State Space Modeling of Time Series

In this book, the author adopts a state space approach to time series modeling to provide a new, computer-oriented method for building models for vector-valued time series. This second edition has been completely reorganized and rewritten. Background material leading up to the two types of estimators of the state space models is collected and presented coherently in four consecutive chapters. New, fuller descriptions are given of state space models for autoregressive models commonly used in the econometric and statistical literature. Backward innovation models are newly introduced in this edition in addition to the forward innovation models, and both are used to construct instrumental variable estimators for the model matrices. Further new items in this edition include statistical properties of the two types of estimators, more details on multiplier analysis and identification of structural models using estimated models, incorporation of exogenous signals and choice of model size. A whole new chapter is devoted to modeling of integrated, nearly integrated and co-integrated time series.

1. Introduction.- 2. The Notion of State.- 3. Data Generating Processes.- 3.1 Statistical Data Descriptions.- 3.2 Spectral Factorization.- 3.3 Decomposition of Time Series.- Dynamic Modes.- Two Aggregation Schemes.- Unit Roots.- Regime Shifts and Structural Changes.- 3.4 Minimum-Phase Transfer Function Representation.- 4. State Space and ARMA Models.- 4.1 State Space Models.- 4.2 Conversion to State Space Representation.- Observability Canonical Form.- Vector Models.- Gilbert’s Method.- 4.3 Conversion of State Space Models into ARMA Models.- 5. Properties of State Space Models.- 5.1 Observability.- Observability and Consistency of Least Squares Estimates.- Lyapunov Equations.- 5.2 Orthogonal Projections.- Example: Kaiman Filters.- 6. Hankel Matrix and Singular Value Decomposition.- 6.1 The Hankel Matrix.- 6.2 Singular Value Decomposition.- Sensitivity of Singular Values.- Rank and Singular Values.- Approximate Regression Analysis.- 6.3 Balanced Realization of State Space Model.- Effects of Scaling.- Parametrization.- 6.4 Examples with Exact Covariance Matrices.- 6.5 Hankel Norm of a Transfer Function.- 6.6 Singular Value Decomposition in the z-Domain.- 7. Innovation Models, Riccati Equations, and Multiplier Analysis.- 7.1 Innovation Models.- Forward Innovation Models.- Backward Innovation Models.- 7.2 Solving Riccati Equations.- Closed Form Solutions for VAR Models.- Iterative Solution Algorithm.- A Non-Iterative Solution Algorithm.- 7.3 Likelihood Functions.- Identification.- 7.4 Dynamic Multiplier Analysis and Structural Model Identification.- Confidence Interval of Impulse Response Analysis.- Variance Decomposition.- Identification Exercises.- 7.5 Out-of-Sample Forecasts.- 8. State Vectors and Optimality Measures.- 8.1 Canonical Variates.- Mutual Information.- 8.2 Prediction Error.- 8.3 Singular Values and Canonical Correlation Coefficients.- 9. Estimation of System Matrices.- 9.1 Two Classes of Estimators of System Matrices.- Stochastic Realization Estimator.- The Instrumental Variables Estimator.- 9.2 Properties of Balanced Models.- Nesting of System Matrix Estimates and ?.- Stability.- 9.3 Examples with Exact Covariance Matrices.- Models for VAR Processes.- Choices of K.- Models for MA Processes.- Models for Vector-Valued ARMA Processes.- 9.4 Numerical Examples.- 9.5 Monte Carlo Experiments.- AR(1) Models.- Experimental Results.- AR(2) Models.- 9.6 Model Selection.- Examples.- 9.7 Incorporating Exogenous Variables.- Regression Model.- Dynamic Model.- 10. Approximate Models and Error Analysis.- 10.1 Structural Sensitivity.- 10.2 Error Norms.- 10.3 Asymptotic Error Covariance Matrices of Estimators.- Variances of$$\hat{\Delta}$$ and $$\hat{\rm Z}$$.- Errors of System Matrix Estimates.- 10.4 Other Statistical Aspects.- Test for Residuals.- Variability of Sample Correlation Coefficients.- Variances of Sample Covariances.- 11. Integrated Time Series.- 11.1 The Beveridge and Nelson Decomposition.- 11.2 State Space Decomposition.- 11.3 Contents of Random Walk Components.- 11.4 Cointegration, Error Correction, and Dynamic Aggregation.- 11.5 Two-Step Modeling Procedure.- First Step.- Second Step.- 11.6 Dynamic Structure of Seasonal Components.- 11.7 Large Sample Properties.- Drift Term.- 11.8 Drifts or Linear Deterministic Trends?.- 11.9 Regime Shifts.- 11.10 Nearly Integrated Processes.- 12. Numerical Examples.- 12.1 West Germany.- 12.2 United Kingdom.- 12.3 The United States of America.- A Money Stock.- Money Stock and CPI.- US Consumer Price Index.- Real GNP, CPI and M2.- 12.4 The US and West German Real GNP Interaction.- 12.5 The US and West German Real GNP and Unemployment Rate.- 12.6 The US and Japan Real GNP Interaction.- 12.7 The USA, West Germany, and Japan Real GNP Interaction.- 12.8 Further Examples.- Appendices.- A.1 Geometry of Weakly Stationary Stochastic Sequences.- A.2 The z-Transform.- A.3 Discrete and Continuous Time System Correspondences.- A.4 Some Useful Relations for Matrix Quadratic Forms.- A.5 Computation of Sample Covariance Matrices.- A.6 Properties of Symplectic Matrices.- A.7 Common Factors in ARMA Models.- A.8 Singular Value Decomposition Theorem.- A.9 Hankel Matrices.- A. 10 Spectral Factorization.- A.11 Time Series from Intertemporal Optimization.- A. 12 Time Series from Rational Expectations Models.- A. 13 Data Sources.- References.