Elements of Nonlinear Time Series Analysis and Forecasting PDF
Preview Elements of Nonlinear Time Series Analysis and Forecasting
Springer Series in Statistics Jan G. De Gooijer Elements of Nonlinear Time Series Analysis and Forecasting Springer Series in Statistics Series editors Peter Bickel, CA, USA Peter Diggle, Lancaster, UK Stephen E. Fienberg, Pittsburgh, PA, USA Ursula Gather, Dortmund, Germany Ingram Olkin, Stanford, CA, USA Scott Zeger, Baltimore, MD, USA More information about this series at http://www.spring er.com/series/692 Jan G. De Gooijer Elements of Nonlinear Time Series Analysis and Forecasting 123 Jan G. De Gooijer University of Amsterdam Amsterdam, The Netherlands ISSN 0172-7397 ISSN 2197-568X (electronic) Springer Series in Statistics ISBN 978-3-319-43251-9 ISBN 978-3-319-43252-6 (ebook ) DOI 10.1007/978-3-319-43252-6 Library of Congress Control Number: 2017935720 © Springer International Publishing Switzerland 2017 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. 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To Jeanne Preface Empiricaltimeseriesanalysisandmodelinghasbeendeviating,overthelast40years orso, fromthelinearparadigmwiththeaimofincorporatingnonlinearfeatures. In- deed, there are various occasions when subject-matter, theory or data suggests that a time series is generated by a nonlinear stochastic process. If theory could provide some understanding of the nonlinear phenomena underlying the data, the modeling process would be relatively easy, with estimation of the model parameters being all that is required. However, this option is rarely available in practice. Alternatively, a particular nonlinear model may be selected, fitted to the data and subjected to a battery of diagnostic tests to check for features that the model has failed adequately to approximate. Although this approach corresponds to the usual model selection strategy in linear time series analysis, it may involve rather more problems than in the linear case. One immediate problem is the selection of an appropriate nonlinear model or method. However, given the wealth of nonlinear time series models now available, this is a far from easy task. For practical use a good nonlinear model should at least fulfill the requirement that it is general enough to capture some of the nonlinear phenomena in the data and, moreover, should have some intuitive appeal. This implies a systematic account of various aspects of these models and methods. The Hungarian mathematician John von Neumann once said that the study of nonlinear functions is akin to the study of non-elephants.1 This remark illustrates a common problem with nonlinear theory, which in our case is equivalent to non- linear models/methods: the subject is so vast that it is difficult to develop general approachesandtheoriessimilartothoseexistingforlinearfunctions/models. Fortu- nately,overthelasttwotothreedecades,thetheoryandpracticeof“non-elephants” has made enormous progress. Indeed, several advancements have taken place in the nonlinear model development process in order to capture specific nonlinear features of the underlying data generating process. These features include symptoms such as 1AsimilarremarkiscreditedtothePolishmathematicianStanislawM.Ulamsayingthatusing a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals; Campbell, Farmer, Crutchfield, and Jen (1985), “Experimental mathematics: The role of computation in nonlinear science”. Communications of the ACM, 28(4), 374–384. vii viii Preface non-Gaussianity, aperiodicity, asymmetric cycles, multi-modality, nonlinear causal relationships, nonstationarity, and time-irreversibility, among others. Additionally, considerable progress has been made in the development of methods for real, out- of-sample, nonlinear time series forecasting.2 Unsurprisingly, the mass of research and applications of nonlinear time series analysis and forecasting methods is scattered over a wide range of scientific discip- lines and numerous journal articles. This does not ensure easy access to the sub- ject. Moreover, different papers tend to use different notations making it difficult to conceptualize, compare, and contrast new ideas and developments across different scientific fields. This book is my attempt to bring together, organize, extend many of the important ideas and works in nonlinear time series analysis and forecasting, and explain them in a comprehensive and systematic statistical framework. While some mathematical details are needed, the main intent of the book is to provide an overview of the current state-of-the-art of the subject, focusing on practical issues rather than discussing technical details. To reach this goal, the text offers a large number of examples, pseudo-algorithms, empirical exercises, and real-world illustrations, as well as other supporting additions and features. In this respect, I hope that the many empirical examples will testify to the breadth of the subject matter that the book addresses. Some of the material presented in the book is my own or developed with co-authors, but a very large part is based on the contributions made by others. Extensive credit for such previously published work is given throughout the book, and additional bibliographic notes are given at the end of every chapter. Who is this book for? The text is designed to be used with a course in Nonlinear Time Series Analysis, Statistical System Processing orwithacoursein Nonlinear Model Identification that would typically be offered to graduate students in system engineering, mathematics, statistics, and econometrics. At the same time, the book will appeal to researchers, postgraduates, and practitioners in a wide range of other fields. Finally, the book should be of interest to more advanced readers who would like to brush up on their present knowledge of the subject. Thus, the book is not written toward a singleprototypicalreaderwithaspecificbackground,anditislargelyself-contained. Nevertheless, it is assumed that the reader has some familiarity with basic linear time series ideas. Also, a bit of knowledge about Markov chains and Monte Carlo simulation methods is more than welcome. The book is selective in its coverage of subjects, although this does not imply that a particular topic is unimportant if it is not included. For instance, Bayesian approaches – that can relax many assumptions commonly made on the type and nature of nonlinearity – can be applied to all models. Of course, the extensive list of 2Throughout the book, I will use the terms forecast and prediction interchangeably, although notquiteprecisely. Thatis,predictionconcernsstatementsaboutthelikelyoutcomeofunobserved events, not necessarily those in the future. Preface ix references allows readers to follow up on original sources for more technical details on different methods. As a further help to facilitate reading, each chapter concludes with a set of key terms and concepts, and a summary of the main findings. What are the main features? Here are some main features of the book. • The book shows concrete applications of “modern” nonlinear time series ana- lysis on a variety of empirical time series. It avoids a “theorem-proof” format. • Thebookpresentsatoolboxofdiscrete-timenonlinearmodels,methods,tests, andconcepts. Thereisusually,butnotinallcases,adirectfocusonthe“best” available procedure. Alternative procedures that boast sufficient theoretical and practical underpinning are introduced as well. • The book uses graphs to explore and summarize real-world data, analyze the validity of the nonlinear models fitted and present the forecasting results. • The book covers time-domain and frequency-domain methods both for the analysis of univariate and multivariate (vector) time series. In addition, the book makes a clear distinction between parametric models on the one hand, and semi- and nonparametric models/methods on the other. This offers the reader the possibility to concentrate exclusively on one of these ways of time series analysis. • One additional feature of the book are the numerous algorithms in pseudo codeformwhichstreamlinemanyideasandmaterialinasystematicway. Thus readerscanrapidlyobtainthegeneralgistofamethodortechnique. Moreover, it is relatively easy to convert a pseudocode to programming language. Real data It is well known that real data analysis can reduce the gap between theory and practice. Hence,throughoutthebookabroadsetofempiricaltimeseries,originating from many different scientific fields, will be used to illustrate the main points of the text. ThisalreadystartsoffinChapter1whereIintroducefiveempiricaltimeseries which will be used as “running” examples throughout the book. In later chapters, other concrete examples of nonlinear time series analysis will appear. In each case, I provide some background information about the data so that the general context becomes clear. It may also help the reader to get a better understanding of specific nonlinear features in the underlying data generating mechanism. About the chapters The text is organized as follows. Chapter 1 introduces some important terms and concepts from linear and nonlinear time series analysis. In addition, this chapter offers some basic tools for initial data analysis and visualization. Next, the book is structured into two tracks. x Preface The first track (Chapters 2, 3, 5 – 8, and 10) mainly includes parametric non- linear models and techniques for univariate time series analysis. Here, the overall outline basically follows the iterative cycle of model identification, parameter es- timation, and model verification by diagnostic checking. In particular, Chapter 2 concentrates on some important nonlinear model classes. Chapter 3 introduces the concepts of stationarity and invertibility. The material on time-domain linearity testing (Chapter 5), model estimation and selection (Chapter 6), tests for serial dependence (Chapter 7), and time-reversibility (Chapter 8) relates to Chapter 2. Although Chapter 7 is clearly based on nonparametric methods, the proposed test statisticstrytodetectstructurein“residuals”obtainedfromfittedparametricmod- els, and hence its inclusion in this track. If forecasting from parametric univariate timeseriesmodelsistheobjective,Chapter10providesahostofmethods. Asapart of the entire forecasting process, the chapter also includes methods for the construc- tion of forecast intervals/regions, and methods for the evaluation and combination of forecasts. When sufficient data is available, the flexibility offered by many of the semi- and nonparametric techniques in the second track may be preferred over parametric models/methods. A possible starting point of this track is to test for linearity and Gaussianity through spectral density estimation methods first (Chapter 4). In some situations, however, a reader can jump directly to specific sections in Chapter 9 which contain extensive material on analyzing nonlinear time series by semi- and nonparametric methods. Also some sections in Chapter 9 discuss forecasting in a semi-andnonparametricsetting. Finally, bothtrackscontainchaptersonmultivari- ate nonlinear time series analysis (Chapters 11 and 12). The following exhibit gives a rough depiction of how the two tracks are interrelated. Univariate Multivariate 8 Parametric 2 5 6 7 10 11 3 1 Semi- and nonparametric 4 9 12 Each solid directed line, denoted by a → b, represents a suggestion that Chapter a be read before Chapter b. The medium-dashed lines indicate that some specific chapters can be read independently. Chapters 2, 7, and 9 are somewhat lengthy, but the dependence among sections is not very strong. At the end of each chapter, the book contains two types of exercises. Theory exercises illustrate and reinforce the theory at a more advanced level, and provide results that are not available in the main text. The chapter also includes empir-
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