TIME SERIES ANALYSIS, COINTEGRATION,
AND APPLICATIONS
Nobel Lecture, December 8, 2003
by
Clive W.J. Granger
Department of Economics, University of California, San Diego, La Jolla, CA
92093-0508, USA.
The two prize winners in Economics this year would describe themselves as
“Econometricians,” so I thought that I should start by explaining that term.
One can begin with the ancient subject of Mathematics which is largely
concerned with the discovery of relationships between deterministic variables
using a rigorous argument. (A deterministic variable is one whose value is
known with certainty.) However, by the middle of the last millennium it became
clear that some objects were not deterministic, they had to be described
with the use of probabilities, so that Mathematics grew a substantial sub-field
known as “Statistics.” This later became involved with the analysis of data and
a number of methods have been developed for data having what may be
called “standard properties.”
However, in some areas of application, the data that they generated were
found to be not standard, and so special sub-sub-fields needed to be developed.
For example, Biology produced Biometrics, Psychology gave us Psychometrics,
and Economics produced Econometrics.
There are many types of economic data, but the type considered by Rob
Engle and myself is know as time series. Consider the measurement of unemployment
rates which is an important measure of the health of the economy.
Figures are gathered by a government agency and each month a new number
is announced. Next month there will be another value, and so forth. String
these value together in a simple graph and you get a “time series.”
Rather than show a diagram, I would rather you use internal visualization
(I think that you learn more that way). Suppose that you have a loosely strung
string of pearls which you throw down, gently, onto a hard table top with the
string of pearls roughly stretched out. You will have created a time series with
time represented by the distance down the table, the size of the variable as
the distance from the bottom edge of the table to a point, and the set of
pearls giving the points in the series. As the placement of one pearl will impact
where the next one lies, because they are linked together, this series will
appear to be rather smooth, and will not have big fluctuations in value for
one term to the next.
Time series can vary in many ways, some are gathered very often, others
less frequently. Values for many important financial variables are known not
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merely daily, but can be found within seconds, if they change, such as highly
traded stock prices or exchange rates. These are called “high frequency data”
and are the input for Rob Engle’s studies.
At the other extreme, some aspects of the overall, or “macro,” economy,
such as national income, consumption, and investment, may be available only
quarterly for many countries, and only annually for others. Population data is
also available only annually or less frequently. Many of these series are rather
smooth, moving with local trends or with long swings, but the swings are not
regular. It is this relative smoothness that makes them unsuitable for analysis
with standard statistical procedures, which assumes data to have a property
know as “stationarity.” Many series in economics, particularly in finance and
macroeconomics, do not have this property and can be called “integrated” or,
sometimes incorrectly, “non-stationary.” However, when expressed in terms of
changes or rates of returns, these derived series appear closer to
being stationary. The string of pearls would be “integrated” as it is a smooth
series.
Methods to analyze a single integrated series had been proposed previously
by Box and Jenkins (1970) and others, but the joint analysis of pairs, or
more, of such series was missing an important feature. It turns out that the
difference between a pair of integrated series can be stationary, and this property
is known as “cointegration.” More formally, two smooth series, properly
scaled, can move and turn, but slowly, in similar but not identical fashions,
but the distance between them can be stationary.
Suppose that we had two similar chains of pearls and we threw each on the
table separately, but for ease of visualization, they do not cross one another.
Each would represent smooth series but would follow different shapes and
have no relationship. The distances between the two sets of pearls would also
give a smooth series if you plotted it.
However, if the pearls were set in small but strong magnets, it is possible
that there would be an attraction between the two chains, and that they would
have similar, but not identical, smooth shapes. In that case, the distance between
the two sets of pearls would give a stationary series and this would give
an example of cointegration.
For cointegration, a pair of integrated, or smooth series, must have the
property that a linear combination of them is stationary. Most pairs of integrated
series will not have the property, so that cointegration should be considered
as a surprise when it occurs. In practice, many pairs of macroeconomic
series seem to have the property, as is suggested by economic theory.
Once we know that a pair of variables has the cointegration property it follows
that they have a number of other interesting and useful properties. For
example, they must both be cointegrated with the same hidden common factor.
Further, they can be considered to be generated by what is know as the
“error-correction model,” in which the change of one of the series is explained
in terms of the lag of the difference between the series, possibly after
scaling, and lags of the differences of each series. The other series will be represented
by a similar dynamic equation. Data generated by such a model is
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sure to be cointegrated. The error-correction model has been particularly important
in making the idea of cointegration practically useful. It was invented
by the well-known econometrician Dennis Sargan, who took some famous
equations from the theory of economic growth and made them stochastic.
The early development of the cointegration idea was helped greatly by colleagues
and friends in the Scandanavian countries, including Søren Johansen
and Katerina Juselius in Copenhagen who developed and applied sophisticated
testing procedures, Svend Hylleberg in Århus who extended the theory
to seasonal data, and Eilev Jansen and his colleagues at the Bank of Norway,
who successfully applied it to a large econometric model of Norway. To complete
the set, Timo Teräsvirta, who is from Finland but now lives in
Stockholm, helped develop models that were useful in nonlinear formulations
of cointegration. I am delighted that they are all here as my guests.
The modern macro economy is large, diffuse, and difficult to define, measure,
and control. Economists attempt to build models that will approximate
it, that will have similar major properties so that one can conduct simple experiments
on them, such as determining the impacts of alternative policies or
the long-run implications of some new institution. Economic theorists do this
using constraints suggested by the theory, whereas the econometrician builds
empirical models using what is hopefully relevant data and which captures
the main properties of the economy in the past. All models simply assume
that the model is correct and extrapolate from there, but hopefully with an
indication of uncertainty around future values.
Error-correction models have been a popular form of macro model in recent
years, and cointegration is a common element. Applications have been
considered using almost all major variables including investment, taxes, consumption,
employment, interest rates, government expenditure, and so
forth.
It is these types of equations that central banks, the Federal Reserve Bank,
and various model builders have found useful for policy simulations and other
considerations.
A potentially useful property of forecasts based on cointegration is that
when extended some way ahead, the forecasts of the two series will form a
constant ratio, as is expected by some asymptotic economic theory. It is this
asymptotic result that makes this class of models of some interest to economic
theorists who are concerned with “equilibrium.” Whether the form of equilibria
suggested by the models is related to that discussed by the theorists is
unclear.
These ideas and models are fairly easily extended to many variables. Once
the idea of cointegration (a name that was my own invention, incidentally)
became public it was quickly picked up and used by many other econometricians
and applied economists. There are now numerous citations and applications
based on it. Rob Engle and I quickly realized that the concept of cointegration
and its extensions could be used to explain and remove a variety of
difficulties that we had observed in our own research and that of others. It
seemed to be the missing piece in our approach to modeling groups of series.
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An example is a problem known as “spurious regressions.” It had been observed,
by Paul Newbold and myself in a small simulation published in 1974,
that if two independent integrated series were used in a regression, one chosen
as the “dependent variable” and the other the “explanatory variable,” the
standard regression computer package would very often appear to “find” a relationship
whereas in fact there was none. That is, standard statistical methods
would find a “spurious regression.” This observation lead to a great deal of
reevaluation of empirical work, particularly in macroeconomics, to see if apparent
relationships were correct or not. Many editors had to look again at
their list of accepted papers. Putting the analysis in the form of an error-correction
model resolves many of the difficulties found with spurious regression.
I am often asked how the idea of cointegration came about; was it the result
of logical deduction or a flash of inspiration? In fact, it was rather more
prosaic. A colleague, David Hendry, stated that the difference between a pair
of integrated series could be stationary. My response was that it could be
proved that he was wrong, but in attempting to do so, I showed that he was
correct, and generalized it to cointegration, and proved the consequences
such as the error-correction representation. I do not always agree with the
philosopher Karl Popper, but in his book “The Logic of Scientific Discovery,”
according to Hacohen (2000), page 244, Popper believed the “discovery was
not a matter of logic” but rather the application of methodology, which fits
the discovery of cointegration. This insight intrigues me partly because
Popper’s book appeared almost exactly at the time of my birth, in September
1934. At this same time Popper was debating Heisenberg on the relevance of
probability theory in physics. It happens to be the case that echoes of that debate
still persists, but relating to economics. My position is that it is clear that
we can best describe many of the movements of economic variables, and the
resulting data, using probabilistic concepts. I should also point out that 1934
was also the year the J.M. Keynes finished the first draft of “The General
Theory of Employment, Interest, and Money” although it was very many years
before I became aware of this book.
As an aside, I wrote this lecture whilst visiting the Department of
Economics of the University of Canterbury in New Zealand, where Karl
Popper also spent some years after World War II.
Before considering the usefulness of the new methods of analysis, I would
like to take a personal detour. This Prize has climaxed a year which started
with me being named a Distinguished Fellow of the American Economic
Association. Previously in my career, I have been Chair of two economics departments,
yet I have received very little formal training in economics. One
third of my first year as an undergraduate at the University of Nottingham was
in economics, with introductions to micro and in national accounts, and that
was it. Whatever other knowledge I have, it has come from living amongst
economists for about forty years, by osmosis, attending seminars, having discussions
with them, and general reading. My question is: does this say something
about me, or something about the field of economics? I think it is true
to say that I am not the first Nobel Prize winner in economics to have little
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formal training in economics. I wonder if economics has less basic core material
than is necessary for fields such as mathematics, physics, or chemistry,
say. Economics does have a multitude of different aspects, applications, and
viewpoints which has to each form their own basis, at least in practice.
Economic theory does seem to maintain common concepts and features but
these may be quite simplistic and are not necessarily realistic.
Possibly because it is not tied down by too many central concepts but certainly
because economics involves a myriad of topics, both theoretical and applied,
it is a hot-bed of new ideas, concepts, approaches, and models. The
availability of more powerful computing at low cost has just increased this activity
even more.
In my reading I came across a statement (unfortunately I have forgotten who
the author was) noting that “economics is a decision science, concerned with
decision makers, such as consumers, employers, investors, and policy makers,
in various forms of government, institutions, and corporations.” I fully accept
this viewpoint as it follows that the “purpose of economics is to help decision
makers make better decisions.” That statement is useful because it gives us a
foundation with which we can compare and evaluate specific pieces of economic
analysis. We can ask “how will a decision maker find this result useful?”
As I stated before, the main uses for the economic techniques that I helped
develop, such as cointegration, was to build statistical models linking major
economic variables that both fit the available data better and agree with the
preconceptions of the model constructors about what the construction
should look like. A major use of these models has been to provide short and
medium term forecasts for important macro variables, such as consumption,
income, investment, and unemployment, all of which are integrated series.
The derived growth rates are found to be somewhat forecastable. Much less
forecastable are inflation rates and returns from speculative markets, such as
stocks, bonds, and exchange rates.
There are a number of stages to the forecasting process; getting the central
forecast and then uncertainty bounds around it to give some idea of the risks
involved in using this forecast. Finally, previous forecasts have to be gathered
and evaluated. Hopefully any tendencies, trends, or swings in the errors can
be detected so that one can learn and produce better forecasts in the future.
The process of forecast evaluation, plus the use of combinations of forecasts
from different series, is an on-going research project at the University of
California, San Diego.
Forecasts do not just come from time series, but also from panels of data,
which can be thought of as a group of series of a similar nature measured
from different sources. An example would be monthly inflation rates from
each of the Scandinavian countries. Once one is clear what is the purpose of
the analysis, suitable techniques can be formulated.
I have recently been involved in such a project where the purpose is to
study the future of the Amazon rainforest in Brazil. This forest covers an area
larger than all the countries in the European Union put together, but it is being
cut down quite rapidly. I was one of five authors who produce a report
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(Anderson, et al., Cambridge University Press (2003)) which includes a model
that could forecast the decline of the forest under various policy scenarios.
The forest is not being cut down for its timber, but to get at the land that the
timber stands upon to produce food. Unfortunately, unlike European ex-forest
land, its useful life span is often rather short, often becoming “fallow”
within five years of being forested.
The advantage of being an academic econometrician is the possibility of
working on data from many areas. I have run pricing experiments in real super
markets, I have analyzed data from stock markets, commodity prices –
particularly gold and silver prices, interest rates, considered electricity demand
in small regions, the female labor force participation, river flooding,
and even sun spots. All data present their own unique problems and I continue
to find data analysis fascinating, particularly in economics.
There are plenty of disadvantages being a statistician working with economic
data without very much training in the area, but occasionally it allows
one to approach a problem from a different direction than that considered
by most economists. As a statistician I am intrigued by the pure magnitude of
some of the major economies, although economists pay little attention to this
aspect of the real world. For example, in the United States there are about
one hundred million households, so total consumption is the sum of all these
household’s consumptions. The sum over such a large number of families
should have very special statistical properties, given various well-known limit
theorems. If these properties are not observed this also has important implications.
I think that these, and many other topics concerning aggregation,
are worth further study.
An earlier concept that I was concerned with was that of causality. As a postdoctoral
student in Princeton in 1959–1960, working with Professors John
Tukey and Oskar Morgenstern, I was involved with studying something called
the “cross-spectrum,” which I will not attempt to explain. Essentially one has
a pair of inter-related time series and one would like to know if there are a
pair of simple relations, first from the variable X explaining Y and then from
the variable Y explaining X. I was having difficulty seeing how to approach
this question when I met Dennis Gabor who later won the Nobel Prize in
Physics in 1971. He told me to read a paper by the eminent mathematician
Norbert Wiener which contained a definition that I might want to consider. It
was essentially this definition, somewhat refined and rounded out, that I discussed,
together with proposed tests in the mid 1960’s. The statement about
causality has just two components:
1. The cause occurs before the effect; and
2. The cause contains information about the effect that that is unique, and is
in no other variable.
A consequence of these statements is that the causal variable can help forecast
the effect variable after other data has first been used. Unfortunately, many
users concentrated on this forecasting implication rather than on the original
definition.
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At that time, I had little idea that so many people had very fixed ideas
about causation, but they did agree that my definition was not “true causation”
in their eyes, it was only “Granger causation.” I would ask for a definition
of true causation, but no one would reply. However, my definition was
pragmatic and any applied researcher with two or more time series could apply
it, so I got plenty of citations. Of course, many ridiculous papers appeared.
When the idea of cointegration was developed, over a decade later, it became
clear immediately that if a pair of series was cointegrated then at least
one of them must cause the other. There seems to be no special reason why
there two quite different concepts should be related; it is just the way that the
mathematics turned out.
As a brief aside for those of you with more technical training, what I have
been telling you about so far has mostly been for concepts using linear models.
Everything can be generalized to the nonlinear situation and recently efforts
have been pushing into using similar concepts in conditional distributions,
which is a very general form. It appears that causality will play a basic role in
the generalization of the error-correction model, but that is still a work-inprogress.
I am not sure if the empirical studies on causation have proved to be so
useful, although the debate relating to money supply and prices was interesting.
The concept does help with the formulation of dynamic models in more
useful ways.
I started this lecture talking about econometrics. We econometricians love
numbers, so let me end with a few. The first two Nobel Prizes in Economics
were to econometricians, Ragnar Frisch and Jan Tinbergen, for which we are
very proud. In all there are now eight of us with the Prize, representing 15%
of the Economics winners. However, in the current millennium, we represent
about 44% of the winners, which I view as a healthy local trend.
Over my career and before today, I have met twenty-one Nobel Laureates:
one in Physics (Dennis Gabor, 1970), one in Peace (Phillip Noel Baker,
1959), one in Chemistry (Harold Urey, 1934), plus 18 Prize winners in
Economics. Without exception I have found them to be both very fine scholars
and also having excellent personalities, willing to help a younger, inexperienced
worker when seeking their advice or meeting them socially. I hope
that I am able to live up to their very high standard.
REFERENCES
Lykke Andersen, C.W.J. Granger, Eustaquio Reis, Diana Weinhold, and Sven Wunder: “The
Dynamics of Deforestation and Economic Growth in the Brazilian Amazon.” Cambridge
University Press, 2003.
M.H. Hacohen: “Karl Popper: The Formative Years, 1902–1945.” Cambridge University
Press, 2000.
K. Popper: “The Logic of Scientific Discovery.” Hutchinson: London, 1959.
G. Box and G. Jenkins: “Time Series Analysis, Forecasting, and Control.” Holden Day: San
Francisco, 1970.
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