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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo


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Volume 15, Issue 1

Issues

Volume 13 (2015)

Quasi-maximum likelihood estimator of Laplace (1, 1) for GARCH models

Haiyan Xuan
  • Corresponding author
  • School of Mathematical Sciences, Dalian University of Technology, Dalian, China
  • School of Economics and Management, Lanzhou University of Technology, Lanzhou, China
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/ Lixin Song / Muhammad Amin / Yongxia Shi
Published Online: 2017-12-29 | DOI: https://doi.org/10.1515/math-2017-0131

Abstract

This paper studies the quasi-maximum likelihood estimator (QMLE) for the generalized autoregressive conditional heteroscedastic (GARCH) model based on the Laplace (1,1) residuals. The QMLE is proposed to the parameter vector of the GARCH model with the Laplace (1,1) firstly. Under some certain conditions, the strong consistency and asymptotic normality of QMLE are then established. In what follows, a real example with Laplace and normal distribution is analyzed to evaluate the performance of the QMLE and some comparison results on the performance are given. In the end the proofs of some theorem are presented.

Keywords: Asymptotic normality; GARCH model; Laplace (1,1); Quasi-maximum likelihood estimator; Strong consistency

MSC 2010: 62M10; 91G70

1 Introduction

The ARCH model has been widely used ever since it was first proposed by Engle (1982)[1] because this model was able to address the volatility in the forecasting of Britain’s inflation rate. In many statistical applications, particularly finance, the ARCH model is the leading way to explain changes in the conditional variance of the error term over time. In recent years, the ARCH model has been extended to the generalized-ARCH (GARCH) model (see Bollerslev (1986)[2]). Since the GARCH model can explain the phenomena of volatility convergence and the thick tail of the rate of return (see David (2014)[3] and Yang (2008)[4]) it has drawn widespread concern from many scholars and has many applications.

Recently, some advances have been made for the structure and parameter estimation of the GARCH model. Weiss (1986)[5] established some results on the asymptotic properties of the QMLE depending on assumptions of moment conditions. Lee (1994)[6] and Lumsdaine (1996)[7] studied the asymptotic properties of the QMLE for the GARCH model. Berkes et al. (2003)[8] studied the structure and estimator of GARCH. Berkes and Horvath (2003, 2004)[9, 10] provided consistency convergence rate that is QMLE and validity of parameter estimation for general GARCH(r, s). Francq and Zakoian (2004)[11] studied the QMLE for GARCH(r, s). Straumann (2006)[12] presented the QMLE by a stochastic recurrence method, which includes GARCH(r, s). Ling (2007)[13] proposed a self-weighted QMLE, and Zhu (2011)[14] investigated the local QMLE for IGARCH models under a fractional moment condition only. The theoretical properties of the QMLE in the GARCH model need to be developed further, especially in statistical applications, to include situations where these sorts of moment conditions are not satisfied. Han and Kristensen (2014)[15] applied the asymptotic properties of Gaussian QMLE to the GARCH model with an additional explanatory variable, and showed that the QMLE of the parameters for the volatility equation is consistent and mixed-normally distributed in large samples.

Although the literature on classical GARCH models is quite rich, most of it is based on residuals of GARCH model, which follow a normal distribution, as noted by Francq and Zakoian (2004)[11]. Nelson (1991)[16] used other distributions to investigate the GARCH model. In this paper, we consider the Laplace distribution since this distribution is worthy of being studied because it describes the fat-tail feature of financial market data. This paper mainly investigates the QMLE for the GARCH model based on Laplace distribution. The theoretical results on strong consistency and asymptotic normality of the QMLE are established. A performance comparison between Laplace distribution and normal distribution is made to show that the former is superior to the later.

The article is organized as follows. The main results for the QMLE of GARCH(r, s) are given based on Laplace distribution in the second section. In the third section a practical instance is described. The proofs of two theorems are in the end.

2 Main results

In this section we investigate the quasi-maximum likelihood estimator (QMLE) for the generalized autoregressive conditional heteroscedastic (GARCH) model based on the Laplace (1,1) residuals to propose some theoretical results.

The GARCH (r,s) model has the following form: εt=ηtσt,σt2=α0+i=1rαiεti2+j=1sβjσtj2,(1)

Here, ηt is a sequence of independent and identical distributed (i.i.d.) random variables, α0 > 0, αi ≥ 0, i = 1, 2, ⋯, r; βj ≥ 0, j = 1, 2, ⋯, s. According to Bougerol and Picard(1992)[17], the sufficient and necessary condition for the strictly stationary solution of GARCH (r, s) model is i=1rαi+j=1sβj<1.(2)

Assume that θ = (α0, α1, ⋯, αr, β1, β2, ⋯, βs)′ is the parameter vector of formula (1) and its true parameter vector is θ0. Let l = r + s + 1, then θ is l dimension vector. The parameter vector space is Θ, ΘR0r+s+1, R0 = [0, ∞). By assumption that a Laplace distribution has the density f(x) = 0.5e−|x−1| for ηt and conditionally on initial values ε0, ⋯, ε1−r, σ02,,σ1s2, then the Laplace quasi-likelihood is Ln(θ)=Ln(θ;ε0,,εn)=t=1n12σt2exp(|εt1|σt2)

with regard to t ≥ 1, where σt2=σt2(θ)=α0+i=1rαiεti2+j=1sβjσtj2.

We select the initial values ε0==ε1r=ω,σ02==σ1s2=ω(3)

or ε0==ε1r=ε1,σ02==σ1q2=ε12.(4)

As a result, θ̂n is named the QMLE for θ and has the following form θ^n=argmaxθΘLn(θ)=argminθΘIn(θ),(5)

where In(θ)=n1t=1nlt,lt=lt(θ)=logσt(θ)+|εt1|σt(θ).(6)

Denote α(z)=i=1rαizi,β(z)=1j=1sβjzj. If r = 0, α(z) = 0; if s = 0, β(z) = 1. Before providing main results, we introduce firstly the following assumptions.

Assumption 1

θ0Θ, Θ is compact and θ0 is an inner dot.

Assumption 2

j=1rβj<1.

Assumption 3

E[|εt1|σt]=1.

Assumption 4

If s > 0, there are no common roots for α(z) and β(z), α(1) ≠ 0, αr + βs ≠ 0.

Assumption 5

τ=E[|εt1|σt]2<.

Assumption 1 ensures the parameter vector space is compact and is required to prove asymptotic normality. Assumption 2 and 4 are the identifiability conditions for model (1). Assumption 3 is a necessary condition to prove the strong consistency and Assumption 5 is asymptotic normality.

Actually, the initial values of εt and σt2 are unknown when t ≤ 0. Let ε̃t(θ) and σ~t2 (θ) be εt(θ) and σt2 (θ), respectively, when εt and σt2 (θ) are constants when t ≤ 0. The formula (6) can be modified as I~n(θ)=I~n(θ;εn,εn1,)=n1t=1nl~t,(7) l~t=l~t(θ)=logσ~t2(θ)+|εt1|σ~t2(θ).(8)

In what follows we establish the main results of this paper.

Theorem 2.1

Under the initial values (3) or (4), if the Assumptions 1-5 hold, then there exists a sequence of minimizers θ̂n of In(θ) such that θ^nθ0a.s.,asn.

Theorem 2.2

If the Assumptions 1-5 hold, then n(θ^nθ0)N(0,MJ1)asn,

where M=τ14,J=Eθ02lt(θ0)θθT=Eθ01σt4(θ0)σt2(θ0)θσt2(θ0)θT.(9)

3 Applications and comparison

In this section, the China Securities Index 800 (CSI 800) from January 12, 2007 to December 31, 2008 is studied. There are 482 data points. The descriptive statistics of data subjected to differential, denoted by {yt}t=1481 are shown in Figure 1.

Log-return descriptive statistics of CSI 800
Figure 1

Log-return descriptive statistics of CSI 800

As shown in Figure 1, the mean was near 0. At the 0.05 significance level, the value of J-Bera statistic is greater than the critical value. This indicates that the regression may not follow the normal distribution. It can be initially determined that the distribution of the return presents “fat tail” feature.

When the significance level is 1%, 5% and 10%, the value of the test statistic t is smaller than the critical value in Table 1. In this way, the sequence rejects the null hypothesis, which the unit root exists. It is also a stationary series. As shown in Table 2, n * R2 = 3.273568 > χ0.12(1), which indicates the sequence has heteroskedasticity.

Table 1

ADF unit root test

Table 2

ARCH test

In this way, the GARCH(1,1) model was established according to {yt}. When ηt obeyed the Laplace (1, 1), the estimation of the parameters vector was performed by using the QMLE at MATLAB.

The results were α0 = 0.0701, α1 = 0.1381, β1 = 0.4605. As α1 + β1 = 0.5986 < 1, which satisfies the strictly stationary condition. Next, the sample biases, the sample standard deviations (SD) and the asymptotic standard deviations (AD) of the estimation on the parameter vector were given when ηt obeyed the N(0, 1) and ηt obeyed the Laplace(1, 1).

The bias is a technical index that reflects the degree of violation of the stock price and the moving average in the process of fluctuation. The computational formula is: bias=ClosingpriceMovingaveragepriceinNdaysMovingaveragepriceinNdays×100%

SD is the square root of the arithmetic mean of deviation from the mean square. AD is the standard deviation of the asymptotic distribution of deviation. They all reflect the degree of dispersion among individuals in the group. Therefore, these indexes can be used to analyze the accuracy of the estimation on the parameter vector. In general, the estimation is much more accurate only if the values of bias, AD and SD are much smaller.

As shown in Table 3, when ηt ∼ Laplace(1, 1), the values of the bias, the SD, and the AD were all smaller than the ones when ηt ∼ N(0,1). This indicated that the fitting effect of Laplace distribution is better than that of normal distribution. It is hereby suggested that instead of the Normal distribution the Laplace distribution is much more effective for the data from financial markets.

Table 3

Estimators for GARCH(1, 1)

4 Proofs

In this section we will present the proof of Theorem 2.1 and Theorem 2.2.

Proof of Theorem 2.1

The formula σt2=α0+i=1rαiεti2+j=1sβjσtj2 in model (1) can be rewritten in vector form as σ~_t2=c~_t+Bσ~_t12,(10)

where σ~_t2=σt2σt12σts+12,c~t_=α0+i=1rαiεti200,B=β1β2βr100010.(11)

Then following intermediate results can be used to prove Theorem 2.1.

  1. limn→∞ supθΘ|In (θ) − Ĩn(θ)| = 0 a.s.

  2. tZ makes σt2 (θ) = σt2 (θ 0) Pθ0 a.s. ⇒ θ = θ0.

  3. Eθ0 | lt(θ0)|<∞, and if θθ0, Eθ0lt(θ) > Eθ0lt(θ0).

  4. For θθ0, there is a neighbourhood V(θ) making lim infn→∞ infθ*V(θ) Ĩn(θ*)>Eθ0l1(θ0) a.s.

  1. As Assumption 1, supθΘρ(B)<1.(12)

    Iterating (10) produces the following: σ_t2=c_t+Bc_t1+B2c_t1++Bt1c_1+Btσ_02=k=0Bkc_tk.(13)

    It is supposed here that σ~_t2 may be the vector obtained by replacing σti2 by σ~ti2. Let be the vector obtained by replacing ε02,,ε1p2 with the initial values (3) or (4). We have σ~_t2=c_t+Bc~_t1++Btr1c~_r+1+Btrc~_r++Bt1c~_1+Btσ~_02.(14)

    Through (12)-(14), it is almost certain the following is true: supθΘσ_t2σ~_t2=supθΘk=1rBtk(c_kc~_k)+Bt(σ_02σ~_02)Kρt,t.(15)

    Hence, supθΘ|In(θ)I~n(θ)|n1t=1nsupθΘσ~tσtσtσ~t(εt1)+12log1+σt2σ~t2σ~t2n1t=1nsupθΘσ~t2σt2σ~t2σt2(εt1)2+12log1+σt2σ~t2σ~t2KnsupθΘ1α0t=1nρt(εt1)2+K2nsupθΘ1α0t=1nρt.

    By the Markov inequality, the following equation can be determined: t=1P(ρt(εt1)2>ϵ)t=1E(ρt(εt1)2)mϵm<.

    From the Borel-Cantelli lemma, (i) is obtained.

  2. It’s obvious that the result (ii) can be easily proved with Assumption 2 and Assumption 4.

  3. Because of Eθ0lt(θ)Eθ0lnσt2max{0,lnω}<. It remains to be shown that Eθ0lt+(θ) < ∞. By Jenson inequality, Eθ0logσt(θ0)=12Eθ0logσt2(θ0)=12Eθ01mlog{σt2(θ0)}m12mlogEθ0{σt2(θ0)}m<,

    Thus, Eθ0lt(θ0)=Eθ012logσt2(θ0)+|εt1|σt(θ0)=1+12Eθ0logσt2(θ0)<.

    Therefore, Eθ0lt(θ)Eθ0lt(θ0)=Eθ0lnσt(θ)σt(θ0)+Eθ0σt(θ0)σt(θ)1.

    For all x > 0, log xx − 1, where the equality is true if and only if x = 1. Thus, it is true that Eθ0lt(θ)Eθ0lt(θ0)Eθ0logσt(θ)σt(θ0)+logσt(θ0)σt(θ)=0(16)

    It is noted that the equality in (16) holds if σt(θ0) = σt(θ).

  4. From result (i), lim infninfθVk(θ)ΘI~n(θ)lim infninfθVk(θ)ΘIn(θ)lim supnsupθΘ|I~n(θ)In(θ)|lim infn1nt=1ninfθVk(θ)Θlt(θ)

    Based on the ergodic theorem, Beppo-Levi theorem and the formula (16), the result (iv) can be proved.

By compactness theory, the proof of Theorem 2.1 is finished.□

Proof of Theorem 2.2

Through a Taylor expansion at θ0, 0=n1/2t=1nθlt(θ^n)=n1/2t=1nθlt(θ0)+1nt=1n2θiθjlt(θij)n(θ^nθ0),

which indicates that both n1/2t=1nθlt(θ0)N(0,(τ14)J)(17)

and n1t=1n2θiθjlt(θij)J(i,j)inprobability.(18)

hold. The proof of Theorem 2.2 is divided into the following six conclusions.

  1. Eθ0∥(∂lt(θ0)/∂θ)(∂lt(θ0)/∂θT)∥<∞, Eθ02lt(θ0)/∂θ∂θT∥<∞.

  2. J is nonsingular, varθ0 {∂lt(θ0)/∂θ} = MJ.

  3. There exists a neighborhood V(θ0) of θ, with regard to i, j, k ∈ {1, ⋯, r + s + 1}, such that Eθ0supθV(θ0)3lt(θ)θiθjθk<.

  4. n−1/2t=1n {∂lt(θ0)/θt(θ0)/θ} ∥ → 0 when n → ∞, supθV(θ0)n−1 t=1n {2lt(θ)/θ∂θT2t(θ)/∂θ∂θT} ∥ → 0 in probability.

  5. n−1/2 t=1n ∂lt(θ0)/∂θN(0, MJ).

  6. n−1 t=1n 2lt(θij)/∂θi∂θjJ(i, j) a.s.

  1. Because of lt(θ) = ln σt2+|εt1|/σt2, it is true that ltθ=12σt2σt2θ+|εt1|21(σt2)3/2σt2θ=121|εt1|σt21σt2σt2θ,(19) 2ltθθT=2σt2θθT12σt2|εt1|21σt6+σt2θσt2θT3|εt1|41σt10121σt4=121σt22σt2θθT1|εt1|σt2+121σt4σt2θσt2θT32|εt1|σt21.(20)

    For θ = θ0, we have Eθ01σt2σt2θ(θ0)<,Eθ01σt22σt2θθT(θ0)<,Eθ01σt4σt2θσt2θT(θ0)<.

    The proof of (i) is finished.

  2. From (i), it have Eθ0lt(θ0)θ=Eθ012|εt1|2σt2Eθ01σt2(θ0)σt2(θ0)θ=0.

    Since (20), (i) and (9), the following must also be determined. varθ0lt(θ0)θ=Eθ0lt(θ0)θlt(θ0)θ=E(1|εt1|σt)24Eθ01σt4(θ0)σt2(θ0)θσt2(θ0)θ=τ14J.(21)

    This shows that J is non-singular. So we establish the conclusion of (ii).

  3. It is shown in (20) that lt(θ) is differentiated. Then 3lt(θ)θiθjθk=121|εt1|σt21σt23σt2θiθjθk+1158|εt1|σt21σt6σt2θiσt2θjσt2θk+121σt42σt2θiθjσt2θk32|εt1|σt21+121σt42σt2θiθkσt2θj32|εt1|σt21+121σt42σt2θjθkσt2θi32|εt1|σt21.

    Since Eθ0supθV(θ0)|εt1|σt<,Eθ0supθV(θ0)1σt23σt2θiθjθk<,Eθ0supθV(θ0)|1σt2σt2θi|<,Eθ0supθV(θ0)1σt42σt2θiθj<,

    we have Eθ0supθV(θ0)3lt(θ)θiθjθk<,

    which shows that the conclusion of (iii) is true.

  4. It follows from (3), (4), (13) and (14) that supθΘσt2θσ~t2θ<Kρt,supθΘ2σt2θθT2σ~t2θθT<Kρt,t,(22)

    which yields 1σt21σ~t2=σ~t2σt2σt2σ~t2Kρtσt2,σt2σ~t21+Kρt.(23)

    Because of lt(θ)θ=121|εt1|σt21σt2σt2θ,l~t(θ)θ=12(1|εt1|σ~t2)(1σ~t2σ~t2θ),

    it is true that lt(θ0)θl~t(θ0)θ=12||εt1|σ~t2|εt1|σt21σt2σt2θi+1|εt1|σ~t21σt21σ~t2σt2θi+1|εt1|σ~t21σ~t2σt2θiσ~t2θi|(θ0)12Kρt(1+|εt1|σt)1+1σt2(θ0)σt2(θ0)θi.

    Thus, n1/2t=1nlt(θ0)θl~t(θ0)θK2n1/2t=1nρt(1+|εt1|σt)1+1σt2(θ0)σt2(θ0)θi.

    Similarly to the proof (i), according to the Markov inequality and the independent relationship between ηt and σt2(θ0), for all ε>0 we have Pn1/2t=1nρt(1+|εt1|σt)1+1σt2(θ0)σt2(θ0)θ>ε2ε1+Eθ01σt2(θ0)σt2(θ0)θn1/2t=1nρt0.(24)

    Thus, the first part of (iv) was obtained. Due to (20), (22), and (23), we have supθV(θ0)n1t=1n2lt(θ)θiθj2l~t(θ)θiθjn12t=1nsupθV(θ0)||εt1|σ~t2|εt1|σt21σt22σt2θiθj+1|εt1|σ~t21σt21σ~t22σt2θiθj+1σ~t22σt2θiθj2σ~t2θiθj+32|εt1|σt232|εt1|σ~t21σt2σt2θi1σt2σt2θj+32|εt1|σ~t211σt21σ~t2σt2θi+1σ~t2σt2θiσ~t2θi1σt2σt2θj+32|εt1|σ~t211σ~t2σ~t2θi1σt21σ~t2σt2θj+1σ~t2σt2θjσ~t2θj|12Kn1t=1nρtNt.

    As a consequence, Nt=supθV(θ0)1+|εt1|σt21+1σt22σt2θiθj+1σt2σt2θi1σt2σt2θj.

    By (iii) and Holder inequality, Nt was integrable for some neighbourhood V(θ0). So, it follows from the Markov inequality that the second part of (iv) is true.

  5. The proof of (v) is easily obtained from the central limit theorem for martingale difference. Suppose that 𝓕t is σ-domain generated from varibles {εti, i ≥ 0}. As Eθ0(∂lt(θ0)/∂θ|𝓕t) = 0, varθ0( lt(θ0)/∂θ) exists. From Assumption 3 and (ii), 0 < τ − 1 < ∞ and J is non-singular. Hence, the matrix varθ0(∂lt(θ0)/∂θ) is non-degenerate. Thus, λRp+q+1, {λT( /∂θ)lt(θ0), 𝓕t is a martingale difference sequence. From the central limit theorem and the Wold-Cramer device, the asymptotic normality result (v) is established.

  6. To prove (vi), we firstly prove that the second-order derivatives of lt(θ) exists. For all i, j, n1t=1n2θiθjlt(θij)=n1t=1n2θiθjlt(θ0)+n1t=1nθT{2θiθjlt(θ~ij)}(θijθ0),(25)

    Here, θ̃ij locates between θij and θ0. As θ̃ij almost certainly converges to θ0, it follows from the ergodic theorem and (iii) that limnsupn1t=1nθT2θiθjlt(θ~ij)limnsupn1t=1nsupθV(θ0)θT2θiθjlt(θ)=Eθ0supθV(θ0)θT2θiθjlt(θ)<.

    Since ∥θijθ0∥ → 0 a.s., the second term on the right-hand side of (25) converges to 0 with probability 1. The first term on the right-hand side of (25) is also proved by the ergodic theorem. As a result, the conclusion of (vi) is obtained immediately.

Finally, the Slutsky lemma, (iv), (v), and (vi) are used to produce (17) and (18), i.e. the conclusion of Theorem 2.2 is established. Here, we complete the proof.□

Conflict of interest statement

We declare that we have no commercial or associative interest conflicts of interest in this work, and we have no financial and personal relationships with other people or organizations which can inappropriately influence our work, the manuscript which have no conflict of interest, entitled “Quasi-maximum Likelihood Estimator of Laplace (1, 1) for GARCH Models”.

Acknowledgement

The authors would like to thank the reviewers for their valuable comments and suggestions. This work was supported by National Natural Science Foundation of China (No.11371077).

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About the article

Received: 2017-04-20

Accepted: 2017-11-09

Published Online: 2017-12-29


Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 1539–1548, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0131.

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