Limit Theory for Moderate Deviation from Integrated GARCH Processes
aa r X i v : . [ m a t h . S T ] D ec Limit Theory for Moderate Deviation from Integrated GARCH Processes
Yubo Tao
90 Stamford Rd, Singapore Management University
Abstract
This paper develops the limit theory of the GARCH(1,1) process that moderately deviates from IGARCHprocess towards both stationary and explosive regimes. The GARCH(1,1) process is defined by equations u t = σ t ε t , σ t = ω + α n u t − + β n σ t − and α n + β n approaches to unity as sample size goes to infinity.The asymptotic theory extends Berkes et al. (2005) by allowing the parameters to have a slower rate ofconvergence. The results can be applied to unit root test for processes with mildly-integrated GARCHinnovations (e.g. Boswijk (2001), Cavaliere & Taylor (2007, 2009)) and deriving limit theory of estimatorsfor models involving mildly-integrated GARCH processes (e.g. Jensen & Rahbek (2004), Francq & Zako¨ıan(2012, 2013)). Keywords:
Central Limit Theorem, Limiting Process, Localization, Explosive GARCH, Volatility Process
1. Introduction
The model considered in this paper is a GARCH(1,1) process:(Return Process) u t = σ t ε t , (Volatility Process) σ t = ω + α n u t − + β n σ t − , ω > α n ≥
0, and β n ≥ , where { ε t } nt =0 is a sequence of independent identically distributed (i.i.d) variables such that Eε = 0 and Eε = 1.Unlike conventional GARCH(1,1) process, the innovation process considered in this paper is a mildly-integrated GARCH process whose key parameters, α n and β n , are changing with the sample size, viz. α n = O ( n − p ) , β n = 1 + O ( n − q ) , where p, q ∈ (0 , , and γ n = α n + β n − O ( n − κ ) , κ = min { p, q } . I would like to thank the co-editor, an associate editor and the referees for helping improve the paper. All possible errorsare mine. Yubo Tao, School of Economics, Singapore Management University, 90 Stamford Road, Singapore 178903. Email:[email protected].
Preprint submitted to Statistics and Probability Letters December 21, 2018 he limiting process of this GARCH process is first derived in Berkes et al. (2005) by imposing the assump-tion κ ∈ (1 / , ,
2. Main Results
The main results are summarized in the following one proposition and three theorems. The first propo-sition modifies the additive representation for σ t in Berkes et al. (2005) to accommodate κ ∈ (0 , σ t and u t underthe cases γ n S σ t , we make the following assumptions on the distribution ofthe innovations { ε t } nt =0 and the convergence rate of the GARCH coefficients, α n and β n . Assumption 1. { ε t } nt =0 is an i.i.d sequence with Eε = 1 and E | ε | δ < ∞ , for some δ > . Assumption 2. α n log log n → , nα n → ∞ and β n → . Assumption 1 imposes a non-degeneracy condition on the distribution of ε t and thus ensures its applica-bility to the central limit theorem. Assumption 2 bounds the convergence rate of α n so that the normalizedsequence could converge to a proper limit. Based on these assumptions, we obtain a modified additiverepresentation for σ t in Proposition 1 on the top of Berkes et al. (2005). Proposition 1 (Additive Representation) . Under Assumption 1 and 2, we have the additive representationfor σ t as σ t = σ t t/ e √ tγ n α n √ t t X j =1 ξ t − j + R (1) t + ω t X j =1 t j/ e jγn √ t α n √ t j X i =1 ξ t − i + R (2) t,j ! (cid:16) R (3) t,j (cid:17) where ξ t = ε t − and the remainder terms satisfy (cid:12)(cid:12)(cid:12) R (1) t (cid:12)(cid:12)(cid:12) = O p (cid:0) α n + γ n (cid:1) , max ≤ j ≤ t (cid:12)(cid:12)(cid:12) R (2) t,j (cid:12)(cid:12)(cid:12) = O p (cid:0) α n (cid:1) max ≤ j ≤ t j log log j (cid:12)(cid:12)(cid:12) R (2) t,j (cid:12)(cid:12)(cid:12) = O p (cid:18) α n t (cid:19) , max ≤ j ≤ t j (cid:12)(cid:12)(cid:12) R (3) t,j (cid:12)(cid:12)(cid:12) = O p (cid:18) α n + γ n t (cid:19) Remark 1.
The key difference between our results and Berkes et al. (2005) is the convergence rate of theapproximation errors. In Berkes et al. (2005), the approximation error | R ( p ) t | , ∀ p = { , , } is of order t ( α n + γ n ) or tα n asymptotically. Hence, these errors are negligible only when κ ∈ (1 / , . We relax thisrestrictive assumption by normalizing the original terms with √ t . Under this new normalization, all theapproximation errors remains negligible when κ ∈ (0 , .
2o formulate the theorems below, I introduce the following notations. For 0 < t < t < · · · < t N < k ( m ) = ⌊ nt m ⌋ , 1 ≤ m ≤ N . Further, we need the assumptions for relative convergence rate between α n and γ n to regulate the asymptotic behaviours of returns and volatilities for near-stationary case. Assumption 3. p | γ n | α n n / → ∞ , while p | γ n | α n n / → , as n → ∞ . Assumption 3 imposes a rate condition on the localized parameters α n and γ n . This condition is lessrestrictive than that in Berkes et al. (2005) in the sense that instead of requiring | γ n | / /α n to converge to0, we allow it to diverge slowly at a rate of n / . The relaxation of the assumption also attributes to thechange of the normalization. Theorem 1 (Near-stationary Case) . Suppose γ n < , then under Assumption 1-3, the random variables p | γ n | α n k ( m ) / p Eξ σ k ( m ) ωk ( m ) k ( m ) / − k ( m ) − X j =1 e jγn √ k ( m ) d −→ N (0 , . In addition, the random variables | γ n | ωk ( m ) ( k ( m )+1) / ! / u k ( m ) are asymptotically independent, each with the asymptotic distribution equals to that of ε . Theorem 2 (Integrate Case) . Suppose γ n = 0 , then under Assumption 1 and 2, the volatility has theasymptotic distribution k ( m ) / n / α n p Eξ σ k ( m ) ωk ( m ) k ( m ) / − k ( m ) ! d −→ Z t m xdW ( x ) . In addition, the random variables (cid:16) ωk ( m ) k ( m ) / (cid:17) − / u k ( m ) are asymptotically independent, each with the asymptotic distribution equals to that of ε . Similar to the near-stationary case, we have to impose additional assumption on the relative speed ofconverging to zero between α n and γ n . Assumption 4. γ n /α n → , as n → ∞ . Theorem 3 (Near-explosive Case) . Suppose γ n > , then under Assumption 1, 2 and 4, the volatility hasthe asymptotic distribution γ n e − √ k ( m ) γ n α n p k ( m ) 1 p Eξ σ k ( m ) ωk ( m ) k ( m ) / − k ( m ) − X j =1 e jγn √ k ( m ) ⇒ W ( t m ) . In addition, the random variables γ n e − √ k ( m ) γ n ωk ( m ) ( k ( m )+1) / ! / u k ( m ) are asymptotically independent, each with the asymptotic distribution equals to that of ε . emark 2. As one may notice, the rate of convergence for both volatility process and return process in allthree cases decreases to 0 asymptotically. These seemingly awkward results are reasonable in the sense thatthe convergence rate is a part of the normalization which reflects the order of the process. In other words,when we compute a partial sum of X s in form of P ni =1 a i X i , the normalization just plays the role of a i whichis usually required to decrease to 0 for applying a central limit theorem.
3. Proofs
In this section, I present detailed proofs for all the propositions and the theorems listed in the previoussection. For readers’ convenience, I provide a roadmap for understanding the proofs of the theorems. Ingeneral, the proofs are done in three steps:
Step 1:
We decompose the volatility process into 4 components, σ k,s , s = 1 , · · · ,
4, by expanding themultiplicative form provided in Proposition 1.
Step 2:
We show the first 3 volatility components are negligible after normalization, and the last termconverges to a proper limit by using Cramer-Wold device and Liapounov central limit theorem or Donsker’stheorem.
Step 3:
We figure out a normalization to make the normalized volatility converges to 1. Then, applyingthis normalization to the return process, we complete the proof.
Proof of Proposition 1 . First, note the GARCH(1,1) model can be written into the following multiplica-tive form: σ t = σ t Y i =1 (cid:0) β n + α n ε t − i (cid:1) + ω t − X j =1 j Y i =1 (cid:0) β n + α n ε t − i (cid:1) = σ t t/ t Y i =1 (cid:0) β n + α n ε t − i (cid:1) √ t + ω t t/ t − X j =1 j Y i =1 (cid:0) β n + α n ε t − i (cid:1) √ t . Note that max ≤ i ≤ t (cid:12)(cid:12) β n + α n ε t − i − (cid:12)(cid:12) √ t ≤ | γ n |√ t + α n max ≤ i ≤ t | ε t − i − |√ t = | γ n |√ t + α n max ≤ i ≤ t − | ε i − |√ t . Then by Assumption 1 and Chow & Teicher (2012), we have the almost sure convergence ofmax ≤ j ≤ t − | ε i − | = O ( √ t ) . Therefore, the term above is max ≤ i ≤ t (cid:12)(cid:12) β n + α n ε t − i − (cid:12)(cid:12) √ t = o p (1) . Now consider the sequence of events A n = (cid:26) max ≤ i ≤ t | β n + α n ε t − i − |√ t ≤ (cid:27) . n →∞ P ( A n ) = 1. Then by Taylor expansion, | log(1 + x ) − x | ≤ x , | x | ≤ / A n , which implies (cid:12)(cid:12)(cid:12) R (3) t,j (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j X i =1 log (cid:0) β n + α n ε t − i (cid:1) √ t − j X i =1 ( γ n + α n ξ t − i ) √ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j X i =1 log ( γ n + α n ξ t − i + 1) √ t − j X i =1 ( γ n + α n ξ t − i ) √ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ j X i =1 (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) γ n + α n ξ t − i √ t + 1 (cid:19) − ( γ n + α n ξ t − i ) √ t (cid:12)(cid:12)(cid:12)(cid:12) ≤ j X i =1 ( γ n + α n ξ t − i ) t ≤ jγ n t + 4 α n P ji =1 ξ t − i t . By Assumption 1 and law of large numbers (LLN), we knowmax ≤ j ≤ t j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j X i =1 ξ t − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∼ max ≤ j ≤ t j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j X i =1 ξ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (1) . Then by the equation above, we have max ≤ i ≤ j j | R (3) t,j | = O p (cid:18) γ n + α n t (cid:19) . Now by direct plugging into the key multiplicative term we care about, we have j Y i =1 (cid:0) β n + α n ε t − i (cid:1) √ t = exp ( j X i =1 log (cid:18) β n + α n ε t − i √ t (cid:19)) = exp (cid:26) jγ n √ t (cid:27) exp ( α n P ji =1 ξ t − i √ t ) exp n R (3) t,j o = e jγn √ t exp ( α n P ji =1 ξ t − i √ t ) (cid:16) R (3) t,j (cid:17) . Further, note { ξ t } nt =1 is an i.i.d sequence with Eξ < ∞ , then we knowmax ≤ j ≤ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j X i =1 ξ t − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p ( √ t ) , which implies max ≤ j ≤ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α n √ t j X i =1 ξ t − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p ( α n ) = o p (1) . Similarly, we define the sequence of events B n = ( max ≤ j ≤ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α n √ t j X i =1 ξ t − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ) , which is known to have the property lim n →∞ P ( B n ) = 1. Then by Taylor expansion, | exp( x ) − (1+ x ) | ≤ √ ex / | x | ≤ /
2, on the event B n (cid:12)(cid:12)(cid:12) R (2) t,j (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp ( α n √ t j X i =1 ξ t − i ) − α n √ t j X i =1 ξ t − i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ √ e α n √ t j X i =1 ξ t − i ! = O p (cid:0) α n (cid:1) , ≤ j ≤ t j log log j α n √ t j X i =1 ξ t − i ! = O p (cid:18) α n t (cid:19) . Combining the results above, we have thus showed that j Y i =1 (cid:18) β n + α n ε t − i √ t (cid:19) = e jγn √ t α n √ t j X i =1 ξ t − i + R (2) t,j ! (cid:16) R (3) t,j (cid:17) . Lastly, by the equation above, we know t Y i =1 (cid:18) β n + α n ε t − i √ t (cid:19) = e tγn √ t α n √ t t X i =1 ξ t − i + O p ( α n ) ! (cid:0) O p ( γ n + α n ) (cid:1) = e √ tγ n α n √ t t X i =1 ξ t − i + O p ( γ n + α n ) ! , and this establishes R (1) t . Proof of Theorem 1 . First, we focus on the volatilities. Denote k = ⌊ nt ⌋ , 0 < t ≤ σ k = ω + σ k k/ e √ kγ n α n √ k k X j =1 ξ k − j + R (1) k + ωk k/ k − X j =1 e jγn √ k α n √ k j X i =1 ξ k − i + R (2) k,j ! R (3) k,j + ωk k/ k − X j =1 e jγn √ k R (2) k,j + ωk k/ k − X j =1 e jγn √ k α n √ k j X i =1 ξ k − i ! = ω + σ k, + σ k, + σ k, + σ k, . For σ k, , note k − / P kj =1 ξ k − j is asymptotically normal, then by Proposition 1, α n √ k k X j =1 ξ k − j + R (1) k = o p (1) , and this implies (cid:12)(cid:12) σ k, (cid:12)(cid:12) = O p (cid:16) k k/ e √ kγ n (cid:17) . For σ k, , note by Lemma 4.1 in Berkes et al. (2005), we have k X j =1 je jγn √ k ∼ k | γ n | Γ(2) , (1)and note that max ≤ j ≤ k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α n √ k j X i =1 ξ k − i + R (2) k,j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o p (1) . (2)6hen by equation (1), (2) and Proposition 1 we have (cid:12)(cid:12) σ k, (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ωk k/ k − X j =1 je jγn √ k α n √ k j X i =1 ξ k − i + R (2) k,j ! j R (3) k,j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (1) ωk k/ α n + γ n k k | γ n | = O p k k/ (cid:0) α n + γ n (cid:1) γ n ! . For σ k, , similarly, by Proposition 1 and Lemma 4.1 in Berkes et al. (2005), we have (cid:12)(cid:12) σ k, (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ωk k/ k − X j =1 e jγn √ k R (2) k,j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (1) ωk k/ α n k k − X j =1 je jγ √ k log log j = O p k k/ (cid:0) α n log log k (cid:1) γ n ! . Lastly, for σ k, , by Lemma 4.1 in 1 we have σ k, = ωk k/ k − X j =1 e jγn √ k + ωk k/ α n √ k k − X j =1 e jγn √ k j X i =1 ξ k − i = O p (cid:18) k k/ k / | γ n | (cid:19) + ωk k/ α n √ k k − X j =1 e jγn √ k j X i =1 ξ k − i . Therefore, we only have to consider the last term in the above equation. Define τ m = k ( m ) − / k ( m ) − X j =1 e jγn √ k ( m ) ξ k ( m ) − j , ≤ m ≤ N, and τ ∗ m = k ( m ) − / k ( m ) − X j =1 e jγn √ k ( m ) j X i =1 ξ k ( m ) − i , ≤ m ≤ N. Then by Cramer-Wold device (Theorem 29.4 of Billingsley (1995)), we have N X m =1 µ m τ m = k (1) − X i =1 N X m =1 µ m k ( m ) / e ( k ( m ) − i ) γn √ k ( m ) + k (2) − X i = k (1) N X m =2 µ m k ( m ) / e ( k ( m ) − i ) γn √ k ( m ) + · · · + k ( N ) − X i = k ( N − µ N k ( N ) / e ( k ( N ) − i ) γn √ k ( N ) = S + S + · · · + S N . ES = Eξ k (1) − X i =1 N X m =1 k ( m ) − / µ m e ( k ( m ) − i ) γn √ k ( m ) = Eξ N X m =1 µ m p k ( m ) k (1) − X i =1 e k ( m ) − i ) γn √ k ( m ) + Eξ X ≤ m = l ≤ N ( k ( m ) k ( l )) − / µ m µ l k (1) − X i =1 e ( k ( m ) − i ) γn √ k ( m ) + ( k ( l ) − i ) γn √ k ( l ) = Eξ µ p k (1) k (1) − X i =1 e k (1) − i ) γn √ k (1) + Eξ N X m =2 µ m p k ( m ) k (1) − X i =1 e k ( m ) − i ) γn √ k ( m ) + Eξ X ≤ m = l ≤ N ( k ( m ) k ( l )) − / µ m µ l k (1) − X i =1 e ( k ( m ) − i ) γn √ k ( m ) + ( k ( l ) − i ) γn √ k ( l ) = Eξ µ p k (1) k (1) − X i =1 e iγn √ k (1) + Eξ N X m =2 µ m p k ( m ) e k ( m ) − k (1)) γn √ k ( m ) k (1) − X i =1 e iγn √ k ( m ) + Eξ X ≤ m = l ≤ N ( k ( m ) k ( l )) − / µ m µ l e ( k ( m ) − k (1)) γn √ k ( m ) + ( k ( l ) − k (1)) γn √ k ( l ) k (1) − X i =1 e iγn √ k ( m ) + iγn √ k ( l ) ∼ Eξ µ | γ n | + Eξ N X m =2 µ m e k ( m ) − k (1)) γn √ k ( m ) | γ n | + Eξ X ≤ m = l ≤ N µ m µ l ( p k ( m ) + p k ( l )) | γ n | e ( k ( m ) − k (1)) γn √ k ( m ) + ( k ( l ) − k (1)) γn √ k ( l ) = Eξ µ | γ n | + o (cid:18) | γ n | (cid:19) , we then have E N X m =1 µ m τ m ! = N X m =1 µ m ! Eξ | γ n | + o (cid:18) | γ n | (cid:19) . Observe also that, for some c i , 1 ≤ i ≤ k ( N ) −
1, we have N X m =1 µ m τ m = k ( N ) − X i =1 c i ξ i , and by Jensen’s inequality, we know for some δ > | c i | δ = (cid:12)(cid:12)(cid:12)(cid:12) k (1) − / µ e ( k (1) − i ) γn √ k (1) + k (2) − / µ e ( k (2) − i ) γn √ k (2) + · · · + k ( N ) − / µ e ( k ( N ) − i ) γn √ k ( N ) (cid:12)(cid:12)(cid:12)(cid:12) δ ≤ C ( N ) (cid:20) | µ | δ k (1) / δ/ e ( k (1) − i )(2+ δ ) γn √ k (1) + · · · + | µ N | δ k ( N ) / δ/ e ( k ( N ) − i )(2+ δ ) γn √ k ( N ) (cid:21) . This implies that k ( N ) − X i =1 | c i | δ ∼ C ( N ) | µ | δ k (1) δ/ (2 + δ ) | γ n | + O (cid:18) k (2) δ/ | γ n | (cid:19) = o (cid:18) | γ n | (cid:19) . Now we can easily check the Liapounov’s condition, where (cid:16)P k ( N ) − i =1 | c i | δ E | ξ i | δ (cid:17) / (2+ δ ) (cid:16)P k ( N ) − i =1 c i Eξ i (cid:17) / = o (cid:16) | γ n | / − / (2+ δ ) (cid:17) = o p (1) . p | γ n | [ τ , τ , · · · , τ N ] d −→ q Eξ [ η , η , · · · , η N ] , where η , η , · · · , η N are independent standard normal random variables.Now we have to check the relationship between τ m and τ ∗ m . Note by k − / (cid:16) e γn √ k − (cid:17) − = ( γ n + o (1)) − ,we have 1 √ k k − X j = i e jγn √ k − | γ n | − e iγn √ k = 1 √ k e kγn √ k − e iγn √ k e γn √ k − − | γ n | − e iγn √ k = ( γ n + o (1)) − (cid:16) e kγn √ k − e iγn √ k (cid:17) − | γ n | − e iγn √ k = (cid:0) γ − n + O (1) (cid:1) e kγn √ k − e iγn √ k O (1) . Then, we know E hp | γ n | τ ∗ m − p | γ n | τ m i = 2 | γ n | √ k E √ k k − X i =1 k − X j = i e jγ √ k ξ k − i − | γ n | − k − X i =1 e iγn √ k ξ k − i = 2 | γ n | √ k Eξ k − X i =1 √ k k − X j = i e jγn √ k − | γ n | − e iγn √ k ∼ | γ n | √ k Eξ kγ − n e √ kγ n + √ k | γ | − γ − n e √ kγ n √ k | γ | ! = 2 Eξ O (cid:16) √ k | γ n | e √ kγ n (cid:17) + o p (1)= o p (1) , where the last equality comes from the well known limits of xe − x ,lim x →∞ xe x = lim x →∞ e x = 0 and lim x → xe x = 0 . Therefore, we have p | γ n | [ τ ∗ , τ ∗ , · · · , τ ∗ N ] d −→ q Eξ [ η , η , · · · , η N ] , Now combine the results above, we have, for each k = ⌊ nt m ⌋ , m = 1 , · · · , N p | γ n | α n k / p Eξ σ k ωk k/ − k − X j =1 e jγn √ k d −→ N (0 , . Now, for returns, we know from the above result that | γ n | σ k ωk ( k +1) / − O p α n n / p | γ n | ! = o p (1) . Therefore, by the return equation, we have (cid:18) | γ n | ωk ( k +1) / (cid:19) / u k = (cid:18) | γ n | σ k ωk ( k +1) / (cid:19) / ε k ∼ ε k . roof of Theorem 2 . Similar to Theorem 1, when γ n = 0, the volatility admits the decomposition. Then,for σ k, , by central limit theorem, we know α n √ k k X j =1 ξ k − j = O p ( α n ) = o p (1)which, combining with Proposition 1, implies that (cid:12)(cid:12) σ k, (cid:12)(cid:12) = O p (cid:16) k k/ (cid:17) For σ k, , note that we have established equation (2), then by Proposition 1, we have (cid:12)(cid:12) σ k, (cid:12)(cid:12) = O p (cid:16) k k/ α n (cid:17) . For σ k, , by Proposition 1 we have (cid:12)(cid:12) σ k, (cid:12)(cid:12) = O p ( k k/ α n ) . Lastly, for σ k, , note by Lemma 5.1 in Berkes et al. (2005), for k = ⌊ nt ⌋ , t ∈ (0 ,
1) , we have1 n / ⌊ nt ⌋− X j =1 j X i =1 ξ k − i d −→ q Eξ Z t xdW ( x ) , where W ( x ) is a Wiener process.Therefore, for k ( m ) = ⌊ nt m ⌋ , m = 1 , · · · , N , we have k ( m ) / n / α n σ k ( m ) ωk ( m ) k ( m ) / − k ( m ) ! = 1 n / ⌊ nt m ⌋− X j =1 j X i =1 ξ k ( m ) − i + o p (1) d −→ q Eξ Z t m xdW ( x ) . Further, note the results above implies that σ k ( m ) ωk ( m ) k ( m ) / − O p (cid:18)(cid:16) nk (cid:17) / α n (cid:19) = o p (1) . Hence, by return equation, we obtain ωk ( m ) k ( m ) / ! / u k ( m ) = σ k ( m ) ωk ( m ) k ( m ) / ! / ε k ( m ) d −→ ε k ( m ) . Proof of Theorem 3 . Similar to proof of Theorem 1, when γ n >
0, the volatility admits the additiverepresentation. For σ k, , similar to that in Theorem 1, (cid:12)(cid:12) σ k, (cid:12)(cid:12) = O p (cid:16) k k/ e √ kγ n (cid:17) . σ k, , by Proposition 1 and equation (2), we have the relation (cid:12)(cid:12) σ k, (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ωk k/ k − X j =1 je jγn √ k (1 + o p (1)) 1 j R (3) k,j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (1) ωk k/ ( α n + γ n ) e kγn √ k − e γn √ k e γn √ k − < O p k k/ (cid:0) α n + γ n (cid:1) √ ke √ kγ n γ n ! , where the last inequality comes from the fact that e kγn √ k − e γn √ k e γn √ k − < e √ kγ n γ n / √ k . For σ k, , by Proposition 1, we have (cid:12)(cid:12) σ k, (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ωk k/ k − X j =1 e jγn √ k ( j log log j ) 1 j log log j R (2) k,j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (1) ωk k/ ( k log log k ) α n k e kγn √ k − e γn √ k e γn √ k − < O p k k/ (cid:0) α n log log k (cid:1) √ ke √ kγ n γ n ! . Lastly, for σ k, , we have σ k, = ωk k/ k − X j =1 e jγn √ k + ωk k/ α n √ k k − X j =1 e jγn √ k j X i =1 ξ k − i . Now, we introduce the following lemma to assist the proof.
Lemma 1.
If Assumption 1 and 2 hold, then γ n k e − √ kγ n E √ k k − X j =1 e jγn √ k j X i =1 ξ k − i − e √ kγ n γ n k − X i =1 ξ i → . Then by Lemma 1, we have γ n e −√ kγ n √ kα n σ k, ωk k/ − k − X j =1 e jγn √ k = γ n e −√ kγ n √ k √ k k − X j =1 e jγn √ k j X i =1 ξ k − i + o p (1) = 1 √ k k − X i =1 ξ i + o p (1) . Therefore, by Donsker’s theorem, we obtain that, for k ( m ) = ⌊ nt m ⌋ , t m ∈ (0 ,
1) and m = 1 , , · · · , N , γ n e − √ k ( m ) γ n p k ( m ) α n p Eξ σ k ( m ) ωk ( m ) k ( m ) / − k ( m ) − X j =1 e jγn √ k ( m ) ⇒ W ( t m ) , where W ( t ) is a finite dimensional Wiener process. 11urther, note that γ n √ k e −√ kγ n k − X j =1 e jγn √ k − √ ke √ kγ n γ n = o (1) , then by the result above we know γ n e − √ k ( m ) γ n p k ( m ) σ k ( m ) ωk ( m ) k ( m ) / − k ( m ) − X j =1 e jγn √ k ( m ) = O p ( α n ) = o p (1) . Hence, by return equantion, we derive γ n e −√ kγ n ωk ( k +1) / ! / u k = γ n e −√ kγ n ωk ( k +1) / σ k ! / ε k ∼ ε k . Proof of Lemma 1.
Note that1 √ k k − X j =1 e jγn √ k j X i =1 ξ k − i = 1 √ k k − X i =1 k − X j = i e jγn √ k ξ k − i and k − X i =1 ξ i = k − X i =1 ξ k − i , Then, E √ k k − X j =1 e jγn √ k j X i =1 ξ k − i − e √ kγ n γ n k − X i =1 ξ i = Eξ k − X i =1 √ k k − X j = i e jγn √ k − √ ke √ kγ n γ n = Eξ k k − X i =1 e kγn √ k − e iγn √ k e γn √ k − − √ ke √ kγ n γ n ! . Note by Taylor expansion, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e kγn √ k − e iγn √ k e γn √ k − − √ ke √ kγ n γ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C √ ke iγn √ k γ n + e √ kγ n ! , which implies that k − X i =1 e kγn √ k − e iγn √ k e γn √ k − − √ ke √ kγ n γ n ! ≤ C k − X i =1 ke iγn √ k γ n + ke √ kγ n ! = O (1) kγ n e kγn √ k − e γn √ k e γn √ k − ke √ kγ n ! = O (1) (cid:18) kγ n e √ kγ n + ke √ kγ n (cid:19) . Now we can see that γ n k e − √ kγ n E √ k k − X j =1 e jγn √ k j X i =1 ξ k − i − e √ kγ n γ n k − X i =1 ξ i = O (1) γ n k e − √ kγ n Eξ k (cid:18) kγ n e √ kγ n + ke √ kγ n (cid:19) = o p (1) . eferences Berkes, I., Horv´ath, L., & Kokoszka, P. (2005). Near-integrated garch sequences.
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