Introduction to Time Series Modeling

Genshiro Kitagawa is the Director-General of the Institute of Statistical Mathematics in Tokyo, Japan.

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