DECOMPOSABLE MEASURES OF ECONOMIC INSTABILITY
| DOI | http://doi.org/10.1111/j.1468-0084.1980.mp42004005.x |
| Date | 01 November 1980 |
| Author | David A. Brodsky |
| Published date | 01 November 1980 |
DECOMPOSABLE MEASURES OF
ECONOMIC INSTABILITY
By DAVID A. BRODSKY*
For more than thirty years, measures of economic instability have been used to
analyze the instability in a wide range of variables, perhaps the most notable
example being that of export earnings. The economic quantities whose instabilities
have been the focus of such studies have in many cases been composite variables,
i.e. those which can be decomposed into the sum or product of more elementary
time series. In analyzing the instability of such composite variables, a number of
researchers have attempted to find a quantitative relationship among the
instabilities of the components and that of the composite itself. While it is well
known that the variance of a composite can be decomposed in this manner, there
does not appear to be general recognition of the fact that a quantitative
decomposition holds as well for a number of more sophisticated measures of
instability.
This paper will develop the notions of the trend-corrected covariance and the
trend-corrected correlation coefficient of two tune series. Using these concepts, it
will then be shown that there is a large class of measures which do indeed permit a
quantitative decomposition of the instability of a composite variable. Sufficient
conditions for a measure to satisfy a decomposition property will be derived; in so
doing, it will be seen that several frequently used measures of instability, which do
not permit such a decomposition, can easily be modified so that they will.
The organization of the paper is as follows. The first section will present an
analytical framework for classifying the large majority of commonly used measures
of instability. Using as an analogy the decomposition provided by the variance,
the second section will define two decomposition properties applicable to general
measures of instability and will derive sufficient conditions for a measure to satisfy
such properties. It will also be shown that a meaningful decomposition property
cannot exist for a class of measures which includes several of the most widely
used. The third section will illustrate the potential usefulness of such decompositions
with respect to three areas of enquiry in which instability analysis has played an
important role, and the paper will conclude with a discussion of the practical
implications of the findings.
I. ANALYTICAL FRAMEWORK
Nearly all measures of instability are based on a comparison of the actual values
of a time series with the values which would be 'predicted' (e.g. by a linear or
exponential trend). We will therefore define a prediction operator p to be a function
* The author is a member of the secretariat of the United Nations Conference on Trade and
Development, Geneva, Switzerland. He gratefully acknowledges the helpful comments of Ho Dac
Tuong. The views expressed in this paper are those of the author and not necessarily the United
Nations. 361
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