**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**