Developments in the Nonlinear Analysis of Economic Series

Abstract
Various aspects of the analysis of nonlinearities are surveyed in this paper. A possibility of
distinguishing between a (low-dimensional) deterministic chaotic process and a white noise
stochastic process using estimates of the correlation dimension is discussed. It is concluded
that there is no evidence of chaos -as opposed to nonlinearity -in the economic data. The
modes of testing for nonlinearity are briefly surveyed, with particular attention paid to a new
test based on a neural network specification. It is found that aggregation can reduce
nonlinearity and a definition of long memory is proposed that suggests a nonlinear generalization
of cointegration.
I. Introduction
There seems to be a strong belief among economists that relationships
between economic variables are nonlinear, production functions being an
example. For a specific, given parametric relationship, such as the
Cobb-Douglas or CES production functions, standard econometric
techniques provide methods of estimation of the parameters and
asymptotic properties of these estimates. It is also possible to compare two
alternative models and to select the best, using likelihood ratio or
encompassing tests. However, as theory does not always provide a precise
specification, an important question might be whether or not the correct
nonlinear specification has been achieved or whether there still remains
some neglected nonlinearity in the estimated relationship. Most specification
tests are for linear models and consider what variables to include in
the model and what lags to use. Two questions to be considered here are:
(i) is the relationship nonlinear rather than linear?
and
(ii) if so, what is a useful nonlinear specification?
Of course, if one has a comprehensive theory giving a complete specification,
these questions do not arise. In recent years a third question has been
added:
(iii) if the process is nonlinear, is the relationship deterministic or
stochastic?