Time Series Analysis and Its Applications With R Examples

Preface to the Third Edition -The goals of this book are to develop an appreciation for the richness and versatility of modern time series analysis as a tool for analyzing data, and still maintain a commitment to theoretical integrity, as exemplified by the seminal works of Brillinger (1975) and Hannan (1970) and the texts by Brockwell and Davis (1991) and Fuller (1995). 

The advent of inexpensive powerful computing has provided both real data and new software that can take one considerably beyond the fitting of simple time domain models, such as have been elegantly described in the landmark work of Box and Jenkins (1970). This book is designed to be useful as a text for courses in time series on several different levels and as a reference work for practitioners facing the analysis of timecorrelated data in the physical, biological, and social sciences. We have used earlier versions of the text at both the undergraduate and graduate levels over the past decade. Our experience is that an undergraduate course can be accessible to students with a background in regression analysis and may include §1.1–§1.6, §2.1–§2.3, the results and numerical parts of §3.1– §3.9, and briefly the results and numerical parts of §4.1–§4.6. At the advanced undergraduate or master’s level, where the students have some mathematical statistics background, more detailed coverage of the same sections, with the inclusion of §2.4 and extra topics from Chapter 5 or Chapter 6 can be used as a one-semester course. Often, the extra topics are chosen by the students according to their interests. Finally, a two-semester upper-level graduate course for mathematics, statistics, and engineering graduate students can be crafted by adding selected theoretical appendices. For the upper-level graduate course, we should mention that we are striving for a broader but less rigorous level of coverage than that which is attained by Brockwell and Davis (1991), the classic entry at this level. The major difference between this third edition of the text and the second edition is that we provide R code for almost all of the numerical examples. In addition, we provide an R supplement for the text that contains the data and scripts in a compressed file called tsa3.rda; the supplement is available on the website for the third edition, http://www.stat.pitt.edu/stoffer/tsa3/,or one of its mirrors. On the website, we also provide the code used in each example so that the reader may simply copy-and-paste code directly into R. Specific details are given in Appendix R and on the website for the text. Appendix R is new to this edition, and it includes a small R tutorial as well as providing a reference for the data sets and scripts included in tsa3.rda. So there is no misunderstanding, we emphasize the fact that this text is about time series analysis, not about R. R code is provided simply to enhance the exposition by making the numerical examples reproducible. We have tried, where possible, to keep the problem sets in order so that an instructor may have an easy time moving from the second edition to the third edition. However, some of the old problems have been revised and there are some new problems. Also, some of the data sets have been updated. We added one section in Chapter 5 on unit roots and enhanced some of the presentations throughout the text. The exposition on state-space modeling, ARMAX models, and (multivariate) regression with autocorrelated errors in Chapter 6 have been expanded. In this edition, we use standard R functions as much as possible, but we use our own scripts (included in tsa3.rda) when we feel it is necessary to avoid problems with a particular R function; these problems are discussed in detail on the website for the text under R Issues. We thank John Kimmel, Executive Editor, Springer Statistics, for his guidance in the preparation and production of this edition of the text. We are grateful to Don Percival, University of Washington, for numerous suggestions that led to substantial improvement to the presentation in the second edition, and consequently in this edition. We thank Doug Wiens, University of Alberta, for help with some of the R code in Chapters 4 and 7, and for his many suggestions for improvement of the exposition. We are grateful for the continued help and advice of Pierre Duchesne, University of Montreal, and Alexander Aue, University of California, Davis. We also thank the many students and other readers who took the time to mention typographical errors and other corrections to the first and second editions. Finally, work on the this edition was supported by the National Science Foundation while one of us (D.S.S.) was working at the Foundation under the Intergovernmental Personnel Act. Davis, CA Robert H. Shumway Pittsburgh, PA David S. Stoffer September 2010

Characteristics of Time Series 

1.1 Introduction 

The analysis of experimental data that have been observed at different points in time leads to new and unique problems in statistical modeling and inference. The obvious correlation introduced by the sampling of adjacent points in time can severely restrict the applicability of the many conventional statistical methods traditionally dependent on the assumption that these adjacent observations are independent and identically distributed. The systematic approach by which one goes about answering the mathematical and statistical questions posed by these time correlations is commonly referred to as time series analysis. The impact of time series analysis on scientific applications can be partially documented by producing an abbreviated listing of the diverse fields in which important time series problems may arise. For example, many familiar time series occur in the field of economics, where we are continually exposed to daily stock market quotations or monthly unemployment figures. ments. An epidemiologist might be interested in the number of influenza cases observed over some time period. In medicine, blood pressure measurements traced over time could be useful for evaluating drugs used in treating hypertension. Functional magnetic resonance imaging of brain-wave time series patterns might be used to study how the brain reacts to certain stimuli under various experimental conditions. Many of the most intensive and sophisticated applications of time series methods have been to problems in the physical and environmental sciences. This fact accounts for the basic engineering flavor permeating the language of time series analysis. One of the earliest recorded series is the monthly sunspot numbers studied by Schuster (1906). More modern investigations may center on whether a warming is present in global temperature measurementsor whether levels of pollution may influence daily mortality in Los Angeles. The modeling of speech series is an important problem related to the efficient transmission of voice recordings. Common features in a time series characteristic known as the power spectrum are used to help computers recognize and translate speech. Geophysical time series such as those produced by yearly depositions of various kinds can provide long-range proxies for temperature and rainfall. Seismic recordings can aid in mapping fault lines or in distinguishing between earthquakes and nuclear explosions. The above series are only examples of experimental databases that can be used to illustrate the process by which classical statistical methodology can be applied in the correlated time series framework. In our view, the first step in any time series investigation always involves careful scrutiny of the recorded data plotted over time. This scrutiny often suggests the method of analysis as well as statistics that will be of use in summarizing the information in the data. Before looking more closely at the particular statistical methods, it is appropriate to mention that two separate, but not necessarily mutually exclusive, approaches to time series analysis exist, commonly identified as the time domain approach and the frequency domain approach. The time domain approach is generally motivated by the presumption that correlation between adjacent points in time is best explained in terms of a dependence of the current value on past values. The time domain approach focuses on modeling some future value of a time series as a parametric function of the current and past values. In this scenario, we begin with linear regressions of the present value of a time series on its own past values and on the past values of other series. This modeling leads one to use the results of the time domain approach as a forecasting tool and is particularly popular with economists for this reason