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1 epub WU Institutional Repository Michael A. Hauser Maximum Likelihoo Estimators for ARMA an ARFIMA Moels. A Monte Carlo Stuy. Paper Original Citation: Hauser, Michael A. (998) Maximum Likelihoo Estimators for ARMA an ARFIMA Moels. A Monte Carlo Stuy. Preprint Series / Department of Applie Statistics an Data Processing, 22. Department of Statistics an Mathematics, Abt. f. Angewante Statistik u. Datenverarbeitung, WU Vienna University of Economics an Business, Vienna. This version is available at: Available in epub WU : July 26 epub WU, the institutional repository of the WU Vienna University of Economics an Business, is provie by the University Library an the IT-Services. The aim is to enable open access to the scholarly output of the WU.

2 Maximum Likelihoo Estimators for ARMA an ARFIMA Moels: A Monte Carlo Stuy Michael Hauser Department of Applie Statistics an Data Processing Wirtschaftsuniversität Wien Preprint Series Preprint 22 May 998

3 Maximum Likelihoo Estimators for ARMA an ARFIMA Moels: A Monte Carlo Stuy Michael A. Hauser University of Economics an Business Aministration, Department of Statistics, Vienna First raft December 997. This version May 998. Forthcoming Journal of Statistical Planning an Inference Abstract We analyze by simulation the properties of two time omain an two frequency omain estimators for low orer autoregressive fractionally integrate moving average Gaussian moels, ARFIMA (p; ; q). The estimators consiere are the exact maximum likelihoo for emeane ata,, the associate moie prole likelihoo,, an the Whittle estimator with,, an without tapere ata,. Length of the series is. The estimators are compare in terms of pile-up eect, mean square error, bias, an empirical conence level. The tapere version of the Whittle likelihoo turns out to be a reliable estimator for ARMA an ARFIMA moels. Its small losses in performance in case of \well-behave" moels are compensate suciently in more \icult" moels. The moie prole likelihoo is an alternative to the but is computationally more emaning. It is either equivalent to the or more favorable than the. For fractionally integrate moels, particularly, it ominates clearly the. The has serious eciencies for large ranges of parameters, an so cannot be recommene in general. The, on the other han, shoul only be use with care for fractionally integrate moels ue to its potential large negative bias of the fractional integration parameter. In general, one shoul procee with caution for ARMA(,) moels with almost canceling roots, an, in particular, in case of the an the for inference in the vicinity of a moving average root of +. Keywors: Fractional integration, Whittle likelihoo, moie prole likelihoo, ata taper, pile-up eect Acknowlegments: I want to express my thanks to Peter Bloomel, Sastry Pantula, Beneikt Potscher, an iscussants at the International Workshop on Long Range Depenence, Brisbane, January 997, an two anonymous referees. Short title: Maximum Likelihoo Estimators for ARFIMA Moels Aress of the author: Department of Statistics, University of Economics an Business Aministration, Augasse 2-6, A-9 Vienna, Austria. Tel.: /4759, FAX: /738, Michael.Hauser@wu-wien.ac.at

4 Introuction In this paper we analyze, by means of simulation, the properties of four versions of maximum likelihoo estimators for tting autoregressive fractionally integrate moving average, ARFIMA, time series moels. The comparison of the small sample properties is of particular interest since ierent estimators may lea to substantial istinct conclusions. E.g. if the Whittle estimate or the moie prole likelihoo is use for tting a class of low orer ARFIMA moels to the frequently investigate postwar quarterly U.S. GNP growth rates Akaike's information criterion will choose a non fractional moel. On the other han, if the same proceure is unertaken with the exact Gaussian likelihoo, it results in a fractionally integrate moel with a highly signicant fractional integration parameter. See Sowell(992b) an Hauser(995). Thus, espite the fact that the estimators are asymptotically equivalent the question remains which estimator is more reliable in small samples. We assume that the series y t is a sample of length n of a weakly stationary process (in iscrete time) which amits a Wol representation y t = + (L)u t = + ( + L + 2 L )u t : where L enotes the backwar shift operator. The impulse response coecients satisfy P = 2 <, an the u t are Gaussian white noise with variance 2 u. An, y t obeys to an ARFIMA(p; ; q) moel with 2 [ 2 ; 2 ). (L)( L) (y t ) = (L)u t ; where (L) = L p L p an (L) = L q L q ; are polynomials in the backwar shift operator an is the fractional integration parameter. The roots of the polynomials (z) an (z) are assume to lie outsie the unit circle. Then the process (y t ) is stationary for < 2 an invertible for > (Bloomel, 985). The spectrum f() = 2 u j(exp( i))j 2 2 j(exp( i))j 2 j exp( i)j 2 ; is innite at the origin for > an zero for <. Long memory is associate with >. If <, the process is sai to have intermeiate memory.

5 Two generic maximum likelihoo proceures for stationary Gaussian series for the parameter vector = ( ; ; p ; ; ; ; q ) are available: the (approximative) spectral (Whittle) maximum likelihoo, an the exact Gaussian maximum likelihoo metho. Both methos yiel p n-consistent, asymptotically normal an asymptotically ecient parameter estimates. See Fox an Taqqu(986), Dahlhaus(989) an Giraitis an Surgailis(99) for the Whittle estimator, an Dahlhaus(989) an An, Bloomel an Pantula(992) for the exact maximum likelihoo. The small sample properties of four estimators will be investigate: the exact Gaussian likelihoo, the moie prole likelihoo erive by An an Bloomel(993) accoring to the proposal of Cox an Rei(987), the Whittle likelihoo an two versions thereof using tapere ata. Having economic applications in min we restrict our stuy to low orer ARMA an ARFIMA moels. The criteria for comparison are the mean square error, bias, an the empirical conence level of the 95% conence interval. In aition, the estimators will be investigate with respect of the occurrence of the pile-up eect, which has been intensively iscusse in the literature for the exact maximum likelihoo metho. Our stuy is a signicant extension to the former simulation experiments performe by Ansley an Newbol(98), Boes et al.(989) an Cheung an Diebol(994). No Monte Carlo stuy seems to be available which compares the small sample properties of maximum likelihoo estimators even for ARMA moels. An, estimators using tapere ata have been applie almost exclusively to AR(p) or FI() moels. Section 2 enes the estimators an summarizes their small sample properties as far as they are known an relevant for our stuy. Section 3 summarizes the Monte Carlo results. Computational aspects are iscusse in Section 3.. For the Monte Carlo results of the pile-up eect see Section 3.2. The ARFIMA moels chosen an the criteria for the comparison of the estimators are given in Section 3.3. The properties of the four estimators are presente in Section 3.4. Section 3.5 gives a heuristic explanation of the systematic negative bias of the exact maximum likelihoo estimates of. Section 4 conclues. 2

6 2 The estimators The exact maximum likelihoo proceure, The exact Gaussian log-likelihoo is given by log L E (; ; 2 u) = 2 [n log(2) + log et() + (Y l) (Y l)] () where = (; 2 u) is the n n covariance matrix epening on an 2 u. n enotes the sample size, Y = (y ;... ; y n ) an l = (;... ; ). We will investigate the reuce prole likelihoo where is replace by the sample mean y, an where the solution of the maximization with respect to 2 u is obtaine analytically. The reuce (or concentrate) likelihoo is then with 2 u = 2 u log L E() = [n log(2) + n log(2 u ) + log et( ) 2 ~ + n] (2) an 2 u = n (Y yl) ~ (Y yl). ~ is ene by the relation = 2 u ~, so that ~ = ~() is a function of alone. The resulting estimator is enote by. Two small sample properties of the may be of importance: The pile-up eect arises when the unerlying MA moel has a root \close" to the unit circle. Then the estimates yiel roots on the unit circle with a positive probability. (Cp. Cryer an Leolter(98) an Anerson an Takemura(986).) On the other han, it is known that the estimate tens to give a negatively biase parameter for pure fractionally integrate processes (Li an McLeo, 986). This is not in eect if the true mean is known an is use for the mean correction of the ata (Cheung an Diebol, 994). The moie prole likelihoo, The moie prole likelihoo is base on the iea to correct the parameter estimates of interest (^a) for secon orer eects ue to nuisance parameters (b) in the moel. (Cp. Cox an Rei(987).) Thereby a transformation is sought which makes b orthogonal to a. For ARFIMA moels the nuisance parameters are chosen as b = (; y), 2 an the parameters of interest as a = = ( ; ; p ; ; ; ; q ). An an Bloomel(993) give the solution for the moie prole likelihoo base on the Gaussian exact maximum likelihoo. The orthogonal vector results as = (; ) with = 2 y (et R) =n, where R = = 2 y is the 3

7 correlation matrix. Remarkably, turns out to be orthogonal to. The without constants is then (in a notation similar to An an Bloomel) log L M(; ^ 2 u; ^) = ( n 2 ) log et (R) 2 log (l R l) + ( + 2 n 2 )(Y ^l) R (Y ^l)] (3) Our actual implementation replaces ^ by the sample mean y, an ^ 2 y by the maximum likelihoo estimator ^ 2 y = (Y n yl) R (Y yl). ^ 2 u results from the relation 2 y = 2 P 2 u i i. An an Bloomel also oer a small Monte Carlo stuy illustrating for some selecte low orer ARFIMA moels that the successfully eliminates the bias in the estimates. The Whittle likelihoo, We ene the Whittle likelihoo as mx log L W (; u) 2 = j= log f( j j ; 2 u) 2 mx j= I( j ) f( j j ; 2 u) (4) where I( j ) enotes the perioogram at the j-th Fourier frequency, j ( j = 2 j n, j = ;... ; m) I( j ) = n j nx m is the largest integer containe in (n )=2. y t exp( it j )j 2 (5) t= It is the iscrete time version of the Whittle function (cp. Dahlhaus(988) an Robinson(99)). In the ARMA case it may be interprete e.g. as the likelihoo associate with the asymptotic istribution of the perioogram (cp. Brockwell an Davis, 99, p.347f). Other interpretations are given in Dahlhaus(988) an Parzen(983, Sec. 3). On the other han, if the term P m j= log f( j j ; 2 u) is roppe the asymptotic properties remain the same (Fox an Taqqu, 986), an it becomes the Yule-Walker estimator for AR(p) processes (Dzhaparize, 986, p.6f). The reuce form of L W with respect to the error variance 2 u is with 2 u = 2 u log L W () = m log(2) m log[ I( j ) m j= 2 u mx = m j= 4 mx g( j ) ] mx I( j ) g( j ) log g( j ) m (6) j= (7)

8 where f() = 2 u g()=(2) with g() = g( j ). This estimator is enote by. The pile-up eect has not been consiere for the Whittle estimate before. In the line with Anerson an Takemura(986) we prove that the likelihoo of a MA() process exhibits a local maximum at =. The same hols contrary to the or for AR() processes at =. The local maximum may turn out as a global one in nite samples, so that parameter estimates of are obtaine with a positive probability. These probabilities are etermine by Monte Carlo simulations in Sec Actually, we only investigate the rst orer conitions. They are for a moel with a single parameter ( may be or ) (@ log L W )=(@) =, or more explicitly X I X j u 2 j g 2 j g = with g j = g( j ) an I( j ) = I j. After some manipulations we obtain X I j j=@ X ] = g j j j g j I j g j [ X k =@ g k ]: So, if the terms in brackets are constant an equal, the rst orer conition hols. In case of MA() processes g() = 2 cos() + 2, = 2[ cos()], g = 8 if =. Both bracket terms are for = either +m or m, an may be cancele. The rst orer conition hols for =. In case of AR() processes g() = =[ 2 cos()+ 2 = ( 2)[ cos()] = [ 2 cos() + 2 ] 2, an g = 8 if =. An, the same hols as above. It is known that Yule-Walker estimates are rather ba for short series, an if the roots of the corresponing characteristic equation are close to the unit circle (Priestley, 98, p.35). So, one may expect similar properties to hol for the Whittle estimates. On the other han, Hauser(995) reports an avantage of the with respect to the for pure fractionally integrate moels an some higher orer ARFIMA moels in small samples. The gives essentially unbiase estimates an smaller mean square errors for in most cases. 5

9 The Whittle likelihoo for tapere ata, Dahlhaus(988) shows that tapering reuces the leakage eect of the perioogram as estimate of the true spectrum. He ns that the new estimate competes well with the Burg estimate for an AR(4) moel where roots of the characteristic equation are complex an close to the unit circle. A tapere series is ene by y T t = h t y t where h t is the ata taper, in our case the Tukey-Hanning taper h t = 8 >< >: 2 [ cosf(t 2 )=lg] t = ;... ; l t = l + ;... ; n l : (8) 2 [ cosf(n t + 2 )=lg] t = n l + ;... ; n The proportion of the ata which is altere by this taper is 2 = 2 l=n. We choose the variable taper of Zhang(99) who proposes to use = 2= p n. The tapere time series is then use to construct moie perioogram orinates, I T ( j ), I T ( j ) = H 2 n j nx h t (y t y) exp( i t j )j 2 (9) t= with H 2 = P t h 2 t. Replacing I( j ) in the function by I T ( j ) yiels the Whittle likelihoo for tapere ata, enote as. There are several stuies concerne with the use of ata tapers ealing essentially only with AR moels: Pukkila an Nyquist(985), Kang(987), Hurvich(988), or Zhang(99), among others. The general conclusions are that the type of taper is not of much importance, nor oes the amount of tapering aect the results consierably. However, some tapering shoul be performe. Cheung an Diebol(994), on the other han, investigate the small sample properties of the approximative Whittle estimate (without the log f-term) for pure FI() moels. They n it to be slightly inferior to the regarless whether tapere or non tapere ata are use. 3 Small sample behavior of the estimators 3. Computational aspects All simulate series throughout the paper are generate via the Durbin-Levinson algorithm. Thereby the true autocovariances (cp. Sowell, 992a) an Gaussian innovations 6

10 with unit variance (Hoermann an Deringer, 99) are use. This metho is exact. For the generation of the uniformly istribute input variates the TT8, a twiste GFSR generator (Matsumoto an Kurita, 994) is use. It has a perio of 2 8 an excellent equiistribution properties up to imension 25. Due to numerical problems of the an when the parameters approach the non stationarity region we have to restrict the estimate AR an parameters. We set max j=z AR j < :9965, where z AR is the root of the characteristic equation for the AR polynomial. The maximal is The program use for calculating the estimates of the, an so also of the, is base on the FORTRAN coe supplie by F. Sowell. The spectral estimates, on the other han, are not boune away from the unit circle. If estimate roots lie insie the unit circle, the inverse of the roots are use to make the moels stationary an invertible. The conence intervals of the an estimates are calculate only when an interior maximum is foun, i.e. if j=^z AR j :99 an : The pile-up eect The probability that parameter estimates of are observe when tting MA() or AR() moels to small ata sets are obtaine by Monte Carlo simulation. Table gives the relative frequencies (with respect to replications) that the estimates ier from + or no more than.35 (cp. the numerical restrictions given above) for series of the length of 25, 5 an. E.g., P( ^ = ) stans for P( ^ 2 [:9965; ]). ** inclue TABLE ** The results for the (), the where the true mean is use to emean the ata, compare to the theoretical results of Cryer an Leolter(98). The an the results are very similar to the theoretical ones in case of the MA() moel with = :9. However, for = :9 the pile-up eect is smaller, especially for the. The, which is inepenent of the mean correction, exhibits small frequencies for = :9, an is comparable to the for = :9. For the AR() processes the pile-up eect is less pronounce. The probability of estimates of the are somewhat larger than those for the. 7

11 3.3 Moels an criteria We will compare four estimators for low orer ARFIMA moels with respect to the mean square error,, of the parameter estimates, the bias, an the percentage that the true parameter lies in the 95% conence interval base on the asymptotic normal istribution. This percentage is enote by, empirical conence level. The values given for the are scale by a factor of. The length of the series consiere is. replications are performe. *** INCLUDE somewhere below FIGURES to an TABLE 2 The moels consiere in the simulation stuy are as follows: AR() with parameter values = :99; :95;... ; :95; :99. The results are summarize in Figure not incluing the estimator. The is neither presente graphically nor in tables (except in Fig. 3) but escribe verbally at the en of this section ue to its limite applicability. MA() with = :; :99; :95;... ; :95; :99; : (Figure 2). FI() with values ranging form :5 to :45; :49 (Figure 3). AR(2) moels with real roots, z ;2, close together as in Zhang(99). For z < the relation =z + :5 = =z 2 is assume, for z > =z :5 = =z 2, with =z = :95; :9;... ; :9; :95. See Figure 4. The same parameter values are implemente in MA(2) moels yieling troughs in the spectrum where the AR moels have peaks (Figure 5). These moels are of interest, since the small sample behavior of the Whittle estimator using tapere ata has essentially been applie only to AR moels. In aition to real roots, we also consier complex roots, z ;2 = r e i, in AR(2) moels (Figure 6). We choose a constant moulus close to the unit circle of r = =:95, an uniformly istribute frequencies over [; ]: = j with j = ;... ;. Both ens, j = ;, of the interval correspon to ouble real roots. Due to the similarity of the behavior of both involve parameter estimates an to space restrictions only the of the secon parameter estimate are given in Fig. 4 to 6. Further, ARMA(,) processes are investigate (Figure 7). The an parameters are chosen to take the values :99; :95;... ; :95; :99. For the list of values is augmente by :. The minimal istance between the AR an MA parameter is chosen to be :5, j j :5. Aitional moels with almost canceling roots are specie as 8

12 pairs ( ; :5) an ( :5; ) for ; = :8; :6;... ; :. The contour lines of the surfaces in Fig. 7.a are rawn to inicate the 5:; :;... levels. (The computations were performe with the triangle contour plot algorithm of Preusser(984).) In orer to accentuate a bias of null the corresponing contour line is rawn fat. The other bias levels are otte inicating levels of :5; :;.... In a similar way the empirical conence levels are marke. The fat lines inicate a level of 95%, the otte ones ; 9; 85;.... Simulation results for ARFIMA(,,) processes are summarize in Figure 8. The values for the fractional integration parameter are chosen between :5 an.45. The autoregressive parameter varies between :95 an.95. However contrary to Fig. 7, the contour levels in the plots are 2:5; 5:;.... Finally, the results for ARFIMA(,,) processes with parameters also ranging from :5 to.45, an from :95 to.95 are given. See Figure. In aition to the graphical representation, the maximal an minimal values of each criterion for moels with 2-imensional parameter spaces, together with the points where they are assume, are collecte in Table 2. Since an estimator may exhibit the largest maximum in a small area but performs excellent for most other moels, while another estimator may be rather ba over the whole class of moels, the overall maximum may be no aequate inicator. So we construct a measure of net avantage of estimator A over estimator B base on the maximal ierences of the surfaces. We ene it as max j A() B ()j max j A() B ()j : B>A A>B A () enotes the of the parameter estimate of obtaine by estimator A. The maximal avantage (measure in ) using estimator A instea of B is so relate to the maximal avantage of estimator B. If our measure is positive, estimator A is to be preferre over estimator B. If it is negative, B is to be preferre. Table 3 summarizes the results for invertible ARMA(,), ARFIMA(,,) an ARFIMA(,,) moels. 3.4 Small sample behavior of the estimators Our numerical results compare - as far as values are available - well to previous stuies like Zhang(99) an Cheung an Diebol(994). However, they ier to some extent from Ansley an Newbol(98). They emean the ata by the population mean while 9

13 we use the sample mean as is commonly one leaing to a more or less pronounce asymmetric behavior of the time omain ML estimates at parameter values associate with real roots close to + on one han, an on the other. In the following we survey the small sample properties of pairs of estimators by rst stating the main properties an then backing up the conclusion by referencing to the gures an tables provie. We compare an, an, give a verbal escription of the, an iscuss the eects of tapering the ata. For non fractionally integrate moels the turns out to be either equivalent to or more favorable than the. In case of AR() moels with negative the behavior of an are essentially ientical. For larger positive the an bias of the estimates are smaller, an the is closer to the theoretical 95% level. The same relation hols for the MA() moels (with the exception of close to +). In case of the AR(2) (Fig. 4 an 6) an the MA(2) moels (Fig. 5) ierences between both estimators are not visible. For the ARMA(,) moel the minimal of the is the same as that of the (.6). The maximum of 54.67, however, is clearly below the corresponing value of the, The criterion of the net avantage between an (see Tab. 3) rates both estimators as essentially equivalent. For fractionally integrate moels the is clearly ominate by the. This hols particularly for mixe autoregressive or moving average fractionally integrate moels. Compare Fig. 3, 8 to an Tab. 2. The systematic negative bias in the estimate of the is enlarge by the inclusion of the AR an MA parameter. E.g., in case of the ARFIMA(,,) moel with true values of (; ) = (:5; :2) the meian estimate moel is me( ) = :5 an me( )=.426! The associate true an estimate spectral ensities are plotte in Figure 9. It turns out that the cannot etect the true sign of the parameter for a rather large set of moels. Those moels are marke by crosses in the bias plots. Compare both Fig. 8.b an Fig..b. The large bias of the estimate is reecte in an. The, on the other han, is not aecte by that systematic bias an has goo an acceptable properties. The criterion of net avantage with respect to the conrms our conclusions. See Tab. 3. The, the Whittle likelihoo applie to tapere ata, exhibits for \well-behave"

14 moels slightly larger than the. However, for more \icult" moels this isavantage seems to be compensate suciently. The turns out to be a reliable estimator. The favorable properties of the o not come into eect for the single parameter moels AR() an FI(). In those cases the is larger than that of the over the whole parameter region. But bias an properties are essentially equivalent. See Fig. an 3. For most values this relation also hols in the MA() moels. However close to = :,, bias an are more favorable with respect to the (Fig. 2). Similarly, as in the MA() case the ominates the close to =z = :95 for the AR(2) moels in Fig. 4, close to =z = :95 for the MA(2) moels in Fig. 5, an close to = for the AR(2) moels with conjugate complex roots in Fig. 6. For ARMA(,) moels the avantage of the is evient (Fig. 7 an Tab. 2, 3). Moels with almost canceling polynomials can be estimate with the more precisely. The sprea of the bias an the maximal are consierably smaller for the for both ^ an ^, than for the. The net avantage as given in Tab. 3 is also in favor of the. The of the for ^ at the true parameters ( ; ) = ( :99; :) is extremely low (6.9), while the minimal value for the is Fig. 8.c also shows that the conence intervals base on or estimates for values close to + are too small. (We have also investigate the sensitivity of our conclusion ue to the inclusion of moels with possibly too strong canceling eects in relation to the length of the series. However, the avantage of the remains if the moel class is restricte to j j :2. In this case the maximal values of an are rather the same, but net avantage, bias an are still in favor of the. The corresponing values are not tabulate.) Accoring to visual inspection of Fig. 8, minimal an maximal, the net avantage with respect to the (cp. Tab. 3), an sprea of the bias (see Tab. 2), the is to be preferre for the ARFIMA(,,) moels. However, its for may be rather low for moels close to (; ) = ( :5; :95). On the contrary, in case of the ARFIMA(,,) moels all criteria apart from the net avantage are in favor of the. The estimates may exhibit serious eciencies when the roots of the moels are \close" to real unit roots, both for AR an MA polynomials. Unacceptable high values are observe together with large biases in relative large parameter ranges. So, the

15 cannot be recommene in general, an a etaile graphical presentation is omitte. The most important simulation results are ocumente in the following in a verbal way. The efects of the may harly be observe for AR() moels. For MA() moels, AR(2) an MA(2) moels with real roots, however, the curves aopt a pronounce trough shape with a rather at bottom. In the AR(2) moels with complex roots a U- shape is observe (with a minimal value at = =2 which competes well with the ). The surface for ^ of the ARMA(,) moels exhibits a W-like prole with the rige generate by the moels with almost canceling roots in the mile. (The surface for ^ is not aecte an similar to the one observe for the.) The estimate of the fractional integration parameter, contrary to the, oes not exhibit a systematic bias. This is most evient for the FI() moels, where the is the secon best estimator being only slightly inferior to the, an more favorable than the. See Fig. 3. Incluing an AR parameter in the FI() moel the of both parameters become rather high at (; ) = (:45; :95), while for the ARFIMA (,,) the overall properties are inferior to the but compete with the. Tapering the ata increases the but not the bias for \well-behave" moels an lowers both, an bias, impressively for all problematic moels investigate. 3.5 An explanation for the bias in the estimate of The potential negative bias in the estimate obtaine by the, in contrast to the where no systematic bias is observe, seems worth to be investigate more closely. A heuristic explanation for this behavior is oere below for fractionally integrate processes with >. We observe that both estimators ier in the treatment of the mean, the frequency zero in the spectral representation respectively. While frequency zero is exclue in the estimation explicitly, it is implicitly inclue in the through the autocovariance function (h). (h) = Z f() e i h () where the covariance matrix is = () = [(ji jj)] i;j with = ( ;... ; p ; ; ;... ; q ). In case of emeane ata the perioogram as estimator of the spectrum is null at frequency zero inepenently on the process. However, for > the spectrum is innite at zero. As the parameter essentially escribes only the slope of the spectrum close to frequency 2

16 zero, it is very sensitive to changes in the low frequency region (but insensitive to high frequency eects). Now consier the replacement of f in () by the perioogram for emeane ata. Then the estimate of will try to moel the upwar slope of the true spectrum for low frequencies, but will have to take into account the value of null at zero. So, a negative bias will result. Not surprisingly, the negative bias in case of the ARFIMA(,,) an (,,) moels may be consierably larger than in pure FI moels. The aitional ARMA parameter (which moels the spectrum at low an high frequencies) oers more exibility to to capture the suen ecrease at frequency zero. The consequence is also a consierable bias in ^, ^ respectively, ue to a compensating eect in the estimate spectrum at low frequencies (cp. Fig. 9). Our explanation is in line with the negative bias in the sample autocovariance function as observe by Newbol an Agiakloglou(993). It is also compatible with the nings of Cheung an Diebol(994) that the exact maximum likelihoo estimator for ata correcte with the true mean yiels unbiase estimates. 4 Summary We investigate the pile-up eect for the exact maximum likelihoo metho, the moi- e prole likelihoo, the Whittle estimator an the Whittle estimator for tapere ata. Some conclusions are: The pile-up eect of the applie to emeane ata is somewhat smaller than the theoretical values for known mean. Contrary to the, the exhibits the pile-up eect also for the AR() moels, but to a lesser extent. An extensive simulation stuy is performe to investigate the properties of the estimators in low orer ARMA an ARFIMA moels. The length of the series is. Despite the asymptotic equivalence of the estimators we n that their small sample properties may ier substantially. The exact maximum likelihoo exhibits a rather ba performance in mixe ARFI or FIMA moels. The estimates ten to be seriously negatively biase leaing to large mean square errors an low empirical conence levels. Also, for ARMA(,) moels with almost canceling roots the use of the may lea to extremely large mean square errors an biases. The moie prole likelihoo exhibits for ARMA moels slightly better properties than the exact maximum likelihoo estimator, 3

17 an ominates it clearly for fractionally integrate ones. The Whittle likelihoo, while yieling no systematically biase estimates, has serious eciencies for large parameter ranges especially \close" to real roots of +=, an so cannot be recommene in general. Base on a comparison of the an we oer a heuristic explanation for the bias in the estimates of in long memory moels. The Whittle likelihoo with tapere ata performs well not only for AR moels but also for MA, ARMA(,) an fractionally integrate moels. It turns out to be an overall reliable estimator. The small losses in performance in case of \well-behave" moels seem to be compensate suciently in more \icult" moels. Its computational simplicity is also attractive. The computationally more emaning alternative with certain avantages is the moie prole likelihoo. Caution is requeste for ARMA(,) moels with almost canceling roots in general, an, in particular, in case of the an the for inference in the vicinity of a moving average root of +. 4

18 References An, S., P. Bloomel an S. Pantula, 992, Asymptotic properties of the MLE in fractional ARIMA moels, Institute of Statistics Mimeograph Series No. 2228, North Carolina State University. An, S. an P. Bloomel, 993, Cox an Rei's moication in regression moels with correlate errors, Department of statistics, North Carolina State University, Raleigh. Anerson, T.W. an A. Takemura, 986, Why o noninvertible estimate moving averages occur?, Journal of Times Series Analysis, 7, Ansley, C.F. an P. Newbol, 98, Finite sample properties of estimators for autoregressive moving average moels, Journal of Econometrics, 3, Bloomel, P., 985, On series representations for linear preictors, The Annals of Probability, 3, Boes, D.C., R.A. Davis an S.N. Gupta, 989, Parameter estimation in low orer fractionally ierence ARMA processes, Stochastic Hyrology an Hyraulics, 3, 97-. Brockwell, P.J. an R.A. Davis, 99, Time series: Theory an methos (Springer, New York). Cheung, Y.-W. an F.X. Diebol, 994, On maximum-likelihoo estimation of the ierencing parameter of fractionally-integrate noise with unknown mean, Journal of Econometrics, 62, Cox, D.R. an N. Rei, 987, Parameter orthogonality an approximate conitional inference (with iscussion), Journal of the Royal Statistical Society Series B, 49, -39. Cryer, J. an J. Leolter, 98, Small-sample properties of the maximum likelihoo estimator in the rst-orer moving average moel, Biometrika, 68, Dahlhaus, R., 988, Small sample eects in time series analysis: A new asymptotic theory an a new estimate, The Annals of Statistics, 6, Dahlhaus, R., 989, Ecient parameter estimation for self-similar processes, The Annals of Statistics, 7, Dzhaparize, K., 986, Parameter estimation an hypothesis testing in spectral analysis of stationary time series (Springer, New York). Fox, R. an M.S. Taqqu, 986, Large-sample properties of parameter estimates for strongly epenent stationary gaussian time series, The Annals of Statistics, 4, Giraitis, L. an D. Surgailis, 99, A central limit theorem for quaratic forms in strongly epenent linear variables an its application to asymptotic normality of Whittle's estimate, Probability Theory an Relate Fiels, 86,

19 Hauser, M.A., 995, Long range epenence in international output series: A reexamination, (Worl Congress of the Econometric Society, Tokyo) Hoermann, W. an G. Deringer, 99, The ACR metho for generating normal ranom variables, OR Spektrum, 2, Hurvich, C.M., 988, A mean square criterion for time series ata winows, Biometrika, 75, Kang, H.-J., 987, The tapering estimation of the rst-orer autoregressive parameters, Biometrika, 74, Li, W.K. an A.I. McLeo, 986, Fractional time series moelling, Biometrika, 73, Matsumoto, M. an Y. Kurita, 994, Twiste GFSR generators, II, ACM Transactions on Moelling an Computer Simulation, 4, 994, Newbol, P. an Ch. Agiakloglou, 993, Bias in the sample autocorrelation of fractional noise, Biometrika, 8, Parzen, E., 983, Autoregressive spectral estimation. In Hanbook of Statistics (Brillinger, D.R. an Krishnaiah, P.R., es.) 3, , (North-Hollan, Amsteram). Priestley, M.B., 98, Spectral analysis of time series, Volume : Univariate Series (Acaemic Press, Lonon). Preusser, A., 984, Triangle contour plotting, ACM-Trans. Math. Software,, (Algorithm 626 Collecte algorithms from ACM). Pukkila, T., an H. Nyquist, 985, On the frequency omain estimation of the innovation variance of a stationary univariate time series, Biometrika, 72, Robinson, P.M., 99, Time series with strong epenence, Paper presente at the Worl Congress of the Econometric Society in Barcelona, August 99. Sowell, F., 992a, Maximum likelihoo estimation of stationary univariate fractionally integrate time series moels, Journal of Econometrics, 53, Sowell, F., 992b, Moeling long-run behavior with the fractional ARIMA moel, Journal of Monetary Economics, 29, Whittle, P., 962, Gaussian estimation in stationary time series, Bulletin of the International Institute of Statistics, 39, Zhang, H.-C., 99, Reuction of the asymptotic bias of autoregressive an spectral estimators by tapering, Journal of Time Series Analysis, 3,

20 TABLE : Pile-up eect in MA() an AR() moels for n=25, 5,. The relative frequency that ^ ( = ; ) is in the one-sie interval [ :; :9965] or [.9965,.]. The estimators are,,,, an (), the exact Gaussian ML with ata correcte for the true mean. The theoretical values obtaine by Cryer an Leolter (98) are enote by th. The number of replications for our simulations is. P(^ = ) P(^ = ) MA(): = :9 = :9 n th () AR(): = :9 = :9 n Remark: Distances from the theoretical values which excee.3,.2 or.9 epening on the theoretical values.53,.333 an.36 are signicant at the % level. 7

21 TABLE 2: Minima an maxima of, bias an empirical conence level for the estimators,, in ARMA(,), ARFIMA(,,) an ARFIMA(,,) moels, an where they are assume. ARMA(,) ^ ^ (, ) min (, ) max (, ) min (, ) max ARFIMA(,,) ^ (, ) min (, ) max (, ) min (, ) max ARFIMA(,,) ^ (, ) min (, ) max (, ) min (, ) max

22 TABLE 3: The measure of net avantage in of estimator A over estimator B for invertible ARMA(,), ARFIMA(,,) an ARFIMA(,,) moels: max j A() B ()j max j A() B ()j B>A A>B with A () as the of ^ using the estimator A. The estimators consiere are:, an. ARMA(,) = = Q QQ A B Q Q QQ A Q ARFIMA(,,) = = B Q QQ A Q ARFIMA(,,) = = B Remark: A positive sign inicates an avantage of estimator A over estimator B, a negative an avantage of estimator B over A. 9

23 * ^ ^ =z ^ 2= 4= ^ 6= 8= ^ ^ ^ FIGURE : Mean ^ square error, bias an empirical conence level for the AR() moels. PSfrag replacements ^ ^ ^ ^ ^ =z ^.5 ^ 2= ^ 4= ^ 6= ^ 8= * ^ ^ ^ ^ ^ ^ ^ ^ ^.5 =z ^ 2= 4= ^ ^ 6= 8= ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^. ^ * ^ ^ ^ ^ =z ^ 2= 4= ^ -. 6= 8= * ^ ^ ^ ^ ^ ^ ^ ^ 9 ^ ^ ^ ^ ^ ^ *

24 * =z 2= 4= 6= 8= ^ ^ FIGURE 2: Mean square error, bias an empirical conence level for the MA() moels. ^ ^ ^ ^.5 ^ ^ ^ ^ PSfrag replacements ^ ^.5 =z ^ 2= 4= ^ 6= 8= ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ PSfrag replacements ^ ^ =z ^ 2= 4= ^ 6= 8= * ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ *

25 * =z 2= 4= 6= 8= ^ ^ FIGURE 3: Mean square error, bias an empirical conence level for the FI() moels. ^ ^ ^ ^.5 ^ ^ ^ ^ PSfrag replacements ^ ^.5 =z ^ 2= 4= ^ 6= 8= ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ PSfrag replacements ^ ^ =z ^ 2= 4= ^ 6= 8= * ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ *

26 PSfrag replacements * * * 2= 4= 6= 8= FIGURE 4: Mean square ^ error for the AR(2) moels with real roots z;2 with =z = :95; : : : ; :5, ^ an =z2 = =z + :5; =z = :5; : : : ; :95, an =z2 = =z :5. ^ ^ ^ ^ ^ 2 ^ ^ ^ ^ ^ ^ ^ PSfrag replacements =z 2= 4= FIGURE 6= 5: Mean square error for the MA(2) moels with real roots z;2 with 8= =z = :; ^ :95; : : : ; :5, an =z2 = =z + :5; =z = :5; : : : ; :95; :, an ^ =z2 = =z :5. ^ ^ ^ ^ ^ 2 ^ ^ ^ ^ ^ ^ ^ PSfrag replacements =z FIGURE 6: Mean square ^ error for the AR(2) moels with complex roots z;2 = re i with =r = :95 ^ an = ; ; 2 ; : : : ; 9 ;. ^ ^ ^ ^ ^ 2 ^ ^ ^ ^ ^ ^ ^.5.5 =z 2= 4= 6= 8= 23

27 =z ^ * IAS ^ ^ ^ ^ =z ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ =z ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ =z ^ ^ ^ IAS ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * IAS ^ ^ ^ ^ =z ^ IAS ^ ^ ents * * -.5 ^ ^ ^ ^ ^ ^ * =z ^ 2= * 4= 6= 8= ^ ^ ^ FIGURE 7: Results for the ARMA(,) ^ moels. =z ^ 2= ^ FIGURE 7.a: Mean square error for the 4= 6= ARMA(,) ^ moels. 8= ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ PSfrag replacements ^ ^ * ^ ^ =z ^ 7 2= ^ 6 5 * 4= 6= ^ 8= 4 ^ 3 2 ^ ^ ^ ^ ^ -.99 ^ -.5 =z ^ - 2= ^ = 6= ^.99 8= ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * ^ ^ ^ 7 ^ 6 5 * ^ 4 ^ 3 2 ^ ^ -.99 ^ -.5 =z - 2= = 6=.99 8= ^ ^ ^ ^ ^ ^ ^ *.5.5 ^.99 ^.99 ^ ^ ^ ^ ^ ^ ^ IAS ^ ^ ^ ^ ^ -.99 ^

28 IAS ^ ^ ^ ^ ^ ^ ^ =z ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ ^ ^ =z ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ =z ^ ^ ^ * ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * IAS ^ ^ ^ ^ =z ^ * ^ ^ ^ ^ ^ ^ ^ ^ 6= 8= ^ ^ ^ ^ ^ ^ ^ =z ^ 2= ^ ^ 4= FIGURE 7.b: Bias for the ARMA(,) ^ 6= ^ moels. 8= ^ ^ ^ ^ ^ ^ * ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ =z ^.6 2= ^.4 * 4= 6= ^.2 8= ^ ^ -.2 ^ ^ -.4 ^ ^ -.99 ^ -.5 =z ^ - 2= ^ * 4= 6= ^.99 8= ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^.6 ^.4 * ^.2 ^ -.2 ^ -.4 ^ -.99 ^ -.5 =z - 2= * 4= 6=.99 8= ^ ^ ^ ^ ^ ^ ^.5.5 ^.99 ^.99 ^ ^ ^ ^ ^ ^ ^ * ^ ^ ^ ^ ^ -.99 ^ -.5 *

29 IAS ^ ^ ^ ^ ^ ^ ^ =z ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ ^ ^ =z ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ =z ^ ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * IAS ^ ^ ^ ^ =z ^ * IAS ^ ^ ^ ^ ^ ^ ^ ^ 6= 8= ^ ^ ^ ^ ^ ^ ^ =z ^ 2= ^ ^ 4= FIGURE 7.c: Empirical conence level for 6= the ^ ARMA(,) ^ moels. 8= ^ ^ ^ ^ ^ ^ * ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ =z ^ 2= ^ 9 * 4= 6= ^ 8= 8 ^ 7 ^ ^ 6 ^ ^ ^ -.99 ^ -.5 =z ^ - 2= ^ * 4= 6= ^.99 8= ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 9 * ^ 8 ^ 7 ^ 6 ^ -.99 ^ -.5 =z - 2= * 4= 6=.99 8= ^ ^ ^ ^ ^ ^ ^.5.5 ^.99 ^.99 ^ ^ ^ ^ ^ ^ ^ * IAS ^ ^ ^ ^ ^ -.99 ^ -.5 *

30 =z ^ ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ ^ =z ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ ^ ^ =z ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ =z ^ ^ ^ IAS ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * IAS ^ ^ ^ ^ =z ^ IAS ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ IAS * * * =z ^ ^ 2= ^ * 4= 6= ^ 8= ^ ^ ^ ^ ^ FIGURE 8: Results for the ARFIMA(,,) ^ ^ moels. =z ^ 2= ^ ^ FIGURE 8.a: Mean square error for the ARFIMA(,,) 4= ^ 6= ^ moels. 8= ^ ^ ^ ^ ^ ^ * ^ ^ ^ ^ ^ ^ ^ * ^ ^ =z ^ 25 2= ^ 2 * 4= 6= ^ 5 8= ^ ^ 5 ^ ^ ^ ^ =z ^ ^ = ^ 4= = ^ = ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * ^ ^ ^ 25 ^ 2 * ^ 5 ^ 5 ^ ^ =z ^ = 4= = = ^ ^ ^ ^ ^ ^ * ^ 25 ^ 2 ^ 5 ^ 5 ^ ^ ^ ^ ^ ^

31 IAS 6= ^ 8= ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ =z ^ =z ^ ^ ^ 2= ^ ^ ^ FIGURE 8.b: Bias for the ARFIMA(,,) moels. A cross 4= inicates ^ ^ 6= ^ a wrong sign of the average. ^ ^ 8= ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ =z ^.4.4 =z ^ ^.2 ^.2 2= * IAS ^ * 4= 6= ^ 8= ^ -.2 ^ -.2 ^ ^ -.4 ^ ^ -.4 ^ ^ ^ ^ ^ ^ =z ^ ^ ^ =z ^ ^ 2= ^ * * 4= ^ = ^.25 ^ = ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^.4 ^.4 ^.2 ^.2 * IAS ^ * ^ ^ -.2 ^ -.2 ^ -.4 ^ -.4 ^ ^ =z ^ ^ =z = * * 4= = = ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * ^ ^ ^ ^ ^ ^ ^ ^.95 ^ *

32 IAS ^ ^ ^ ^ ^ ^ ^ =z ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ ^ ^ =z ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ =z ^ ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * IAS ^ ^ ^ ^ =z ^ * IAS ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * IAS 6= 8= ^ ^ ^ ^ ^ ^ ^ =z ^ 2= ^ ^ 4= FIGURE 8.c: Empirical conence level for the 6= ARFIMA(,,) ^ ^ moels. 8= ^ ^ ^ ^ ^ ^ * ^ ^ ^ ^ ^ ^ ^ ^ ^ =z ^ 9 2= ^ 9 8 * 4= 6= ^ 8 8= 7 ^ 7 6 ^ ^ 6 ^ ^ ^ =z ^ ^ = ^ * 4= = ^ = ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 9 ^ 9 8 * ^ 8 7 ^ 7 6 ^ 6 ^ =z ^ = * 4= = = ^ ^ ^ ^ ^ ^ ^ 9 ^ 9 8 ^ 8 7 ^ 7 6 ^ 6 ^ ^ * ^ ^ ^

33 ^ ^ ^ ^ ^ FIGURE 9: Spectrum of the ARFIMA(,,) moel with = :5, an = :2, an spectrum of the ^ meian moel estimate by with me( ) = :5 an me(^) = :426 (ashe line). * =z 2= 4= 6= 8= 7 ^ ^ 6 ^ ^ ^ 5 ^ ^ 4 ^ ^ ^ ^ ^ ^ ^ *

34 =z ^ ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ ^ =z ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ ^ ^ =z ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ =z ^ ^ ^ IAS ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * IAS ^ ^ ^ ^ =z ^ IAS ^ ^ ^ ^ ^ ^ ^ ^ * * * =z ^ ^ 2= ^ * 4= 6= ^ 8= ^ ^ ^ ^ ^ FIGURE : Results for the ARFIMA(,,) ^ ^ moels. =z ^ 2= ^ ^ FIGURE.a: Mean square error for the ARFIMA(,,) 4= ^ 6= ^ moels. 8= ^ ^ ^ ^ ^ ^ * ^ ^ ^ ^ ^ ^ ^ * ^ ^ =z ^ 25 2= ^ 2 * 4= 6= ^ 5 8= ^ ^ 5 ^ ^ ^ ^ =z ^ ^ = ^ 4= = ^ = ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * ^ ^ ^ 25 ^ 2 * ^ 5 ^ 5 ^ ^ =z ^ = 4= = = ^ ^ ^ ^ ^ ^ ^ * ^ ^ ^ ^ ^ ^ ^ ^ ^ IAS ^ ^ ^ ^ ^.95 ^

35 IAS 6= ^ 8= ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ =z ^ =z ^ ^ ^ 2= ^ ^ ^ FIGURE.b: Bias for the ARFIMA(,,) moels. A cross 4= inicates ^ ^ 6= ^ a wrong sign of the average. ^ ^ 8= ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ =z ^.4.4 =z ^ ^.2 ^.2 2= * IAS ^ * 4= 6= ^ 8= ^ -.2 ^ -.2 ^ ^ -.4 ^ ^ -.4 ^ ^ ^ ^ ^ ^ =z ^ ^ ^ =z ^ ^ 2= ^ * * 4= ^ = ^.25 ^ = ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^.4 ^.4 ^.2 ^.2 * IAS ^ * ^ ^ -.2 ^ -.2 ^ -.4 ^ -.4 ^ ^ =z ^ ^ =z = * * 4= = = ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * ^ ^ ^ ^ ^.95 ^ *

36 IAS ^ ^ ^ ^ ^ ^ ^ =z ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ ^ ^ =z ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ =z ^ ^ ^ * IAS ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * IAS ^ ^ ^ ^ =z ^ * IAS ^ ^ ^ ^ ^ ^ ^ ^ 6= 8= ^ ^ ^ ^ ^ ^ ^ =z ^ 2= ^ ^ 4= FIGURE.c: Empirical conence level for the ^ 6= ARFIMA(,,) ^ moels. 8= ^ ^ ^ ^ ^ ^ * ^ ^ ^ ^ ^ ^ ^ ^ ^ =z ^ 9 2= ^ 9 8 * 4= 6= ^ 8 8= 7 ^ 7 6 ^ ^ 6 ^ ^ ^ =z ^ ^ = ^ * 4= = ^ = ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 9 ^ 9 8 * ^ 8 7 ^ 7 6 ^ 6 ^ =z ^ = * 4= = = ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * IAS ^ ^ ^ ^ ^.95 ^ *