Indirect estimation of ARFIMA and VARFIMA models

Journal of Econometrics 93 (1999) 149}175

Indirect estimation of ARFIMA and VARFIMA models Vance L. Martin , Nigel P. Wilkins
* Department of Economics, The University of Melbourne, Parkville, Vic 3052, Australia
Department of Econometrics and Business Statistics, Monash University, Clayton, Vic 3168, Australia Received 1 September 1996; accepted 1 December 1998

Abstract Indirect estimation methods are proposed for estimating ARFIMA, as well as more complex VARFIMA models. A general framework for conducting indirect estimation of fractional models is developed that covers simulation methods, choice of auxiliary model and estimation algorithm. Special attention is given to comparing the "nite sampling properties of the indirect estimator with Sowell's (1992a) exact time domain maximumlikelihood estimator, the spectral maximum-likelihood estimator of Fox and Taqqu (1986) and the Geweke and Porter-Hudak (1983) spectral regression estimator. The indirect estimator can be computationally faster than the exact time domain maximumlikelihood estimator while generating similar small sample properties. The computational gains of the indirect estimator over maximum likelihood increase as the complexity of the data generating process increases.  1999 Elsevier Science S.A. All rights reserved. JEL classixcation: C13; C22 Keywords: Fractional integration; Persistence; Long memory; Frequency domain; Multivariate ARFIMA; Auxiliary models; Simulation

1. Introduction The class of ARFIMA models introduced by Granger and Joyeux (1980) and Hosking (1981) provides a convenient framework for simultaneously modelling * Corresponding author. Tel.: 61-3-99052456; fax: 61-3-99055474. E-mail address: [email protected] (N.P. Wilkins) 0304-4076/99/$ - see front matter  1999 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 0 7 - X

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short-run and long-run relations in time-series data; for a review of this literature see Brockwell and Davis (1991), Sowell (1992b) and Baillie (1996). An important advantage of the ARFIMA model is that it provides a parsimonious parameterization for modelling long memory in processes without the problems associated with estimating near non-stationary models. The two most widely used estimation procedures are the GPH spectral regression estimator introduced by Geweke and Porter-Hudak (1983) and the exact time domain maximum-likelihood estimator (EMLE) introduced by Sowell (1992a). The GPH estimator has the advantage that it is computationally simple, but has the disadvantages that it only estimates the level of fractional integration, it is relatively ine$cient and it can su!er from an identi"cation problem in the very low frequencies of the spectrum; see for example Agiakloglou et al. (1992), Robinson (1993), and Hurvich and Beltrao (1993). In contrast, EMLE is asymptotically e$cient, but is computationally burdensome for higherorder ARMA speci"cations and for moderate sample sizes; see for example Cheung and Diebold (1994). To circumvent the computational problems associated with EMLE, Chung and Baillie (1993) adopt an approximate MLE based on a conditional sum of squares estimator while Tieslau et al. (1996) consider a minimum-distance estimator that compares the theoretical autocorrelations of an ARFIMA process with the sample autocorrelations. In contrast to time domain approaches, frequency domain approximations to EMLE are considered by Fox and Taqqu (1986) and Boes et al. (1989). The "nite sampling performance of the time domain and frequency domain MLEs are compared by Cheung and Diebold (1994). While EMLE of higher-order ARFIMA models is computationally di$cult, for multivariate ARFIMA models, denoted as VARFIMA, it is even less attractive. Multivariate analogues of many of the univariate procedures which approximate EMLE represent natural solutions. This is the approach of Hosoya (1996) who proposes a quasi MLE in the frequency domain which is the multivariate extension of the univariate frequency domain estimators mentioned above. The approach adopted in this paper is to use the indirect estimation framework of Smith (1993), Gourieroux et al. (1993), GMR hereafter, and Gallant and Tauchen (1996), GT hereafter. For a general review of indirect estimation methods see Gourieroux and Monfort (1994). Indirect estimation provides a computationally convenient procedure for obtaining parameter estimates of both ARFIMA and VARFIMA models without necessarily sacri"cing the optimality properties of MLE. In particular, the indirect parameter estimates are both consistent and asymptotically normal under very general conditions.  In addition, Koop et al. (1997) have recently suggested a Bayesian approach for analyzing ARFIMA models. However, the computational demands of this approach are comparable to EMLE.  Gourieroux et al. (1995) also identi"y the conditions for the indirect estimator to deliver superior small sample properties to MLE.

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The key underlying principle of indirect estimation is that correct inferences can be made using a misspeci"ed model. The misspeci"ed model, also referred to as the auxiliary model, is the model that is estimated. This model need not provide a good approximation to the data generating process (DGP). However, the better the approximation, the more e$cient the indirect parameter estimates will be. In the limit, the indirect estimator achieves the same level of e$ciency as the EMLE. For this reason, Gallant and Tauchen (1996) also refer to indirect estimation as e$cient method of moments estimation. The only demand placed on the theoretical model, that is the model that contains the parameters of primary interest, is that it is easily simulated. In the case of ARFIMA models, both of these conditions are satis"ed. An early example of the indirect estimator in econometrics is the indirect least-squares estimator used for exactly identi"ed simultaneous equation systems. Here the structural equation represents the theoretical model and the reduced form represents the auxiliary model. The reduced form equation is in a sense misspeci"ed, in that the structural equation is the correct model. However, it is well known that by estimating the reduced form, consistent parameter estimates of the structural model can still be derived even though this model is misspeci"ed. The underlying principle of indirect estimation is the same, although the framework and applicability are far more general than this example suggests; see for example, Pastorello et al. (1994), Frachot et al. (1995), Pagan et al. (1996), Tauchen (1995) and Pagan and Martin (1996). The computational advantages of the indirect estimator stem from a demarcation between the theoretical model and the auxiliary model; see Pagan and Martin (1996). For relatively large sample sizes, the computational cost of time domain EMLE of ARFIMA models increases polynomially as it is necessary to invert a ¹;¹ autocovariance matrix, whereas for the indirect estimator it is only necessary to extend the length of the sample path of the simulated data set; see for example Hosking (1984, p. 1902). By extending the modelling framework to VARFIMA models, the computational advantages of the indirect estimator over EMLE are enhanced as simulating a system of ARFIMA models is relatively straightforward using the techniques proposed in later sections. In this paper, the sampling properties of the indirect estimator are investigated for univariate and multivariate VARFIMA models using Monte Carlo simulation methods, with the results compared, where appropriate, to the sampling performance of EMLE, frequency domain MLE, FDMLE hereafter, and the GPH estimator. Special attention is given to investigating

 Although the focus of this paper is on estimating the ARFIMA model, it is straight forward to extend the analysis to other models exhibiting long memory characteristics such as Gegenbauer processes as investigated by Gray et al. (1989), and the class of FIGARCH models introduced by Baillie et al. (1996) and discussed by Bollerslev and Mikkelsen (1996).

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computationally e$cient and accurate methods for simulating ARFIMA models as well as exploring the choice of optimal auxiliary models. The paper is structured as follows. Section 2 provides a brief discussion of the properties of ARFIMA processes. Indirect estimation methods for estimating ARFIMA models are discussed in Section 3. The sampling performance of the indirect estimator is investigated in Section 4 and is compared to the time and frequency domain MLEs and the GPH estimator. Section 5 contains the conclusions of the paper.

2. The ARFIMA model The autoregressive fractionally integrated moving average model, denoted ARFIMA(p, d, q), can be formally expressed as U(¸) By "d#H(¸)e , R R

(1)

where the level of integration, d, belongs to the set of real numbers, d is a constant re#ecting some mean value and e & iid(0, p). The lag R C operator polynomials U(¸)"1! ¸! ¸!2! ¸N and H(¸)"1#   N h ¸#h ¸#2#h ¸O have roots outside the unit circle that are distinct so   O that (1) is both stationary and invertible provided that, in addition, d3(!0.5, 0.5). The fractional di!erencing operator B in (1) is de"ned as  C( j!d)

B" ¸H, C(!d)C( j#1) H

(2)

where C( ) ) is the gamma function. Note that by substituting (3) into (1), and inverting the moving average process, enables an ARFIMA model to be written as an in"nite-order autoregression  y "k # k y #e . R  H R\H R H

(3)

For d'0, long memory is displayed by an in"nite-valued spectrum at a frequency of zero or a hyperbolically decaying autocorrelation function. The spectrum of (1), assuming d3(!0.5, 0.5), is given by




I(j)"I夹(j) 2 sin

j 2

\B ,

(4)

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where p"H(e\ H)" I夹(j)" C , 2p "U(e\ H)"

(5)

is the spectrum of a purely stationary ARMA process. As j approaches zero, (4) obeys I(j)PCj\B,

(6)

where C is a constant value determined according to the ARMA parameters in the DGP. It can be seen from (6) that the behaviour of the spectrum at the very low frequencies is characterized by d. The autocovariance function of (1) is given in Hosking (1981) and Sowell (1992a); but see also Chung (1994) for a concise summary. Following Chung (1994), the autocovariance function of (1) can be expressed in terms of three components; N c "cH f A Q, H H Q HM Q

(7)

for j"0, 1, 2,2, ¹!1. The "rst term cH, is the autocovariance function for H fractional noise which is given by C(1!2d)C( j#d) cH" . H C(d)C(1!d)C(1!d#j)

(8)

The second term f , contains the AR parameters, and is obtained as the partial Q fraction decomposition of the characteristic equation for an AR(p) process with roots o , for s"1, 2,2, p, Q





\ N N f " o “ (1!o o ) “ (o !o ) . Q G Q Q K Q G K$Q Finally, the third term A Q, contains the moving average parameters, and is HM derived as O A Q" t B [oNC #C !1], HM G HN>G Q N>G\HMQ H\N\GMQ G\O

(9)

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where O\G t" h h , G I I>G I for i"0, 1, 2,2, q, is the autocovariance function for an MA(q) process C(1!d!j)C(d!j#h) B " HF C(d!j)C(1!d!j#h) and "nally  C(1!d#h)C(d#h#k) C Q"F(d#h,1; 1!d#h; o )" oI, (10) FM Q C(d#h)C(1!d#h#k) Q I where F( ) ) is the hypergeometric function. See Chung (1994) for an alternative expression for A Q and a statement of the autocovariances for the HM ARFIMA(p, d, q) model where p, q"1, 2, 3. As a minimum, these expressions highlight the degree of complexity of the autocovariance function for even parsimonious ARFIMA models. Given that time domain EMLE relies on the autocovariance matrix, it provides insight into the potential computational problems of undertaking EMLE based on the approach of Sowell (1992a). In an analogous manner to that in (1), the k-dimensional vector ARFIMA, or VARFIMA(p, d, q), model can be formally expressed as U(¸) dY "d#H(¸)e , R R

(11)

where Y is a k;1 vector process, d is a k;1 fractional di!erencing operator R with dⴝ+d , d ,2, d ,, e &iid(0, I r) is a k;1 vector error process,   I R I U(¸)"1!U ¸!U ¸!2!U ¸N with U as a k;k matrix of autoregres  N G sive parameters and H(¸)"1#H ¸#H ¸#2#H ¸O with H as a k;k   O G matrix of moving average parameters. Clearly, by attempting to generalize the analysis to multivariate VARFIMA models, the complexity of the model increases dramatically, thereby making time domain EMLE extremely di$cult, if not infeasible for general model speci"cations.

3. The indirect estimator The indirect estimation methods of Smith (1993) and generalised by Gourieroux et al. (1993) and Gallant and Tauchen (1996), o!er a way of

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circumventing the problems associated with the complicated likelihood function arising from ARFIMA and VARFIMA models, whilst generating consistent and asymptotically normal parameter estimates under fairly general conditions. In this section, the emphasis is on computational issues arising from the application of the indirect estimator to ARFIMA and VARFIMA models. For the theoretical issues underlying the indirect estimator see Gourieroux and Monfort (1994). Implementation of the indirect estimator involves generating a simulated process using the ARFIMA model for a given set of parameter values. In the case of the GMR indirect estimator, the actual data and the simulated data are then used to estimate an auxiliary, or indirect, model. The two sets of parameter values obtained for the indirect model using the two data sets are then calibrated by choosing the parameters of the ARFIMA model. The GT indirect estimator di!ers from this in that the calibration of the ARFIMA or VARFIMA parameters occurs on the scores of the auxiliary model. To formalize the idea of indirect estimation for the ARFIMA model, let W"+d; d; , i"1, 2,2, p; h , i"1, 2,2, q; p ,, G G C

(12)

represent the set of parameters for the ARFIMA model. The GMR indirect estimator is given by









1 &  1 & WK "Argmin PK !2! PK 1'+ X PK !2! PK 1'+ , %+0 F F H H W F F

(13)

where PK !2 and PK 1'+, are vectors of the parameter estimates of the auxiliary F model using the actual and simulated data sets, respectively, X is a weighting matrix given in Gourieroux et al. (1993) and H denotes the number of simulation paths, where a path consists of an independent drawing of random numbers with which the direct ARFIMA model is simulated. This function can be minimized using standard gradient optimization procedures. Eq. (13) shows that the GMR indirect estimates are chosen when the derived simulation paths yield estimates of the auxiliary model similar to those that are obtained using the actual data.

 Indirect estimation of the VARFIMA model follows immediately from the discussion of the ARFIMA model by simply letting the parameters in (12) represent the set of VARFIMA parameters.  For certain types of problems, Pastorello et al. (1994) "nd that algorithms based on direct search methods perform better in computing the indirect estimates.

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The criterion of the Gallant and Tauchen (1996) indirect estimator when there are H simulation paths is









1 & RL  1 & RL WK "Argmin (PK !2, y1'+) N (PK !2, y1'+) , %2 RF RF H RP H RP W F F (14) where ¸ is the log of the likelihood function of the auxiliary model with parameter vector P, and N is the weighting matrix de"ned in Gallant and Tauchen (1996) as well as in Gourieroux et al. (1993). The term R¸/RP(PK !2, y1'+) represents the score vector of the auxiliary model for the RF simulated data y1'+ evaluated at PK !2, the maximum-likelihood parameter RF estimates of the auxiliary model using the actual data y!2. Note that by R construction, the scores when evaluated using the actual data are such that R¸/RP(PK !2, y!2)"0. This suggests that the GT indirect estimator in (14) R chooses WK so as to generate a simulated process that drives the score vector towards zero. The indirect estimators based on (13) and (14) are asymptotically equivalent so that the choice between the two is governed by computational issues. For linear auxiliary models, the computational demands of both indirect estimators are approximately the same. For the case of nonlinear auxiliary models, the GT approach is more convenient because it is only necessary to evaluate the scores of the log of the likelihood. This contrasts with the GMR approach where it is necessary to compute the estimates of the auxiliary model PK 1'+, using an additional nonlinear algorithm inside the indirect estimation algorithm. 3.1. Auxiliary model choices 3.1.1. Time domain Eq. (3) expresses an ARFIMA process as an in"nite-order autoregression. This suggests that a natural choice of auxiliary model is given by the AR(k) model I y "n # n y #v , R  H R\H R H

(15)

where v & iid(0, p). The adoption of an AR auxiliary model follows the work of R T Smith (1993) who suggests the use of such a model in the general indirect estimation context. The choice of the lag length k, in part depends on the dimension of W and the desired precision of the indirect parameter estimates. The dimension of P"+n ; i"0, 1,2, k,, needs to be at least greater than or G equal to the dimension of W.

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When the dimension of W and P are the same, the criterion for the GMR indirect estimator simpli"es to PK !2"1/H & PK 1'+, so that the value of the F F minimized objective function in (13) is zero, implying that the solution is independent of X. If the number of simulation paths is allowed to go to in"nity then PK !2"E[PK 1'+], where the expectation is taken with respect to the true F DGP. This shows that the indirect estimator can be interpreted as a minimum distance estimator where W is chosen to equate the coe$cients from an estimated "nite AR model with the theoretical autoregressive parameters from an ARFIMA model. Galbraith and Zinde-Walsh (1997) propose such an estimator. A minimum distance estimator is also proposed by Tieslau et al. (1996) except that the comparison is based on the autocorrelation function. Note that even though (15) represents a misspeci"cation of the true model as given by the representation in (3), the indirect estimates based on this auxiliary model are still consistent. It can be anticipated however, that the precision of the indirect estimator will increase on average as the lag length of the auxiliary model in (15) increases. For the multivariate model in (11), it is natural to choose a vector autoregression (VAR) as the auxiliary model. In either case, the parameters of the auxiliary model are estimated using OLS. 3.1.2. Frequency domain An alternative choice of auxiliary models which is motivated by the GPH estimator, is to equate the periodogram with the theoretical spectrum of an ARFIMA(p, d, q) process as given by (4). For the ARFIMA(0, d, 0) process this amounts to estimating the conventional GPH regression equation




j ln z(j )"n #n ln 4 sin H H   2

#g , H

(16)

where g is an error term de"ned in Geweke and Porter-Hudak (1983), H j "2pj/¹, for j"1, 2,2, ¹/2, represent the ¹/2 Fourier frequencies and the H periodogram is de"ned as





 1 2 z(j )" y e\ HHR . (17) H R 2p¹ R The regression is estimated over a truncated sample of size m"¹?, where a"0.5 is commonly used. The estimate of d is then simply equal to !n in (16).  When there are nonzero ARMA parameters to be estimated, it is necessary to expand the GPH equation to ensure that the dimension of the parameter space of the auxiliary model is at least as large as the dimension of W in (12). One way to do this is to augment the standard GPH regression equation in (16) by the frequency domain parametric representation of the ARMA speci"cation as given by (5). In the case of the ARFIMA(1, d, 1) model, for example, the

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parametric GPH regression equation becomes


 

j z(j )"n #n ln 4 sin H H   2

#ln



1#h#2h cos j H #v0+, R 1# !2 cos j H

(18)

where v0+ is de"ned as the error term of this parametric auxiliary model. In R estimating (18), the entire spectrum can be exploited as allowance is made for higher-frequency behaviour and accordingly all ¹/2 Fourier frequencies can be used in the regression. This regression equation is referred to as an augmented parametric GPH auxiliary model. As estimation of (18) requires the use of nonlinear least squares, the GT indirect estimator is relatively more attractive than the GMR estimator for the reasons discussed earlier. Potentially, from a direct estimation point of view, the augmented GPH estimators should be an improvement over the standard GPH estimator in small samples when ARMA terms are present. It can be expected that the size of the small sample bias will be larger for the standard GPH estimator when the spectrum of the ARMA component contributes signi"cantly to the spectrum of the process around the zero frequency. This may occur when there is strong positive "rst-order autocorrelation; see, for example, Agiakloglou et al. (1992). 3.2. Simulating ARFIMA models An important condition for the successful application of the indirect estimator is that the theoretical model can be easily and quickly simulated. Two simulation methods for the ARFIMA model are investigated; the Cholesky decomposition of the autocovariance function approach and the truncated autoregression approach. 3.2.1. Cholesky autocovariance function method The simulation algorithm using the autocovariance function consists of multiplying a standardized Gaussian white noise process by the Cholesky decomposition of the Toeplitz autocovariance matrix. This can be expressed as y "d#Cf , R R

(19)

 A further variation on the frequency domain auxiliary models is motivated by noting that these estimators are based on equating the sample spectrum with the theoretical spectrum. This suggests the need for consistent estimates of the spectrum which can be achieved by using a smoothed estimate of the periodogram.  Alternative simulation methods are discussed in Hosking (1984). In a subsequent Monte Carlo study it would be of interest to compare the accuracy and speed of these alternative methods in the context of indirect estimation in more detail.

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where C is a ¹;¹ lower triangular matrix representing the Cholesky decomposition of the ¹;¹ autocovariance matrix V which is speci"ed according to (7), f &iid(0, 1) is a ¹;1 white noise sequence, y is the ¹;1 ARFIMA process and R R d is as de"ned in (1). Simulation of random numbers using this method will be referred to as the Cholesky simulation method to emphasize the Cholesky decomposition that is required. The Cholesky decomposition algorithm has been used extensively; see, for example, Geweke and Porter-Hudak (1983), Diebold and Rudebusch (1991), Newbold and Agiakloglou (1993) and Cheung and Diebold (1994). The extensive use of gamma functions in the expressions for the autocovariances causes this simulation procedure to become time consuming when large sample sizes are considered and unmanageable for general ARFIMA and VARFIMA models. However, the computational demands of the indirect estimator using the Cholesky decomposition to simulate the model will still constitute a saving in computational time relative to the EMLE as it is not necessary to invert the ¹;¹ autocovariance matrix in the estimation procedure. Alternative methods of evaluating the autocovariance function are possible using recursive methods, see for example Chung (1994), but even these can present problems for large sample sizes. 3.2.2. Truncated autoregression method A potentially simpler and faster procedure than the autocovariance approach for simulating an ARFIMA process is based on a truncated autoregression. Given the model speci"cations in (1) and (2), the simulation model can be written as






J C( j!d) U(¸) ¸H y "d#H(¸)u , R R C(!d)C( j#1) H

(20)

where u &iid(0, 1) and l is the truncation parameter for the in"nite-order R di!erencing "lter. The two polynomials on the left-hand side of (20) can be easily multiplied to compute the coe$cients of an AR(p#l) model, and an ARMA(p#l, q) model can then be simulated using N(0, 1) random numbers for

 Another approach for computing the autocovariance function is to compute the inverse Fourier transform of the spectrum of an ARFIMA process using standard numerical integration procedures. For example, to compute the jth autocovariance, the pertinent Fourier integral is



c" H

L

e HHI(j) dj. \L

Given that it is easy to evaluate I(j) for general ARFIMA processes, evaluation of this Fourier integral numerically is computationally attractive.

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u . Simulation of random numbers using this method will be referred to as the R truncated simulation method. In choosing the lag length, l, not only is it necessary that it be long enough to ensure the long-memory characteristics of the model are captured adequately, but for the indirect parameter estimates to be consistent it is also necessary that l increases as the sample size increases. Of course, as simulation of (20) is not constrained by data availability, the lag length can be made as long as necessary to ensure that the contribution of the last lag is negligible; see also Hosking (1984) for alternative ways of choosing the optimal lag length. To begin the algorithm, it is necessary to choose the l#p starting values for simulating y . The simplest approach is to set these values equal to the uncondiR tional mean of the process. This approach may be inaccurate if the long memory property of the model takes a long time to forget these values; see for example Granger and Joyeux (1980). There are two possibilities for overcoming this start-up problem. The "rst is to simulate the model for ¹#q observations and truncate the "rst q observations. The second is motivated by Granger and Joyeux (1980), and choose the l#p starting values for y from the autoR covariance function for fractional noise thus capturing the long memory component of the time series. Whilst this does not constitute a great saving in terms of computational time over the Cholesky method when simulating an ARFIMA(0, d, 0) process for moderate size ¹, the real savings are achieved when simulating general ARFIMA processes as it is no longer necessary to evaluate a number of hypergeometric functions. Naturally, even greater savings are achieved when simulating VARFIMA models.

3.3. Algorithmic details To compute the indirect estimates of an ARFIMA(p, d, q) model using the GT or the GMR indirect estimator with a "nite AR auxiliary model in (15) and the truncated simulator in (20), the algorithm proceeds as follows: 1. Estimate the auxiliary model in (15) for given lag length k"kH, using the actual data y "y!2, and compute PK !2. R R 2. Choose an initial set of parameter estimates for the ARFIMA model: W"+d; d; , i"1, 2,2, p; h, i"1, 2,2, q; p,. G G C 3. Draw a set of random numbers w , from a N(0, 1) distribution. R  This problem is akin to the application of indirect estimation methods to estimating the parameters of stochastic di!erential equations whereby the degree of inconsistency can be controlled by making the time discretization smaller; see Gourieroux and Monfort (1994).  Note that similar stopping rules are necesary when computing the autocovariances in the Cholesky simulation procedure because of the in"nite summation in (10).

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4. Simulate the ARFIMA model (1! ¸! ¸!2! ¸N) By1'+   N R "d#(1#h¸#h¸#2#h¸O)1 , R   O

(21)

where 1 "pw . R C R 5. For the GMR indirect estimator, the auxiliary model in (15) is estimated for a given lag length k"kH, and PK 1'+ is computed for the hth simulation path. F While for the GT indirect estimator, the score is evaluated using the simulated data y "y1'+ R RF RL 2\I Rv (PK !2, y1'+) R RF v (PK !2, y1'+) (PK !2, y1'+)"2 RF R RF Rp RP R 2\I "!2 (1, y1'+ ,2, y1'+ ) R\F R\IF R






I y1'+!n( !2! n( !2y1'+ R  H R\HF H

(22)

for the hth simulation path. 6. Repeat Steps 4 and 5, h"1, 2,2, H times using the same set of random numbers each time from Step 3. 7. Calibrate the parameter vector W, to satisfy the criterion in (13) for the GMR indirect estimator or (14) for the GT indirect estimator. Computing GMR indirect estimates using one of the frequency domain auxiliary models requires (15) in Steps 1 and 5 to be replaced by (16) or (18), while for the GT indirect estimator it is necessary to replace (22) with the score of (16) or (18). In the case where the Cholesky simulator is used, (21) is replaced by (19) with all parameters evaluated at W. Finally, for multivariate models, the AR auxiliary model is replaced by a VAR auxiliary model.

4. Monte Carlo experiments This section presents the results of a range of simulation experiments to determine the sampling properties of the indirect estimator. Both univariate and bivariate VARFIMA models are considered. Special attention is given to comparing the accuracy of the truncated autoregression simulation method to the theoretically correct, but computationally more burdensome Cholesky decomposition simulation method. Where possible, the sampling properties of the

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indirect estimator are compared to the sampling properties of the EMLE, and the FDMLE as well as the GPH estimator. Given that MLE of ARFIMA models is asymptotically e$cient, the sampling properties of the MLEs provides a suitable benchmark for comparison with the indirect estimator. The use of the FDMLE is motivated by the simulation results of Cheung and Diebold (1994) who "nd that whilst the e$ciency of EMLE in small samples of ¹"100 observations for the ARFIMA(0, d, 0) model is superior to FDMLE, both estimators achieve comparable levels of e$ciency for moderate sample sizes of ¹"500 observations. The results of the Monte Carlo experiments are reported in terms of small sample bias and root-mean-squared error (RMSE) statistics. 4.1. Univariate experiments The univariate Monte Carlo experiments are based on the "rst-order ARFIMA(1, d, 1) model (1! ¸) By "(1#h¸)e , R R

(23)

where e &N(0, 1). As no intercept term is included in (23), the population mean R of y is set equal to zero and is assumed to be known in all estimation R procedures. All computations are performed using GAUSS. The normal random number generator RNDN is used to generate e in (23). The optimization algorithm used R to compute the various estimators is the quasi-Newton algorithm of Broyden, Fletcher, Goldard and Shanno which is contained in the GAUSS OPTMUM library. The true parameters are used as the starting values in the optimization algorithm. For all experiments, the number of replications is set at R"1000. 4.1.1. ARFIMA(0, d, 0) The small sample properties of the indirect estimator are reported in Table 1 for the ARFIMA(0, d, 0) DGP, with a sample size of ¹"100 observations. The parameter values for the DGP in (23) are d"+!0.4,!0.2, 0.0, 0.2, 0.4,, which correspond to the case of y being both stationary and invertible, and "h"0. R The sampling performance of the indirect estimator for the ARFIMA(0, d, 0) DGP is evaluated using three alternative auxiliary models. The "rst two auxiliary models are AR(1) and AR(4) time domain auxiliary models as given by (15), while the third is the GPH frequency domain auxiliary model given by (16). To gauge the accuracy of the truncated autoregression simulator in (20), results are

 An extension of the "nite sample analysis conducted in the present paper would be to follow Cheung and Diebold (1994) and allow the mean of the ARFIMA process to be unknown.  The GAUSS code is available from Wilkins on request.

!0.010 !0.003 !0.009 !0.012 0.009 0.004

!0.004 !0.011 !0.004 !0.007 !0.007 !0.014

0.006 0.003 0.009 0.011 0.007 0.016 0.009 0.004 0.007 0.010 0.019 0.004

0.070 0.077 0.084 0.095 0.291 0.293

0.124 0.123 0.123

0.118 0.118 0.115

0.110 0.109 0.118

0.109 0.119 0.113

0.125 0.118 0.115

0.118 0.123 0.109

Note: Details of the simulation DGP are given in Section 4.1.

EMLE(C) 0.003 EMLE(T) !0.004 FDMLE(C) !0.002 FDMLE(T) !0.008 GPH(C) 0.029 GPH(T) 0.025 Indirect: AR(1) auxiliary model H"10 C/C 0.034 C/T 0.029 T/T 0.031 H"100 C/C 0.027 C/T 0.028 T/T 0.021 Indirect: AR(4) auxiliary model H"10 C/C 0.027 C/T 0.027 T/T 0.034 H"100 C/C 0.032 C/T 0.032 T/T 0.046 Indirect: GPH auxiliary model H"10 C/C 0.037 C/T 0.033 T/T 0.031 H"100 C/C 0.034 C/T 0.045 T/T 0.029

Bias

Bias

RMSE

d"!0.2

d"!0.4

0.138 0.139 0.138

0.147 0.141 0.141

0.106 0.110 0.105

0.112 0.113 0.113

0.129 0.130 0.131

0.136 0.138 0.134

0.083 0.091 0.084 0.095 0.281 0.292

RMSE

0.008 0.013 0.002

0.008 0.001 0.006

0.004 0.002 0.004

0.002 !0.004 0.002

!0.008 !0.010 !0.013

!0.012 !0.014 !0.005

!0.012 !0.014 !0.012 !0.014 0.000 0.001

Bias

d"0.0

Table 1 Bias and RMSE of the estimators for an ARFIMA(0, d, 0) process, ¹"100

0.138 0.144 0.141

0.144 0.144 0.142

0.090 0.100 0.089

0.098 0.104 0.096

0.101 0.106 0.107

0.111 0.112 0.108

0.082 0.089 0.085 0.095 0.283 0.283

RMSE

0.007 0.014 0.002

0.005 0.003 0.004

!0.005 0.001 0.001

!0.006 !0.007 !0.006

!0.010 !0.005 !0.012

!0.011 !0.010 !0.005

!0.015 !0.016 !0.012 !0.013 0.003 0.006

Bias

d"0.2

0.135 0.142 0.139

0.138 0.140 0.141

0.082 0.094 0.086

0.090 0.097 0.091

0.074 0.085 0.088

0.085 0.090 0.087

0.079 0.085 0.086 0.095 0.283 0.281

RMSE

!0.016 !0.006 !0.015

0.018 !0.016 !0.016

!0.030 0.007 0.007

!0.030 !0.001 !0.027

!0.028 0.015 !0.023

!0.026 0.008 !0.017

!0.024 !0.023 !0.008 !0.009 0.008 0.018

Bias

d"0.4

0.110 0.113 0.114

0.110 0.117 0.116

0.077 0.086 0.091

0.080 0.098 0.094

0.067 0.077 0.083

0.072 0.083 0.081

0.068 0.072 0.085 0.095 0.289 0.284

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presented for both this simulator and the Cholesky simulator in (19). The results for the EMLE where the data are generated using the Cholesky simulator are denoted as EMLE(C), whereas the results for the EMLE where the data are generated using the truncated simulator are denoted as EMLE(T). Similar naming conventions apply to the FDMLE and GPH estimators. Three combinations of the random number generators are adopted for the indirect estimator; 1. C/C indicates that the Cholesky simulator is used for the DGP as well as in simulating the model to compute the indirect estimates according to Step 4 in the indirect estimation algorithm in Section 3.3. 2. C/T indicates that the Cholesky simulator is used for the DGP but that the truncated simulator is used in Step 4 of the indirect estimation procedure. 3. T/T indicates that the truncated simulator is used for both the DGP and in Step 4 of the indirect estimation procedure. For the truncated simulator in (20), a lag length of l"1000 is chosen to capture the long memory characteristics of the ARFIMA model with the "rst 1000 observations truncated to counter the e!ects of any initialization problem. For each combination of random number generators, H"10 and 100 simulation paths are considered. As Gourieroux et al. (1993, p. S92) show, the asymptotic covariance matrix is proportional to (1#H\). Therefore improvements in e$ciency are achieved for larger values of H. For all simulations, the GT version of the indirect estimator is used with N in (14) set equal to the identity matrix when dim(P)"dim(W), while N is set equal to the outer product of the "rst derivatives of the log-likelihood function based on the actual data when dim(P)'dim(W). The results in Table 1 show that the sampling performance of the indirect estimator in terms of both bias and RMSE generally improves as the dimension of the auxiliary model expands from an AR(1) to an AR(4) model, and as the number of simulation paths increases from H"10 to 100, albeit marginally in some cases. For values of d*!0.2, the indirect estimator tends to achieve similar levels of e$ciency as EMLE but with a lower bias level. The FDMLE tends to exhibit slightly better e$ciency than the indirect estimator, whereas the indirect estimator performs better in terms of bias. The bias of the indirect estimator using a GPH auxiliary model is similar to the biases obtained using AR auxiliary models. However, the use of a GPH auxiliary model in the indirect estimation procedure tends to result in less e$cient estimates than the indirect estimator using an AR auxiliary model. This result suggests that the ine$ciency associated with the conventional GPH estimator in small samples is being inherited by the indirect estimator.  In practice this was not a problem as comparable simulation results were also obtained for much shorter lag lengths of say l"500 or even 100.

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165

The "nal observation to note about Table 1 is that there is very little di!erence in the bias and RMSE statistics associated with each estimator when using the di!erent simulators. This is true regardless of whether the Cholesky or truncated simulators are used in the DGP or in the indirect estimation procedure. These results provide strong support for the use of the truncated simulator when computing the indirect estimator. 4.1.2. ARFIMA(1, d, 1) The "nite sampling properties of the alternative estimators for the ARFIMA(1, d, 1) DGP are reported in Table 2 for a sample of size ¹"100 observations. The GPH procedure does not estimate ARMA parameters and so the GPH results are not reported for this particular DGP. Three ARFIMA models are considered with DGP parameter values of d"+!0.3, 0.0, 0.3, and

"0.7 and h"0.5 in (23). To conserve space, the sampling properties of the indirect estimator are reported only for the AR(3) auxiliary model. The indirect estimator is once again computed for H"10 and 100 simulation paths. As with the results in Table 1, alternative combinations of the simulators are used in the DGP as well as in the indirect estimation procedure to determine the accuracy of the truncated simulator. The results in Table 2 show that the indirect estimator achieves comparable levels of e$ciency to EMLE for the parameters d and , and in some cases surpasses the small sample e$ciency of EMLE. However, for the parameter h, the indirect estimator consistently delivers RMSEs above those obtained for EMLE. For all parameters, the small sample bias of the indirect estimator is lower than that for the EMLE. The main di!erence in the results between Tables 1 and 2 is that the FDMLE is consistently less e$cient than the indirect estimator of d while also achieving higher bias levels. Similar sampling properties occur for the parameter . However, in the case of the parameter h, the FDMLE is relatively more e$cient than the indirect estimator. A comparison of the results for the alternative random number generators shows that the truncated autoregression simulator produces comparable results  Other auxiliary models were also experimented with. For example, the augmented parametric GPH model in (18) was also used but yielded inferior small sampling properties relative to the indirect estimator based on an AR(3) auxiliary model.  A number of additional experiments using higher-order AR auxiliary models were also undertaken. These results showed that by increasing the auxiliary model from an AR(3) to an AR(4) or an AR(5), allowed the performance of the indirect estimator to approach that of EMLE in Table 2. For example, the RMSE for h using an AR(4) auxiliary model was 0.128 with a bias of !0.006 when the truncated simulator is used in the DGP and in the indirect estimator. This result compares favourably with the RMSE of 0.123 obtained for FDMLE reported in Table 2, and is an improvement over the RMSE of 0.148 obtained for the indirect estimator using an AR(3) auxiliary model.

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Table 2 Bias and RMSE of the estimators for an ARFIMA(1, d, 1) process, ¹"100

"0.7

d"!0.3 Bias

h"0.5

RMSE

Bias

RMSE

Bias

RMSE

EMLE(C) EMLE(T) FDMLE(C) FDMLE(T)

0.019 0.030 0.042 0.011

0.165 0.166 0.256 0.265

!0.051 !0.081 !0.079 !0.063

0.190 0.207 0.246 0.262

0.020 0.025 0.007 0.024

0.112 0.111 0.123 0.123

Indirect: H"10 C/C C/T T/T H"100 C/C C/T T/T

0.015 0.008 0.011 0.024 0.015 0.010

0.140 0.147 0.134 0.133 0.144 0.137

!0.033 !0.002 !0.034 !0.038 !0.032 !0.029

0.176 0.164 0.170 0.160 0.181 0.171

!0.002 !0.006 0.011 0.006 0.010 0.003

0.135 0.139 0.148 0.133 0.141 0.132

"0.7

d"0.0

h"0.5

Bias

RMSE

Bias

RMSE

EMLE(C) EMLE(T) FDMLE(C) FDMLE(T)

!0.025 !0.040 !0.001 !0.010

0.182 0.196 0.248 0.265

!0.023 !0.031 !0.050 !0.048

0.175 0.191 0.237 0.245

0.029 0.036 0.004 0.015

0.118 0.111 0.125 0.122

Indirect: H"10 C/C C/T T/T H"100 C/C C/T T/T

!0.035 !0.029 !0.023 !0.018 !0.024 !0.031

0.148 0.154 0.153 0.138 0.152 0.149

!0.020 !0.027 !0.036 !0.042 !0.032 !0.021

0.179 0.184 0.195 0.181 0.192 0.179

0.036 0.034 0.041 0.042 0.043 0.033

0.153 0.150 0.169 0.142 0.150 0.143

"0.7

d"0.3

Bias

RMSE

h"0.5

Bias

RMSE

Bias

RMSE

Bias

RMSE

EMLE(C) EMLE(T) FDMLE(C) FDMLE(T)

!0.073 !0.093 !0.060 !0.046

0.166 0.196 0.259 0.260

0.019 0.034 !0.008 !0.023

0.140 0.136 0.203 0.224

0.034 0.015 !0.060 !0.029

0.112 0.111 0.161 0.143

Indirect: H"10 C/C C/T T/T H"100 C/C C/T T/T

!0.053 !0.064 !0.071 !0.063 !0.047 !0.058

0.138 0.145 0.147 0.135 0.135 0.141

!0.022 0.020 0.011 !0.017 0.009 0.003

0.135 0.133 0.128 0.125 0.125 0.122

0.036 !0.017 !0.002 0.055 !0.020 0.002

0.155 0.205 0.192 0.151 0.196 0.174

Note: Details of the simulation DGP are given in Section 4.1. The auxiliary model for the indirect estimator is an AR(3).

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to the Cholesky simulator. This is true for di!erent estimators and for di!erent parameterizations. For example, the RMSEs for any given estimator are often within 0.01 units of each other for alternative simulators. The largest di!erences occur for the indirect estimator of the parameter h for the d"0.3, "0.7, h"0.5 parameterization reported in the bottom block of Table 2. Here the di!erence in RMSE is approximately 0.05 units for the alternative simulators. 4.1.3. Larger sample behaviour The e!ect of increasing the sample size from ¹"100 to 500 observations on the sampling performance of the alternative estimators is illustrated in Tables 3 and 4 for the ARFIMA(0, d, 0) and ARFIMA(1, d, 1) models, respectively. All simulations are based on the truncated simulator given the computational burden of using the Cholesky simulator and the results of Tables 1 and 2. To allow for the increase in the sample size, the lag length in (20) is increased from l"1000 to 5000. The results reported for the indirect estimator are based on H"10 simulation paths, and the auxiliary model is an AR(1) for the ARFIMA(0, d, 0) model and an AR(3) for the ARFIMA(1, d, 1) model. For the ARFIMA(0, d, 0) process in Table 3, the e$ciency of the estimators is improved by a factor of approximately 50% in comparison to the smaller sample size. With the exception of the GPH estimator, all estimators display similar bias levels. The e$ciency ranking of the alternative estimators is not changed by increasing the sample size. As before, the best performer is EMLE followed by FDMLE and then the indirect estimator with the GPH estimator coming in last. The results for the ARFIMA(1, d, 1) model in Table 4 show that there is no clear ordering in terms of e$ciency between the FDMLE and the indirect estimator when estimating the parameters d and . In contrast, the indirect estimator is consistently less e$cient than FDMLE in estimating the moving average parameter h. 4.2. Bivariate experiments In this section, the sampling properties of the indirect estimator are reported for two bivariate VARFIMA model speci"cations. These are the  The RMSE of 0.038 for FDMLE in Table 3 across all parameterizations of d, compares very favourably to a typical value of (0.0015 reported in Cheung and Diebold (1994, Table 1). This result not only shows that the simulation framework of Cheung and Diebold is comparable to the present paper, but as Cheung and Diebold use the Cholesky simulator this provides further support for the accuracy of the truncated simulator used in this paper.  As was the case for the ARFIMA(1, d, 1) model with ¹"100 observations, further experiments using higher-order AR auxiliary models were tried for the ¹"500 case. These additional results showed that the performance of the indirect estimator once again approached EMLE with the largest improvement being obtained for h.

0.002

Indirect: H"10 T/T

0.070

0.036 0.038 0.172

!0.005

!0.002 !0.001 0.010

0.064

0.036 0.038 0.168

RMSE

!0.005

!0.002 !0.002 0.006

Bias

d"0.0

0.048

0.036 0.038 0.168

RMSE

!0.005

!0.004 !0.001 0.007

Bias

d"0.2

0.038

0.036 0.038 0.166

RMSE

Note: Details of the simulation DGP are given in Section 4.1. The auxiliary model for the indirect estimator is an AR(1).

!0.004 0.003 0.019

EMLE(T) FDMLE(T) GPH(T)

Bias

Bias

RMSE

d"!0.2

d"!0.4

Table 3 Bias and RMSE of the estimators for an ARFIMA(0, d, 0) process, ¹"500

!0.010

!0.013 0.003 0.012

Bias

d"0.4

0.045

0.036 0.038 0.166

RMSE

168 V.L. Martin, N.P. Wilkins / Journal of Econometrics 93 (1999) 149}175

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169

Table 4 Bias and RMSE of the estimators for an ARFIMA(1, d, 1) process, ¹"500

"0.7

d"!0.3 Bias

h"0.5

RMSE

Bias

RMSE

Bias

RMSE

EMLE(T) FDMLE(T)

0.019 0.068

0.098 0.128

!0.025 !0.070

0.096 0.129

!0.007 !0.003

0.045 0.047

Indirect: H"10 T/T

!0.012

0.116

0.000

0.099

0.014

0.062

"0.7

d"0.0

h"0.5

Bias

RMSE

Bias

RMSE

Bias

EMLE(T) FDMLE(T)

!0.015 !0.001

0.112 0.106

!0.001 !0.013

0.093 0.095

0.007 0.004

0.052 0.047

Indirect: H"10 T/T

!0.020

0.134

!0.011

0.130

0.038

0.076

"0.7

d"0.3 Bias

RMSE

EMLE(T) FDMLE(T)

!0.052 !0.018

0.102 0.137

Indirect: H"10 T/T

!0.024

0.110

Bias

RMSE

h"0.5 RMSE

Bias

RMSE

0.030 0.013

0.078 0.112

0.012 !0.010

0.050 0.056

!0.005

0.091

0.032

0.103

Note: Details of the simulation DGP are given in Section 4.1. The auxiliary model for the indirect estimator is an AR(3). The results for the EMLE are based on R"100 replications.

VARFIMA(0, d, 1) and VARFIMA(1, d, 1) models. The DGP being considered is a special case of (11) and can be expressed as


 I!

0



0

By h h R " I#   ¸

By h h R  

 




¸




 
 e e

R , R

(24)

where (e , e )&N(0, I). The parameter values considered are +d , d ," R R   +0.3,!0.3, and + , , h , h , h , h ,"+0.6, 0.3, 0.6, !0.2, !0.3, 0.8,.       A bivariate VAR with two lags and no intercept terms is chosen as the auxiliary model. For the VARFIMA(0, d, 1) experiments where there are no autoregressive components in (24), " "0 and dim(P)"8'dim(W)ⴝ6. For the  

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Table 5 Bias and RMSE of the indirect estimators for a VARFIMA(0, d, 1) process d "0.3 

d "!0.3 

h "0.6 

h "!0.2 h "!0.3 h "0.8   

¹"100 H"10 Bias RMSE H"100 Bias RMSE

!0.013 0.105

0.025 0.134

!0.091 0.199

0.077 0.161

0.067 0.161

!0.129 0.205

!0.013 0.098

0.021 0.129

!0.091 0.194

0.079 0.149

0.070 0.160

!0.121 0.197

0.030 0.171

0.004 0.137

!0.050 0.185

¹"500 H"10 Bias RMSE

!0.015 0.055

0.005 0.168

0.006 0.188

Note: Details of the simulation DGP are given in Section 4.2. The auxiliary model is a bivariate VAR(2).

full bivariate VARFIMA(1, d, 1) model, dim(P)"dim(W)"8. The sample size is set at ¹"100 and 500 observations. The number of simulation paths are H"10 and 100 for the ¹"100 sample size, whereas for ¹"500 only H"10 simulation paths are used in computing the indirect estimator. For computational reasons the truncated autoregression simulator is used in the DGP as well as in simulating the VARFIMA model in the indirect estimation procedure. 4.2.1. VARFIMA(0, d, 1) The sampling performance of the indirect estimator is presented in Table 5 for the VARFIMA(0, d, 1) experiments. The results for a sample size of ¹"100 observations show that there is very little change in the "nite sample distribution from an increase in the number of simulation paths from H"10 to 100. There is, in general, a reduction in the bias as the sample is increased from ¹"100 to 500 with the exception of the parameter d "0.3 where the (abso lute) bias increases by 0.002. There is also a general reduction in RMSE due to the increase in sample size. The few exceptions show a possible trade-o! between e$ciency and bias. 4.2.2. VARFIMA(1, d, 1) Table 6 shows that the indirect estimator displays similar properties for the VARFIMA(1, d, 1) model as it does for the VARFIMA(0, d, 1) model. In particular, there is very little di!erence to the "nite sample distribution when ¹"100 from increasing the number of simulation paths from H"10 to 100. Increasing

!0.021 0.199

!0.080 0.195

!0.067 0.154

H"10 Bias RMSE 0.026 0.139

0.006 0.229

0.032 0.167

0.053 0.133

¹"500

0.019 0.238

¹"100

"0.3 

0.032 0.171

"0.6 

!0.018 0.138

!0.178 0.272

!0.180 0.275

h "0.6 

0.030 0.180

0.069 0.188

0.078 0.199

h "!0.2 

Note: Details of the simulation DGP are given in Section 4.2. The auxiliary model is a bivariate VAR(2).

!0.025 0.108

!0.023 0.204

!0.081 0.198

d "!0.3 

H"10 Bias RMSE H"100 Bias RMSE

d "0.3 

Table 6 Bias and RMSE of the indirect estimators for a VARFIMA(1, d, 1) process

!0.004 0.121

0.051 0.179

0.049 0.183

h "!0.3 

!0.069 0.177

!0.119 0.227

!0.138 0.241

h "0.8 

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the sample size from ¹"100 to 500 observations generally results in a reduction in both bias and RMSE. The main exception to this is for the parameter

"0.3, where there is an improvement in e$ciency partly at the expense of  an increase in bias. 5. Conclusion This paper has provided a framework for performing indirect estimation of both univariate ARFIMA and multivariate VARFIMA models. The approach consisted of simulating ARFIMA and VARFIMA processes and calibrating the simulated series with the actual time series via an auxiliary model. Alternative auxiliary models were discussed. Special attention was given to identifying the small sample properties of the indirect estimator using Monte Carlo methods although some simulation experiments were reported for relatively larger samples to gain some indication as to the asymptotic behaviour of the indirect estimator. Attention was also paid to comparing the accuracy of alternative random number generators for simulating ARFIMA models. The main result of the paper is that the small sample properties of the indirect estimator for a broad range of parameterizations are similar to the sampling properties obtained with the EMLE suggested by Sowell (1992a), and superior to the sampling properties of the GPH estimator introduced by Geweke and Porter-Hudak (1983). The FDMLE suggested by Fox and Taqqu (1986) yields similar small sample properties to EMLE for ARFIMA(0, d, 0) DGPs. For more complicated models the relative performance of the indirect estimator and FDMLE are mixed. For example, the indirect estimator tends to yield better results in terms of bias and RMSE when estimating the fractional di!erencing parameter and the autoregressive parameter, whereas FDMLE tends to yield better results for the moving average parameter. The simulation results show that low-dimensional AR models can provide suitable approximations to the true model as the associated indirect estimators achieve comparable levels of e$ciency to EMLE. These results complement the original work of Smith (1993) who also found that VARs acted as a suitable auxiliary model in an indirect estimation procedure. This suggests that the use of additional moment information as based on the SNP class of auxiliary models introduced by Gallant and Tauchen (1996) and implemented by Gallant and Long (1997) and Anderson and Lund (1997), for example, can be expected to achieve at most modest improvements in e$ciency. Inferior results were  Liu and Zhang (1996) highlight the dangers of over"tting when redundant moments are used in the auxiliary model. Their suggestion is to conduct a range of tests on the moments and exclude those moments which are found to be insigni"cant. An advantage of this framework is that it formalizes the original suggestion of Gourieroux et al. (1993) and Gallant and Tauchen (1996) which was to choose the auxiliary model which "ts the data best.

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obtained using frequency-domain-based estimators such as the GPH regression as an auxiliary model. A comparison of the computational times for the alternative estimators showed that FDMLE was faster than the indirect estimator and EMLE for the speci"cations used in the simulations in Section 4. In the case of the ARFIMA(0, d, 0) model, it is possible to design the indirect estimator to achieve similar computational times as FDMLE by reducing the number of simulation paths and the lag length in the truncated simulator, and yet still achieve comparable levels of e$ciency and bias. For the ARFIMA(1, d, 1) experiments, the indirect estimator could also achieve comparable computational times as FDMLE while still delivering smaller RMSEs for the d and parameters, but larger RMSEs for the h parameter. In comparison to EMLE and once again adopting the same speci"cations of the indirect estimator used in Section 4, the main computational gains of the indirect estimator over EMLE occurred for the larger sample size of ¹"500. For example, the indirect estimator was approximately 30 times faster than EMLE for an ARFIMA(1, d, 1) model with H"10 simulation paths and l"1000 initial values in the truncated simulator. Increasing the truncation parameter to l"5000 resulted in the indirect estimator being approximately nine times faster than EMLE. Similar results were obtained for the ARFIMA(0, d, 0) model. The last important advantage of the indirect estimator for ARFIMA models is that the framework is easily generalized to multivariate models. To highlight this feature, bivariate fractional models with bivariate moving average structures and own autoregressive components were estimated using indirect methods. The relative bias of the parameter estimates was small and the sampling variance consistent with the univariate results, whilst the computational time was exceedingly fast. For example, on a Pentium II 266, the average time it took to estimate the bivariate VARFIMA(0, d, 1) model for H"10 simulation paths was less than 1 min for a sample size of ¹"100 observations, and less than "ve minutes for a sample size of ¹"500 observations.

Acknowledgements We thank an Associate Editor of this journal and two referees for comments and suggestions made that have played a signi"cant role in improving an earlier version of this paper. We also thank conference participants for comments made at the 1996 Australasian Meetings of the Econometrics Society, Perth, Western Australia and at the 1997 European Meetings of the Econometric Society, Toulouse, France and seminar participants at the University of New South Wales.

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