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

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:

Here, η_t is a sequence of independent and identical distributed (i.i.d.) random variables, α₀ > 0, α_i ≥ 0, i = 1, 2, ⋯, r; β_j ≥ 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

Assume that θ = (α₀, α₁, ⋯, α_r, β₁, β₂, ⋯, β_s)′ 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, Θ ⊂ R0r+s+1, R₀ = [0, ∞). By assumption that a Laplace distribution has the density f(x) = 0.5e^−|x−1| for η_t and conditionally on initial values ε₀, ⋯, ε_1−r, σ02,⋯,σ1−s2, then the Laplace quasi-likelihood is

with regard to t ≥ 1, where

We select the initial values

or

As a result, θ̂_n is named the QMLE for θ and has the following form

where

Denote α(z)=∑i=1rαizi,β(z)=1−∑j=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 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 ε_t and σt2 are unknown when t ≤ 0. Let ε̃_t(θ) and σ~t2 (θ) be ε_t(θ) and σt2 (θ), respectively, when ε_t and σt2 (θ) are constants when t ≤ 0. The formula (6) can be modified as

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

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.

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.

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.

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 η_t obeyed the N(0, 1) and η_t 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:

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 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 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.

4 Proofs

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