System-Theoretic Methods in Economic Modelling I

184 Pages·1989·17.214 MB·English

by  S. Mittnik

International Series in
MODERN APPLIED MATHEMATICS AND COMPUTER SCIENCE
General Editor Ε. Y. Rodin
Volume 19
Pergamon Titles of Related Interest
Avula et al Mathematical Modelling in Science and Technology
Bellman et al Mathematical Aspects of Scheduling and Applications
Cercone Computational Linguistics
Cooper and Cooper Introduction to Dynamic Programming
Giarini The Emerging Service Economy
Hargittai Symmetry 2: Unifying Human Understanding
Martos et al Dynamic Modelling & Control of National Economies
Paenson Systematic Glossary of the Terminology of Statistical Methods
Pugachev et al Probability Theory and Mathematical Statistics for Engineers
Roos Economics and Artificial Intelligence
Saaty and Alexander Thinking with Models
Schultz and Lee Computational Ocean Acoustics
Suri Resource Management Concepts for Large Systems
Witten Hyberbolic Partial Differential Equations, Volumes I, II, III and IV
Witten Diseases and Epidemics
Witten Mathematical Models in Medicine, Volume 2
Yavin and Pachter Pursuit-Evasion Differential Games
Related Journals*
Computers & Mathematics with Applications
Journal of Applied Mathematics and Mechanics
Journal of Physics and Chemistry of Solids
Journal of Quantitative Spectroscopy and Radiative Transfer
Mathematical and Computer Modelling
Nonlinear Analysis
Progress in Crystal Growth and Characterization
Topology
*Full details of all Pergamon publications/free specimen copy of any Pergamon journal available on
request from your nearest Pergamon office.
SYSTEM-THEORETIC METHODS IN
ECONOMIC MODELLING I
Guest Editor
S. MITTNIK
Department of Economics, State University of New York at Stony Brook,
Stony Brook, NY 11794-4384, U.S.A.
General Editor
Ε. Y. RODIN
Department of Systems Science and Mathematics, Washington University,
St Louis, MO 63130, U.S.A.
PERGAMON PRESS
OXFORD NEW YORK · BEIJING · FRANKFURT
SÂO PAULO · SYDNEY TOKYO · TORONTO
U.K. Pergamon Press pic, Headington Hill Hall,
Oxford OX3 OBW, England
U.S.A. Pergamon Press Inc., Maxwell House, Fairview Park,
Elmsford, New York 10523, U.S.A.
PEOPLE'S REPUBLIC Pergamon Press, Room 4037, Qianmen Hotel, Beijing,
OF CHINA People's Republic of China
FEDERAL REPUBLIC Pergamon Press GmbH, Hammerweg 6,
OF GERMANY D-6242 Kronberg, Federal Republic of Germany
BRAZIL Pergamon Editora Ltda, Rua Eça de Queiros, 346,
CEP 04011, Paraiso, Sâo Paulo, Brazil
AUSTRALIA Pergamon Press Australia Pty Ltd., P.O. Box 544,
Potts Point, N.S.W. 2011, Australia
JAPAN Pergamon Press, 5th Floor, Matsuoka Central Building,
1-7-1 Nishishinjuku, Shinjuku-ku, Tokyo 160, Japan
CANADA Pergamon Press Canada Ltd., Suite No. 271,
253 College Street, Toronto, Ontario, Canada M5T 1R5
Copyright © 1989 Pergamon Press pic
All Rights Reserved. No part of this publication may be reproduced, stored in a
retrieval system or transmitted in any form or by any means: electronic, electro-
static, magnetic tape, mechanical photocopying, recording or otherwise, without
permission in writing from the publisher.
ISBN 0 08 037228 7
Published as a special issue of the journal Computers & Mathematics with Applications, Volume 17,
Number 8/9, and supplied to subscribers as part of their normal subscription. Also available to non-
subscribers. In the interest of economics and rapid publication this edition has not been re-paginated.
Printed in Great Britain by BPCC Wheatons Ltd, Exeter
Computers Math. Applic. Vol. 17, No. 8/9, pp. vii-viii, 1989 0097-4943/89 $3.00 + 0.00
Printed in Great Britain Pergamon Press pic
PREFACE
S. MlTTNIK
Department of Economics, State University of New York at Stony Brook, Stony Brook, NY 11794-4384, U.S.A.
Although there had been attempts in the mid 1950s and in the latter half of the 1960s considerable
progress had been made in connection with economic growth problems, it has been only since the
early 19701s that system-theoretic concepts became increasingly applied to economic modelling
problems. The attractiveness of mathematical system theory arose from the fact that it offers a
unifying framework for modelling dynamic systems. In addition to this powerful conceptual frame-
work, it provides a wide range of tools useful in applied work. System-theoretic techniques enter
predominantly two stages of economic modelling efforts: the stage of model construction and the
stage of model application in accordance with the modelling objectives. It was, in particular, the
latter stage which led to joint research efforts between economists and engineers. In May 1972 the
first NBER Stochastic Control Conference was held at Princeton University. The engineering side
followed in July 1973 by organizing the first IFAC/IFORS International Conference on Dynamic
Modeling and Control of National Economies which was held in Warwick, England. Even then, as
the motto of the conferences and the presented papers indicate, econo2mists were primarily attracted
by the optimal control methods developed by "control scientists". The interest in these dynamic
optimization techniques appeared to recede somewhat with the adv3ent of Lucas's critique, arguing
that agents' decision rules are not invariant under interventions. Despite4 this criticism, dynamic
optimization represents now an important tool in economic modelling. The significant impact
of system theory on this area of research may explain the fact that economists frequently equate
system theory with dynamic optimization theory.
In the early 1970s, there was an increasing number of contr5ibutions emphasizing the potential
of system-theoretic techniques in the model construction stage and, eventually, triggering a second
wave of influx after the "control theory wave". In particular, the advantages of Kalman filtering
methods in constructing empirical economic models were soon recognized. Although the Kalman
filter has become almost a standard tool in econometrics, and algorithms can be found in many
graduate econometrics textbooks, new areas of applications are still being discovered. Only very
recently, the econometric model building toolbox has been enriched by another system-theoretic
concept, na6m ely stochastic realization theory, which integrates model selection and parameter
estimation.
The objective of this and subsequent volumes of special issues on System-theoretic Methods in
Economic Modelling is to initiate and/or intensify dialogs between researchers and practitioners
within and across the disciplines involved. In view of the growing spectrum of promising system-
theoretic concepts and techniques as well as specific economic applications, the following statement
made by K. D. 7W all and J. H. West in 1974 (p. 873) is—in a slightly altered form—as valid today
as it was then:
"Most [systems] engineers and theorists are not sufficiently cognizant of the special
economic issues involved. Conversely, most economists or econometricians do not
fully understand the generality and unified approach to dynamic systems afforded
by [system] theory. Both areas have considerable amount of mutual interest and
much can be learned from the other."
This first volume brings together papers exhibiting a wide range of system-theoretic techniques
and applications to economic problems. The papers have been divided into two groups, following
roughly—but not necessarily—the above classification into the construction and application stages
of economic modelling. The first group focuses on the identification of dynamic and static systems,
while the papers in the second group address dynamic optimization problems.
I would like to thank the contributors for their support and timely responses. I am indebted to
Ervin Y. Rodin, Editor-in-Chief of Computers & Mathematics with Applications, for his encourage-
vii
viii S. MITTNIK
ment and advice, to Patricia A. Busch, Editorial Assistant, for her professional handling of this
project, and Marie Traylor for her patient secretarial assistance.
Notes
'For a brief review of the earlier attempts and their comeback almost twenty years later, see M. Aoki, Optimal Control
and System Theory in Dynamic Economic Analysis, pp. 4-6, Elsevier, New York (1976). Control-theoretic approaches
to economic growth problems, as well as further references on this topic, can be found in A. R. Dobell, Some
characteristic features of optimal control problems in economic theory, IEEE Trans, autom. Control AC14, 39-48
(1969). A review of more recent applications in econometric modelling is given in E. J. Moore, On system-theoretic
2
methods and econometric modelling, Int. econ. Rev. 26, 87-110 (1985).
3
Selected papers presented at this conference appeared in the October issue of Ann. econ. soc. Measur. 1, No. 4.
R. E. Lucas, Econometric policy evaluation: a critique, in The Phillips Curve and Labor Markets (Eds K. Brunner and
A. H. Meltzer), pp. 19-46, North-Holland, Amsterdam (1976). For related discussions see also F. E. Kydland and
E. C. Prescott, Rules rather than discretion: the inconsistency of optimal plans, J. polit. Econ. 85, 473^491 (1977),
and G. A. Calvo, On the time consistency of optimal policy in a monetary economy, Econometrica 46, 1411-1428
4
(1978).
5
See, for example, T. J. Sargent, Dynamic Macroeconomic Theory, Harvard University Press, Cambridge (1987).
See, for example, R. K. Mehra, Identification in control and econometrics; similarities and differences, Ann. econ. soc.
Measur. 3, 21-47 (1974).
^he following monographs are largely devoted to this topic: M. Aoki, Notes on Economic Time Series Analysis: System
Theoretic Perspectives, Springer, Berlin (1983); P. W. Otter, Dynamic Feature Space Modelling, Filtering and
Self-Tuning Control of Stochastic Systems, Springer, Berlin (1985); and M. Aoki, State Space Modeling of Time Series,
7
Springer, Berlin (1987).
K. D. Wall and J. H. West, Macroeconomic modeling for control, IEEE Trans, autom. Control AC19, 862-873 (1974).
Computers Math. Applic. Vol. 17, No. 8/9, pp. 1165-1176, 1989 0097-4943/89 $3.00 + 0.00
Printed in Great Britain. All rights reserved Copyright © 1989 Pergamon Press pic
A TWO-STEP STATE SPACE TIME SERIES
MODELING METHODf
M. AOKI
4731 Boelter Hall, University of California, Los Angeles, CA 90024-1600, U.S.A.
Abstract—A state space method for building time series models without detrending each component
of data vectors is presented. The method uses the recent algorithm based on the singular value
decomposition of the Hankel matrix and a two step sequential procedure suggested by the notion of
dynamic aggregation.
1. INTRODUCTION
Time series are usually decomposed into trends and the remainders (consisting of cyclical, seasonal,
and residual components, or simply cyclical and residual) because trends convey information
distinct from that to be culled from cyclical components. In macroeconomic time series, for
example, policy-makers may be primarily interested in their trend behavior, while those concerned
with business cycles are more interested in cyclical components, such as phases of business cycles
than trends. Another reason for singling out trends is that they may have simpler dynamic structure
than cyclical components in the sense that a small number of "common" factors are responsible
for a larger number of trend components, as in macroeconomic time series where there are reasons
to suspect or expect from economic theory for a set of macroeconomic variables to behave generally
in the same way, at least in longer-run time horizon, i.e., aside from short-run (individual)
variations. Here again one needs to separate "trend" components, and seek a set of a small number
of common factors thay may "cause" a larger number of macroeconomic variables to change, and
to extract "common" trend components from these macroeconomic time series. Granger's notion
of co-integration [1] is one way to formalize this idea of common factors.
Time series are often transformed to render them weakly stationary for a technical reason that
currently available modeling methods can more efficiently handle weakly stationary time series
than nonstationary ones. One transformation takes differences of the logarithms of data series.
A serious drawback of this common practice is that longer-run information of time series is
lost in the process of rendering them weakly stationary. Recent interests in modeling economic time
series without prior detrending is sparked by the seminal works of Beveridge and Nelson [2]
and Nelson and Plosser [3] who posited a model of time series with separate and explicit equations
for random trends. Harvey [4] also used models with explicit random trend dynamics. Random
trends are provided for by specifying that the first difference is weakly stationary. In other words,
random trend dynamics are hypothesized to have a unit root. By now a number of studies is
available which examines the question of unit roots in the economic time series, such as the U.S.
real GNP series [e.g. 5,6]. In multivariate time series, however, this approach posits the same
number of unit roots as the number of component series with "trends", which often results in too
many unit roots. A transformation which extracts a smaller number of common trends than this
approach is needed.
This paper proposes an alternative modeling procedure for separating out trend dynamics from
those for the cyclical and residual components without constraining the components of time series
to have unit roots from the beginning, and thus allow for easy determination of the presence (and
the number) of common factors. The idea is based on the notion of dynamic aggregation which
was originally suggested as a way for building simplified dynamic models for control purposes [7].
We build time series models in two sequential steps. In the first step state space models for trends
fResearch supported in part by a grant from the National Science Foundation.
1165
1166 Μ. Αοκι
are built followed by a second step in which state space models for the residuals of the first step
are constructed. In each of the two steps, state space modeling algorithms recently proposed by
Aoki [8,9] is employed. Aoki [10] has recently pointed out that Granger's co-integration and the
idea of error correction mechanism, originally proposed by Sargan [11] are derivable from the
common notion of aggregation of dynamic models.
The procedure will not require prior detrending as in Stock and Watson [12] and will determine
co-integrating factors, when some of the components of the vector-valued time series contain
common trends. This paper also discusses why this two-step procedure may be superior to a single
state modeling strategy, especially when trend components contain random walk components.
Section 2 is a brief description of the dynamic aggregation procedure originally employed in Aoki
[7]. Section 3 describes how to construct state space models in two stages following the suggested
scheme in Section 2. Section 4 clarifies the differences in the extraction of trends in Beveridge-
Nelson and the state space models. Some examples are presented in Section 5 and the concluding
Section 6 elaborates on the reasons why one might wish to employ the suggested two stage
procedure.
2. DYNAMIC AGGREGATION
The dynamic aggregation procedure in Aoki [7] starts by classifying dynamic modes of a model
x = Ax,+ v,, i.e. eigenvalues of the transition matrix A into two classes and transforming the
/ 1+
coordinates to put A into a block triangular representation. Although many dichotomized
classificatons are possible, here put all eigenvalues with magnitude greater than some critical
number into class C and the rest in class C. Thus C contains unit roots and those roots of the
x 2 x
characteristic polynomial near the unit circle in the complex plane. Suppose that A is η χ η and
that there are k eigenvalues in C (counting multiplicities). Let η χ k matrix Τ form a basis for the
x
right invariant subspace of linearly independent columns of A associated with the eigenvalues in
C,. If At, = ^λι, i — 1,..., k, then Τ = [t,,..., t ] and A = diag(A ..., k), for example. Let S be
k 1? k
an η χ (η —k) matrix of linearly independent columns forming a left invariant subspace of A
associated with class C . They satisfy
2
AT = TA,
SA = NS', (1)
and we normalize Τ and S by TT = I* and S'S = I„_ . These two matrices are orthogonal:
k
ST = 0,
, , , ,
because SAT = STA and S AT = NS T implies 0 = S'TA-NST and A and Ν have no
eigenvalues in common [13]. Express the state vector x, using the columns of Ρ and S as basis
vectors, i.e. let
x, = Sz, + Ττ,
in the model x, , = Ax, + v,, i.e. Sz,+ + Ττ, = ΤΛτ, + ASz, + v,, where the first equation in (1)
+ x + x
is used. The vector τ, is the set of new coordinates related to slower dynamic modes and z, refers
to the coordinates representing faster dynamic modes. Multiply this relation from the left by T'
to see that
,
t,+, = AT, + Τ ASz, + Tv,. (2)
Multiplication from the left by S' yields
z, = Nz, + S'v,. (3)
+ 1
The matrix Ν has eigenvalues of class C only, i.e. they are all asymptotically stable eigenvalues
2
by choice. Jointly written, the state space model has the recursive structure
Two-step state space time series modeling method 1167
Note that the term T'AS explicitly shows how the state vector for short-run dynamics affect longer
run dynamics. The model specification is completed by specifying that the data vector y, is related
to x, by y, = Cx, + e,. The data vector y, is related to the new vectors τ, and z, by
y, = CTt, + CSz, + e,. (5)
Equation (5) shows how the data is decomposed into slower modes, i.e. trend (-like) movements
CTc, and the rest CSz,+ e,, i.e. cyclical component plus innovations on observations.
3. MODELING PROCEDUREf
The previous section suggests a procedure to construct a model with a block triangular transition
matrix. Since the dynamic matrix of time series in unknown, we do not know how many eigenvalues
are in C. The data determines the dimension of the vector τ,. In the algorithm of Aoki [9],
x
the ratio of singular values of certain Hankel matrix is one important indication of the size of n.
First, trend models is estimatedî
|τ,+ ι = AT, + GU,
where u, stands for CSz, + e, in (5), followed by a model for short-run behavior
fz,+ 1= Fz,+ Je, ] 1
\ u, = Hz, + e,.
Note that u, is weakly stationary since the dynamics for z, are stable by construction. In (6) u, is
usually (highly) serially correlated but e, in (7) are not serially correlated.
When τ, is chosen to be scalar, (6) is
K+i = At, + g'u,
{ y, = dx, + u,
where g and d are ρ -dimensional column vectors where /?=dim y,. The connection with
Granger's notion of co-integration is now clerly seen from (8). Any vector ν orthogonal to d will
nullify the dynamic mode with eigenvalue λ since v'y, = v'u, is governed by the dynamics (7) and
has no eigenvalue in C i.e. v'y, is weakly stationary, even when some components of y, have unit
x
roots.
If the dimension 2 is tried, then the matrix Λ in (6) is 2 χ 2. When it has two real eigenvalues
one can decide then whether the trend dynamics has one dominant eigenvalue or two. When the
data y, contains a single common trend variable, this fact becomes apparent when the eigen-
values of the matrix Λ is calculated. The matrix Λ can be put into Schur form to exhibit this fact
explicitly as
A=U0U, UU = I
2
where
fFor completeness the model matrix estimation method in Aoki [8,9] is outlined in the Appendix.
JA seemingly more general model χ, + =x Ax, + u,, y, = Cx, + w,, where u, and w, are serially uncorrelated, can be put in
the assumed form where
C O V= A ( G / I )
G) (I) '
and Δ = cov e by spectral decomposition. See Aoki [9, p. 67].
1168 Μ. AOKI
If λ is judged to be significantly larger than A, then there is one trend factor. If λ and λ are
χ 2 χ 2
sufficiently close to each other, then there are two common factors. Define /i, = U'T,. Model (6)
is transformed into
y, = DU/i, + u,.
The first column vector of DU now corresponds to the vector d of (8) since it shows how the first
component of /i, is distributed among the components of the vector y. Usually dim y, is larger than
dim /ι,. Thus DU is a disaggregation matrix which distributes the effects of μ among y,. To allow
ί
for the possibility of a real eigenvalue and a pair of complex eigenvalues in the trend dynamics
dim τ, = 3 should also be considered. Too high an initial choice of the dimension of τ, causes no
harm since the Schur decomposition tells us if the eigenvalues are all equally large and the column
vectors of DU tells us if the components of the vector τ, are equally important in y,. If not, some
of the modes in (6) can be easily lumped together with (7). We return to this point in Section 5
where an example is discussed.
Equations (6) and (7) imply that the transfer function from e, to y, can be factored as
l _1
y, = [I + D(?I - Ar'GJÛ = [I + D(?I - \yG][l + H(?I - F) J]ê (9)
x x
where q~ is the lag operator q~y = y _ i. The first factor of this factorization corresponds to the
t f
slower dynamics, i.e. lower frequency factor, the second to the factor dynamics, i.e. fast frequency
factor. This modeling method in effects factors the transfer function into a low frequency factor
and a high frequency one as shown.
Since the residuals in (6) are usually correlated, unlike the modeling situations for weakly
stationary, u, is not an innovation vector. To show that the Riccati equation used in the algorithm
of Aoki is well-defined, consider the model with a scalar τ, as an example.
Rewrite (8) as
T, =XT,-hg>,
+ 1
by substituting u, out from the first equation, where X = λ — g'd. If
00
converges for all t, where R = E(y y') is the covariance matrix of the data vector, then
t +kJ t+kt
the covariance matrix of τ, is well-defined and Aoki's algorithm [9] can be applied to estimate
λ and g'.|
Similarly, rewrite (7) as
z, = (F- JH)z, + Ju,.
+ 1
Then, the covariance matrix Π = cov z, is well-defined, and the subscript t is dropped from Π
because ζ process is weakly stationary if all the eigenvalues of F — JH lie strictly inside the unit
disk. Let F = F — JH. The matrix F is the dynamic matrix in the Kalman filter for (7). It is known
that if (7) is observable F is asymptotically stable. To see the importance of this condition, consider
solving the Riccati equation for Π by an iterative procedure, where Π = FIIF' + JU J' and where
0
U = cov u = ΗΠΗ' + Δ, and where Δ = cov e,. Supposing that Π exists, denote Π* - Π by P^.
0
Then P* =FP^F' or
+1
vecP = (F®F)vecP , 1, 2,...
w +1 m
Therefore, the equation converges as m is increased if and only if all the eigenvalues of F lie
inside the unit disk as claimed.
2
fFor example, if y, is a pure random walk, then Rt+ kJ= Rw + ka. The sum Σ*&£* converges if the magnitude of X is less
than one.
Two-step state space time series modeling method 1169
The above description shows that the proposed two-step procedure will construct state space
models even when the data vector contain unit root components, i.e. even when λ in (7) is one,
provided λ is less than one in magnitude. A sufficient condition is that the unit root is an observable
mode of the model.!
4. DECOMPOSITION INTO TRENDS AND CYCLICAL COMPONENTS
Beveridge and Nelson [2] posit that a univariate y, is governed by
l Υ, = Υ,-ι + Α(<Γ>, (10)
where q ~e = e, _ and
t x ] l 2
A(q~)= 1 + aq~ +aq~ + ···
{ 2
such that
00
0
The coefficients in the Wold decompositioln representation (10) are the impulse (dynamic multiplier)
responses. For example, a in Ay, = A(q~)e tells us how much the shock three period earlier, e,_ ,
3 t 3
still affects Ay,. This class of models has been proposed by Beveridge and Nelson [2] and used by
Nelson and Plosser [3], Cochrane [14] and others. This section demonstrates the difference in the
decomposition of time series by this and state space methods for a univariate {>>,}·
By rewriting (10) as
l
y-y_=A(\)e + [A(q-)-A(l)]e
t t x t n
where
oo
A(l) = X«,<oo
ο
is assumed, one can integrate this equation. Decompose y, into y„ -I- y , where
2
Ay„ = A(l)e, (11)
and
Δν* = [Μ9-ι)-ΜΦ„ (12)
where Ay = y — y,-,_ / = 1, 2. Equation (11) immediately shows that y„ is a random walk
it it l5
because
yif-yu-i =0 = A(l)e„
or
A}( 1
yn = , - , = A(l)(e, + e,_, + e,_ + · · ·)· (13)
1 - q % 2
It is the integral of past disturbances. This term represents the random trend in Beveridge-Nelson
decomposition.
Solve y, from (12) as
2
„ag=Mi> .
% (14)
When the spectral density function of Ay, is rational, Ay, can be regarded as an output of a
finite-dimensional state space model driven by a white noise sequence. The impulse responses
{a,} are then characterized by finite parameter combinations. When α, = 1ιΤ'"^, ι1, a = 1,
0
we use
l - 1
A(q~) = I + h'(ql — F) g.
|For more precise analysis see [17].