1 Introduction.- 2 The Notion of State.- 3 Time-invariant Linear Dynamics.- 3.1 Continuous time systems.- 3.2 Inverse systems.- 3.3 Discrete-time sequences.- 4 Time Series Representation.- 5 Equivalence of ARMA and State Space Models.- 5.1 AR models.- 5.2 MA models.- 5.3 ARMA models.- Examples.- 6 Decomposition of Data into Cyclical and Growth Components.- 6.1 Reference paths and variational dynamic models.- 6.2 Log-linear models as variational models.- 7 Prediction of Time Series.- 7.1 Prediction space.- 7.2 Equivalence.- 7.3 Cholesky decomposition and innovations.- 8 Spectrum and Covariances.- 8.1 Covariance and spectrum.- 8.2 Spectral factorization.- 8.3 Computational aspects.- Sample covariance Matrices.- Example.- 9 Estimation of System Matrices: Initial Phase.- 9.1 System matrices.- 9.2 Approximate model.- 9.3 Rank determination of Hankel matrices: singular value decomposition theorem.- 9.4 Internally balanced model.- example.- construction..- properties of internally balanced models.- principal component analysis.- 9.5 Inference about the model order.- 9.6 Choices of basis vectors.- 9.7 State space model.- example.- 9.8 ARMA (input-output) model.- 9.9 Canonical correlation.- 10 Innovation Processes.- 10.1 Orthogonal projection.- 10.2 Kaiman filters.- 10.3 Innovation model.- causal invertibility.- 10.4 Output statistics Kaiman filter.- 10.5 Spectral factorization.- 11 Time Series from Intertemporal Optimization.- 11.1 Example: dynamic resource allocation problem.- 11.2 Quadratic regulation problems.- discrete-time systems.- 11.3 Parametric analysis of optimal solutions.- choice of weighting matrices.- 12 Identification.- 12.1 Closed-loop systems.- 12.2 Identifiability of a closed-loop system.- 13 Time Series from Rational Expectations Models 140.- 13.1 Moving Average processes.- 13.2 Autoregressive processes.- 13.3 ARMA models.- 13.4 Examples.- example.- example.- example.- case of common information pattern.- case of differential information set.- 14 Numerical Examples.- Mathematical Appendices.- A.1 Solutions of difference equations.- A.2 Geometry of weakly stationary stochastic sequences.- A.3 Principal components.- A.4 Fourier transforms.- A.5 The z-transform.- A.6 Some useful relations for quadratic forms.- A.7 Calculation of the inverse, (z I-A)-1.- A.8 Sensitivity analysis of optimal solutions: scalar-valued case.- A.9 Common factor in ARMA models and controllability.- A.10 Non-controllability and singular probability distribution.- A.11 Spectral decomposition representation.- A.12 Singular value decomposition theorem.- A.13 Hankel matrices.- A.14 Dual relations.- A.15 Quadratic regulation problem: continuous time systems.- A.16 Maximum principle: discrete-time dynamics.- A.17 Policy reaction functions, stabilization policy and modes.- A.18 Dynamic policy multipliers.- References.
