Non-Linear Time Series

Transcription

1 Non-Linear Time Series

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3 Kamil Feridun Turkman Manuel González Scotto Patrícia de Zea Bermudez Non-Linear Time Series Extreme Events and Integer Value Problems 123

4 Kamil Feridun Turkman Departmento de Estatística e Investigação Operacional Faculdade de Ciências Universidade de Lisboa Lisboa, Portugal Manuel González Scotto Departamento de Matemática Universidade de Aveiro Aveiro, Portugal Patrícia de Zea Bermudez Departmento de Estatística e Investigação Operacional Faculdade de Ciências Universidade de Lisboa Lisboa, Portugal ISBN ISBN (ebook) DOI / Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 Preface Linear processes have been one of the most fundamental tools in modeling serially dependent data. These models and methods heavily depend on Gaussian processes and their properties and therefore second-order moments have played the central role in this toolbox. Linear Gaussian models do not allow for large fluctuations and the stationary distribution when exists has fairly fast decaying tails, thus are not particularly adequate for modeling high variability. We are more and more aware that data coming from many scientific fields as diverse as telecommunications, hydrology or economics show heavy-tailed phenomena which are not compatible with Gaussian assumption. Therefore, such serially dependent data need richer parametrization based on nonlinear relations; consequently there is need to specify and model adequately observations coming from the tails. For such models, inferential methods based on second-order properties are no longer adequate and likelihood-based methods frequently cannot be used, since analytical expressions of likelihood often are not available. Least squares method, which is equivalent to likelihood-based inference for Gaussian processes, looses its nice properties as models deviate from Gaussian structure. Quasi-likelihood methods such as estimating functions and composite likelihood methods seem to work for some cases, whereas recent advances in sequential Markov Chain Monte Carlo (MCMC) methods and particle filters, as well as likelihood free methods such as approximate Bayesian computation (ABC), are quite promising alternatives in dealing with inference for nonlinear models. The objective of this book is to give an overall view of all these problems, including the consequences of nonlinearity on tails, some nonlinear models and adequate inferential methods. There are many excellent books on nonlinear time series, responding adequately to these issues. Our approach is to introduce diverse topics that appear in different sources under one title which can serve as a reference source, without entering into details. We believe that the book may be particularly useful to graduate students and other scientists who want to have a starting source for further detailed study of the subject. Chapter 1 gives many examples of nonlinear time series and different sources of nonlinearity. Chapter 2 introduces the basic notions of nonlinearity, and a v

6 vi Preface collection of nonlinear models. There are many ways a process can be nonlinear, and correspondingly there are many different classes of models to cover such diverse sources. We do not claim that the models we introduce are exhaustive, but we hope this selection covers most sources of nonlinearity. Chapter 3 is on the tail behavior of nonlinear processes. The objective of this chapter is to show how nonlinear relationships generate heavy tails and to quantify the effect of nonlinearity on tails. Much is known on the extremal properties of linear and nonlinear time series and the objective of this chapter is to give a quick reference to these results. Chapter 4 gives inferential methods available for nonlinear processes, including several tests for nonlinearity. We hope that this chapter will be particularly useful as an integrated introductory text on inferential techniques, including Bayesian and simulation-based methods. Finally, in Chap. 5 we give linear models based on thinning operators for integer-valued time series. Although these models are linear in structure, technically they are nonlinear models. Many would favor observationdriven or parameter-driven state space models for dealing with such data sets. However, recently there has been a lot of interest for linear models based on thinning operators and there is an accumulation of a rich and diverse information on this subject. To our knowledge, this chapter is the first attempt to put together in a coherent manner these advances spread out in many journals and other sources. Several institutions provided support for this work. We would like to thank FCT (Fundação para a Ciência e a Tecnologia) of Portugal, which supported our work through the projects PEst-OE/MAT/UI0006/2011, PEst-C/MAT/UI4106/2011, PEst-OE/MAT/UI4106/2014, and PTDC/MAT/118335/2010. The third author was also partially supported by a FCT Sabbatical Grant (ref FRH/BSAB/1138/2011) from 01 April 2011 to 29 July We also thank CEAUL Center of Statistics and its Applications of the University of Lisbon and CIDMA Center for Research & Development in Mathematics and Applications for their role in this initiative. Finally, we would like to thank Profs. John McDermott and Maria Antónia Amaral Turkman and Dr. Sónia Gouveia for reading the manuscript carefully and suggesting many useful comments which improved the content. Lisbon and Aveiro, Portugal May 2014 K. F. Turkman M. G. Scotto P. de Zea Bermudez

7 Contents 1 Introduction Why Do We Need NonlinearModels? Content of the Book Some Examplesof Time Series Some Simulated Time Series References Nonlinear Time Series Models Some Probabilistic Aspects of Nonlinear Processes Linear RepresentationsandLinear Models Linear andnonlinearoptimal Predictions NonlinearRepresentations Sensitive Dependence on Initial Conditions, Lyapunov Exponents Limit Cycles Time Reversibility Invertibility A Selection of NonlinearTime Series Models ParametricModels for the ConditionalMean Exponential Autoregressive Models Polynomial-Type Models Bilinear Models ParametricModels for the ConditionalVariance MixedModels for the ConditionalMean andvariance GeneralizedState Space Models Max-StableMovingAverageProcesses Nonparametric Methods References vii

8 viii Contents 3 Extremes of Nonlinear Time Series Tail Behavior ExtremeValue Theory Tail Behavior oflinear Processes Bivariate Regular Variation and the Coefficient of Tail Dependence ConnectionBetween Nonlinearityand Heavy Tails Extremal Properties of Certain Types of Nonlinear DifferenceEquations Tails of Bilinear Processes The Relation Between the Extremes of Finite-Order Volterra Series and Bilinear Processes Linear Modelsat the Presence of Heavy Tails References Inference for Nonlinear Time Series Models Identificationof Nonlinearity Graphical Methods and Other Informal Tests for Types of Nonlinearity Statistical Tests forlinearityand Gaussianity Checking for Stationarity and Invertibility Tail IndexEstimation Methods of Inference Least Squares and Likelihood Methods Estimating Functions Composite Likelihood Methods Bayesian Methods Simulation-Based Methods Sequential MCMC Methods for Generalized State Space Models ParameterEstimation forgarch-type Processes Conditional Maximum Likelihood Whittle Estimation Bayesian Approach References Models for Integer-Valued Time Series Introduction Integer-ValuedARMA Models INAR.1/ Model INAR.p/ Model INMA(q) Model ParameterEstimation Moment-BasedEstimators ConditionalLeast Squares

9 Contents ix Conditional Maximum Likelihood Estimation Bayesian Approach Model Selection ExtremalBehavior Some Extensions References Data Sets Reference

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11 List of Abbreviations ABC ACF A-FIGARCH APARCH AR ARCH ARMA BL cdf CINAR CLS CMLE DSD DSINAR EAR EGARCH EM EXPAR FIAPARCH FIGARCH GARCH GGS GINARS GJR-GARCH GPD i.i.d. IGARCH INAR INARMA Approximate bayesian computation Autocorrelation function Adaptative FIGARCH Asymmetric power ARCH Autoregressive Autoregressive conditionally heteroskedastic Autoregressive moving average Bilinear model Cumulative distribution function Combined INAR Conditional least squares Conditional maximum likelihood estimator Discrete self-decomposable Doubly stochastic integer-valued autoregressive Exponential autoregressive Exponential GARCH Expectation-maximization algorithm Exponential autoregressive Fractionally integrated asymmetric power ARCH Fractionally IGARCH Generalized ARCH Griddy-Gibbs sampler Integer-valued autoregressive model with signed generalized power series thinning operator Special case of the APARCH model Generalized Pareto distribution independent and identically distributed Integrated GARCH Integer-valued autoregressive Integer-valued ARMA xi

12 xii List of Abbreviations INARS Integer-valued autoregressive based on the signed binomial thinning INMA Integer-valued moving average INARMA Integer-valued ARMA INE Instituto Nacional de Estatística INGARCH Integer-valued GARCH LADE Least absolute deviations estimator LGARCH Logarithmic GARCH log-arch Special case of the APARCH model MA Moving average MAR Markov switching autoregressive MCMC Markov chain Monte Carlo MEF Mean excess function M-H Metropolis-Hastings algorithm ML Maximum likelihood NARCH Special case of the APARCH model PACF Partial autocorrelation function pdf Probability density function PINAR Periodic integer-valued autoregressive PMCMC Particle MCMC PoINAR Poisson INAR POT Peaks over threshold PSI20 Portuguese Stock Market Index 20 RCA Random coefficient autoregressive r.v s Random variables SETAR Self-exciting threshold autoregressive S-HY-APARCH Seasonal hyperbolic APARCH SMCMC Sequential MCMC STAR Smooth transition autoregressive TAR Threshold autoregressive TARCH Special case of the APARCH model US GNP United States Gross National Product WN White noise YW Yule-Walker ZTPINAR Zero-truncated Poisson INAR 4NLGMACH Fourth order nonlinear generalized moving average conditional heteroskedasticity