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Functional central limit theorems for ARCH(∞) models
Communications for Statistical Applications and Methods 2017;24:443-455
Published online September 30, 2017
© 2017 Korean Statistical Society.

Seunghee Choia, and Oesook Lee1,a

aDepartment of Statistics, Ewha Womans University, Korea
Correspondence to: 1Corresponding author: Department of Statistics, Ewha Womans University, 52 Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Korea. E-mail: oslee@ewha.ac.kr
Received March 28, 2017; Revised July 28, 2017; Accepted September 14, 2017.

In this paper, we study ARCH(∞) models with either geometrically decaying coefficients or hyperbolically decaying coefficients. Most popular autoregressive conditional heteroscedasticity (ARCH)-type models such as various modified generalized ARCH (GARCH) (p, q), fractionally integrated GARCH (FIGARCH), and hyperbolic GARCH (HYGARCH). can be expressed as one of these cases. Sufficient conditions for L2-near-epoch dependent (NED) property to hold are established and the functional central limit theorems for ARCH(∞) models are proved.

Keywords : functional central limit theorem, ARCH(∞) model, L2-NED property
1. Introduction

Introduced by the seminal work of Engle (1982) and Bollerslev (1986), autoregressive conditional heteroscedasticity (ARCH)-type processes are widely used for modelling dynamics in different fields such as in econometric studies. A number of modifications of the classical generalized ARCH (GARCH) model such as various modified-GARCH(p, q) models, integrated GARCH (IGARCH), fractionally IGARCH (FIGARCH), hyperbolic GARCH (HYGARCH), and fractionally integrated asymmetric power ARCH (FIAPARCH) models were proposed to account for long memory property, asymmetry, leverage effect, and other stylized facts. ARCH(∞) models was first introduced by Robinson (1991) in the context of testing for strong serial correlation. It can often be helpful to view a GARCH(p, q) process as an ARCH(∞) processes. In particular, from the ARCH(∞) representation we can easily read off the conditional variance given its infinite past. All of these aforementioned models can be represented as ARCH(∞) models. When we consider a time series model as a data generating process, one of the important properties to show is the (functional) central limit theorem (CLT). Functional central limit theorem (FCLT) is applied for statistical inference in time series to establish the asymptotics of various statistics concerned. For example, the FCLT has been employed in the theory of detecting structural breaks in GARCH-type models. Probabilistic and statistical properties of ARCH(∞) models have been studied by many authors (e.g., Davidson, 2004; Giraitis et al., 2000; Kazakevičius and Leipus, 2002; Zaffaroni, 2004) and the references therein.

A process yt is said to obey the FCLT if

Yn(ξ)=σn-1t=1[nξ](yt-E(yt)),         0ξ1,

converges weakly to standard Brownian motion, where σn2=Var(t=1nyt). If ξ = 1, then this convergence implies the CLT. There are numerous literatures considering the CLT and FCLT for various GARCH family models (Berkes et al., 2008; Billingsley, 1968; Davidson, 2002; De Jong and Davidson, 2000; Herrndorf, 1984; Lee, 2014a, 2014b).

This paper is to find some sufficient conditions under which the FCLT holds for the partial sums processes of the given ARCH(∞) models. The typical approach to obtaining the FCLT for time series models is to show that a specific dependence property such as various mixing conditions, Lp-NED (near-epoch dependent), association or θ, ℒ or ψ-weak dependence holds. In order to prove such dependence properties, rather restrictive conditions such as distributional assumptions on errors and higher order moment are required (Dedecker et al., 2007; Doukhan and Wintenberger, 2007).

Our proof is based on the L2-NED condition among various dependence conditions. The proofs, in part, rely on results in Davidson (2004) and Giraitis et al. (2000).

Definition 1

ytis said to be L2-NED on {et} of sizeλ0if


where dtis a sequence of positive constants andν(m) = O(mλ) forλ > λ0. Ifν(m) = O(eδm) for someδ > 0, we say that the process is geometrically L2-NED. Here we defineEt-mt+m(yt)=E(ytσ(et-m,,et-1,et,et+1,,et+m)).

Theorem 1

(Davidson, 2002)Suppose that the following Assumptions (1)–(3) hold: (1) ytis L2-NED of size −1/2 on the underlying i.i.d. process {et}. (2) suptE|ytE(yt)|r < ∞ for some r ≥ 2. (3)σn2/nσ2>0as n → ∞. Then the FCLT holds for yt.

2. Functional central limit theorem for ARCH(∞) model

In this section, we consider a nonnegative coefficients ARCH(∞) model defined as follows.




where ω > 0, θi ≥ 0 for all i ≥ 1 are constants and {et} is a sequence of independent and identically distributed random variables with mean 0 and variance 1.

Repeated substitution of the equations (2.1) and (2.2) leads to, for given m,


For notational simplicity, we let S=i=1θi, μn = E(|et|n), and Mn=E(σtn) for n = 1, 2, …. For a process yt, define Et-mt+m(yt)=E(ytt-mt+m), where st=σ(es,,et) is a sigma field generated by {ej, sjt}.

We first write the lemma due to Giraitis et al. (2000) and Zaffaroni (2004) in which the stationarity and finite moment condition of the process is obtained.

Lemma 1

Assume S < 1. Thenσt2given by


is the unique nonanticipative strictly stationary solution to (2.1) and (2.2) with finite first momentE(σt2). If, in addition,μ41/2S<1, then (2.4) is also unique weakly stationary solution to (2.1) and (2.2).

The following lemma will be used to derive some sufficient conditions for L2-NED property and then to prove the FCLT for the process ut.

Lemma 2

Consider the processes utandσt2given by (2.1) and (2.2). Then we obtain thatut-Et-mt+m(ut)22E|σt2-Et-mt+m(σt2)|.


We have that


The first inequality in (2.5) follows from the relation: ||YE(Y|X)||2 ≤ ||Yg(X)||2 for any measurable function g. For the second inequality in (2.5), we apply the relation (a-b)2a-b(a>0,b>0).

Next, we define for some mp ≥ 1,


where IA(·) is the indicator function of a set A. Then combining (2.3) and (2.6), we can easily show that (Davidson, 2004, p.26)




Now, we first consider the processes which have hyperbolically decaying lag coefficients.

  • (A1) 0 ≤ θjCj1δ for j ≥ 1, C > 0, δ > 0, and S < 1.

Theorem 2

  • (1-1) If the Assumption (A1) holds, then utgiven by (2.1) and (2.2) is L2-NED on {et}, of sizeλ/2, withδ > λ > 0.

  • (1–2) If the assumption (A1) andμ41/2S<1hold, then ut, ut2, andσt2are L2-NED on {et} of sizeλ, withδ > λ > 0.


After simple calculation, we derive that


and then use mathematical induction to obtain that


where [x] denotes the largest integer which is less than or equal to x.

Moreover, if the Assumption (A1) holds, then


Combining the Assumption (A1), the equations (2.9) and (2.10) yields that

TpO(m-δpδ+1Sp-1),         p1.

Proof of Theorem 2(1-1): Note that


Then the equations (2.7), (2.11), and (2.12) give that


Use Lemma 2 and the equation (2.13) to obtain that


and the conclusion follows.

Proof of Theorem 2(1–2): Combine, similarly the equations (2.8), (2.11), and (2.12) to have that


since μ41/2S<1. Also, we have


Hence ut2 and σt2 hold the L2-NED property of size −λ with δ > λ > 0.

On the other hand, using the inequalities σt2ω,Et-mt+mσt2ω, and a-ba-b if a, b ≥ 1 yields that


Then from (2.16) and the first inequality in (2.5), we obtain that


Therefore, L2-NED property of ut follows from (2.14) and (2.17).

Next, consider the case where the process has geometrically decaying coefficients. We make the Assumption (A2):

  • (A2) 0 ≤ θjCrj, for j ≥ 1, C > 0, 0 < r < 1, and S < 1.

Theorem 3

  • (2-1) If the Assumption (A2) holds, then utis L2-NED on {et} of size −1/2.

  • (2-2) If the Assumption (A2) andμ41/2S<1hold, thenut2andσt2are L2-NED on {et} of size −1/2.

  • (2–3) If the Assumption (A2) and 0 < rC < 1 hold, then utis geometrically L2-NED on {et}.

  • (2–4) If the Assumption (A2),μ41/2S<1, 0 < rC < 1, andμ41/2rC<1hold, thenut2andσt2are geometrically L2-NED on {et}.


Proof of Theorem 3(2-1): From the Assumption (A2) and the equation (2.9), we have


Throughout this paper, K denotes a generic constant. There is no loss of generality in setting r > S. Choose 0 < ε< 1. Then we have that


Define f (p) = (m/p) + εp (1 ≤ pm). Then the minimum value of f(p)=2ɛm if m > 1/ε. Thus we have that, for sufficiently large m,




for some η > 0 where r=r2ɛ<1 and ab = max{a, b}. The equality in the equation (2.20) is obtained from (1/2)mlog(rS)(-1/2-η)logm for large enough m. Combining Lemma 2 and the equation (2.20) yields the conclusion.

Proof of Theorem 3(2-2): Let S0=μ41/2S<1. Without loss of generality we assume that r > S0 and choose 0 < ε < 1. Then from (2.8) and (2.18)


Then by the same method used to prove the equation (2.19) and (2.20), we obtain that


for some η > 0 and sufficiently large m. From (2.15) and (2.21), ut2 and σt2 are L2-NED of size −1/2. Proof of Theorem 3(2–3): The baseline of the proof of Theorem 3(2–3) is the same as that of Theorem 2(a) in Davidson (2004). Choose ε > 0 such that S˜=j=1θj1-ɛ<1. Then


By using (2.7) and (2.22), we have that


where α = rε ∨ (rC)εS < 1 and ρ = − log α > 0. To prove the first equality in (2.23), note that

rɛmCɛm(C-ɛm-S˜m){2rɛm,if C-ɛS˜,2(rC)ɛm,if C-ɛ<S˜.

Then apply Lemma 2 to get ||ut-Et-mt+m(ut)||2O(e-(1/2)ρm), which implies that ut holds the geometric L2-NED property.

Proof of Theorem 3(2–4): Choose ε > 0 such that S˜=j=1θj1-ɛ<1, and μ4(1-ɛ)/2S˜<1. Now use the equations (2.8) and (2.22) to have that


where β=rɛ(μ41/2rC)ɛμ41/2S<1 and ρ* = − log β > 0. Combine the equation (2.15) and (2.24) to obtain the geometric L2-NED property of ut2 and σt2. Note that the first equality in (2.24) is obtained from the following inequality:

rɛm(1-(Cɛμ412S˜)m){2rɛm,if (Cμ412)-ɛμ41-ɛ2S˜,2(rCμ412)ɛm,if (Cμ412)-ɛ<μ41-ɛ2S˜.
Remark 1

Compared to the results in Davidson (2004), Theorems 2 and 3 weaken sufficient conditions for L2-NED property of ut, ut2, and σt2.

Theorem 4

If one of the following conditions (a)–(c) is satisfied then the FCLT holds for the process utgiven by (2.1) and (2.2):

  • the Assumption (A1) withδ > 1,

  • the Assumption (A1) withδ > 1/2 andμ41/2S<1,

  • the Assumption (A2).


Lemma 1 ensures that S < 1 implies the strict stationarity of σt2 with E(σt2)<. In Theorems 2 and 3, it is shown that ut is either L2-NED of size −1/2 or geometrically L2-NED under one of the above assumptions (a)–(c). Also, σn2=Var(t=1nut)=nE(σt2). Apply Theorem 1 to obtain the FCLT for ut.

Theorem 5
  • (3-1) If the Assumption (A1) withδ > 1/2 andμ41/2S<1, then the FCLT holds forut2andσt2.

  • (3-2) If the Assumption (A2) andμ41/2S<1, then the FCLT holds forut2andσt2.


Lemma 1 shows that the condition μ41/2S<1 is sufficient for the existence of E(ut4) and the existence of weakly stationary solution of the process ut2. Moreover, by Proposition 3.1 in Giraitis et al. (2000), μ41/2S<1 implies that


Also, from weak stationarity of ut2,


From (2.25) and (2.26), as n → ∞,


Proof of Theorem 5(3-1): Theorem 2(1–2) shows that under the assumptions, ut2 and σt2 are L2-NED of size −1/2. Therefore, the FCLT for ut2 and σt2 follows from (2.14), (2.15), (2.27) and Theorem 1.

Proof of Theorem 5(3-2): In Theorem 3(2-2), it is shown that under the given assumptions, ut2 and σt2 are L2-NED of size −1/2. Then the FCLT for ut2 and σt2 are obtained from (2.15), (2.21), (2.27), and Theorem 1.

Remark 2

Assume μ41/2S<1. It is known that if the exponential decay of the coefficient θj in (2.2) implies the exponential decay of the covariance function of the sequence {ut2}. On the other hand, if θjC j1δ, δ > 0, then the hyperbolic decay of the covariance function of ut2 is proved, that is, there exists K > 0 such that for t ≥ 1, Cov(ut2,u02)Kt-1-δ (Giraitis et al., 2000; Zaffaroni, 2004).

Example 1

Under proper constraints, conditional variance σt2 of various GARCH-type process can be rewritten as an ARCH(∞) model. The FCLT for various GARCH-type model including augmented GARCH, asymmetric power GARCH (APGARCH), vector GARCH (VGARCH), exponential GARCH (EGARCH) as well as the classical GARCH model is studied in Lee (2014a). For the classical GARCH model

ut=σtet,         σt2=ω+i=1pαiut-i2+j=1qβjσt-j2         (ω>0,   αi0,   βj0),

recall that if ∑αi + ∑βj < 1, then the process satisfies the Assumption (A2) and S < 1. Thus, Theorem 3(2–3) ensures that ∑αi + ∑βj < 1 is sufficient for ut to be L2-NED of size −1/2 and the FCLT holds for ut. Lee (2014a) shows that αiet2+βi2<1 is sufficient for the FCLT for ut2 and σt2. Note that μ4 ≥ 1 and αiet2+βi2(μ41/2αi+βi)<1 if μ41/2S<1.

Example 2

Results obtained in this section can be easily extended to a general ARCH(∞) model. Consider the following process

ut=σtet,         σtd=ω+j=1θjut-jd         (d>0,   ω>0,   θj0).

If E|e0|2d < ∞ and (E|e0|2d)1/2θj < 1, then a unique strictly stationary and weak stationary solution to (2.28) with E|ut|2d < ∞ exists. If θj in (2.28) satisfies the condition (A1) (or (A2)) and (E|e0|2d)1/2θj < 1, then the FCLT holds for |ut|d and σtd. If θj satisfies the condition (A2), then the FCLT holds for |ut|d/2.

Example 3

Consider the HYGARCH model which is given by

ut=σtet,         σt2=ω+θ(L)ut2,         (ω>0)

where θ(L) = 1−(δ(L)/β(L))(1+α((1− L)δ −1)) (α ≥ 0, δ ≥ 0). Here L is the lag operator defined by Lyt = yt1. HYGARCH model given by (2.29) includes IGARCH, FIGARCH, and classical GARCH models depending on the values of α and δ. If δ > 0, then S = 1 − (δ(1)/β(1))(1 − α). When δ in (2.29) is not too large, then this model will correspond closely to the following case

θ(L)=1-δ(L)β(L)(1-αϕ(L)),         ϕ(L)=ζ(1+δ)-1j=1j-1-δLj,         (δ>0)

and ζ(·) is the Riemann zeta function (Davidson, 2004). Note that δ > 1 in (2.29) gives rise to negative coefficients where as δ in (2.30) can take any positive values. Let δ > 1 in (2.30) and S = 1 − (δ(1)/β(1))(1 − α) < 1, then Theorem 4(a) yields the FCLT for ut in (2.29) with θ(L) given by (2.30).

Example 4

For an ARCH(∞) model in order to σt20 with probability 1, all its coefficients are expected to be nonnegative. In general, nonnegative coefficients condition for HYGARCH model are more complicated than those of FIGARCH (Conrad and Haag, 2006; Conrad, 2010). Li et al. (2015) suggests the following so called HGARCH process

ut=σtet,         σt2=γβ(1)+ω{1-δ(L)β(L)(1-L)δ}ut2,         (0<δ1,ω>0,γ>0).

The process given by (2.31) allows the existence of finite variance as in HYGARCH models, while it has a form nearly as simple as FIGARCH models. σt2 in (2.31) can be rewritten as σt2=γ/β(1)+j=1θjut-j2. When ω < 1, S = ∑θj = ω < 1 and there exists a unique strictly stationary solution ut2 to (2.31) with E(ut2)<. If in addition μ41/2ω<1, then applying Theorem 5 yields the FCLT for ut2 and σt2.

3. Simulations

3.1. Structural breaks of the ARCH(∞) model

As an application of the FCLT, we consider the cumulative sum (CUSUM) tests for mean break and variance break.

H0 : no structural breaks versus H1 : not H0.

The following CUSUM statistics are the most often used statistics to test for the stability of {f (ut) : 1 ≤ tn}:

QnM=1σ^nnmax1kn|1ikf(ui)-kn1inf(ui)|,         f(ui)=ui


QnV=1σ^nnmax1kn|1ikf(ui)-kn1inf(ui)|,         f(ui)=ui2,


σ^n2=1nj=1n(f(uj)-f(un)¯)2+2nj=1q(1-jq+1)i=1n-j(f(ui)-f(un)¯)(f(ui+j)-f(un)¯),         q<n

and f(un)¯=(1/n)i=1nf(ui), 0 ≤ in. According to Theorem 4 and 5, asymptotic null distributions of QnM and QnV are all standard Brownian bridges (Csörgő and Horváth, 1997; Hwang and Shin, 2013).

3.2. A Monte-Carlo study

We conduct a simulation to examine the finite sample sizes and powers of the CUSUM test for breaks. In this simulation study, we perform a test at a nominal level α = 0.05. The empirical sizes and powers are calculated as the rejection number of the null hypothesis out of 1,000 repetitions. In order to see the performance of Qn, we generate data by approximating ARCH(∞) by ARCH(10) model

ut=σtet,         σt2=ω+i=110θiut-i2,

where {et} is a sequence of independent standard normal errors. We evaluate Qn with sample sizes n = 1,000, 2,000, and 4,000. For power study of mean break tests, we add 0.002 to ut for all t > n/2. For power study of variance breaks test, we multiply 1.1 to et for all t > n/2. The parameters for the ARCH model are chosen as in Table 1: D1, D2, and D3 for ARCH(10) models with j=110θj=0.86,j=110θj=0.84, and j=110θj=0.90, respectively which are estimation results for three data sets that will be analyzed in Subsection 3.3 below.

The finite sample performance depends on the sample size n as well as the bandwidth parameter q used to estimate the long-run variance and covariance. Since the optimal bandwidth is O(n1/3) for the Bartlett kernel and the tests are very sensitive to q, we consider wide range of q values that are 1/3-order bandwidth: q1 = [2n1/3] and q2 = [4n1/3]. Table 2 summarizes the empirical sizes and powers of mean break tests.

Table 2 show that QnM has no severe size distortions in most cases. The empirical sizes are reasonably close to the nominal level 0.05 as n increases. Meanwhile, we can see that the powers are close to 0.9 when the sample size n is over 2000.

In Table 3, the size block shows that QnV has unstable sizes. In addition, the power values susbstantially decrease as q increases in ARCH(10) model. Since f (ut) is strongly autocorrelated when f(ut)=ut2, it is important to estimate long-run variance. The performance of estimator is sensitive to bandwidth q which is used to estimate σ2 and represents another research area in selecting an optimal bandwidth.

3.3. Real data analysis

In this section, we apply our tests to three real data sets: log-returns of the KOSPI, the S&P500 index, and the KRW/USD exchange rate during the period from January 2, 2007 to December 29, consisting of 2480, 2480, 2518 observations.

In Figure 1, we observe that the log-returns rapidly fluctuate and spike to a peak around the year 2009. It shows the volatility change during global financial crisis of 2008. Through the graphs, we find that three log-returns might have some breaks: in 2008 and in 2011.

We first apply the goodness-of-fit test to examine whether the ARCH(10) model fits the data well. Since the obtained p-values are 0.9467, 0.7265, and 0.8580, respectively, we conclude that these three data sets are well fitted to ARCH(10) model. We perform the CUSUM tests and CUSUMSQ tests for these data sets.

We see significant CUSUM test for the S&P500 index with p-values 2.1%, which implies the presence of at least one mean break. However, the KOSPI and the KRW/USD exchange rate have no significant p-values for the CUSUM tests. Therefore, the CUSUM test does not provide us statistical evidence for mean break for the KOSPI and the KRW/USD exchange rate (Table 4).

We now perform the CUSUMSQ tests for the KOSPI and the KRW/USD exchange rate, in which no mean shifts exist. In these cases we see significant CUSUMSQ tests for the KOSPI and the KRW/USD exchange rate with p-values 0.01% and 1.1%, respectively. The two data sets have at least one variance break; however, the result does not involve the number of breaks and the dates for the break times (Table 5).


This research was supported by Basic Science Research Program through the NRF funded by the Ministry of Education, Science and Technology (No. 2014R1A1A2039928 ).

Fig. 1. Time series plots of log-returns for 2007–2016.

Table 1

Parameters for DGP


Table 2

Size (%) and power (%) of CUSUM test QnM



Nominal level is 5%; number of replication is 1,000. CUSUM = cumulative sum.

Table 3

Size (%) and power (%) of CUSUMSQ test QnV



Nominal level is 5%; number of replication is 1,000. CUSUMSQ = cumulative sum of squares.

Table 4

CUSUM test QnM for log-returns for 2007–2016


CUSUM = cumulative sum.

Table 5

CUSUMSQ test QnV for log-returns for 2007–2016


CUSUMSQ = cumulative sum of squares.

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