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This is page i Printer: Opaque this Jianqing Fan Qiwei Yao Nonlinear Time Series: Nonparametric and Parametric Meth- ods

This is page ii Printer: Opaque this TO THOSE WHO EDUCATE US; WHOM WE LOVE; AND WITH WHOM WE COLLABORATE

Back-cover iii This book is on the contemporary statistical methods and theory of nonlinear time series analysis. The principal focus is on nonparametric and semiparametric techniques developed in the last decade. It covers the techniques for modelling in state-space, in frequency-domain as well as in time-domain. To reﬂect the integration of parametric and nonparametric methods in analyzing time series data, the book also presents an up-to-date exposure on some parametric nonlinear models, including ARCH/GARCH models and threshold models. A compact view on linear ARMA models is also provided. Data arising in real applications are used throughout to show how nonparametric approaches may help to reveal local structure in high-dimensional data. Important technical tools are also introduced. The book is designed to appeal to graduate students, application-oriented time series analysts, new and experienced researchers. It will have the value both within the statistical community and across a broad spectrum of other ﬁelds such as econometrics, empirical ﬁnance, population biology and ecol- ogy. The prerequisites are merely sound basic courses in probability and statistics. Jianqing Fan, coauthor of the highly regarded book Local Polynomial Modeling, is Professor of Statistics at the University of North Carolina at Chapel Hill and the Chinese University of Hong Kong. His published work on nonparametric modeling, nonlinear time series, ﬁnancial econometrics, analysis of longitudinal data, model selection, wavelets and other aspects of methodological and theoretical statistics has been recognized with the Presidents’ Award from the Committee of Presidents of Statistical Soci- eties, the Hettleman Prize for Artistic and Scholarly Achievement from the University of North Carolina, and by his election as a fellow of the Ameri- can Statistical Association and the Institute of Mathematical Statistics. Qiwei Yao is Professor of Statistics at the London School of Economics and Political Science. He is an elected member of the International Sta- tistical Institute, has served on editorial board for Journal of the Royal Statistical Society (Series B) and Australian and New Zealand Journal of Statistics.

Preface This is page v Printer: Opaque this Among many exciting developments in statistics over the last two decades, Nonlinear Time Series and Data-analytic Nonparametric Methods have greatly advanced along seemingly unrelated paths. In spite of the fact that the application of nonparametric techniques in time series can be backdated to as early as 1940’s, there still exists healthy and justiﬁed skepticism about the capability of nonparametric methods in analyzing time series data. As enthusiastic explorers of the modern nonparametric tool-kit, we feel obliged to assemble together the newly developed relevant techniques in one place. The aim of this book is to advocate those modern nonparametric techniques which have proved useful for analyzing real time series data, and to provoke further research in both methodology and theory for nonparametric time series analysis. Exponential increases in computing power (Moore’s law) and falling costs have had a profound impact on the development of statistics including time series. Modern computers and the information age have brought us oppor- tunities with challenges. Technological inventions have lead to the explosion in data collection (e.g. daily grocery sales, stock market trading, microar- ray data). The internet makes big data warehouses readily accessible. It has become commonplace to collect and analyze time series data of un- precedented size and complexity. Whilst classic parametric models, which postulate global structures for underlying systems, are still very useful, large sizes of data sets invite the search for more reﬁned structures, which leads to better understandings and approximations of the real world. Be- yond postulated parametric models, there are inﬁnite other possibilities. Nonparametric techniques provide useful exploratory tools to this venture,

vi PREFACE including the suggestion of new and the validating of existing parametric models. In this book, we present an up-to-date picture of techniques for analyz- ing time series data. Although we have tried to maintain a good balance among methodology, theory, and numerical illustration, our primary goal is to present a comprehensive and self-contained account for each of the key methodologies. For practical relevant time series models, we aim for exposure with deﬁnition, probability properties (if possible), statistical in- ference methods, and numerical examples with real data sets. We also in- dicate where to ﬁnd our (only our!) favorite computing codes to implement these statistical methods. When soliciting real-data examples, we attempt to maintain a good balance among diﬀerent disciplines, though our per- sonal interests in quantitative ﬁnance, risk management, and biology can be easily seen. It is our hope that readers can apply these techniques to their own data sets. We trust that the book will be of interest to those coming to the area for the ﬁrst time, and to readers more familiar with the ﬁeld. Application- oriented time series analysts will also ﬁnd this book useful as it focuses on methodology and includes several case studies with real data sets. We be- lieve that nonparametric methods must go hand in hand with parametric methods in applications. In particular, parametric models provide explana- tory power and concise descriptions of the underlying dynamics, which, when used sensibly, is an advantage over nonparametric models. For this reason, we have also provided a compact view on the parametric methods for both linear and selected nonlinear time series models. This will also give new comers suﬃcient information on the essence of the more classical approaches. We hope that this book will reﬂect the power of the integration of nonparametric and parametric approaches in analyzing time series data. The book has been prepared for a broad readership — the prerequisites are merely sound basic courses in probability and statistics. Whilst advanced mathematics has provided valuable insights into nonlinear time series, the methodological power of both nonparametric and parametric approaches can be understood without sophisticated technical details. Due to the in- nate nature of the subject, it is inevitable that we occasionally appeal for more advanced mathematics; such sections are marked with a ‘*’. Most technical arguments are collected in ‘Complements’ section at the end of each chapter, but key ideas are left within text. The introduction in Chapter 1 sets the scene for the book. Chapter 2 deals with basic probabilistic properties of time series processes. The high- lights include strict stationarity via ergodic Markov chains (§2.1) and mix- ing properties (§2.6). We also provide a generic central limit theorm for kernel-based nonparametric regression estimation for α-mixing processes. A compact view of linear ARMA models is given in Chapter 3, including Gaussian MLE (§3.3), model selection criteria (§3.4), and linear forecasting with ARIMA models (§3.7). Chapter 4 introduces three types of paramet-

PREFACE vii ric nonlinear models. An introduction on threshold models that emphasizes developments after Tong (1990) is provided. ARCH and GARCH models are presented in detail, as they are less exposed in statistical literature. The chapter concludes with a brief account of bilinear models. Chapter 5 introduces the nonparametric kernel density estimation. This is arguably the simplest problem for understanding nonparametric techniques. The re- lation between “localization” for nonparametric problems and “whitening” for time series data is elucidated in §5.3. Applications of nonparametric techniques for estimating time trends and univariate autoregressive func- tions can be found in Chapter 6. The ideas in Chapter 5 and §6.3 provide a foundation for the nonparametric techniques introduced in the rest of the book. Chapter 7 introduces spectral density estimation and nonparametric procedures for testing whether a series is white noise. Various high-order au- toregressive models are highlighted in Chapter 8. In particular, techniques for estimating nonparametric functions in FAR models are introduced in §8.3. The additive autoregressive model is exposed in §8.5, and methods for estimating conditional variance or volatility functions are detailed in §8.7. Chapter 9 outlines approaches to testing a parametric family of models against a family of structured nonparametric models. The wide applicabil- ity of the generalized likelihood ratio test is emphasized. Chapter 10 deals with nonlinear prediction. It highlights the features that distinguish non- linear prediction from linear prediction. It also introduces nonparametric estimation for conditional predictive distribution functions and conditional minimum volume predictive intervals. 1 2 ppppppppppppppppp- @ @ @R I @ @ @ 6 - - 4 3 pppppppppppppppppppppppppppppppppppppppppppppppppppU ppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppp 7 5 6 8 ppppppppppppppppppppppppppppppppppppppppppp 10 - - 6 @ @ ? 9 @ @@R - The interdependence of the chapters is depicted above, where solid di- rected lines indicate prerequisites and dotted lines indicate weak associ- ations. For lengthy chapters, the dependence among sections is not very strong. For example, the sections in Chapter 4 are fairly independent, and so are those in Chapter 8 (except that §8.4 depends on §8.3, and §8.7 de- pends on the rest). They can be read independently. Chapter 5 and §6.3 provide a useful background for nonparametric techniques. With under-

viii PREFACE standing of these, readers can jump directly to sections in Chapters 8 and 9. For readers who wish to obtain an overall impression of the book, we suggest reading Chapters 1, §2.1, §2.2, 3, §4.1, §4.2, 5, §6.3, §8.3, §8.5, §8.7, §9.1, §9.2, §9.4 and §9.5. These core materials could even serve as the text of a graduate course on nonlinear time series. Although the scope of the book is wide, we have not achieved complete- ness. The nonparametric methods are mostly centered around kernel/local polynomial based smoothing. Nonparametric hypothesis testing with struc- tured nonparametric alternatives is mainly conﬁned to the generalized like- lihood ratio test. In fact, many techniques that are introduced in this book have not been formally explored mathematically. State-space mod- els are only mentioned brieﬂy within the discussion on bilinear models and stochastic volatility models. Multivariate time series analysis is untouched. Another noticeable gap is the lack of exposure of the variety of paramet- ric nonlinear time series models listed in Chapter 3 of Tong (1990). This is undoubtedly a shortcoming. In spite of the important initial progress, we feel that the methods and theory of statistical inference for some of those models are not as well-established as, for example, ARCH/GARCH models or threshold models. Their potential applications should be further explored. Extensive eﬀort was expended in the composition of the reference list, which, together with the biographical notes, should guide readers to a wealth of available materials. Although our reference list is long, it merely reﬂects our immediate interests. Many important papers which do not ﬁt our presentation have been omitted. Other omissions and discrepancies are inevitable. We apologize for their occurrence. Although we both share the responsibility for the whole book, Jianqing Fan was the lead author for Chapters 1 and 5–9, and Qiwei Yao for Chapters 2-4 and 10. Many people have been of great help to our work on this book. In par- ticular we would like to thank Hong-Zhi An, Peter Bickel, Peter Brockwell, Yuzhi Cai, Zongwu Cai, Kung-Sik Chan, Cees Diks, Rainer Dahlhaus, Li- udas Giraitis, Peter Hall, Wai-Keung Li, Jianzhong Lin, Liang Peng, Stathis Paparodistis, Wolfgang Polonik, John Rice, Peter Robinson, Richard Smith, Howell Tong, Yingcun Xia, Wenyang Zhang and anonymous reviewers. We are also grateful to all of our collaborators for their friendly and stimulating collaboration. Many of their ideas have been reﬂected in this book. Jianqing Fan’s research was partially supported by the National Science Foundation and National Institute of Health of the USA and the Research Grant Council of the Hong Kong Special Autonomic Region. Qiwei Yao’s work was partially supported by the Engineering and Physical Sciences Research Council and the Biotechnology and Biological Sciences Research Council of UK. This book was written while Jianqing Fan was employed by the University of California at Los Angeles, the University of North Carolina at Chapel Hill, and the Chinese University of Hong Kong; and

PREFACE ix Qiwei Yao was employed by the University of Kent at Canterbury and the London School of Economics and Political Science. We would like to acknowledge gratefully for the generous support and inspirations of our colleagues. May 2002 Jianqing Fan Qiwei Yao

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