A general asymptotic theory for time-series models

Transcription

1 97 Statistica Neerladica 200 Vol. 64, r., pp. 97 doi:0./j x A geeral asymptotic theory for time-series models Shiqig Lig* Departmet of Mathematics, Hog Kog Uiversity of Sciece ad Techology, Hog Kog, Chia Michael McAleer Ecoometric Istitute, Erasmus School of Ecoomics, Erasmus Uiversity Rotterdam ad Tiberge Istitute, The Netherlads ad Ceter for Iteratioal Research o the Japaese Ecoomy CIRJE Faculty of Ecoomics, Uiversity of Tokyo, Tokyo, Japa This paper develops a geeral asymptotic theory for the estimatio of strictly statioary ad ergodic time series models. Uder simple coditios that are straightforward to check, we establish the strog cosistecy, the rate of strog covergece ad the asymptotic ormality of a geeral class of estimators that icludes LSE, MLE ad some M-type estimators. As a applicatio, we verify the assumptios for the log-memory fractioal ARIMA model. Other examples iclude the GARCH, model, radom coefficiet AR model ad the threshold MA model. Keywords ad Phrases: asymptotic ormality, estimatio, rate of strog covergece, strog cosistecy, time-series models. Itroductio The three mai results that ca be used for the asymptotic theory of the estimators i time-series models are Basawa, Feig ad Heyde 976, Amemiya 985 ad Tjøstheim 986. While ot specific to time-series models, Basawa et al. 976 ad Amemiya 985 provide the coditio for the weak cosistecy of the estimated parameters. For asymptotic ormality, the result i Basawa et al. 976 requires the expectatio of the third derivatives of the objective fuctio OF. The coditio i Amemiya 985 does ot give a specific method for the covergece of the sample iformatio matrix to prove asymptotic ormality. The result i Tjøstheim 986 holds for strictly statioary ad ergodic time-series models, ad implies that there exists a sequece of strogly cosistet estimators to imize the OF. However, this sequece of estimators may ot be the global *malig@ust.hk michael.mcaleer@gmail.com 200 The Authors. Joural compilatio 200 VVS. Published by Blackwell Publishig, 9600 Garsigto Road, Oxford OX4 2DQ, UK ad 350 Mai Street, Malde, MA 0248, USA.

2 98 S. Lig ad M. McAleer imizer of the OF. His result also requires the expectatio of the third derivatives of the OF, but the third derivatives of the OF ca be extremely complicated i some models, as i the case of the likelihood fuctio for ARMA-GARCH models. Jeatheau 998 also gives the coditio for strog cosistecy of the imum likelihood estimator MLE for a class of GARCH models. However, the results i each of these papers do ot discuss the iitial value problem, which eeds to be addressed for each idividual model. As may time-series models have bee developed i the last two decades, a uified ad simple asymptotic theory of estimatio for time-series models should have wide applicability. This paper establishes a geeral asymptotic theory for the estimatio of strictly statioary ad ergodic time-series models. The estimators, icludig LSE, MLE ad some M-type estimators except for LAD estimator, amog others, are the global imizers of the respective OFs. We establish the strog cosistecy, the rate of strog covergece ad the asymptotic ormality of the estimated parameters. The rate of strog covergece of the estimated parameters has ot previously appeared i the literature i a geeral settig. The coditios, icludig the iitial coditios, are simple ad easy to check, ad third derivatives are ot required. Some related refereces are Huber 967 ad Pfazagl 969. This paper proceeds as follows. Sectio 2 presets the model ad the mai results. Sectio 3 examies the log-memory FARIMA model. The proofs are give i Appedices A ad B. 2 Model ad mai results Assume that the real p vector time series {y t : t = 0, ±,...} is F t measurable, strictly statioary ad ergodic, ad its coditioal distributio is give by y t F t Gθ, Y t, where F t is the σ-field geerated by {y t, y t,...}, Y t = y t,..., y t p + or Y t = y t, y t,..., ad θ is a m ukow parameter vector. The structure of the time series {y t } is characterized by the distributio G ad the parameter θ. We assume that the parameter space is a compact subset of R m, ad the true value θ 0 of θ is a iterior poit i. We use the followig OF with the iitial value Y 0 to estimate θ 0 : L θ = l t θ, where l t θ = ly t, θ is a measurable fuctio with respect to Y t ad is almost surely a.s. ad cotiuously twice differetiable i terms of θ. Deote D t θ = l t θ/ θ, P t θ = 2 l t θ/ θ θ, Σ = E[P t θ 0 ] ad Ω = E[D t θ 0 D tθ 0 ]. Let V 0 η = {θ : θ θ 0 < η}. The followig assumptio is made. 200 The Authors. Joural compilatio 200 VVS.

3 Asymptotic theory for time-series models 99 Assumptio. i E θ [l t θ] <, ad E[l t θ] has a uique imizer at θ 0 ; ii D t θ 0 is a martigale differece i terms of F t with 0 < Ω < ; iii Σ > 0 ad E θ V0 η P t θ < for some η > 0. Whe model reduces to the class of models: y t = f θ, Y t + η t hθ, Yt with {η t } beig idepedet ad idetically distributed i.i.d. with mea zero ad Eη 2 t =, ad the Quasi-MLE is used, we have l t θ = [ ] log h t θ + ε2 t θ, 2 h t θ where f λ, Y t ad hλ, Y t > 0 are measurable fuctios i terms of Y t. I this case, Assumptio i is esured by the coditios: a E θ [ε 2 t θ h t θ] < ad E θ log h t θ < ad b [ε t θ, h t θ] = [ε t θ 0, h t θ 0 ] a.s. if ad oly if θ = θ 0. See Jeatheau 998 ad Lig ad McAleer Whe the dimesio of the iitial value Y 0 is ifiite, it eeds to be replaced by some costat Ỹ 0. We deote l t θ with the iitial value Ỹ 0 by l t θ. Similarly, defie D t θ ad P t θ. The iitial coditio is give as follows. Assumptio 2. For some ν > 0, it follows that i E l t θ l t θ = O t ; ν ii E D t θ 0 D t θ 0 = O t ad /2 + ν iii E P t θ P t θ = O t. ν The decay rates i Assumptio 2.2 are very low ad are satisfied by most of timeseries models. For log-memory time series, Assumptio 2ii ca be replaced by: Assumptio 2.2. ii. For ay ɛ > 0, lim P [D t θ 0 D t θ 0 ] l > ɛ = 0. The correspodig OF is modified as L θ = l t θ. Whe the dimesio of Y 0 is fiite, Assumptio 2 is redudat. I what follows, L deotes covergece i distributio. We ow state our mai result as follows: 200 The Authors. Joural compilatio 200 VVS.

4 00 S. Lig ad M. McAleer Theorem. Let ˆθ = arg Lθ. a If Assumptios i ad 2i hold, the ˆθ θ 0 a.s. b Furthermore, if Assumptios ii, iii ad 2ii, iii hold, the [ log ] /2 log ˆθ = θ 0 + O a.s. ad ˆθ θ 0 L N0, Σ ΩΣ. Remark. For a, we oly eed l t θ to be cotiuous i terms of θ a.s., while twice differetiability is redudat. ˆθ ca be LSE, MLE ad some M-type estimators, amog others. May oliear time-series models i Tog 990 satisfy Assumptios ad 2, such as TAR, biliear ARMA, GARCH ad radom coefficiet AR models. The compactess of is ot a serious restrictio i practice as the true value θ 0 is a iterior poit i the parameter space ad we ca always get a compact to iclude it. Compared with the assumptios metioed i Sectio, our assumptios are simple, clear ad easy to check i practice. Σ ad Ω ca be estimated, respectively, by ˆΣ = P t ˆθ ad ˆΩ = D t ˆθ D t ˆθ. By Lemma A2, ˆΣ = Σ + o a.s. As P t θ /2 is strictly statioary ad has a fiite variace, we kow that P t θ /2 / = o p. t By Taylor s expasio, we have D t ˆθ D t θ 0 / [ P tθ /] ˆθ θ 0 = o p uiformly i t. Furthermore, by Assumptio 2ii, we kow that D t ˆθ D t θ 0 / = o p uiformly i t. By Taylor s expasio ad Lemma A2, we have Thus, D t ˆθ / = P t ˆθ ˆθ θ 0 = O p. [ D t ˆθ D t θ 0 ] D t ˆθ ] = o p. Fially, by the ergodic theorem, we ca see that ˆΩ = Ω + o p. Thus, uder Assumptios ad 2 ˆΣ ad ˆΩ are cosistet estimators of Σ ad Ω respectively. 200 The Authors. Joural compilatio 200 VVS.

5 Asymptotic theory for time-series models 0 Example. Cosider the GARCH, model: y t = η t ht ad h t = α 0 + αy 2 t + βh t, where η t i.i.d. N0,, α 0 > 0, α > 0 ad β > 0. Assume that Elβ + αη 2 t < 0. Let θ = α 0, α, β. Whe usig MLE to estimate θ 0, we take l t θ = logh t θ y2 t h t θ, where h t θ = α 0 + αyt 2 + βh t θ,,..., ad y 0 = 0 ad h 0 =. From the proof of theorem 2. i Fracq ad Zakoa 2004, we see that Assumptios i ad 2i hold. From the proof of Theorem 2.2 i Fracq ad Zakoia 2004, we kow that Assumptios ii, iii ad 2ii, iii hold, see also Lee ad Hase 994 ad Lig ad McAleer We do ot eed to study the third derivative of l t θ as doe i these papers. Example 2. Cosider the radom coefficiet AR model: y t = + ψ t y t + ε t, where {ψ t } ad {ε t } are i.i.d. sequeces with zero mea ad variace α > 0 ad σ 2 > 0, ad they are mutually idepedet. Assume E l + ψ t < 0 ad let θ =, α, σ 2. Whe we use QMLE to estimate θ 0, we take l t θ = 2 logσ2 + αyt 2 y t y t 2 2σ 2 + αyt 2. By exactly followig the proof of Lemmas A., A.2 ad A.3i i Lig 2004, we ca show that Assumptio holds. Assumptio 2 holds automatically. Example 3. Cosider the first-order threshold MA [TMA ] model: y t = [ + ψiy t r]ε t + ε t, where {ε t } is a sequece of i.i.d. radom variables, with mea zero, variace 0 < σ 2 < ad a desity fuctio f. Assume that <, + ψ < ad ψ x xf x <. This assumptio esures that the TMA model is strictly statioary ad ergodic, ad ivertible, see Lig, Tog ad Li Let θ =, ψ. Whe usig the Coditioal LSE to estimate θ 0, we take l t θ = ε 2 t θ, where ε t θ = y t [ + ψiy t r]ε t θ,,..., ad ε t θ = 0ast 0. Furthermore, assume that the delay parameter r is kow ad Eε 4 t <. By the very mior modificatio of Lemmas i Lig ad Tog 2005, we ca show that Assumptio holds. Similarly, a mior modificatio of Lemma 6.6 i Lig ad Tog 2005 shows that Assumptio 2 holds. We should metio that this is a ew 200 The Authors. Joural compilatio 200 VVS.

6 02 S. Lig ad M. McAleer result for the TMA model. Whe r is ukow, the asymptotic theory o the TMA model remais ope. 3 Applicatio to log-memory FARIMA models The process {y t } is said to follow the log-memory LM-ARFIMA model if where B B d y t = ψbε t, 2 B = B d = p i B i, ψb = + i = q ψ i B i, i = a k B k with a k = k d!/k! d!, k = 0 B is the backward-shift operator, ad {ε t } is i.i.d. with zero mea ad variace 0 < σ 2 <. θ = d,,..., p, ψ,..., ψ q ad its true value is θ 0. We assume that the parameter space is a compact subset of R p + q +, θ 0 is a iterior poit i, ad Assumptio holds. Assumptio 3. d 0, /2, all the roots of B ad ψb are outside the uit circle, p = 0, ψ q = 0, ad B ad ψb have o commo root. Give {y,..., y }, we cosider the coditioal LSE of θ 0, which is defied as ˆθ = argmi ε2 t θ, where ε t θ isε t θ = ψ B B B d y t, with iitial value Ỹ 0. We have the followig results. Theorem 2. If Assumptio 3 holds, the [ log a ˆθ /2 ] = log θ 0 + O a.s., b ˆθ [ θ 0 L N 0, σ 2 E εt θ 0 θ ε t θ 0 θ ]. Remark 2. Model 2 has the log-memory property ad has bee widely applied i hydrology ad ecoomics. Some related refereces are Grager ad Joyeux 980, Hoskig 98, Li ad McLeod 986, Robiso 994 ad Bera 995, amog others. Whe ε t follows the GARCH model, model 2 was studied by Baillie 996, Lig ad Li 997 ad Lig However, the paper is the first to provide the rate of strog covergece of ˆθ, as i a. From the proof i Appedix B, we ca see that the iitial coditio is crucial i this developmet. 200 The Authors. Joural compilatio 200 VVS.

7 Asymptotic theory for time-series models 03 Ackowledgemets The first author thaks the Hog Kog Research Grats Commissio for Grat #HKUST60607 ad The secod author is most grateful for the fiacial port of the Australia Research Coucil ad the Natioal Sciece Coucil, Taiwa. Appedix A: Proof of Theorem Lemma A. If Assumptios i ad 2i hold, the for ay η > 0, lim P l Proof. Let θ θ 0 η [ l t θ l t θ 0 ] 0 = 0. V η = { θ : θ θ η} ad X t η = l t θ l t θ. θ V η By Assumptio i, EX t η 0 as η 0. Thus, for ay ɛ > 0, there is η > 0 such that EX t η < ɛ/2. As X t η is strictly statioary ad ergodic, by Lemma i Chow ad Teicher 978, p. 66 ad the ergodic theorem, for ay ɛ > 0, we have P [X t η EX t η] ɛ < ɛ, 2 as l is large eough. Thus, for ay ɛ, ɛ > 0, there exists a costat η > 0 such that P X t η ɛ P [X t η EX t η] ɛ < ɛ.3 2 By Assumptio i ad the ergodic theorem, for each θ ad ay ɛ > 0, lim l P [l t θ El t θ] ɛ = 0. 4 As is compact, we ca choose a collectio of balls of radius Δ > 0 coverig, ad the umber of such balls is a fiite iteger N. Itheith ball, we take a poit ξ i ad deote this ball by V ξ i. For ay ɛ > 0, 200 The Authors. Joural compilatio 200 VVS.

8 04 S. Lig ad M. McAleer P P [l t θ El t θ] ɛ j N θ V ξ j + P + P P + + j N θ V ξ j j N [l t θ l t ξ j ] ɛ 3 E[l t θ l t ξ j ] ɛ 3 ξ i θ V ξ j N P j = N P j = θ V ξ j [l t ξ j El t ξ j ] ɛ 3 [l t θ l t ξ j ] ɛ 3 E[lt θ l t ξ j ] ɛ 3 [l t ξ j El t ξ j ] ɛ < ɛ, 3 as l is large eough ad Δ is small eough, where the last iequality holds by Equatios 3 ad 4 ad the uiform cotiuity of El t θ. As E[l t θ] has a uique imum at θ 0, is compact, ad El t θ is cotiuous, there exists a costat c > 0, such that E[l t θ l t θ 0 ] c, 6 θ θ 0 > η 5 for ay η > 0. By Equatios 5 ad 6, it follows that P θ θ 0 η = P θ θ 0 η { } [l t θ l t θ 0 ] + c > 0 2 { [l t θ El t θ] [l t θ 0 El t θ 0 ] + [El t θ El t θ 0 ] + c } > 0 { 2 P 2 [l t θ El t θ] } c + c > 0 2 { P [l t θ El t θ] } > c 0, 4 7 as l. By Assumptio 2i ad Markov s iequality, it follows that 200 The Authors. Joural compilatio 200 VVS.

9 Asymptotic theory for time-series models 05 lim l P lim l P lim l ɛl 0.5ν t 0.5ν E t 0.5ν l t θ l t θ > ɛ l t θ l t θ > ɛl 0.5ν l t θ l t θ = lim l O ɛl 0.5ν for ay ɛ > 0. By Equatios 7 ad 8, we have P [ l t θ l t θ 0 ] + c 4 > 0 θ θ 0 η P θ θ 0 η + 2 P + P θ θ 0 η [l t θ l t θ 0 ] l t θ l t θ + c 4 > 0 [l t θ l t θ 0 ] + c 2 > 0 l t θ l t θ > c 0, 8 as l. By this equatio, we ca see that the coclusio holds. = 0, t + 0.5ν 8 Lemma A2. If Assumptios iiiad 2iii hold, the for ay ɛ > 0, lim P l V 0 η [ P t θ Σ] ɛ = 0. Proof. Let X t η = V0 η P t θ P t θ 0. By Assumptio iii, as η is small eough, EX t η < ɛ/4. As {X t η} is strictly statioary ad ergodic, by Lemma i Chow ad Teicher 978, p. 66 ad the ergodic theorem, we have lim P l X t η ɛ lim l P X t η EX t η ɛ = 0, 2 for ay ɛ > 0. By Assumptio iii ad the ergodic theorem, it follows that lim P l [P t θ 0 Σ] ɛ = 0, for ay ɛ > 0. By the precedig two equatios, there is η > 0 such that 200 The Authors. Joural compilatio 200 VVS.

10 06 S. Lig ad M. McAleer lim P l V 0 η [P t θ Σ] ɛ = 0. 9 Let X t = V0 η P t θ P t θ. By Assumptio 2iii, we ca show that P X t / ɛ 0asl, for ay ɛ > 0. X t. Furthermore, by Equatio 9, the co- Note that [ P t θ P t θ] V 0 η clusio holds. Proof of Theorem 2.. By Lemma A, for ay ɛ > 0, we have lim P ˆθ θ 0 > ɛ l { } = lim P ˆθ θ 0 > ɛ, [ l t ˆθ l t θ 0 ] 0 l { } lim P [ l t θ l t θ 0 ] 0 = 0. l θ θ 0 > ɛ Thus, a holds. Applyig Taylor s expasio to l t θ / θ ad usig Lemma A2, [ ] ˆθ θ 0 = P t ˆθ D t θ 0 = [Σ + o] D t θ 0 a.s., where ˆθ lies betwee ˆθ ad θ 0 ad ˆθ θ 0 a.s. By Assumptio 2ii ad usig a similar method as for Equatio 8, we ca show that lim P [ D t θ 0 D t θ 0 ] l ɛ = 0, for ay ɛ > 0. Thus, we have ˆθ θ 0 = [Σ + o] D t θ 0 / + o /2 a.s. By the law of iterated logarithm, we ca claim that ˆθ θ 0 = Olog log / /2 a.s. By Assumptio ii ad the cetral limit theorem, b holds. 200 The Authors. Joural compilatio 200 VVS.

11 Asymptotic theory for time-series models 07 Appedix B: Proof of Theorem 3. Proof. We verify Assumptios ad 2 with l t θ = εt 2 θ. For simplicity, we oly cosider the case with p = q = 0, while the geeral case ca be similarly verified. First, Assumptio 3 esures that {y t } is strictly statioary ad ergodic with Eyt 2 <, ad the followig expasios hold: y t = c 0i ε t i ad ε t θ = B d y t = a i θy t i, 0 i = 0 where c 00 = a 0 θ =, c 0i = Oi + d 0 ad a i θ = Oi d. As is compact, there are d ad d such that 0 < d d d < 0.5. Thus, we have ε t θ = i = 0 i = 0 a i θ = Oi d, θ ad hece it follows that a i θy t i y t + O i = i + d y t i. By the Cauchy Schwarz iequality, we have E ε t θ 2 <. It is ot difficult to show that E[εt 2 θ] has a uique imum o. Thus, Assumptio i holds, ad where D t θ = 2ε t θ ε tθ ad P t θ = 2 ε t θ/ = log B B d y t = 2 ε t θ/ 2 = log 2 B B d y t = [ ] 2 ε t θ + 2ε t θ 2 ε t θ 2, a i θy t i i = a 2i θy t i, i = ad with a ji θ = Oi d as j =, 2. θ Usig these, it is straightforward to show that Assumptio ii ad iii holds. We ext cosider Assumptio 2. For simplicity, let Ỹ 0 = 0, 0,... By Equatio 0, E[ [ ] 2 ε t θ ε t θ ] 2 = E a i θy t i CE 2 i + d y t i = Ot 2d. It is readily show that E ε 2 t θ is bouded uiformly i t. Thus, by the Cauchy Schwarz iequality, we have E ε 2 t θ ε 2 t θ {E[ ε t θ + ε t θ ] 2 E[ ε t θ ε t θ ] 2 } /2 = Ot d, so that Assumptio 2i holds. Similarly, we ca show that Assumptio 2iii holds. 200 The Authors. Joural compilatio 200 VVS.

12 08 S. Lig ad M. McAleer We ow verify Assumptio 2ii. Deote A t = ε t θ 0 ε t θ 0 = a i θ 0 y t i, where A t = ε tθ 0 A 2t = ε tθ 0 t v t = ε t i /i. i = ε tθ 0, v t = i ε t i, We first make the followig decompositio: D t θ 0 D t θ 0 = 2ε t θ 0 ε tθ 0 2 ε t θ 0 ε tθ 0 = 2ε t θ 0 A t + 2 ε tθ 0 A t = 2ε t θ 0 A t + 2A t v t + 2A t A 2t 2A t A t. As Ey t y t + r = O r + 2d 0, we have EA 2 t = ai 2 θ 0 Eyt i a i θ 0 a i + r θ 0 Ey t i y t i r = O O O [ [ [ r = i 2 + d r = i 2 + d i 2 + d i + d 0 i 2 Usig a similar method, we ca show that i + d 0 i + r + d 0 r 2d 0 0 ] ] i + x + d dx 0 x 2d 0 + z + d 0 z 2d 0 dz ] = Ot. 2 EA 2 it = Ot ad EA t A it = Ot as i =, 2. 3 As i the proof of Equatio 8, usig Equatio 3, we ca show that lim P A t A it > ɛ = 0, as i =, 2. 4 l We ext show that ε t A t / = o which is equivalet to 200 The Authors. Joural compilatio 200 VVS. a.s. as,

13 Asymptotic theory for time-series models 09 lim l P ε t A t > ɛ = 0. 5 By the Kroecker lemma i Hall ad Heyde 980, p. 3, it is sufficiet to show that S k = k By Equatio 3, it follows that E k t = s t A t ε t coverges a.s. 6 k A t ε 2 t = O t t = s k = t 2 O t = s α t + ν, 7 for ay iteger 0 < s k ad some α >, where O holds uiformly i k ad s. Cosider the subsequece {S 2 k : k = 0,,...}. By Equatio 7, we have E S 2 k + S 2 k O 2 k + t = 2 k + t + ɛ O 2 ɛk, for some ɛ > 0. By this equatio ad the mootoe covergece theorem, we have E lim k = 0 S 2 k + S 2 k = lim E k = 0 S 2 k + S 2 k O Thus, k = 0 S 2 S k + 2k coverges a.s. as, ad hece lim S 2 = X + + lim k = 0 2 ɛk <. S 2 k + S 2 k coverges a.s. 8 k = 0 By Equatio 7 ad theorem 2.2 i Billigsley 968, it follows that, for ay Δ > 0, P S S 2 k Δ O 2 k < 2 k + 2 k + t = 2 k + α = t + ν O 2 ɛ, k for some ɛ > 0. By the Borel Cateli lemma, we ca claim that 2 k < 2 k + S S 2 k 0 a.s. as k. 9 By Lemma 2.3. i Stout 974 ad Equatios 8 ad 9, we kow that Equatio 6 holds. 200 The Authors. Joural compilatio 200 VVS.

14 0 S. Lig ad M. McAleer Note that v s is idepedet of A t as s 2, so that E k A t ν 2 t = E t t = s k t = s t = s k = O t = s k EA t A t Eν t ν t t t 2 k = O t + /2 t = s α, t + ν for ay iteger 0 < s k, where O holds uiformly i k ad s. Usig this equatio ad a similar method as for Equatio 5, we ca show that lim P v t A t > ɛ = l By Equatios, 4, 5 ad 20 we ca show that Assumptio 2.2ii holds. Refereces Amemiya, T. 985, Advaced ecoometrics, Harvard Uiversity Press, Cambridge. Baillie, R. T. 996, Log memory processes ad fractioal itegratio i ecoometrics, Joural of Ecoometrics 73, Basawa, I. V., P. D. Feig ad C. C. Heyde 976, Asymptotic properties of imum likelihood estimators for stochastic processes, Sakhya A 38, Bera, J. 995, Maximum likelihood estimatio of the differecig parameter for ivertible short ad log memory autoregressive itegrated movig average models, Joural of the Royal Statistical Society B 57, Billigsley, P. 968, Covergece of probability measures, Wiley, New York. Chow, Y. S. ad H. Teicher 978, Probability theory. Idepedece, iterchageability, Martigales, Spriger-Verlag, New York ad Heidelberg. Fracq, C. ad J. M. Zakoia 2004 Maximum likelihood estimatio of pure GARCH ad ARMA-GARCH processes. Beroulli 0, Grager, C. W. J. ad R. Joyeux 980, A itroductio to log-memory time series models ad fractioal differeces, Joural of Time Series Aalysis, Hall, P. ad C. C. Heyde 980, Martigale limit theory ad its applicatios, Academic, Sa Diego, CA. Hoskig, J. R. M. 98, Fractioal differecig, Biometrika 68, Huber, P. J. 967, The behavior of imum likelihood estimates uder ostadard coditios, i: Berkley, Calif Proceedigs of the fifth Berkeley symposium of mathematical statistics ad Probability, Vol., Uiv. Califoria Press, Berkeley, CA pp Jeatheau, T. 998, Strog cosistecy of estimators for multivariate ARCH models, Ecoometric Theory 4, Lee, S-W. ad R. E. Hase 994, Asymptotic theory for GARCH, quasi-imum likelihood estimator, Ecoometric Theory 0, Li, W. K. ad A. I. McLeod 986, Fractioal time series modellig, Biometrika 73, Lig, S. 2003, Adaptive estimators ad tests of statioary ad o-statioary short ad log memory ARIMA-GARCH models, Joural of the America Statistical Associatio 98, Lig, S Estimatio ad testig of statioarity for double autoregressive models, Joural of the Royal Statistical Society: Series B 66, Lig, S. ad W. K. Li 997, Fractioal ARIMA-GARCH time series models, Joural of the America Statistical Associatio 92, The Authors. Joural compilatio 200 VVS.

15 Asymptotic theory for time-series models Lig, S. ad M. McAleer 2003, Asymptotic theory for a vector ARMA-GARCH model, Ecoometric Theory 9, Lig, S. ad H. Tog 2005, Testig for a liear MA model agaist threshold MA models, Aals of Statistics 33, Lig, S., H. Tog ad D. Li 2007, Ergodicity ad ivertibility of threshold MA models. Beroulli, 3, Pfazagl, J. 969, O the measurability ad cosistecy of miimum cotrast estimates, Metrika 4, Robiso, P. M. 994, Time series with strog depedece, i: C. Sims ed., Advaces i ecoometrics, sixth world cogress, Cambridge, Cambridge Uiversity Press Stout, W. F. 974, Almost sure covergece. Academic Press, New York ad Lodo. Tjøstheim, D. 986, Estimatio i oliear time series models. Stochastic Processes ad Applicatios 2, Tog, H. 990, Noliear time series. A dyamical system approach, Claredo Press ad Oxford Uiversity Press, New York. Received: October Revised: October The Authors. Joural compilatio 200 VVS.