The book "Time Series Analysis by State Space Methods" by J. Durbin and S. J. Koopman, published by Oxford University Press, is a comprehensive guide to state space methods in time series analysis. The second edition covers both linear and non-Gaussian, nonlinear state space models, providing detailed explanations and practical applications. **Part I: Linear State Space Model** - **Introduction**: Basic concepts, linear models, and notation. - **Local Level Model**: Filtering, smoothing, forecasting, and parameter estimation. - **Linear State Space Models**: Univariate and multivariate structural time series models, ARMA and ARIMA models, exponential smoothing, regression models, dynamic factor models, and continuous-time models. - **Filtering, Smoothing, and Forecasting**: Kalman filter, smoother, and other smoothing algorithms. - **Initialisation of Filter and Smoother**: Exact initial Kalman filter, state smoothing, disturbance smoothing, and simulation smoothing. - **Further Computational Aspects**: Regression estimation, square root filter and smoother, univariate treatment of multivariate series, collapsing large observation vectors, and filtering and smoothing under linear restrictions. - **Maximum Likelihood Estimation of Parameters**: Likelihood evaluation, parameter estimation, goodness of fit, and diagnostic checking. **Part II: Non-Gaussian and Nonlinear State Space Models** - **Special Cases of Nonlinear and Non-Gaussian Models**: Models with exponential family densities, heavy-tailed distributions, stochastic volatility, and financial models. - **Approximate Filtering and Smoothing**: Extended Kalman filter, unscented Kalman filter, nonlinear smoothing, and approximation via data transformation and mode estimation. - **Importance Sampling for Smoothing**: Implementation details, estimating functions, and loglikelihood estimation. - **Particle Filtering**: Sequential importance sampling, bootstrap particle filter, auxiliary particle filter, and Rao-Blackwellisation. - **Bayesian Estimation of Parameters**: Posterior analysis for linear Gaussian and nonlinear non-Gaussian models, and Markov chain Monte Carlo methods. - **Non-Gaussian and Nonlinear Illustrations**: Applications to nonlinear decomposition, Poisson density, heavy-tailed density, volatility, and binary density. The book includes numerous illustrations, exercises, and references to support the theoretical and practical aspects of state space methods in time series analysis.The book "Time Series Analysis by State Space Methods" by J. Durbin and S. J. Koopman, published by Oxford University Press, is a comprehensive guide to state space methods in time series analysis. The second edition covers both linear and non-Gaussian, nonlinear state space models, providing detailed explanations and practical applications. **Part I: Linear State Space Model** - **Introduction**: Basic concepts, linear models, and notation. - **Local Level Model**: Filtering, smoothing, forecasting, and parameter estimation. - **Linear State Space Models**: Univariate and multivariate structural time series models, ARMA and ARIMA models, exponential smoothing, regression models, dynamic factor models, and continuous-time models. - **Filtering, Smoothing, and Forecasting**: Kalman filter, smoother, and other smoothing algorithms. - **Initialisation of Filter and Smoother**: Exact initial Kalman filter, state smoothing, disturbance smoothing, and simulation smoothing. - **Further Computational Aspects**: Regression estimation, square root filter and smoother, univariate treatment of multivariate series, collapsing large observation vectors, and filtering and smoothing under linear restrictions. - **Maximum Likelihood Estimation of Parameters**: Likelihood evaluation, parameter estimation, goodness of fit, and diagnostic checking. **Part II: Non-Gaussian and Nonlinear State Space Models** - **Special Cases of Nonlinear and Non-Gaussian Models**: Models with exponential family densities, heavy-tailed distributions, stochastic volatility, and financial models. - **Approximate Filtering and Smoothing**: Extended Kalman filter, unscented Kalman filter, nonlinear smoothing, and approximation via data transformation and mode estimation. - **Importance Sampling for Smoothing**: Implementation details, estimating functions, and loglikelihood estimation. - **Particle Filtering**: Sequential importance sampling, bootstrap particle filter, auxiliary particle filter, and Rao-Blackwellisation. - **Bayesian Estimation of Parameters**: Posterior analysis for linear Gaussian and nonlinear non-Gaussian models, and Markov chain Monte Carlo methods. - **Non-Gaussian and Nonlinear Illustrations**: Applications to nonlinear decomposition, Poisson density, heavy-tailed density, volatility, and binary density. The book includes numerous illustrations, exercises, and references to support the theoretical and practical aspects of state space methods in time series analysis.