relationship between expected treasury bill and eurodollar interest rates: a fractional...

Review of Quantitative Finance and Accounting, 16, 65–80, 2001©C 2001 Kluwer Academic Publishers. Manufactured in The Netherlands.

Relationship Between Expected Treasury Bill andEurodollar Interest Rates: A FractionalCointegration Analysis

KESHAB SHRESTHAFaculty of Administration, University of Regina, Regina, Saskatchewan, Canada S4S 6A2

ROBERT L. WELCHDepartment of Accounting and Finance, Faculty of Business, Brock University, St. Catharines, Ontario,Canada L2S 3A1, Fax: 905 688 9779E-mail: [email protected]

Abstract. In this paper, we extend Booth and Tse’s (BT) 1995 analysis of fractional cointegration between theexpected Eurodollar and Treasury bill interest rates implied by their respective futures contracts. The definitionof fractional cointegration suggested by Cheung and Lai (1993) and used by BT is refined so that it requires thecointegrating relationship to be stationary as well as mean-reverting. In addition to the Geweke and Porter-Hudakmethod used by BT, a more efficient Maximum Likelihood (ML) method is used to estimate the cointegrat-ing relationship. The LM (Engle (1982)) test indicates the possible existence of a heteroscedastic cointegratingrelationship. Therefore, we use heteroscedastic models (GARCH and Exponential GARCH) to represent the coin-tegrating regression instead of the simple homoscedastic model used by BT. The empirical evidence cannot rejectthe null hypothesis of a stationary fractional cointegration relationship between the Eurodollar and Treasury billinterest rates.

Key words: fractional cointegration of Eurodollar, Treasury bill futures

JEL Classification: M21

1. Introduction

Because the U.S. dollar has served as the primary international currency over many decades,it is important to study and understand the stochastic properties of short-term U.S. dollarinterest rates such as U.S. Treasury bill and Eurodollar interest rates. The importance of thisis evident from the many published papers dealing with these domestic and offshore interestrates. Even though both interest rates are short-term U.S. dollar interest rates, Treasury billsrepresent the domestic U.S. dollar interest rate whereas Eurodollars represent the U.S. dollarinterest rate prevailing in external (offshore) markets. Eurodollars and Treasury bills arenearly identical securities except that Eurodollars trade in the less regulated international(offshore) markets and are not backed by the U.S. government against default. Consequently,economic theory suggests there is a strong positive relationship between these two interestrates and any deviation from the equilibrium relationship should disappear quickly. In other

66 SHRESTHA AND WELCH

words, the domestic and international market for short-term U.S. dollar borrowing shouldbe integrated. The historical difference between these two interest rates is shown in Figure 1.

There are number of studies that analyze the relationship between these two interest ratesas well as other domestic and offshore short-term rates like Certificates of Deposit andCommercial Paper rates. Some of these studies examine the lead-lag relationship betweeninterest rates using the Granger-Sim type causality tests (see e.g., Hendershott (1967),Kwack (1971), Levin (1974), Kaen and Hachey (1983), Hartman (1984) and Swanson(1988) etc.). These studies indicate that in general the Eurodollar (offshore) interest ratelags the Treasury bill (domestic) interest rate. However, there is some (weak) evidence ofreverse causality.

More recent studies use the concept of cointegration and error-correction mechanismin testing the relationship between the two interest rates. For example, Fung, and Isberg(1992) use an error-correction model (ECM) to analyze U.S. and Eurodollar Certificatesof Deposit and find that during the period 1981–1983 there exists unidirectional causalityfrom domestic to external (offshore) markets. However, for the period 1984–1988 a signif-icant reverse causality is also observed. Similarly, Fung and Lo (1995) and Tse and Booth(1995) use cointegration and ECM to analyze the relationship between Treasury bill andEurodollar interest rates implied by their respective (International Money Market—IMM)futures contracts and report that the two implied interest rates are cointegrated. Fung andLo (1995) find that these two interest rates Granger-cause each other. However, Tse andBooth (1995) find that the Treasury bill rate causes the Eurodollar rate through the errorcorrection term.

Another approach to the analysis of offshore and domestic interest rates involves theuse of long memory analysis. For example, Fung and Lo (1993) use a rescaled rangestatistic developed by Mandelbrot (1972) and Lo (1991) to test for the long memory inTreasury bill and Eurodollar futures prices and find that neither of the contracts has longmemory.

Booth and Tse (1995) use the concepts of fractional integration (proposed by Gewekeand Porter-Hudak (1983), Granger and Joyeux (1980) and Hosking (1981)) and fractionalcointegration in analyzing the relationship between the 3-month Treasury bill and Eurodollarinterest rates implied by their respective futures contracts. One of the attractive featuresof the fractional cointegration analysis is that conventional cointegration analysis can beconsidered as a special case of the fractional cointegration analysis. Booth and Tse use thefractional cointegration analysis suggested by Cheung and Lai (1993). The analysis involvesa two-step technique. In the first step, a regression of one interest rate on the other interestrate is estimated. This regression is sometimes referred to as a cointegrating regression. Asthe second step, the fractional (root) parameter of the residual of the cointegrating regressionis estimated. The two interest rates are said to be fractionally cointegrated if both the interestrates consist of unit roots and the fractional (root) parameter (d ) of the residual is less thanone (i.e., d < 1). It can be shown that if the fractional parameter is less than one, the residualprocess will be mean reverting (see Cheung and Lai (1993)).1 Therefore, if the residual ofthe fractional cointegration is mean reverting, then the two interest rates are considered tobe fractionally cointegrated. This is the definition of fractional cointegration suggested byCheung and Lai (1993). Using this definition, Booth and Tse (1995) find the two interest

EXPECTED TREASURY BILL AND EURODOLLAR INTEREST RATES 67

Figure 1. Daily TED spread (ED yield minus TB Yield) (March 1, 1982–April 31, 1997).

rates to be fractionally cointegrated for their entire sample period (the fractional para-meter d is found to be about 0.75). They also report that the two interest rates are fractionallycointegrated during the post-crash (1/4/1988–2/22/1994) period but not during the pre-crash(3/1/1982–9/30/1987) period.

It is important to note that the definition of fractional cointegration suggested by Cheungand Lai (1993) differs in spirit from the conventional definition of cointegration used byEngle and Granger (1987). According to Engle and Granger (1987), two non-stationary(unit root) series are cointegrated if there exists a stationary relationship between the twoseries. This essentially means that the residual series of the cointegrating regression mustbe stationary. However, the definition used by Cheung and Lai (1993) does not requirethe residual to be stationary. This follows because the residuals are (covariance) stationaryonly if the fractional parameter is less than 0.5. If the fractional parameter is greater thanor equal to 0.5, the residual will be not be covariance stationary and the variance of theresidual process will be infinite (see Hosking (1981)).

Based on the above discussion, we can point out two problems associated with theresults obtained and the definition used by Booth and Tse (1995). First, Booth and Tse(1995) report the relationship between the two interest rates is non-stationary because thefractional parameter of the residual is found to be significantly greater than 0.5 even if itis less than 1. This is true for the whole sample period as well as for sub-periods. Thismeans the residual, which represents the deviation of one interest rate from its equilibriumvalue in relation to the other interest rate, has unbounded variance. In other words, thedeviation from equilibrium could grow without bound. This behavior of almost identicalinterest rates is inconsistent with what economic theory would suggest. The second problemis the definition of fractional cointegration used by Booth and Tse (1995). This definitionis not consistent with the conventional definition of cointegration. This relates to the factthat even a non-stationary relationship is considered to be cointegrated so long as it is amean-reverting.

The main motivation of this paper is to address these problems by further analyzing the(cointegrating) relationship between the two interest rates by using more general models, amore efficient estimation technique, a longer time period as well as a refined definition offractional cointegration that is consistent with the conventional definition proposed by Engle


and Granger (1987). This will provide us with a better understanding of the relationshipbetween the two interest rates as well as an opportunity to test for stationarity.

In this paper, we extend the fractional cointegration analysis of Treasury bill and Eu-rodollar interest rates performed by Booth and Tse (1995) in several respects. First, thepaper extends the sample period used by Booth and Tse (1995) from 3/1/1982–2/22/1994to 3/1/1982–5/1/1997. This is expected to incorporate more recent trends in the analysisas well as increase efficiency due to a larger sample size. Second, as discussed above, thedefinition of fractional cointegration is refined so that fractional cointegration implies astationary relationship (d < .5) rather than just mean-reverting (d < 1). Third, in additionto the Geweke and Porter-Hudak (GPH) estimation method used by Booth and Tse (1995),we use an alternative Maximum Likelihood (ML) estimation method which eliminates theproblem of multiple estimates associated with the GPH method. Furthermore, the ML esti-mator is expected to be more efficient because the GPH method leads to a large reductionin effective sample size. Fourth, we test to see if the relationship between the Eurodollarand Treasury bill interest rates is heteroscedastic instead of homoscedastic. We find thatthe relationship is heteroscedastic. Therefore, we estimate the relationship between the twointerest rates using the heteroscedastic models GARCH and Exponential GARCH. Finally,we modify the fractional cointegration test so that it will be suitable for heteroscedasticmodels. The modification involves the application of the fractional cointegration test onthe standardized residuals instead of non-standardized residuals. The empirical evidenceindicates that the null hypothesis of fractional cointegration cannot be rejected. This im-plies that the relationship between the expected Treasury bill and Eurodollar interest ratesis stationary and thus the domestic and international markets for short-term U.S. dollarborrowings are integrated. This result is different from the one obtained by Booth and Tse(1995) who find the relationship between the two interest rates to be non-stationary eventhough it is mean reverting.

The paper is divided into four sections. In Section 2, we discuss the concepts of fractionalintegration and cointegration. The discussion will also include the estimation of the frac-tional parameter. The empirical results are presented in Section 3 and the paper concludeswith Section 4.

2. Long-run analysis and fractional cointegration

2.1. Fractional integration

The fractionally integrated time series process, independently proposed by Granger andJoyeux (1980) and Hosking (1981), can be expressed by the following stochastic equation2:

(1 − L)d Xt = ut , t = 1, . . . , T, (1)

where ut is a stationary process, d is a real number and L is a lag operator, i.e., L Xt = Xt−1.Since the parameter (or root) d can take fractional values, the process Xt is known as a


fractionally integrated process. In general, the process Xt is called an I (d) process. Whend = 1, Xt is known as unit-root (I(1) or random walk) process. When d < 1, the process Xt issaid to be a mean-reverting process in the sense that its infinite cumulative impulse responseis zero, which implies that there is no long-run impact of an innovation on Xt (see Cheung andLai (1993)). However, when 0.5 < d < 1, the process Xt is still nonstationary. On the otherhand, when 0 < d < 0.5 the process Xt is (covariance) stationary and its autocorrelationfunction ρ(τ) for large lags (τ ) is proportional to τ 2d−1. Since the autocorrelation functionof a nonstationary process approaches zero hyperbolically (instead of the faster geometricdecay of a stationary process) this series is known as a long memory process. It is importantto note that when d < 0.5 the process is stationary as well as mean-reverting and when0.5 ≤ d the process is non-stationary even if the fractional parameter is significantly lessthan 1 (i.e., even if the process is mean-reverting).

2.2. Estimation of the fractional parameter using Geweke and Porter-Hudak method

Geweke and Porter-Hudak (1983) (GPH) propose a semi-nonparametric method for theestimation of d, which involves the use of an OLS regression based on a periodogram.Specifically, they consider the following OLS regression equation

ln(I (λ j )) = α − d ln

(4 sin2

(λ j

2

))+ ε j , j = 1, . . . , g(T ), (2)

where I (λ j ) is the periodogram of Xt evaluated at the harmonic ordinates λ j = 2π j/T andg(T ) is an integer such that limT →∞ g(T ) = ∞ and limT →∞[g(T )/T ] = 0. In empiricalanalysis, g(T ) = T µ is used with µ ranging from 0.5 to 0.7. In the calculation of thestandard errors, one can use the theoretical variance of the error term ε, which is given byπ2/6 (see GPH). It is important to note that the sample size for the OLS regression is givenby g(T ) whose value depends on the value of µ. Since one can choose different valuesof µ (ranging from 0.5 to 0.7), we can get different estimates of the fractional parameterfor the same process. For example, Booth and Tse (1995) use two different values for µ

(0.5 and 0.55) and get different estimates of d for the same series. This problem of multipleestimates can be avoided by using a frequency domain maximum likelihood (ML) techniquedeveloped by Dalhaus (1989) and Fox and Taqqu (1986).

Another problem associated with the GPH method is that it uses information from theperiodogram at a small number of ordinates. This leads to a small sample size when usingthe OLS method. For example, suppose that the sample size of the series is 3840, i.e.,T = 3840. This means, for µ = 0.5, the sample size for the OLS method representedby equation (2) will be equal to T µ = 38400.5 ≈ 61. It is well known in statistics that asmaller sample size is expected to lead to a higher standard error (a less efficient estimator).However, for the ML method the sample size is 3840 − 1 = 3839 and therefore it is expectedto yield a more efficient estimator.


2.3. Estimation of the fractional parameter using the maximum likelihood method

The ML technique involves minimizing the negative of the frequency domain log-likelihoodfunction given (apart from an irrelevant constant) by

− ln L(d, σ 2) = 1

2

T −1∑s=1

[ln f (λs) + IT (λs)

f (λs)

], (3)

where λs = 2πs/T and IT (λs) is the periodogram, which, in turn, is given by

IT (λ) = 1

2πT

∣∣∣∣∣T∑

j=1

(y j − y)ei jλ

∣∣∣∣∣2

= 1

2πT

{[T∑

j=1

y j cos( jλs)

]2

+[

T∑j=1

y j sin( jλs)

]2}. (4)

The information matrix is given by

I (θ) = 1

2

T −1∑s=0

1

f (λs)2

[∂ f (λs)

∂θ

][∂ f (λs)

∂θ

]t

, (5)

where

∂ f

∂σ= 2σ

2π2−d(1 − cos(λs))

−d

and

∂ f

∂d= σ 2

2π(2 − 2 cos(λs))

−d ln(2 − 2 cos(λs))(−1)

and θ = (d σ)t is the unknown parameter vector. The ML estimation of the parameters canbe obtained using various numerical techniques that are available. The covariance matrixcan be obtained from the information matrix and can be used for testing various hypothesesregarding the values of the parameters. In testing hypotheses regarding d, one can also usethe well-known asymptotic result that

√T (d − d) ⇒ N

(0,

6

π2

). (6)

Therefore, the asymptotic standard deviation of d is given by√

6/T π2. Note that thismethod only calculates a single estimate of d and therefore eliminates the problem of


different estimates of the fractional parameter (depending on the value of µ chosen) whenusing the GPH method.

2.4. Cointegration and fractional cointegration

The concept of cointegration, as developed by Engle and Granger (1987), refers to theexistence of a stationary relationship between non-stationary variables. For example, Engleand Granger (1987) define two non-stationary (unit root, I (1)) processes to be cointegratedif there exists a linear combination of the two processes, which is a stationary I (0) process.

The conventional concept of cointegration can be described by considering the followingrelationship between two different non-stationary (unit root, I (1)) processes, xt and yt .

yt = α0 + α1xt + zt . (7)

The regression equation (7) is known as cointegrating regression. If there exists two param-eters α0 and α1 such that the residual process zt in equation (7) is stationary (I (0)), thenthe two processes xt and yt are said to be cointegrated in the conventional sense.

Cheung and Lai (1993) define two I (1) processes xt and yt to be fractionally cointegratedif the fractional parameter d of the residual process zt is less than unity (i.e., d < 1). Asmentioned above, d < 1 implies a mean-reverting process but not necessarily a station-ary process, which requires d to be less than 0.5. Therefore, the definition of fractionalcointegration proposed by Cheung and Lai (1993) is not consistent with the conventionaldefinition of cointegration, which implies a stationary instead of simply a mean-reverting,relationship between non-stationary processes. Since Booth and Tse (1995) follow the def-inition used by Cheung and Lai (1993), their test is not consistent with the conventionaldefinition of a cointegrating relationship.

In this paper, we refine the definition proposed by Cheung and Lai (1993) by imposingthe condition that the cointegrating relationship should be stationary, i.e., the fractionalparameter d of the residual process zt should be less than 0.5. In other words, according to ourdefinition, the two non-stationary processes xt and yt are said to be fractionally cointegrated,if the residual process zt is stationary (d < 0.5) in addition to being mean-reverting.

It is important to realize that, in general, the cointegrating relationship between financialvariables is an equilibrium condition. Therefore, the mean-reverting equilibrium relation-ship (as implied by d < 1) is less desirable than a stationary one especially when dealingwith financial variables. This assertion can be supported as follows:

(i) First, if the cointegrating relationship is mean-reverting but not stationary, then adisequilibrium condition, represented by a non-zero zt , in any period will take a verylong (infinite) time to disappear. However, a disequilibrium condition in financialmarkets, in general, implies the existence of some sort of arbitrage opportunity andshould disappear within a very short period of time.

(ii) Second, if the cointegrating relationship is stationary then the cointegrating relationshipwill hold on average with finite variance around the equilibrium condition. This implies


the mean of zt is zero with finite variance. However, if the cointegrating relationshipis only mean-reverting but not stationary, the variance of zt is undefined (infinitelylarge).

(iii) Finally, when analyzing the equilibrium relationship between two financial time series,we are not only interested in finding whether the equilibrium relationship exists butwe are also interested in the specific form of the equilibrium condition. In terms ofequation (7), we might be interested in testing specific restrictions on α0 and α1, such asα0 = 0 and α1 = 1 (the two series xt and yt are equal in equilibrium). However, if thecointegrating relationship is only mean-reverting but not stationary, then conventionaltests (such as the t-test, F-test and χ2-test) on α0 and α1 cannot be applied because thedistributions of these statistics are non-stationary. Furthermore, since no asymptoticdistribution exists, any asymptotic tests will be meaningless.

Therefore, a stationary (rather than simply mean-reverting) relationship between non-stationary variables is a better definition of fractional cointegration when dealing withfinancial time series. Having discussed the concept of fractional cointegration, we nowapproach the issue of testing for fractional cointegration.

Following Engle and Granger (1987), fractional cointegration can be tested using a two-step procedure. In the first step, the cointegrating regression equation (7) can be estimatedusing the OLS method. Once the regression equation (7) is estimated, the second stepinvolves testing for the fractional parameter d of the residual process zt obtained fromequation (7). Either the ML or the GPH method can be used in the estimation of d. Thispaper emphasizes the ML procedure because of the attractive characteristics of this method.

Once the fractional parameter d is estimated, the existence of fractional cointegration canbe tested. Following the refined definition presented above, the two non-stationary processesxt and yt are said to be fractionally cointegrated if the fractional parameter d of its OLSresidual is less than 0.5. Hence, the fractional cointegration test estimates the value of thefractional parameter of the cointegrating regression residual.

Since most financial series are found to be heteroscedastic, there is a possibility that thecointegrating regression between the two interest rates follows a heteroscedastic process in-stead of a homoscedastic process. In other words, the residual of the cointegrating regressionzt follows a stationary heteroscedastic process. If the relationship between the two interestrates is heteroscedastic, regression equation (7) needs to be replaced by heteroscedasticmodels. One widely used heteroscedastic model is known as GARCH(p,q) and is given by(see Bollerslev et al. (1992)):

yt = δ0 + δ1xt + ut , ut | �t−1 ∼ N (0, ht ) (8)

ht = ω0 +p∑

i=1

ωi ht−i +q∑

i=1

ωi ut−i . (9)

Since this GARCH model allows the residual to be heteroscedastic, it is more generalcompared to the homoscedastic model represented by equation (7). The existence of a


heteroscedastic relationship can be tested using the LM test. In this paper, we also use theExponential GARCH (EGARCH) model proposed by Nelson (1991).

However, once we use the GARCH or EGARCH models, care must be taken in estimatingthe fractional parameter of the residual process. It is important to note that both the GPHand the ML methods are designed for a homoscedastic process but the residuals fromGARCH or EGARCH models, by assumption, are heteroscedastic. Therefore, the GPH andML methods cannot be used on the residuals of the GARCH or EGARCH models withoutmodification.

In this paper, we propose to estimate the fractional integration parameter of the standar-dized residual ut/

√ht rather than the non-standardized residual. Since the standardized

residuals are now homoscedastic, both GPH and ML estimation can be used.

3. Empirical results

In this paper we use the daily settlement prices of the 3-month Treasury bill and Eurodollarfutures from March 1, 1982 to May 1, 1997 (sample size equal to 3840). The implied Trea-sury bill and Eurodollar interest rates are computed and converted into the bond equivalentyield (see Booth and Tse (1995) footnote 2). Since these implied interest rates are expectedto prevail in the future, we consider our analysis to be about the relationship between ex-pected interest rates instead of actual spot interest rates. For the results to be comparable toBooth and Tse (1995), the logarithm of the two interest rate series is used in the analysis.Each series is tested for a unit root and for the sake of completeness, both Phillips-Perron(PP) and augmented Dickey-Fuller (ADF) tests are performed even though these tests havelow power against fractional alternatives (see Diebold and Rudebusch (1991) and Sowell(1990)). The results are summarized in table 1. The critical values are obtained from Dickeyand Fuller (1981) and Fuller (1976). The results indicate that both series have a unit rootand it is clear from both the PP and ADF tests that both the Eurodollar and Treasury billinterest rate series are I (1).

Next, the results for estimating the fractional parameter d using the GPH and ML meth-ods are presented in table 2. Here, the unit root (d = 1) is used as the null hypothesis.However, for d = 1, equation (1) represents a non-stationary process and therefore, the usualt-test cannot be applied. In order to get around this problem, the following two equations

Table 1. Results of unit roots tests using ADF and PP tests

Log(TB) Log(ED) � Log(TB) � Log(ED)

ADF with Intercept −2.1549 −2.1324 −7.6562 −7.5250ADF with Intercept and Trend −1.8950 −1.8147 −7.7279 −7.6082PP with Intercept −2.1507 −2.0616 −60.176 −60.358PP with Intercept and Trend −1.9747 −1.7889 −60.224 −60.350

Note: The critical values are −2.8628(−2.5675) at 5%(10%) level of significance for models without trend and−3.4136(−3.1285) at 5%(10%) level of significance with models with trend.


Table 2. Results of the GPH and ML technique

GPH technique

Dependent variable

Log(TB) Log(ED)

µ N = T µ d T -Statistic* d T -Statistic

0.50 61 1.0259 0.282 1.0659 0.7170.55 93 1.0407 0.563 1.0467 0.6460.60 141 1.0083 0.144 1.0634 1.1010.65 213 0.9952 0.104 1.0497 1.0780.70 322 0.9802 0.532 1.0167 0.449

ML technique

d T -Statistic* d T -Statistic

1.0180 0.948 1.0260 1.429

*The unit root (i.e., d1 = 0) is used as the null hypothesis.

are estimated:

(1 − L)d1�Log(TB) = u1t and (1 − L)d2�Log(ED) = u2t .

Thus, the fractional parameters of Log(TB) and Log(ED) are given by (d1 +1) and (d2 +1)

respectively. The null hypothesis of d = 1 can be tested by the null hypothesis of d1 = 0.The results summarized in table 2 do not reject the null hypothesis that d = 1 for bothseries. Hence, as in the PP and ADF tests above, the same conclusion of non-stationarity(unit root) is obtained. The standard errors of GPH estimates of d for µ = 0.5 are about5 times larger than the standard errors of the ML estimates of d. In addition, the standarderrors of GPH estimates of d for µ = 0.7 are approximately 2 times larger than the standarderrors of the ML estimates. This confirms our earlier statement that the ML estimator isexpected to be more efficient than the GPH estimator because of the reduced sample size.

Since both series are found to be non-stationary, our next step is to perform the fractionalcointegration test. As discussed above, this requires the estimation of the cointegrating re-gression (7) as the first step of a two-step procedure. Since there is no specific reason forchoosing either Log(TB) or Log(ED) as the dependent (LHS) variable, two cointegratingregressions, one with Log(TB) as dependent variable and another with Log(ED) as thedependent variable, are estimated. Following Booth and Tse (1995), two more cointegrat-ing regressions with a trend term are estimated. Therefore, a total of four cointegratingregressions are estimated. The results are summarized in table 3 and indicate that all theparameters including the trend term are highly significant.

In the second step, the fractional parameters of the four different residual series (arisingfrom the four different cointegrating regressions) are estimated. Both the GPH and ML


Table 3. Results of OLS estimation of cointegrating regression equation (7)

Dependent variable

Log(TB) Log(ED)

γ LM test γ LM testα0 α1 (10−5) (T R2) R2 α0 α1 (10−5) R2 (TR2)

No trend −0.234 0.955 3478.6 0.989 0.214 1.036 0.989 3495.3(0.0043) (0.0016) (0.0049) (0.0018)

With −0.163 0.995 1.769 3462.3 0.991 0.118 0.985 −2.23 0.991 3462.5trend (0.0051) (0.0023) (0.0758) (0.0054) (0.0023) (0.072)

Note: LM Test is the test for the conditional heteroscedasticity, which is given by T R2 (see Engle (1982)). TheLM statistic follows a χ2 distribution with one degrees of freedom and the critical values are 3.84(2.71) at5%(10%) level of significance. The figures inside the parentheses are standard errors.

Table 4. Results of the fractional cointegration tests on the residuals from equation (7). Using GPH andML techniques

GPH technique

Dependent variable

Log(TB) Log(ED)

t-Statistic t-Statistic

µ T µ d d = 1 d = 0.5 d d = 1 d = 0.5

Without trend 0.50 61 0.7059 −3.204 2.242 0.7090 −3.170 2.2760.55 93 0.7189 −3.887 3.027 0.7272 −3.772 3.1410.60 141 0.7632 −4.115 4.575 0.7669 −4.051 4.6390.65 213 0.7805 −4.760 6.084 0.7824 −4.719 6.1250.70 322 0.7963 −5.489 7.985 0.7997 −5.398 8.076

With trend 0.50 61 0.6865 −3.414 2.032 0.6978 −3.292 2.1540.55 93 0.6780 −4.452 2.461 0.6967 −4.193 2.7200.60 141 0.7242 −4.794 3.896 0.7412 −4.498 4.1920.65 213 0.7571 −5.269 5.575 0.7689 −5.013 5.8310.70 322 0.7797 −5.936 7.538 0.7875 −5.726 7.748

ML technique

Without trend 0.8943 −5.816 21.692 0.8930 −5.927 21.777With trend 0.8904 −6.199 22.081 0.8895 −6.335 22.323

techniques are used in the estimation. These estimation results are summarized in table 4.As mentioned earlier, when using the GPH method multiple estimates of the fractionalparameter d can be a problem. In some cases, the difference between the smallest andthe largest estimates is approximately 0.1. However, all of the estimates indicate that allfour residual processes are mean-reverting but non-stationary (i.e., 0.5 ≤ d < 1). The ML


estimates for the fractional parameters also indicate that the residual processes have frac-tional parameters significantly less than 1 (mean-reverting). However, all the fractionalparameters are significantly greater than 0.5 (non-stationary).

Obviously, our GPH results are very similar to those obtained by Booth and Tse (1995).As to the ML estimates, even though they are numerically different (higher) than thoseobtained from the GPH method, they lead to the same conclusion that the residual seriesare mean reverting but non-stationary.

However, the non-stationarity between these two nearly identical interest rates couldbe due to model misspecification. The cointegrating relationship could be heteroscedasticinstead of homoscedastic as assumed in regression equation (7). In order to test for theexistence of a heteroscedastic cointegrating regression, the LM statistic, as suggested byEngle (1982), is estimated for all four cointegrating regressions. The LM statistics, whichfollow χ2 distributions with one degree of freedom, are reported in table 3 and are all highlysignificant. This indicates that the cointegrating relationship is heteroscedastic instead of ahomoscedastic.

In order to incorporate the heteroscedasticity, we use two popular heteroscedas-tic models—GARCH and Exponential GARCH (EGARCH).3 We estimate variousGARCH(p,q) models represented by equations (8) and (9) with p ranging from 1 to 5 andq ranging from 1 to 5.4 This involves the estimation of twenty-five different GARCH(p,q)models with Log(ED) as the dependent variable and another twenty-five GARCH(p,q)models with Log(TB) as the dependent variable. The estimation is again repeated with atrend term included in equation (8). This involves the estimation of another fifty GARCHmodels. Based on the values of the log-likelihood function, Akaike Information Criterion(AIC) and Schwartz Bayesian Criterion (SBC), and the significance of parameters, the mostsuitable values of p and q are chosen for each of the four different cases—(i) Log(TB) asdependent variable without a trend term in equation (8), (ii) Log(TB) as dependent vari-able with a trend term included, (iii) Log(ED) as dependent variable without a trend termand (iv) Log(ED) as dependent variable with the trend term included.5 Then the fractionalparameter of the standardized residual is computed for each of the four cases. The resultsare reported in Panel A of table 5 for the GARCH model. The whole process is repeatedusing the EGARCH model instead of the GARCH model. The results for the EGARCHmodel are presented in Panel B of table 5. Due to the problem of multiple estimates as wellas inefficiency associated with the GPH method, we only use the ML method in order toestimate the fractional parameter of the standardized residual.

There are significant differences between the estimates of the fractional parameters ofhomoscedastic residuals (reported in table 4) and the estimates of the fractional parametersof the standardized residuals derived from heteroscedastic models (reported in table 5).The values of the fractional parameters of the standardized residuals are much closer to thecritical value of 0.5.

Since the estimates of the fractional parameter are close to 0.5 (the boundary point thatseparates stationarity from non-stationarity) our conclusion will depend on the selection of anull hypothesis. If we choose the null hypothesis to be d = 0.5 (non-stationary), then we willnot reject the null hypothesis of non-stationarity. However, for all cases (with Log(TB) andLog(ED) as the dependent variable, with or without the trend terms), the 95% confidence


Table 5. The maximum likelihood estimation of the fractional parameters of the standardized residuals fromGARCH(p,q) and EGARCH(p,q) model

Panel A: GARCH(p,q) model

Dependent variable

Log(TB) Log(ED)

T -Statistic T -StatisticWith or withouttrend term (p,q) d H0 : d = 1 H0 : d = 0.5 (p,q) d H0 : d = 1 H0 : d = 0.5

Without trend (2,2) 0.4957* 35.204 −0.300 (1,2) 0.4946* 35.195 −0.376With trend (2,1) 0.5204* 39.755 1.691 (2,2) 0.5188* 37.956 1.483

Panel B: Exponential GARCH(p,q) model

Dependent variable

Log(TB) Log(ED)

T -Statistic T -StatisticWith or withouttrend term (p,q) d H0 : d = 1 H0 : d = 0.5 (p,q) d H0 : d = 1 H0 : d = 0.5

Without trend (4,4) 0.4950* 37.947 −0.376 (5,3) 0.4911* 38.088 −0.666With trend (5,4) 0.5196* 41.414 1.690 (5,3) 0.5064* 41.243 0.535

*Estimate of d for which the 95 percent confidence interval includes part of the stationary region (i.e., the lowerlimit is less than 0.5).

interval includes the stationary region (d < 0.5). Therefore, the nature of our conclusiondepends on the choice of the null hypothesis. For example, if we choose the null hypothesisto be d = 0.499 (stationary), we will not reject the null hypothesis of stationarity at a 5%level of significance.

In any event it is traditional, except in the case of cointegration and unit root tests, that thenull hypothesis is derived from the equilibrium conditions. Here, economic theory suggeststhat the two series should have a stationary long run equilibrium relationship and this isimplied by a cointegrating relationship. Therefore, if we take this equilibrium condition asthe null hypothesis, then stationarity cannot be rejected. This implies that the domestic andinternational markets for short-term U.S. dollar borrowings are integrated.

4. Conclusions

In this paper, we extend Booth and Tse’s (1995) analysis of the fractional cointegratingrelationship between the expected Eurodollar and Treasury Bill interest rates implied bytheir respective futures contracts. The fractional cointegration test is related to the fractional


parameter d of the residual process obtained from a cointegrating regression. Booth and Tse(1995) use the definition of fractional cointegration suggested by Cheung and Lai (1993),where fractional cointegration requires only a mean-reverting (d < 1), instead of stationary(d < 0.5), relationship between the two interest rates. This definition is not consistent withthe conventional definition of cointegration. Another problem associated with the Booth andTse (1995) study is that they find the relationship between the two interest rates to be non-stationary (i.e., d is significantly greater than 0.5). This is contrary to what economic theorywould predict. Our main motivation is to address the issue of non-stationarity between thesetwo interest rates and to refine the definition of fractional cointegration so that it is consistentwith the conventional definition by extending the study of Booth and Tse (1995).

Following the conventional approach to the definition of cointegration (see Engle andGranger (1987)), we refine the definition suggested by Cheung and Lai (1993). Specifically,two interest rates are fractionally cointegrated if the fractional parameter of the residual isstationary (d < 0.5). This definition is consistent with the definition proposed by Engle andGranger (1987).

Booth and Tse (1995) use the Geweke and Porter-Hudak (GPH, 1983) method in esti-mating the fractional parameter of the residual from a cointegrating regression. However,this method leads to multiple estimates as well as a significant loss of sample size. In thispaper, a more efficient Maximum Likelihood (ML) estimation method is used. The MLmethod is not only more efficient but also has a unique (rather than multiple) estimate of thefractional parameter. Results from both GPH and ML methods indicate that a simple linearhomoscedastic relationship (assumed by Booth and Tse (1995)) between the two interestrates is mean reverting but non-stationary. This implies, according to the refined definition,that the two interest rates are not cointegrated.

However, we find that these results indicating a lack of cointegration are caused by theexistence of a heteroscedastic cointegrating relation instead of the homoscedastic relationas assumed by Booth and Tse (1995). The LM tests strongly suggest the use of a condi-tional heteroscedastic cointegrating regression instead of a homoscedastic one. Therefore,we extend the model in Booth and Tse (1995) by utilizing the well known conditionalheteroscedastic models of GARCH and Exponential GARCH to represent the cointegratingregression. However, once the heteroscedastic models are used to describe the cointegratingregression, the fractional cointegration test, which is designed for a homoscedastic process,cannot be applied to the residuals. In this paper, we suggest the use of a fractional cointegra-tion test on a standardized residual which satisfies the homoscedasticity condition requiredby the fractional cointegration test.

These standardized residuals yield a significant reduction in the fractional parameter(d) towards stationarity. As a result, stationarity of the residuals cannot be rejected as anull hypothesis. In other words, the 95% confidence interval for d includes the stationaryregion (d < 0.5). Therefore, if we take stationarity as the null hypothesis, then the empiricalevidence does not reject the hypothesis that the expected Eurodollar and Treasury Billinterest rates are fractionally cointegrated.

However, it is important to note that the 95% confidence interval for d also includesthe non-stationary region (d ≥ 0.5). Therefore, if we choose non-stationarity as the nullhypothesis, it cannot be rejected either. As a result, the issue of fractional cointegration be-


tween the two yields is not completely settled. We suspect that the nature of the relationshipmight have changed over time in the form of a structural shift. This can be analyzed bylooking at different sub-periods rather than the entire sample period. Another possibility isthat the cointegration relationship could be non-linear instead of the linear type analyzedin this paper. We intend to pursue this matter in the future.

Notes

1. A mean-reverting process means that the infinite cumulative impulse response is zero. This mean-revertingcharacteristic, when applied to the relationship between the two interest rates, implies that any deviation fromequilibrium at time t will disappear eventually.

2. See Baillie (1996) for a comprehensive discussion on the long memory process.3. We would like to thank the anonymous referee for suggesting the use of GARCH(p,q) and EGARCH(p,q)

models with suitable choice of values for p and q.4. Since we are analyzing daily data, we expect these values for p and q to be sufficient to describe the cointegrating

relationship.5. Here we are dealing with non-nested models. For example, GARCH(3,1) and GARCH(1,3) are non-nested.

Similarly, all the GARCH and EGARCH models are non-nested models. Therefore, we are using a combinationof criteria instead of one single criterion to select the best model.

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