Time Series : A First Course with Bootstrap Starter book cover
1st Edition

Time Series
A First Course with Bootstrap Starter

ISBN 9781032083308
Published June 30, 2021 by Chapman and Hall/CRC
586 Pages

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Book Description

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.

Table of Contents

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

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

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

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

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

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

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

8. Information and Entropy
   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

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

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

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

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
    Block Bootstrap Methods

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

B. Mathematical Statistics
   Sampling Distributions
   Con_dence Intervals
   Hypothesis Testing

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

D. Fourier Series
   Complex Random Variables
   Trigonometric Polynomials

E. Stieltjes Integration
   Deterministic Integration
   Stochastic Integration

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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."