Local asymptotic powers of nonparametric and semiparametric tests for fractional integration

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1 Stochastic Processes and their Applications 117 (2007) Local asyptotic powers of nonparaetric and seiparaetric tests for fractional integration Xiaofeng Shao, Wei Biao Wu Departent of Statistics, University of Illinois, Chapaign, IL 61820, United States Departent of Statistics, University of Chicago, Chicago, IL 60637, United States Received 3 February 2006; received in revised for 19 June 2006; accepted 21 August 2006 Available online 15 Septeber 2006 Abstract The paper concerns testing long eory for fractionally integrated nonlinear processes. We show that the exact local asyptotic power is of order O[(log n) 1 ] for four popular nonparaetric tests and is O( 1/2 ), where is the bandwidth which is allowed to grow as fast as n κ, κ (0, 2/3), for the seiparaetric Lagrange ultiplier (LM) test proposed by Lobato and Robinson [I. Lobato, P.M. Robinson, A nonparaetric test for I (0), Rev. Econo. Stud. 68 (1998) ]. Our theory provides a theoretical justification for the epirical findings in finite saple siulations by Lobato and Robinson [I. Lobato, P.M. Robinson, A nonparaetric test for I (0), Rev. Econo. Stud. 68 (1998) ] and Giraitis et al. [L. Giraitis, P. Kokoszka, R. Leipus, G. Teyssiére, Rescaled variance and related tests for long eory in volatility and levels, J. Econoetrics 112 (2003) ] that nonparaetric tests have lower power than LM tests in detecting long eory. c 2006 Elsevier B.V. All rights reserved. Keywords: Fractional integration; KPSS test; Lagrange ultiplier test; Local Whittle estiation; Long eory; R/S test 1. Introduction Consider the I (d) (fractionally integrated process of order d R) odel (1 B) d (X t µ) = u t, t Z, (1) Corresponding address: Departent of Statistics, University of Illinois, Urbana-Chapaign 104A, Illini Hall, 725 S Wright St., Chapaign, IL 61820, United States. Tel.: ; fax: E-ail addresses: xshao@uiuc.edu (X. Shao), wbwu@galton.uchicago.edu (W.B. Wu) /$ - see front atter c 2006 Elsevier B.V. All rights reserved. doi: /j.spa

2 252 X. Shao, W.B. Wu / Stochastic Processes and their Applications 117 (2007) where B is the backward shift operator, µ is an unknown ean and {u t } t Z is a ean zero covariance stationary short-range dependent (or short eory) process. Loosely speaking, a stationary process is short-range dependent if its autocovariances are absolutely suable. The process {X t } is short eory if d = 0, long eory if d (0, 1/2) and negative eory (antipersistence) if d ( 1/2, 0). A priary issue in studying such processes is testing the existence of long eory. We forulate it as the hypothesis testing proble: I (0) versus I (d), d (0, 1/2), or ore generally I (0) versus I (d), d ( 1/2, 0) (0, 1/2). In the literature various paraetric, nonparaetric and seiparaetric tests have been proposed. Paraetric tests include the Lagrange ultiplier (LM) test in the frequency doain [15] and LM and Wald tests in the tie doain [20]. Recently [13] incorporated the likelihood ratio test and conducted a thorough study of paraetric tests. All these tests assue certain paraetric fors of the spectral density of {u t } and have a local power O(n 1/2 ), where n is the saple size. Here we say that a test has a local power O(a n ) (a n 0 as n ) if it has nontrivial power against the local alternative d = δa n, δ 0. To allow robustness to odel isspecifications of the short eory coponent {u t }, Lobato and Robinson [12] proposed a frequency doain seiparaetric LM test. As an iportant feature, their test does not ipose paraetric assuptions on the spectral density of {u t }. It is expected that the local power of their test is O( 1/2 ), where is the bandwidth which typically grows no faster than n 4/5. However Lobato and Robinson did not give theoretical justification of this order of local power. Another type of test is of nonparaetric nature. Four popular nonparaetric tests have been well studied both epirically and theoretically: the odified R/S test [11], KPSS test [8], K/S test [28] and V/S test [4]. Wright [22] showed that R/S and KPSS tests only have trivial powers against the local alternative d = δn 1/2 in the sense that the asyptotic distributions under null and local alternatives are the sae. This is not surprising since nonparaetric tests are expected to be inferior to paraetric ones in ters of local powers. On the other hand, however, for a given stretch of tie series, the paraetric for of the short eory coponent is often unknown. Both nonparaetric and seiparaetric tests are iportant alternatives if one wants to avoid possible odel isspecifications of {u t } in conducting paraetric tests. In this paper, we show that all the four nonparaetric tests entioned above have a local power O[(log n) 1 ], which refines and iproves Wright s result. Further we prove that the local power of the LM test is O( 1/2 ), where can grow as fast as n κ, κ (0, 2/3). Therefore, nonparaetric tests are inferior to the LM test so far as the local asyptotic power is concerned. Our theory confirs the findings in finite saple siulations by Lobato and Robinson [12] and Giraitis et al. [4] that the seiparaetric LM test is superior to nonparaetric ones with respect to both size and power. Now we introduce soe notation. For a rando variable ξ, write ξ L p (p > 0) if ξ p := [E( ξ p )] 1/p < and let = 2. For two sequences (a n ), (b n ), write a n b n if a n /b n 1 as n. The sybols D and P stand for convergence in distribution and in probability, respectively. The sybols O P (1) and o P (1) signify being bounded in probability and convergence to zero in probability. Let B( ) be the standard Brownian otion; let D[0, 1] be the space of functions on [0, 1] which are right continuous and have left liits, endowed with the Skorokhod topology [1]. Denote weak convergence by. Let N(µ, σ 2 ) be a noral distribution with ean µ and variance σ 2. The paper proceeds as follows. Section 2 introduces four nonparaetric tests and states their asyptotic distributions under both null and local alternatives. Section 3 presents the seiparaetric LM test and its asyptotic distribution under local alternatives. Section 4 concludes and technical details are gathered in Appendices A and B.

3 X. Shao, W.B. Wu / Stochastic Processes and their Applications 117 (2007) Nonparaetric tests Here we review four well-known nonparaetric tests for long eory. Let X n = n 1 n X j be the saple ean and S k = k (X j X n ) the centered partial su. Modified R/S statistic [11]: Q n = 1 { } ax S k in S k, w n,l 1 k n 1 k n where wn,l 2 is the long-run variance estiator. Following Lo [11], wn,l 2 = 1 n l ( (X j X n ) j ) ˆγ j, (2) n l + 1 where ˆγ j = n 1 n j i=1 (X i X n )(X i+ j X n ), 0 j < n. Here the bandwidth satisfies l = l(n) and l/n 0 as n. KPSS statistic [8]: K n = (wn,l 2 n2 ) 1 n k=1 S k 2. V/S statistic [4]: V n = (wn,l 2 n2 ) 1 { n k=1 S k 2 ( n 1 n ) 2}. k=1 S k K/S statistic [28]: G n = (n 1/2 w n,l ) 1 ax 1 k n S k. Aong the above four tests, KPSS and K/S tests were originally proposed to test trend stationarity versus unit root nonstationarity. They have been used by Lee and Schidt [9] and Lia and Xiao [10] respectively for tests of fractional integration. Proposition 1 below describes their asyptotic null distributions. The proof is oitted since it easily follows fro the continuous apping theore [1]. Let σ 2 = 2π f u (0), where f u ( ) is the spectral density function of {u t }. Throughout the paper, we assue without loss of generality that µ = 0 and 0 < f u (0) <. Proposition 1. Let d = 0. Assue that wn,l 2 nt P σ 2 and (nσ 2 ) 1/2 X j B(t) in D(0, 1). (3) Let B(t) = B(t) tb(1) be the Brownian bridge. Then we have (a) n 1/2 Q n D sup 0 t 1 B(t) inf 0 t 1 B(t); 1 (b) K n B(t) D 0 2 dt; 1 (c) V n B(t) D 0 2 dt ( 1 B(t)dt) 0 2 ; (d) G n D sup 0 t 1 B(t). Giraitis et al. [4] provided sufficient conditions for (3) and derived liiting distributions for R/S, KPSS and V/S statistics under both the short eory null hypothesis and long eory alternatives. All of these tests are consistent against both long eory and antipersistent alternatives; see [18] for the treatent of R/S and KPSS tests. The consistency was obtained for fixed d, d ( 1/2, 0) (0, 1/2). Since we are interested in the local power, we allow d to be dependent on the saple size n. Let d n = c/ log n, where c is a fixed constant. Define (1 B) d n (X t µ) = u t, t = 1,..., n. (4)

4 254 X. Shao, W.B. Wu / Stochastic Processes and their Applications 117 (2007) Strictly speaking, the series {X t } n t=1 for a triangular array of type {X tn : t = 1,..., n; n = 1, 2,...}. For the convenience of presentation, we shall use {X t } n t=1 and no confusion will arise. Throughout the paper, we assue u t = F(..., ε t 1, ε t ), where ε t are independent and identically distributed (iid) rando variables and F is a easurable function such that u t is well defined. Then {u t } is a stationary causal ergodic process. The class of processes that (5) represents is large. It includes a nuber of widely used nonlinear tie series odels such as bilinear odels, threshold odels, GARCH and ARMA-GARCH odels; see [25] and [19] for ore exaples. As in [21,14] and [24], (5) can be interpreted as a physical syste with F t = (..., ε t 1, ε t ) being the input, F being a filter and u t being the output. Let {ε t } t Z be an iid copy of {ε t } t Z. For ξ L 1 define projection operators P k ξ = E(ξ F k ) E(ξ F k 1 ). Let g k (F 0 ) = E(u k F 0 ) and w q (k) := g k (F 0 ) g k (F0 ) q, where F0 = (F 1, ε 0 ); let δ q(k) := u k u k q, where u k = F(F 0, ε 1,..., ε k ). In [24], w q ( ) and δ q ( ) are called the predictive dependence easure and physical dependence easure respectively. Intuitively, w q (k) easures the contribution of ε 0 in predicting u k, while δ q (k) quantifies the dependence of u k on ε 0 by easuring the distance between u k and its coupled version u k. In any applications physical and predictive dependence easures are easy to work with since they are directly related to data-generating echaniss. For ore details see [24]. Theore 1. Let {X t } n t=1 be generated fro (4). Assue u t L q, q > 2, and (5) w q (k) <. k=0 Then n 1/2 nt X j σ e c B(t) in D(0, 1). (6) Reark 1. Since P 0 u k q w q (k) 2 P 0 u k q [24], (6) is equivalent to P 0 u k q <. k=0 In the special case of the linear process u t = k=0 a k ε t k, P 0 u k = a k ε 0, so (7) holds if k=0 a k < and ε 1 L q. See Reark 3 for nonlinear tie series. (7) Theore 2. Let {X t } n t=1 be generated fro (4). Assue u t L q, q > 2, (6) and l = n α, α (0, 1). Then w 2 n,l P σ 2 e 2cα. Consequently, we have (a) n 1/2 Q n D e c(1 α) {sup 0 t 1 B(t) inf 0 t 1 B(t)}; (b) K n D e 2c(1 α) 1 B(t) 0 2 dt; (c) V n D e 2c(1 α) { 1 B(t) 0 2 dt ( 1 B(t)dt) 0 2 }; (d) G n D e c(1 α) sup 0 t 1 B(t). See Appendix A for the proofs of Theores 1 and 2. Reark 2. The long-run variance estiator (2) is equivalent to the nonparaetric spectral density estiator evaluated at zero frequency with a Bartlett window. The lack of power of

5 X. Shao, W.B. Wu / Stochastic Processes and their Applications 117 (2007) nonparaetric tests is not due to the choice of the window or the estiator of σ 2 itself. Even if we know σ 2 and replace w 2 n,l by σ 2 in these test statistics, by Theore 1 the exact local power is still O[(log n) 1 ]. Clearly our ethods are applicable to other test statistics that can be expressed as continuous functionals of partial su processes. Theore 2 shows that nonparaetric tests have a nontrivial asyptotic power against the local alternative of the for d n = c/ log n, c 0. Their asyptotic distributions under the local alternative d n = o[(log n) 1 ] are the sae as the asyptotic null distributions (cf. Reark 4). In other words, nonparaetric tests have no power to detect the local alternative, which is as close as o[(log n) 1 ] to zero integration. Interestingly, the asyptotic distributions in (a) (d) under null and local alternatives only differ by a ultiplicative factor. 3. Seiparaetric LM test Lobato and Robinson [12] introduced a frequency doain LM test of I (0) in a ultivariate context. The idea is closely related to the local Whittle estiation of the long eory paraeter [7,17]. For a process {X t } t Z, denote the periodogra by I X (λ) = w X (λ) 2, where w X (λ) = 1 2πn n X t e itλ. The local Whittle estiator ˆd is obtained by iniizing the local objective function ( ) R(d) = log 1 λ 2d j I X (λ j ) 2d log λ j, where λ j = 2π j/n and = (n) is the bandwidth satisfying 1 + /n 0 as n. Let R (d) = R(d)/ d and R (d) = 2 R(d)/ 2 d. The standard LM test statistics can be written as R (0)R (0) 1 R (0). Define t = 1/2 1 v j I X (λ j ) I X (λ j ) t=1, (8) where v j = log j 1 log j. Theore 1 of [12] asserts that the standard LM test statistic is equal to t 2 + o P(1). In fact, Lobato and Robinson s LM test statistic is defined to be t 2 in the univariate case. Note that the local Whittle estiator ˆd is asyptotically equivalent to t /2 if d = 0 (cf. [19]). Under the null hypothesis, t D N(0, 1). We reject the null hypothesis in favor of the long eory alternative if t falls into the upper tails of N(0, 1), or in favor of the antipersistent alternative if t falls into the lower tails of N(0, 1). Theore 2 of [12] shows that the LM test is consistent against the fractional alternatives d ( 1/2, 0) (0, 1/2). They entioned that the LM test is expected to have good power against local alternatives of the for d = δ 1/2, where δ is a fixed constant. But no proof was given in their paper. To investigate the local power, we define (1 B) d (X t µ) = u t, t = 1,..., n, (9)

6 256 X. Shao, W.B. Wu / Stochastic Processes and their Applications 117 (2007) where u t is defined in (5) as before. Again, we avoid using the double array notation to ease the presentation. Theore 3. Let {X t } n t=1 be generated fro (9). Assue that (log n) (10) n2/3 and f u (λ) = f u (0)(1 + O(λ 2 )) as λ 0. Further assue u t L q, q > 4, cu(u 0, u k1, u k2, u k3 ) < (11) and k 1,k 2,k 3 Z kδ q (k) <. k=1 Then t D N(2δ, 1). See Appendix B for the proof of Theore 3. Reark 3. The conditions (10) (12) were originally iposed in [19] to study asyptotic properties of the local Whittle estiator for general fractionally integrated nonlinear processes. The fourth cuulants suability condition (11) is a coon assuption adopted in spectral analysis [3]. For nonlinear processes (5), it is satisfied under a geoetric oent contraction (GMC) condition with order 4 [26]. The process {u t } is GMC(q), q > 0, if there exists a C = C(q) and ρ = ρ(q) (0, 1) such that E( u n u n q ) Cρ n, n N, (13) where u n = F(..., ε 1, ε 0, ε 1,..., ε n ) is a coupled version of u n with all the past innovations {ε t } t 0 coupled. The property (13) indicates that the process {u n } forgets its past exponentially fast and it can be verified for any nonlinear tie series odels [25]. Theore 4.1 of [19] provides another set of sufficient conditions for (11). Specifically, it is shown that k=1 k 3 δ 4 (k) < iplies (11). Note that both (7) and (12) result fro (13). Theore 3 provides a theoretical justification for the clai ade in [12] in the univariate case for general nonlinear processes. It shows that the exact local power of LM test is O( 1/2 ), where is allowed to grow as fast as n κ, κ (0, 2/3). For linear processes u t = k=0 a k ε t k, Theore 3 holds if 5 (log ) 2 = o(n 4 ), k=0 ka k < and ε 1 L 4. In other words, κ is allowed to fall within (0, 4/5). 4. Conclusions In this paper, we consider local asyptotic powers of four nonparaetric tests for fractional integration. The result sees surprising in that these tests only have nontrivial power against local fractional alternatives such as d n = c(log n) 1, where c is a fixed constant. In contrast, Lobato and Robinson s seiparaetric LM test is shown to have a local power O( 1/2 ), where can grow as fast as n κ, κ (0, 2/3), for nonlinear processes. We conclude that, to avoid possible odel isspecifications in conducting paraetric tests, the seiparaetric LM test is preferable to all the nonparaetric tests discussed in this paper, at least in the large saple case. On the (12)

7 X. Shao, W.B. Wu / Stochastic Processes and their Applications 117 (2007) other hand, as entioned in [12,4], a reliable bandwidth selection for the LM test is not yet available. This liits the use of LM test in practice and otivates us to pursue further study along this direction. We further reark that our theoretical investigation is liited to the fraework (1). Recently, there has been a substantial aount of work on testing and estiation of long eory in the volatility of financial tie series (cf. [5,6] and the references therein). It is not clear whether in that setting one can still obtain siilar results for nonparaetric and seiparaetric LM tests. Acknowledgeents We would like to thank the editor and a referee for helpful coents. We are also grateful to Michael Stein for his constructive suggestions. The work is supported in part by NSF grant DMS Appendix A Lea 1. Let f (d, j) := [Γ (d + j)/γ ( j +1) j d 1 ]/j d 2, j N. Then for any fixed M > 0, there exists a finite constant C = C(M) such that sup d [ M,M] sup f (d, j) < C. j N, j>m Proof of Lea 1. The proof follows fro Stirling s forula Γ (z) = e z z z 1/2 2π(1 + 1/(12z) + O(z 2 )) and soe eleentary algebra. By the binoial expansion, we write X t = j=0 φ j (d n )u t j, where φ j (d n ) = Γ (d n + j)/{γ ( j + 1)Γ (d n )}. Note that φ 0 (d) = 1 for any d R. For 1 n, write S = t=1 X t = k=0 A k, u k, where A k, = ψ k ψ k, ψ k = k j=0 φ j (d n ) if k 0 and 0 if k < 0. Let ξ t = k=t P t u k, T n = n t=1 u t and M n = n t=1 ξ t. We approxiate S by S = k=0 A k, ξ k. Let R = S S be the reainder ter. Lea 2. Suppose X t is generated fro (4) and u t L q, q 2. Assuing (7), then for any l = l(n) = n α, α (0, 1], l 1 ES 2 l σ 2 e 2cα. Proof of Lea 2. By Theore 1 of [23], (7) iplies T n M n q = o( n) for q 2. By Karaata s theore [2] and suation by parts, for q 2, 2l R l q A k,l A k 1,l T k M k q + k=0 2l = k dn 1 o( k) + k=0 k=2l+1 k=2l+1 A k,l A k 1,l T k M k q k d n 1 o( k) = o( l). (14) Since {ξ t } are stationary artingale differences, we have k=0 A k,l ξ l k 2 = k=0 A 2 k,l ξ 1 2. Then our conclusion follows fro the fact that ξ 1 2 = σ 2 (cf. [23]), l 1 k=0 ψ2 k l 1 k=0 k2d n Γ (d n + 1) 2 e 2cα l and k=l A 2 k,l k=l (k d n (k l) d n ) 2 l 2d n+1 h=1 [h d n (h 1) d n ] 2 = o(l), where we have applied Lea 1. Reark 4. If d n = o[(log n) 1 ], then it is easy to see that l 1 ES 2 l σ 2.

8 258 X. Shao, W.B. Wu / Stochastic Processes and their Applications 117 (2007) Proof of Theore 1. Set C q = 18q 3/2 (q 1) 1/2. By (14) and Proposition 1 in [23], ax R 2 k q k 2 (k s)/q R 2 s q = s=0 k 2 (k s)/q o(2 s/2 ) = o(2 k/2 ). s=0 Thus it suffices to show n 1/2 S nt σ e c B(t) in D(0, 1). The finite diensional convergence follows fro the arguent in the proof of Lea 4.1 of [18]. By Lea 3 in [25], S n 2 q C 2 q ξ 1 2 k=0 A 2 k,n Cn, where C is a generic constant. Then for any 0 t 1 t t 2 1, we have E( S nt2 S nt q/2 S nt S nt1 q/2 ) S nt2 S nt q q/2 S nt S nt1 q/2 q C( nt 2 nt ) q/4 ( nt nt 1 ) q/4 C(nt 2 nt 1 ) q/2. Therefore the tightness follows fro Theore 15.6 of [1] and the proof is copleted. Proof of Theore 2. The convergence in (a) (d) is a direct consequence of the assertion wn,l 2 P σ 2 e 2cα and Theore 1 by the continuous apping theore. By the sae arguent as in the proof of Theore 3.1 of [18], where the case of fixed d is treated, wn,l 2 P σ 2 e 2cα holds in view of Lea 2. We oit the details. Appendix B For notational convenience, write I X j = I X (λ j ), I u j = I u (λ j ), f u j = f u (λ j ), w X j = w X (λ j ) and w u j = w u (λ j ). Denote by α n (λ) = (1 e iλ ) d and α nj = α n (λ j ). Then it is easily seen that α n (λ) is differentiable in a neighborhood of the origin (0, ɛ) and α n (λ) = O( α n(λ) λ 1 ) as λ 0. Denote by D(w) = D n (w) = n t=1 e itw. Let K (w) = (2πn) 1 D(w) 2 be Fejér s kernel. The following leas correspond to Lea in [19], where the case of fixed d is treated. Let g j = w X j /( α nj f u j ) and h j = w u j / f u j. Lea 3. Suppose {X t } n t=1 is generated fro (9). Under (12), the following relations hold uniforly over 1 k j = o(n): E{g j ḡ j } 1 + E{h j h j } 1 + E{g j h j } α n ( λ j )/ α nj = O(log j/j); (15) E{g j g j } = O(log j/j), E{g j g k } = O(log j/k), E{g j ḡ k } = O(log j/k); E{h j h j } = O(log j/j), E{h j h k } = O(log j/k), E{h j h k } = O(log j/k); E{g j h j } = O(log j/j), E{g j h k } = O(log j/k), E{g j h k } = O(log j/k). Proof of Lea 3. Since E[(u k u k ) F 0] = P 0 u k and P 0 u k P 0 u k q δ q (k) by Jensen s inequality, (12) iplies k=0 k P 0 u k <. Thus we have k=0 kγ (k) <, where γ ( ) is the autocovariance function of {u t }; see Lea 6.1 of [19] for a rigorous proof. Then we have α n (λ) = O(α n(λ)λ 1 ) and f u (λ) = O(λ 1 ) as λ 0. The rest of the proof follows fro the arguent in the proof of Theore 2 of [16], where d is fixed. We oit the details.

9 X. Shao, W.B. Wu / Stochastic Processes and their Applications 117 (2007) Lea 4. Suppose {X t } n t=1 is generated fro (9). Under (12), we have E g j 2 h j 2 = O( j 1/2 ) uniforly over j = 1,...,. (16) Further, if (11) holds, then we have r E ( g j 2 h j 2 ) C(r 1/4 (1 + log r) 1/2 + r 1/2 n 1/4 ), r = o(n), (17) where C is a generic constant independent of r, and n. Proof of Lea 4. For (16), the arguent of Lea 6.3 of [19] applies. The key step is to show that π K (λ λ j ) α n (λ) 2 1 α = O(1/j) uniforly over j = 1,...,. (18) nj π Using the sae arguent as for Lea 3 of [17], by the properties of α n (λ), (18) holds. The assertion (17) corresponds to Lea 6.4 of [19], where the arguent uses (11) and (18). The conclusion follows. Proof of Theore 3. Let b nj = v j α nj 2. We first list soe useful facts. Fact 1. α nj 2 = 1 + O((log n)/ ) uniforly over j = 1,...,. Fact 2. b nj b n( j+1) C j 1 uniforly over j = 1,...,. Here C is a generic constant. Fact 3. Under (12), cov(w u j, w uk ) = f u j 1( j = k) + O(1/n) and cov(w u j, w uk ) = O(1/n) uniforly over j, k = 1,...,. Fact 4. Under (11) and (12), we have cov(i u j, I uk ) = fu 2 j1( j = k) + O(1/n) uniforly over j, k = 1,...,. Facts 1 and 2 follow fro eleentary calculations. By (12), k=0 kγ (k) <, which yields cov(w u j, w uk ) = 1 2πn n γ (t s)e itλ j isλ k = f u j 1( j = k) + O(1/n). t,s=1 So the first assertion of Fact 3 holds and the second assertion siilarly follows. Regarding Fact 4, we have cov(i u j, I uk ) = cu(w u j, w u j, w uk, w uk ) + cov(w u j, w uk )cov( w u j, w uk ) + cov(w u j, w uk )cov( w u j, w uk ) = O(1/n) + fu 2 j1( j = k), which follows fro Fact 3 and (11). We now deal with the denoinator of (8). Fact 1 and (16) iply that 1 I X j = 1 g j 2 α nj 2 f u j = 1 h j 2 α nj 2 f u j + o P (1) = 1 I u j + o P (1) = f u (0) + o P (1). (19)

10 260 X. Shao, W.B. Wu / Stochastic Processes and their Applications 117 (2007) The last equality above is due to Facts 3 and 4. Next we shall treat the nuerator of (8). By Lea 4 and Fact 2, E b nj ( g j 2 h j 2 1 ) r b nr b n(r+1) E ( g j 2 h j 2 ) r=1 + b n E ( g j 2 h j 2 ) = o( ). Then we have 1 v j I X j = 1 = f u(0) Finally by Fact 3, we get 1 b nj EI u j = 1 = 1 b nj g j 2 f u j = f u(0) b nj g j 2 + o P (1) b nj h j 2 + o P (1) = 1 b nj f u j + o(1) b nj I u j + o P (1). (20) b nj f u (0) + o(1) = 2δ f u (0) + o(1), where the last equality above follows fro eleentary calculations. By Theore 2 of [27], 1/2 f u (0) 1 b nj (I u j EI u j ) D N(0, 1). Therefore t D N(2δ, 1) in view of (19) and (20). Reark 5. Condition (10) is needed to prove the asyptotic norality for b nj (I u j EI u j ). By Fact 4, ( ) var 1/2 b nj (I u j EI u j ) = 1 bnj 2 var(i u j) + O( log 2 /n) = 1 bnj 2 f u 2 j + o(1) = [ f u(0)] 2 + o(1). In other words, t = O P (1) under the less restrictive assuption 5 (log ) 2 = o(n 4 ). References [1] P. Billingsley, Convergence of Probability Measures, Wiley, [2] N.H. Bingha, C.M. Goldie, J.L. Teugels, Regular Variation, Cabridge University Press, New York, [3] D.R. Brillinger, Tie Series: Data Analysis and Theory, Holden-Day, San Francisco, [4] L. Giraitis, P. Kokoszka, R. Leipus, G. Teyssiére, Rescaled variance and related tests for long eory in volatility and levels, J. Econoetrics 112 (2003) [5] C.M. Hurvich, P. Soulier, Testing for long eory in volatility, Econo. Theory 18 (2002) [6] C.M. Hurvich, E. Moulines, P. Soulier, Estiating long eory in volatility, Econoetrica 73 (2005) [7] H. Künsch, Statistical aspects of self-siilar processes, in: Yu.A. Prohorov, V.V. Sazonov (Eds.), Proceedings of the 1st World Congress of the Bernoulli Society, vol. 1, Science Press, Utrecht, 1987, pp

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