fractionally integrated models with arch errors: with an application to the swiss 1-month euromarket...

Review of Quantitative Finance and Accounting, 10 (1998): 95–113© 1998 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.

Fractionally Integrated Models With ARCH Errors:With an Application to the Swiss 1-Month EuromarketInterest Rate

MICHAEL A. HAUSERUniversity of Economics and Business Administration, Augasse 2-6, A-1090 Vienna, Austria

ROBERT M. KUNSTInstitute for Advanced Studies, Stumpergasse 56, A-1060 Vienna, Austria and Johannes Kepler University,Linz-Auhof, A-4040 Linz, Austria

Abstract. We introduce ARFIMA-ARCH models, which simultaneously incorporate fractional differencingand conditional heteroskedasticity. We develop the likelihood function and we use it to construct the bias-corrected maximum (modified profile) likelihood estimator. Finite-sample properties of the estimation procedureare explored by Monte Carlo simulation. Backus and Zin (1993) have motivated the existence of fractionalintegration in interest rates by the persistence of the short rate and the variability of the long end of the yieldcurve. An empirical investigation of a daily one-month Swiss Euromarket interest rate finds a difference pa-rameter of 0.72. This indicates non-stationary behavior. In contrast to first-order integrated models, the long-runcumulative response of shocks to the series is zero.

Key words: Conditional heteroskedasticity, autoregressive moving average model, fractional differencing, longmemory

1. Introduction

Models involving fractional differences have recently drawn much attention, both in thefields of financial data (exchange rates (Cheung 1993a; Diebold et al. 1991), interest rates(Backus and Zin 1993), stock prices (Lo 1991)) and of macroeconomics such as businesscycle analysis (Diebold and Rudebusch 1989; Sowell 1992b) or inflation (Hassler andWolters 1995; Baillie et al. 1996). In the latter area, fractional differences are often seenas “bridging the gulf ” between the concepts of trend stationarity and of integrated series(see Robinson 1994). Viewing, e.g., industrial output as being fractionally integrated witha differencing parameter 0 , d , 1 explains the concentration of spectral mass atfrequency 0 and permits to retain the more classical picture of trend reversion and tran-sitory shocks. For a much more complete survey see Baillie (1996).

Unlike macroeconomic aggregates, financial price series on theoretically efficient mar-kets, such as stock prices or exchange rates, are close to random walks and little dynamicstructure can be found. Incited by the lucrative prospects of being able to predict futuremovements of prices on “nearly” efficient markets, many researchers have taken up thechallenge to detect as much structure as possible. Some researchers claim that long-run

Kluwer Journal@ats-ss3/data11/kluwer/journals/requ/v10n1art6 COMPOSED: 11/04/97 10:33 am. PG.POS. 1 SESSION: 8

dependence or long memory is present in some financial series which would allow sys-tematic gains from speculation over sufficiently long time intervals. Such long memorycan be well represented by fractional models of the autoregressive fractionally integratedmoving-average type (in short, ARFIMA(p, d, q); see Granger and Joyeux (1980) andHosking (1981)). Empirical evidence on this phenomenon remains mixed. Cheung(1993b) indicates that failing to incorporate infrequent shifts of the mean in a model mayinduce the spurious finding of long memory.

Whereas short-run dependence of increments or returns is low, serial dependence involatility is a well known empirical fact and has incited research on ARCH models ofvarious types (in particular, see Engle (1982) and Weiss (1984)). Early research in thisdirection considered monthly series (Engle (1982)) but, as daily data appear to be the“natural” frequency of many financial data,—e.g. common stocks, exchange rates, money-market rates—interest has shifted toward high-frequency time series. In daily data, ARCHeffects are typically more pronounced than in weekly or monthly series but also morecomplicated due to the intra-week microstructure of financial markets.

Fractional models for short money-market rates have been suggested by Backus andZin (1993). Using the approach of Vasicek (1977) they derive the appertaining dynamicsfor the term structure of interest rates in discrete time. They find that their processes arecompatible with the observation that even very long-term interest rates display substantialvolatility which does not square with traditional ARMA models. Their empirical analysisleads to the conclusion that inflation (cf. also Baillie et al. 1996) and money growth arethe sources of long memory. The existence of fractional integration in the spot rate alsocauses a type of long-range dependence in their future yield rates that will be the subjectof investigation in our empirical application. Similar ideas are pursued by Shea (1991)who considers fractional models for the term structure and assumes a common d param-eter for all maturities. In weekly and monthly US and Japanese series, his d estimatesrange from 0.7 to nearly 1. Crato and Rothman (1994) apply ARFIMA models to mac-roeconomic data and estimate d as 0.81 in a monthly series of US bond yields. However,none of these studies takes the joint presence of ARCH effects and long memory intoaccount.

Because both non-standard features (ARCH and ARFIMA) have aroused interest in theliterature on financial time series, it seems worth while to study both at the same time andto gauge possible cross-effects between estimation of heteroskedasticity and of modelinglong-run dependence (see also, e.g., Cheung (1993a, p.95) or Lo (1991, p. 1283)). Con-sequently, we suggest the use of ARFIMA-ARCH models and develop a maximum-likelihood estimation procedure. In related but independent work, Baillie et al. (1996)have identified similar structures in a multi-country study of monthly inflation and labeledthem ARFIMA-GARCH. In line with Backus and Zin (1993), we apply our procedure toa money-market rate, viz. the Swiss yield on Euro-bonds with duration of one month.Although the Swiss economy may seem comparatively small, many applied economistsregard the Swiss money market as influential for major European economies, hence Swissinterest rates are frequently used as ingredients in the construction of leading indicatorsfor other countries.

96 MICHAEL A. HAUSER AND ROBERT M. KUNST


This paper is organized as follows. In Section 2, we present the ARFIMA-ARCH modeland also give stationarity conditions. Section 3 develops the maximum-likelihood estima-tion procedure and presents some evaluation of its small-sample performance by MonteCarlo simulation. In section 4, the maximum likelihood (ML) procedure is applied to firstdifferences of daily data on the Swiss one-month Euromarket rate for the time periodJanuary 1986 to April 1989 and evidence on intermediate memory (for a definition, seeSection 2) in the series as well as on Engle-type ARCH effects is established. Section 5concludes.

2. Model and stationarity conditions

We consider weakly stationary processes with a Wold representation of the form

yt 2 µ 5 (i50

`

ci«t2i

where «t is uncorrelated noise with mean zero. We restrict attention to processes obeyingARFIMA models with conditionally heteroskedastic errors. In their most general form,these models can be written as follows:

F~B!~1 2 B!d~yt 2 µ! 5 Q~B!«t

E~«t2?It21! 5 ht (1)

Here, F(B) 5 1 2 w1B 2 … is the autoregressive (AR) polynomial of order p, and Q(B)5 1 1 u1B 1 … is the moving average (MA) polynomial of order q, with all roots outsidethe unit disk and no common roots. d is the fractional differencing parameter in thehalf-open interval [21/2,1/2). These conditions guarantee stationarity and invertibility(Brockwell and Davis 1991, p. 524f.). In concordance with prevailing usage in the currentliterature, we use the word “long memory” for d . 0 and “intermediate memory” for d ,0. It denotes the information set generated by {yt, yt21, yt22,…} and ht is some functionof t to be specified in more detail.

Depending on the exact specification of the ht function, various heteroskedasticARFIMA models may be considered. Firstly, we consider Engle’s (1982) ARCH modelthat expresses conditional heteroskedasticity via a linear combination of squared pastinnovations:

ht 5 E~«t2?It21! 5 a0 1 a1«t21

2 1…1 as«t2s2 (2)

The model defined by (1) & (2) will be denoted ARFIMA-ARCH(p,d,q/s) in the follow-ing. It is seen that stability properties of the ARCH process in (2) do not depend on thelinear ARFIMA model in (1). Assuming a0 . 0, ai $ 0, i 5 1,…,s, and a conditional

FRACTIONALLY INTEGRATED MODELS 97


normal distribution, Engle (1982) proved that a weakly stationary solution for the «t

process exists iff a1 1…1 as , 1. His result easily generalizes to a broader class ofconditional distributions with finite second moments. Nelson (1990) showed that strictlystationary solutions may still exist if Engle’s condition fails. Such solutions necessarilydisplay infinite variances. Even if a weakly stationary solution exists and a conditionalGaussian law is assumed, the stationary distribution of «t is non-Gaussian and leptokurtic,except in trivial cases.

The conditions for weak stationarity of the ARFIMA-ARCH processes (1) & (2), i.e.,based on Engle-ARCH, can be summarized as follows. If (i) all p roots of the ARpolynomial F(.) have modulus greater than 1, (ii) d , 1/2, and (iii) (i51

s ai , 1, ai $ 0for all i $ 1, and a0 . 0, then a strictly and covariance-stationary solution to the equations(1) & (2) exists. The three conditions guarantee individual stability of the ARMA, frac-tional, and ARCH part, respectively. Then, the variance of the resulting ARFIMA-ARCHprocess can be expressed as

a0

1 2 (i51

s

ai

(i50

`

ci2

where C(B) 5 (1 2 B)2dF21(B)Q(B). In particular, for pure fractional processes, thevariance is given by

a0

1 2 (i51

s

ai

G~1 2 2d!

G2~1 2 d!

Whereas the covariance-stationarity conditions for the ARFIMA-ARCH process are notstronger than for each process separately, the existence of higher moments and otherproperties of the stationary distribution depend on assumptions on the conditional distri-bution of the ARCH innovations. In the simplest case of a conditional Gaussian law,moments have been provided by Milhøj (1985). The mutual independence of conditionson the ARFIMA and the ARCH part hinges critically on the specified ht function (2). Forexample, assuming that conditional error variances depend on squared past observationsof the process Yt itself, Weiss (1984) has shown that the existence of a weakly stationarysolution depends on joint conditions for the linear and the ARCH parameters. In anARFIMA-ARCH model built on Weiss’ specification, stationarity conditions on d and theARCH coefficients interact in a complicated way. For a recent survey of the large varietyof other linear and non-linear ht specifications in current use, see Hentschel (1995).

Within the limits of this paper, we restrict ourselves to the conditional Gaussian speci-fication in the Engle-type ARCH model (2), i.e., we explicitly assume «t.It21 ; N(0,ht).However, the likelihood approach presented in the next section holds for other possibleARCH specifications as well. In practice, one frequently encounters situations where theconditional Gaussian assumption appears to be violated. A theoretical treatment of the



properties of quasi-ML estimation methods for ARFIMA models in the presence ofnon-normality was provided by Giraitis and Surgailis (1990). With respect to ARCH,Weiss (1986) and Lee and Hansen (1994) explicitly view ML estimation in conditionallyGaussian ARCH models as quasi-ML, thus allowing for distributional misspecification.

3. Estimation

There is an increasing number of consistent methods—semiparametric or parametric—forestimating the fractional parameter. For a survey, see Baillie (1996). Specifying a fullmodel we are interested in parametric estimators and decided to consider only ‘exact’maximum likelihood (ML) procedures in spite of their computational complexity. Inparticular, we will use a modified profile likelihood estimation procedure. The reason forthis is that even the exact ML estimate of the fractional parameter exhibits a small samplebias inducing a divergence from the asymptotic distribution of the affected likelihood ratiostatistic as recent studies indicate (Cheung and Diebold 1994; An and Bloomfield 1993;Hauser 1993). Furthermore, the bias may increase considerably if AR and MA compo-nents are included in the model.

The modified profile likelihood estimation derived by An and Bloomfield (1993) per-forms well even in small samples. Cheung and Diebold (1994) consider an approximationto the Whittle (spectral) likelihood to construct an alternative estimator. Their suggestionturned out to be not so successful in improving the mean squared error in small samples.However, Hauser (1993) shows that the Whittle likelihood yields essentially unbiasedestimates in pure fractionally integrated models, with smaller mean squared errors for abroad range of d values. Hence, there are two methods available which do not suffer fromthe small sample bias problem for the linear part of the model. On the other hand, Baillieet al. (1996) implement a conditional sum of squares approximation of the exact maxi-mum likelihood method for the linear part of their ARFIMA-GARCH model to achievecomputational simplifications. It turns out that, for small samples of pure FI processes, theproperties of that method are similar to those of exact ML (see their Table 1a.). We arguethat, in order to use AIC (Akaike’s Information Criterion) for selecting between nonfrac-tional and fractional models—as we will proceed below—it is essential that the imple-mented estimator gives unbiased estimates of d. A biased estimator would go along witha maximized likelihood which is too large, and so the AIC will likely choose the wrongmodel (Hauser, 1993). We observe that Baillie et al. (1996), in comparing inflation ratesacross different countries, consider a fixed model structure.

Maximum likelihood estimation

Estimation can be based on the Gaussian ML algorithm for ARFIMA models by Hosking(1984). This method considers the full likelihood for T successive observations YT 5(y1,…,yT) from an ARFIMA model



,~µ,S;YT! 5 ~2p!2T/2?S?21/2exp~212~YT 2 µT!8S21~YT 2 µT!!

µT 5 µ~1,…,1!8 (3)

where S is a symmetric Toeplitz matrix that depends on the autoregressive parameters F,on the moving average parameters Q, on the fractional differencing parameter d, and onthe innovations variance s«

2. The numerical evaluation of S in (3) for given F and Q isderived in Sowell (1992a) using the hypergeometric function and the roots of the ARpolynomial. Dahlhaus (1989) considered estimating µ by the arithmetic mean and maxi-mizing (3) for the remaining parameters. He showed that this ML estimator is=T-consistent, asymptotically normal and efficient.

If the data are generated from an ARFIMA model, the coefficient parameters F, d, Qtogether with s«

2 uniquely determine the unconditional variance-covariance matrix S. Letus denote its Cholesky decomposition by S 5 LL8. From (3) we can derive the likelihoodof the linear process innovations ET 5 («1,…,«T)8 5 s«L21(YT 2 µT) according to thedensity transformation theorem:

,~s«2; ET! 5 ~2p!2T/2s«

21 expS21

2s«2

ET8 ETD (4)

This representation is valid for the standard case of homoskedastic observations. In pass-ing, we remark that «t are not necessarily the ARFIMA innovations but, leaving the trueinnovations undefined for t , 1, they represent possible innovations that secure a time-constant correlation structure exactly corresponding to S(F, d, Q).

Now consider the ARFIMA-ARCH model where the innovations are conditionallyheteroskedastic. Then, V(«t.It21) 5 ht and ET is heteroskedastic. The correction for het-eroskedasticity is conducted by dividing each element by its conditional standard devia-tion, i.e., by multiplying the inverse of diag(h1,…,hT) into ET.

H21/2ET 5 diag~h121/2,…,hT

21/2!ET 5 ~u1,…,uT!8 5 UT

UT is then homoskedastic and distributed standard normal N(0, IT). ht is a function of pastinformation, its particular form is determined by the selected ARCH model class (e.g.,Engle- or Weiss-type ARCH).

For the ARFIMA-ARCH model, the process likelihood of the innovations ET in (4)changes to

,~H; ET! 5 ~2p!2T/2?H?21/2 expS212

ET8 H21ETD

5 ~2p!2T/2 )t51

T

ht21/2 (

t51

T

expS2«t

2

2htD (5)



Re-transforming the likelihood, we obtain from (5)

,~µ, S, H; YT! 5 ~2p!2T/2?LHL8?21/2

3 exp$212

~YT 2 µT!8~LHL8!21~YT 2 µT!%or, defining SH 5 LHL8,

,~µ, SH; YT! 5 ~2p!2T/2?SH?21/2 exp$21

2~YT 2 µT!8SH

21~YT 2 µT!% (6)

SH is a random matrix, unlike the constant S encountered in the multivariate Gaussianmodel. From SH 5 LHL8, it follows that this randomness is rooted solely in the random-ness of H, as L is constant. H is diagonal, with ht an It21-measurable random variable.Hence, (6) is not the likelihood of a Gaussian distribution problem. One can, however,interpret (6) formally as the likelihood of an ARFIMA process with process-dependentcovariance matrix SH. The numerical evaluation of the likelihood can be conducted via theLevinson algorithm applied to S (Marple 1987, p. 87).

Although a formal proof of the asymptotic properties of the ML estimator is beyond thescope of this paper, we may conjecture that consistency and asymptotic normality followfrom those of the ARFIMA and ARCH models as outlined in the cited literature. (5)corresponds to the log-likelihood given by Engle (1982, p. 990). For the pure ARCHmodel, i.e., yt 5 µ 1 «t, consistency and asymptotic normality of the ML estimator basedon (5) was shown by Engle (1982) and the results were extended to the ARMA-ARCHmodel by Weiss (1984). Lee and Hansen (1994) gave a new proof of the asymptoticproperties, which enabled them to relax the finite fourth-order moments conditions em-ployed by Weiss (1984, 1986). The contribution by Lee and Hansen (1994) is among thevery few results, where finiteness of fourth moments is not needed and it is uncertainwhether it can be derived in the presence of long memory in the linear part of the model.Hence, our conjecture relies on safer grounds if one is able to establish the finiteness offourth moments (kurtosis) in the estimated model.

Possible simplifying modifications

The likelihood (6) could be maximized numerically in all parameters to yield the exactML estimator in a straightforward fashion. This is unattractive for two reasons. Firstly,optimization in the fractional-differencing parameter d requires matrix inversions of orderT, which is computationally burdensome for large samples even for pure fractional noiseand even more so in the presence of the remaining parameters from the ARMA and ARCHpart of the model. Secondly, it has been shown that exact ML methods may yield a rather



sizable bias for d in ARFIMA models, which could be even more critical in ARFIMA-ARCH models. We first consider some suggestions for simplifications of the numericaloptimization step and then turn to the bias problem.

Engle (1982) has shown that, for all combinations of linear and ARCH models obeyingcertain regularity conditions, the information matrix is block diagonal and estimation ofARCH and linear parameters can be conducted by alternating iterations. The ARFIMA-ARCH model is essentially regular in the sense of Engle, with the only caveat beingmoment conditions which, as shown in Section 2, are not stronger than for the pure ARCHmodel as long as the Engle-type model is adopted. Hence, one suggested estimationprocedure could be to iterate between optimization of the linear ARFIMA part of themodel and the ARCH part.

A modification of the algorithm that achieves a considerable reduction of the number ofT 3 T matrix inversions can be based on the Whittle likelihood (for a definition anddescription of the approximate spectral or Whittle likelihood, see Whittle (1951)). ForARFIMA models, Hauser (1993) shows that the estimator for d obtained by this approxi-mate spectral ML is essentially unbiased with reasonably small mean square error,whereas the exact ML based on demeaned data (see Li and McLeod (1986) and Cheungand Diebold (1994)) yields negatively biased estimates in small samples.

We tentatively set up an iterative estimator as follows. We used the Whittle likelihoodto obtain estimates of the linear model parameters F, Q, d, and of the covariance structureS. We then estimate the ARCH coefficients from the estimated linear model innovationsvia the ARCH likelihood (5). Based on these ARCH estimates, a new optimization of theWhittle likelihood follows. The procedure is then iterated between the ARCH and theARFIMA estimation steps until convergence is achieved.

The advantage of this ML modification is that it requires fewer inversions of S. Onlyfor the iterations between the linear and the ARCH part, this computationally intensiveoperation is necessary. Our experience with the outlined procedure is based on simula-tions of the ARFIMA-ARCH(0,d,0/1) model. The algorithm worked satisfactorily for a1

, 0.8, with the number of iterations required for convergence remaining small, typicallyaround 10. For a1 $ 0.8, the algorithm failed to converge in some cases and the numberof non-convergent cases increased with a1. Hence we did not pursue this suggestionfurther.

The modified profile ML estimator by Cox and Reid

Recently, An and Bloomfield (1993) suggested to use the modified profile likelihoodaccording to Cox and Reid (1987, 1992) in order to reduce the bias of ML estimation forthe d parameter in ARFIMA models. A minute description of this method is given in theappendix. For pure FI models, this estimator has smaller finite-sample mean square errorthan the Whittle likelihood estimator. In detail, after estimating µ by y and replacing a0 bythe estimator evolving from



sy2 5 sy

2~F, d, Q, a1,…,as; µ,a0! 51T

~YT 2 y!8RH21~YT 2 y!

5 Sci2s«

2 5 Sci2

a0

1 2 (i51

s

ai

the suggested modified profile likelihood becomes

,M~F, d, Q, a1,…,as; µ, a0! 5 ~1T

212! log~det RH!

212

log~deti8RH21 i! 1 ~1 1

12

2T2!log@~YT 2 y!8RH

21~YT 2 y!#

RH 5 sy22SH 5 RH~F, d, Q, a1,…,as!

i 5 ~1,…,1!8

RH is the process-dependent correlation matrix.To calculate ht for t # s in (2), starting values for «t

2 with t # 0 are required which wespecified to equal the unconditional variance. As a possible alternative, Diebold andSchuermann (1992) propose a rather time-consuming procedure to approximate exact MLfor ARCH situations. However, they report substantial relative gains over our approximatemethod for very small samples only.

Table 1a and 1b give the results of Monte Carlo simulations based on 10000 and 5000replications of ARFIMA-ARCH(0,d,0/1) processes (Engle-type) of length T 5 100 and T5 500, respectively. The processes were simulated by Yt 5 LEt where Et is the vector ofconditionally normal ARCH innovations.

The bias of the d estimate is indeed very small as compared to the bias of the unmodi-fied ML estimate reported by Cheung and Diebold (1994). Except for very pronouncedARCH effects in conjunction with large intermediate memory (d 5 20.4), the perfor-mance of the estimate appears widely unaffected by ARCH. For T 5 500, it appears thatstronger ARCH effects improve the accuracy of d estimates.

Not surprisingly, a1 is positively biased for very small true values due to the non-negativity constraints. For larger values of true a1, a bias toward zero becomes sizable andstandard errors increase. Except for the very long-memory case d 5 0.4, performance ofthe a1 estimate is unaffected by d.

To enhance the empirical relevance of the experiment, comparable simulations werealso conducted based on the estimated ARFIMA-ARCH(0, d, 2/4) model ranked at thefourth place in table 2 (see section 4). This model was chosen due to the “small” numberof parameters to limit computation time. These simulations are not fully reported here dueto lack of space but agree well with the qualitative results seen in table 1b. The maybemost comforting outcome of those simulations is that also the remaining ARMA andARCH coefficients are estimated with very satisfactory accuracy.



Table 1a. Small sample properties of the modified ML estimator for the ARFIMA-ARCH (0,d,0/1) model. (T5 100, 10000 replications)

(a) Mean estimates for d

d

a1 2.4 2.2 2.1 0.0 0.1 0.2 0.4

0.00 20.395 20.205 20.107 20.008 0.091 0.191 0.3880.25 20.391 20.202 20.104 20.005 0.094 0.195 0.3900.50 20.386 20.197 20.100 20.001 0.099 0.199 0.3960.75 20.380 20.190 20.093 0.006 0.106 0.206 0.4030.90 20.377 20.188 20.090 0.008 0.108 0.208 0.4050.95 20.373 20.184 20.087 0.012 0.111 0.211 0.4081.00 20.371 20.182 20.085 0.013 0.112 0.212 0.408

(b) Mean squared errors for d estimates 3 100

d

a1 2.4 2.2 2.1 0.0 .1 .2 .4

0.00 0.850 0.883 0.895 0.899 0.895 0.886 0.7460.25 0.954 0.978 0.988 0.988 0.982 0.970 0.7690.50 0.863 0.842 0.845 0.840 0.836 0.831 0.6410.75 0.853 0.774 0.757 0.734 0.720 0.709 0.5320.90 0.860 0.742 0.720 0.692 0.666 0.654 0.4830.95 0.922 0.760 0.725 0.695 0.669 0.646 0.4651.00 0.954 0.788 0.736 0.704 0.677 0.649 0.465

(c) Mean a1 estimates

d

a1 2.4 2.2 2.1 0.0 .1 .2 .4

0.00 0.039 0.038 0.038 0.038 0.037 0.036 0.0220.25 0.221 0.220 0.220 0.219 0.216 0.208 0.1330.50 0.440 0.440 0.440 0.438 0.433 0.421 0.2880.75 0.631 0.632 0.632 0.630 0.625 0.613 0.4500.90 0.724 0.725 0.725 0.723 0.719 0.708 0.5480.95 0.752 0.753 0.753 0.751 0.747 0.736 0.5761.00 0.778 0.779 0.779 0.777 0.773 0.763 0.606

(d) Mean squared errors of a1 estimates 3 100

d

a1 2.4 2.2 2.1 0.0 .1 .2 .4

0.00 0.659 0.654 0.651 0.643 0.623 0.579 0.3010.25 2.488 2.476 2.469 2.452 2.423 2.379 3.0870.50 3.864 3.843 3.842 3.850 3.885 4.051 8.6360.75 4.955 4.912 4.922 4.961 5.101 5.548 15.5860.90 6.108 6.044 6.061 6.139 6.324 6.900 19.6110.95 6.688 6.609 6.624 6.701 6.936 7.606 21.5181.00 7.456 7.373 7.378 7.474 7.748 8.462 23.104



Table 1b. Small sample properties of the modified ML estimator for the ARFIMA-ARCH (0,d,0/1) model. (T5 500, 5000 replications)

(a) Mean estimates for d

d

a1 2.4 2.2 2.1 0.0 .1 .2 .4

0.00 20.398 20.201 20.102 20.002 0.098 0.198 0.3980.25 20.397 20.200 20.101 20.001 0.099 0.199 0.3990.50 20.395 20.199 20.100 0.000 0.100 0.200 0.4000.75 20.394 20.198 20.099 0.001 0.101 0.201 0.4020.90 20.392 20.196 20.097 0.003 0.103 0.203 0.4040.95 20.391 20.196 20.096 0.003 0.103 0.203 0.4041.00 20.390 20.195 20.096 0.004 0.103 0.203 0.404

(b) Mean squared errors for d estimates 3 100

d

a1 2.4 2.2 2.1 0.0 .1 .2 .4

0.00 0.136 0.137 0.137 0.137 0.137 0.136 0.1330.25 0.145 0.144 0.144 0.143 0.143 0.142 0.1410.50 0.131 0.121 0.121 0.120 0.120 0.119 0.1170.75 0.106 0.093 0.092 0.091 0.090 0.089 0.0900.90 0.106 0.085 0.084 0.080 0.078 0.077 0.0770.95 0.122 0.091 0.085 0.081 0.078 0.077 0.0741.00 0.134 0.098 0.095 0.082 0.078 0.079 0.072

(c) Mean a1 estimates

d

a1 2.4 2.2 2.1 0.0 .1 .2 .4

0.00 0.018 0.018 0.018 0.018 0.017 0.017 0.0120.25 0.243 0.243 0.243 0.243 0.242 0.239 0.1830.50 0.486 0.486 0.486 0.486 0.485 0.481 0.3950.75 0.706 0.707 0.707 0.707 0.706 0.702 0.6230.90 0.811 0.811 0.812 0.812 0.811 0.808 0.7460.95 0.840 0.841 0.841 0.841 0.840 0.837 0.7821.00 0.866 0.867 0.867 0.867 0.866 0.863 0.815

(d) Mean squared errors of a1 estimates 3 100

d

a1 2.4 2.2 2.1 0.0 .1 .2 .4

0.00 0.111 0.111 0.111 0.111 0.110 0.106 0.0580.25 0.537 0.536 0.536 0.536 0.535 0.532 0.9340.50 0.827 0.812 0.812 0.829 0.848 0.846 2.1480.75 1.064 1.050 1.046 1.048 1.062 1.116 3.0710.90 1.501 1.474 1.469 1.471 1.490 1.564 3.6970.95 1.824 1.795 1.788 1.791 1.811 1.888 3.9781.00 2.331 2.306 2.302 2.307 2.331 2.414 4.498



4. Modeling the Swiss 1-month Euromarket interest rate

In accordance with the conjecture of Backus and Zin (1993) that the yields of bonds ofsome future maturity should exhibit long-range dependence, we investigate the Swiss1-month Euromarket series with respect to fractional integration. In a natural way, weextend the models to incorporate ARCH effects. The sample runs from January 3, 1986,to April 12, 1989, which are 826 daily observations—for the differenced series—corre-sponding to 854 weekdays (Monday to Friday) after eliminating missing observations dueto holidays. The interest rate series and its first difference are depicted in figures 1 and 2.The empirical mean of the differences is 0.002421, a very small number indeed, as thedata were measured in percentages. Although the data were available for longer timespans, maximization of the likelihood over longer intervals would have caused problems,as inverting symmetric Toeplitz matrices of dimension T 3 T quickly exceeds the limi-tations of accessible computer resources.

All possible models up to an order of ARFIMA-ARCH(5,d,5/5) were estimated andcompared via the AIC. These maximum orders were considered in order to accommodateintra-weekly effects in daily data, as a business week has 5 days. Models with cancelingroots in the AR and MA polynomials were excluded from consideration. Table 2 gives themain information on the six best models. The first five models all contain a negativefractional d coefficient. The first three models all need an AR order of 5 in order to capturethe correlation patterns within a week of trading. All models need an ARCH order of atleast 4. The model ranked fourth is an interesting “parsimonious” (0,d,2/4) structure thatturned out to be almost equivalent to the optimum model in a prediction experiment (seeHauser and Kunst, 1995). The best non-fractional model selected by AIC (3,0,3/4) appearsat the sixth rank.

In detail, the best model, i.e., ARFIMA-ARCH(5,d,2/4) yielded the following coeffi-cients structure:

Table 2. Summary of six best ARFIMA-ARCH models according to AIC estimated from the Swiss 1-monthEuromarket interest rate series.

rank p d q s AIC Sai

1 5 20.281 2 4 22566.01 0.641[0.111] [0.063]

2 5 20.281 2 5 22568.01 0.640[0.111] [0.063]

3 5 20.365 1 4 22568.11 0.651[0.073] [0.062]

4 0 20.120 2 4 22568.46 0.656[0.056] [0.059]

5 2 20.113 0 4 22568.91 0.665[0.047] [0.058]

6 3 0 3 4 22569.07 0.647[0.062]

NOTE: Numbers in square brackets are estimated standard errors of the coefficient estimates.



~1 2 1.647B 1 1.071B2 2 .360B3 1 .167B4 2 .101B5!~1 2 B!2.281yt 5

@.208# @.239# @.124# @.082# @.045# @.111#

5 ~1 2 1.349B 1 .590B2!«t

@.216# @.177#

ht 5 .013 1 .100«t212 1 .210«t222 1 .014«t232 1 .317«t242

@.046# @.063# @.014# @.060#

The shape of the lag pattern in the ARCH part remains fairly constant across specifica-tions, as does their sum at around 0.65. Hence, it appears reasonable to state that the dataprovide evidence on intermediate memory in the ARFIMA sense as well as on conditionalheteroskedasticity of the Engle-ARCH type.

Figure 1. Swiss 1-month Euromarket interest rate series. The sample runs from January 3, 1986 to April 12, 1989and thus contains 854 observations. Occasional solitary spots indicate missing values caused by holidays.



For brevity we do not display the full correlation matrices of the estimated parametersin the considered models. For the (5,d,2/4) model, correlation between the d estimate andall other parameter estimates is very low, while correlation among some ARMA param-eters is high. This may indicate possible redundancies among the ARMA parameterswhich could be resolved by, e.g., subset modeling. In more parsimonious models withlower lag order, such as (0,d,2/4), the correlation between estimates of d and the ARMAcoefficients increases. Hence the d estimate from the (5,d,2/4) model may be the mostreliable one. Correlation among the ARCH parameter estimates is low, which supports ourfinding that an ARCH order of four is necessary. The correlation between the ARFIMAand the ARCH part is also low, thus approximating well the theoretical asymptotic blockdiagonality of the information matrix.

Our estimated models also fulfill the Milhøj (1985) condition that guarantees the ex-istence of finite fourth moments. Hence, the common condition for the asymptotic nor-mality of the estimator holds. However, one should keep in mind that, unlike the station-arity conditions given in Section 2, that condition is valid for conditional Gaussian ARCHmodels only.

Figure 2. Changes of the Swiss 1-month Euromarket interest rate series. The sample runs from January 4, 1986to April 12, 1989 and thus contains 853 observations. The remark for figure 1 again applies.



Note that the estimated shape of the lag pattern does not correspond to the popularGARCH(1,1) specification.

In order to demonstrate the cross-influence between specification of the ARFIMA andthe ARCH parts of the model, table 3 summarizes the best models for fixed ARCH order,for d 5 0 as well as for the general ARFIMA case. Except for s 5 1, AIC always prefersthe fractional model, with clear evidence of intermediate memory for all s . 1. Table 3also shows that the ARMA lag orders p and q are sensitive to the specification of theARCH order s. These observations enhance the importance of simultaneously capturingARFIMA and ARCH effects in a joint model.

5. Summary and conclusion

We have introduced the ARFIMA-ARCH model that incorporates conditional heterosk-edasticity and fractional differencing simultaneously.

The corresponding Gaussian likelihood function was developed which a ML estimationalgorithm can be based on. It is an exact ML conditional on starting values for the ARCHpart of the model. A Cox/Reid-type modification was applied to the original likelihood inorder to get rid of the inherent bias in the estimate of the fractional integration parameter.Alternatively, we outlined an iterative two-step algorithm based on the Whittle likelihoodthat, however, failed to convince on grounds of its performance.

We simulated ARFIMA-ARCH(0,d,0/1) models and found that, even for T 5 100, themodified ML estimation procedure appears to be unbiased for d. As in the case ofestimation of pure ARCH models, estimates of ARCH parameters are biased upward forvery small ARCH effects but are biased downward for larger effects. Precision of destimates is not much affected by ARCH effects.

Table 3. Summary of best ARFIMA-ARCH (p,0,q/s) and (p,d,q/s) models selected by AIC estimated from theSwiss 1-month Euromarket interest rate series, based on fixed ARCH order.

p,0,q/s p,d,q/s

s model AIC Sai model AIC Sai

0 (0,0,2/0) 22642.36 – (1,2.08,1/0) 22641.46* –[0.035]

1 (4,0,2/1) 22611.64* 0.40 (4,2.02,2/1) 22613.55 0.40[0.088] [0.068] [0.087]

2 (0,0,2/2) 22599.41 0.39 (2,2.38,3/2) 22592.46* 0.44[0.082] [0.112] [0.061]

3 (0,0,2/3) 22600.17 0.43 (2,2.33,3/3) 22592.10* 0.47[0.084] [0.102] [0.069]

4 (3,0,3/4) 22569.07 0.65 (5,2.28,2/4) 22566.01* 0.64[0.062] [0.111] [0.063]

5 (3,0,3/5) 22571.07 0.65 (5,2.28,2/5) 22568.01* 0.64[0.062] [0.111] [0.062]

NOTE: Asterisks (*) indicate whether the model with d 5 0 or the model with freely estimated d is selected byAIC for fixed s. Numbers in square brackets are estimated standard errors of the coefficient estimates.



Our empirical investigation of daily observations on the Swiss one-month Euromarketyield rate shows that our suggested ARFIMA-ARCH model is clearly ranked higheraccording to the AIC than conventional ARIMA-ARCH models. The series may be char-acterized as integrated of order 0.72 with some short run effects and ARCH effects up tolag 4, i.e. slightly less than a week. The unconditional variance of the first difference of theseries is finite.

This model implies mean reversion in the sense that there is no long-run impact of aninnovation on the value of the process (Cheung and Lai 1993) contrary to the first-orderintegrated ARIMA model suggested by, e.g., Campbell and Shiller(1987). In contrast,Backus and Zin (1993) consider a stationary short rate of interest and consequentlystationary future yield rates. However, their implication of long-range dependence in yieldrates is confirmed.

During the search of the best approximating model, all model orders up to (5,d,5/5)were considered. Substantial cross-effects between the choice of the ARCH order and ofthe ARFIMA orders were found. Fixing the ARCH order at an arbitrary value led todifferent ARMA orders or to the exclusion of the fractional parameter. E.g., restricting theARCH order to 1 would entail a nonfractional model. Thus, simultaneous estimation ofthe linear and the ARCH part seems to be necessary in order to obtain reliable models.

Appendix: The procedure by Cox and Reid

Let us assume that the density under investigation belongs to a collection of densitiesparameterized by an (m 1 n) 3 1-vector u. Suppose the vector u can be partitioned intotwo parts as u 5 (c8,l8)8 such that the information matrix is block diagonal with respectto this partition. As outlined by Cox and Reid (1987), such parameterization has a numberof advantageous features, in particular it decreases the influence between the two subsetsof parameters in such a way that the maximum likelihood estimate of e.g. c has asymp-totically the same standard error whether l is treated as known or random. Hence, thisblock diagonality facilitates the application of iterative profile likelihood techniques intothe alternating directions of c and l. Cox and Reid (1987) demonstrate that, assumingregularity conditions such as sufficient smoothness of the densities with respect to pa-rameters, the profile likelihood estimate c(l) varies only slowly with l in the sense that

c~l! 2 c 5 Op~1T!

where c is used to denote the full maximum likelihood estimate of c. The same argumentholds symmetrically with respect to the estimate of l.

In practice, one often is confronted with situations where c and l are not orthogonal.Then, Cox and Reid (1987) suggest to re-parameterize one of the two parts in order toachieve orthogonality. Usually, the selection on which part to re-parameterize is guided by



(in our case economic) theory. In most applications, either c or l will contain the “pa-rameters of interest” whereas the other vector will contain the “nuisance parameters”.Suppose we start from a non-orthogonal parameter vector u 5 (c8,f8)8 with f 5(f1,…,fn) which we want to transform into orthogonal parameter coordinates(c8,l1,…,ln). We then write f1 5 f1(c,l),…,fn 5 fn(c,l) using the notation l 5(l1,…,ln)8, i.e., the original parameters are written in the new parameters. We introducethe notations , and ,* for the likelihood function in the two different parameterizations

,~c, l! 5 ,*$c, f1~c, l!,…,fn~c, l!%

Then, the following system of partial difference equation can be used to establish param-eter orthogonality

(r51,…,n

ifrfs

*]fr

]c5 2icfs

* , s 5 1,…,n

where i* are parts of the information matrices corresponding to l*. Solving these partialdifference equations leads to an orthogonal parameterization, in general not uniquely.Then, a correction term is added to the profile likelihood in the direction of c in order topurge it from all second-order influence by l. This modified profile likelihood is thendenoted as ,M.

For the ARFIMA model, let us take F 5 (µ, a0) and c as the remaining parameters ofthe model. a0 stands for the unconditional errors variance. An and Bloomfield (1993)describe how to solve the differential equation for this case of pure ARFIMA models. Indetail, the modified profile likelihood to be maximized becomes

,M~F, d, Q; µ, a0! 5 ~1T

212!log~det r!

212

log~det i8 R21i! 1 ~1 112

2T2!log@~YT 2 y!8R21~YT 2 y!#

R 5 sy22S 5 R~F, d, Q!

i 5 ~1,…,1!8

In the case of the ARFIMA-ARCH model, the process-dependent matrix RH(F, d, Q,a1,…,as) replaces R and the parameter set is extended to (F, d, Q, a1,…,as; µ, a0). RH

is the process-dependent correlation matrix corresponding to the process-dependent co-variance matrix SH described in the main text. From our simulation experiment, it turnsout that the reparameterization definitely helps in greatly reducing the bias in the estima-tion of the fractional integration parameter and gives reasonably good estimates for theARCH parameters.



Acknowledgments

The authors wish to thank Manfred Deistler, Benedikt Poetscher, Wolfgang Polasek,George Tauchen, and the referees for helpful comments on previous versions of the paper.The usual proviso applies.

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