Reduced Forms and Relationships with ARIMA Models

The purpose of this chapter is to examine the links between the (linear) innovations state space models and autoregressive integrated moving average (ARIMA) models, frequently called “Box-Jenkins models” because Box and Jenkins (1970) proposed a complete methodology for identification, estimation and prediction with these models. We will show that when the state variables are eliminated from a linear innovations state space model, an ARIMA model is obtained. This ARIMA form of the state space model is called its reduced form.

We begin the chapter with a brief summary of ARIMA models and their properties. In Sect. 11.2 we obtain reduced forms for the simple cases of the local levelmodel, ETS(A,N,N), and the local trendmodel, ETS(A,A,N). Then, in Sect. 11.3 we show how to put a general linear innovations state space model into an ARIMA reduced form. (Causal) stationarity and invertibility conditions for the reduced form model are developed in Sect. 11.4, and we explore the links with causal stationarity and stability of the corresponding innovations state space model.