^{a}Department of Statistics, Ewha Womans University, Korea
Correspondence to:^{1}Corresponding 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.
Abstract
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 L_{2}-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, L_{2}-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.
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, L_{p}-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 L_{2}-NED condition among various dependence conditions. The proofs, in part, rely on results in Davidson (2004) and Giraitis et al. (2000).
Definition 1
y_{t}is said to be L_{2}-NED on {e_{t}} of size −λ_{0}if
where d_{t}is a sequence of positive constants andν(m) = O(m^{−}^{λ}) forλ > λ_{0}. Ifν(m) = O(e^{−}^{δ}^{m}) for someδ > 0, we say that the process is geometrically L_{2}-NED. Here we define${E}_{t-m}^{t+m}({y}_{t})=E({y}_{t}\mid \sigma ({e}_{t-m},\dots ,{e}_{t-1},{e}_{t},{e}_{t+1},\dots ,{e}_{t+m}))$.
Theorem 1
(Davidson, 2002)Suppose that the following Assumptions (1)–(3) hold: (1) y_{t}is L_{2}-NED of size −1/2 on the underlying i.i.d. process {e_{t}}. (2) sup_{t}E|y_{t} − E(y_{t})|^{r} < ∞ for some r ≥ 2. (3)${\sigma}_{n}^{2}/n\to {\sigma}^{2}>0$as n → ∞. Then the FCLT holds for y_{t}.
2. Functional central limit theorem for ARCH(∞) model
In this section, we consider a nonnegative coefficients ARCH(∞) model defined as follows.
where ω > 0, θ_{i} ≥ 0 for all i ≥ 1 are constants and {e_{t}} 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 $S={\sum}_{i=1}^{\infty}{\theta}_{i}$, μ^{n} = E(|e_{t}|^{n}), and ${M}_{n}=E({\sigma}_{t}^{n})$ for n = 1, 2, …. For a process y_{t}, define ${E}_{t-m}^{t+m}({y}_{t})=E({y}_{t}\mid {\mathcal{F}}_{t-m}^{t+m})$, where ${\mathcal{F}}_{s}^{t}=\sigma ({e}_{s},\dots ,{e}_{t})$ is a sigma field generated by {e_{j}, s ≤ j ≤ t}.
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.
is the unique nonanticipative strictly stationary solution to (2.1) and (2.2) with finite first moment$E({\sigma}_{t}^{2})$. If, in addition,${\mu}_{4}^{1/2}S<1$, 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 L_{2}-NED property and then to prove the FCLT for the process u_{t}.
Lemma 2
Consider the processes u_{t}and${\sigma}_{t}^{2}$given by (2.1) and (2.2). Then we obtain that$${\Vert {u}_{t}-{E}_{t-m}^{t+m}({u}_{t})\Vert}_{2}^{2}\le E\left|{\sigma}_{t}^{2}-{E}_{t-m}^{t+m}\left({\sigma}_{t}^{2}\right)\right|.$$
The first inequality in (2.5) follows from the relation: ||Y −E(Y|X)||_{2} ≤ ||Y −g(X)||_{2} for any measurable function g. For the second inequality in (2.5), we apply the relation ${(\sqrt{a}-\sqrt{b})}^{2}\le \mid a-b\mid (a>0,b>0)$.
Now, we first consider the processes which have hyperbolically decaying lag coefficients.
(A1) 0 ≤ θ_{j} ≤ Cj^{−}^{1}^{–}^{δ} for j ≥ 1, C > 0, δ > 0, and S < 1.
Theorem 2
(1-1) If the Assumption (A1) holds, then u_{t}given by (2.1) and (2.2) is L_{2}-NED on {e_{t}}, of size −λ/2, withδ > λ > 0.
(1–2) If the assumption (A1) and${\mu}_{4}^{1/2}S<1$hold, then u_{t}, ${u}_{t}^{2}$, and${\sigma}_{t}^{2}$are L_{2}-NED on {e_{t}} of size −λ, withδ > λ > 0.
Hence ${u}_{t}^{2}$ and ${\sigma}_{t}^{2}$ hold the L_{2}-NED property of size −λ with δ > λ > 0.
On the other hand, using the inequalities ${\sigma}_{t}^{2}\ge \omega ,{E}_{t-m}^{t+m}{\sigma}_{t}^{2}\ge \omega $, and $\mid \sqrt{a}-\sqrt{b}\mid \le \mid a-b\mid $ if a, b ≥ 1 yields that
Define f (p) = (m/p) + εp (1 ≤ p ≤ m). Then the minimum value of $f(p)=2\sqrt{\varepsilon}\sqrt{m}$ if m > 1/ε. Thus we have that, for sufficiently large m,
for some η > 0 where ${r}^{\prime}={r}^{2\sqrt{\varepsilon}}<1$ and a ∨ b = max{a, b}. The equality in the equation (2.20) is obtained from $(1/2)\sqrt{m}\text{log}({r}^{\prime}\vee S)\le (-1/2-\eta )\text{log}\hspace{0.17em}m$ for large enough m. Combining Lemma 2 and the equation (2.20) yields the conclusion.
Proof of Theorem 3(2-2): Let ${S}_{0}={\mu}_{4}^{1/2}S<1$. Without loss of generality we assume that r > S_{0} and choose 0 < ε < 1. Then from (2.8) and (2.18)
for some η > 0 and sufficiently large m. From (2.15) and (2.21), ${u}_{t}^{2}$ and ${\sigma}_{t}^{2}$ are L_{2}-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 $\tilde{S}={\sum}_{j=1}^{\infty}{\theta}_{j}^{1-\varepsilon}<1$. Then
Then apply Lemma 2 to get ${\left|\right|{u}_{t}-{E}_{t-m}^{t+m}({u}_{t})\left|\right|}_{2}\le O({e}^{-(1/2)\rho m})$, which implies that u_{t} holds the geometric L_{2}-NED property.
Proof of Theorem 3(2–4): Choose ε > 0 such that $\tilde{S}={\sum}_{j=1}^{\infty}{\theta}_{j}^{1-\varepsilon}<1$, and ${\mu}_{4}^{(1-\varepsilon )/2}\tilde{S}<1$. Now use the equations (2.8) and (2.22) to have that
where $\beta ={r}^{\varepsilon}\vee {({\mu}_{4}^{1/2}rC)}^{\varepsilon}\vee {\mu}_{4}^{1/2}S<1$ and ρ^{*} = − log β > 0. Combine the equation (2.15) and (2.24) to obtain the geometric L_{2}-NED property of ${u}_{t}^{2}$ and ${\sigma}_{t}^{2}$. Note that the first equality in (2.24) is obtained from the following inequality:
Compared to the results in Davidson (2004), Theorems 2 and 3 weaken sufficient conditions for L_{2}-NED property of u_{t}, ${u}_{t}^{2}$, and ${\sigma}_{t}^{2}$.
Theorem 4
If one of the following conditions (a)–(c) is satisfied then the FCLT holds for the process u_{t}given by (2.1) and (2.2):
the Assumption (A1) withδ > 1,
the Assumption (A1) withδ > 1/2 and${\mu}_{4}^{1/2}S<1$,
the Assumption (A2).
Proof
Lemma 1 ensures that S < 1 implies the strict stationarity of ${\sigma}_{t}^{2}$ with $E({\sigma}_{t}^{2})<\infty $. In Theorems 2 and 3, it is shown that u_{t} is either L_{2}-NED of size −1/2 or geometrically L_{2}-NED under one of the above assumptions (a)–(c). Also, ${\sigma}_{n}^{2}=\text{Var}({\sum}_{t=1}^{n}{u}_{t})=nE({\sigma}_{t}^{2})$. Apply Theorem 1 to obtain the FCLT for u_{t}.
Theorem 5
(3-1) If the Assumption (A1) withδ > 1/2 and${\mu}_{4}^{1/2}S<1$, then the FCLT holds for${u}_{t}^{2}$and${\sigma}_{t}^{2}$.
(3-2) If the Assumption (A2) and${\mu}_{4}^{1/2}S<1$, then the FCLT holds for${u}_{t}^{2}$and${\sigma}_{t}^{2}$.
Proof
Lemma 1 shows that the condition ${\mu}_{4}^{1/2}S<1$ is sufficient for the existence of $E({u}_{t}^{4})$ and the existence of weakly stationary solution of the process ${u}_{t}^{2}$. Moreover, by Proposition 3.1 in Giraitis et al. (2000), ${\mu}_{4}^{1/2}S<1$ implies that
Proof of Theorem 5(3-1): Theorem 2(1–2) shows that under the assumptions, ${u}_{t}^{2}$ and ${\sigma}_{t}^{2}$ are L_{2}-NED of size −1/2. Therefore, the FCLT for ${u}_{t}^{2}$ and ${\sigma}_{t}^{2}$ 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, ${u}_{t}^{2}$ and ${\sigma}_{t}^{2}$ are L_{2}-NED of size −1/2. Then the FCLT for ${u}_{t}^{2}$ and ${\sigma}_{t}^{2}$ are obtained from (2.15), (2.21), (2.27), and Theorem 1.
Remark 2
Assume ${\mu}_{4}^{1/2}S<1$. It is known that if the exponential decay of the coefficient θ_{j} in (2.2) implies the exponential decay of the covariance function of the sequence {${u}_{t}^{2}$}. On the other hand, if θ_{j} ≤ C j^{−}^{1}^{–}^{δ}, δ > 0, then the hyperbolic decay of the covariance function of ${u}_{t}^{2}$ is proved, that is, there exists K > 0 such that for t ≥ 1, $\text{Cov}({u}_{t}^{2},{u}_{0}^{2})\le K{t}^{-1-\delta}$ (Giraitis et al., 2000; Zaffaroni, 2004).
Example 1
Under proper constraints, conditional variance ${\sigma}_{t}^{2}$ 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
recall that if ∑α_{i} + ∑β_{j} < 1, then the process satisfies the Assumption (A2) and S < 1. Thus, Theorem 3(2–3) ensures that ∑α_{i} + ∑β_{j} < 1 is sufficient for u_{t} to be L_{2}-NED of size −1/2 and the FCLT holds for u_{t}. Lee (2014a) shows that $\sum {\Vert {\alpha}_{i}{e}_{t}^{2}+{\beta}_{i}\Vert}_{2}<1$ is sufficient for the FCLT for ${u}_{t}^{2}$ and ${\sigma}_{t}^{2}$. Note that μ_{4} ≥ 1 and $\sum {\Vert {\alpha}_{i}{e}_{t}^{2}+{\beta}_{i}\Vert}_{2}\le \sum ({\mu}_{4}^{1/2}{\alpha}_{i}+{\beta}_{i})<1$ if ${\mu}_{4}^{1/2}S<1$.
Example 2
Results obtained in this section can be easily extended to a general ARCH(∞) model. Consider the following process
If E|e_{0}|^{2}^{d} < ∞ and (E|e_{0}|^{2}^{d})^{1}^{/}^{2} ∑θ_{j} < 1, then a unique strictly stationary and weak stationary solution to (2.28) with E|u_{t}|^{2}^{d} < ∞ exists. If θ_{j} in (2.28) satisfies the condition (A1) (or (A2)) and (E|e_{0}|^{2}^{d})^{1}^{/}^{2} ∑θ_{j} < 1, then the FCLT holds for |u_{t}|^{d} and ${\sigma}_{t}^{d}$. If θ_{j} satisfies the condition (A2), then the FCLT holds for |u_{t}|^{d}^{/}^{2}.
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 Ly_{t} = y_{t}_{−}_{1}. 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
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 u_{t} in (2.29) with θ(L) given by (2.30).
Example 4
For an ARCH(∞) model in order to ${\sigma}_{t}^{2}\ge 0$ 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
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. ${\sigma}_{t}^{2}$ in (2.31) can be rewritten as ${\sigma}_{t}^{2}=\gamma /\beta (1)+{\sum}_{j=1}^{\infty}{\theta}_{j}{u}_{t-j}^{2}$. When ω < 1, S = ∑θ_{j} = ω < 1 and there exists a unique strictly stationary solution ${u}_{t}^{2}$ to (2.31) with $E({u}_{t}^{2})<\infty $. If in addition ${\mu}_{4}^{1/2}\omega <1$, then applying Theorem 5 yields the FCLT for ${u}_{t}^{2}$ and ${\sigma}_{t}^{2}$.
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.
H_{0} : no structural breaks versus H_{1} : not H_{0}.
The following CUSUM statistics are the most often used statistics to test for the stability of {f (u_{t}) : 1 ≤ t ≤ n}:
and $\overline{f({u}_{n})}=(1/n){\sum}_{i=1}^{n}f({u}_{i})$, 0 ≤ i ≤ n. According to Theorem 4 and 5, asymptotic null distributions of ${Q}_{n}^{M}$ and ${Q}_{n}^{V}$ 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 Q_{n}, we generate data by approximating ARCH(∞) by ARCH(10) model
where {e_{t}} is a sequence of independent standard normal errors. We evaluate Q_{n} with sample sizes n = 1,000, 2,000, and 4,000. For power study of mean break tests, we add 0.002 to u_{t} for all t > n/2. For power study of variance breaks test, we multiply 1.1 to e_{t} for all t > n/2. The parameters for the ARCH model are chosen as in Table 1: D_{1}, D_{2}, and D_{3} for ARCH(10) models with ${\sum}_{j=1}^{10}{\theta}_{j}=0.86,{\sum}_{j=1}^{10}{\theta}_{j}=0.84$, and ${\sum}_{j=1}^{10}{\theta}_{j}=0.90$, 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(n^{1}^{/}^{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: q_{1} = [2n^{1}^{/}^{3}] and q_{2} = [4n^{1}^{/}^{3}]. Table 2 summarizes the empirical sizes and powers of mean break tests.
Table 2 show that ${Q}_{n}^{M}$ 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 ${Q}_{n}^{V}$ has unstable sizes. In addition, the power values susbstantially decrease as q increases in ARCH(10) model. Since f (u_{t}) is strongly autocorrelated when $f({u}_{t})={u}_{t}^{2}$, it is important to estimate long-run variance. The performance of estimator is sensitive to bandwidth q which is used to estimate σ^{2} 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).
Acknowledgments
This research was supported by Basic Science Research Program through the NRF funded by the Ministry of Education, Science and Technology (No. 2014R1A1A2039928 ).
Figures
Fig. 1. Time series plots of log-returns for 2007–2016.
TABLES
Table 1
Parameters for DGP
DGP
θ_{1}
θ_{2}
θ_{3}
θ_{4}
θ_{5}
θ_{6}
θ_{7}
θ_{8}
θ_{9}
θ_{10}
D_{1}
0.054
0.130
0.071
0.098
0.153
0.038
0.074
0.048
0.105
0.084
D_{2}
0.046
0.135
0.092
0.105
0.074
0.062
0.074
0.093
0.091
0.063
D_{3}
0.156
0.135
0.074
0.102
0.106
0.019
0.049
0.055
0.071
0.131
Table 2
Size (%) and power (%) of CUSUM test ${Q}_{n}^{M}$
n
q
D_{1}
D_{2}
D_{3}
Size
Power
Size
Power
Size
Power
1,000
20
4.0
84.3
3.9
96.3
3.7
92.1
1,000
40
2.9
83.1
2.7
93.4
3.9
90.4
2,000
25
4.3
92.6
4.4
98.7
4.1
96.8
2,000
50
4.1
93.7
3.7
96.2
3.3
98.2
4,000
31
3.9
97.4
3.8
99.3
4.2
98.9
4,000
63
4.5
98.9
3.9
99.7
4.3
99.8
Nominal level is 5%; number of replication is 1,000. CUSUM = cumulative sum.
Table 3
Size (%) and power (%) of CUSUMSQ test ${Q}_{n}^{V}$
n
q
D_{1}
D_{2}
D_{3}
Size
Power
Size
Power
Size
Power
1,000
20
21.9
40.4
23.4
47.0
27.6
44.1
1,000
40
4.3
13.2
5.6
23.4
6.5
16.8
2,000
25
23.4
45.9
20.9
54.5
25.9
44.7
2,000
50
6.2
35.3
5.7
30.1
7.0
24.1
4,000
31
21.3
72.1
15.0
66.7
20.7
51.8
4,000
63
5.6
51.8
5.5
62.5
5.3
29.4
Nominal level is 5%; number of replication is 1,000. CUSUMSQ = cumulative sum of squares.
Table 4
CUSUM test ${Q}_{n}^{M}$ for log-returns for 2007–2016
${Q}_{n}^{M}$
p-value(%)
KOSPI
0.768
59.7
S&P500
1.511
2.1
KRW/USD
1.179
12.4
CUSUM = cumulative sum.
Table 5
CUSUMSQ test ${Q}_{n}^{V}$ for log-returns for 2007–2016
${Q}_{n}^{V}$
p-value(%)
KOSPI
2.289
0.01
KRW/USD
1.610
1.10
CUSUMSQ = cumulative sum of squares.
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