ARIMA Processes with ARIMA Parameters

Carlo Grillenzoni
lnstituto Universitario di Architettura di Venezia, Venice, Italy, and Department of Economics, University of
Modena, 41 100 Modena, Italy
This article introduces a general class of nonlinear and nonstationary time series models whose
basic scheme is an autoregressive integrated moving average (ARIMA). The main feature is
that the parameters are assumed to behave like a vector ARlMAx model in which the exogenous
(x) component is represented by the regressors of the observable process. For this class a
general algorithm of identification-estimation is outlined, based on the sampling information
alone. The initial estimation, in particular, consists of an iterative procedure of nonlinear regressions
on recursive parameter estimates generated with the extended Kalman filter. An empirical
example on real economic data illustrates the method and compares alternative criteria of
estimation.
KEY WORDS: Approximate maximum likelihood; Doubly stochastic processes; Extended
Kalman filter; Global sum of squares.

1. INTRODUCTION
The problem of the time variability by the coefficients
in dynamic regression models is particularly relevant
for social and economic applications. Here, the corresponding
systems have large dimensions and complex
internal relationships and are subject to the influence
of structural factors that evolve over time. In classical
econometrics (e.g., Raj and Ullah 1981; Wolff 1987),
several elements have been proposed to explain the
phenomenon, such as omitted variables, dynamic misspecification,
economic interventions, market instabilities,
and so forth. In recent literature these factors seem
of relative importance with respect to two groups of
endogenous and exogenous causes, (1) the approximation
of nonlinear dynamic relationships (e.g., of bilinear
type) with linear combinations of variables, and
(2) shocks due to the change of structural qualitative
factors, such as customs, technologies, and institutional
relationships. The mutual interaction of these elements
makes the quantitative models intrinsically nonstationary
in covariance so that a substantial change in the
way of conceiving the dynamic modeling is needed.