1st Edition

Time Series A First Course with Bootstrap Starter

586 Pages
by Chapman & Hall

586 Pages
by Chapman & Hall

300 Pages
by Chapman & Hall

Also available as eBook on:

Time Series: A First Course with Bootstrap Starter provides an introductory course on time series analysis that satisfies the triptych of (i) mathematical completeness, (ii) computational illustration and implementation, and (iii) conciseness and accessibility to upper-level undergraduate and M.S. students. Basic theoretical results are presented in a mathematically convincing way, and the methods of data analysis are developed through examples and exercises parsed in R. A student with a basic course in mathematical statistics will learn both how to analyze time series and how to interpret the results.

The book provides the foundation of time series methods, including linear filters and a geometric approach to prediction. The important paradigm of ARMA models is studied in-depth, as well as frequency domain methods. Entropy and other information theoretic notions are introduced, with applications to time series modeling. The second half of the book focuses on statistical inference, the fitting of time series models, as well as computational facets of forecasting. Many time series of interest are nonlinear in which case classical inference methods can fail, but bootstrap methods may come to the rescue. Distinctive features of the book are the emphasis on geometric notions and the frequency domain, the discussion of entropy maximization, and a thorough treatment of recent computer-intensive methods for time series such as subsampling and the bootstrap. There are more than 600 exercises, half of which involve R coding and/or data analysis. Supplements include a website with 12 key data sets and all R code for the book's examples, as well as the solutions to exercises.

1. Introduction
Time Series Data
Cycles in Time Series Data
Spanning and Scaling Time Series
Time Series Regression and Autoregression
Overview
Exercises

2. The Probabilistic Structure of Time Series
Random Vectors
Time Series and Stochastic Processes
Marginals and Strict Stationarity
Autocovariance and Weak Stationarity
Illustrations of Stochastic Processes
Three Examples of White Noise
Overview
Exercises

3. Trends, Seasonality, and Filtering
Nonparametric Smoothing
Linear Filters and Linear Time Series
Some Common Types of Filters
Trends
Seasonality
Trend and Seasonality Together
Integrated Processes
Overview
Exercises

4. The Geometry of Random Variables
Vector Space Geometry and Inner Products
L2(; P;F): The Space of Random Variables with Finite Second Moment
Hilbert Space Geometry
Projection in Hilbert Space
Prediction of Time Series
Linear Prediction of Time Series
Orthonormal Sets and Infinite Projection
Projection of Signals
Overview
Exercises

5. ARMA Models with White Noise Residuals
Definition of the ARMA Recursion
Difference Equations
Stationarity and Causality of the AR(1)
Causality of ARMA Processes
Invertibility of ARMA Processes
The Autocovariance Generating Function
Computing ARMA Autocovariances via the MA Representation
Recursive Computation of ARMA Autocovariances
Overview
Exercises

6. Time Series in the Frequency Domain
The Spectral Density
Filtering in the Frequency Domain
Inverse Autocovariances
Spectral Representation of Toeplitz Covariance Matrices
Partial Autocorrelations
Application to Model Identification
Overview
Exercises

7. The Spectral Representation
The Herglotz Theorem
The Discrete Fourier Transform
The Spectral Representation
Optimal Filtering
Kolmogorov's Formula
The Wold Decomposition
Spectral Approximation and the Cepstrum
Overview
Exercises

8. Information and Entropy
Introduction
Events and Information Sets
Maximum Entropy Distributions
Entropy in Time Series
Markov Time Series
Modeling Time Series via Entropy
Relative Entropy and Kullback-Leibler Discrepancy
Overview
Exercises

9. Statistical Estimation
Weak Correlation and Weak Dependence
The Sample Mean
CLT for Weakly Dependent Time Series
Estimating Serial Correlation
The Sample Autocovariance
Spectral Means
Statistical Properties of the Periodogram
Spectral Density Estimation
Refinements of Spectral Analysis
Overview
Exercises

10. Fitting Time Series Models
MA Model Identification
EXP Model Identification
AR Model Identification
Optimal Prediction Estimators
Relative Entropy Minimization
Computation of Optimal Predictors
Computation of the Gaussian Likelihood
Model Evaluation
Model Parsimony and Information Criteria
Model Comparisons
Iterative Forecasting
Applications to Imputation and Signal Extraction
Overview
Exercises

11. Nonlinear Time Series Analysis
Types of Nonlinearity
The Generalized Linear Process
The ARCH Model
The GARCH Model
The Bi-spectral Density
Volatility Filtering
Overview
Exercises

12. The Bootstrap
Sampling Distributions of Statistics
Parameters as Functionals and Monte Carlo
The Plug-in Principle and the Bootstrap
Model-based Bootstrap and Residuals
Sieve Bootstraps
Time Frequency Toggle Bootstrap
Subsampling
Block Bootstrap Methods
Overview
Exercises

A. Probability
Probability Spaces
Random Variables
Expectation and Variance
Joint Distributions
The Normal Distribution
Exercises

B. Mathematical Statistics
Data
Sampling Distributions
Estimation
Inference
Con_dence Intervals
Hypothesis Testing
Exercises

C. Asymptotics
Convergence Topologies
Convergence Results for Random Variables
Asymptotic Distributions
Central Limit Theory for Time Series
Exercises

D. Fourier Series
Complex Random Variables
Trigonometric Polynomials

E. Stieltjes Integration
Deterministic Integration
Stochastic Integration

Biography

Tucker S. McElroy is Senior Time Series Mathematical Statistician at the U.S. Census Bureau, where he has contributed to developing time series research and software for the last 15 years. He has published more than 80 papers and is a recipient of the Arthur S. Flemming award (2011).

Dimitris N. Politis is Distinguished Professor of Mathematics at the University of California at San Diego, where he is also serving as Associate Director of the Halıcıoğlu Data Science Institute. He has co-authored two research monographs and more than 100 journal papers. He is a recipient of the Tjalling C. Koopmans Econometric Theory Prize (2009-2011) and is Co-Editor of the Journal of Time Series Analysis.

"The authors should be congratulated for providing many concise and compact proofs for various technical assertions in time series. (There are many seemingly inconspicuous but intriguing technical details in time series!) The authors' strength and perhaps also their preference in frequency domain methods are well-reflected in the treatments in Chapters 6, 7 and 9, and also some parts of Chapters 10 and 11. Chapter 12 introduces several of the most popular bootstrap methods for time series, including AR-sieve bootstrap, block bootstrap and frequency domain bootstrap. In terms of the mathematical level, the book is for students with a solid mathematical background. The style of the presentation would also better suit courses offered in statistics, mathematics or engineering programmes for which spectral analysis is pertinent."
~International Statistical Review

"The first eight chapters of this book mainly focus on understanding the structure of time series. From the ninth chapter onwards, they discuss statistical inference based on time series data…Since the book includes a large number of exercises, teachers of a course on time series may find this book useful. Overall, researchers working in the area of time series may also find this book a useful reference. Finally, applied researchers involved with time series data may also find this book helpful." ~ISCB News

"This new monograph by McElroy (US Census Bureau) and Politis (Univ. of California, San Diego) is a timely publication, whereas the more well-known time series monographs were published long ago (in the 1980s and 1990s).. this volume stands out as an ideal source for readers exploring time series analysis both theoretically and empirically…Some unique topics are introduced, for example, information entropy in time series, time-series-specific statistical inference, and dependent data bootstrapping. The latter represents an important recent advancement in time series analysis."
~CHOICE