model's 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 Akaike's 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.

1 Introduction.- 2 The Notion of State.- 3 Representation of Time Series.- 3.1 Time Domain Representation.- Preprocessing of Data.- Parameter Shifts and Structural Changes.- Time Invariant Linear Models.- 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.- Scalar Models.- Observability Canonical Form.- Conversion of State-Space Models into ARMA Models.- Vector Models.- 5 Properties of State Space Models.- 5.1 Observability.- Observability and Consistency of Least Squares Estimates.- Grammian.- 5.2 Covariance and Impulse Response Matrices.- 5.3 The Hankel Matrix.- Covariance and Hankel Matrices.- 5.4 System Parameters and Innovation Models.- Spectral Factorization.- 5.5 Singular Value Decomposition.- Sensitivity of Singular Values.- Rank and Singular Values.- Approximate Regression Analysis.- 5.6 Balanced Realization of State Space Model.- Properties of Balanced Models.- Stability.- Parametrization.- Analytical Example.- 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.- Innovation Models.- 7.4 Likelihood Functions and Identification.- 7.5 A Non-Iterative Algorithm for Riccati Equations.- 7.6 Forecasting Equations.- Out-of-Sample Forecasts.- ARMA Forecasting Models.- 8 State Vectors and Optimality Measures.- 8.1 State Vectors.- 8.2 Optimality Measures.- Mutual Information.- Prediction Error.- Hankel Norm.- 9 Compution of System Matrices.- 9.1 System Matrices.- 9.2 Balanced Models for Scalar Time Series.- Examples.- 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.- Bivariate Series.- Adding Another Time Series.- 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.