Notes on Economic Time Series Analysis: System Theoretic Perspectives

ISBN-13: 9783540126966 / Angielski / Miękka / 1983 / 249 str.

Masanao Aoki

Opis

In seminars and graduate level courses I have had several opportunities to discuss modeling and analysis of time series with economists and economic graduate students during the past several years. These experiences made me aware of a gap between what economic graduate students are taught about vector-valued time series and what is available in recent system literature. Wishing to fill or narrow the gap that I suspect is more widely spread than my personal experiences indicate, I have written these notes to augment and reor- ganize materials I have given in these courses and seminars. I have endeavored to present, in as much a self-contained way as practicable, a body of results and techniques in system theory that I judge to be relevant and useful to economists interested in using time series in their research. I have essentially acted as an intermediary and interpreter of system theoretic results and perspectives in time series by filtering out non-essential details, and presenting coherent accounts of what I deem to be important but not readily available, or accessible to economists. For this reason I have excluded from the notes many results on various estimation methods or their statistical properties because they are amply discussed in many standard texts on time series or on statistics.

Spis treści

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.