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

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.


Introduction
The ARCH model has been widely used ever since it was rst proposed by Engle (1982) [1] because this model was able to address the volatility in the forecasting of Britain's in ation rate. In many statistical applications, particularly nance, 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.  [8] studied the structure and estimator of GARCH. 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 satis ed. 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 nancial 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.

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: Here, η t is a sequence of independent and identical distributed (i.i.d.) random variables, α > , α i ≥ , i = , , ⋯, r; β j ≥ , j = , , ⋯, s. According to Bougerol and Picard(1992) [17], the su cient and necessary condition for the strictly stationary solution of GARCH (r, s) model is Assume that θ = (α , α , ⋯, α r , β , β , ⋯, β s ) ′ is the parameter vector of formula (1) and its true parameter vector is θ . Let l = r + s + , then θ is l dimension vector. The parameter vector space is Θ, Θ ⊂ R r+s+ , R = [ , ∞). By assumption that a Laplace distribution has the density f (x) = . e − x− for η t and conditionally on initial values ε , ⋯, ε −r , σ , ⋯, σ −s , then the Laplace quasi-likelihood is We select the initial values or As a result,θ n is named the QMLE for θ and has the following form θ n = arg max θ∈Θ L n (θ) = arg min θ∈Θ I n (θ), where Before providing main results, we introduce rstly the following assumptions. Assumption 1. θ ∈ Θ, Θ is compact and θ is an inner dot.
Assumption 1 ensures the parameter vector space is compact and is required to prove asymptotic normality. Assumption 2 and 4 are the identi ability 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 ε t and σ t are unknown when t ≤ . Letε t (θ) andσ t (θ) be ε t (θ) and σ t (θ), respectively, when ε t and σ t (θ) are constants when t ≤ . The formula (6) can be modi ed as In what follows we establish the main results of this paper.

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 di erential, denoted by {y t } t= are shown in Figure 1.
As shown in Figure 1, the mean was near 0. At the 0.05 signi cance 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 signi cance 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 * R = .
In this way, the GARCH(1,1) model was established according to {y t }. When η t obeyed the Laplace (1, 1), the estimation of the parameters vector was performed by using the QMLE at MATLAB.
< , which satis es 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 η t obeyed the N(0, 1) and η t obeyed the Laplace(1, 1).
The bias is a technical index that re ects the degree of violation of the stock price and the moving average in the process of uctuation. The computational formula is: 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 re ect 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 η t ∼ Laplace (1, 1), the values of the bias, the SD, and the AD were all smaller than the ones when η t ∼ N(0,1). This indicated that the tting e ect 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 e ective for the data from nancial markets.

Proofs
In this section we will present the proof of Theorem 2.1 and Theorem 2.2. (1) can be rewritten in vector form Then following intermediate results can be used to prove Theorem 2.1.
Iterating (10) produces the following: It is supposed here thatσ t may be the vector obtained by replacing σ t−i byσ t−i . Letc be the vector obtained by replacing ε , ⋯, ε −p with the initial values (3) or (4). We havẽ Through (12)- (14), it is almost certain the following is true: Hence, By the Markov inequality, the following equation can be determined: From the Borel-Cantelli lemma, (i) is obtained.
(ii) It's obvious that the result (ii) can be easily proved with Assumption 2 and Assumption 4.
It remains to be shown that E θ l + t (θ) < ∞. By Jenson inequality, Thus, Therefore, For all x > , log x ≤ x − , where the equality is true if and only if x = . Thus, it is true that It is noted that the equality in (16) holds if σ t (θ ) = σ t (θ).
(iv) From result (i), 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 nished.
Proof of Theorem 2.2. Through a Taylor expansion at θ , and hold. The proof of Theorem 2.2 is divided into the following six conclusions.
(iii) There exists a neighborhood V(θ ) of θ, with regard to i, j, k ∈ { , ⋯, r + s + }, such that For θ = θ , we have The proof of (i) is nished.
This shows that J is non-singular. So we establish the conclusion of (ii). (iii)It is shown in (20) that l t (θ) is di erentiated. Then we have which shows that the conclusion of (iii) is true.
(iv)It follows from (3), (4), (13) and (14) that which yields Because of it is true that Similarly to the proof (i), according to the Markov inequality and the independent relationship between η t and σ t (θ ), for all ε > we have Thus, the rst part of (iv) was obtained. Due to (20), (22), and (23), we have As a consequence, By (iii) and Holder inequality, N t was integrable for some neighbourhood V(θ ). So, it follows from the Markov inequality that the second part of (iv) is true.
(vi) To prove (vi), we rstly prove that the second-order derivatives of l t (θ) exists. For all i, j, n − n t= ∂ ∂θ i ∂θ j l t (θ * ij ) = n − n t= ∂ ∂θ i ∂θ j l t (θ ) + n − n t= ∂ ∂θ T { ∂ ∂θ i ∂θ j l t (θ ij )}(θ * ij − θ ), Here,θ ij locates between θ * ij and θ . Asθ ij almost certainly converges to θ , it follows from the ergodic theorem and (iii) that Since θ * ij − θ → a.s., the second term on the right-hand side of (25) converges to 0 with probability 1. The rst 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 con icts of interest in this work, and we have no nancial and personal relationships with other people or organizations which can inappropriately in uence our work, the manuscript which have no con ict of interest, entitled "Quasi-maximum Likelihood Estimator of Laplace (1, 1) for GARCH Models".