Geweke, J., and S. Porter-Hudak (1983): The estimation and application of long memory time series models, Journal of Time Series Analysis, 4(4), 221238.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Gaussian AR(1) process, X t = 0.9X t 1 # t . In both case the number of samples is n = 10 . Prior to taking the log, the periodogram ordinates has been averaged (pooled) over m = 5 bins. This idea has been pushed forward in an early work by Geweke and Porter Hudak (hence the acronym GPH) [8]. Of course, the relation log f(#) log(C) 2d log(#) is valid only in a neighborhood of the zero frequency, and thus the regression line should be computed using only a subset of the log periodogram ordinates log I(# k ) # 1, M , where # k are the Fourier frequencies. The choice ....

J. Geweke and S. Porter-Hudak, "The estimation and application of long memory time series models," J. of Time Series Analysis, vol. 4, pp. 221-238, 1983.

.... exists and it is given by S 2# DeltaJ ( 6) 2H ; 1) 2(1 ;H) sin(H)c 0 oe 2 0 fi fi e i ; 1 fi fi 2 1 X k= 1 fi fi fi (k) fi fi fi ;1;2H # for 2 (0# 2) where (k) 2k, k 2 Z, the same as the spectrum of the Discrete Fractional Gaussian Noise (DFGN) See [8] for the latter spectrum. ffl The bispectrum of the process DeltaJ (t) exists and it is real valued and positive, namely, S 3# DeltaJ i (2) j = 9ic 0 oe 4 0 2 sin(H) 3 ; 2H) 3 Y j=1 (1;e i j ) 7) sym i (k# ) 3) j 1 X k= 1 1 X = 1 2 6 4 fi fi fi (k) 1 ....

S. Geweke, J.and Porter-Hudak. The estimation and application of long memory time series models. Journal of Time Series Analysis, 4:221--237, 1983.

....the empirical distribution is more pronounced for data generated with ff = 1:5 than for data generated with ff = 4: This fact should not come as a surprise given the moment assumptions made by Lo (1991) Other tests of the long memory hypothesis are available in the literature. This set includes Geweke and Porter Hudak (1983) hereafter GPH, the locally optimal and beta optimal tests of Davies and Harte (1987) the Lagrange multiplier tests developed by Robinson (1991a) and Agiakloglou, 12 Left tail rejections correspond to rejection of the null hypothesis H = 1=2 against the alternative H 1=2 (anti persistent ....

....the expansion (1 Gamma B) d = 1 X j=0 Gamma(j Gamma d) Gamma(j 1) Gamma( Gammad) B j : 3.1) See Brockwell and Davis (1991) for a detailed treatment of this model. The spectral density of an ARFIMA(p; d; q) model is proportional to C j j Gamma2d as j j 0; for C 0: The Geweke and Porter Hudak (1983) test for long memory is based on this fact: regress the logarithm of the periodogram at low frequencies on some function of those frequencies and estimate d by the slope of this least squares regression. 14 GPH argued that the resulting estimator of d could capture the long memory behavior ....

Geweke, J. and S. Porter-Hudak (1983),"The estimation and application of long memory time series models," Journal of Time Series Analysis 4, 221-238.

....that the estimated values of the long memory parameter was largely affected by the parametric specification of the short run dynamics part of the model, i.e. the ARMA component. Hence, an important reason whichgave further impetus to the importance of LRD in economics, was the introduction by[40] of a semiparametric estimator (hereafter GPH) for the estimation of the degree of LRD in a time series. In fact, GPH is robust to many forms of complicated short run dynamics insofar as it is based on a frequency domain ordinary least squares regression of periodogram ordinates in a shrinking ....

....samples when the mean is unknown ( 21] Parametric exact and approximate maximum likelihood estimation, however, relies heavily on a correct specification of the short run dynamics, i.e. the a(z) and b(z) polynomials in 2.2. The semiparametric approachadvocated by[51] and [68] and developed by [40], 69] 99] 102] 101] and others, relies only on (2.1) or, more generally, on (2.4) which only specifies the spectral densityin a neighbourhood of zero, the frequency of interest. Like the Whittle likelihood, they are periodogram based, but, in accordance with the local specification, only ....

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Geweke, J., and S. Porter-Hudak (1983): "The estimation and application of long memory time series models," Journal of Time Series Analysis, 4, 221--238.

....time series together. Examples might be capital appropriations and expenditures or household income and expenditures. A generalised notion of cointegration, called fractional cointegration is examined. Some tests on (fractional) cointegration have been developed. 13] employ a t test based on [4] estimates of the long memory parameter of the regression residuals. 7] summarises di#erent semi parametric and nonparametric tests on long memory, e.g. tests based on the trimmed Whittle Likelihood (see [11] or based on Robinson s estimator, see [10] Several other estimators for long memory ....

J. Geweke and S. Porter-Hudak. The estimation and application of long memory time series models. Journal of Time Series Analysis, 4(4):221-- 238, 1983.

....need not be smooth, and can have poles and or zeroes. Indeed, Robinson(1995a,b) showed that two leading semiparametric estimates of H, based on an observed sequence x t , t =1#: #n,have desirable asymptotic properties in such a broad setting these are the log periodogram which originated in Geweke and Porter Hudak (1983), and the semiparametric Gaussian or local Whittle estimate which originated in Kunsch (1987) Both estimates depend on a bandwidth parameter m,thenumber of low frequency periodogram ordinates employed in the estimation, and both are p m consistent, where m increases as n increases, but more ....

.... Aside from the M estimation and higher order kernel aspects discussed above, we also allow for different implementations of the estimates corresponding to alternative asymptotically equivalentversions of (1) a leading one being f( Cj1 ; e i j 1;2H # as 0: 4) For example, the Geweke and Porter Hudak (1983) original version of the log periodogram estimate is based on (4) while Robinson (1995b) is based on (1) but both have the same first order asymptotic properties (see Robinson (1995b) However, the errors in the approximations in (1) and (4) can differ from each other, and this can affect ....

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Geweke, J., and S. Porter-Hudak (1983): "The estimation and application of long memory time series models," Journal of Time Series Analysis, 4, 221--238.

....of ARFIMA models was still in its infancy, researchers usually assumed that the fractional di erencing parameter was known in advance. Recently, there has been a tremendous amount of promising research on the estimation of d via various methods. Estimator after estimator has been proposed. Geweke and Porter Hudak (1983) developed the well known GPH estimator for estimating d of a stationary and invertible fractionally integrated process at low frequencies. Hurvich and Ray (1995) have extended the GPH estimator to the case of a non stationary, non invertible process. Maximum likelihood estimation has been ....

Geweke, J., Porter-Hudak, S., 1983. The estimation and application of long memory time series models. Journal of Time Series Analysis 4, 221}238.

....p s = h 2 2 p c . Since m t is independent of e s for all t and s the spectrum of y t in (1) is then (11) 0 , 2 1 2 1 1 2 2 2 w s p s p = w w = w e h e p z f f f m y It follows that (12) c c f y w = w 2 , for w small. where p s = h 2 2 p c , p s = e 2 2 c . Geweke and Porter Hudak (1983) (henceforth GPH) show that, when attention is confined to frequencies near zero, the differencing parameter can be estimated consistently from the least square regression since 10 ( d f z 2 ln ln = w w , with some z t I(d) series. If we apply this to (12) then ( ....

Geweke, J., Porter-Hudak, S., 1983, The Estimation and Application of Long Memory Time Series Models, Journal of Time Series Analysis, 4, 15-39.

....usually thought of as having a spectral pole at zero frequency, with spectral density behaving like # 2d nearby, where # indicates frequency, and 0 # 1 2 . Methods of estimating # based on a band of frequencies around zero that degenerates slowly as sample size increases were considered by Geweke and Porter Hudak (1983), Kunsch (1986, 1987) and Robinson (1994a,b, 1995a,b) the asymptotic theory of the latter author imposing essentially no conditions on spectral behaviour away from zero frequency and thereby demonstrating a signal advantage of such semiparametric methods. The main theoretical concern of ....

Geweke, J. and Porter-Hudak, S. (1983). The estimation and application of long memory time series models, Journal of Time Series Analysis, 4, 221-238.

....(IFS) to create an annual series that goes from 1884 1996. 6 The exact maximum likelihood estimator can lead to incorrect inference if the orders of the ARFIMA model are not correctly specified. A possible alternative to the exact MLE would be one of the semiparametric estimation methods of Geweke and Porter Hudak (1983), Jensen (1999a) and Jensen (1999b) 11 Country Variable d OE 1 OE 2 1 L Argentina Deltay 0.1550 0.5931 121.6397 (0.2102) 0.1836) Deltam 0.4504 0.6411 0.3698 12.7642 (0.0607) 0.1034) 0.0898) Australia Deltay 0.0040 138.5244 (0.0853) Deltam 0.2169 140.7538 (0.0822) ....

Geweke, J. and S. Porter-Hudak (1983) "The Estimation and Application of Long Memory Time Series Models," Journal of Time Series Analysis, 4, 221-238.

....95 bound Lower 95 bound Figure 8: GPH estimates of the degree of fractional integration, d, as a function of the number of periodogram ordinates, n # , used in their calculations. the logarithm of the periodogram estimate of the spectral density against ln(#) over a range of frequencies # (Geweke and Porter Hudak, 1983). We use frequencies 2#j n , with j = 1, 2, n # , # = 0.8 and n = 2075 observations, to obtain an estimated d equal to 0.43 with a standard error of 0.031. Figure 8 shows the estimated degree of fractional integration d as a function of the number of periodogram ordinates n # , and ....

Geweke, J. and S. Porter-Hudak (1983). The estimation and application of long memory time series models. Journal of Time Series Analysis 4, 221--238.

.... rst and second order methods [16] 18] the Higuchi method [19] the Detrended Fluctuation Analysis (DFA) 13] 20] and various wavelet based estimators [14] 21] 22] 23] 24] 25] Other commonly used estimators are the R=S method [26] and the estimators based on the periodogram [27]. Let us note that con dence intervals have only been obtained for the aggregated rst and second order estimators [16] Taqqu, Teverovsky and Willinger [13] have compared numerically most of these estimators except the ones based on wavelets. They have used them for estimating the scaling ....

J. Geweke and S. Porter-Hudak, \The estimation and application of long-memory time series models," J. Time Ser. Anal., vol. 4, pp. 221-238, 1983.

.... asset return volatility may be conveniently captured by a long memory, or fractionally integrated, process (e.g. Ding, Granger and Engle, 1993, and Andersen and Bollerslev, 1997) Hence in the last column of Table 2 we report estimates of the degree of fractional integration, obtained using the Geweke and Porter Hudak (1983) (GPH) log periodogram regression estimator as formally developed by Robinson (1995) The three estimates of d are all significantly greater than zero and less than one half when judged by the 21 Consistent with the simulation evidence in Bollerslev and Wright (2000) the corresponding ....

Geweke, J. and S. Porter-Hudak (1983), "The Estimation and Application of Long Memory Time Series Models," Journal of Time Series Analysis, 4, 221-238.

....volatility models Rohit S. Deo and Clifford M. Hurvich New York University January 2000 Abstract: We consider semi parametric estimation of the memory parameter in a long memory stochastic volatility model. We study the estimator based on a log periodogram regression as originally proposed by Geweke and Porter Hudak (1983). Expressions for the asymptotic bias and variance of the estimator are obtained and the asymptotic distribution is shown to be the same as that obtained in recent literature for a Gaussian long memory series. The theoretical result does not require omission of a block of frequencies near the ....

....on a parametric specification for Y t . Several authors (Breidt et al. (1998) Andersen and Bollerslev (1997) have estimated d semiparametrically using either squared returns or some transformation such as absolute or log squared returns. The semi parametric estimator they have used is the Geweke Porter Hudak (GPH, 1983) estimator based on a log periodogram regression. However, the asymptotic behaviour of this estimator is known only under the assumption that the observations used in computing the periodogram are Gaussian. See Robinson (1995a) and Hurvich, Deo and Brodsky (HDB, 1998) As Andersen and Bollerslev ....

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Geweke, J. and Porter-Hudak, S. (1983). The estimation and application of long memory time series models. Journal of Time Series Analysis, 4, 221-238.

....when is stationary, with d being between 0 and 1. An ARFIMA (1 L) d y t process would admit autoregressive and moving average components after having been applied the fractional difference (1 L) d , with a white noise innovation [see Hosking (1981) and Granger and Joyeux (1980) Mills(1990) Geweke and Porter Hudak (1983)] The process is stationary and ergodic for 0.5 d 0.5, with a bounded and positively valued spectrum at all frequencies. When 0 d 0.5, the sum of the absolute values of the autocorrelation coefficients goes to infinity and the ARIMA(0,d,0) process is said to have a long memory. When 0.5 d 0, ....

....regressions do not have a standard t distribution, but attempts to estimate by maximum likelihood a multivariate system for the interest rates involved failed, due to high multicollinearity. Regarding the second test, the last panel in Table 4 shows our estimates of the d parameter using the Geweke and Porter Hudak (1983) method (GPH) for the residual of the regression (left column) as well as for the differenced residual (right column) A process y t with the representation , where , can be written as, 1 L) d y t u t u t I (0) 17 f (T) y 1 e i T 2d f (T) u (8) log [ f y (T j ....

Geweke, J., and Porter-Hudak, S. (1983), The estimation and application of long memory time series models, Journal of Time Series Analysis, 4, 221-238.

....fractionally integrated moving average process. It was introduced independently by Granger and Joyeux (1980) and by Hosking (1981) and have been 7 proved as a valuable tool in various areas of econometric modeling (e.g. Diebold, Husted, and Rush 1991; Sowell 1992) forecasting (e.g. Geweke and PorterHudak 1983; Ray 1993) and financial time series analysis (e.g. Shea 1992; Cheung 1993) An ARFIMA(p,d,q) model is nonstationary when d # 1 2. The existence of a simple (integer) unit root in the autoregressive polynomial corresponds to the particular case d = 1. ARFIMA models have an interesting ....

....performed by using the finite sample counterpart of the spectrum, the periodogram. For robustness of the results, we have used various truncations for choosing the number m of lower periodogram ordinates to use. The standard deviations were computed from the regression results. For details, see Geweke and Porter Hudak (1983) or Brockwell and Davis (1991) All the estimates shown on Table 1 are compatible with processes with a unit root and strongly reject the stationarity hypothesis at all conventional levels of significance. INSERT TABLE 1 ABOUT HERE Next, we estimated the parameter # in equation ....

Geweke, John and Porter-Hudak, Susan (1983) The estimation and application of long memory time series models, Journal of Time Series Analysis, 4, 4, 221--238.

....hydrology and economics exhibit both short memory and long memory behavior, which may be modeled by the class of autoregressive fractionally integrated moving average (ARF IMA) processes. Applications of these processes to model 1 time series include work by Diebold and Rudebusch (1989) Geweke and Porter Hudak (1983), Hosking (1984) Lo (1989) and Sowell (1992) A time series fz t g is generated by an autoregressive fractionally integrated moving average (ARFIMA) process (Granger and Joyeaux, 1980 and Hosking, 1981) if OE(B) 1 0B) d z t = B) t ; 1) where OE(B) 1 0 OE 1 B 0 1 1 1 0 OE p B p and (B) ....

Geweke, J.F. and Porter-Hudak, S. (1983) The estimation and application of long memory time series models. J. of Time Series Anal., 4(4), 221-238.

....the log linear explosion of the low frequency spectrum, are distinguishing features of a covariance stationa ry fractionally integrated, or I(d) process with 0 d . The low frequency spectral behavior also forms the ba sis for t he log periodogram regression estimation procedure pr oposed by Geweke and Porter Hudak (1983) and refined by Robinson (1994, 1995) Hurvich and Beltrao (1994) and Hurvich, Deo and Brodsky (1998) In particular, let I(# j ) denote the s ample periodogr am at the jth Fourier frequency, # = 2#j T, j = 1, 2, T 2] The log periodogram estimator of d is then based on the leas t squares ....

Geweke, J., and Porter-Hudak, S. (1983), "The Estimation and Application of LongMemory Time Series Models," Journa l of Time Se ries Analysis, 4, 221-238.

.... or, equivalently, the log linear explosion of the low frequency spectrum, are distinguishing features of a covariance stationary fractionally integrated, or I(d) process with 0 d , The low frequency spectral behavior also forms the basis for the log periodogram regression procedure of Geweke and Porter Hudak (1983) and later refinements by Robinson (1994, 1995) Hurvich and Beltrao (1994) and Hurvich, Deo and Brodsky (1998) In particular, let I(T j ) denote the sample periodogram at the jth Fourier frequency, T j = 2Bj T, j = 1, 2, T 2] The log periodogram estimator of d is then based on the OLS ....

Geweke, J. and Porter-Hudak, S. (1983), "The Estimation and Application of Long Memory Time Series Models," Journal of Time Series Analysis, 4, 221-238.

....b ( fbk ) to log 10 (p b ( fbk ) In both these cases we are invoking ATS: average, transform, and then smooth. In the rst case the smoothing is accomplished by loess, and in the second by model tting. The averaging before taking logs is important; if we proceed without it, as is done in [13], then estimates based on the logs are inecient, that is, they do not use full information in the data [6] We t on the log scale because the large change in the power spectrum due to long range persistence is smoother than on the original scale, so estimation methods based on the power spectrum ....

J. Geweke and S. Porter-Hudak. The Estimation and Application of Long Memory Time Series Models. Journal of Time Series Analysis, 4:221-238, 1983.

....they claim to have found persistent long memory in a significant group of futures contracts. This paper reexamines the memory of futures returns using a modified version of the R S statistic developed by Lo (1991) as well as a test based on the estimator of the long memory parameter due to Geweke and Porter Hudak (1983), henceforth GPH. Our results indicate no long memory behavior in futures returns. However, a similar analysis applied to the volatility of the returns finds overwhelming evidence of persistence in volatility. This finding is consistent with recent work by Ding, Granger and Engle (1993) de Lima ....

....length n is simply J = log Q(n,q) log n . 5) No distributional results are currently available for this estimate of J. 5 Another method of determining the existence of long range dependence in a time series is based on the spectral form of a long range dependent process, as given in (2) Geweke and Porter Hudak (1983) suggested regressing the log of the estimated spectrum of the series on the log of the frequency values themselves for a set of Fourier frequencies close to zero, where the slope of the log spectrum relative to the frequency is directly dependent on the long memory parameter d. They argued that ....

Geweke, J. and Porter-Hudak, S. (1983): "The estimation and application of long memory time series models", Journal of Time Series Analysis 4, 4, 221-238.

....Periodogram points are evaluated at Fourier frequencies 2#j T ,j =1, iTrunc. In the notation of Robinson (1995b) c.f. Beran (1994, 4.6) and Robinson (1995a) n = T , m = iTrunc, l =1. EstimateGPH implements the log periodogram regression method for estimating d as discussed in Geweke and Porter Hudak (1983). Zero periodogram points are omitted, see Ooms and Hassler (1997) EstimateGSP implements the Gaussian semi parametric method for estimating d as discussed in Robinson and Henry (1998) Arfima: FixAR, Arfima: FixD, Arfima: FixMA FixAR(const iOrder) FixD(const dD) FixMA(const iOrder) iOrder ....

....z t = y t t . When either the sample mean or a specified (known, possibly zero) mean is used: t = If regressors are used, take t = f (x t ,#) In the linear case # is obtained by regression. 1) For the fractional integration parameter the (frequency domain) log periodogram regression of Geweke and Porter Hudak (1983) is used, yielding d 0 .Weuse[T 1 2 ] nonzero periodogram points, except when p = q =0when we use all available points. The initial time domain residuals are then obtained using the Ox function diffpow: u t = t # j=0 # d 0 # j j z t j . 7) 2) Next, AR starting values are ....

Geweke, J. F., and Porter-Hudak, S. (1983). The estimation and application of long memory time series models.

....data. See, e.g. Hamilton (1994) p. 447. 7 Examples of such criticism include DeJong et al. 1992) Diebold and Rudebusch (1991) Hassler and Wolters (1994) and Sowell (1990) 3 the popular nonparametric, spectral regression based procedure, called the GPH estimator after its developers, Geweke and Porter Hudak (1983). They show that based on series of length n T a where T is the number of observations and a is the power, the differencing parameter d can be consistently estimated from the least squares regression n j d c I j j j , 1 2 sin 4 ln ) ln 2 = e l l (3) where c is a ....

....for long memory by determining whether the estimated d value from the first differenced data is significantly different from zero. In the following Table I present three alternative estimates of d using n = T 0.45 , n = T 0.50 and n = T 0.55 , respectively. Table 2: Estimates of d Using the Geweke and Porter Hudak (1983) Procedure Estimate of d Variable Power = 0.45 Power = 0.50 Power = 0.55 DU t 0.36 (1.8) 1.4] 0.36 (1.8) 1.4] 0.23 (1.2) 1.1] Notes: The dependent variable is the first difference of the quarterly U.S. unemployment rate over the sample period 1957Q2 1997Q4. The OLS t values are ....

Geweke, J. and S. Porter-Hudak (1983) "The Estimation and Application of Long Memory Time-Series Models", Journal of Time Series Analysis 4, 221-238.

....volatility models Rohit S. Deo and Clifford M. Hurvich New York University August 1999 Abstract: We consider semi parametric estimation of the memory parameter in a long memory stochastic volatility model. We study the estimator based on a log periodogram regression as originally proposed by Geweke and Porter Hudak (1983). Expressions for the asymptotic bias and variance of the estimator are obtained and the asymptotic distribution is shown to be the same as that obtained in recent literature for a Gaussian long memory series. The theoretical result does not require omission of a block of frequencies near the ....

....on a parametric specification for Y t . Several authors (Breidt et al. (1998) Andersen and Bollerslev (1997) have estimated d semiparametrically using either squared returns or some transformation such as absolute or log squared returns. The semi parametric estimator they have used is the Geweke Porter Hudak (GPH, 1983) estimator based on a log periodogram regression. However, the asymptotic behaviour of this estimator is known only under the assumption that the observations used in computing the periodogram are Gaussian. See Robinson (1995a) and Hurvich, Deo and Brodsky (HDB, 1998) As Andersen and Bollerslev ....

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Geweke, J. and Porter-Hudak, S. (1983). The estimation and application of long memory time series models. Journal of Time Series Analysis, 4, 221-238.

....Metropolis algorithm is run for 1000 iterations. The Hastings step for generating d is repeated twenty times within each iteration. Initial parameter values are obtained as follows: an initial estimate of d is obtained by smoothing the original series and applying the spectral regression method of Geweke Porter Hudak (GPH) 1983) to each smoothed series, s ti , i = 1, 2. The average of the estimated d for each series is used 7 as the initial d. An initial # value is obtained as the ratio of # n t=1 s 2 t1 to # n t=1 s t1 s t2 . Initial values for # and # 2 # i are obtained by least squares estimation on y ....

....stations and a detailed space time analysis of the data. We analyze wind speeds measured at Birr and Dublin. These stations are located in central Ireland; Dublin is northeast of Birr, on the eastern coast of Ireland. Initial estimates of d for the two series obtained using the method of Geweke and Porter Hudak (GPH) 1983) with [n .55 ] 125 periodogram ordinates (and first periodogram ordinate dropped from the regression) are 0.2248 and 0.1984 respectively, with standard deviations of 0.0617 and 0.0660. However Breidt, Crato, and de Lima (1998) show that the GPH estimator of the long memory parameter may have ....

Geweke, J.F. and Porter-Hudak, S. (1983). The estimation and application of long memory time series models, J. Time Ser. Anal., 4, 221--238.

....and the log linear explosion of the low frequency spectrum are distinguishing features of a covariance stationary fractionally integrated, or I(d) process with 0 d . 26 The low frequency spectral behavior also forms the basis for the logperiodogram regression estimation procedure proposed by Geweke and Porter Hudak (1983) and later formalized by Robinson (1994a, 1995) and Hurvich, Deo and Brodsky (1998) In particular, let I(T j ) denote the sample periodogram at the jth Fourier frequency, T j = 2Bj T, j = 1, 2, T 2] The logperiodogram estimator of d is then based on the least squares regression, log[ I(T ....

.... regression, log[ I(T j ) 0 1 log(T j ) u j , 16) 27 The calculations in Hurvich and Beltrao (1994) suggest that the estimator proposed by Robinson (1994a, 1995) which leaves out the very lowest frequencies in the regression in equation (16) has larger MSE than the original Geweke and Porter Hudak (1983) estimator defined over all of the first m Fourier frequencies. For that reason, we include periodogram ordinates at all of the first m Fourier frequencies. 28 While the earlier proofs for consistency and asymptotic normality of the log periodogram regression estimator relied on normality, Deo ....

Geweke, J. and S. Porter-Hudak (1983), "The Estimation and Application of Long Memory Time Series Models," Journal of Time Series Analysis, 4, 221-238.

....Hurst coefficent is biased for some values of the coefficient and has large sampling variability. Granger and Joyeux (1980) approximated the ARFIMA model by a high order autoregressive process and estimated the difference parameter d by comparing variances for each different choice of d. Geweke and Porter Hudak (1983) and Kashyap and Eom (1988) used a regression procedure for the logarithm of the periodogram to estimate d. Hassler (1993) suggested an estimator of d based on the regression procedure for the logarithm of the smoothed periodogram. Chen et al. 1993) and Reisen (1994) also considered estimators of ....

....the smoothed periodogram regression estimator of d and concluded that this test is superior to the test based on the periodogram regression estimator of d, for discriminating between ARFIMA(p, 0, 0) and ARFIMA(p, d, 0) processes. Reinsen (1994) considered tests based on the asymptotic results of Geweke and Porter Hudak (1983), of the periodogram regression estimators of d, as well as asymptotic results based smoothed periodogram regression estimators of d. He concluded from simulated results that smoothed periodogram regression may be superior to periodogram regression for discriminating between ARFIMA(p, 0, q) and ....

[Article contains additional citation context not shown here]

Geweke, J. and Porter-Hudak, S. (1983) The estimation and application of long memory time series models. Journal of Time Series Analysis 4, 221-238.

....to reevaluate the memory of future returns. Studying a large new data set, they claim to have found persistent long memory in a significant group of future contracts. In a later work (Barkoulas, Labys, and Onochie, 1998) they use a di#erent statistical tool, the spectral regression method of Geweke and Porter Hudak (1983), henceforth GPH, to estimate the parameter characterizing long memory and again claim to find evidence of long memory behavior. In this paper, we reexamine the memory of futures returns using a modified version of the R S statistic developed by Lo (1991) as well as a test based on the GPH ....

....length n is simply J = log(R(n) S(n) log n . 5) No distributional results are currently available for this estimate of J. Another method of determining the existence of long range dependence in a time series is based on the spectral form of a long range dependent process, as given in (2) Geweke and Porter Hudak (1983) suggested regressing the log of the estimated spectrum of the series on the frequency values themselves for a set of Fourier frequencies close to zero, where the slope of the log spectrum relative to the frequency is directly dependent on the long memory parameter d. They argued that their ....

Geweke, John and Porter-Hudak, Susan (1983). "The estimation and application of long memory time series models", Journal of Time Series Analysis 4, 4, 221-238.

....result presented is readily extendable to other kernel estimates of the spectral density. As for the estimation of long memory, two other semiparametric estimates are available in the literature: the local Whittle or Gaussian estimate of Robinson (1995a) and the log periodogram estimate of Geweke and Porter Hudak (1983) and Robinson (1995b) However, unlike the local Whittle estimate, also considered in this long memory conditional heteroscedasticity framework by Robinson and Henry (1999) the averaged periodogram estimate is available in closed form. As for the more popular log peri2 odogram estimate, no ....

Geweke, J., and S. Porter-Hudak (1983): "The estimation and application of long memory time series models," Journal of Time Series Analysis, 4, 221--238.

....differencing parameter d, the precise determination of d is very important in applied work. Under the presumption of correct model specification, Cheung (1990) demonstrates that various maximum likelihood methods display a superior performance to the semiparametric procedure suggested by Geweke and Porter Hudak (1983). However, in contrast to the semiparametric procedures, in the actual application of maximum likelihood procedures the model has to be specified completely. It is to be expected that model selection is plagued by the same small sample problems as estimaton itself. Yet, this issue has not ....

Geweke, J. and Porter-Hudak, S. (1983). `The estimation and application of long memory time series models', Journal of Time Series Analysis, Vol. 4,pp. 221 -- 238.

....the theoretical properties of Whittle type quasi maximum likelihood estimators for multivariate long memory processes which are not necessarily Gaussian. Robinson (1991) discussed a spectral regression method of estimating d for a general multivariate long memory process analogous to the Geweke Porter Hudak (1983) method in the univariate case. In an application of multivariate long memory models to several related series of wind speeds measured in northern Ireland, Haslett Raftery (1989) used an approximate time domain maximum likelihood approach to fit a contemporaneous ARFIMA model, i.e. one in ....

GEWEKE, J.F. & PORTER-HUDAK, S. (1983). The estimation and application of long memory time series models, Journal of Time Series Anal., 4(4), 221-238.

....mean square convergence of the parameter estimates is established and is shown to slow as fl approaches one. Technical Report EE9351, Department of Electrical and Computer Engineering, University of Newcastle,AUSTRALIA 1 Introduction Recently there has been significant interest in long memory [18] and multi scale stochastic processes [4, 3, 27] and their overlap with work on Fractals and Wavelet Analysis [13] particularly through the study of fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) 15, 14, 33, 38, 41, 5, 11] A large part of the impetus for such work has ....

.... discussed, the term 1=f fl process will be used 2 Modelling of 1=f Processes There are a number of technical difficulties in the modelling of random processes with 1=f fl type spectra, the main difficulty being that for fl 1 (so that the process is a so called long memory process [18], although other authors reserve this for the case fl 2) the spectrum is non integrable so that no stationary process can be associated with the spectrum. The first attempts at solving this conundrum [1] involved using the idea of fractional integrals [34] to extend the usual definition of the ....

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J. Geweke and S. Porter-Hudak, The estimation and application of long memory time series models, Journal of Time Series Analysis, 4 (1983), pp. 221-- 238.

....developments. We apply the modified locally best invariant (MLBI) test of Wu (1992) to formally test the unitroot hypothesis against other nonstationary alternatives. We use the maximum likelihood (ML) approach of Sowell (1992) complemented with Robinson s (1991) asymptotic bias correction of the Geweke and Porter Hudak (1983) spectral regression method (GPH) in order to estimate 3 directly the degree of integration and thus to test for the stationarity or nonstationarity of the series. Our results reject the stationarity of the series for the current period of flexible exchange rates. But the results also suggest ....

....model. If d = 0, then (3) is an ARMA(p,q) model. The ARFIMA model was introduced independently by Granger and Joyeux (1980) and by Hosking (1981) and has proved a valuable tool in various areas of macroeconometrics, see, e.g. Diebold and Rudebush (1989) and in forecasting, see, e.g. Geweke and Porter Hudak (1983). An ARFIMA(p,d,q) model is nonstationary when d 0.5. The existence of a simple integer unit root corresponds to the particular case d = 1. There is a significant difference between nonstationary series with d = 1 and nonstationary series with 0.5 d 1. This can be clearly seen in the impulse ....

[Article contains additional citation context not shown here]

Geweke, John and Porter-Hudak, Susan (1983). "The Estimation and Application of Long Memory Time Series Models", Journal of Time Series Analysis, 4, 221-38.

....alternate, a property known as anti dependence or anti persistence. For H = 1=2, X(t) is serially uncorrelated. The estimation of H as a constant has been extensively studied, predominantly in the context of long memory where it is assumed that H 2 (1=2; 1) Relevant references include Geweke Porter Hudak (1983), Taylor Taylor (1991) Constantine Hall (1994) Chen et al. 1995) Robinson (1995) Abry Sellan (1996) Comte (1996) McCoy Walden (1996) Hall et al. 1997) Kent Wood (1997) and Jensen (1998) In certain modeling applications, treating the self similarity parameter H as a constant ....

....regarding the behavior of the process. It is therefore desirable to develop a procedure for estimating H(t) In what follows, we propose, describe, and investigate such a procedure. Techniques for estimating a constant self similarity parameter H are often founded on loglinear regression (e.g. Geweke Porter Hudak, 1983; Taylor Taylor, 1991; Constantine Hall, 1994) Such methods take advantage of an approximate log linear relationship between either the spectrum of X(t) or the variogram of Y (t) and the time index t, employing least squares regression to obtain the estimate of H. With a locally self similar ....

Geweke, J. & Porter-Hudak, S. (1983). The estimation and application of longmemory time series models. Journal of Time Series Analysis 4, 221--37.

....f u ) is of the form u t = a Gamma1 (z)b(z) t ; t ) white noise and oe 2 = E 2 t . Several approaches for estimation and testing for ARFIMA models have been used. From (7) we obtain log(f y ( log(f u (0) Gamma d log(4 sin 2 ( 2) log(f u ( f u (0) In Geweke Porter Hudak [9] a semiparametric estimation procedure for d has been suggested inspired by the formula above. They estimate d from a regression, using in a log log scale the periodogram of y t and the frequencies in a neighborhood of zero. The asymptotic properties of this estimator have been investigated e.g. ....

Geweke, J. and Porter-Hudak, S. (1983). The Estimation and Application of Long Memory Time Series Models. Journal of Time Series Analysis 4, 221--238.

....mean. The empirical autocorrelations, shown inFigure 3, decrease very slowly with increasing lags and the corresponding periodogram in Figure 4 clearly has a pole at the origin. These figures therefore suggest that the regression residuals of Lufthansa have long memory. Table 3 displays three Geweke Porter Hudak (1983) estimates of the long memory parameter d of regression residuals t for each of the seven pairs of stocks. The estimates are calculated from a periodogram regression which runs across the first T n smallest Fourier frequencies, where T is the number of observations and n is 0.5, 0.55 and ....

GEWEKE, JOHN AND SUSAN PORTER-HUDAK (1983): "The estimation and application of long memory time series models", Journal of Time Series Analysis, 4, 221-238.

....; j = 0; 1; Delta Delta Delta ; N Gamma 1 and plot log(I 2X ( versus log( k ) between two frequencies l = 2 l N and u = 2 u N such that l 0 and u 0. The slope (l) of the straight line is obtained using least squares estimation. The estimate of d is obtained as d = l 2 [10]. In case of an ARMA process, we know that the raw periodogram is an asysmptotically unbiased estimate of the spectral density function. Further, we use techniques such as windowing, periodogram averaging and smoothing to obtain consistent estimates [11] However, these variance reduction methods ....

J. Geweke and S. Porter-Hudak. The Estimation and Application of Long Memory Time Series Models. Journal of Time Series Analysis, 4(4):221-238, 1983.

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Geweke, J., and S. Porter-Hudak (1983): The estimation and application of long memory time series models, Journal of Time Series Analysis, 4(4), 221238.

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J. Geweke and S. Porter-Hudak, "Estimation and application of long memory time series models", J. Time Ser. An., 1983.

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J. Geweke and S. Porter-Hudak. The estimation and application of long memory time series models. J. Time Ser. Anal., 4:221--238, 1983.

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Geweke, J. and Porter{Hudak, S. (1983) The estimation and application of long memory time series models. J. Time Series Analysis 4, 221-238.

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Geweke, J. & S. Porter-Hudak (1983) The Estimation and Application of Long Memory Time Series Models, Journal of Time Series Analysis,4, 15-39.

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Geweke, J. and S. Porter-Hudak (1983) "The Estimation and Application of Long Memory Time Series Models", Journal of Time Series Analysis, 4, 221-238.

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Geweke, J., and S. Porter Hudak (1983): "The Estimation and Application of Long Memory Time Series Models", Journal of Time Series Analysis, 4, 221-238.

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GEWEKE, J. & S. PORTER-HUDAK (1983): \ The Estimation and Application of LongMemory Times Series Models", Journal of Times Series Analysis, #, 221-238.

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Geweke, J., and S. Porter-Hudak (1983): "The Estimation and Application of Long Memory Time Series Models", Journal of Time Series Analysis, 4, 221-38.

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Geweke, J.F., and Porter-Hudak, S. (1983). The estimation and application of long memory time series. Journal of Time Series Analysis 4, 221--238.

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J. Geweke and S. Porter-Hudak, The Estimation and Application of Long Memory Time Series Models, Journal of Time Series Analysis 4 (1983) 221-238.

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Geweke, J.F. and Porter-Hudak, S. (1983). The estimation and application of long memory time series models, Journal of Time Series Analysis, 4, 221--238. 20

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Geweke, J. and S. Porter-Hudak (1983): "The Estimation and Application of Long Memory Time Series Models", Journal of Time Series Analysis, 4, 221-238.

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