适应不对称性的金融时间序列建模：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）模型来分析具有不对称性的金融时间序列数据的研究论文。作者Philippe Lambert和Sébastien Laurent在2002年提出了一种适应性方法，通过引入偏斜度动态到偏斜位置尺度分布中，以捕捉数据中的偏斜性。他们还提出了一个通用的动态模型，用于衡量下一个观察值超过条件模式的概率比，从而量化偏斜度。该模型应用到了1980年至1996年的DEM-USD汇率数据上，关键词包括ARMA、Power GARCH、变化的偏斜度、峰度、稳定分布和偏斜学生分布。" 详细说明: 1. **GARCH模型**：GARCH模型是一种统计模型，用于预测金融时间序列（如股票价格波动、汇率等）的方差，它结合了自回归模型（ARMA）和条件异方差的概念，能捕捉到数据中随机波动的变化性。 2. **ARMA-Power GARCH模型**：这是一种扩展的GARCH模型，它同时考虑了条件均值（由ARMA模型描述）和条件方差（由GARCH模型描述）。ARMA部分处理序列的线性趋势，而Power GARCH部分则处理方差的非线性变化。 3. **不对称性**：金融时间序列中经常出现正负收益不对称的现象，即负向异常事件通常带来更大的波动。ARMA-Power GARCH模型被调整以适应这种不对称性，通过引入动态到偏斜度和分散参数中。 4. **偏斜位置尺度分布**：这是一种能够表示数据偏斜性的概率分布，它允许分布的形状不仅围绕均值对称，而是可以偏向一侧。通过这种方式，模型可以更好地拟合具有明显偏斜的金融数据。 5. **时间变化的偏斜度和峰度**：在金融数据中，偏斜度和峰度（数据分布的尖峰程度）可能会随时间变化。文档中提出的模型可以捕捉这些动态特性，提供更准确的分析。 6. **稳定分布和偏斜学生分布**：稳定分布是一类广泛用于金融数据分析的连续概率分布，因为它可以处理极端值和变化的波动率。偏斜学生分布是学生t分布的偏斜版本，同样适用于描述具有厚尾部的金融数据。 7. **示例应用**：DEM-USD汇率数据分析展示了该模型的实际应用。通过对1980年至1996年的数据进行建模，作者展示了如何利用提出的模型分析历史汇率波动的偏斜动态。总结来说，这篇论文提供了对金融时间序列分析的深入见解，特别是针对具有不对称特性的数据。通过使用ARMA-Power GARCH模型和动态偏斜度模型，研究人员可以更精确地理解和预测金融市场的波动行为。

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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