适应金融时间序列偏斜分布的ARMA-Power GARCH模型

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"这篇文档是关于使用GARCH（广义自回归条件异方差）模型来模拟具有偏斜学生分布的金融时间序列的研究。作者Philippe Lambert和Sébastien Laurent在2002年10月8日的论文中探讨了如何适应性地运用ARMA-Power GARCH模型来分析显示不对称性的金融数据时间序列。他们引入动态机制到偏斜位置标度分布的定位和分散参数中，与ARMA-APARCH模型中的条件均值和条件方差结构相呼应。此外，他们还提出了一种通用的动态模型来衡量由下一个观察值超过条件模式的概率比（赔率比）所表示的偏斜度。这个工具通过分析1980年至1996年的DEM-USD汇率数据进行了具体展示。关键词包括：ARMA，Power GARCH，时间变化的偏斜度，尖峰度，稳定分布，偏斜学生分布。" 本文的核心知识点： 1. **GARCH模型**：GARCH（广义自回归条件异方差）模型是一种统计模型，用于描述和预测时间序列数据的波动性，特别是金融市场的波动。它考虑了过去波动性对当前波动性的影响，即条件方差的动态变化。 2. **ARMA-Power GARCH模型**：这是一种扩展的GARCH模型，其中不仅包含条件均值的ARMA（自回归滑动平均）结构，还包含条件方差的GARCH结构，并考虑了方差的幂律行为。这使得模型能够更好地捕捉到金融时间序列中极端事件的影响。 3. **偏斜学生分布**：学生分布通常用于金融数据分析，因为它能更好地描述异常值和非正态分布的数据。偏斜的学生分布进一步增加了分布的不对称性，以反映金融市场中普遍存在的正偏斜性（收益通常偏向于负值）。 4. **动态偏斜度**：论文提出了引入动态机制到偏斜度的建模中，使模型能够捕捉到偏斜度随时间的变化。这对于理解金融市场的不稳定性至关重要，因为偏斜度可以指示市场情绪和风险预期。 5. **赔率比**：赔率比是衡量偏斜度的一个指标，它反映了未来观测值大于条件模式（即预期值）的概率。这为分析市场情绪和预测未来极端事件提供了工具。 6. **应用实例**：DEM-USD汇率分析展示了所提出的模型在实际问题中的应用，通过对历史数据的分析，验证了模型的有效性和解释力。 7. **关键词**：还包括了时间变化的偏斜度、尖峰度（数据分布的尖锐程度，反映异常值的频率）和稳定分布。稳定分布是一类包含许多常见分布（如高斯分布和学生分布）的广义分布，对于描述金融数据的非正态特性特别有用。总结来说，这篇论文是金融时间序列分析的深度研究，通过GARCH模型和偏斜学生分布的结合，提供了一种有效工具来理解和预测金融市场波动，特别是在考虑非对称性和动态偏斜度的情况下。

by adding a skewness parameter. Their procedure allows the introduction of skewness in any

continuous unimodal and symmetric (about γ) distribution g(y; γ) by changing the scale at each

side of the mo de. More speciﬁcally,

f(y|γ, ξ) =

ξ +

g(ξ(y − γ); 0)I

(−∞,γ)

(y) + g((y − γ) /ξ; 0)I

[γ,∞)

(y)

(2)

is a unimodal density with the same mode as g(y; γ) and a skewness parameter ξ > 0 such that

the ratio of probability masses above and below the mode is

Pr(Y ≥ γ|ξ)

Pr(Y < γ|ξ)

= ξ

(3)

Note that the density f (y|γ, 1/ξ) is the symmetric of f(y|γ, ξ) with respect to the mode. Therefore,

working with ξ

= log(ξ) might be preferable to indicate the sign of the skewness. If we take

for g(y; γ) the Student density g

(y|µ = γ, σ

) in Equation (1), we obtain the four parameter

skewed Student distribution in Fern´andez and Steel (1998). These parameters all have a clear

interpretation:

• µ, as the mode, models the location,

• σ

> 0 (which is not the variance anymore) models the dispersion,

• ξ > 0 models the skewness,

• υ > 0 models the tail thickness.

Four important aspects of the distribution can thus be independently speciﬁed. The skewed

normal distribution directly obtained by applying Equation (2) to the symmetric normal density

with mean γ and variance σ

is a limiting case ( υ → ∞) of the skewed Student with the same tail

properties than the traditional normal.

2.3 The asymmetric stable distribution

Stable distributions are by deﬁnition the only possible limiting distributions for the sum of in-

dependent, identically distributed random variables (see for example, Samorodnitsky and Taqqu,

1994; Adler, Feldman, and Taqqu, 1998; Lambert and Lindsey, 1999). Therefore these limiting

distributions generalize the normal that arises in the central limit theorems when the variance

of the summed random variables is ﬁnite. Unfortunately, these four parameter distributions are

only known through their characteristic function φ(t). A possible parameterization of the latter

is given by

log φ(t) = iγt − |t|

exp

−iβ

sign(t)

(4)

= min(α, 2 − α ) = 1 − |1 − α|

where γ, δ > 0, β ∈ [−1, 1] and α ∈ (1, 2] are location, dispersion, skewness and tail parameters.

The cases α = 1, α = 2 and (β, α) = (1, 0.5) respectively correspond to the Cauchy, the normal

and L´evy distributions. For other parameter values, the density g

(y|γ, δ, β) can only be obtained

using numerical approximations (see Part VII in Adler, Feldman, and Taqqu, 1998 ; Lambert and

Lindsey, 1999 ; Hoﬀmann-Jørgensen , 1994, I, pp. 406–411). The density is unimodal Yamazato

(1978) and bell-shaped Gawronski (1984). It is symmetric when β = 0 and right (left) skewed

when β is negative (positive). Stable distributions (α 6= 2) are said heavy tailed because the

distribution and survivor functions show a power decay

(x) ∝ (−x)

−α

as x → −∞

(x) = 1 − G

(x) ∝ x

−α

as x → +∞ (5)

as in the Pareto case. This can be contrasted to the normal distribution (α = 2) where an

exponential decay is observed

α=2

(x) ∝

exp(−x

/2)

One important aspect of stable distributions is that the four parameters can be speciﬁed in an

independent manner (the space parameter is simply the cartesian product of the four parameter

spaces). This is a remarkable result for statisticians used to the exponential family where the

speciﬁcation of the mean often aﬀects in an important manner other aspects of the shape of the

considered distribution. However, the range of possible skewness introduced by β in the stable

distribution highly depends on the value of the tail parameter α as illustrated in Figure 1. If we

measure skewness by the log of the ratio ξ

of probability masses above and below the mode as

given in Equation (3), we see that the asymmetry introduced by non-zero values of β becomes

negligible as α approaches 2. It presents at least two drawbacks:

• the analyst might think, for example, that the data present severe skewness because the

estimated value of β is close to −1 or 1. However, the last ﬁgure shows that he might be

completely wrong if the ﬁtted stable distribution is characterized by an α parameter close

to 2. A re-parameterization of the stable distribution is thus desirable to reﬂect reliably

the skewness introduced by the skewness parameter. This parameter would obviously be a

function of β and α.

• the analyzed dataset might present (severe) skewness and have heavy tails (cf. Equation (5))

with an α close to 2. The stable family of distributions cannot adequately describe such a

phenomenon.

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