I would also like to thank Hali Edison, Neil Ericcson and the participants of the Federal Reserve System's October 1989 Conference on International Tirade and Finance for valuable discussion.

2. Turnovsky [1989] provides a detailed discussion of the relationship between macroeconomic policies and the term structure.

3. In contrast to Campbell and Clarida’s [1987] results, Campbell [1987] rejects this model for a broad range of domestic excess returns (including the term premium) in the United States. His rejection may reflect the heterogeneity of the assets chosen. A more homogeneous set of assets is chosen in this paper in order to focus on the issue oi; international capital market integration by eliminating one competing source of rejection of the model.

The next section of the paper discusses models of the term structure of interest rates and currency risk, and introduces the latent variable model. The empirical implications of the model are discussed in Section 3, and it is tested in Section 4. A summary is presented in Section 5. Some econometric issues and alternative test

statistic formulations are discussed in the appendix. 2. Models of the Term Structure

Both the pure expectations hypothesis of the term structure and the hypothesis of uncovered interest rate parity narrowly constrain an expected risky return to equal a nominally riskfree one: the expectations hypothesis equates the expected return to holding a series of short-term assets with a known long-term return, and uncovered interest parity equates a default-free domestic return with the expected home-currency return to an otherwise identical asset denominated in a foreign currency.“ Repeated empirical rejections of these hypotheses and their slightly less restrictive variants that allow for a non-zero constant risk premium imply that both the term premium and the exchange rate risk premium are non-zero and frequently

time-varying.> The latent variable model nests both hypotheses by

4. Shiller and McCulloch [1986] present simple relationships between numerous empirical formulations of the expectations hypothesis, including the roll-over term premium described here. Campbell, Shiller and Shoenholtz [1983] derive a discrete-time linearized version with which several theoretical formulations are consistent.

5. Tests of the expectations theory of the term structure are surveyed in Shiller and McCulloch [1986]. Hodrick [1987] surveys tests of the hypothesis that forward exchange rates are unbiased predictors of future spot rates. Both theories are rejected in most of the tests surveyed.

treat:iing them as restrictions on specific asset returns in a general mode]. of asset prices.

A number of representative-agent capital asset pricing models give rise to Euler equations restricting the relationships between asset prices and the marginal rate of substitution of nominal consumption between periods.° These relationships have been used to characterize returns to assets of differing maturities to give a generalized vers:.on of the expectations theory of the term structure. ’ Relating real and nominal variables, this version and the corresponding theory for exchange risk premia have been difficult to evaluate empirically. The Latent variable model provides an alternative specification using the construct of an unobservable return to derive testable rest:rictions on the relationships among only the observable asset returns, themselves.

The solution to the representative consumer's intertemporal optimization problem requires the expected marginal utility from the purchase of an asset to equal the marginal utility of the consumption foregone for its purchase. The Euler equation giving the relationship between the marginal rate of intertemporal substitution and an asset return embodies this requirement in the following expected product:

R la.) = 1.

E ¢ Qe tin * i t,tin t?

6. In these models, the representative agent maximizes some form of:

E( ,3) s*u(e

0 tri) | 9%).

subject to an intertemporal budget constraint, where & is a discount factor, 0 < 6 <1, and u is a concave utility function of period t consumption, ch: Lucas [1978] provides a general equilibrium version of the representative-agent asset pricing model, and Hansen and Hodrick [1983] explain its relation to the latent variable model in detail.

7. The version discussed here is a discrete-time version of Cox, Ingersol, and Ross [1985].

Here, Q ten is the intertemporal marginal rate of substitution of >

money; .R is the nth,

: asset's gross nominal return from period t i-t,ttn

to period t+n; and, a is the agent’s information at period t.° Hansen and Hodrick [1983] point out that for a pair of expected returns, the difference between the products equals zero. In terms of

a nominally risk-free return, R and any other asset i,

t,t+n’ subtracting one Euler equation from the other gives: EIQ tain’ ¢ ae tan” Re can) 9] - 9.

Denoting the excess return by ivt tun = iRe tin” R. tin’ the above e] ? ?

expression may be rewritten to show that the excess return is proportional to the return'’s covariance with the intertemporal

marginal rate of substitution of money: ”

ECTe tn!) = -cov, (Q )-R

t,tin’ it, ten? “et, tin’ (1)

where the factor of proportionality is the risk-free rate, R, tin’

Equation 1 characterizes the term premium and the exchange risk premium since both are examples of an expected difference between a risky and a nominally riskless return. The term premium, denoted Poni? is the expected difference between the return on the risky

strategy of rolling over n consecutive l-period assets and the

8. Qe tin weights the marginal rate of substitution of consumption by

the relative purchasing power of money. More specifically, Q = 6" - [u'(c,, y/u'(e_)] - 1/(p,,_/p,)

where: u'(c) fattfe marginal utifiky of cOnsumption, “8nd‘p, is the t period price of a unit of consumption

9. Since the risk-free rate, R. ttn’ is known, the original Eule: equation defines it implicitly, Ry ton= L/E(Q. an! 2) - This definition and the definition of d’Covariance give Expression 1.

th

nominally riskless, n-period domestic rate, R, ean?

) = EC ,fR R | a,).

t,n,l t+i-l,t+i “t,tin

Likewise, the exchange risk premium, denoted jit tin’ is the ’ expectied difference between the return to a domestic asset and the net return to exchanging domestic currency for foreign currency at the

spot rate, purchasing a foreign asset with return Re tin’ then re- ?

exchanging the gross amount into domestic currency at the period t+n

spot rate, Stan’

r = E [GR R Re

t,ttn t, ten’ St/Sttn? ~ St, tn tl:

Substituting these definitions into Equation 1 gives general

expressions for the term and the exchange risk premia:

)°R

ton,l 7 7OO% Ce ten? Hy’ eaa-1, t4i t,tn’ (2)

jit, tin ~ Cove (Qe tan » gic, cen’t/Stin) Re, cen’ (3)

The pure expectations hypothesis and uncovered interest parity restrict the covariances in Equation 2 and 3 to equal zero. More common empirical variants of the hypotheses restrict the covariances to be only time-invariant, rather than zero. Most empirical tests of both versions of the term premium hypothesis reject it, finding that

the term spread, R R known at period t, has predictive

t,ttn ~t,t+l’

value for the term premium. Similarly, uncovered interest parity is

often rejected empirically.

10. ‘the use of a nominally riskless rate is more restrictive than is

necessary. More generally, R tan, Bay have a zero-f return, where £

is the ratio of the return’s cdévariance with Q ten to its variance. ?

Without these rejected restrictions, direct empirical tests of

Equations 1 through 3 require evaluation of the nominal marginal rate

: . : 11 : of substitution of consumption, Q. tan’ The latent variable model 3

reformulates the Euler equations in terms of asset returns only, so that measures of consumption are not required. The model uses the

construct of a benchmark asset (or portfolio) with an unobservable

12

return, , perfectly correlated with Q Perfect

qet tin t,ttn’

correlation implicitly defines this asset’s return. }3 This allows the expected excess return of Equation 1 to be expressed as in terms of

asset returns. 4 Specifically, the excess return is proportional to

the expected excess return perfectly correlated with Qe ten’ Letting L = =R - R gives? t,tin q t,ttm t,ttn’ EGTe con!%) = Be iq eee, con lO? (4) where: Beri = COVE tan ght, tan? /8Fe6 GRe tan? and

Similarly, the term premium and the exchange risk premium are

proportional to the latent expected excess return, E(L |9,):

t,ttn

11. Hansen and Singleton [1983] and others have used consumption data to determine Q. ttn’ However, like most measures of real economic activity, observations of consumption are typically available infrequently and measured imprecisely relative to observations of asset prices.

12. Campbell [1987] uses the more general framework of k hedge portfolios rather than a single benchmark asset.

13. Equation 1 and the requirement that Corr(Q

together define R

Y=1 q t,ttn~ Qe cen EC? tan!)

t,tt+n’ qet, ten

14. With the riskfree rate, R

=1/E(Q |Q.), excess returns may be expressed: E(,r t,ttn! ot

t,t+n |.) = -éov, Gr qt tan) / EC Re tan! Me) -

t,t+n i-t,tin

15. Note R

get, tan ike, tin ? var Re cam GR, can Be, ton EGR e, c+ le) -

-7-

nl ~ Bacte.n,1),4 E(Le tan !9,), (3) gre, tin ~ Brig) .q (Le tan!) (6) where: Pacesn.1).q 7 COVE ab Resi-a, tai? qRt,cen ) / V@%e(qRe, tun’? 2nd Prii).q = cove GT tanSt/Stan’? get, ten ? 7 Vr GRe, t+?

Equations 4, 5, and 6 parallel the structure of Equations 1,2, and 2, where the expected excess return and the term premium are

Proportional to covariances with Q However, in the first

t,t+n’

equations, the factors of proportionality, R are equal; while,

t,t+n’ the factors of proportionality in the subsequent equations depend on the covariances of the risky returns with the unobservable return, qe

™

3. Observable Implications of the Latent Variable Model

As discussed above, the latent variable model implies that excess returns move in proportion to an unobservable return. Hansen and Hodrick [1983] give empirical content to this by characterizing the restrictions in terms of observable variables, Xe available to the

investor at period t, x,€ Q.. If expectations are formed rationally,

t

Equation 4 becomes:

itt, ten 7 Feiiyg BCL | a) +

t,ttn t where E(u, |Q,) = 0. The latent variable may itself be linearly projected onto the

‘ . : . _ ' + : : : information variables, X,3 So, Le ten aX, €, Substituting this

: : . . : .th projection into the above expression gives the i excess return as a . . 16 function of observable variables:

: = : ' +e i™t,tin Beit *t t

where, eo B. €, tov

This may be directly estimated in the case where B. i is a constant, by defining the reduced form parameters:

O, = Big Q@.

Then: ikt,ten = 05°, + e: (7)

The definition of 6, and Equation 7 together describe the ith excess return as a linear combination of the information variables, where the weights, @, have particular restrictions. The coefficients, O55 for each excess return must be proportional to the coefficients for any other excess return. That is, for any two assets, i and j, the coefficients are a scalar multiple of each other: O. = ke, where k=B5/B;- In all, for j asset returns and h information variables, this provides (j-1)(h-1) over-identifying restrictions on the estimation of Equation 7 for j assets.

While Equation 7 is written in terms of an expected excess return, the restrictions of the model also apply to differences between excess returns. For the foreign term premium, the first risky return is the n-period foreign return evaluated in the domestic currency, and the

second risky return is the return from rolling over consecutive

. om ’ 16. Collect terms from: iTt, ten Be ig’ x, + €,) + v,.

-9.-

l-period foreign assets, also evaluated in domestic currency at the final period: (5+ Sean) GRe, can ita jBevi-1, c+i — Bs gr By Dee + 5%: Or, more simply, 4?t,1,n = 48 aX, + i,jet’ where: 48 = 371,47 Pn, a"

In its reduced form,

j?t,1,n = gO Xe + 4et (8) Unrestricted, this equation simply predicts the term premia in all the countries on the basis of a common set of information variables, XE: The power of the test to reject the restrictions imposed by the latent variable model (or by any of the versions of the expectations hypothesis) depends on the predictive value of these information

variables in explaining the term premia. The choice of information

variables used here is discussed in the following section. 4. Testing the Latent Variable Model

This section discusses the estimation of the term premia as well as tests of the latent variable model and the expectations hypothesis. The estimated coefficients and test statistics are given in Tables l through 4.

The. roll-over premium is estimated for each country using 1-month and 3-month assets. With each time period being one month long, and n=3, the domestic term premium is defined as follows:

® = E(R

t,3,1 a).

3 t,t+3 ~ iy Resi-1, teil t

- 10 -

While, the foreign term premium ist?

3 gPe,3,27 ELC pRe cas > aba pReea-a, coi) Se/5 Sea lel -

This definition is used to estimate Equation 8. Weekly observations of annual yields on 30-day and 90-day certificates of deposit are used to construct the series of returns for the United States and Japan. For Germany, interbank rates are used. The sample period extends from October, 1986 to July, 1988.

The information variables, x, in Equation 8, are all available to

t investors at period t. They include: the annualized U.S. term spread,

R - R

t.t43 the annualized spread between the yield on U.S.

t, ttl’ 90-day C.D.s and the yield on the German 3-month interbank rate, ust t43 “weet, 43) and, the spread between the annualized yields on

the U.S. C.D. and the Japanese 3-month C.D., uske, 43 “jake te:

These variables are the domestic analogues of those used by Campbell and Clarida [1987], where they show that the differences between the 3-month nominal returns across currencies, along with the term spread, help to predict eurocurrency term premia. The use of the onshore term spread is also consistent with the many domestic studies rejecting a constant term premium by showing that the term spread predicts the term premium.

The fact that the observations are drawn weekly results in the

overlapping observations structure modeled by Hansen and

Hodrick [1980] and in the corresponding moving average error process.

17. Recall poeay

is the spot rate at period t+j (not the forward rate). J

-ll-

Weekly observations of the domestic and foreign realized returns give

rise to error processes that are MA(8) and MA(13), respectively .1®

Hansen and Hodrick’s [1980] procedure for consistent estimation of the

variance-covariance matrix addresses this problem, and a modification

1

allows for conditional heteroskedasticity as well. 9 The tests of both

the expectations hypothesis and the latent variable model use this

heteroskedastic consistent version, modified as in Newey and

West [1985] to ensure positive definiteness. 7°

Table 1 gives the unrestricted estimates of the parameters of Equation 6 and the heteroscedastic-serial correlation consistent esttimates of their standard errors. Unrestricted parameter estimates are given, rather than the constrained estimates, because of some undesirable small sample properties inherent in the usual constrained estimation procedures. 7+ As shown on the table, the coefficient estimates are not large, but their corrected standard errors (all much larger than the corresponding uncorrected standard errors) are small, and a number of the coefficients are individually statistically

significant. In each of the three countries, the coefficients are

18. In the case of the domestic asset, agents’ expectational errors remain unresolved for 2 months, at which time R. 0 t+ becomes known. In the case of the foreign assets, the uncertainty’ remains until the asset matures in 3 months and the exchange rate uncertainty is resolved.

19. Cumby and Obstfeld [1984], Hodrick and Srivastava [1984], and Giovannini and Jorion [1987] all find strong evidence of conditional heteroskedasticity.

20. The heteroskedastic version of Hansen and Hodrick's [1980] variance-covariance estimator used here is (X'X) “X’WX(X‘'X) ', where X is a (Txk) matrix of information variables. The (i,j)th element of W equals €,e.g, 5" when (|i-j|+1) is less than,the order of the moving average Iag, Kk, and equals zero otherwise; e. is a consistent estimate of e,. The weight g. .is chosen to équal [ 1 - |i-j|/(k+1) ] by Newey and West [1985]. *+J

21. The GMM estimators are not unique in this case, and using Maximum Likelihood, convergence is difficult.

jointly statistically significant (rendering the expectations hypothesis rejected).

Table 2 gives the test statistics for evaluating the latent variable model and for the various versions of the expectations hypothesis. As above, the xy? statistic presented is constructed using a Wald test, rather than using Generalized Method of Moments (GMM) to constrain the estimates and calculate the test statistic. Because the correct formulation of the Wald statistic is ambiguous, a number of different formulations are presented in the appendix, as is the GMM test statistic, As the upper panel of the table shows, the hypothesis that the coefficients in the term premium equations are proportional to one another cannot be rejected. In the context of the latent variable model, this provides evidence of a high degree of co-movement of the term premia across the onshore markets of the United States, Germany, and Japan.

This result extends that of Campbell and Clarida [1987], where they find that the latent variable model cannot be rejected for returns to such assets within the Euromarket, while it contrasts with Campbell's [1987] rejection of the latent variable model for the term premium in the United States. Campbell’s [1987] rejection may, in part, be due to the wide range of assets included in his study. An attempt has been made here to choose assets that differ in only three ways: maturity, currency of denomination, and political jurisdiction of issue. By examining the behavior of the term premia across countries, one is asking whether a particular relative price, that of capital for one period in terms of capital for another period, behaves

similarly across these markets. In the sense of the constant-f latent

variable model, the relative prices examined here do appear to move together.

The joint hypotheses of the expectations theory and uncovered interest parity are tested using the same parameter estimates given in Table 1, and the resulting test statistics are given on Table 2. As nested within the model, these hypotheses together imply that for each country, = 0 for all i. These constraints are strongly rejected for the three countries together and are rejected for each country, including the United States, where the test of the pure expectations hypothesis cannot be rejected in a simpler test, given in Table 3, that does not include these information variables. The somewhat less restrictive hypothesis of constant exchange risk and term premia implies that, for each country, 95= 0 for i not equal to 1. This hypothesis is rejected jointly and is also strongly rejected for assets in the United States. But, it cannot be rejected individually for yields on assets in Japan at standard confidence levels or in Germany, at the 95 percent confidence level; though, it can be rejected in Germany at the 90 percent level.

Table 3 presents the results of the estimation of the term premia restricting the coefficients on the information variables, except the constant, to equal zero: O5= 0, ixl. The top panel gives the term premia estimates for the three countries, their corrected standard errors, and the probability values associated with the tests that each term premia estimate is not zero. The point estimates of the term premia are all positive, though small: between 10 and 25 basis points, at an annual rate. Their corrected standard errors are also fairly small, though they are much larger than the uncorrected ones.

As a result, among individual countries, the pure

expectations/rational expectations hypothesis is rejected at the 95 percent confidence level for both German and Japanese assets. At the same time, the expectations hypothesis cannot be rejected for U.S. assets without the information variables at the 95 percent confidence level, but it can be rejected at the 90 percent level.

The test statistic for the hypothesis that rational expectations and the pure expectations hypothesis hold in all three countries is shown in the middle panel of the table. In this case, the combined hypothesis for all three countries can also only be rejected at the 90 percent level .?? The final panel of the table gives the hypothesis that the three term premia are identical. It, too, cannot be rejected

without information variables.

7. Summary

This paper uses the framework of the latent variable model to investigate the extent of term-structure co-movement across onshore markets in the United States, Germany, and Japan. The term premia of these markets are evaluated in terms of a single domestic currency so that the their co-movement may be viewed as reflecting the extent: to which the premia, from the domestic investor’s point of view, behave as if assets are trading within a single market. The restrictions of

a constant-8 latent variable model are not rejected for the onshore

22. One alternative to invoking rational expectations is to use survey data to separate the forward rate into an expected future rate and a term premium. Froot [1987] does this and finds, for short-term maturities in the United States (such as those examined here), that expectations of future rates appear to be formed rationally and that the term premium is time-varying. Froot and Frankel [1987] find the opposite for exchange rates.

marxets in these countries, and the term premia appear to move together.

Nested within this framework is a test of uncovered interest parity and the pure expectations hypothesis of the term structure of interest rates. More restrictive than the latent variable model, this joint hypothesis is rejected for the three countries as a whole, and individually for the United States and Japan at the 95 percent significance level. For German assets, it is rejected at the 90

percent confidence level.

- 16 - Appendix: Formulation of the Wald Statistic

This Appendix discusses the construction of the Wald statistic. Gregory and Veall [1985] show that the statistic is not unique in any finite sample when non-linear transformations of the null hypothesis are allowed, despite their asymptotic equivalence .*> For tests of the latent variable model, the conclusions of Section 4 are sensitive to the particular chosen formulation of the hypothesis. Some guicance in choosing a formulation is taken from the scalar results of bott. simulations by Gregory and Veall [1985] and the analytical work of Phillips and Park [1988]. In addition, Monte Carlo simulations are used. Finally, the model is tested using GMM.

Gregory and Veall’s [1985] classic example is the hypothesis that one parameter, a, is the inverse of another, b. That is H,: a = l/b. This may be written either as Hj: a - 1/b = 0, or as Hy: ab-1=0. The two formulations yield distinct test statistics. Gregory and Veall [1985] provide Monte Carlo evidence showing that the differences can be numerically substantial and depend on the parameter values. This point is amplified by the analytical work of Phillips and Park [1988] and by Lafontaine and White [1986].

The latent variable model similarly restricts parameters to be inversely related. From Equation 6, the hypothesis may be writen as either:

j?t,1,3/5° - aX, = 0, (A2) or,

j@t,1,3 7 98 2%, 7 O (Al)

23. The problem stems from the fact that the asymptotic properties of the Wald statistic rely on a Taylor approximation of the null hypothesis. The approximation can be a bad one when the hypothesis or its reparameterization is extremely non-linear.

-17-

While two formulations are algebraically equivalent, they are nonlinear reparameterizations of each other and yield numerically different Wald statistics. Table 4 gives the xy? statistics calculated from Al. Three x? statistics are given, since the choice of the country for normalization is also arbitrary. In contrast to the results presented in Section 4, the single-beta latent variable model is rejected at any standard significance level in all three cases. The Wald statistics constructed using the A2, the multiplicative formulation, are given in Table 5. The latent variable model cannot be rejected in any of the three multiplicative formulations.

The intuition of Gregory and Veall [1985] provides some insight into why these results might differ so sharply and guidance regarding the choice of parameterization. They point out that approximations of inverse functions that rely on derivatives might be very bad when the value of the function is close to zero, suggesting the multiplicative alternative in that case.°4 Many of the coefficient estimates in Equation 6 are small. Phillips and Park [1988] analytically compare alternative parameterizations similar to those of Gregory and Veall [1985] by evaluating the higher order terms of the Taylor apprceximations. While their work has not been extended to multivariate cases, it reinforces the conclusions of Gregory and Veall.

4. Monte Carlo experiment provides additional support for choosing the multiplicative formulation of the hypothesis. Using the unrestricted parameter estimates, given in Table 2, for the

coefficients in the U.S. term premium equation and for the constants

24. Their Monte Carlo work, as well as the work of Phillips and Park, substantiates this.

- 18 -

in the term premium equations for Germany and Japan, the Wald statistics were calculated with the hypothesis as true using both the multiplicative and the proportionate formulations.

Table 6 gives the results of this experiment. At the 95 percent confidence level, the Wald statistic constructed from A2 rejects the latent variable model (true by construction) 29 times in 1000 draws using normal random variables. In contrast, Al resulted in 845 rejections of the true model. At the 50 percent confidence level, there were 627 rejections using the A2, and 957 rejections using Al. These results are entirely consistent with the interpretation that the highly non-linear reparameterization changes the results dramatically, and that approximations will be particularly poor when values close to zero appear in the denominator the hypothesis formulation.

Finally, Table 7 provides the GMM estimation. *>

The results of this estimation are given in Table 1. The test statistics given. at the bottom of the table show that, in both cases, the latent variable

model of the term premia cannot be rejected at any standard levels of

se ee 26 significance.

25. The J-k sample poment conditions are: tz (®, - aB'x,) ® x= (0 0 ...0). Starting values are takes from the unréstricted estimates of the term structure equation given Table 1. 26. This confirms the results of Section 4.

- 19 -

Table 1

Latent Variable Model of the Term Structure Unrestricted Coefficient Estimates

jP c+, t43 6, + §,ts, + §,dms . + §4jas, + e, j=

United States _Germany -_Japan_ $ - .5697* 1462 .1818* s.e.(9,) (.1673) (2365) (.0911) pval(9,~0) 9993 4637 9540 $, 3076 .6392% 3045 s.e.(#,) (.4163) (.4158) (2239) pval(9,~0) 5400 8757 8262 # -.0317 -.2990 0022 s.e. (Pq) (.1334) (.1335) (0416) pval(9,~0) 1879 9749 0423 $, .2941% 2989 - 0552 s.e.(9,) (.1290) (.1871) (.0612) pval(9,*0) 9774 8898 6329

pre ernn

Notes: 2 Le Pear t43™ (Se/Se43) fre,t+3 ~ iLo presi, teiei ? See Re tas 7 Re ta dm

SF ucke 43 7 were 43

J48e = uske t43 7 jake, t+3°

2. This table presents estimates based on weekly

observations from October 22, 1986 to July 27, 1988.

3. Heteroskedastic and moving-average consistent standard error estimates and xy? statistics are given here.

4, An asterisk indicates a coefficient statistically different from zero.

- 20 -

Table 2

Tests of the Latent Variable Model and the Expectations Hypothesis of the Term Structure

6, + 6,ts, + 9,dms, + O,jas, + e

jPt+1, t4+3 t t

eee The Latent Variable Model eee

Hy: 574 = TLS LALORELED i=2to4, j = Germany, apan

x? (6) 9.40 prob . 848

eS The Expectations Hypothesis eee

H,: .8,= 0, Vi ji United States Germany Japan x2 (4) 18.35 9.63 59.04 prob .9989 .9529 .9999 Combined x? (12) 87.01 prob .9999 Hy: .0,= 0, Viel ji United States Germany Japan x2 (3) 16.83 6.64 2.60 prob .9992 .9155 .5431 Combined

x? (12) 26.07 prob .

Notes: See Table 1.

- 21 -

Table 3 Tests of the Expectations Hypothesis when 65= 0, Viel « 2 1/3 b g5t/jhe+3 © GReee3 ~ io GRetajesind 7 G2 + Ue meee Coefficient Estimates j =

reenter

United States Germany Japan 3? 0.1380 0.2147 0.1006 s.e.(;®) (0.1024) (0.1036) (0.0343) pval(,&*0) .9111 . 9809 . 9983 Pure Expectations Hypothesis Hy : us? = wo? = ga® = 0 eee x? (3) 14.721 prob .936 Single Term Premium Hypothesis Ho = ys® = wo® = ga? eee x? (2) 1.1533 prob -562

-_ OO eee

Notes:

1. This table presents estimates based on weekly observations from October 10, 1987 to May 18,

1988.

2. Heteroskedastic and moving-average consistent standard error estimates and yx? statistics are given in the table. The uncorrected standard errors of ys?) we? and ga® are:

0.0479, 0.04462 and 0.0242, respectively.

- 22 -

Table 4

Reparameterizations of Tests

of

the

Latent Variable Model

rt 6,dms, + O,jas, + e

t

t

Wald Statistics using the Proportionate Hypothesis Formulation

Hy: Ki - «1° 4 594/521 ) = 0 i=2to4 j = U.S.; k = Germany, Japan: x? (6) 30.77 prob 9999 j = Germany; k = U.S., Japan: x? (6) 22.94 prob .9992 j = Germany; k = U.S., Japan: x? (6) 39.39 prob .9999 Notes: See Table 3.

- 23 -

Table 5

Reparameterizations of Tests of the Latent Variable Model

6, + O,ts, + O,dms, + O,jas, + e

jP t+, +3 t

Wald Statistics using the Multiplicative Hypothesis Formulation

Hy: C594 49) - Le ee = 0 i=2to4 j = U.S.; k = Germany, Japan: x? (6) 9.40 prob . 848 j = Germany; k = U.S., Japan: x? (6) 5.65 prob .536 j = Germany; k = U.S., Japan: x? (6) 5.66 prob .938

Notes: See Table 3.

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Table 6

Reparameterizations of Wald Tests of the Latent Variable Model of the Term Structure of Interest Rates

Monte Carlo Simulations

= 6, + O6,ts, + @,dms, + 6,jas, + e

jPttl ,t+3 t t t

Multiplicative Formulation of Hypothesis:

Ho: (29, +10,) - (,6,°,6.) = 0, i=2 to4 ° jiokl jik *3 = U.S., Germany, Japan

Proportionate Formulation of Hypothesis:

= 0, i=2to4

H,: .6. - .6,°(,6-/,6,) ov ji jl ‘ki’k 15 = U.S., Germany, Japan

Confidence Number of Rejections from 1000 Trials Level Multiplicative Proportionate Formulation Formulation 95 percent 29 845 90 percent 72 881 75 percent 263 926 50 percent 627 957 25 percent 907 982 Notes:

1. 93 draws in each sample.

2. Simulations are made using the unconstrained parameter estimates for the U.S. term premium equation and the estimated constants for the other two equations.

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Table 7 GMM Estimation of the Latent Variable Model

Alternate Specifications

E,(®, - ap’ x, ) ® x = (0 0... QO).

Hy 9: a=(1, a5, Ay ) a=(a,,'Q, a, )

Parameter Estimates

U.S. _ C»efficients Constant -0.059 -0.015 Term Spread 0.358 0.086 DM Spread -0.058 -0.016 Yea Spread 0.104 0.125 Proportionality Factors Germany 1.0027 1.066 Japan 1.169 1.017

Test Statistics

x2 (6) 6.162 5.830 (probability) .5995 .5957 Notes:

1. This estimation used modified versions of a very useful GAUSS GMM proceedure written by Lars P. Hansen, John C. Heaton, and Masao Ogaki, under NSF Grant No. SES-8512371.

2. See Table 1.

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References

Campbell, J. 1987. Stock Returns and the Term Structure, Journal of Financial Economics 18, 373-399.

Campbell and Clarida. 1987. The Term Structure of Euromarket

Interest Rates, an Empirical Investigation, Journal of Monetary Economics 19, 25-44.

Cox, J., J. Ingersoll, and S. Ross. 1985. A Theory of the Term Structure of Interest Rates, Econometrica 53, 385-408.

Cumby, R. and M. Obstfeld. 1984. International Interest Rate and Price Level Linkages Under Flexible Exchange Rates, in J. Bilson and R. Marston, eds., Exchange Rate Theory and Practice, University of Chicago Press, Chicago, 121-151.

Evans, M. 1989. Interpreting the Term Structure Using the Intertermporal Capital Asset Pricing Model: an Application of the Non-Linear ARCH-M model, Working Paper No. 514, Graduate School of Business Administration, NYU.

Frankel, J. and K. Froot. 1987. Findings of Forward Discount Bias Interpreted in Light of Exchange Rate Survey Data. NBER Working Paper No. 1963.

Froot, K. 1987. New Hope for the Expectations Hypothesis of the Term Structure of Interest Rates, Working Paper, Sloan School of Management, M.I.T.

Giovannini A., and Jorion P. 1987. Interest Rates and Risk Premia in the Stock Market and the Foreign Exchange Market, Journal of International Money and Finance, 6:1, 107-123.

Gregory, Allan and Michael Veall, 1985. "Formulating Wald Tests of Nonlinear Restrictions," Econometrica, Vol. 53, No. 6, p.1465.

Hansen, L. and R. Hodrick. 1980. Forward Exchange Rates as Optimal Predictors of Future Spot Rates, Journal of Political Economy 88, 829-853.

Hansen, L. and R. Hodrick. 1983. Risk Averse Speculation in the Forward Foreign Exchange Market, in J. Frenkel, ed., Exchange Rates and International Macroeconomics, University of Chicago Press, Chicago, 115-152.

Hansen, L. and K. Singleton. 1983. Stochastic Consumption, Risk Aversion and the Temporal Behavior of Asset Returns, Journal of Political Economy, 91, 249-65.

Hodrick, R. 1987. The Empirical Evidence on the Efficiency of Forward and Futures Foreign Exchange Markets, Harwood Academic Publishers, London.

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Hedrick, R. and Srivastava. 1984. An Investigation of Risk and Return in Forward Foreign Exchange, Journal of International Mcney and Finance, 3, 5-29.

Hsieh, D. 1984. Tests of Rational Expectations and No Risk Premium in Forward Foreign Exchange Markets, Journal of Ir.ternational Economics 17, 173-184.

Lafontaine and Kenneth White, 1986. "Obtaining Any Wald Statistic You Want," Economics Letters 21, p. 35-40.

Lucas, R. 1978. Asset Prices in an Exchange Economy, Econometrica 46:6, 1429-45.

Newey, W. and K. West. 1987. A Simple, Positive Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix, Econometrica 55:3, 703-708.

Phillips, P.C.B. end J. Park. 1988. On the Formulation of Wald Tests of Nonlineai Restrictions, Econometrica 56:5, 1065-1083.

Shiller, R. and McCulloch. 1987. The Term Structure of Interest Rates, NBER Working paper No. 2341.

Shiller, F., J. Campbell and K. Shoenholtz. 1983. Forward Rates and Future Policy: Interpreting the Term Structure of Interest Rates, Brookings Papers on Economic Activity 1, 173-217.

Turnovsky, S.J. 1989. The Term Structure of Interest Rates and the Effects of Macroeconomic Policy, Journal of Mondy, Credit, and Banking, 21:3, 321-347.

White, H. 1980. A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity, Econometrica, 48:4, 817-838.

IFDP NUMBER

382

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376

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International Finance Discussion Papers

TITLES 1990

The Term Structure of Interest Rates in the Onshore Markets of the United States, Germany, and Japan

Financial Structure and Economic Development

Foreign Currency Operations: An Annotated Bibliography

The Global Economic Implications of German Unification

_Computers and the Trade Deficit: The Case of the Falling Prices

Evaluating the Predictive Performance of Trade-Account Models

Towards the Next Generation of Newly Industrializing Economies: The Roles for Macroeconomic Policy and the Manufacturing Sector

The Dynamics of Interest Rate and Tax Rules in a Stochastic Model

Stock Markets, Growth, and Policy Prospects for Sustained Improvement in U.S. External Balance: Structural Change versus

Policy Change

International Financial Markets and the U.S. External Imbalance

Why Hasn't Trade Grown Faster Than Income? Inter-Industry Trade Over the Past Century

Contractionary Devaluation with Black Markets for Foreign Exchange

1989

Exchange Rate Variability and the Level of International Trade

A Substitute for the Capital Stock Variable in Investment Functions

e——_—_oOoOoO

Please address requests for co

AUTHOR(s)

Helen Popper

Ross Levine

Hali J. Edison Lewis S. Alexander Joseph £.. Gagnon Ellen E. Meade Jaime Marquez

Neil R. Ericsson

Catherine L. Mann

Eric M. Leeper

Ross Levine

Catherine L. Mann

Deborah Danker Peter Hooper

Joseph E. Gagnon Andrew K. Rose

Steven B. Kamin

Joseph E, Gagnon

Guy V.G. Stevens

pies to International Finance Di.scussion

Papers, Division of International Finance, Stop 24, Board of Governors of the

Federal Reserve System, Washington, D.C.

20551.

IFDP NUMBER 367 366

365

364

363

362

361

360

359

358

357

356

354

353

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351

- 29 .

International Finance Discussion Papers

TITLES

An Empirical Assessment of Non-Linearities In Models of Exchange Rate Determination

Equilibrium in a Production Economy with an Income Tax

Tariffs and the Macroeconomy: Evidence from the USA

European Integration, Exchange Rate Management, and Monetary Reform: A Review of the Major Issues

Savings Rates and Output Variability in Industrial Countries

Determinants of Japanese Direct Investment in U.S. Manufacturing Industries

The U.S. and U.K. Activities of Japanese Banks: 1980-1988

Policy Rules, Information, and Fiscal Effects in a "Ricardian" Model

A Forward-Looking Multicountry Model: Mx3

Implications for Future U.S. Net Investment Payments of Growing U.S Net International Indebtedness

U.S. Policy on the Problems of International Debt

International Economic Policy: The Role of Exchange Rates

An Econometric Analysis of UK Money Demand in Monetary Trends in the United States and the United Kingdom by Milton Friedman and Anna J. Schwartz

Encompassing and Rational Expectations: How Sequential Corroboration Can Imply Refutation

The United States as a Heavily Indebted Country

External Debt and Developing Country Growth

An Algorithm to Solve Dynamic Models

AUTHOR(s

Richard A. Meese Andrew K. Rose

Wilbur John Coleman II Andrew K. Rose Jonathan D. Ostry

Garry J. Schinasi

Garry J. Schinasi Joseph E. Gagnon

Catherine L. Mann Henry S. Terrell Robert S. Dohner

Barbara R. Lowrey

Eric M. Leeper Joseph E. Gagnon Lois E. Stekler William L. Helkie Edwin M. Truman

Edwin M. Truman

David F. Hendry Neil R. Ericsson

Neil R. Ericsson David F. Hendry

David H. Howard

Steven B. Kamin Robert B. Kahn Ross Levine

Wilbur John Coleman IT