GARCH模型详解：结构、统计推断与金融应用

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GARCH（Generalized Autoregressive Conditional Heteroskedasticity）模型是一种统计方法，主要用于处理金融领域中的非线性时间序列数据，特别是对于那些具有条件异方差性的数据。该模型由Christian Francq和Jean-Michel Zakoïan两位学者在他们的著作《GARCH模型：结构、统计推断与金融应用》中详细介绍。该书由John Wiley & Sons Ltd出版社于2010年首次出版，版权受到严格的保护。 GARCH模型的核心思想是将过去的误差（残差）平方的观测值作为对未来误差方差的预测因子，从而捕捉到金融市场的波动性随时间变化的特性。它区分了均值（长期方差）和短期波动（条件方差），使得模型能够适应市场的动态变化，而不仅仅依赖于历史平均值。这种模型被广泛应用于风险管理和金融市场分析，例如股票价格预测、期权定价、信贷风险评估以及宏观经济指标的波动性研究。 GARCH模型主要分为两个基本类型：GARCH(p,q)模型，其中p代表滞后误差项的数量，q代表滞后条件方差的数量。这意味着模型考虑了p个前一期的误差平方以及q个前期条件方差的影响。此外，还有扩展版本如EGARCH（Exponential GARCH）、TGARCH（Threshold GARCH）和VGARCH（Variable GARCH）等，它们引入了不同的非线性结构来更准确地反映市场变化。在统计推断方面，GARCH模型通常涉及到估计参数、诊断模型假设（如正态分布和无自相关性）、以及通过似然比检验或信息准则（如Akaike信息准则AIC或Bayesian信息准则BIC）来选择最优模型。对于金融应用，模型的预测能力和解释能力至关重要，因此，估计出的GARCH模型参数可以帮助金融机构做出风险管理决策，并在市场波动性加剧时调整投资策略。 GARCH模型是金融时间序列分析的重要工具，它的理论框架、参数估计方法以及在实际中的应用都是本书的重点内容。通过阅读《GARCH模型：结构、统计推断与金融应用》，读者可以深入理解这种模型的工作原理，掌握如何在实践中运用它来理解和预测金融市场动态。

xiv NOTATION

→ convergence in distribution

a.s. almost surely

= o

→ 0 in probability

(1)

= baequals b up to the stochastic order o

(1)

Estimation

I Fisher information matrix

(κ

− 1)J

−1

asymptotic variance of the QML

true parameter value

 parameter set

θ element of the parameter set

n,f

, ... estimators of θ

= σ

(θ) volatility built with the value θ

˜σ

= ˜σ

(θ) as σ

but with initial values



= 

(θ) −2log(conditional variance of 

)



(θ) approximation of 

, built with initial values

Var

, Cov

asymptotic variance and covariance

Some abbreviations

ES expected shortfall

FGLS feasible generalized least squares

OLS ordinary least squares

QML quasi-maximum likelihood

RMSE root mean square error

SACR sample autocorrelation

SACV sample autocovariance

SPAC sample partial autocorrelation

VaR value at risk

2 GARCH MODELS

Deﬁnition 1.2 (Second-order stationarity) The process (X

) is said to be second-order

stationary if:

(i) EX

< ∞, ∀t ∈ Z;

(ii) EX

= m, ∀t ∈ Z;

(iii) Cov(X

t+h

) = γ

(h), ∀t,h ∈ Z.

The function γ

(·) (ρ

(·) := γ

(·)/γ

(0)) is called the autocovariance function (autocorrelation

function) of (X

The simplest example of a second-order stationary process is white noise. This process is

particularly important because it allows more complex stationary processes to be constructed.

Deﬁnition 1.3 (Weak white noise) The process (

) is called weak white noise if, for some pos-

itive constant σ

(i) E

= 0, ∀t ∈ Z;

(ii) E

= σ

, ∀t ∈ Z;

(iii) Cov(

,

t+h

) = 0, ∀t,h ∈ Z,h= 0.

Remark 1.1 (Strong white noise) It should be noted that no independence assumption is made

in the deﬁnition of weak white noise. The variables at different dates are only uncorrelated and

the distinction is particularly crucial for ﬁnancial time series. It is sometimes necessary to replace

hypothesis (iii) by the stronger hypothesis

(iii



) the variables 

and 

t+h

are independent and identically distributed.

The process (

) is then said to be strong white noise.

Estimating Autocovariances

The classical time series analysis is centered on the second-order structure of the processes. Gaus-

sian stationary processes are completely characterized by their mean and their autocovariance

function. For non-Gaussian processes, the mean and autocovariance give a ﬁrst idea of the tem-

poral dependence structure. In practice, these moments are unknown and are estimated from a

realization of size n of the series, denoted X

,...,X

. This step is preliminary to any construction

of an appropriate model. To estimate γ(h), we generally use the sample autocovariance deﬁned,

for 0 ≤ h<n,by

ˆγ(h)=

n−h



j=1

− X)(X

j+h

− X) := ˆγ(−h),

where

X = (1/n)



j=1

denotes the sample mean. We similarly deﬁne the sample autocorrela-

tion function by ˆρ(h) = ˆγ(h)/ˆγ(0) for |h| <n.

The previous estimators have ﬁnite-sample bias but are asymptotically unbiased. There are

other similar estimators of the autocovariance function with the same asymptotic properties (for

instance, obtained by replacing 1/n by 1/(n − h)). However, the proposed estimator is to be

preferred over others because the matrix ( ˆγ(i − j)) is positive semi-deﬁnite (see Brockwell and

Davis, 1991, p. 221).

It is of course not recommended to use the sample autocovariances when h is close to n, because

too few pairs (X

j+h

) are available. Box, Jenkins and Reinsel (1994, p. 32) suggest that useful

estimates of the autocorrelations can only be made if, approximately, n

50 and h ≤ n/4.

4 GARCH MODELS

The class of ARMA models is the most widely used for the prediction of second-order

stationary processes. These models can be viewed as a natural consequence of a fundamental

result due to Wold (1938), which can be stated as follows: any centered, second-order stationary,

and ‘purely nondeterministic’ process

admits an inﬁnite moving-average representation of the form

= 

∞



i=1



t−i

, (1.3)

where (

) is the linear innovation process of (X

), that is



= X

− E(X

(t − 1)), (1.4)

where H

(t − 1) denotes the Hilbert space generated by the random variables X

t−1

t−2

,....

and E(X

(t – 1)) denotes the orthogonal projection of X

onto H

(t – 1). The sequence of

coefﬁcients (c

) is such that



< ∞. Note that (

) is a weak white noise.

Truncating the inﬁnite sum in (1.3), we obtain the process

(q) = 



i=1



t−i

called a moving average process of order q,orMA(q).Wehave

X

(q) − X



= E



→ 0, as q →∞.

It follows that the set of all ﬁnite-order moving averages is dense in the set of second-order

stationary and purely nondeterministic processes. The class of ARMA models is often preferred

to the MA models for parsimony reasons, because they generally require fewer parameters.

Deﬁnition 1.4 (ARMA( p, q) process) A second-order stationary process (X

) is called

ARMA(p, q),wherep and q are integers, if there exist real coefﬁcients c, a

,...,a

,...,b

such that,

∀t ∈ Z,X



i=1

t−i

= c + 



j=1



t−j

, (1.5)

where (

) is the linear innovation process of (X

This deﬁnition entails constraints on the zeros of the autoregressive and moving average poly-

nomials, a(z) = 1 +



i=0

and b(z) = 1 +



i=0

(Exercise 1.9). The main attraction of

this model, and the representations obtained by successively inverting the polynomials a(·) and

b(·), is that it provides a framework for deriving the optimal linear predictions of the process, in

much simpler way than by only assuming the second-order stationarity.

Many economic series display trends, making the stationarity assumption unrealistic. Such

trends often vanish when the series is differentiated, once or several times. Let X

= X

−

t−1

denote the ﬁrst-difference series, and let 

= (

d−1

) (with 

= X

) denote the

differences of order d.

A stationary process (X

) is said to be purely nondeterministic if and only if



∞

n=−∞

(n) ={0},where

(n) denotes, in the Hilbert space of the real, centered, and square integrable variables, the subspace consti-

tuted by the limits of the linear combinations of the variables X

n−i

, i ≥ 0. Thus, for a purely nondeterministic

(or regular) process, the linear past, sufﬁciently far away in the past, is of no use in predicting future values.

See Brockwell and Davis (1991, pp. 187–189) or Azencott and Dacunha-Castelle (1984) for more details.

In this representation, the equivalence class E(X

(t − 1)) is identiﬁed with a random variable.

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