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Contents
Introduction
1 Preliminaries
1.1 Markov Chains
1.1.1 Strict Stationarity and Stationarit
1.1.2 Invariant Measure
1.1.3 Irreducibility, Small Sets and Aperiodic Chains
1.1.4 Petite Set
1.1.5 Feller Chain
1.1.6 Transience, Recurrence and Harris Recurrence
1.1.7 Ergodicit
1.1.8 β - Mixin
1.1.9 Criterion for Ergodicity and β - Mixing
1.2 Algebraic Geometr
1.2.1 Semi-algebraic and Algebraic Set
1.2.2 Regular Points and Dimension of Algebraic Varieties
1.2.3 Regular Map
2 Autoregressive Processes defined by a Composition of a Regular Map and a Diffeomorphism
2.1 Introductio
2.2 Properties of the Image Measur
2.3 Semi-polynomial Markov Chain
2.3.1 Model and Assumption
2.3.2 Algebraic Variety of State
2.3.3 Harris Recurrence, Ergodicity and β - Mixing
3 Multivariate GARCH Models
3.1 Introduction and Notation
3.2 The vec and BEKK Model
3.3 Stationarity of Multivariate GARCH Model
3.3.1 Autoregressive Representatio
3.3.2 Some Results from Linear Algebr
3.3.3 Verification of Assumption (A
3.3.4 Verification of Assumption (A
3.3.5 Foster - Lyapounov Condition (FL
3.3.6 Harris Recurrence, Ergodicity and β - Mixing
Introduction
The origin of time series analysis can be found within the development of ARMA (autoregressive moving average) processes. However, the linearity supposed for such processes is a constraint which gave rise to the design of non-linear models. The most simple ones are Markov models in the form
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where (Xt)t∈N is a sequence of random variables with values in a topological state space S equipped with its Borel σ-field B(S), F maps S × Rd into S and (et)t∈N_ is an Rd-valued i.i.d. sequence of random variables independent of X0.
It is often helpful for modelization if one can find strictly stationary solutions to the equation (1), i.e. if one can find invariant probabilities for the Markov chain.
In general, there are two well-known concepts to deduce stationarity. The first method is to show certain Lipschitz properties for the map F. More precisely, if there is a norm k·k on S and a real function α such that, for all (x, y) ∈ S2, kF(x, et) − F(y, et)k ≤ α(et) kx − yk (2) with E[α(et)] < 1 and E[kF(0, et)k] < ∞, then there is a strictly stationary solution for the model (cf. [7]).
Another approach, developed in the book of Meyn and Tweedie [12], is based upon the irreducibility of the Markov chain (Xt)t∈N and a so-called Lyapounov function V ≥ 1 which satisfies E[V (Xt)|Xt−1 = x] ≤ αV (x) + b1C(x) (3) where C is a suitable set on which V is bounded and α < 1, b < ∞ are positive constants.
This function permits to control the behavior of the chain and yields the existence of a strictly stationary solution for (1), too. Since our aim in this project is to study stationarity of multivariate GARCH models, where the first approach is not applicable, we will concentrate on the second method (3) during our work. Our proceeding will be the following:
The first chapter will recall important definitions and results which we will use in this project thesis. It consists of two parts where the first one considers the principal notions and results for Markov chains on general state spaces and establishes a Foster - Lyapounov condition which ensures stationarity and ergodicity for irreducible and aperiodic chains.
Markov chains on general state spaces in discrete time can be seen as an analogy to chains on discrete state spaces where the future value of the chain only depends on the present state and transitions are specified by stochastic matrices. The “only” difference is that we cannot use these matrices any longer but have to switch to so-called transition probability kernels, an appropriate “generalization” of stochastic matrices.
The second part of the first chapter will concentrate on algebraic geometry and in particular on algebraic varieties and regular maps. The motivation to introduce these concepts is clear: since we will use the Foster - Lyapounov method described above in (3) for the study of stationarity of multivariate GARCH models and since this approach requires irreducibility which in general cannot be shown for such processes on the whole state space, we will restrict the chain to a suitable subspace on which the process is irreducible. It will turn out that this subspace is an algebraic variety. Since we are going to use the Zariski topology in addition to the standard topology, we note that if not otherwise stated wealways consider the “normal” topology.
In the second chapter, in view of an application to GARCH processes, we study strict stationarity and ergodicity of semi-polynomial Markov chains which are defined by (1) whereas F is a composition of a regular map and a diffeomorphism. In particular, we will construct using the Zariski topology a subspace on which these chains are irreducible. Therefore Chapter two will use the methods of algebraic geometry introduced in Chapter one. The chapter can be considered a generalization of Mokkadem [13]. The third chapter will start with a repetition of vec and BEKK models for multivariate GARCH processes. In the main part of this chapter we are going to prove that a GARCH process in the BEKK representation satisfying some additional properties is positive Harris recurrent and geometrically ergodic on the Zariski closure W of an orbit of a well chosen point T and that the strictly stationary solution is β - mixing.
To show this we will suppose that the distribution of every et of the i.i.d. innovation sequence (et)t∈N_ is absolutely continuous with respect to the Lebesgue measure, that the point zero is in the interior of the domain of positivity of the density of the distribution of et and that the spectral radius of the matrix
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than 1, where the matrices Ai and Bj occur in the vech representation of a BEKK model (i.e. the conditional covariance matrix _t of the GARCH process can be written as vech(_t) = vech(C) +
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we will obtain the uniqueness of the strictly stationary solution. On the other hand, we will also show that whenever there exists a weak stationary solution the spectral radius of
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The main idea and structure of this work can be found in Boussama [3]. However, we changed subtle details in our proceeding so that attentive readers will easily find the difference to the work of Boussama.
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