A new estimator of the fractionally integrated stochastic volatility model

发布于:2021-10-14 11:34:59

Economics Letters 63 (1999) 295–303

A new estimator of the fractionally integrated stochastic volatility model
Jonathan H. Wright*
University of Virginia, Charlottesville, VA 22903, USA Received 9 June 1998; accepted 4 December 1998

Abstract Many recent papers have considered models of fractional integration in the second moments of time series. But none proposes an estimator of the fractionally integrated stochastic volatility model with a known asymptotic distribution. In this paper I propose a GMM estimator of the fractionally integrated stochastic volatility model and prove that it is T 1 / 2 consistent and asymptotically normal if the order of fractional integration is less than 0.25. I provide calculations of asymptotic standard errors and Monte Carlo evidence on its ?nite sample performance. ? 1999 Elsevier Science S.A. All rights reserved.
Keywords: Fractional Integration; Stochastic Volatility; GMM JEL classi?cation: C22

1. Introduction Ever since the seminal paper of Engle (1982), an enormous literature has developed on models with time varying heteroskedasticity. Researchers have found these models very useful for characterizing the persistence in volatility and the fat tails of time series of asset returns. Much of the literature has considered ARCH / GARCH models in which the variance of the time series at date t is known, conditional on information dated t 2 1 and earlier. More recently, researchers have considered stochastic volatility models in which the variance at date t is random, even after conditioning on information dated t 2 1 and earlier, which may provide a better ?t to the data. Squared or log squared asset returns are however time series which have autocovariances that decay very slowly. These series are correlated over horizons of perhaps a year or more (see for example Ding et al., 1993). Fractional integration is a model that has proved useful for time series which have
*Tel.: 11-804-924-3177; fax: 11-804-982-2904. E-mail address: jw7x@virginia.edu (J.H. Wright) 0165-1765 / 99 / $ – see front matter PII: S0165-1765( 99 )00046-4 ? 1999 Elsevier Science S.A. All rights reserved.

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very slowly decaying autocovariances. Motivated by these considerations, a number of authors have examined fractional integration both in the context of models in which the conditional volatility is time varying but nonrandom and in the context of stochastic volatility models (e.g. Baillie et al., 1996; Breidt et al., 1998; Harvey, 1993). Breidt et al. (1998) propose a frequency-domain estimator for the fractionally integrated stochastic volatility (FISV) model. They show that it is strongly consistent, but provide no asymptotic distribution theory. Though fractional integration in stochastic volatility models is a relatively new topic it has potentially important implications, such as for option pricing. Despite the considerable interest in this topic, I am not aware of any paper which proposes an estimator of the stochastic volatility model with fractional integration and provides an asymptotic distribution theory for it. This paper proposes a generalized method of moments (GMM) estimator for the FISV model that is related to the estimator proposed by Tieslau et al. (1996) for the simple model of fractional 1 integration. I show that it is T 1 / 2 -consistent and asymptotically normal, provided that d , ] . The plan 4 of the remainder of this paper is as follows. In Section 2, I brie?y review the model of fractional integration. Section 3 describes the FISV model and the proposed estimator and contains calculations of its asymptotic standard errors. Section 4 reports Monte Carlo evidence on its ?nite sample behavior.

2. Fractional integration In this section, I describe the key properties of a fractionally integrated time series. A recent and comprehensive review is provided by Baillie (1996). De?ne a series as being I(0) if it is a stationary and invertible ?nite order ARMA model. A time series hh t j T51 is said to be fractionally integrated of t order d if (1 2 L)d h t is an I(0) series where L denotes the lag operator, d [ R and the fractional differencing operator is de?ned by the usual binomial expansion. This time series can be written in the form a(L)(1 2 L)d h t 5 b(L)u t
2

(1)

where u t is i.i.d. with mean zero and variance s and a(L) and b(L) denote the autoregressive and moving average polynomials, respectively. This is referred to as an ARFIMA( p, d, q) model. Other de?nitions of an I(0) time series could be used, but are not required for this paper. The covariances of an ARFIMA( p, d, q) model may be calculated by the algorithm proposed by Sowell (1992). The time 1 1 series is stationary iff d , ] and invertible iff d . 2 ] . 2 2 Many estimators of d have been considered. These include Gaussian maximum likelihood (Sowell, 1992) and various frequency domain approximations to it as well as a number of semiparametric estimates. Of special relevance to this paper is the GMM estimate proposed by Tieslau et al. (1996). This minimizes the distance between the correlation function of the fractionally integrated time series and its sample counterpart. It involves a simple application of the standard GMM principle, given that the sample autocorrelations of a fractionally integrated time series are T 1 / 2 -consistent for their 1 population counterparts and are asymptotically normally distributed, for d , ] (Hosking, 1996). The 4 1/2 1 GMM estimate of d is hence T -consistent and asymptotically normal. The restriction that d , ] is 4 1 1 indeed necessary for these asymptotic properties to hold. If ] # d , ] , Hosking shows that the sample 4 2 autocorrelations have a nonnormal distribution and converge at a rate slower than T 1 / 2 . In the context

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of the FISV model, introduced in the next section, I propose a related estimator and derive its asymptotic distribution.

3. The model and proposed estimator
T Consider the time series hy t j t51 generated by the FISV model:

y t 5 exp(( m 1 h t ) / 2)?t where h t is an ARFIMA( p,d,q) model as de?ned by Eq. (1) and ?t and u t are mutually independent random variables which are both i.i.d. and zero-mean with variances one and s 2 , respectively. The object is to estimate the parameters of the model, especially d. In this paper, I propose an estimator with a known asymptotic distribution. To motivate and describe the estimator, note that log( y 2 ) 5 m * 1 h t 1 j t t where m * 5 m 1 E(log(? 2 )) and j t 5 log(? 2 ) 2 E(log(? 2 )). So the log of the squared time series is the t t t sum of a fractionally integrated Gaussian process and an i.i.d. non-normal random variable. The 1 1 cumulant generating function of log(? 2 ) is t log( ] ) 1 logG (t 1 ] ), where G (.) is the gamma function t 2 2 2 and so it has mean 2 1.27, variance p / 2 and fourth cumulant kj 5 97.41(Abramowitz and Stegun, 1970). The zero-mean variable j t has the same variance and fourth cumulant. Clearly, the covariance function of log( y 2 ) can be written as g (k) 5 gh (k) 1 (p 2 / 2)1(k 5 0), where gh (k) is the covariance t function of the ARFIMA process h t . Let u [ Q be the p31 vector of unknown parameters consisting of d, s 2 and the coef?cients in the autoregressive and moving average lag polynomials a(L) and b(L), for some compact set Q with u in its interior. The estimator for u that I propose involves selecting u so as to minimize a measure of the 1 distance between this covariance function and its sample counterpart. For d , ] , a result of Hannan 4 (1976) gives the asymptotic distribution of the sample covariance function of the vector time series ? (h t , j t )9 from which the asymptotic distribution of g(k) may be deduced. Standard GMM arguments then give the limiting distribution of the proposed estimate of the parameters. The proposed estimator is ? ? ? u 5 argminu ( g 2 g )9W( g 2 g ) ? ? ? where g 5 (g (0), . . . , g (K))9, g 5 ( g(0), . . . g(K))9 for some ?xed K and W is a weighting matrix. 1/2 ? Theorem 1 gives the limiting distribution of T ( g 2 g ).
1 1 Theorem 1. As T → `, if 2 ] , d , ] , T 2 4 ` 1/2

? ( g 2 g ) → N(0,V ) where V 5 [vij ] and
2 2 4 h h

vij 5

u 52`

1 O hg (u)g (u 1 i 2 j) 1 g (u)g (u 1 i 1 j)j 1 p g (i 2 j) 1 p g (i 1 j) 1 ]p 1(i 5 j) 4
h h h h

1 1 kj 1 ]p 4 1 (i 5 j 5 0). 4 ? The proof of Theorem 1 is given in Appendix A. The limiting distribution of u is given in Theorem 2, which is a direct consequence of Theorem 1.

S

D

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J.H. Wright / Economics Letters 63 (1999) 295 – 303

1 1 Theorem 2. As T → `, if 2 ] , d , ] , then 2 4

T

1/2

21 21 ? ( u 2 u ) → d N(0,(D9WD) D9WVWD(D9WD) )

where D is the (K 1 1) 3 p matrix of derivatives of g with respect to u. The ef?cient choice of the weighting matrix is W 5V 21 and this choice leads the covariance matrix in Theorem 2 to simplify to (D9V 21 D)21 . This is not a feasible estimate, since V is unknown, but an asymptotically equivalent feasible estimate may be obtained by the usual two-step strategy. A ? preliminary estimate of u is found by setting W to the identity matrix, this is used to obtain V, an 21 ? estimate of V and then in the second step, W is set equal to V . This estimator is closely related to the estimator proposed in the context of the standard ARFIMA( p, d, q) model by Tieslau et al. (1996) and discussed in the previous section 1 . Given the 1 1 ? nonstandard behavior of their estimator for ] # d , ] in the standard ARFIMA model, I expect that u 4 2 1/2 1 1 will be neither T -consistent nor asymptotically normal for ] # d , ] in this FISV model. 4 2 ? The asymptotic standard errors of each of the three elements of T 1 / 2 ( u 2 u ), when the optimal weight matrix is used, are given in Table 1 in the leading case in which h t is an ARFIMA(1,d, 0) process with autoregressive parameter f. The results are reported for some values of d, s 2 , f and K ? (the number of moment conditions is K 1 1). To obtain the approximate standard errors of d, the 1/2 entries in this table should be divided by T . Not surprisingly, increasing the number of moment conditions reduces the asymptotic standard errors, though the gains from increasing K beyond 50 are generally small. The standard errors for all the parameters fall as d rises, then start to rise again when 1 d gets very close to ] . For d , 0, the standard errors are very large and enormous sample sizes are 4 required to obtain accurate inference. But for non-negative values of d (presumably most relevant empirically), while precise estimation remains dif?cult, the large sample sizes that are available for applications of the FISV model make it possible to do much better. For example, in the case f 5 0.9, d 5 0.2 and s 2 5 0.1, if K5200, the approximate standard errors of the estimate of d, s 2 and f with a sample of size 10 000 are 0.06, 0.023 and 0.018 respectively.

4. Monte Carlo evidence In this section, I report Monte Carlo evidence on the ?nite sample behavior of the proposed estimator in the FISV model, where the volatility is an ARFIMA(1,d,0) process with autoregressive 1 parameter f. The ?nite sample properties of the estimator may be considered even if d $ ] , although 4 in this case no asymptotic distribution theory for the estimator is provided in this paper. Given the computational cost of these simulations (the algorithm for computing autocovariances of ARFIMA processes must be run for each evaluation of the objective function), the simulations refer to the model with s 2 5 0.1, K 5 50, d 5 0,0.2,0.4, f 5 0.8,0.9 and samples of size 1000 and 4000. Table 2 reports the simulated bias and standard errors of the estimates of all three parameters. I also report the bias
Tieslau et al. (1996) minimize the distance between the population and sample autocorrelations: in this way estimation of the variance of the ARFIMA innovations is avoided. In the FISV model, both the autocovariances and autocorrelations of log( y 2 )depend on s 2 and so no such simpli?cation is achieved by working with autocorrelations rather than autocovariances. t
1

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Table 1 ? Asymptotic standard errors of T 1 / 2 ( u 2 u )a K525 K550 109.72 (12.46) [59.62] 7.68 (55.31) [30.93] 15.74 (4.28) [11.91] 8.95 (3.50) [8.19] 5.86 (2.93) [6.04] 6.08 (2.75) [5.71] 11.36 (2.78) [7.63] 62.72 (13.30) [19.20] 38.74 (8.18) [11.62] 13.97 (3.14) [4.83] 9.19 (2.21) [3.47] 6.56 (1.77) [2.59] 6.29 (1.72) [2.41] 9.22 (2.27) [2.91] K5100 109.68 (12.45) [59.60] 7.65 (54.18) [30.26] 13.99 (4.27) [10.89] 7.92 (3.50) [7.51] 5.56 (2.91) [5.72] 6.06 (2.72) [5.60] 9.89 (2.69) [7.25] 61.02 (13.08) [18.52] 31.50 (6.99) [9.51] 10.76 (2.77) [3.85] 7.59 (2.09) [2.92] 6.07 (1.75) [2.35] 6.10 (1.72) [2.27] 9.13 (2.22) [2.89] K5150 109.61 (12.45) [59.58] 7.65 (54.11) [30.21] 13.52 (4.27) [10.62] 7.64 (3.50) [7.32] 5.52 (2.91) [5.66] 6.04 (2.71) [5.60] 8.89 (2.67) [6.88] 61.00 (13.08) [18.50] 30.56 (6.85) [9.23] 10.03 (2.70) [3.64] 7.24 (2.07) [2.79] 5.99 (1.75) [2.31] 6.09 (1.72) [2.26] 8.88 (2.15) [2.86] K5200 109.57 (12.45) [59.56] 7.65 (54.11) [30.21] 13.29 (4.27) [10.50] 7.51 (3.50) [7.23] 5.51 (2.90) [5.64] 6.00 (2.71) [5.60] 8.25 (2.67) [6.63] 61.00 (13.07) [18.50] 30.30 (6.81) [9.15] 9.70 (2.67) [3.54] 7.10 (2.06) [2.74] 5.97 (1.75) [2.30] 6.09 (1.72) [2.26] 8.62 (2.09) [2.82]

s 5 0.1, f 5 0.9 d5 20.4 112.70 (12.59) [61.69] d5 20.2 8.12 (66.20) [37.01] d50 22.03 (4.29) [15.70] d50.1 12.45 (3.52) [10.62] d50.2 7.34 (3.00) [7.44] d50.24 6.74 (2.84) [6.65] d50.249 11.64 (2.97) [7.88] s 2 5 0.1,f 5 0.9 d5 20.4 68.22 (14.24) [21.06] d5 20.2 65.73 (13.40) [19.06] d50 29.64 (5.57) [9.60] d50.1 17.81 (3.10) [6.51] d50.2 9.76 (1.90) [4.01] d50.24 8.16 (1.76) [3.40] d50.249 10.33 (2.29) [3.67]
a

2

? Note: The numbers not in brackets are the asymptotic standard errors of T 1 / 2 (d 2 d). The numbers in round and square ? ? brackets are the asymptotic errors of T 1 / 2 ( f 2 f ) and T 1 / 2 ( s 2 2 s 2 ), respectively.

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Table 2 Finite sample standard errors (biases) of estimates a d T51000: GMM Estimator d 5 0,f 5 0.8 0.317 (20.170) d 5 0,f 5 0.9 0.219 (20.074) d 5 0.2,f 5 0.8 0.225 (20.158) d 5 0.2,f 5 0.9 0.202 (20.156) d 5 0.4,f 5 0.8 0.227 (20.261) d 5 0.4,f 5 0.9 0.167 (20.247) T51000: Frequency d 5 0,f 5 0.8 d 5 0,f 5 0.9 d 5 0.2,f 5 0.8 d 5 0.2,f 5 0.9 d 5 0.4,f 5 0.8 d 5 0.4,f 5 0.9 Domain Estimator 0.320 (20.094) 0.227 (20.027) 0.274 (20.091) 0.266 (20.057) 0.257 (20.070) 0.257 (20.036)

s2
0.263 (0.168) 0.259 (0.159) 0.270 (0.182) 0.212 (0.140) 0.231 (0.148) 0.167 (0.142)

f
0.367 (20.129) 0.262 (20.088) 0.317 (20.087) 0.146 (20.114) 0.180 (0.036) 0.078 (0.027)

0.279 (0.138) 0.263 (0.136) 0.279 (0.163) 0.246 (0.137) 0.288 (0.162) 0.235 (0.118)

0.408 (20.146) 0.283 (20.097) 0.340 (20.103) 0.246 (20.071) 0.333 (20.115) 0.242 (20.071)

T54000: GMM Estimator d 5 0,f 5 0.8 0.216 (20.099) d 5 0,f 5 0.9 0.159 (20.056) d 5 0.2,f 5 0.8 0.179 (20.105) d 5 0.2,f 5 0.9 0.144 (20.094) d 5 0.4,f 5 0.8 0.115 (20.113) d 5 0.4,f 5 0.9 0.110 (20.136) T54000: Frequency d 5 0,f 5 0.8 d 5 0,f 5 0.9 d 5 0.2,f 5 0.8 d 5 0.2,f 5 0.9 d 5 0.4,f 5 0.8 d 5 0.4,f 5 0.9
a

0.125 (0.077) 0.094 (0.050) 0.101 (0.052) 0.069 (0.046) 0.058 (0.029) 0.061 (0.050)

0.235 (20.051) 0.096 (20.012) 0.143 (20.095) 0.043 (0.011) 0.065 (0.037) 0.027 (0.025)

Domain Estimator 0.220 (20.068) 0.169 (20.039) 0.158 (20.043) 0.157 (20.041) 0.121 (20.022) 0.136 (20.025)

0.133 (0.070) 0.110 (0.048) 0.123 (0.051) 0.083 (0.036) 0.111 (0.035) 0.053 (0.023)

0.238 (20.050) 0.123 (20.020) 0.183 (20.034) 0.083 (20.007) 0.155 (20.028) 0.053 (20.005)

Note: The FISV model is as described in text. 500 replications were conducted for each experiment.

and standard errors of the frequency-domain estimator proposed by Breidt et al. (1998). No asymptotic distribution theory is available for this estimator, though Breidt et al. prove that it is strongly consistent. The most useful property of the GMM estimator is its known asymptotic distribution and the standard errors reported in Table 1 are reasonably close to those predicted by this distribution theory, particularly for T54000. The standard errors for the GMM estimate of s 2 are higher than predicted by Theorem 1, particularly in the case T51000. Both the GMM estimator and the frequency domain estimator suffer from considerable bias, especially in the smaller sample size. The estimates of d are substantially biased downwards. This is a recurrent problem in long-memory time series analysis which is well known in the simple ARFIMA model (Baillie (1996)). Large sample sizes are clearly necessary for useful inference in long memory stochastic volatility models: fortunately these are often available in applications of this model. In terms of ?nite sample properties, neither the frequency

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domain nor the GMM estimator dominates the other. Generally the GMM estimator has somewhat smaller standard errors but more bias for the estimation of d. The GMM estimate of f has, in some cases, much smaller ?nite sample standard errors than the frequency domain estimate of that parameter and additionally has smaller bias. I conclude from this Monte Carlo simulation that the GMM estimator is competitive with the frequency domain estimate. The development of alternative, more ef?cient estimates and less biased of the FISV model, preferably with an associated asymptotic distribution theory, is an important area for future research.

Acknowledgements I am grateful to Tim Bollerslev for helpful comments on earlier versions of this manuscript. All errors are the sole responsibility of the author.

Appendix A. Proof of Theorem 1 Hannan (1976) proves a central limit theorem for the autocovariances of an L-vector times series x t 5 o `50 Bj vt 2j where vt are i.i.d. mean zero random variables that are uncorrelated across time periods, j o j`50 uuBj uu 2 , ` and uu.uu denotes the matrix norm. It requires all the elements of x t to have square integrable spectra. Let the abth element of Bj be bab ( j) and let v a and x a denote the ath element of vt t t b and x t , respectively. The result states that if gab (i) is the covariance between x ta and x t1i and if c ab (i) is 1/2 its usual sample counterpart, then hT (c ab (i) 2 gab (i))j a51,...L;b 51,...L;i50,...K are jointly normally distributed with mean zero such that lim Cov(T
v 1/2

T →`

(c ab (i) 2 gab (i)),T
v v pqrs

1/2

(c cd ( j) 2 gcd ( j))) 5
bq cr

u52`

O g (u)g (u 1 i 2 j) 1 g (u)g (u 1 i
` ac bd ab bc ds p s q r

1 j) 1

p 51 q 51 r51 s51 p q r s

O O O O k O b (k)b (k 1 i)b (k 1 u)b (k 1 u 1 j)
v ap p q r s p r q s

wherekpqrs 5 E[v t v t v t v t ] 2 E[v t v t ][v t v t ] 2 E[v t v t ][v t v t ] 2 E[v t v t ][v t v t ]. 1 Turning to the model considered in this paper, since d , ] , the spectra of h t and j t are both square 4 ? ? integrable. So Hannan’s theorem applies to the unobservable vector time series (h t , j t )9. Let h and j denote the sample means of h t and j t , respectively. De?ne the sample autocovariances and cross-autocovariances of h t and j t as c hh (k) 5 T 5T and
T2k 21 t 51 21

O (h 2 h? )(h O ( j 2 j? )(h
t T2k t t51

T2k 21

? t1k 2 h ), c h j (k) 5 T ? 2h)

t51

O (h 2 h? )( j
t

t1k

? 2 j ), cj h (k)

t1k

302
T2k

J.H. Wright / Economics Letters 63 (1999) 295 – 303

cj j (k) 5 T

21

t51

O ( j 2 j? )( j
t

t 1k

? 2 j ).

Since h t and j t are independent, Hannan’s result implies that the 4(K 1 1)31 vector

vec

1

T 1 / 2 (c hh (0) 2 gh (0)) T 1 / 2 c h j (0) T 1 / 2 cj h (0)

T 1 / 2 (c hh (1) 2 gh (1)) T 1 / 2 c h j (1) T 1 / 2 cj h (1)

T 1 / 2 (c hh (K) 2 gh (K)) .. T 1 / 2 c h j (K) T 1 / 2 cj h (K) T 1 / 2 cj j (K)

T 1 / 2 (cj j (0) 2 p 2 / 2) T 1 / 2 cj j (1)

2
h h 1/2

is asymptotically normally distributed with mean zero and

T →`

lim Cov(T
T →`

1/2

(c hh (i) 2 gh (i)), T
1/2

1/2

(c hh ( j) 2 gh ( j))) 5
1/2

u52`

O hg (u)g (u 1 i 2 j) 1 g (u)g (u 1 i
` h h 1/2

1 j)j lim Cov(T
T →`

(c hh (i) 2 gh (i)), T

c h j ( j)) 5 0 lim Cov(T
T →`

(c hh (i) 2 gh (i)), T

cj h ( j))

5 0 lim Cov(T 1 / 2 (c hh (i) 2 gh (i)), T 1 / 2 (cj j ( j) 2 1( j 5 0)p 2 / 2)) 5 0 lim Cov(T
T →` 2 1/2

c h j (i), T

1/2

c h j ( j)) 5 gh (i 2 j)p / 2 lim Cov(T
T →`

2

1/2

c h j (i), T

1/2

cj h ( j))

5 gh (i 1 j)p / 2 lim Cov(T
T →`

1/2

c h j (i), T

1/2

(cj j ( j) 2 1( j 5 0)p / 2))

2

5 0 lim Cov(T
T →`

1/2

cj h (i), T

1/2

cj h ( j))

5 gh (i 2 j)p 2 / 2 lim Cov(T 1 / 2 cj h (i), T 1 / 2 (cj j ( j) 2 1( j 5 0)p 2 / 2))
T →`

1 5 0 lim Cov(T 1 / 2 (cj j (i) 2 1(i 5 0)p 2 / 2), T 1 / 2 (cj j ( j) 2 1( j 5 0)p 2 / 2)) 5 ]p 4 1(i 5 j) 4 T →` 1 1 ]p 4 1 kj 1(i 5 j 5 0) 4

H

J

2 ? Noting that g(k) 2 g (k) 5 [c hh (k) 2 gh (k)] 1 c h j (k) 1 cj h (k) 1 [cj j (k) 2 1(k 5 0)p / 2], the required result follows immediately.

References
Abramowitz, M., Stegun, N., 1970. Handbook of Mathematical Functions, Dover Publications, New York. Baillie, R.T., 1996. Long memory processes and fractional integration in econometrics. Journal of Econometrics 73, 5–59. Baillie, R.T., Bollerslev, T., Mikkelsen, H.O., 1996. Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 74, 3–30. Breidt, F.J., Crato, N., de Lima, P., 1998. The detection and estimation of long memory in stochastic volatility. Journal of Econometrics 83, 325–348. Ding, Z., Granger, C.W.J., Engle, R.F., 1993. A long memory property of stock returns and a new model. Journal of Empirical Finance 1, 83–106.

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Engle, R.F., 1982. Autoregressive conditional heteroskedasticity with estimates of the variance of UK in?ation. Econometrica 50, 987–1007. Hannan, E.J., 1976. The asymptotic distribution of serial covariances. Annals of Statistics 4, 396–399. Harvey, A.C. (1993). Long memory in stochastic volatility. Unpublished manuscript. Hosking, J.R.M., 1996. Asymptotic distributions of the sample mean, autocovariances and sample autocovariances of long memory time series. Journal of Econometrics 73, 261–284. Sowell, F.B., 1992. Maximum likelihood estimation of stationary univariate fractionally integrated time series models. Journal of Econometrics 53, 165–188. Tieslau, M.A., Schmidt, P., Baillie, R.T., 1996. A minimum distance estimator for long memory processes. Journal of Econometrics 71, 249–264.


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