# Limit Theory for Moderate Deviation from Integrated GARCH Processes

This paper develops the limit theory of the GARCH(1,1) process that moderately deviates from IGARCH process towards both stationary and explosive regimes. The GARCH(1,1) process is defined by equations u_t = σ_t ε_t, σ_t^2 = ω + α_n u_t-1^2 + β_nσ_t-1^2 and α_n + β_n approaches to unity as sample size goes to infinity. The asymptotic theory developed in this paper extends Berkes et al. (2005) by allowing the parameters to have a slower convergence rate. The results can be applied to unit root test for processes with mildly-integrated GARCH innovations (e.g. Boswijk (2001), Cavaliere and Taylor (2007, 2009)) and deriving limit theory of estimators for models involving mildly-integrated GARCH processes (e.g. Jensen and Rahbek (2004), Francq and Zakoïan (2012, 2013)).

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## 1 Introduction

The model considered in this paper is a GARCH(1,1) process:

 (Return Process) ut=σtεt, (Volatility Process) σ2t=ω+αnu2t−1+βnσ2t−1,ω>0, αn≥0, and βn≥0,

where is a sequence of independent identically distributed (i.i.d) variables such that and .

Unlike conventional GARCH(1,1) process, the innovation process considered in this paper is a mildly-integrated GARCH process whose key parameters, and , are changing with the sample size, viz.

 αn=O(n−p),βn=1+O(n−q), where p,q∈(0,1),

and

 γn=αn+βn−1=O(n−κ),κ=min{p,q}.

The limiting process of this GARCH process is first derived in Berkes et al. (2005) by imposing the assumption . Extending their results, we obtain the limiting process that applies to parameter values that covers the whole range of . This is a non-trivial extension because when the process deviates further from the integrated GARCH process, the approximation errors in Berkes et al. (2005) diverges and thus a different normalization is needed.

## 2 Main Results

The main results are summarized in the following one proposition and three theorems. The first proposition modifies the additive representation for in Berkes et al. (2005) to accommodate . Based on the proposition, we establish three theorems to describe the asymptotic behaviours of and under the cases respectively.

To establish the additive representation of , we make the following assumptions on the distribution of the innovations and the convergence rate of the GARCH coefficients, and .

###### Assumption 1.

is an i.i.d sequence with and , for some .

###### Assumption 2.

, and .

Assumption 1 imposes a non-degeneracy condition on the distribution of and thus ensures its applicability to the central limit theorem. Assumption 2 bounds the convergence rate of so that the normalized sequence could converge to a proper limit. Based on these assumptions, we obtain a modified additive representation for in Proposition 1 on the top of Berkes et al. (2005).

###### Proposition 1 (Additive Representation).

Under Assumption 1 and 2, we have the additive representation for as

 σ2t =σ20tt/2e√tγn(1+αn√tt∑j=1ξt−j+R(1)t)+ω[1+t∑j=1tj/2ejγn√t(1+αn√tj∑i=1ξt−i+R(2)t,j)(1+R(3)t,j)]

where and the remainder terms satisfy

 ∣∣R(1)t∣∣=Op(α2n+γ2n), max1≤j≤t∣∣R(2)t,j∣∣=Op(α2n) max1≤j≤t1jloglogj∣∣R(2)t,j∣∣=Op(α2nt), max1≤j≤t1j∣∣R(3)t,j∣∣=Op(α2n+γ2nt)
###### Remark 1.

The key difference between our results and Berkes et al. (2005) is the convergence rate of the approximation errors. In Berkes et al. (2005), the approximation error , is of order or asymptotically. Hence, these errors are negligible only when . We relax this restrictive assumption by normalizing the original terms with . Under this new normalization, all the approximation errors remains negligible when .

To formulate the theorems below, I introduce the following notations. For define , . Further, we need the assumptions for relative convergence rate between and to regulate the asymptotic behaviours of returns and volatilities for near-stationary case.

###### Assumption 3.

, while , as .

Assumption 3 imposes a rate condition on the localized parameters and . This condition is less restrictive than that in Berkes et al. (2005) in the sense that instead of requiring to converge to 0, we allow it to diverge slowly at a rate of . The relaxation of the assumption also attributes to the change of the normalization.

###### Theorem 1 (Near-stationary Case).

Suppose , then under Assumption 1-3

, the random variables

 √2|γn|3αnk(m)1/41√Eξ20⎛⎝σ2k(m)ωk(m)k(m)/2−k(m)−1∑j=1ejγn√k(m)⎞⎠d→N(0,1).

In addition, the random variables

 (|γn|ωk(m)(k(m)+1)/2)1/2uk(m)

are asymptotically independent, each with the asymptotic distribution equals to that of .

###### Theorem 2 (Integrate Case).

Suppose , then under Assumption 1 and 2, the volatility has the asymptotic distribution

 k(m)1/2n3/2αn1√Eξ20⎛⎝σ2k(m)ωk(m)k(m)/2−k(m)⎞⎠d→∫tm0xdW(x).

In addition, the random variables

 (ωk(m)k(m)/2+1)−1/2uk(m)

are asymptotically independent, each with the asymptotic distribution equals to that of .

Similar to the near-stationary case, we have to impose additional assumption on the relative speed of converging to zero between and .

, as .

###### Theorem 3 (Near-explosive Case).

Suppose , then under Assumption 1, 2 and 4, the volatility has the asymptotic distribution

 γne−√k(m)γnαn√k(m)1√Eξ20⎛⎝σ2k(m)ωk(m)k(m)/2−k(m)−1∑j=1ejγn√k(m)⎞⎠⇒W(tm).

In addition, the random variables

 (γne−√k(m)γnωk(m)(k(m)+1)/2)1/2uk(m)

are asymptotically independent, each with the asymptotic distribution equals to that of .

###### Remark 2.

As one may notice, the rate of convergence for both volatility process and return process in all three cases decreases to 0 asymptotically. These seemingly awkward results are reasonable in the sense that the 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 s in form of , the normalization just plays the role of which is 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 previous section. For readers’ convenience, I provide a roadmap for understanding the proofs of the theorems. In general, the proofs are done in three steps:

Step 1: We decompose the volatility process into 4 components, , , by expanding the multiplicative form provided in Proposition 1.

Step 2: We show the first 3 volatility components are negligible after normalization, and the last term converges to a proper limit by using Cramer-Wold device and Liapounov central limit theorem or Donsker’s theorem.

Step 3: We figure out a normalization to make the normalized volatility converges to 1. Then, applying this 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 multiplicative form:

 σ2t =σ20t∏i=1(βn+αnε2t−i)+ω[1+t−1∑j=1j∏i=1(βn+αnε2t−i)] =σ20tt/2t∏i=1(βn+αnε2t−i)√t+ω[1+tt/2t−1∑j=1j∏i=1(βn+αnε2t−i)√t].

Note that

 max1≤i≤t∣∣βn+αnε2t−i−1∣∣√t≤|γn|√t+αnmax1≤i≤t|ε2t−i−1|√t=|γn|√t+αnmax1≤i≤t−1|ε2i−1|√t.

Then by Assumption 1 and Chow & Teicher (2012), we have the almost sure convergence of

 max1≤j≤t−1|ε2i−1|=O(√t).

Therefore, the term above is

 max1≤i≤t∣∣βn+αnε2t−i−1∣∣√t=op(1).

Now consider the sequence of events

 An={max1≤i≤t|βn+αnε2t−i−1|√t≤12}.

From the previous result we know . Then by Taylor expansion, , on the event , which implies

 ∣∣R(3)t,j∣∣ =∣∣ ∣∣j∑i=1log(βn+αnε2t−i)√t−j∑i=1(γn+αnξt−i)√t∣∣ ∣∣ =∣∣ ∣∣j∑i=1log(γn+αnξt−i+1)√t−j∑i=1(γn+αnξt−i)√t∣∣ ∣∣ ≤j∑i=1∣∣ ∣∣log(γn+αnξt−i√t+1)−(γn+αnξt−i)√t∣∣ ∣∣ ≤2j∑i=1(γn+αnξt−i)2t≤4jγ2nt+4α2n∑ji=1ξ2t−it.

By Assumption 1

and law of large numbers (LLN), we know

Then by the equation above, we have

 max1≤i≤j1j|R(3)t,j|=Op(γ2n+α2nt).

Now by direct plugging into the key multiplicative term we care about, we have

 j∏i=1(βn+αnε2t−i)√t =exp{j∑i=1log(βn+αnε2t−i√t)} =exp{jγn√t}exp{αn∑ji=1ξt−i√t}exp{R(3)t,j} =ejγn√texp{αn∑ji=1ξt−i√t}(1+R(3)t,j).

Further, note is an i.i.d sequence with , then we know

 max1≤j≤t∣∣ ∣∣j∑i=1ξt−i∣∣ ∣∣=Op(√t),

which implies

 max1≤j≤t∣∣ ∣∣αn√tj∑i=1ξt−i∣∣ ∣∣=Op(αn)=op(1).

Similarly, we define the sequence of events

 Bn={max1≤j≤t∣∣ ∣∣αn√tj∑i=1ξt−i∣∣ ∣∣≤12},

which is known to have the property . Then by Taylor expansion, when , on the event

 ∣∣R(2)t,j∣∣=∣∣ ∣∣exp{αn√tj∑i=1ξt−i}−(1+αn√tj∑i=1ξt−i)∣∣ ∣∣≤√e2(αn√tj∑i=1ξt−i)2=Op(α2n),

and by law of iterated logarithm, we know

 max1≤j≤t1jloglogj(αn√tj∑i=1ξt−i)2=Op(α2nt).

Combining the results above, we have thus showed that

 j∏i=1(βn+αnε2t−i√t)=ejγn√t(1+αn√tj∑i=1ξt−i+R(2)t,j)(1+R(3)t,j).

Lastly, by the equation above, we know

 t∏i=1(βn+αnε2t−i√t) =etγn√t(1+αn√tt∑i=1ξt−i+Op(α2n))(1+Op(γ2n+α2n)) =e√tγn(1+αn√tt∑i=1ξt−i+Op(γ2n+α2n)),

and this establishes . ∎

###### Proof of Theorem 1.

First, we focus on the volatilities. Denote , ,

 σ2k =ω+σ20kk/2e√kγn(1+αn√kk∑j=1ξk−j+R(1)k)+ωkk/2k−1∑j=1ejγn√k(1+αn√kj∑i=1ξk−i+R(2)k,j)R(3)k,j +ωkk/2k−1∑j=1ejγn√kR(2)k,j+ωkk/2k−1∑j=1ejγn√k(1+αn√kj∑i=1ξk−i) =ω+σ2k,1+σ2k,2+σ2k,3+σ2k,4.

For , note is asymptotically normal, then by Proposition 1,

 αn√kk∑j=1ξk−j+R(1)k=op(1),

and this implies

 ∣∣σ2k,1∣∣ =Op(kk/2e√kγn).

For , note by Lemma 4.1 in Berkes et al. (2005), we have

 k∑j=1jejγn√k∼k|γn|2Γ(2), (1)

and note that

 max1≤j≤k−1∣∣ ∣∣αn√kj∑i=1ξk−i+R(2)k,j∣∣ ∣∣=op(1). (2)

Then by equation (1), (2) and Proposition 1 we have

 ∣∣σ2k,2∣∣ =∣∣ ∣∣ωkk/2k−1∑j=1jejγn√k(1+αn√kj∑i=1ξk−i+R(2)k,j)1jR(3)k,j∣∣ ∣∣ =Op(1)ωkk/2α2n+γ2nkk|γn|2 =Op(kk/2(α2n+γ2n)γ2n).

For , similarly, by Proposition 1 and Lemma 4.1 in Berkes et al. (2005), we have

 ∣∣σ2k,3∣∣ =∣∣ ∣∣ωkk/2k−1∑j=1ejγn√kR(2)k,j∣∣ ∣∣ =Op(1)ωkk/2α2nkk−1∑j=1jejγ√kloglogj =Op(kk/2(α2nloglogk)γ2n).

Lastly, for , by Lemma 4.1 in 1 we have

 σ2k,4 =ωkk/2k−1∑j=1ejγn√k+ωkk/2αn√kk−1∑j=1ejγn√kj∑i=1ξk−i =Op(kk/2k1/2|γn|)+ωkk/2αn√kk−1∑j=1ejγn√kj∑i=1ξk−i.

Therefore, we only have to consider the last term in the above equation. Define

 τm=k(m)−1/4k(m)−1∑j=1ejγn√k(m)ξk(m)−j,1≤m≤N,

and

 τ∗m=k(m)−1/2k(m)−1∑j=1ejγn√k(m)j∑i=1ξk(m)−i,1≤m≤N.

Then by Cramer-Wold device (Theorem 29.4 of Billingsley (1995)), we have

 N∑m=1μmτm =k(1)−1∑i=1N∑m=1μmk(m)1/4e(k(m)−i)γn√k(m)+k(2)−1∑i=k(1)N∑m=2μmk(m)1/4e(k(m)−i)γn√k(m) +⋯+k(N)−1∑i=k(N−1)μNk(N)1/4e(k(N)−i)γn√k(N) =S1+S2+⋯+SN.

Observe that

 ES21 =Eξ20⎛⎝k(1)−1∑i=1N∑m=1k(m)−1/4μme(k(m)−i)γn√k(m)⎞⎠2 =Eξ20N∑m=1μ2m√k(m)k(1)−1∑i=1e2(k(m)−i)γn√k(m)+Eξ20∑1≤m≠l≤N(k(m)k(l))−1/4μmμlk(1)−1∑i=1e(k(m)−i)γn√k(m)+(k(l)−i)γn√k(l) =Eξ20μ21√k(1)k(1)−1∑i=1e2(k(1)−i)γn√k(1)+Eξ20N∑m=2μ2m√k(m)k(1)−1∑i=1e2(k(m)−i)γn√k(m) =Eξ20μ21√k(1)k(1)−1∑i=1e2iγn√k(1)+Eξ20N∑m=2μ2m√k(m)e2(k(m)−k(1))γn√k(m)k(1)−1∑i=1e2iγn√k(m) +Eξ20∑1≤m≠l≤N(k(m)k(l))−1/4μmμle(k(m)−k(1))γn√k(m)+(k(l)−k(1))γn√k(l)k(1)−1∑i=1eiγn√k(m)+iγn√k(l) ∼Eξ20μ2112|γn|+Eξ20N∑m=2μ2me2(k(m)−k(1))γn√k(m)12|γn| +Eξ20∑1≤m≠l≤Nμmμl(√k(m)+√k(l))|γn|e(k(m)−k(1))γn√k(m)+(k(l)−k(1))γn√k(l)

we then have

 E(N∑m=1μmτm)2 =(N∑m=1μ2m)Eξ012|γn|+o(1|γn|).

Observe also that, for some , , we have

 N∑m=1μmτm=k(N)−1∑i=1ciξi,

and by Jensen’s inequality, we know for some ,

 |ci|2+δ =∣∣ ∣∣k(1)−1/4μ1e(k(1)−i)γn√k(1)+k(2)−1/4μ1e(k(2)−i)γn√k(2)+⋯+k(N)−1/4μ1e(k(N)−i)γn√k(N)∣∣ ∣∣2+δ ≤C1(N)[|μ1|2+δk(1)1/2+δ/4e(k(1)−i)(2+δ)γn√k(1)+⋯+|μN|2+δk(N)1/2+δ/4e(k(N)−i)(2+δ)γn√k(N)].

This implies that

 k(N)−1∑i=1|ci|2+δ∼C1(N)|μ1|2+δ1k(1)δ/4(2+δ)|γn|+O(1k(2)δ/4|γn|)=o(1|γn|).

Now we can easily check the Liapounov’s condition, where

 (∑k(N)−1i=1|ci|2+δE|ξi|2+δ)1/(2+δ)(∑k(N)−1i=1c2iEξ2i)1/2=o(|γn|1/2−1/(2+δ))=op(1).

Then by Liapounov central limit theorem (Theorem 27.3, p.362 of Billingsley (1995)), we have

 √2|γn|[τ1,τ2