Continuous-time ARMA processes

Abstract

Continuous-time autoregressive (CAR) processes have been of interest to physicists and engineers for many years (see e.g., Fowler, 1936 ). Early papers dealing with the properties and statistical analysis of such processes, and of the more general continuous-time autoregressive moving average (CARMA) processes, include those of Doob (1944) , Bartlett (1946) , Phillips (1959) and Durbin (1961) . In the last ten years there has been a resurgence of interest in continuous-time processes partly as a result of the very successful application of stochastic differential equation models to problems in finance, exemplified by the derivation of the Black-Scholes option-pricing formula and its generalizations ( Hull and White, 1987 ). Numerous examples of econometric applications of continuous-time models are contained in the book of Bergstrom (1990) . Continuous-time models have also been utilized very successfully for the modelling of irregularly-spaced data ( Jones, 1981 , Jones, 1985 , Jones and Ackerson (1990) ). At the same time there has been an increasing realization that non-linear time series models provide much better representations of many empirically observed time series than linear models. The threshold ARMA models of Tong, 1983 , Tong, 1990 , have been particularly successful in representing a wide variety of data sets, and the ARCH and GARCH models of Engle (1982) and Bollerslev (1986) respectively have had great success in the modelling of financial data. Continuous-time versions of ARCH and GARCH models have been developed by Nelson (1990) . In this paper we discuss continuous-time ARMA models, their basic properties, their relationship with discrete-time ARMA models, inference based on observations made at discrete times and non-linear processes which include continuous-time analogues of Tong's threshold ARMA models.