理解多元时间序列分析新指南：VARMA模型与多变量GARCH详解

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《新引入多时间序列分析》是一本经典的多变量时间序列分析教材，由赫尔穆特·吕克波尔教授撰写。本书全面涵盖了VARMA模型的估计与协整分析，多变量GARCH模型，以及状态空间方法等核心内容。共计49幅图表和36个表格，为读者提供了深入理解这些复杂技术的直观指导。在VARMA（Vector Autoregressive Moving Average）模型部分，本书介绍了如何通过自回归和移动平均过程来建模多个时间序列的动态关联，这对于经济、金融等领域理解变量间的长期和短期关系至关重要。作者详细解释了模型的识别、估计参数、诊断检验和预测等方面的方法，并通过实例展示了其在实际问题中的应用。多变量GARCH（Generalized Autoregressive Conditional Heteroskedasticity）模型则涉及到了如何处理金融市场的波动性问题，它考虑了不同变量之间共同的异方差性，对于风险管理及市场动态建模具有重要作用。书中讲解了GARCH模型的不同形式，如标准GARCH、EGARCH（Exponential GARCH）、T-GARCH（Threshold GARCH）等，并讨论了模型选择和估计策略。此外，状态空间方法是现代时间序列分析中的一个重要工具，通过将非线性模型转化为线性动态系统的形式，便于数值求解和统计推断。书中对隐含Markov模型（UnobservedComponents Model）和Kalman滤波器等内容进行了详尽的阐述，这对于宏观经济预测、状态估计和政策制定具有实用价值。《新引入多时间序列分析》不仅适合高级研究生和专业研究人员，也为那些希望深入了解多变量时间序列分析的从业者提供了深入学习的资源。书中丰富的实证案例和理论框架，使得读者能够掌握这些关键理论并将其应用于实际问题中。同时，版权信息表明该书遵循德国版权法，复制或使用须获得Springer-Verlag许可。

XX Contents

Appendix

A Vectors and Matrices

......................................645

A.1 BasicDeﬁnitions ........................................645

A.2 BasicMatrix Operations .................................646

A.3 TheDeterminant........................................647

A.4 TheInverse, the Adjoint,andGeneralizedInverses...........649

A.4.1 Inverse and Adjoint ofaSquareMatrix ..............649

A.4.2 Generalized Inverses ...............................650

A.5 TheRank ..............................................651

A.6 Eigenvalues and -vectors – Characteristic Values and Vectors . . 652

A.7 TheTrace ..............................................653

A.8 SomeSpecial MatricesandVectors ........................653

A.8.1 Idempotent and NilpotentMatrices..................653

A.8.2 Orthogonal Matrices and Vectors and Orthogonal

Complements .....................................654

A.8.3 Deﬁnite MatricesandQuadraticForms ..............655

A.9 Decomposition and Diagonalization of Matrices .............656

A.9.1 The Jordan CanonicalForm ........................656

A.9.2 Decomposition ofSymmetricMatrices ...............658

A.9.3 The Choleski Decomposition of a Positive Deﬁnite

Matrix...........................................658

A.10PartitionedMatrices .....................................659

A.11TheKronecker Product ..................................660

A.12 The vec and vech Operators and Related Matrices . . . .......661

A.12.1 The Operators . . . .................................661

A.12.2 Elimination, Duplication, and Commutation Matrices . . 662

A.13VectorandMatrixDiﬀerentiation .........................664

A.14Optimizationof VectorFunctions..........................671

A.15Problems...............................................675

B Multivariate Normal and Related Distributions ............677

B.1 Multivariate NormalDistributions.........................677

B.2 RelatedDistributions ....................................678

C Stochastic Convergence and Asymptotic Distributions .....681

C.1 Conceptsof StochasticConvergence........................681

C.2 Order in Probability . ....................................684

C.3 Inﬁnite Sums of Random Variables ........................685

C.4 Lawsof Large NumbersandCentralLimit Theorems ........689

C.5 Standard Asymptotic Properties of Estimators and Test

Statistics . . .............................................692

C.6 MaximumLikelihood Estimation ..........................693

C.7 Likelihood Ratio, Lagrange Multiplier, and Wald Tests .......694

2 1 Introduction

it makes sense to use their possible additional information content in forecast-

ing consumption expenditures. In other words, denoting the related variables

by y

,...,y

, the forecast of y

1,T +h

at the end of period T may be of

the form

y

1,T +h

= f

1,T

2,T

,...,y

K,T

1,T −1

2,T −1

,...,y

K,T −1

1,T −2

,...).

Similarly, a forecast for the second variable may be based on past values of

all variables in the system. More generally, a forecast of the k-th variable may

be expressed as

y

k,T+h

= f

1,T

,...,y

K,T

1,T −1

,...,y

K,T −1

,...). (1.1.2)

A set of time series y

, k =1,...,K, t =1,...,T, is called a multiple time

series and the previous formula expresses the forecast y

k,T+h

as a function

of a multiple time series. In analogy with the univariate case, it is one ma-

jor objective of multiple time series analysis to determine suitable functions

,...,f

that may be used to obtain forecasts with good properties for the

variables of the system.

It is also often of interest to learn about the dynamic interrelationships

between a number of variables. For instance, in a system consisting of invest-

ment, income, and consumption one may want to know about the likely impact

of a change in income. What will be the present and future implications of

such an event for consumption and for investment? Under what conditions

can the eﬀect of an increase in income be isolated and traced through the sys-

tem? Alternatively, given a particular subject matter theory, is it consistent

with the relations implied by a multiple time series model which is developed

with the help of statistical tools? These and other questions regarding the

structure of the relationships between the variables involved are occasionally

investigated in the context of multiple time series analysis. Thus, obtaining

insight into the dynamic structure of a system is a further objective of multiple

time series analysis.

1.2 Some Basics

In the following chapters, we will regard the values that a particular economic

variable has assumed in a speciﬁc period as realizations of random variables. A

time series will be assumed to be generated by a stochastic process. Although

the reader is assumed to be familiar with these terms, it may be useful to

brieﬂy review some of the basic deﬁnitions and expressions at this point, in

order to make the underlying concepts precise.

Let (Ω, F,Pr)beaprobability space,whereΩ is the set of all elementary

events (sample space), F is a sigma-algebra of events or subsets of Ω and Pr

is a probability measure deﬁned on F.Arandom variable y is a real valued

function deﬁned on Ω such that for each real number c, A

= {ω ∈ Ω|y(ω) ≤

1.2 Some Basics 3

c}∈F. In other words, A

is an event for which the probability is deﬁned in

terms of Pr. The function F : R → [0, 1], deﬁned by F (c)=Pr(A

), is the

distribution function of y.

A K-dimensional random vector or a K-dimensional vector of random

variables is a function y from Ω into the K-dimensional Euclidean space R

that is, y maps ω ∈ Ω on y(ω)=(y

(ω),...,y

(ω))



such that for each

c =(c

,...,c

)



∈ R

= {ω|y

(ω) ≤ c

,...,y

(ω) ≤ c

}∈F.

The function F : R

→ [0, 1] deﬁned by F (c)=Pr(A

) is the joint distribution

function of y.

Suppose Z is some index set with at most countably many elements like, for

instance, the set of all integers or all positive integers. A (discrete) stochastic

process is a real valued function

y : Z × Ω → R

such that for each ﬁxed t ∈ Z, y(t, ω) is a random variable. The random

variable corresponding to a ﬁxed t is usually denoted by y

in the following.

The underlying probability space will usually not even be mentioned. In that

case, it is understood that all the members y

of a stochastic process are

deﬁned on the same probability space. Usually the stochastic process will also

be denoted by y

if the meaning of the symbol is clear from the context.

A stochastic process may be described by the joint distribution functions

of all ﬁnite subcollections of y

’s, t ∈ S ⊂ Z. In practice, the complete system

of distributions will often be unknown. Therefore, in the following chapters, we

will often be concerned with the ﬁrst and second moments of the distributions.

In other words, we will be concerned with the means E(y

)=µ

, the variances

E[(y

− µ

)

] and the covariances E[(y

− µ

)(y

− µ

)].

A K-dimensional vector stochastic process or multivariate stochastic pro-

cess is a function

y : Z × Ω → R

where, for each ﬁxed t ∈ Z, y(t, ω)isaK-dimensional random vector. Again

we usually use the symbol y

for the random vector corresponding to a ﬁxed

t ∈ Z. For simplicity, we also often denote the complete process by y

. The par-

ticular meaning of the symbol should be clear from the context. With respect

to the stochastic characteristics the same applies as for univariate processes.

That is, the stochastic characteristics are summarized in the joint distribution

functions of all ﬁnite subcollections of random vectors y

. In practice, inter-

est will often focus on the ﬁrst and second moments of all random variables

involved.

A realization of a (vector) stochastic process is a sequence (of vectors)

(ω), t ∈ Z, for a ﬁxed ω. In other words, a realization of a stochastic process

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