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The rational expectations hypothesis and the cross-section of bond yieldsRichard D. F. Harris aa School of Business and Economics, University of Exeter , Exeter EX4 4PU, UK E-mail:Published online: 21 Aug 2006.

To cite this article: Richard D. F. Harris (2004) The rational expectations hypothesis and the cross-section of bond yields,Applied Financial Economics, 14:2, 105-112, DOI: 10.1080/0960310042000176371

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The rational expectations hypothesis and

the cross-section of bond yields

RICHARD D. F. HARRIS

School of Business and Economics, University of Exeter, Exeter EX4 4PU, UKE-mail: R.D.F.Harris@exeter.ac.uk

In the context of the bond market, empirical tests of the rational expectationshypothesis (REH) have without exception been tests of the time-series propertiesof interest rates. However, the REH also imposes restrictions on the cross-section ofbond yields at each point in time. This study tests these restrictions using the Famaand MacBeth repeated cross-section regression procedure. Specifically, a long seriesof monthly cross-section regressions is estimated using zero coupon bond yield datafor maturities from two months to thirty-five years. The REH is tested using thetime-series average of the estimated slope parameter in the cross-section regressions.The maturity-specific risk premium is proxied by the time-series volatility of excessreturns for each bond maturity. Time-variation in the risk premium is allowed forthrough time-variation in the volatility of excess returns, and in the market price ofrisk. While the risk premium proxy is significant in explaining the cross-section ofexcess returns, the REH is very strongly rejected.

I . INTRODUCTION

There have been many tests of whether the movement of

bond yields satisfies the rational expectations hypothesis

(REH). Most have been conducted in the framework of

the expectations hypothesis of the term structure of interest

rates. The expectations hypothesis implies that the spread

between short and long bonds yield should forecast the

change in the long bond yield in the following period. In

particular, when the long yield exceeds the short yield, the

long yield should be expected to rise in the following period

in order to generate the capital loss required to offset the

initial yield premium. When combined with the REH, this

hypothesis can be formulated as a regression of the actual

one period change in the long yield on the current spread

between short and long yields. When the spread is appro-

priately scaled, the estimated slope coefficient should be

unity under the expectations hypothesis. Empirical evi-

dence based on this regression weighs heavily against the

REH, particularly at the long end of the maturity spec-

trum. The estimated slope coefficient is typically found to

be not only significantly less than unity, but also signifi-

cantly less than zero for all but the shortest maturity

bonds, implying that bond yields actually move in a

direction opposite to that predicted by the REH.1

There are a number of potential explanations for this

apparent failure of the REH. One possibility is that tests

based on the expectations hypothesis fail to account for

a time varying risk premium that is correlated with the

yield spread, leading to a downward bias in the estimated

coefficient (see, for instance, Fama (1984), Mankiw and

Miron (1996) and Evans and Lewis (1994)). Tests that

have allowed for the possibility of a time varying risk pre-

mium go some way towards rescuing the REH, although

the results are sensitive to the choice of proxy for the risk

premium, the bond maturities considered and the sample

period used, and are generally confined to the short end of

the maturity spectrum (see, for example, Fama (1984),

Shiller et al. (1983), Jones and Roley (1983), Froot

(1989), Simon (1989) and Tzavalis and Wickens (1997)).

In the context of the bond market, empirical tests of the

REH have almost without exception been tests of the

1 See, for instance, Shiller (1979), Shiller et al. (1983), Campbell and Shiller (1984), Mankiw and Summers (1984), Mankiw (1986),Campbell and Shiller (1991) and Campbell (1995).

Applied Financial Economics ISSN 0960–3107 print/ISSN 1466–4305 online # 2004 Taylor & Francis Ltd 105

http://www.tandf.co.uk/journalsDOI: 10.1080/0960310042000176371

Applied Financial Economics, 2004, 14, 105–112

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time-series properties of interest rates. However, the REHhas implications not just for the movement of a single longyield through time relative to the movement of the shortyield, but also the relationship between yield movements onbonds of all maturities over a single period of time. Theaim of this study is to test these implications by investigat-ing whether the REH holds in cross-section, that is whetherdifferences in yields on bonds of different maturities reflectdifferences in the movement of those yields over thesubsequent period.

A potential difficulty in testing the REH using cross-section data is the significant cross-section correlation andheteroscedasticity that characterizes the movement of bondyields, which would seriously distort the asymptotic distri-bution of the standard test statistics that are used to test theREH. In order to circumvent this problem, this paperadopts the approach of Fama and MacBeth (1973), whichhas been widely used to test hypotheses about the cross-section of stock returns, such as the Capital Asset PricingModel or the Arbitrage Pricing Theory. Specifically, a seriesof cross-section regressions is estimated and the averagevalue of the estimated parameter is computed. Under thenull hypothesis that the error term in the cross-sectionregressions is serially uncorrelated (as implied by theREH), the series of estimated parameters will itself beserially uncorrelated and thus the time-series standarddeviation of each estimated parameter may be used tomake inference about its population value.

An alternative to using the Fama-MacBeth procedurewould be to pool the data from different cross-sectionsand to use panel data techniques to estimate the relation-ships implied by the REH. However, unlike the Fama–MacBeth approach, this requires the estimation of theerror covariance matrix of the panel, which is not straight-forward. Estimation of an unrestricted error covariancematrix is prohibited by the dimensions of the panel andthus the only solution is to impose some arbitrary structureon the variance-covariance matrix by specifying the func-tional form of the cross-section correlation and hetero-scedasticity, or by assuming that the variance-covariancematrix is constant over time (see, for instance, Bams andWolff (2000) and Harris (2000)). The advantage of theFama and MacBeth procedure is that it allows the estima-tion of the cross-section relationship implied by the REHwithout the need to impose any structure on the errorcovariance matrix, other than the absence of serial correla-tion which is already precluded under the null hypothesis.

A second advantage of the repeated cross-sectionapproach over both the time-series approach and thepanel data approach is that it is possible to include bondmaturities for which there are only short time-series of dataavailable. The data used in this paper are the US zerocoupon bond data estimated by McCulloch and Kwon(1993). These comprise estimated zero coupon yields for54 bond maturities between one month and 40 years over

the period December 1946 to February 1991. However,complete time-series of data are only available for bondmaturities up to 13 years. The time-series for longer matu-rities are incomplete, with only small amounts of dataavailable for very long maturities, particularly for thosein excess of 30 years. While there are insufficient data toestimate a time-series regression for these maturities, theycan be easily included in a cross-section regression for theperiods in which data are available.However, a problem with both cross-section tests and

panel data tests of the REH is that even if risk premiawere not time-varying, they will almost certainly varywith maturity. Omission of the maturity specific risk pre-mium will lead to the same omitted variable bias that ariseswhen a time-varying risk premium is omitted from a time-series regression. In particular, since the spread is positivelycorrelated with the risk premium, the omission of the riskpremium will tend to bias the slope coefficient on the yieldspread downwards. Following the time series literature,this paper assumes that risk-premia are positively relatedto the volatility of excess returns. Specifically, it is assumedthat at any point in time, there is a linear cross-sectionrelationship between maturity-specific risk premium andthe logarithm of the conditional variance of excess holdingperiod returns.It is assumed that volatility in the bond market is driven

by a limited number of factors, so that increases in vola-tility affect the conditional variance of all bond maturitiesbut to varying degrees. Time-variation in the conditionalvariance of bond returns is introduced by specifying anexponential cross-section relationship between the condi-tional and unconditional variance of excess returns, andallowing the parameters of this relationship to vary overtime. Although on average this relationship must be linear(since, on average, the conditional volatility of excessreturns is equal to the unconditional volatility), the expo-nential relationship allows the relationship to be non-linearin any particular period. Time-variation in the maturity-specific risk premium therefore arises both from time-variation in the parameters of the function that relatesrisk premia to the conditional variance of excess returns(i.e. time-variation in the market price of risk), andfrom time-variation in the conditional variance itself (i.e.time-variation in the quantity of risk).Thus, a further advantage of the cross-section approach

over both the time-series and panel data approaches, is thatby exploiting the cross-section correlation in bond yields,one can allow for time-variation in conditional volatilitywithout the need to explicitly specify its dynamic process.Not only does it allow conditional volatility to be affectedby past volatility and past shocks to excess returns, as ina GARCH model, but it also implicitly accommodatesthe influence of exogenous variables, such as monetarypolicy or regulatory changes. Moreover, it allows fortime-variation both in the quantity of risk (the conditional

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variance of excess returns) and in the market price of risk(the parameters which relate excess returns to conditionalvariance). This makes the cross-section approach consider-ably more flexible than the GARCH models that aretypically used in the time-series literature.

The analysis is conducted for the full sample of bondmaturities from two months to 35 years, and for sub-samples of short, medium and long maturity bonds.Consistent with evidence from time series tests, the REHis strongly rejected, with an estimated yield spread coeffi-cient that is significantly less than unity in all cases, andoften significantly less than zero. The level of excess returnsis shown to be strongly correlated with the volatility ofexcess returns in cross-section, suggesting that the volatilityof excess returns is an effective proxy for the maturity-specific risk premium. However, in the tests of the REH,the risk premium proxy, while statistically significant, hasthe incorrect sign.

The organization of the paper is as follows. The follow-ing section describes the methodology and data that isused in the econometric analysis. Section III presents theempirical results while Section IV concludes.

II . METHODOLOGY AND DATA

The expectations hypothesis of the term structure of inter-est rates, in its log-linear form, states that the expected oneperiod continuously compounded holding period return foran n period bond is equal to the log yield on a one periodbond plus a risk premium that may vary with maturity, butis constant over time. A more general framework thatallows for the possibility that the maturity-specific risk pre-mium is time-varying, and which nests the expectationshypothesis as a special case, may be written as

Et hn, tþ1

� �¼ y1, t þ �n, t ð1Þ

where hn, tþ1 ¼ ln pn�1, tþ1 � ln pn, t ¼ nyn, t � ðn� 1Þyn�1, tþ1

is the holding period return on an n period zero couponbond between t and tþ 1, pn,t and yn,t are the price andcontinuously compounded yield on the bond at time t, y1,tis the continuously compounded yield on a one period bond,�n,t is the equilibrium risk premium of the n period bondover the one period bond and Et :½ � is the expectation condi-tional on the information set at time t. Under the expecta-tions hypothesis, �n,t¼ �n. In the more general model,however, �n,t is free to vary over both maturity and time.By replacing the expectation in Equation 1 with its real-ization and rearranging, the relationship between actualbond yields may be written as

�yn, tþ1 ¼1

n� 1sn, t �

1

n� 1�n, t þ en, tþ1 ð2Þ

where�yn,tþ 1¼ yn�1,tþ 1�yn,t, sn,t¼yn,t �y1,t and en,tþ 1 is azero mean random shock that is orthogonal to the

information set at time t. Equation 2 forms a basis fortests of the REH. Regressions based on Equation 2 aretypically estimated by ordinary least squares (OLS) usingtime-series data over the period t¼ 1, . . . ,T, for eachmaturity, n¼ 1, . . . ,N. Tests of the REH based on theexpectations hypothesis exclude the risk premium variable�n,t (allowing the time-invariant risk premium to be cap-tured by the regression constant), while tests of the REHunder the more general assumption of a time-varying riskpremium replace �n,t with a suitable proxy. The nullhypothesis that the coefficient on sn,t/(n�1) is equal tounity is tested in the conventional way for each individualmaturity, n, using either the OLS estimates of the para-meter standard errors, or standard errors that are correctedfor heteroscedasticity.In this study, the relationship implied by Equation 2 is

instead estimated for a cross-section of bond maturities,n¼ 1, . . . ,N, at a number of different dates t¼ 1, . . . ,T.However, in order to test the REH, it is necessary to specifya proxy for the maturity-specific risk premium, �n,t,whether or not it is time-varying, since in cross-section,its omission would lead to the same downward biasthat would arise in a time-series regression in which atime-varying risk premium has been omitted.In the time-series approach, there are a number of vari-

ables that have been used as a proxy for the risk premium,almost all of which are measures of yield or return volatil-ity. These include a moving average of the absolute valueof yield changes (Fama (1976), Mishkin (1982), Shiller et al.(1983) and Jones and Roley (1983)), the conditional var-iance of excess holding period returns (Engle, Lilien andRobins (1987)) and the squared excess holding periodreturn on the long bond over the short bond (Simon(1989)). In most cases, these measures are found to be sig-nificant in explaining excess returns (supporting the use ofreturn volatility as a proxy for the risk premium) and inmany cases, go some way towards rescuing the REH,although evidence is limited largely to the short end ofthe maturity spectrum.Following the time series literature, this study assumes

that risk-premia are positively related to the volatility ofexcess returns. Specifically, it is assumed that at any pointin time, there is a linear cross-section relationship betweenmaturity-specific risk premia and the logarithm of the con-ditional variance of excess holding period returns. Theparameters of this linear relationship are allowed to varyover time, implicitly allowing for time-variation in themarket price of risk. The risk premium in Equation 2 cantherefore be written as

�n, t ¼ at þ bt ln �2n, t ð3Þ

where �2n, t ¼ vartðhn, tþ1Þ. While the choice of functional

form of the relationship between the risk premium andconditional volatility is inevitably somewhat arbitrary,the use of the logarithm, as opposed to the level or the

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square-root of the conditional variance is consistent withthe fact that volatility increases approximately exponen-tially with maturity while average excess returns do not(see Campbell et al. 1997). Moreover, there is evidencefrom time-series studies that supports the use of the loga-rithm, as opposed to the level or square-root of the condi-tional variance as a proxy for the risk premium (see, forinstance, Engle, Lilien and Robins, 1987). The followingsection shows that in the present sample, excess returns areindeed significantly correlated in cross-section with thelogarithm of the conditional variance of excess returns.Further analysis revealed a weaker association with thestandard deviation or variance.

Rather than estimating the conditional variance directlyfor each maturity, as would be appropriate in the timeseries approach, it is assumed instead that volatility inexcess returns is driven by a limited number of market-wide factors, so that shocks to volatility affect the condi-tional variance of all bond maturities but to varyingdegrees. A parsimonious way of capturing this form oftime-variation in volatility is to specify a cross-section rela-tionship between the conditional variance of excess returnsand the unconditional variance of excess returns, but allowthe parameters of this relationship to vary through time.Specifying a linear relationship would imply that volatilityshocks have a proportionate affect on conditional volatility(i.e. the increase in the conditional volatility of a bond’sreturn that arises from a volatility shock is proportional tothe unconditional volatility of the bond’s return). Whilethis must be true on average (since, on average, the condi-tional volatility of a bond’s return is equal to its uncondi-tional volatility), it may be non-linear in any particularperiod. It is therefore assumed instead that conditionalvariance is an exponential function of unconditional var-iance. The exponential function allows volatility shocks tohave an effect on conditional variance of each bond that isproportional, more-than-proportional or less-than-propor-tional to its unconditional variance. With this specification,the conditional variance of excess returns can be written as

�2n, t ¼ ct�

dtn ð4Þ

where �2n ¼ Eð�2

n, tÞ ¼ varðhn, tþ1Þ, with the expectation andvariance taken over t¼ 1, . . . ,T.

Time-variation in risk premia therefore arises both fromtime-variation in the parameters of function that relatesrisk premia to the conditional variance of excess returns(i.e. time-variation in the market price of risk), and alsofrom time-variation in the conditional variance itself (i.e.time-variation in the quantity of risk). In the present sam-ple, the unconditional variance of excess returns increasesalmost monotonically with maturity, implying that thetime-varying maturity-specific risk premium does also.This is consistent with studies such as McCulloch (1987)and Richardson et al. (1992).

Combining Equations 3 and 4, the time-varying, matur-ity-specific risk premium in Equation 2 can be written as

�n, t ¼ at þ bt ln ct þ btdt ln �n

¼ At þ Bt ln �nð5Þ

With this specification of the risk premium and the condi-tional variance of excess returns, the risk premium isa linear function of the logarithm of the unconditionalvariance of excess returns. Differences in the unconditionalvariance of excess returns determine the cross-sectional dif-ferences in risk premia, while time-variation in risk premiaarises from changes in both the intercept and the slope ofthe linear function. Note that the specification of the riskpremium in this way is similar to the single factor riskpremium employed by Heston (1992) and Tzavalis andWickens (1997). The differences are that first, the matur-ity-specific component of the risk premium is explicitlyspecified to be the logarithm of the unconditional varianceof excess returns and, second, it allows for an additionaltime-varying shift parameter in the level of all risk premia.In order to test the REH using the above specification of

the risk premium, the following regression is estimated ateach date, t¼ 1, . . . ,T.

�yn, tþ1 ¼ a0, t þ a1sn, tn� 1

þ a2, tln �nn� 1

þ en, tþ1 ð6Þ

Under the null hypothesis of rational expectations, �1¼ 1.Although the OLS estimator of the parameter �1 in regres-sion Equation 6 is unbiased and consistent, conventionalhypothesis testing is not possible because the error term,"n,tþ 1, is correlated and heteroscedastic across bonds ofdifferent maturities and so the OLS estimate of the stan-dard error of ��1, t is biased and inconsistent. However,under the null hypothesis of rational expectations, "n,tþ 1

is serially uncorrelated and so the parameter estimate ��1, t isalso serially uncorrelated. This enables one to test thehypotheses that �1¼ 1 using the estimated time-series stan-dard deviation of ��1, t. Specifically, the test statisticz1 ¼ ð �����1 � 1Þ

� ffiffiffiffiffiffiffiffiffiffiffi��21=T

q, where �����i ¼ 1=T

PTt¼1 ��1, t and

��2i ¼ 1=ðT � 1Þ

PTt¼1 ð��i, t �

�����iÞ2 has a t-distribution with

T�1 degrees of freedom (see, for instance, Campbell et al.1997). This is the procedure introduced by Fama andMacBeth (1973), and used widely in cross-section empiricaltests of asset pricing models such as the Capital AssetPricing Model or the Arbitrage Pricing Theory.The Fama and MacBeth procedure outlined above is

inefficient in the sense that equal weight is attached toeach parameter estimate irrespective of its variance.A more efficient estimator can be obtained by weightingeach monthly estimate of the parameter by a measure ofits precision (see Litzenberger and Ramaswamy (1979)and Chan et al. (1991)). In this paper, the precision ofthe parameter estimate is measured by the inverse of its

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estimated standard deviation. This then yields the teststatistic.

zi ¼ ð �~��~��i � 1Þ=ffiffiffiffiffiffiffiffiffiffiffiffi~ss2i =T

q

where

�~��~��i ¼ ð1=TÞXTt¼1

wi, t��i, t,

~ss2i ¼ ð1=ðT � 1ÞÞ

XT

t¼1

ðwi, t��i, t ��~��~��iÞ

2,

wi, t ¼ ð1=ssi, tÞ=ð1=TÞXTt¼1

1=ssi, t

and

ssi, t

is the ordinary least squares estimate of the standarddeviation of ��i, t in the regression for month t.

The data that are used to estimate the cross-sectionregressions are the monthly zero coupon bond data com-puted by McCulloch (1989) and McCulloch and Kwon(1993) for the period December 1946 to February 1991.The quality of the estimated data improves significantlyafter the Treasury Accord of 1951, and so only data fromJanuary 1952 is used, as recommended by McCulloch andKwon. For this period, full time-series of data are availablefor the one month yield and 31 longer maturities up to13 years. Incomplete time-series are available for a further23 maturities up to 40 years, although there is a sharpreduction in the number of observations available formaturities of 30 years and over. Table 1 gives the dataavailability for each of the maturities.

Each set of cross-section regressions at date t requires theregression variables dated t and tþ 1, so that a minimum oftwo consecutive observations must be available fora maturity to be included in a single regression. Thisrequirement is satisfied by all but the longest maturity of40 years, for which only a single observation is available.The minimum number of maturities in the full sample istherefore 31, while the maximum is 53. A total of 469monthly cross-section regressions are estimated fromJanuary 1952 to January 1991. McCulloch and Kwonreport monthly maturities only up to 18 months, and sofor maturities in excess of 18 months it is necessary toapproximate the one month change in the long bondyield, yn�1,tþ 1�yn,t, by yn,tþ 1�yn,t.

III . EMPIRICAL RESULTS

The results of the empirical analysis are reported in Tables2 and 3. In order to evaluate the effectiveness of the loga-rithm of the conditional variance as a proxy for the riskpremium, Table 2 presents the results of a regression of theexcess holding period return, hn, tþ1 � y1, t, on the risk pre-mium proxy, ln �n, using the Fama and MacBeth regres-sion methodology outlined in the previous section. Resultsare reported for the full sample of 53 bond maturities andsub-samples of 11 short, 12 medium and 31 long maturitybonds. For the full sample and the sub-sample of longmaturity bonds, results are also reported for those matu-rities for which complete time series of data are available.The table reports the time-series average of each estimated

Table 1. Data availability

Maturity(months)

Number ofobservations

Maturity(months)

Number ofobservations

Maturity(months)

Number ofobservations

1 470 24 470 240 4352 470 30 470 252 4213 470 36 470 264 4184 470 48 470 276 4115 470 60 470 288 3806 470 72 470 300 3407 470 84 470 312 3248 470 96 470 324 3079 470 108 470 336 28110 470 120 470 348 24911 470 132 470 360 7012 470 144 470 372 5013 470 156 470 384 3814 470 168 463 396 3715 470 180 444 408 3716 470 192 442 420 3617 470 204 442 480 118 470 216 44221 470 228 442

Notes: The table gives the number of months for which data are available for each maturity for the 470 monthsbetween January 1952 and February 1991.

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coefficient, together with their estimated standard errors.The unweighted average estimates are reported in eachtable in Panel A, and the weighted average estimates inPanel B.

In all cases, the average estimated slope coefficient hasthe expected positive sign. Using the unweighted averageestimates, the risk premium proxy is not significant in thefull sample, and significant only for the sub-sample of 11short maturity bonds. As expected, using the weighted esti-mates reported in Panel B substantially increases the sta-tistical significance of the average estimated parameters,for all six samples. The risk premium proxy is statisticallysignificant in the full sample and in all of the sub-samples at

the 10% significance level. For the full sample, and for thesub-sample of 11 short maturity bonds, it is significant atthe 1% level. The results of Table 2 therefore suggest thatthe logarithm of the variance of excess returns is indeed aneffective proxy for the maturity-specific risk premium, andare consistent with the findings of time-series studies, suchas Engle et al. (1987).Table 3 reports the estimation results for the cross-

section regression test of the REH, using the logarithm ofthe variance of excess returns as a proxy for the risk pre-mium. From Panel A, the unweighted average slope coeffi-cient on the yield spread is not only significantly less thanunity – the value implied by the REH – but also

Table 3. Cross-section regression tests of the REH

All 53maturities(2 mths–35 yrs)

All 31 completematurities(2 mths–13 yrs)

11 shortmaturities(2 mths–12 mths)

12 mediummaturities(13 mths–5 yrs)

31 longmaturities(5 yrs–35 yrs)

8 completelong maturities(5 yrs–13 yrs)

Panel A: unweighted time-series averages of cross-section regression parametersIntercept 0.01 (0.01) 0.02 (0.02) 0.04 (0.03) 0.00 (0.02) �0.03 (0.03) �0.05 (0.04)Yield spread �1.20 (0.24)*** �1.17 (0.22)*** �1.32 (0.30)*** �0.21 (1.31) �17.55 (6.44)*** �22.00 (9.11)***Risk premium 0.06 (0.02)*** 0.06 (0.02)*** 0.05 (0.02)*** 0.11 (0.28) 3.75 (1.54)*** 2.90 (1.76)*

Panel B: weighted time-series averages of cross-section regression parametersIntercept 0.02 (0.01) 0.02 (0.01) 0.03 (0.02) 0.02 (0.01) �0.01 (0.01) �0.01 (0.01)Yield spread �0.62 (0.09)*** 0.64 (0.09)*** �0.72 (0.11)*** �2.12 (0.68)*** �16.11 (3.12)*** �12.27 (2.83)***Risk premium 0.05 (0.01)*** 0.04 (0.01)*** 0.03 (0.01)*** 0.11 (0.11) 2.60 (0.45)*** 1.73 (0.43)***

Notes: The cross-section regression ðn� 1Þ�yn;tþ1 ¼ �0;t þ �1;tsn;t=ðn� 1Þ þ �2;t lnsn=ðn� 1Þ þ "n;tþ1 is estimated in each of 469 monthsfrom January 1952 to January 1991. Estimation is by ordinary least squares. Panel A reports the unweighted time-series averagesof the estimated coefficients, �����i ¼ 1=T

PTt¼1 ��i;t, and standard errors of the average estimated coefficients,

SEð �����iÞ ¼ ��i=ffiffiffiffiT

p¼ 1=T ½

PTt¼1 ð��i;t �

�����iÞ2�1=2. Panel B reports the weighted time-series average of the estimated coefficients, �~��~��i ¼

1=TPT

t¼1 wi;t��i;t, and standard errors of the average estimated coefficients, SEð �~��~��iÞ ¼ ~��i=ffiffiffiffiT

p¼ 1=T ½

PTt¼1 ðwi;t��i;t �

�~��~��iÞ2�1=2, where

wi;t ¼ ð1=ssi;tÞ=PT

t¼1 1=ssi;t and ssi;t is the ordinary least squares estimate of the standard deviation of ��i;t. *, ** and *** denote a yieldspread coefficient that is significantly different from unity, or a risk premium coefficient that is significantly different from zero, at the10%, 5% and 1% significance level, respectively.

Table 2. The cross-section relationship between excess holding period returns and the risk premium proxy

All 53maturities(2 mths–35 yrs)

All 31 completematurities(2 mths–13 yrs)

11 shortmaturities(2 mths–12 mths)

12 mediummaturities(13 mths–5 yrs)

31 longmaturities(5 yrs–35 yrs)

8 completelong maturities(5 yrs–13 yrs)

Panel A: unweighted time-series averages of cross-section regression parametersIntercept �0.32 (1.71) 0.42 (0.82) 0.68 (0.13) 0.30 (1.30) �1.39 (3.33) �2.60 (4.54)Risk premium 2.26 (2.65) 0.88 (0.91) 0.36 (0.12)*** 2.27 (3.56) 21.00 (28.73) 38.98 (49.96)

Panel B: weighted time-series averages of cross-section regression parametersIntercept �0.39 (0.45) 0.23 (0.24) 0.40 (0.03) �0.21 (0.73) �1.79 (12.00) �3.26 (1.89)Risk premium 1.60 (0.65)*** 0.67 (0.26)*** 0.18 (0.03)*** 3.15 (1.81)* 18.93 (12.00)* 28.21 (18.38)*

Notes: The cross-section regression hn;tþ1 ¼ �0;t þ �1;t ln �n þ "n;tþ1 is estimated in each of 469 months from January 1952 to January1991. Estimation is by ordinary least squares. Panel A reports the unweighted time-series averages of the estimated coefficients,�����i ¼ 1=T

PTt¼1 ��i;t, and standard errors of the average estimated coefficients, SEð �����iÞ ¼ ��i=

ffiffiffiffiT

p¼ 1=T ½

PTt¼1 ð��i;t �

�����iÞ2�1=2. Panel B reports

the weighted time-series average of the estimated coefficients, �~��~��i ¼ 1=TPT

t¼1 wi;t��i;t, and standard errors of the average estimated

coefficients, SEð �~��~��iÞ ¼ ~��i=ffiffiffiffiT

p¼ 1=T ½

PTt¼1 ðwi;t��i;t �

�~��~��iÞ2�1=2, where wi;t ¼ ð1=ssi;tÞ=ð1=TÞ

PTt¼1 1=ssi;t and ssi;t is the ordinary least squares

estimate of the standard deviation of ��i;t. *, ** and *** denote that the coefficient on the risk premium is significantly different from unityat the 10%, 5% and 1% significance level, respectively.

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significantly less than zero both for the full sample and forall but one of the sub-samples. For the sample of 12medium maturity bonds, the average estimated coefficientis negative, but estimated with low precision. The largestandard error of the estimated parameters for long matur-ity bonds is consistent with the evidence from time seriesstudies (see Campbell, 1995, and also Schotman, 1996).The risk premium proxy is again significant for all exceptthe sub-sample of medium maturity bonds, but has thewrong sign.

As with Table 2, the use of the weighted estimatesincreases the statistical significance of the average esti-mated parameters. The estimated slope coefficient on theyield spread is now significantly less than both unity andzero in all cases. The risk premium proxy is significant forall except the sub-sample of medium maturity bonds. In allcases, it has the incorrect sign. Table 3 therefore providesa very strong rejection of the REH.

IV. SUMMARY AND CONCLUSION

Theories of the term structure of interest rates, togetherwith the assumption of rational expectations, are centralto the implementation of monetary policy, providing thelink between the short term interest rate that is underthe control of the monetary authorities, and the longterm interest rates that the authorities may ultimatelywant to influence. Empirical tests of the REH in thebond market have without exception been tests of thetime-series properties of interest rates. In contrast, thisstudy tests the REH using cross-section data on bondyields.

The cross-section approach offers two distinct advan-tages over the time series approach. First, it is possible toinclude bond maturities for which only a few time-seriesobservations are available. Second, by exploiting the cross-section correlation bond yields, it is possible to allow fortime-variation in their conditional volatility without theneed to specify the exact nature of this time-variation.The cross-section approach also has the advantage overthe panel data approach that it is not necessary to specifythe variance-covariance matrix of the regression errorterms to control for the very significant heteroscedasticityand cross-section correlation that characterizes themovement of bond yields.

It is found that while the risk premium proxy is signifi-cant in explaining the cross-section of excess returns, theREH is very strongly rejected using cross-section data.Consistent with results of time series studies, it is foundthat not only are interest rates movements inconsistentwith the REH, they actually move in the wrong direction,even once allowance is made for a time-varying riskpremium.

ACKNOWLEDGEMENTS

I am grateful to George Bulkley, Kaddour Hadri andElias Tzavalis, and to seminar participants at theUniversity of Auckland, the University of Otago, theUniversity of Strathclyde, the University of Waikato,Victoria University of Wellington, the Reserve Bank ofNew Zealand, and ESAM99, Sydney, for their usefulcomments and suggestions.

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