Introduction to Time Series Modeling

In time series modeling, the behavior of a certain phenomenon is expressed in relation to the past values of itself and other covariates. Since many important phenomena in statistical analysis are actually time series and the identification of conditional distribution of the phenomenon is an essential part of the statistical modeling, it is very important and useful to learn fundamental methods of time series modeling. Illustrating how to build models for time series using basic methods, Introduction to Time Series Modeling covers numerous time series models and the various tools for handling them. The book employs the state-space model as a generic tool for time series modeling and presents convenient recursive filtering and smoothing methods, including the Kalman filter, the non-Gaussian filter, and the sequential Monte Carlo filter, for the state-space models. Taking a unified approach to model evaluation based on the entropy maximization principle advocated by Dr. Akaike, the author derives various methods of parameter estimation, such as the least squares method, the maximum likelihood method, recursive estimation for state-space models, and model selection by the Akaike information criterion (AIC). Along with simulation methods, he also covers standard stationary time series models, such as AR and ARMA models, as well as nonstationary time series models, including the locally stationary AR model, the trend model, the seasonal adjustment model, and the time-varying coefficient AR model. With a focus on the description, modeling, prediction, and signal extraction of times series, this book provides basic tools for analyzing time series that arise in real-world problems. It encourages readers to build models for their own real-life problems.

Table of Contents:
Introduction and Preparatory Analysis Time Series Data Classification of Time Series Objectives of Time Series Analysis Preprocessing of Time Series Organization of This Book The Covariance Function The Distribution of Time Series and Stationarity The Autocovariance Function of Stationary Time Series Estimation of the Autocovariance Function Multivariate Time Series and Scatterplots Cross-Covariance Function and Cross-Correlation Function The Power Spectrum and the Periodogram The Power Spectrum The Periodogram Averaging and Smoothing of the Periodogram Computational Method of Periodogram Computation of the Periodogram by Fast Fourier Transform Statistical Modeling Probability Distributions and Statistical Models K-L Information and the Entropy Maximization Principle Estimation of the K-L Information and Log-Likelihood Estimation of Parameters by the Maximum Likelihood Method Akaike Information Criterion (AIC) Transformation of Data The Least Squares Method Regression Models and the Least Squares Method Householder Transformation Method Selection of Order by AIC Addition of Data and Successive Householder Reduction Variable Selection by AIC Analysis of Time Series Using ARMA Models ARMA Model The Impulse Response Function The Autocovariance Function The Relation between AR Coefficients and the PARCOR The Power Spectrum of the ARMA Process The Characteristic Equation The Multivariate AR Model Estimation of an AR Model Fitting an AR Model Yule–Walker Method and Levinson’s Algorithm Estimation of an AR Model by the Least Squares Method Estimation of an AR Model by the PARCOR Method Large Sample Distribution of the Estimates Yule–Walker Method for MAR Model Least Squares Method for MAR Model The Locally Stationary AR Model Locally Stationary AR Model Automatic Partitioning of the Time Interval Precise Estimation of a Change Point Analysis of Time Series with a State-Space Model The State-Space Model State Estimation via the Kalman Filter Smoothing Algorithms Increasing Horizon Prediction of the State Prediction of Time Series Likelihood Computation and Parameter Estimation for a Time Series Model Interpolation of Missing Observations Estimation of the ARMA Model State-Space Representation of the ARMA Model Initial State of an ARMA Model Maximum Likelihood Estimate of an ARMA Model Initial Estimates of Parameters Estimation of Trends The Polynomial Trend Model Trend Component Model—Model for Probabilistic Structural Changes Trend Model The Seasonal Adjustment Model Seasonal Component Model Standard Seasonal Adjustment Model Decomposition Including an AR Component Decomposition Including a Trading-Day Effect Time-Varying Coefficient AR Model Time-Varying Variance Model Time-Varying Coefficient AR Model Estimation of the Time-Varying Spectrum The Assumption on System Noise for the Time-Varying Coefficient AR Model Abrupt Changes of Coefficients Non-Gaussian State-Space Model Necessity of Non-Gaussian Models Non-Gaussian State-Space Models and State Estimation Numerical Computation of the State Estimation Formula Non-Gaussian Trend Model A Time-Varying Variance Model Applications of Non-Gaussian State-Space Model The Sequential Monte Carlo Filter The Nonlinear Non-Gaussian State-Space Model and Approximations of Distributions Monte Carlo Filter Monte Carlo Smoothing Method Nonlinear Smoothing Simulation Generation of Uniform Random Numbers Generation of Gaussian White Noise Simulation Using a State-Space Model Simulation with Non-Gaussian Model Appendix A: Algorithms for Nonlinear Optimization Appendix B: Derivation of Levinson’s Algorithm Appendix C: Derivation of the Kalman Filter and Smoother Algorithms Appendix D: Algorithm for the Monte Carlo Filter Bibliography