Time Series A First Course with Bootstrap Starter 1st edition by Tucker McElroy, Dimitris Politis ISBN 0429527225 9780429527227
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ISBN 10: 0429527225
ISBN 13: 9780429527227
Author: Tucker McElroy, Dimitris Politis
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.
Time Series A First Course with Bootstrap Starter 1st Table of contents:
1 Introduction
1.1 Time Series Data
1.2 Cycles in Time Series Data
1.3 Spanning and Scaling Time Series
1.4 Time Series Regression and Autoregression
1.5 Overview
1.6 Exercises
2 The Probabilistic Structure of Time Series
2.1 Random Vectors
2.2 Time Series and Stochastic Processes
2.3 Marginals and Strict Stationarity
2.4 Autocovariance and Weak Stationarity
2.5 Illustrations of Stochastic Processes
2.6 Three Examples of White Noise
2.7 Overview
2.8 Exercises
3 Trends, Seasonality, and Filtering
3.1 Nonparametric Smoothing
3.2 Linear Filters and Linear Time Series
3.3 Some Common Types of Filters
3.4 Trends
3.5 Seasonality
3.6 Trend and Seasonality Together
3.7 Integrated Processes
3.8 Overview
3.9 Exercises
4 The Geometry of Random Variables
4.1 Vector Space Geometry and Inner Products
4.2 L2(Ω, P, 𝓕) The Space of Random Variables with Finite Second Moment
4.3 Hilbert Space Geometry [*]
4.4 Projection in Hilbert Space
4.5 Prediction of Time Series
4.6 Linear Prediction of Time Series
4.7 Orthonormal Sets and Infinite Projection
4.8 Projection of Signals [*]
4.9 Overview
4.10 Exercises
5 ARMA Models with White Noise Residuals
5.1 Definition of the ARMA Recursion
5.2 Difference Equations
5.3 Stationarity and Causality of the AR(1)
5.4 Causality of ARMA Processes
5.5 Invertibility of ARMA Processes
5.6 The Autocovariance Generating Function
5.7 Computing ARMA Autocovariances via the MA Representation
5.8 Recursive Computation of ARMA Autocovariances
5.9 Overview
5.10 Exercises
6 Time Series in the Frequency Domain
6.1 The Spectral Density
6.2 Filtering in the Frequency Domain
6.3 Inverse Autocovariances
6.4 Spectral Representation of Toeplitz Covariance Matrices
6.5 Partial Autocorrelations
6.6 Application to Model Identification
6.7 Overview
6.8 Exercises
7 The Spectral Representation [*]
7.1 The Herglotz Theorem
7.2 The Discrete Fourier Transform
7.3 The Spectral Representation
7.4 Optimal Filtering
7.5 Kolmogorov’s Formula
7.6 The Wold Decomposition
7.7 Spectral Approximation and the Cepstrum
7.8 Overview
7.9 Exercises
8 Information and Entropy [*]
8.1 Introduction
8.2 Events and Information Sets
8.3 Maximum Entropy Distributions
8.4 Entropy in Time Series
8.5 Markov Time Series
8.6 Modeling Time Series via Entropy
8.7 Relative Entropy and Kullback-Leibler Discrepancy
8.8 Overview
8.9 Exercises
9 Statistical Estimation
9.1 Weak Correlation and Weak Dependence
9.2 The Sample Mean
9.3 CLT for Weakly Dependent Time Series [*]
9.4 Estimating Serial Correlation
9.5 The Sample Autocovariance
9.6 Spectral Means
9.7 Statistical Properties of the Periodogram
9.8 Spectral Density Estimation
9.9 Refinements of Spectral Analysis
9.10 Overview
9.11 Exercises
10 Fitting Time Series Models
10.1 MA Model Identification
10.2 EXP Model Identification [*]
10.3 AR Model Identification
10.4 Optimal Prediction Estimators
10.5 Relative Entropy Minimization
10.6 Computation of Optimal Predictors
10.7 Computation of the Gaussian Likelihood
10.8 Model Evaluation
10.9 Model Parsimony and Information Criteria
10.10 Model Comparisons
10.11 Iterative Forecasting
10.12 Applications to Imputation and Signal Extraction
10.13 Overview
10.14 Exercises
11 Nonlinear Time Series Analysis
11.1 Types of Nonlinearity
11.2 The Generalized Linear Process [*]
11.3 The ARCH Model
11.4 The GARCH Model
11.5 The Bi-spectral Density
11.6 Volatility Filtering
11.7 Overview
11.8 Exercises
12 The Bootstrap
12.1 Sampling Distributions of Statistics
12.2 Parameter Functionals and Monte Carlo
12.3 The Plug-In Principle and the Bootstrap
12.4 Model-Based Bootstrap and Residuals
12.5 Sieve Bootstraps
12.6 Time Frequency Toggle Bootstrap
12.7 Subsampling
12.8 Block Bootstrap Methods
12.9 Overview
12.10 Exercises
A Probability
A.1 Probability Spaces
A.2 Random Variables
A.3 Expectation and Variance
A.4 Joint Distributions
A.5 The Normal Distribution
A.6 Exercises
B Mathematical Statistics
B.1 Data
B.2 Sampling Distributions
B.3 Estimation
B.4 Inference
B.5 Confidence Intervals
B.6 Hypothesis Testing
B.7 Exercises
C Asymptotics
C.1 Convergence Topologies
C.2 Convergence Results for Random Variables
C.3 Asymptotic Distributions
C.4 Central Limit Theory for Time Series
C.5 Exercises
D Fourier Series
D.1 Complex Random Variables
D.2 Trigonometric Polynomials
E Stieltjes Integration
E.1 Deterministic Integration
E.2 Stochastic Integration
Index
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Tucker McElroy,Dimitris Politis,Time Series,Bootstrap Starter