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Lecture Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and W. Krelle
Econometrics
220
Masanao Aoki
Notes on Economic
Time Series Analysis:
System Theoretic Perspectives
Springer-Verlag
Berlin Heidelberg New York Tokyo 1983
Editorial Board
H.Albach A.v. Balakrishnan M. Beckmann (Managing Editor)
P.Dhrymes G. Fandel J. Green W Hildenbrand W Krelle (Managing Editor)
H.P.KOnzi G.L.Nemhauser K.Ritter R.Sato U.Schittko P.Schonfeld R.Selten
Managing Editors
Prof. Dr. M. Beckmann
Brown University
Providence, RI 02912, USA
Prof. Dr. W. Krelle
Institut fOr Gesellschafts- und Wirtschaftswissenschaften
der Universitat Bonn
Adenauerallee 24-42, 0-5300 Bonn, FRG
Author
Prof. Masanao Aoki
The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567, Japan
Current Address:
4531 Boelter Hall
University of California, Los Angeles
Los Angeles, CA 90024, USA
ISBN-13: 978-3-540-12696-6 e-ISBN-13: 978-3-642-45565-0
001: 10.1007/978-3-642-45565-0
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payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1983
To Chieko and
Thursday's children
Preface
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.
The notes naturally divide into three parts: Chapters 1 through 6 are pre
paratory to the main part of the notes. The notion of state, which is basic in
representing time series by Markovian models, is introduced early in Chapter 2.
Chapter 3 describes time-invariant dynamic systems, i.e., systems whose properties
remain invariant with respect to shift of the origin of the time axis, which we
mostly use to represent time series after suitable processings of data if necessary.
Here, locations of zeros of the numerator and denominator polynomials of transfer
functions are related to the notions of inverse systems, stable systems and minimum
phase systems, the last appearing prominently in our later chapters. Several ways
to represent time series are taken up in Chapters 4 and 5. Chapter 6 considers
preliminary processings of time series data to fit economic data series into a
common framework of mean zero, finite covariance weakly stationary stochastic
processes. Chapters 7 through 10 constitute the main part of these notes. There
I use singular value decomposition of certain matrices made up of covariances of
data vectors to produce Markovian models that generate time-indexed data vectors.
These models, after further refinement by maximum likelihood procedures if necessary,
can be used to predict future values of the data vectors. Connection of this method
with the canonical correlation method of Akaike is also explained. Chapter 11 on
time series from intertemporal optimization may be of particular interest to some
macroeconomists in view of recent research interests in explaining business cycles
using equilibrium macroeconomic models. Identification of closed-loop systems
and time series generated by dynamic models incorporating rational expectation are
the final two topics of the lecture notes. Chapter 14 is the third part of the
VI
notes and contain several numerical examples mostly drawn from Japanese economic
time series.
"To help bridge the gap or barrier faced by someone who is not versed in the
system theoretic language I have collected a number of brief but mostly self-contained
accounts of the facts I use in the main body as mathematical appendices.
In preparing these notes, the author received help from many friends and col
leagues. Sean Becketti of University of California, Los Angeles and Hiroshi Yoshikawa
of the Institute of Social and Economic Research, Osaka University commented on an
earlier draft. Leonard Silverman of Department of Electrical Engineering, University
of Southern California showed me his unpublished report. Jorma Rissanen of IBM,
San Jose told me of several important recent works on the time series analysis. I
owe Quirino Paris of University of California, Davis a reference. Dr. Hirotsugu
Akaike of the Institute of Statistical Mathematics, Tokyo made available to the
author the computer programs that implement his AIC criterion. Arthur Havenner of
University of California, Davis was instrumental in organizing a series of seminars
at which some of the material in preliminary form was tried out. He also provided
most useful comments on an earlier draft. Axel Leijohhufvud of University of
California, Los Angeles helped the author by arranging for his visits to the
Department of Economics at University of California, Los Angeles where a preliminary
version of the notes was tried out at a graduate level economics course.
These notes were typed expertly and expeditiously by Ms. Y. Ishida, T. Kawata,
K. Uto and G. Nystrom. Computations were carried out by Messrs. H. Ebara, S. Tateishi,
K. Nakagawa and Ms. C. Baden.
Osaka
M.A.
TABLE OF CONTENTS
1 Introduction 1
2 The Notion of State 5
3 Time-invariant Linear Dynamics 7
3.1 Continuous time systems 8
3.2 Inverse systems 11
3.3 Discrete-time sequences 12
4 Time Series Representation 15
5 Equivalence of ARMA and State Space Models 22
5.1 AR models 23
5.2 MA models 24
5.3 ARMA models 25
examples 29
6 Decomposition of Data into Cyclical and Growth Components 33
6.1 Reference paths and variational dynamic models 33
6.2 Log-linear models as variational models 35
7 Prediction of Time Series 38
7.1 Prediction space 38
7.2 Equivalence 44
7.3 Cho1esky decomposition and innovations 45
8 Spectrum and Covariances 48
8.1 Covariance and spectrum 48
8.2 Spectral factorization 51
8.3 Computational aspects 57
sample covariance matrices 57
example 58
9 Estimation of System Matrices: Initial Phase 60
9.1 System matrices 60
9.2 Approximate model 64
VIII
9.3 Rank determination of Hankel matrices: singular value
decomposition theorem 67
9.4 Internally balanced model 68
example 69
construction. 70
properties of internally balanced models 72
principal component analysis 74
9.5 Inference about the model order 75
9.6 Choices of basis vectors 77
9.7 State space model 80
example 81
9.8 ARMA (input-output) model 83
9.9 Canonical correlation 85
10 Innovation Processes 90
10.1 Orthogonal projection 90
10.2 Kalman filters 93
10.3 Innovation model 98
causal invertibility 101
10.4 Output statistics Kalman filter 102
10.5 Spectral factorization 103
11 Time Series from Intertemporal Optimization 106
11.1 Example: dynamic resource allocation problem 108
11.2 Quadratic regulation problems 116
discrete-time systems 117
11.3 Parametric analysis of optimal solutions 121
choice of weighting matrices 122
12 Identification 132
12.1 Closed-loop systems 134
12.2 Identifiability of a closed-loop system 137
13 Time Series from Rational Expectations Models 140
13.1 Moving Average processes 141
IX
13.2 Autoregressive processes 143
13.3 ARMA models 145
13.4 Examples 147
example 1 147
example 2 147
example 3 149
case of common information pattern 150
case of differential information set 152
14 Numerical Examples 154
Mathematical Appendices 178
A.l solutions of difference equations 178
A.2 Geometry of weakly stationary stochastic sequences 190
A.3 Principal components 194
A.4 Fourier transforms 197
A.5 The z-transform 202
A.6 Some useful relations for quadratic forms 208
A.7 Calculation of the inverse, (ZI_A)-l 211
A.8 Sensitivity analysis of optimal solutions:
scalar-valued case 212
A.9 Common factor in ARMA models and controllability 215
A.lO Non-controllability and singular probability distribution 217
A.ll Spectral decomposition representation 218
A.12 Singular value decomposition theorem 219
A.13 Hankel matrices 221
A.14 Dual relations 228
A.15 Quadratic regulation problem: continuous time systems 231
A.16 Maximum principle: discrete-time dynamics 235
A.17 policy reaction functions, stabilization policy and modes 237
A.18 Dynamic policy multipliers 240
References 242
Index 248
1 INTRODUCTION
Time series arise when data are collected over time, either continously or
at discrete time instants, and usually on several related variables. Together
they produce vector-valued, and time-indexed data which record economic activities.
Economic data are often collected at regular intervals such as daily, weekly,
monthly etc.
We analyze data jointly rather than singly, i.e., as vectors rather than as
scalars;
(1) to uncover dynamic or structural relations among them, because some series
may lead or lag other series, and there may be feedbacks between them,
and ultimately
(2) to forecast better, because modeling of a collection of time series as vector
valued use related information in data jointly.
Study of time series has history much older than modern system theory.
Probabilists, statisticians and econometricians all have contributed to advance
our understanding of time series over the past several decades. Many well estab
lished books record their contributions. One may wonder what new results system
theory can add to this well-established field and doubt if any new perspective
or insight can be gained by this relative newcomer to the field. History of
science shows us, however, that same problems can and have been examined with
advantage by different disciplines, partly because implications of alternative
assumptions are explored by researchers with different backgrounds or interests,
and partly because new techniques developed elsewhere are brought in to explore
areas left untouched by the discipline in which the problem originated. Although
a latecomer to the field of time series analysis, system theory has brought a
