ARMA Memory Index Modeling of Economic Time Series

HERMANJ. BIERENS
Free University, Amsterdam
In this paper, it will be shown that if we condition a k-variate rational-valued
time series process on its entire past, it is possible to capture all relevant information
on the past of the process by a single random variable. This scalar random
variable can be formed as an autoregressive moving average of past
observations: Since economic data are usually reported in a finite number of
digits, this result applies to virtually all economic time series. Therefore, economic
time series regressions generally take the form of a nonlinear function
of an autoregressive moving average of past observations. This approach is
applied to model specification testing of nonlinear ARX models.

1, INTRODUCTION
In econometric time series modeling, the aim of the analyst is often to specify
and estimate the conditional expectation function, where the conditioning is
on the entire past of the process. The reason for this is the well-known fact
that this conditional expectation function is the best forecasting scheme in
terms of quadratic loss, i.e., its forecast error variance is minimal. This conditional
expectation function, however, is essentially a function of infinitely
many variables, which renders its specification and estimation cumbersome.
In the time series literature various parsimonious specifications have been
proposed, such as the well-known ARMA and ARIMA models, and the
index models advocated by Sargent and Sims [19] and Sims [21]. Also the
unrestricted vector autoregressions approach of Sims [20] belongs to the class
of parsimonious specifications of infinite-dimensional conditional expectation
functions. Of course, these parsimonious specifications are nothing
more than convenient approximations of the true conditional expectation
function.