Time Series: A First Course with Bootstrap Starter, 1st Edition (Hardback) book cover

Time Series

A First Course with Bootstrap Starter, 1st Edition

By Tucker S. McElroy, Dimitris N. Politis

Chapman and Hall/CRC

560 pages

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Hardback: 9781439876510
pub: 2019-11-25
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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

About the Authors

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 Halicioglu 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.

About the Series

Chapman & Hall/CRC Texts in Statistical Science

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General