Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA–GARCH/IGARCH models

# Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA–GARCH/IGARCH models

## Abstract

This paper investigates the asymptotic theory of the quasi-maximum exponential likelihood estimators (QMELE) for ARMA–GARCH models. Under only a fractional moment condition, the strong consistency and the asymptotic normality of the global self-weighted QMELE are obtained. Based on this self-weighted QMELE, the local QMELE is showed to be asymptotically normal for the ARMA model with GARCH (finite variance) and IGARCH errors. A formal comparison of two estimators is given for some cases. A simulation study is carried out to assess the performance of these estimators, and a real example on the world crude oil price is given.

[
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\doi

10.1214/11-AOS895 \volume39 \issue4 2011 \firstpage2131 \lastpage2163 \newproclaimasmAssumption[section] \newproclaimremRemark[section]

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QMELE for ARMA–GARCH/IGARCH models

{aug}

A]\fnmsKe \snmZhulabel=e1]mazkxaa@gmail.com and A]\fnmsShiqing \snmLing\corref\thanksreft1label=e2]maling@ust.hk

\thankstext

t1Supported in part by Hong Kong Research Grants Commission Grants HKUST601607 and HKUST602609.

class=AMS] \kwd62F12 \kwd62M10 \kwd62P20. ARMA–GARCH/IGARCH model \kwdasymptotic normality \kwdglobal self-weighted/local quasi-maximum exponential likelihood estimator \kwdstrong consistency.

## 1 Introduction

Assume that is generated by the ARMA–GARCH model

 yt = μ+p∑i=1ϕiyt−i+q∑i=1ψiεt−i+εt, (1) εt = ηt√htandht=α0+r∑i=1αiε2t−i+s∑i=1βiht−i, (2)

where , and is a sequence of i.i.d. random variables with . As we all know, since Engle (1982) and Bollerslev (1986), model (1)–(2) has been widely used in economics and finance; see Bollerslev, Chou and Kroner (1992), Bera and Higgins (1993), Bollerslev, Engel and Nelson (1994) and Francq and Zakoïan (2010). The asymptotic theory of the quasi-maximum likelihood estimator (QMLE) was established by Ling and Li (1997) and by Francq and Zakoïan (2004) when . Under the strict stationarity condition, the consistency and the asymptotic normality of the QMLE were obtained by Lee and Hansen (1994) and Lumsdaine (1996) for the GARCH model, and by Berkes, Horváth and Kokoszka (2003) and Francq and Zakoïan (2004) for the GARCH model. Hall and Yao (2003) established the asymptotic theory of the QMLE for the GARCH model when , including both cases in which and . Under the geometric ergodicity condition, Lang, Rahbek and Jensen (2011) gave the asymptotic properties of the modified QMLE for the first order AR–ARCH model. Moreover, when for some , the asymptotic theory of the global self-weighted QMLE and the local QMLE was established by Ling (2007) for model (1)–(2).

It is well known that the asymptotic normality of the QMLE requires and this property is lost when ; see Hall and Yao (2003). Usually, the least absolute deviation (LAD) approach can be used to reduce the moment condition of and provide a robust estimator. The local LAD estimator was studied by Peng and Yao (2003) and Li and Li (2005) for the pure GARCH model, Chan and Peng (2005) for the double AR(1) model, and Li and Li (2008) for the ARFIMA–GARCH model. The global LAD estimator was studied by Horváth and Liese (2004) for the pure ARCH model and by Berkes and Horváth (2004) for the pure GARCH model, and by Zhu and Ling (2011a) for the double AR() model. Except for the AR models studied by Davis, Knight and Liu (1992) and Ling (2005) [see also Knight (1987, 1998)], the nondifferentiable and nonconvex objective function appears when one studies the LAD estimator for the ARMA model with i.i.d. errors. By assuming the existence of a -consistent estimator, the asymptotic normality of the LAD estimator is established for the ARMA model with i.i.d. errors by Davis and Dunsmuir (1997) for the finite variance case and by Pan, Wang and Yao (2007) for the infinite variance case; see also Wu and Davis (2010) for the noncausal or noninvertible ARMA model. Recently, Zhu and Ling (2011b) proved the asymptotic normality of the global LAD estimator for the finite/infinite variance ARMA model with i.i.d. errors.

In this paper, we investigate the self-weighted quasi-maximum exponential likelihood estimator (QMELE) for model (1)–(2). Under only a fractional moment condition of with , the strong consistency and the asymptotic normality of the global self-weighted QMELE are obtained by using the bracketing method in Pollard (1985). Based on this global self-weighted QMELE, the local QMELE is showed to be asymptotically normal for the ARMA–GARCH (finite variance) and –IGARCH models. A formal comparison of two estimators is given for some cases.

To motivate our estimation procedure, we revisit the GNP deflator example of Bollerslev (1986), in which the GARCH model was proposed for the first time. The model he specified is an AR(4)–GARCH model for the quarterly data from 1948.2 to 1983.4 with a total of 143 observations. We use this data set and his fitted model to obtain the residuals . The tail index of is estimated by Hill’s estimator with the largest data of , that is,

 ^αη(k)=k∑kj=1(log~η143−j−log~η143−k),

where is the th order statistic of . The plot of is given in Figure 1. From this figure, we can see that when , and when . Note that Hill’s estimator is not so reliable when  is too small. Thus, the tail of is most likely less than 2, that is, . Thus, the setup that has a finite forth moment may not be suitable, and hence the standard QMLE procedure may not be reliable in this case. The estimation procedure in this paper only requires . It may provide a more reliable alternative to practitioners. To further illustrate this advantage, a simulation study is carried out to compare the performance of our estimators and the self-weighted/local QMLE in Ling (2007), and a new real example on the world crude oil price is given in this paper.

This paper is organized as follows. Section 2 gives our results on the global self-weighted QMELE. Section 3 proposes a local QMELE estimator and gives its limiting distribution. The simulation results are reported in Section 4. A real example is given in Section 5. The proofs of two technical lemmas are provided in Section 6. Concluding remarks are offered in Section 7. The remaining proofs are given in the Appendix.

## 2 Global self-weighted QMELE

Let be the unknown parameter of model (1)–(2) and its true value be , where and . Given the observations and the initial values , we can rewrite the parametric model (1)–(2) as

 εt(γ) = yt−μ−p∑i=1ϕiyt−i−q∑i=1ψiεt−i(γ), (3) ηt(θ) = εt(γ)/√ht(θ)and (4) ht(θ) = α0+r∑i=1αiε2t−i(γ)+s∑i=1βiht−i(θ).

Here, , and . The parameter space is , where , , and . Assume that and are compact and is an interior point in . Denote , , and . We introduce the following assumptions: {asm} For each , and when , and and have no common root with or . {asm} For each , and have no common root, and . {asm} has a nondegenerate distribution with .

Assumption 2 implies the stationarity, invertibility and identifiability of model (1), and Assumption 2 is the identifiability condition for model (2). Assumption 2 is necessary to ensure that is not almost surely (a.s.) a constant. When follows the standard double exponential distribution, the weighted log-likelihood function (ignoring a constant) can be written as follows:

 Lsn(θ)=1nn∑t=1wtlt(θ)andlt(θ)=log√ht(θ)+|εt(γ)|√ht(θ), (5)

where and is a measurable, positive and bounded function on with . We look for the minimizer, , of on , that is,

 ^θsn=argminΘLsn(θ).

Since the weight only depends on itself and we do not assume that  follows the standard double exponential distribution, is called the self-weighted quasi-maximum exponential likelihood estimator (QMELE) of . When is a constant, the self-weighted QMELE reduces to the weighted LAD estimator of the ARMA model in Pan, Wang and Yao (2007) and Zhu and Ling (2011b).

The weight is to reduce the moment condition of [see more discussions in Ling (2007)], and it satisfies the following assumption: {asm} for any , where .

When , the is the global QMELE and it needs the moment condition for its asymptotic normality, which is weaker than the moment condition as for the QMLE of in Francq and Zakoïan (2004). It is well known that the higher is the moment condition of , the smaller is the parameter space. Figure 2 gives the strict

stationarity region and regions for of the GARCH model: and , where . From Figure 2, we can see that the region for is very close to the region for strict stationarity of , and is much bigger than the region for .

Under Assumption 2, we only need a fractional moment condition for the asymptotic property of as follows: {asm} for some .

The sufficient and necessary condition of Assumption 2 is given in Theorem 2.1 of Ling (2007). In practice, we can use Hill’s estimator to estimate the tail index of and its estimator may provide some useful guidelines for the choice of . For instance, the quantity can be any value less than the tail index . However, so far we do not know how to choose the optimal . As in Ling (2007) and Pan, Wang and Yao (2007), we choose the weight function according to . When (i.e., ), we can choose the weight function as

 wt=(max{1,C−1∞∑k=11k9|yt−k|I{|yt−k|>C}})−4, (6)

where is a constant. In practice, it works well when we select as the 90% quantile of data . When (AR–ARCH model), for any , the weight can be selected as

 wt=(max{1,C−1p+r∑k=11k9|yt−k|I{|yt−k|>C}})−4.

When and or , the weight function need to be modified as follows:

 wt=(max{1,C−1∞∑k=11k1+8/ι|yt−k|I{|yt−k|>C}})−4.

Obviously, these weight functions satisfy Assumptions 2 and 2. For more choices of , we refer to Ling (2005) and Pan, Wang and Yao (2007). We first state the strong convergence of in the following theorem and its proof is given in the Appendix.

###### Theorem 2.1

Suppose has a median zero with . If Assumptions 22 hold, then

 ^θsn→θ0a.s., as n→∞.

To study the rate of convergence of , we reparameterize the weighted log-likelihood function (5) as follows:

 Ln(u)≡nLsn(θ0+u)−nLsn(θ0),

where . Let . Then, is the minimizer of on . Furthermore, we have

 Ln(u)=n∑t=1wtAt(u)+n∑t=1wtBt(u)+n∑t=1wtCt(u), (7)

where

 At(u) = 1√ht(θ0)[|εt(γ0+u1)|−|εt(γ0)|], Bt(u) = log√ht(θ0+u)−log√ht(θ0)+|εt(γ0)|√ht(θ0+u)−|εt(γ0)|√ht(θ0), Ct(u) = [1√ht(θ0+u)−1√ht(θ0)][|εt(γ0+u1)|−|εt(γ0)|].

Let be the indicator function. Using the identity

 |x−y|−|x| = −y[I(x>0)−I(x<0)] (8) +2∫y0[I(x≤s)−I(x≤0)]ds

for , we can show that

 At(u)=qt(u)[I(ηt>0)−I(ηt<0)]+2∫−qt(u)0Xt(s)ds, (9)

where , with

 q1t(u)=u′√ht(θ0)∂εt(γ0)∂θandq2t(u)=u′2√ht(θ0)∂2εt(ξ∗)∂θ∂θ′u,

and lies between and . Moreover, let and

 ξt(u)=2wt∫−q1t(u)0Xt(s)ds.

Then, from (9), we have

 n∑t=1wtAt(u)=u′T1n+Π1n(u)+Π2n(u)+Π3n(u), (10)

where

 T1n = n∑t=1wt√ht(θ0)∂εt(γ0)∂θ[I(ηt>0)−I(ηt<0)], Π1n(u) = n∑t=1{ξt(u)−E[ξt(u)|Ft−1]}, Π2n(u) = n∑t=1E[ξt(u)|Ft−1], Π3n(u) = n∑t=1wtq2t(u)[I(ηt>0)−I(ηt<0)] +2n∑t=1wt∫−qt(u)−q1t(u)Xt(s)ds.

By Taylor’s expansion, we can see that

 n∑t=1wtBt(u)=u′T2n+Π4n(u)+Π5n(u), (11)

where

 T2n = n∑t=1wt2ht(θ0)∂ht(θ0)∂θ(1−|ηt|), Π4n(u) = u′n∑t=1wt(38∣∣∣εt(γ0)√ht(ζ∗)∣∣∣−14)1h2t(ζ∗)∂ht(ζ∗)∂θ∂ht(ζ∗)∂θ′u, Π5n(u) = u′n∑t=1wt(14−14∣∣∣εt(γ0)√ht(ζ∗)∣∣∣)1ht(ζ∗)∂2ht(ζ∗)∂θ∂θ′u,

and lies between and .

We further need one assumption and three lemmas. The first lemma is directly from the central limit theorem for a martingale difference sequence. The second- and third-lemmas give the expansions of for and . The key technical argument is for the second lemma for which we use the bracketing method in Pollard (1985). {asm} has zero median with and a continuous density function satisfying and .

###### Lemma 2.1

Let . If Assumptions 22.1 hold, then

 1√nTn→dN(0,Ω0)as n→∞,

where denotes the convergence in distribution and

 Ω0=E(w2tht(θ0)∂εt(γ0)∂θ∂εt(γ0)∂θ′)+Eη2t−14E(w2th2t(θ0)∂ht(θ0)∂θ∂ht(θ0)∂θ′).
###### Lemma 2.2

If Assumptions 22.1 hold, then for any sequence of random variables such that , it follows that

 Π1n(un)=op(√n∥un∥+n∥un∥2),

where in probability as .

###### Lemma 2.3

If Assumptions 22.1 hold, then for any sequence of random variables such that , it follows that:

where

 Σ1=g(0)E(wtht(θ0)∂εt(γ0)∂θ∂εt(γ0)∂θ′)

and

 Σ2=18E(wth2t(θ0)∂ht(θ0)∂θ∂ht(θ0)∂θ′).

The proofs of Lemmas 2.2 and 2.3 are given in Section 6. We now can state one main result as follows:

###### Theorem 2.2

If Assumptions 22.1 hold, then:

where .

{pf}

(i) First, we have by Theorem 2.1. Furthermore, by (7), (10) and (11) and Lemmas 2.2 and 2.3, we have

 Ln(^un)=^u′nTn+(√n^un)′Σ0(√n^un)+op(√n∥^un∥+n∥^un∥2). (12)

Let be the minimum eigenvalue of . Then

 Missing or unrecognized delimiter for \biggl

Note that . By the previous inequality, it follows that

 √n∥^un∥≤[λmin+op(1)]−1[∥∥∥1√nTn∥∥∥+op(1)]=Op(1), (13)

where the last step holds by Lemma 2.1. Thus, (i) holds.

(ii) Let . Then, by Lemma 2.1, we have

 √nu∗n→dN(0,14Σ−10Ω0Σ−10)as n→∞.

Hence, it is sufficient to show that . By (12) and (13), we have

 Ln(^un) = (√n^un)′1√nTn+(√n^un)′Σ0(√n^un)+op(1) = (√n^un)′Σ0(√n^un)−2(√n^un)′Σ0(√nu∗n)+op(1).

Note that (12) still holds when is replaced by . Thus,

 Ln(u∗n) = (√nu∗n)′1√nTn+(√nu∗n)′Σ0(√nu∗n)+op(1) = −(√nu∗n)′Σ0(√nu∗n)+op(1).

By the previous two equations, it follows that

 Ln(^un)−Ln(u∗n) = (√n^un−√nu∗n)′Σ0(√n^un−√nu∗n)+op(1) (14) ≥ Missing or unrecognized delimiter for \bigr

Since a.s., by (14), we have . This completes the proof. {rem} When , the limiting distribution in Theorem 2.2 is the same as that in Li and Li (2008). When (ARMA model), it reduces to the case in Pan, Wang and Yao (2007) and Zhu and Ling (2011b). In general, it is not easy to compare the asymptotic efficiency of the self-weighted QMELE and the self-weight QMLE in Ling (2007). However, for the pure ARCH model, a formal comparison of these two estimators is given in Section 3. For the general ARMA–GARCH model, a comparison based on simulation is given in Section 4.

In practice, the initial values are unknown, and have to be replaced by some constants. Let , and be , and , respectively, when are constants not depending on parameters. Usually, are taken to be zeros. The objective function (5) is modified as

 ~Lsn(θ)=1nn∑t=1~wt~lt(θ)and~lt(θ)=log√~ht(θ)+|~εt(γ)|√~ht(θ).

To make the initial values ignorable, we need the following assumption. {asm} , where .

Let be the minimizer of , that is,

 ~θsn=argminΘ~Lsn(θ).

Theorem 2.3 below shows that and have the same limiting property. Its proof is straightforward and can be found in Zhu (2011).

###### Theorem 2.3

Suppose that Assumption 2 holds. Then, as ,

 i(i) if the assumptions of Theorem 2.1 hold ~θsn→θ0a.s., (ii) if the assumptions of Theorem 2.2 hold √n(~θsn−θ0)→dN(0,14Σ−10Ω0Σ−10).

## 3 Local QMELE

The self-weighted QMELE in Section 2 reduces the moment condition of , but it may not be efficient. In this section, we propose a local QMELE based on the self-weighted QMELE and derive its asymptotic property. For some special cases, a formal comparison of the local QMELE and the self-weighted QMELE is given.

Using in Theorem 2.2 as an initial estimator of , we obtain the local QMELE through the following one-step iteration:

 ^θn=^θsn−[2Σ∗n(^θsn)]−1T∗n(^θsn), (15)

where

 Σ∗n(θ) = n∑t=1{g(0)ht(θ)∂εt(γ)∂θ∂εt(γ)∂θ′+18h2t(θ)∂ht(θ)∂θ∂ht(θ)∂θ′}, T∗n(θ) = n∑t=1{1√ht(θ)∂εt(γ)∂θ[I(ηt(θ)>0)−I(ηt(θ)<0)] +12ht(θ)∂ht(θ)∂θ(1−|ηt(θ)|)}.

In order to get the asymptotic normality of , we need one more assumption as follows: {asm} or

 Eη2tr∑i=1α0i+s∑i=1β0i=1

with having a positive density on such that for all and for some .

Under Assumption 3, there exists a unique strictly stationary causal solution to GARCH model (2); see Bougerol and Picard (1992) and Basrak, Davis and Mikosch (2002). The condition is necessary and sufficient for under which model (2) has a finite variance. When , model (2) is called IGARCH model. The IGARCH model has an infinite variance, but for all under Assumption 3; see Ling (2007). Assumption 3 is crucial for the ARMA–IGARCH model. From Figure 2 in Section 2, we can see that the parameter region specified in Assumption 3 is much bigger than that for which is required for the asymptotic normality of the global QMELE. Now, we give one lemma as follows and its proof is straightforward and can be found in Zhu (2011).

###### Lemma 3.1

If Assumptions 22, 2.1 and 3 hold, then for any sequence of random variables such that , it follows that: