Bootstrap-Based Test for Volatility Shifts in GARCH against Long-Range Dependence

Transcription

1 Communicaions for Saisical Applicaions and Mehods 015, Vol., No. 5, DOI: hp://dx.doi.org/ /csam Prin ISSN / Online ISSN Boosrap-Based Tes for Volailiy Shifs in GARCH agains Long-Range Dependence Yu Wang a, Cheolwoo Park a, Taewook Lee 1,b a Deparmen of Saisics, Universiy of Georgia, USA; b Deparmen of Saisics, Hankuk Universiy of Foreign Sudies, Korea Absrac Volailiy is a variaion measure in finance for reurns of a financial insrumen over ime. GARCH models have been a popular ool o analyze volailiy of financial ime series daa since Bollerslev (1986) and i is said ha volailiy is highly persisen when he sum of he esimaed coefficiens of he squared lagged reurns and he lagged condiional variance erms in GARCH models is close o 1. Regarding persisence, numerous mehods have been proposed o es if such persisency is due o volailiy shifs in he marke or naural flucuaion explained by saionary long-range dependence (LRD). Recenly, Lee e al. (015) proposed a residual-based cumulaive sum (CUSUM) es saisic o es volailiy shifs in GARCH models agains LRD. We propose a boosrap-based approach for he residual-based es and compare he sizes and powers of our boosrap-based CUSUM es wih he one in Lee e al. (015) hrough simulaion sudies. Keywords: Boosrap, CUSUM es, GARCH models, Long-range dependence, Volailiy shifs 1. Inroducion In finance, volailiy is a variaion measure for reurns of a financial insrumen over ime. Measuring volailiy has been one of he mos criical seps in financial modeling. A volailiy model can be used o forecas he absolue magniude of reurns or he densiy of he volailiy in risk managemen, asse pricing, and porfolio selecion. Engle (198) inroduced he auoregressive condiional heeroscedasic (ARCH) process and menioned several aracive characerisics of ARCH models. A more general process, generalized auoregressive condiional heeroscedasiciy (GARCH) models, have been a popular ool o analyze volailiy of financial ime series daa since Bollerslev (1986). GARCH (p, q) models saisfy: x = σ ϵ, σ = ω + α i x i + β j σ j, (1.1) where he innovaion {ϵ } Z is a sequence of sandard i.i.d. random variables. I is assumed ha ω > 0, α i 0 for all i = 1,..., q and β j 0 for all j = 1,..., p. I is said ha he volailiy is highly persisen when he sum of he esimaed coefficiens of he squared lagged reurns and he lagged condiional variance erms in GARCH models is close o 1. This research was suppored by Basic Science Research Program hrough he Naional Research Foundaion of Korea (NRF) funded by he Minisry of Educaion, Science and Technology (NRF-013R1A1A006796). 1 Corresponding auhor: Deparmen of Saisics, Hankuk Universiy of Foreign Sudies, Yongin 17035, Korea. wlee@hufs.ac.kr Published 30 Sepember 015 / journal homepage: hp://csam.or.kr c 015 The Korean Saisical Sociey, and Korean Inernaional Saisical Sociey. All righs reserved.

2 496 Yu Wang, Cheolwoo Park, Taewook Lee An alernaive way o deal wih he high persisence of volailiy is o use long-range dependen (LRD) modeling on ransformed daa such as squared reurns. Beran (1994) poined ou ha scieniss in various areas have observed correlaions beween observaions ha decay o zero a a slow rae. This long memory phenomenon has been documened long before suiable models were developed. Ding e al. (1993) invesigaed he long-memory propery of he sock marke reurn series. They found ha he power ransformaion of he absolue reurns has quie high auocorrelaions for long lags. A LRD ime series {x } Z is defined as a second-order saionary ime series model wih a slowly decaying auocovariance funcion, γ(h) = Cov(x, x +h ) Ch d 1 = Ch H, as h, (1.) where C > 0 is a consan, d ( 0, 1 ) is he long-range dependence parameer, and H = d + 1 ( 1, 1 ) is he self-similariy (SS) parameer. Based on (1.) for LRD series, he auocovariances are no absolue summable. Tha is, h= γ(h) = +. If, on he oher hand, he auocovariances are absolue summable, h= γ(h) <, he series is shor-range dependence (SRD). Firs proposed by Efron (1979), he boosrap mehod is anoher key ool in his paper. Boosrap has been developed o overcome inaccuracies caused by small sample sizes in saisical inference. There are many differen ypes of boosrap mehods, bu no all of hem are easy o implemen or work well in cerain cases. In his paper, we use he sandard residual-based boosrap mehod wih he procedures presened in Secion 3. This paper focuses on he persisence in volailiy of financial ime series. Numerous mehods have been proposed o es if such persisency is due o volailiy shifs in he marke or naural flucuaion explained by saionary long-range dependence. Lee e al. (015) proposed a residual-based cumulaive sum (CUSUM) es saisic o es volailiy shifs in GARCH models agains LRD. In his paper, we propose a boosrap-based approach for he residual-based es and compare he sizes and powers of our boosrap-based CUSUM es wih he one in Lee e al. (015) hrough simulaion sudies. The paper is organized as follows. Secion inroduces he ess on volailiy shif versus LRD in Lee e al. (015). Secion 3 proposes he boosrap-based CUSUM mehod for he residual-based es. Simulaion resuls are repored in Secion 4 and he conclusion in Secion 5.. Residual-Based CUSUM Tes for Volailiy Shifs agains LRD Many saisical ess o disinguish LRD and volailiy shif divide he procedure ino wo classes according o he null hypoheses: eiher LRD as he null hypohesis (Qu, 011; Kuswano, 011), or he volailiy-shif model as he null (Jach and Kokoszka, 008). In his paper, we use he laer one as he null hypohesis. The ess consider: H (R) 0 : VS-R model versus H 1 : LRD model,

3 Boosrap-Based Tes for Volailiy Shifs in GARCH agains Long-Range Dependence 497 where VS-R represens ha volailiy shifs occurred a R unknown locaions. We fi each of he (R+1) regimes using GARCH models. Researchers proposed various mehods o esimae he unknown change locaions for he volailiy. The mos popular is he CUSUM ype of es saisic firs considered by Kokoszka and Leipus (000), where hey use he squared asse reurns for a single volailiy shif. Andreou and Ghysels (00) argue ha his es suffers from size disorions under he finie sample because of he srong dependence on observaions. The Barle long-run variance esimaor can be used o correc he size disorion (Berkes e al., 006; Zhang e al., 007). However, he Barle long-run variance esimaor is sensiive o choice of he kernel bandwidh. To overcome such defecs, Lee e al. (015) adoped he residualbased CUSUM es. The es uses he sandardized residuals from an esimaed GARCH model ha imiae he innovaion series in GARCH models and reduce he dependence among observaions. Unlike he saisic proposed by Lee e al. (004), he residual-based CUSUM es saisic does no require uning parameers for runcaion and can be used for various ypes of volailiy models. The power ransformaion of reurns, such {x } Z, displays recognizable auocorrelaions; consequenly, he series {x } Z saisfies (1.) and will be used for he alernaive hypohesis. One of he popular models for LRD is he fracionally inegraed GARCH (FIGARCH) model of Baillie e al. (1996). We use his as he alernaive model in he power of he es and described i more deails in he simulaion secion. The firs es we consider is: H (0) 0 : The observed daa {x } Z follow he VS-0 model, H 1 : no H 0. In order o conduc he es, consider he CUSUM es saisic based on {x } Z : T n = 1 max nsn 1 k n x k x n, where n is he sample size, and s n is he esimaor of he long-run variance σ = h= Cov(x, x+h ). Noe ha he CUSUM es for GARCH models suffers from size disorions and low powers under finie sample size due o he srong correlaion beween observaions. Therefore, Lee e al. (015) consider he CUSUM es based on he i.i.d. innovaion series {ϵ } in equaion (1.1), which are uncorrelaed observaions. Tha is, ˆT n = 1 nτ max 1 k n ϵ k n where τ = Var(ϵ ). Since {ϵ } is unobservable, one can esimae he innovaion series by ϵ, := x ), = 1,..., n, (.1) σ (ˆθ where ˆθ = ( ˆω 0, ˆα 1,..., ˆα q, ˆβ 1,..., ˆβ p ) is he esimaed parameer vecor of GARCH models, and σ (θ) is calculaed recursively from σ (θ) = ω + α i x i + β j σ j (θ),

4 498 Yu Wang, Cheolwoo Park, Taewook Lee where θ = (ω, α 1,..., α q, β 1,..., β p ) is he parameer vecor. Noe ha calculaing σ recursively requires he iniial values for x0,..., x 1 q and σ 0,..., σ 1 p. The fgarch package in R provides he iniial values o be he sample average of x1,,..., x n. Finally, he residual-based CUSUM saisic is defined as T n = 1 max nˆτ 1 k n k n, (.) where ˆτ = 1 n 4 1 n is a mehod of momen esimaor of Var(ϵ1 ). Lee e al. (015) also proved ha under cerain assumpions and condiions, and under H (0) 0, as n, d T n sup B (u), 0 u 1 where B (u) is a sandard Brownian bridge. We will use his as he asympoic disribuion of he residual-based CUSUM es saisic T n. According o Resnick (199), for a sandard Brownian moion, he formula for he disribuion of T n can be derived as Pr [ T n υ ] = 1 + ( 1) k e k υ, υ > 0 (.3) k=1 and he p-values will be calculaed based on (.3) in he simulaion secion. Consider a es on volailiy shif for known R number of imes wih unknown locaions agains LRD afer he null hypohesis of no volailiy shif is rejeced. Therefore, he hypohesis would become: H (R) 0 : The observed daa {x } Z follow he VS-R model, H 1 : The observed daa {x } Z follow he LRD model. We follow Lee e al. (015) and only consider he simple case when R = 1 in his paper, which is he single-volailiy-shif case. The esing procedure can be easily exended o muliple volailiy shifs. Suppose he whole series consiss of wo differen GARCH(p, q) models, namely {x 1, } Z and {x, } Z, wih he same sandard i.i.d. innovaions. Tha is, x 1,, if 1 k, x = x,, if k (.4) < n, where k is an unknown change poin and x 1, = σ 1, ϵ, σ 1, = ω 1 + α 1,i x 1, i + β 1, j σ 1, j, (.5) x, = σ, ϵ, σ, = ω + α,i x, i + β, j σ, j. ϵ

5 Boosrap-Based Tes for Volailiy Shifs in GARCH agains Long-Range Dependence 499 By Kokoszka and Leipus (000), k can be esimaed by ˆk = arg max k(n k) 1 1 k n n x 1 k n k =k+1 x. (.6) Noe ha in heoreical perspecive, any change-poin esimaor of VS-1 model saisfying ˆk k = O P (1). will be sufficien. Theorem 6 of Kokoszka and Leipus (000) verifies (.7) for he change-poin esimaor in (.6). Assume again ha ω m > 0, α m,i 0 for all i = 1,..., q and β m, j 0 for all j = 1,..., p, for m = 1,. The es saisic is given by Lee e al. (015) as where T n,1 = 1ˆτ ˆk 1 max ϵ 1 1 k ˆk kˆk and ˆτ 1 = 1ˆk ˆ 4 1 ˆk ˆ ˆ M n = max { T n,1, T n, }, (.7), T n, = 1ˆτ, ˆτ = 1 n ˆk ( ) 1 n ˆk max ˆk<k n 4 1 n ˆk ϵ k ˆk n ˆk Le x 1, = x, = 1,..., ˆk, and x, = x, = ˆk + 1,..., n. We define recursively ϵ. ϵ, σ 1, (θ 1) = ω 1 + σ, (θ ) = ω + α 1,i x 1, i + β 1, j σ 1, j (θ 1), α,i x, i + β, j σ, j (θ ), = 1,..., ˆk, = ˆk + 1,..., n wih given fixed iniial vecors ( x 1,0,..., x 1,1 q ), ( x,ˆk,..., x,ˆk q+1 ), ( σ 1,0,..., σ 1,1 p ), and ( σ,ˆk,..., σ ) and parameer vecors θ,ˆk p+1 m = (ω m, α m,1,..., α m,q, β m,1,..., β m,p ), m = 1,. By using hese equaions, Lee e al. (015) hen calculaed {, = 1,..., n} by x ), = 1,..., ˆk, σ 1, (ˆθ 1 = x (.8) ), = ˆk + 1,..., n. σ, (ˆθ The Gaussian QMLEs for θ 1 and θ are given by ˆθ 1 = argmin θ 1 Θ ˆ l (θ 1 ), ˆθ = argmin θ Θ l (θ ), (.9)

6 500 Yu Wang, Cheolwoo Park, Taewook Lee where Θ is a parameer space belongs o [c 1, c ] p+q+1 for some 0 < c 1 < c <, and l (θ m ) = r σ (θ m) + log σ (θ m ), σ (θ m ) = σ 1, (θ m)i ( ˆk ) + σ, (θ m)i ( > ˆk ), m = 1,. Similarly, hey also proved ha if cerain condiions hold, for he VS-1 model (.5) under H (1) 0, as n, { d M n max sup B 1 (u), 0 u 1 } sup B (u), where B 1 (u) and B (u) are independen sandard Brownian bridges. Afer calculaing M n, we can hen use equaion (.3) o calculae he p-value of he es. Noe ha we consider only he case when he single volailiy shif is known in priori. The binary segmenaion mehod (Bai, 1997) can be used o esimae he number of change poins if he number of volailiy shif is unknown. 0 u 1 3. Boosrap Tess for Volailiy Shifs agains LRD The use of he asympoic es saisics for he finie sample migh give size disorion and lower power. Efron (1979) inroduced he boosrap mehod o sudy he disribuions of esimaors (sampling disribuion) and es saisics by resampling he finie sample daa. In his paper, we compare he size and power of he ess presened in Secion using he asympoic and boosrap-based mehods. If he daa {x 1,..., x n } are independen observaions, a commonly used boosrap procedure is o sample m observaions, {x1,..., x m}, from he original daa wih replacemen, and calculae he esimaor or es saisic based on he sampled observaions, say. This procedure is repeaed B imes, and {1,..., B } is obained. Then he disribuion of boosrap replicaion can approximae he disribuion of he esimaor or es saisic under he null hypohesis. If one performs boosrap for he es saisic, hen he righ-sided boosrap p-value can be calculaed as ˆp = 1 B B I ( b ), b=1 where I( ) is he indicaor funcion and is he es saisic evaluaed from he original daa. Time series daa are auocorrelaed, and herefore are no independen o each oher. Simply resampling he original daa migh yield even worse resuls han he asympoic esimaion does. One mehod o deal wih such a ime series problem is o boosrap he esimaed i.i.d. innovaion series { }, and hen generae recursively a boosrap sample {x }. The algorihm for he boosrap residual-based CUSUM es for no volailiy shif agains LRD is: 1. Obain he esimaed GARCH model parameers and calculae he innovaion series { } according o equaion (.1) and compue he es saisic T n by equaion (.).. Consruc he boosrap innovaion series { ϵ } wih size equal o he original sample size n, by drawing independenly wih replacemen from { }.

7 Boosrap-Based Tes for Volailiy Shifs in GARCH agains Long-Range Dependence Generae recursively a boosrap sample {x } using he esimaed parameers and he boosrap innovaion series from Seps 1 and, respecively as: x = σ, σ = ˆω + ˆα i x i + wih fixed iniial values for x0,..., x 1 q and σ 0,..., σ 1 p. ˆβ j σ j, 4. Compue he boosrap es saisic T n from he boosrap sample {x } as follows: T n = 1 max nˆτ 1 k n k n, where { } is defined hrough he boosrap sample {x } by = x / σ for = 1,,..., n. Here, ˆθ = ( ˆω, ˆα 1,..., ˆα q, ˆβ 1,..., ˆβ p) is he esimaed parameer vecor of GARCH models for {x } and σ = ˆω + ˆα i x i + wih fixed iniial values for x0,..., x 1 q and σ 0,..., σ 1 p. 5. Repea Seps 4 B imes o obain T n 1,..., T n B. 6. Esimae he boosrap p-value by where I( ) is he indicaor funcion. ˆp = 1 B B I ( ) T n b T n, b=1 ˆβ j σ j For he es saisic M n for one volailiy shif agains LRD, he boosrap procedure is given as: 1. Ge he esimaed change poin ˆk by equaion (.6) and obain he esimaed parameers of GARCH models as defined by equaion (.9).. Calculae innovaion series { } by equaion (.8) and compue he es saisic M n as in equaion (.7). 3. Consruc he boosrap innovaion series { ϵ } wih size equal o he original sample size, by drawing independenly wih replacemen from { }. 4. Generae recursively a boosrap sample {x } by using he esimaed change poin ˆk, esimaed parameers ˆθ m = ( ˆω m, ˆα m,1,..., ˆα m,q, ˆβ m,1,..., ˆβ m,p ) for m = 1, of GARCH models and he boosrap innovaion series { ϵ } from Seps 1 and 3, respecively as: x = σ ϵ,

8 50 Yu Wang, Cheolwoo Park, Taewook Lee where σ = ˆω 1 + ˆω + ˆα 1,i x i + ˆα,i x i + ˆβ 1, j σ 1, j, ˆβ, j σ, j, = 1,..., ˆk, = ˆk + 1,..., n wih fixed iniial values for x0,..., x 1 q and σ 0,..., σ 1 p. 5. Compue he boosrap es saisic M n from he boosrap sample {x } as: M n = max { T n,1, T n,}, where T n,1 = 1ˆτ 1 ˆk 1 max 1 k ˆk ϵ kˆk ˆ, T n, = 1ˆτ ( n ˆk ) 1 max ˆk<k n ϵ k ˆk n ˆk ϵ, and ˆτ 1 = 1ˆk ˆ ϵ 4 1 ˆk ˆ, ˆτ = 1 n ˆk ϵ 4 1 n ˆk ϵ, where { } is defined hrough he boosrap sample {x } by = x / σ for = 1,,..., n. Here, ˆθ m = ( ˆω m, ˆα m,1,..., ˆα m,q, ˆβ m,1,..., ˆβ m,p) for m = 1, is he esimaed parameer vecor of GARCH models, obained by (.9) based on {x } and σ = ˆω 1 + ˆω + ˆα 1,i x i + ˆα,i x i + ˆβ 1, j σ 1, j, wih fixed iniial values for x0,..., x 1 q and σ 0,..., σ 1 p. 6. Repea Seps 3 5 B imes o obain M n 1,..., M n B. 7. Esimae he boosrap p-value by where I( ) is he indicaor funcion. ˆp = 1 B = 1,..., ˆk, ˆβ, j σ, j, = ˆk + 1,..., n, B I ( ) Mn b M n, b=1

9 Boosrap-Based Tes for Volailiy Shifs in GARCH agains Long-Range Dependence Simulaion Sudy For empirical sizes of ess for volailiy shifs agains LRD, we consider he univariae GARCH(1, 1) models as x = σ ϵ, σ = ω + αx 1 + βσ 1, where {ϵ } Z is he he innovaion series following specific disribuions. In he es for no volailiy shifs agains LRD, we consider wo ypes of innovaion disribuions: i.i.d. sandard normal and - disribuion wih degrees of freedom 5, denoed by (5), following he seing from Lee e al. (015). We also follow he 6 ses of GARCH(1, 1) parameers (ω, α, β) wih α + β ranging from 0.5 o very close o 1. In he es for one volailiy shif agains LRD, i.i.d. sandard normal is only considered as he innovaion series. The GARCH(1, 1) parameers shif a he midpoin of he samples from (ω, α, β) = (0.1, 0.1, 0.8) o 4 oher ses wih α + β ranging from 0.5 o very close o 1. For he empirical powers, we consider only he models, proposed by Baillie e al. (1996), given as x = σ ϵ, (1 βl)σ = ω + ( (1 βl) + (1 ϕl)(1 L) d) x. Here, L is he backshif operaor. In he models, he power d of he fracional differencing operaor, (1 L) d, is allowed o be a non-ineger and can be expanded as (1 L) d = π i L i, i=0 where π i = 0 k i k 1 d, k and d ( 1/, 1/). For d > 0, he process is long range dependen. In he simulaion of empirical powers, we se parameers as (ω, β, ϕ) = (0.6, 0.1, 0.) and (0., 0., 0.), each wih hree differen values of d = 0.5, 0.35, All resuls are based on 1000 replicaions on sample sizes 500, 750, 1000 and 000 wih significan level 5%, and he boosrap replicaion is se o be B = 100. Table 1 presens he empirical sizes of he es for no volailiy shif agains LRD under differen sample sizes and differen ypes of innovaions. I conains he asympoic (columns under Asympoic ) and boosrap (columns under Boosrap ) sizes for he es saisic T n in (.) and he firs boosrap algorihm in Secion 3, respecively. According o Table 1, he boosrap mehod yields beer sizes wihin accepable range han he asympoic ones, even α + β are very close o 1. This difference is more eviden when he innovaion follows a heavy-ailed disribuion; in addiion, he boosrap size is closer o he nominal 5% significance level when he sample size is small, unlike asympoic ones. When he sample sizes ge larger, he boosrap size ges similar o he asympoic size when he innovaion is normally disribued, bu sill beer han he asympoic ones when he innovaion follows heavy-ailed disribuion. Hence, he boosrap mehod migh be a good choice

10 504 Yu Wang, Cheolwoo Park, Taewook Lee Table 1: Empirical sizes of es for no volailiy shifs under es saisic T n n = 500 n = 750 n = 1000 n = 000 (ω, α, β) N(0, 1) (5) Asympoic Boosrap Asympoic Boosrap (0.1, 0.1, 0.8) (0.1, 0.1, 0.6) (0.1, 0.1, 0.4) (0.1, 0., 0.6) (0.3, 0.1, 0.8) (0.3, 0.1, 0.89) (0.1, 0.1, 0.8) (0.1, 0.1, 0.6) (0.1, 0.1, 0.4) (0.1, 0., 0.6) (0.3, 0.1, 0.8) (0.3, 0.1, 0.89) (0.1, 0.1, 0.8) (0.1, 0.1, 0.6) (0.1, 0.1, 0.4) (0.1, 0., 0.6) (0.3, 0.1, 0.8) (0.3, 0.1, 0.89) (0.1, 0.1, 0.8) (0.1, 0.1, 0.6) (0.1, 0.1, 0.4) (0.1, 0., 0.6) (0.3, 0.1, 0.8) (0.3, 0.1, 0.89) Table : Empirical powers of es for no volailiy shifs under es saisic T n (ω = 0.6, β = 0.1) (ω = 0., β = 0.) (ω = 0.6, β = 0.1) (ω = 0., β = 0.) d n = 500 n = 750 Asympoic Boosrap Asympoic Boosrap d n = 1000 n = 000 Asympoic Boosrap Asympoic Boosrap FIGARCH = fracionally inegraed generalized auoregressive condiionally heeroskedasic when eiher daa show he evidence of heavy ailed innovaion or he sample size is small. Table summarizes he power comparison in he boosrap mehod ha is more powerful han he asympoic one for he cases wih small samples; however, he power of boh mehods increases dramaically as he sample size increases and he advanage of he boosrap diminishes. Table 3 presens he empirical sizes of he es for one volailiy shif agains LRD. The resuls show ha he boosrap size is more assuring under he small sample sizes for es M n. The asympoic size ges close o he nominal significance level as he sample size increases. Table 4 shows he empirical

11 Boosrap-Based Tes for Volailiy Shifs in GARCH agains Long-Range Dependence 505 Table 3: Empirical sizes of es for one volailiy shif under es saisic M n (0.1, 0.1, 0.8) n = 500 n = 750 o (ω, α, β ) Asympoic Boosrap Asympoic Boosrap (0.1, 0.1, 0.6) (0.1, 0.1, 0.4) (0.1, 0., 0.6) (0.3, 0.1, 0.8) (0.1, 0.1, 0.8) n = 1000 n = 000 o (ω, α, β ) Asympoic Boosrap Asympoic Boosrap (0.1, 0.1, 0.6) (0.1, 0.1, 0.4) (0.1, 0., 0.6) (0.3, 0.1, 0.8) Table 4: Empirical powers of es for one volailiy shif under es saisic M n (ω = 0.6, β = 0.1) (ω = 0., β = 0.) (ω = 0.6, β = 0.1) (ω = 0., β = 0.) d n = 500 n = 750 Asympoic Boosrap Asympoic Boosrap d n = 1000 n = 000 Asympoic Boosrap Asympoic Boosrap FIGARCH = fracionally inegraed generalized auoregressive condiionally heeroskedasic power of es for one volailiy shif. Overall, he power increases significanly when he sample size increases and he long range dependence of he series increases; in addiion, he boosrap mehod yields superior powers han asympoic mehod when he sample size is small. This difference is barely observed as he sample size increases. In conclusion, boosrap ess are shown o be more preferable for one volailiy shif agains LRD when sample sizes are small. 5. Conclusion Lee e al. (015) proposed he es saisics for volailiy shifs agains LRD based on he CUSUM saisic using residual series from he fied GARCH models. In his paper, we exended i o he boosrap mehods and compared he empirical sizes and powers of he es saisics of Lee e al. (015) and our boosrap ess. We concluded ha our boosrap mehods are more promising han he asympoic ones for volailiy shifs agains LRD in small sample sizes, since empirical sizes are closer o he nominal significan level and empirical powers are higher. References Andreou, E. and Ghysels, E. (00). Deecing muliple breaks in financial marke volailiy dynamics, Journal of Applied Economerics, 17, Bai, J. (1997). Esimaion of a change poin in muliple regression models, Review of Economics and

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