State Space Modeling of 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.