APPENDIX A4.3 ARIMA PROCESSES WITH ADDED NOISE

In this appendix we consider the effect of added noise (e.g., measurement error) to a general ARIMA(p, d, q) process. The results are also relevant to determine the nature of the reduced form ARIMA model of an observed process in structural component models (see Section 9.4), in which an observed series Z_t is presumed to be represented as the sum of two unobservable component processes that follow specified ARIMA models.

A4.3.1 Sum of Two Independent Moving Average Processes

As a necessary preliminary to what follows, consider a stochastic process w_t, which is the sum of two independent moving average processes of orders q₁ and q₂, respectively. That is,

where ?₁(B) and ?₂(B) are polynomials in B, of orders q₁ and q₂, and the white noise processes a_t and b_t have zero means, variances and , and are mutually independent. Suppose that q = max(q₁, q₂); then since