Table Of ContentANALYSIS OF
ECONOMIC TIME SERIES
A Synthesis
Marc Nerlove
Department of Economics
Northwestern University
Evanston, Illinois
David M. Grether
Division of the Humanities and Social Sciences
California Institute of Technology
Pasadena, California
José L. Carvalho
Fundaçâo Getulio Vargas-EPGE
Rio de Janeiro-R.J., Brazil
ACADEMIC PRESS New York San Francisco London 1979
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Library of Congress Cataloging in Publication Data
Nerlove, Marc, Date
Analysis of economic time series.
(Economic theory and mathematical economics)
Bibliography: p.
1. Economics—Statistical methods. 2. Time-
series analysis. I. Grether, David M.Joint author.
II. Carvalho, Jose^L. Joint author. III. Title.
HB137.N47 330\01'51 78-26059
ISBN 0-12-515750-9
PRINTED IN THE UNITED STATES OF AMERICA
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For
Mary Ellen, Susan, and Gelda
PREFACE
In this book my coauthors and I attempt to integrate several topics in
time-series analysis: (1) the formulation and estimation of distributed-lag
models of dynamic economic behavior; (2) the application of the tech-
niques of spectral analysis in the study of the behavior of economic time
series; and (3) unobserved-components models for economic time series
and the closely related problem of seasonal adjustment. The research
underlying the book began a long time ago, and I would like to take the
opportunity afforded by this preface to set down some of the intellectual
antecedents of the book, as well as to acknowledge many debts and much
assistance.
My interest in the formulation and estimation of distributed-lag models
began with my work on agricultural supply analysis in 1954-1956; in 1970,
however, I began to realize the possibilities inherent in combining time-
series models with models of dynamic optimization. Many ideas in this
connection were developed in the course of a Workshop on Lags in
Economic Behavior held under the auspices of the Mathematical Social
Science Board at the University of Chicago in the summer of 1970. I am
grateful to the workshop members and to the coleaders of the workshop,
G. S. Maddala and David Grether, for their important contributions in this
connection. I must also acknowledge with gratitude the many discussions
I had with Dale W. Jorgenson in the course of preparing some of this
material for presentation as the Henry Schultz Memorial Lecture at the
Second World Congress of the Econometric Society.
Spectral analysis of economic time series is now a fashionable topic,
xiii
xiv Preface
but in the early 1960s spectral analysis was little known among econ-
omists. In the late 1950s Milton Friedman was a fellow of the Center
for Advanced Study in the Behavioral Sciences at Stanford in the same
year as John Tukey, a key figure in the development of modern time-
series techniques. Tukey introduced Friedman to spectral-analytic tech-
niques, with which the latter experimented in his studies of the demand
for money. In discussions in 1959, Friedman told me of this work and
convincingly argued that spectral analysis was a sensitive tool for uncov-
ering data anomalies, such as cycles due to the number of trading days in a
month. Some time later, Robert Aaron Gordon became chairman of the
President's Commission to Appraise Employment and Unemployment
Statistics and asked whether I would consult with the staff with respect to
the problem of seasonal adjustment. Particularly troublesome were the
issues of defining seasonality and of assessing the effects of seasonal
adjustment procedures in terms of that definition. Recalling my dis-
cussions with Friedman, it occurred to me that spectral techniques might
be a way of resolving both the issue of definition and the issue of effect.
Thus, much of the concern of this book with both seasonality and spectral
analysis may be traced to the influence of Friedman and Gordon. I should
also like to acknowledge many helpful discussions with, and much assis-
tance in data acquisition from, Margaret E. Martin, who was at that time
the Staff Director of the Commission. I must, as well, express my great
debt to Emanuel Parzen, who taught me the elements of spectral analysis
and patiently explained its somewhat arcane terminology, and to the
spectral-analysis study group that I organized at Stanford, which included
David Grether, Joseph B. Kadane, and G. S. Maddala.
Seasonal adjustment rests in part on the idea that a time series can be
decomposed into two or more separate but not directly observable com-
ponents. Moving-average or autoregressive time-series models originated
in the work of Slutsky and Yule in the early part of this century. The
current popularity of the work of G. E. P. Box and Gwilym M. Jenkins on
time series rests principally on their discovery of practical methods of
formulating and estimating mixed moving-average autoregressive models
for time series and the use of such results in forecasting. The idea of
combining two or more such mixed moving-average autoregressive mod-
els to obtain a new kind of "unobserved-components" model I owe to
interchanges in 1962-1963 with Henri Theil, who was then Director of the
Econometric Institute at the Netherlands School of Economics. Working
with his assistant S. Wage, I extended a simple forecasting model that
Theil had developed and derived an explicit form for the forecasts as a
weighted average of past errors. Theil's influence has been profound, and
the ideas formed during the year at the Institute recur again and again
throughout this book.
Preface xv
Many of my students and colleagues have greatly assisted in the re-
search which underlies this volume, but none more so than my coauthors,
David M. Grether and José Luiz Carvalho. Grether's Stanford University
Ph.D. dissertation (1968) forms the basis for Chapters I, VI, parts of XIII,
and Appendices A, D, and G. His paper on distributed lags and signal
extraction (1977) reports on related research also discussed in Chapter
XIII. Carvalho's University of Chicago Ph.D. dissertation (1972) forms
the basis for Chapter XIV and parts of Chapter XIII. His work also led us
into the research on multiple time-series analysis, described in Chapters
VII (Section 7) and XI, in which he actively participated during two visits
to Evanston in 1974 and 1975.
I have benefited greatly from the aid of, and interchange with, a number
of extraordinarily able research assistants. At Stanford University,
George Fishman, Bridger Mitchell, Karl Shell, David Couts, and Kenneth
F. Wallis contributed to early work on this book, and at Yale University, I
had the assistance of Elizabeth Boekelman. At the University of Chicago,
Uri Ben-Zion and Michael P. Ward assisted and helped to shape many of
my ideas about the analysis of economic time series. In the final stages of
the book, my coauthors and I were fortunate to have the able assistance of
Seiichi Kawasaki, George Sweeney, and Bruce Vavrichek at Northwest-
ern University.
Our greatest debt has been the continuing support of the National
Science Foundation through a series of grants to Stanford University
(G-16114 and GS-142), to Yale University (GS-1721), to the University of
Chicago (GS-2670 and GS-39872), and to Northwestern University (SOC
74-21194). In addition, a senior postdoctoral fellowship from the Founda-
tion in 1971-1972 enabled me to begin work on the book itself. I also wish
to acknowledge gratefully the John Simon Guggenheim Memorial Found-
ation and the Program for the International Exchange of Persons of the
U.S. Department of State, whose fellowships enabled me to spend the
academic year 1962-1963 at the Netherlands School of Economics, and
the Federal Reserve Board, which supported some of the work underlying
Chapters VI and IX. Needless to say, neither the National Science Foun-
dation, the U.S. Department of State, the Guggenheim Foundation, nor
the Federal Reserve Board are responsible for the ideas expressed in this
book.
Andrew Melczer and Quang Vuong have proofread the entire manu-
script and the galley and page proofs and have ably assisted in the
preparation of the index. Gloria Feigenbaum and Stina L. Hirsch have
typed our manuscript. Mrs. Hirsch has also assisted in the several revi-
sions that have been undertaken since 1975.
Many friends and colleagues have been good enough to read our work,
or parts of it, and to give us the benefits of their comments and criticism.
xvi Preface
An inevitable dimness of memory prevents me from thanking all those
who deserve our thanks, but I gratefully acknowledge the more recent
comments of C. W. J. Granger and Christopher Sims. Three patient
colleagues, T. Amemiya, E. J. Hannan, and K. F. Wallis, have com-
mented extensively on the entire manuscript. Time constraints and the
desire to have a published, if not perfect, book have prevented us from
taking account of their strenuous efforts to save us from error as fully or
as completely as they—and we—would have liked. We, not they, bear the
responsibility for the final text.
Marc Nerlove
Chapter I
A HISTORY OF THE IDEA
OF UNOBSERVED COMPONENTS
IN THE ANALYSIS
OF ECONOMIC TIME SERIES
1. Introduction
In the statistical literature dealing with the analysis of economic time series
it is common practice to classify the types of movements that characterize a
time series as trend, cyclical, seasonal, and irregular. The idea that a time series
may best be viewed as being composed of several unobserved components is
by no means universal, but it plays a fundamental role in certain areas of
application, e.g., the choice of methods for seasonal adjustment. In contem-
porary discussions of the unobserved-components model, the components
typically are taken as given, and relatively little attention is paid to their exact
interpretation. Indeed, it is sometimes said that such components cannot be
precisely defined.
In this chapter we review the literature on the trend, cyclical, seasonal,
irregular model to gain some understanding of the current uses of the model ;
in particular, we explore the origins of the model and how it came into use in
economics. The survey that follows is not intended to be a complete historical
review of the uses of the unobserved-components model; much of the literature
on the subject consists of discussions of operational techniques for estimating
the various components, and no attempt is made to cover that part of the
literature. The literature dealing with the existence of long cycles is also not
treated in any detail.1 The survey that follows is somewhat episodic, emphasiz-
ing the earliest literature and that of the past thirty years.
1 For a summary of the writings on long cycles, see Fellner (1956). See also Kondratieff (1935)
and Schumpeter (1939).
1
2 I. A History of the Idea of Unobserved Components
2. Background
The idea of unobserved components is basically a way of looking at data.
Throughout history man has observed various phenomena, noted regularities,
and formulated "laws" based upon observed regularities; hence, the exact
beginning of any particular attitude toward empirical observation is difficult
to date. A society subject to changes of seasons or in which a calendar is in
general use must recognize more or less formally the approximately periodic
occurrence of certain events. Deviations from regular periodic occurrence must
somehow be explained, although the explanation may be different in different
cultures; for example, when a solar eclipse predicted by a Babylonian astrono-
mer failed to occur, the failure might be explained as the result of the interven-
tion of some supernatural power. Later, small unexpected deviations in the
orbits of planets might be attributed to random shocks. Here, we concentrate
on only those developments that appear to have influenced economics and
the analysis of economic data.
The notion of unobserved components appears to have become common
in economics during the period 1825-1875. In tracing the history of this idea,
it is important to keep in mind the nature of scientific inquiry during this
period, in particular the role of statistics. In this connection, the remarks of
Stanley (1856) are relevant.
The axiom on which ... [statistics] is based may be stated thus: that the laws by which nature
is governed, and more especially those laws which operate on the moral and physical condi-
tion of the human race, are constant, and are, in all cases best discoverable—in some cases
only discoverable—by the investigation and comparison of phenomena extending over a
very large number of individual instances. In dealing with the individual human being every-
thing is uncertainty; in dealing with MAN in the aggregate, results may be calculated with
the precision and accuracy of a mathematical problem ....
This, then, is the first characteristic of statistics as a science: that it proceeds wholly by the
accumulation and comparison of registered facts;—that from these facts alone, properly
classified, it seeks to deduce general principles, and that it rejects all a priori reasoning, em-
ploying hypothesis, if at all, only in a tentative manner, and subject to future verification. 2
3. Origins
The notion that a series of observations is composed of separate unobserved
components was important in the calculation of planetary orbits by seventeenth
century astronomers. In the course of the eighteenth century more accurate
instruments for observing the movements of the planets became available; it
2 The quotation is from Lord Stanley's presidential address to Section F of the British Association
for the Advancement of Science. The speech was given the day after a resolution was passed "which
has enlarged the scope of our duties so as to include, in addition to statistics, properly so called,
Economic Science, in general" (1856, p. 305); the remarks quoted in the text were prepared before
the resolution was passed.
3. Origins 3
was discovered that Kepler's laws did not hold exactly. The discrepancies
between observations and predictions based on Kepler's laws were minor so
that the laws were approximately correct. Thus the notion that Kepler's laws
gave the mean position of a planet and not its exact position, i.e., that the
position actually observed was the mean plus some irregular fluctuations,
became common.3
Comparison of the contemporary observations with previous records later
revealed that slow persistent changes in planetary orbits were taking place.
Thus, a distinction was made between secular and periodic movements. Slow
changes were especially apparent in the orbits of the moon, Jupiter, and Saturn.
Explanation of these changes occupied some of the great mathematicians of
the eighteenth and early nineteenth centuries, including Laplace, Euler, and
Lagrange. In discussions of these problems "secular change" was frequently
mentioned. In 1787 Laplace showed that the secular change in the behavior
of Jupiter and Saturn was in fact periodic with a period in excess of 900 years.
Similarly, the secular movement in the behavior of the moon was also shown
to be of a very long-term periodicity. 4 Explanations of the orbits of Jupiter and
Saturn and of the moon were major triumphs of celestial mechanics and gave
further strength to Newton's theory of gravitation. 5
The use of distinctions among different types of changes was not confined
to discussion of planetary orbits. For example, Cannon and Jensen (1975)
state that "The secular decrease of the earth's rotation rate due to tidal inter-
action with the sun and moon was known to nineteenth century physicists,
who theorized about the existence of short-term fluctuations in the rotation
rate of the solid earth due to daily, seasonal, decadal, and irregular changes
in its moment of inertia as well as exchanges of angular momentum between
the solid earth and the oceans, atmosphere, and liquid core."
Early writers on economic subjects, when dealing with periodicities, occa-
sionally made explicit references to astronomy as the source of their ideas. For
example, in 1838 Cournot said, "as in astronomy, it is necessary to recognize
the secular variations which are independent of the periodic variations." 6
Jevons (1884, p. 4) remarks that his studies of short-term fluctuations use the
methods of astronomy and meteorology.
One of the earliest writers in economics to mention cycles and to suggest a
statistical correction for them was Sir William Petty (1899).
3 See Pannekoek (1961, p. 280).
4 See Whewell (1847, Vol. II, pp. 232-235).
5 Newton's laws are relatively easy to solve for orbits if only two bodies are involved. If three
bodies are involved, one obtains a set of second order differential equations that cannot be inte-
grated. The solution can only be obtained as an approximation, and the calculations are enormous.
To obtain the accuracy needed often required years of computation. See Pannekoek (1961, pp.
299-307) and Whewell (1847, pp. 101-107, 213-252).
6 Cournot (1927, p. 25); also cited by Mitchell (1928).
