State Space Modeling of Time Series 1 Introduction 2 The Notion of State 3 Representation of Time Series 3.1 Time Domain Representation 3.2 Frequency Domain Representation 4 State Space and ARMA Representation 4.1 State Space Models 4.2 Unit Roots 4.3 Conversion to State Space Representation 5 Properties of State Space Models 5.1 Observability 5.2 Covariance and Impulse Response Matrices 5.3 The Hankel Matrix 5.4 System Parameters and Innovation Models 5.5 Singular Value Decomposition 5.6 Balanced Realization of State Space Model 5.7 Hankel Norm of a Transfer Function 5.8 Singular Value Decomposition in the z Domain 6 Innovation Processes 6.1 Cholesky Decomposition and Innovations 6.2 Orthogonal Projections 7 Kalman Filters 7.1 Innovation Models 7.2 Kalman Filters 7.3 Causal Invertibility and Innovation 7.4 Likelihood Functions and Identification 7.5 A Non Iterative Algorithm for Riccati Equations 7.6 Forecasting Equations 8 State Vectors and Optimality Measures 8.1 State Vectors 8.2 Optimality Measures 9 Compution of System Matrices 9.1 System Matrices 9.2 Balanced Models for Scalar Time Series 9.3 Prediction Error Analysis 9.4 Non Stationary Models 9.5 Rescaling and Other Transformation of Variables 9.6 Dynamic Multipliers 9.7 Numerical Examples 10 Approximate Models and Error Analysis 10.1 Structural Sensitivity 10.2 Error Norms 10.3 Error Propagation 10.4 Some Statistical Aspects 11 Numerical Examples 11.1 Chemical Process Yields 11.2 IBM Stock Prices 11.3 Canadian GNP and Money Data 11.4 Germany 11.5 United Kingdom 11.6 Combined Models for the United Kingdom and Germany 11.7 Japan 11.8 Japan US Interactions 11.9 The United States of America 11.10 Comparison with VAR Models Appendices A.1 Differences Equations First Order Stable Equations First Order Unstable Equations Second Order Equations State Space Method A.2 Geometry of Weakly Stationary Stochastic Sequences A.3 The z Transform A. 4 Discrete and Continuous Time System Correspondences A.5 Calculation of the Inverse A. 6 Some Useful Relations for Matrix Quadratic Forms A.7 Spectral Decomposition Representation A. 8 Computation of Sample Covariance Matrices A.9 Vector Autoregressive Models A. 10 Properties of Symplectic Matrices A. 11 Common Factors in ARMA Models A. 12 Singular Value Decomposition Theorem A. 13 Hankel Matrices A. 14 Spectrum and Factorization A. 15 Intertemporal Optimization by Dynamic Programming A. 16 Solution of Scalar Riccati Equations A. 17 Time Series from Intertemporal Optimization A. 18 Time Series from Rational Expectations Models A. 19 Data Sources References. models predictive capability These are some of the questions that need to be answered in proposing any time series model construction method. This book addresses these questions in Part II. Briefly the covariance matrices between past data and future realizations of time series are used to build a matrix called the Hankel matrix. Information needed for constructing models is extracted from the Hankel matrix. For example its numerically determined rank will be the di­ mension of the state model. Thus the model dimension is determined by the data after balancing several sources of error for such model construction. The covariance matrix of the model forecasting error vector is determined by solving a certain matrix Riccati equation. This matrix is also the covariance matrix of the innovation process which drives the model in generating model forecasts. In these model construction steps a particular model representation here referred to as balanced is used extensively. This mode of model representation facilitates error analysis such as assessing the error of using a lower dimensional model than that indicated by the rank of the Hankel matrix. The well known Akaikes canonical correlation method for model construc­ tion is similar to the one used in this book. There are some important differ­ ences however. Akaike uses the normalized Hankel matrix to extract canonical vectors while the method used in this book does not normalize the Hankel ma­ trix.