Communications in Statistics—Simulation and Computation® , 35: 789–802, 2006 Copyright © Taylor & Francis Group, LLC ISSN: 0361-0918 print/1532-4141 online DOI: 10.1080/03610910600716928
Time Series Analysis
Correlated Errors in the Parameters Estimation of the ARFIMA Model: A Simulated Study M. R. SENA Jr.1 , V. A. REISEN2 , AND S. R. C. LOPES3 1
Department of Statistics, Federal University of Pernambuco, Recife, Brazil 2 Department of Statistics, Federal University of Espírito Santo, Vitória, Brazil 3 Department of Statistics, Federal University of Rio Grande do Sul, Porto Alegre, Brazil Processes with correlated errors have been widely used in economic time series. The fractionally integrated autoregressive moving-average processes— ARFIMA(p d q)—(Hosking, 1981) have been explored to model stationary and non stationary time series with long-memory property. This work uses the Monte Carlo simulation method to evaluate the performance of some parametric and semiparametric estimators for long and short-memory parameters of the ARFIMA model with conditional heteroskedastic (ARFIMA-GARCH model). The comparison is based on the empirical bias and the mean squared error of each estimator. Keywords Fractional differencing; Long memory and GARCH models. Mathematics Subject Classification 62M10; 62M15.
1. Introduction Recently, the analysis of time series with long memory has been widely argued in the literature. Since the ARFIMAp d q model was introduced by Granger and Joyeux (1980) and Hosking (1981), many estimation methods of the memory parameter d have been proposed and some corrections of the estimators have been suggested to improve their statistical properties. Amongst the usually socalled parametric approaches we mention the works of Fox and Taqqu (1986), Dahlhaus (1989), and Sowell (1992) which are estimation methods based on the maximum likelihood theory. Among the semiparametric approaches we cite the works of Geweke and Porter-Hudak (1983), Robinson (1994, 1995), Reisen (1994), Received February 27, 2004; Accepted November 23, 2005 Address correspondence to V. A. Reisen, Department of Statistics, Federal University of Espírito Santo, Vitória, Brazil; E-mail:
[email protected]
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and Lobato and Robinson (1996). Bootstrap estimation procedures for d has been the focus of Franco and Reisen (2004) and references therein. The estimation methods for memory parameters of seasonal fractionally ARMA models have been discussed and investigated empirically by Reisen et al. (2006a,b). In this article, the authors have also presented the invertibility and casual parameters conditions of the model. The main goal of this article is to evaluate, through a simulation study, the bias and the robustness of six estimation methods for the fractional differencing parameter d in ARFIMA processes with heteroskedastic errors. This research topic has recently received considerable attention in a variety of studies in time series and econometric areas. Ling and Li (1997), Henry (2001a,b), and Jensen (2004) have provided a good survey of the literature. The ARFIMA-GARCH model and its parameter estimators have been considered in Ling and Li (1997). The authors have derived some sufficient conditions for stationarity and ergodicity, and an algorithm for approximate maximum likelihood estimation has been also presented. Henry (2001a) has focused his research on the estimation of the memory parameter of a time series with long-memory conditional heteroskedasticity (ARFIMA-GARCH model) by using the average periodogram estimator suggested in Robinson (1994). In this context, Henry (2001a) has shown that the average periodogram estimator remains consistent. This semiparametric memory parameter estimation method is also considered here. The choice of the optimal bandwidth for some semiparametric estimation methods, under a general form of the errors such as GARCH errors, has been the focus of Henry (2001b). The author has derived optimal bandwidths and has shown asymptotically that the estimators considered in his work are not affected by conditional heteroskedasticity property of the innovations. Jensen (2004) has given contributions based on Bayesian approach to estimate the memory parameter in long-memory stochastic volatility processes. An application of the useful ARFIMA-GARCH model in inflation data has been the focus of Baillie et al. (1996). The results reported here increase our understanding of the finite sample properties of some fractional memory parameter estimators of the ARFIMA model with heteroscedastic errors. The plan of this article is as follows. Section 2 outlines the use of ARFIMA model with GARCH errors. Section 3 presents a summary on the estimation methods for the parameter d. Section 4 discusses the set-up of the Monte Carlo experiment and the results. Some conclusions are drawn in Sec. 5.
2. The ARFIMA Model with GARCH Errors Let Xt t∈Z be a time series presented in a set of information available at an instant t − 1. If t−1 is a set of evaluated available information at an instant t − 1, we can represent Xt t∈Z by the form Xt = gt−1 b + t
(2.1)
where g· · is a function, b is a vector of the parameters to be estimated, and t is a random perturbation. Equation (2.1) is sufficiently general and it has been studied and shaped by many authors. The most common specification for it is the
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autoregressive ARp model and the moving average MAq model that can be mixed to have the ARMAp q model, described hereof by Xt − = t t ∈ Z
(2.2)
where is the operator such that k Xt = Xt−k , = 1 − pj=1 j j , q backward
= 1 − i=1 i i , p and q are positive integers, is the mean of the process, and t t∈Z is a white noise process with zero mean and constant variance 2 . and are polynomials with all roots outside the unit circle and share no common factors. Studies in economic time series have shown that the dependent variable— returns in the interest rates, for instance—presents significant autocorrelation, even for lags largely separated in time. Time series with this behavior is said to have long-memory property and it may be modeled by the ARFIMAp d q model described as 1 − d Xt − = t t ∈ Z
(2.3)
where d is a real number. When d ∈ −05 05, the ARFIMAp d q process is said to be invertible and stationary and its spectral density function fX · · is given by −2d w fX w = fU w 2 sin w ∈ − 2
(2.4)
where fU w is the spectral density function of an ARMAp q process and and are the unknown parameter vectors of the ARMA and ARFIMA models, respectively. Hosking (1981), Beran (1994), and Reisen (1994) have described the ARFIMA models with more details. Engle (1982) has defined the process conditional autoregressive heteroskedastic (ARCH) when t is of the form t = zt t
(2.5)
where zt is an independent and identically distributed process with Ezt = 0 and Varzt = 1 and t , varying in time, is a function of the set t−1 . By definition, the variables t are not autocorrelated, for any t ∈ Z, but its conditional variance depends on time, opposing to what is assumed for the usual least squares estimation method. The ARCHs model or its generalization GARCHs r model can be summarized here, through the form of the innovation variance for the time t, according to Eq. (2.1) and with the assumption that t t−1 ∼ 0 t2 , where
t2 = 0 +
s i=1
i 2t−i +
r
2 j t−j
(2.6)
j=1
where s and r are positive integers, i ≥ 0, for i ∈ 0 1 s, and j ≥ 0, for j ∈ 1 2 r. Bollerslev (1986) has process is stationary if 1 + shown that a GARCH 1 < 1, where 1 = si=1 i and 1 = rj=1 j , whenever Et = 0, Vart = 0 /1 − 1 − 1 and the innovation process is not autocorrelated. We remark
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here that if one excludes the last sum in Eq. (2.6), one has the simplest model, an ARCHs model. The combination of models in expressions (2.3), (2.5), and (2.6) yields to the ARFIMA p d q-GARCHr s model. Under the parameter conditions for ,
, 1, and 1, as previously described, the ARFIMAp d q-GARCHr s process is stationary and invertible if −05 ≤ d ≤ 05, see Theorem 2.3 in Ling and Li (1997). The authors have also demonstrated the condition for ergodicity and derived the existence of higher-order moments. To estimate the parameters, an algorithm for approximate maximum likelihood (ML) estimation has been also presented by the authors. The autocorrelation function of ARFIMAp d q-GARCHr s model behaves asymptotically with the similar form of the stationary ARFIMA model. This means that the dependency between observations decays hyperbolically as a function of d. Based on this autocorrelation behavior, we are here interested in studying the estimation of the parameter d for some parameter order specifications of the ARFIMAp d q-GARCHr s model. This empirical study will provide an evidence whether the memory estimators are robust or not under conditional heteroskedasticity. It is worth noting that all estimation methods focused here have been previously implemented to the estimation of the stationary and non stationary ARFIMA processes see, for instance, Lopes et al. (2004) and Reisen et al. (2001a) for an overview.
3. Estimation of d We consider six estimators for the parameter d and provide a summary of each method below. The first four approaches are based on the linear regression method built from Eq. (2.4) and are considered semiparametric methods. The averaged periodogram estimator also belongs to the semiparametric class. The remaining two estimators are the parametric methods proposed by Fox and Taqqu (1986) and Sowell (1992). The first estimator, denoted hereafter by dˆ Per , has been proposed by Geweke and Porter-Hudak (1983). They use the periodogram function I· as an estimator for the spectral density function, presented in Eq. (2.4). The number of regressors used in the regression equation is a function of the sample size n and it is denoted here by gn = n , for 0 < < 1. The second estimator, denoted by dˆ Per S , has been proposed by Reisen (1994). The author has made a modification in the regression equation, substituting the periodogram function by its smoothing version based on the Parzen lag window. In this estimator the function gn = n , for 0 < < 1, is also chosen to represent the number of regressors in the regression equation. The truncation point in the Parzen lag window is denoted by m = n , for 0 < < 1. The appropriate choices of and values had been investigated by Geweke and Porter-Hudak (1983) and Reisen (1994), respectively, among several other authors. The third estimator, denoted hereafter by dˆ Lob , has been considered by Robinson (1994) and Lobato and Robinson (1996). It is a weighted average of the logarithm of the periodogram function. This estimator is based on a number of frequencies and on a constant q ∈ 00 10. Lobato and Robinson (1996) have presented a Monte Carlo simulated experiment to investigate the sensitivity on the choice of and q values.
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The fourth estimator, denoted by dˆ Rob , has been introduced by Robinson (1995). This method is also a modified version of the Geweke and Porter-Hudak’s estimator. This estimator regresses lnIwj on ln2 sinwj /22 , for j = l l + 1 gn, where l is the smallest truncated point that tends to infinity smoother than gn. Robinson (1995) has developed some asymptotic results for dˆ Rob , when d ∈ 00 05, and has showed that this estimator is asymptotically less efficient than the Gaussian maximum likelihood estimator and in this situation the efficiency is considered to be zero. The number of regressors gn can take different forms and its appropriate choice has been studied by several authors, such as Robinson (1995), Hurvich et al. (1998), and Hurvich and Deo (1999), to name just a few. The fifth estimator, hereafter denoted by dˆ Fox , is a parametric procedure that has been considered by Fox and Taqqu (1986), based on the work of Whittle (1953). The estimates are obtained by minimizing the finite and discrete form given by n =
Iwj 1 n−1 ln fX wj + 2n j=1 fX wj
(3.1)
where denotes the vector of unknown parameters and fX w is the spectral density function (see Dahlhaus, 1989; Fox and Taqqu, 1986). The ML estimator (see Sowell, 1992) is the sixth method focused here. Assuming that the process Xt is Gaussian, the log-likelihood function may be expressed as 1 1 n = − log det TfX w − X TfX w −1 X 2 2
(3.2)
where is the parameter vector and T is the variance–covariance matrix of the process Xt t∈Z with j kth element given by Tjk fX w =
1 f w expiwjkdw 2 − X
see for example Beran (1994, Sec. 5.3). The ML estimates are obtained by maximizing (3.2), that is, ˆ = arg max n . Following the estimator notations adopted here the ML fractional estimator is denoted by dˆ ML . Fractional integration has been applied in models like GARCH, including the FIGARCH model by Chung (1999), the FIEGARCH model by Baillie et al. (1994), and the FIAPARCH model by Tse (1998). All of them have in common some variations in the estimation process through the maximization of the likelihood function. The comparison among parameter estimation procedures have been widely studied and they are sources of research for several authors. However, it is not our main goal. This article focuses on the robustness of the estimators for the fractional parameter d, in view of violating the normality assumption for the innovation process and also in the study of how they are affected by the existence of heteroskedastic errors.
4. Monte Carlo Simulation Study In order to investigate the robustness of the estimators described previously, in the context of ARFIMA models with GARCH errors, a number of Monte
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Carlo experiments were carried out. Realization of the Gaussian white noise sequence zt , defined in (2.5), t = 1 n, with unit variance, were generated by IMSL-FORTRAN subroutine DRNNOR. The ARFIMA processes were generated according to Hosking (1984) with errors t given by (2.5) and t2 by (2.6). The models and the parameter values are specified in the tables which also give the empirical mean and mean squared error (MSE) of the estimation methods based on 1,000 replications of series with length n = 300. For comparison purpose, the tables also give the estimation results when the errors of the ARFIMA process were generated from 0 1 distribution. In the semiparametric methods two bandwidths were used, and their values are also specified in the tables. The trimming number was fixed l = 2 for the dˆ Rob estimator, and the constant q = 05 for the averaged periodogram estimator. Note that we did not concentrate on the sensitivity of the estimate with the choice of the constant q. A review of this method and the appropriated choice of q are the main focus of Lobato and Robinson (1996). In the parametric group, the estimates of dˆ Fox were obtained by using the subroutine DBCONF-IMSL. For the same data set, the ML estimates were obtained by using the Ox-program (Doornik, 1999). Here, the finite sample performance of the estimators described previously was examined under different structures of the ARFIMA-GARCH model with two memory parameter values: d = 02 (moderate long memory) and d = 045 (strong long memory). The choice of d = 02 is motivated by the theoretical results of the averaged periodogram estimator given in Lobato and Robinson (1996). Results from ARFIMA(0 d 0 model are in Tables 1a and b. From these tables we see that all methods perform well. When the process is generated by Gaussian
Table 1a ARFIMA0 d 0 model with Gaussian GARCH errors, d = 02 and sample size n = 300 Estimator gn
dˆ Per 05
n
dˆ Per S 07
n
05
n
dˆ Lob 07
n
07
n
dˆ Rob 08
n
05
n
07
n
dˆ Fox –
dˆ ML –
ˆ d(mean) MSE
d = 02, Gaussian error 02042 02017 01513 01806 01676 01652 02098 02030 01960 01796 00407 00106 00267 00074 00071 00046 00689 00131 00025 00028
ˆ d(mean) MSE
d = 02, ARCH error, with 0 = 1 and 1 = 02 01922 01966 01464 01814 01701 01669 01917 01973 01982 01842 00403 00101 00267 00071 00068 00047 00661 00133 00032 00032
ˆ d(mean) MSE
d = 02, ARCH error, with 0 = 1 and 1 = 05 02016 02004 01462 01804 01662 01641 01946 01981 01939 01836 00399 00106 00258 00079 00078 00068 00670 00137 00051 00051
ˆ d(mean) MSE
d = 02, ARCH error, with 0 = 1 and 1 = 09 01970 01823 01446 01648 01479 01403 01978 01800 01745 01783 00387 00177 00255 00138 00167 00170 00686 00235 00116 00132
ˆ d(mean) MSE
d = 02, GARCH error, with 0 = 03, 1 = 02, and 1 = 05 02009 01974 01527 01807 01638 01668 01983 01960 01971 01810 00417 00118 00261 00080 00083 00053 00716 00149 00032 00034
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Table 1b ARFIMA0 d 0 model with Gaussian and GARCH errors, d = 045 and sample size n = 300 Estimator gn
dˆ Per n05
dˆ Per S n07
n05
dˆ Lob n07
n07
dˆ Rob n08
n05
n07
dˆ Fox –
dˆ ML –
ˆ d(mean) MSE
d = 045, Gaussian error 04623 04592 04008 04394 03487 03552 04594 04581 04869 04194 00405 00101 00294 00070 00125 00105 00631 00128 00038 00024
ˆ d(mean) MSE
d = 045, ARCH error, with 0 = 1 and 1 = 02 04595 04608 04009 04384 03479 03540 04661 04627 04770 04178 00409 00092 00281 00066 00126 00107 00670 00114 00038 00028
ˆ d(mean) MSE
d = 045, ARCH error, with 0 = 1 and 1 = 05 04491 04605 03961 04394 03474 03542 04509 04628 04663 04155 00387 00115 00293 00086 00132 00112 00636 00150 00057 00041
ˆ d(mean) MSE
d = 045, ARCH error, with 0 = 1 and 1 = 09 04502 03945 04393 04196 03325 03354 04575 04396 04377 03959 00418 00300 00176 00136 00189 00188 00722 00229 00136 00100
ˆ d(mean) MSE
d = 045, GARCH error, with 0 = 03, 1 = 02, and 1 = 05 04729 04592 04116 04388 03483 03559 04662 04551 04884 04178 00413 00118 00280 00086 00132 00106 00654 00148 00049 00029
errors, the biases are generally small, specially for d = 02. In the semiparametric class, the regression methods are very competitive where dˆ Per S is an estimator with smaller MSE. In general, the estimates from dˆ Lob present the largest biases, but with the smallest MSE among all semiparametric estimation methods. The MSE is small due to the substantial reduction of its sample variance. The increase of the bandwidth leads to estimates with small MSE. These findings are in accordance with other published works as in Lopes et al. (2004) and Reisen et al. (2001a,b) among others. The estimator dˆ Fox has typically smaller absolute bias than dˆ ML and their MSE are nearly identical. These results are consistent with the findings in Cheung and Diebold (1994). These authors have presented a comprehensive analysis comparing both parametric methods with unknown mean, see also Baillie et al. (1996). When d = 045 (Table 1b), the methods perform similarly to the case d = 02 (Table 1a) except for the average periodogram estimator. The absolute bias of this later method increased substantially and it may be explained by the fact that its asymptotic Normal distribution is only guarantied when 0 < d < 025. In general, it seems that there is no significant change in the estimates when considering the ARCH and GARCH error types. This evidences that the estimators are robust in the presence of heteroskedastic errors. For a process where the correlation of the squares are very strong (ARCH model with 1 = 09), there is an empirical evidence that the bias and the MSE of the parametric estimates increase substantially. By increasing the bandwidth, independently of the error type, the semiparametric estimates become more precise by presenting smaller MSE and are very competitive with the parametric methods. These empirical results reveal that the asymptotic normality results for the investigated estimation methods still
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hold for heteroskedastic errors. Similar findings using the average periodogram and parametric estimators are, respectively, in Henry (2001a) and Baillie et al. (1996). Reisen et al. (2001a) have studied the robustness of these estimators when the errors come from a non normal distribution. Their empirical investigation have evidenced that the estimators have no influence with the violation of normality assumption, and have called the attention of the good performance of the parametric estimator dˆ Fox under correct model specification. Tables 2–6 present the simulation results when short-memory parameters were introduced in the ARFIMA-ARCH model. Here we present the ARMA combinations ±07 0, 0 ±07, 02 ±07, and ±07 ±02. Other parameter values produced similar behavior and are available upon request. The values of the true and the estimated short-memory parameters are also given in the tables. The dˆ Fox and dˆ ML methods estimates all parameters simultaneously. The estimates from the semiparametric methods were obtained from two steps: first the long-memory ˆ parameter estimate dˆ was obtained then, the filter 1 − Bd was applied to the series to obtain the short-memory parameter estimates. From the results we observe that the short-memory parameter affects in the parameter estimates. This is directly related to the type of ARMA model and to the sign and magnitude of the shortmemory coefficients. The estimates are also affected by the size of the bandwidth. In general, the estimators seem to be unaffected by the presence of ARCH errors. For the ARFIMA1 d 0 model (see Tables 2 and 3) the study revels that the bias of d is affected by the sign and the value of the AR coefficient. Results for
Table 2 ARFIMA1 d 0 model with Gaussian and ARCH errors, d = 02, 1 = −07 07, and sample size n = 300 Estimator gn
dˆ Per n05
dˆ Per S n07
n05
dˆ Lob n07
n07
dˆ Rob n08
n05
n07
dˆ Fox –
dˆ ML –
ˆ d(mean) MSE ˆ1 MSE
Gaussian error, d = 02 and 1 = −07 01960 01567 01422 01373 01224 −00182 01924 01486 01991 01754 00429 00120 00281 00107 00132 00540 00708 00158 00040 00036 −06579 −06707 −06484 −06643 −06554 −05514 −06229 −06630 −07117 −06863 00278 00058 00157 00053 00067 00329 00667 00081 00093 00026
ˆ d(mean) MSE ˆ1 MSE
ARCH error with 0 = 1 and 1 = 02, d = 02, and 1 = −07 01977 01571 01430 01380 01212 −00189 02016 01506 02043 01771 00405 00123 00276 00108 00133 00558 00641 00155 00055 00041 −06585 −06693 −06471 −06633 −06538 −05473 −06342 −06632 −07371 −06858 00291 00065 00186 00059 00072 00357 00571 00082 00180 00028
ˆ d(mean) MSE ˆ1 MSE
03226 00583 05697 00521
ˆ d(mean) MSE ˆ1 MSE
03380 00584 05519 00565
05793 01541 03326 01455
Gaussian error, 02735 05614 00292 01374 06179 03487 00273 01307
ARCH error with 05805 02773 01546 00304 03276 06098 01499 00305
d = 02 and 1 = 07 04016 04446 03512 00416 00600 01011 05069 04659 05388 00396 00569 00850
0 = 1 and 1 = 02, d = 02, and 1 05585 04009 04438 03801 01351 00412 00597 00960 03475 05027 04619 05118 01325 00420 00593 00897
06311 01990 02840 01856
02754 00379 06135 00396
01328 00179 07363 00113
= 07 06338 02012 02780 01914
02517 00329 06278 00335
01334 00202 07321 00118
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Table 3 ARFIMA1 d 0 model with Gaussian and ARCH errors, d = 045, 1 = −07 07, and sample size n = 300 Estimator gn
dˆ Per n05
dˆ Per S n07
n05
dˆ Lob n07
n07
dˆ Rob n08
n05
n07
dˆ Fox –
dˆ ML –
ˆ d(mean) MSE ˆ1 MSE
Gaussian error, d = 045 and 1 = −07 04660 04156 04079 03961 03249 02593 04580 04044 05412 04114 00428 00114 00292 00098 00186 00398 00626 00148 00126 00034 −06531 −06688 −06354 −06616 −06177 −05513 −06225 −06579 −08905 −06823 00440 00068 00332 00064 00118 00336 00680 00102 00549 00024
ˆ d(mean) MSE ˆ1 MSE
ARCH error, with 0 = 1 and 1 = 02, d = 045, and 1 = −07 04416 03993 03891 03840 03157 02437 04357 03897 04480 04137 00392 00129 00284 00140 00212 00456 00627 00170 00051 00035 −06700 −06758 −06578 −06754 −06366 −05851 −06459 −06686 −07639 −06816 00624 00505 00525 00496 00496 00588 01061 00593 00708 00029
ˆ d(mean) MSE ˆ1 MSE
05806 00611 05680 00567
ˆ d(mean) MSE ˆ1 MSE
05837 00585 05687 00556
08313 01557 03287 01489
Gaussian error, d = 045 and 1 = 07 05233 08122 04622 04822 06098 00331 01379 00004 00011 01013 06267 03465 06997 06802 05375 00320 01327 00019 00023 00889
ARCH error with 0 = 1 and 1 = 02, d = 045, and 1 08285 05245 08109 04618 04820 06250 01538 00317 01369 00004 00011 00999 03354 06262 03514 07007 06812 05264 01448 00314 01302 00023 00027 00887
08824 02003 02807 01890
08659 01965 02972 02062
03159 00250 07891 00120
= 07 08821 02001 02855 01855
07750 01339 03955 01557
03205 00245 07850 00114
= ±02 are available upon request. By increasing the bandwidth from gn = n05 , the semiparametric estimates overestimate the true parameter and the bias becomes significantly large. This overestimation gives estimate values in the non stationary region. There is an increasing of the MSE of AR estimates which is associated with the large sample variance of the AR estimate. The effect of positive AR coefficient in the estimates may be explained by the fact it produces large contribution to the spectral density for those frequencies away from the zero frequency. In the parametric group, positive and large AR parameter value seems to affect the estimates specially those from the ML estimator. In this method, the absolute bias increases substantially. For negative values of , the biases of both parameters (d and ) are relatively small (negative bias for d and positive for ) while, for positive there is a large positive bias for d and, consequently, large negative bias for . In general, for the process with strong memory (d = 045), there is a similar behavior when d = 02, however, its estimates strongly suggest a non stationary memory parameter value. In general, the presence of heteroskedastic errors in the ARFIMA model does not affect the estimates. The biases seem to be intrinsically related to the AR parameter values. Table 4 shows the results of ARFIMA0 d 1 model. For negative , it is observed similar behavior of the estimates from the previous model when < 0. The biases are generally small and, the empirical values of MSE are nearly identical. In the semiparametric methods, the values of the bandwidth does not affect
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Table 4 ARFIMA0 d 1 model with Gaussian and ARCH errors, d = 02, 1 = −07 07, and sample size n = 300 Estimator gn
dˆ Per n05
dˆ Per S n07
n05
dˆ Lob n07
n07
dˆ Rob n08
n05
n07
dˆ Fox –
dˆ ML –
ˆ d(mean) MSE ˆ 1 MSE
Gaussian error, d = 02 and 1 = −07 02072 01567 02438 02250 02101 02857 02066 02504 02002 01806 00413 00281 00123 00076 00049 00091 00715 00158 00031 00033 −06917 −06805 −07140 −06890 −06956 −06659 −06872 −06778 −06933 −07086 00096 00046 00062 00033 00026 00038 00173 00053 00026 00027
ˆ d(mean) MSE ˆ 1 MSE
ARCH error with 0 = 1 and 1 = 02, d = 02, and 1 = −07 02149 02478 01591 02266 02109 02846 02193 02554 01981 01821 00357 00114 00249 00070 00047 00091 00598 00150 00037 00037 −06869 −06775 −07121 −06862 −06933 −06645 −06798 −06742 −06938 −07073 00102 00048 00060 00038 00032 00042 00179 00056 00027 00024
ˆ d(mean) MSE ˆ 1 MSE
00688 −01807 00561 01555 05507 02668 00662 02074
Gaussian error, d = 02 and 1 = 07 00218 −01968 −03365 −05121 00336 −02333 00549 01645 03074 05200 00889 02014 05111 02471 00353 −02481 04965 01901 00636 02188 04911 09466 01198 02903
01074 00449 06015 00418
00125 00656 05196 00632
ˆ d(mean) MSE ˆ 1 MSE
ARCH error with 0 = 1 and 1 = 02, d = 02, and 1 = 07 00736 −01791 00309 −01953 −03381 −05139 00308 −02344 00515 01536 00506 01624 03076 05252 00894 02017 05555 02635 05205 02440 00232 −02602 04914 01814 00631 02124 00588 02228 05115 09728 01262 03022
01020 00482 05982 00491
00253 00607 05332 00597
the estimates. The parametric methods presented the smallest MSE. There is no deterioration in the performance of the estimators under ARCH errors. When > 0, the picture of the estimation results changes dramatically. The bias of d is very large providing evidence that the biases of the parameters are directly related to the sign of the short-memory model. When dealing with both short-memory parameters the picture is presented in Tables 5 and 6. The performance of the methods are very similar to the models previously considered. The biases are directly related to the sign and magnitude of the ARMA coefficients and the size of the bandwidth. The ML estimator loses a lot of its superiority for some AR and MA parameter combinations. The estimates of the ARFIMA parameters under Gaussian errors are similar to those seen elsewhere. In general, the methods seem to be robust in the presence of ARCH errors and the bias and MSE become large in the presence of AR and/or MA parameters with values close to the boundary stationary conditions. The next two tables show the estimation results from the ARFIMA0 d 0 and ARFIMA1 d 0 models with GARCH1 1 errors. For comparison purpose, our model is the same considered by Ling and Li (1997). From Tables 7 and 8, we notice similar performance with the previous cases. In the semiparametric class with the smallest bandwidth, the bias and MSE are generally small. By increasing the bandwidth, the biases of the empirical estimates increase substantially. The parametric dˆ Fox estimate has the smallest bias and MSE, however, its parametric method competitor performs poorly. The dˆ Fox estimates are very close to the
Parameters Estimation of the ARFIMA Model
799
Table 5 ARFIMA1 d 1 model with Gaussian and GARCH1 0, errors, d = 02, 1 = −07 07, 1 = 02, and n = 300 Estimator gn
dˆ Per n05
dˆ Per S n07
n05
dˆ Lob n07
n07
dˆ Rob n08
n05
n07
dˆ Fox –
dˆ ML –
ˆ d(mean) MSE ˆ1 MSE ˆ 1 MSE
Gaussian error, d = 02, 1 = −07, and 1 = 02 01973 01151 01438 00942 00704 −01728 01918 00990 02167 01490 00372 00158 00275 00175 00251 01474 00694 00217 00206 00108 −06981 −07152 −07113 −07180 −07242 −07537 −06860 −07202 −06984 −07033 00145 00034 00037 00036 00038 00526 00399 00034 00031 00028 01927 00899 01259 00647 00305 −02969 01883 00653 02137 01484 00576 00275 00474 00302 00439 02852 01001 00393 00297 00183
ˆ d(mean) MSE ˆ1 MSE ˆ 1 MSE
ARCH error with 0 = 1 and 1 = 02, d = 02, 1 = −07, and 1 = 02 01794 01092 01295 00906 00677 −01831 01834 00974 01858 01422 00397 00191 00288 00190 00265 01561 00727 00251 00151 00123 −07000 −07162 −07117 −07192 −07237 −07614 −06844 −07194 −07070 −07033 00132 00040 00048 00038 00045 00420 00485 00046 00070 00032 01734 00845 01117 00623 00305 −03165 01828 00663 01805 01437 00611 00334 00514 00330 00471 03014 01040 00436 00265 00211 Gaussian error, d = 02, 1 = 07, and 05304 02722 05145 03854 04201 01196 00313 01058 00355 00489 02685 05826 02966 05063 04634 02386 00511 02059 00554 00782 00847 01504 00983 01829 01735 00511 00222 00470 00249 00290
ˆ d(mean) MSE ˆ1 MSE ˆ 1 MSE
03232 00603 05120 01031 01285 00313
ˆ d(mean) MSE ˆ1 MSE ˆ 1 MSE
ARCH error with 0 = 1 and 1 03245 05371 02662 05172 00581 01239 00297 01076 05179 02549 05944 02961 01007 02589 00492 02131 01386 00802 01586 01023 00326 00587 00244 00523
= 02, d = 02, 1 03869 04199 00361 00488 05098 04736 00571 00763 01899 01855 00296 00336
1 = 02 03674 01048 04675 01472 01244 00419
05782 01571 01875 03204 00487 00622
01649 −00519 00582 01138 06472 07860 00681 00337 01094 00437 00380 00398
= 07, and 1 = 02 03593 05828 01052 −00590 00921 01600 00674 01135 04862 01792 06904 07984 01325 03317 00597 00314 01387 00470 00984 00510 00389 00653 00429 00406
estimated parameter values given by Ling and Li (1997). In general, the estimates by semiparametric and the dˆ Fox approaches are not affected by GARCH error type. This result is consistent with those that have been presented in Ling and Li (1997) and Henry (2001a).
5. Concluding Remarks Through the simulation studies this article shows that a class of parametric and semiparametric memory parameter estimators are robust with the presence of ARCH and GARCH errors. These empirical results evidence that the asymptotic normality results for the investigated estimation methods still hold for heteroscedastic errors. The biases are a consequence of the characteristics of the estimators and they depend on the structure of the ARFIMA model. In this context,
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Table 6 ARFIMA1 d 1 model with Gaussian and GARCH, errors, d = 02, 1 = 02, 1 = −07 07, and n = 300 Estimator gn
dˆ Per n05
dˆ Per S n07
n05
dˆ Lob n07
n07
dˆ Rob n08
n05
n07
dˆ Fox –
dˆ ML –
ˆ d(mean) MSE ˆ1 MSE ˆ 1 MSE
Gaussian error, d = 02, 1 = 02, and 1 = −07 02141 02935 01578 02707 02487 03419 02039 03051 01035 00631 00382 00184 00261 00112 00059 00212 00677 00239 00689 00665 01937 00911 02539 01155 01414 00407 02176 00803 02850 03274 00555 00281 00429 00193 00121 00309 00918 00340 00758 00703 −07018 −07143 −06927 −07094 −07050 −07200 −07005 −07162 −06912 −07022 00041 00037 00040 00038 00036 00038 00046 00039 00038 00032
ˆ d(mean) MSE ˆ1 MSE ˆ 1 MSE
ARCH error with 0 = 1 and 1 = 02, d = 02, 1 = 02, and 1 = −07 02253 03014 01744 02807 02515 03453 02183 03133 00399 00796 00378 00201 00232 00129 00064 00223 00647 00257 01191 00578 01893 00932 02443 01148 01478 00479 02093 00823 03549 03193 00553 00287 00399 00199 00126 00288 00886 00342 01191 00655 −07012 −07116 −06915 −07071 −07031 −07160 −07001 −07133 −06902 −06998 00035 00031 00036 00033 00030 00031 00040 00033 00034 00031
ˆ d(mean) MSE ˆ1 MSE ˆ 1 MSE
Gaussian error, d = 02, 1 = 02, and 00883 −01313 00367 −01509 −02539 −03310 00500 01197 00485 01297 02227 02931 01669 00667 01572 00314 02171 04149 00334 00677 00288 00716 01601 02479 05482 02112 04879 01532 02196 03298 00879 02937 01009 03435 03426 02975
1 = 07 00586 −01767 00835 01544 01738 00923 00553 00922 05204 01797 01146 03394
ˆ d(mean) MSE ˆ1 MSE ˆ 1 MSE
ARCH error with 0 = 1 and 1 = 02, d = 02, 1 00813 −01283 00330 −01492 −02516 −03328 00519 01183 00497 01287 02209 02969 01781 00690 01604 00362 02317 04367 00356 00625 00318 00688 01655 02634 05530 02147 04878 01573 02310 03499 00814 02875 01026 03384 03355 02888
= 02, and 1 = 07 00492 −01730 01313 00864 01524 00621 01893 01038 00988 00592 00934 00666 05266 01929 05342 01076 03292 01719
01741 −00519 00646 01138 00836 07860 00532 00337 05580 00437 01486 00398 00796 00578 03193 00655 06998 00031
Table 7 ARFIMA0 d 0-GARCH1 1 model, with d = 03 and sample size n = 300 Estimator gn ˆ d(mean) bias sd MSE
dˆ Per 05
n
03050 00050 01916 00366
dˆ Per S 07
n
05
n
dˆ Lob n
07
n
07
dˆ Rob 08
n
05
n
n07
GARCH error, with 0 = 04, 1 = 03, and 1 = 03 03099 02565 02912 02546 02629 03067 03111 00099 −00435 −00088 −00454 −00371 00067 00111 00985 01468 00816 00642 00511 02341 01055 00098 00234 00067 00062 00040 00547 00112
dˆ Fox –
dˆ ML –
03089 02909 00089 −01591 00566 00537 00033 00282
Parameters Estimation of the ARFIMA Model
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Table 8 ARFIMA1 d 0-GARCH1 1 model, with d = 03, 1 = 05, and sample size n = 300 Estimator gn ˆ d(mean) bias sd MSE
dˆ Per n05
03227 00227 01898 00364
dˆ Per S n07
n05
dˆ Lob n07
GARCH error, with 0 04581 02734 04422 01581 −00266 01422 00949 01472 00786 00340 00223 00264
ˆ1 (mean) 04697 03362 bias −00303 −01638 sd 01900 01070 MSE 00369 00382
n07 = 04, 1 03538 00538 00399 00045
dˆ Rob n08
n05
n07
dˆ Fox –
dˆ ML –
= 03, and 1 = 03 04161 03247 04798 02862 01455 01161 00247 01798 −00138 −01545 00205 02290 01007 01471 01652 00139 00529 00424 00217 00512
05179 03503 04391 03780 04702 03156 00179 −01497 −00609 −01220 −00298 −01844 01556 00916 00658 00555 02281 01111 00244 00308 00080 00180 00528 00463
05002 00002 01584 00250
06281 01281 01520 00395
the positive parameters of the AR and MA components contribute to a significant increase of the bias when their values are close to non stationary conditions. This can be explained by the shape of the spectral density when the AR and/or MA parameters are present in the model. Some findings presented here are in accordance with other that have recently appeared in the literature of volatility long-memory models.
Acknowledgments The author V. A. Reisen gratefully acknowledges partial financial support from CNPq/ Brazil. S.R.C. Lopes was partially supported by CNPq-Brazil, by Pronex Probabilidade e Processos Estocásticos (Convênio MCT/CNPq/FAPERJ - Edital 2003), and also by Fundação de Amparo à Pesquisa no Estado do Rio Grande do Sul (FAPERGS Foundation). The authors wish to thank the editor and two anonymous referees for their valuable comments on the earlier version of this article.
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