ABSTRACT
A stochastic process Yi is called an ARMA(p,q) process if it has the following representation:
Yi = μ+
p∑
φj(Yi−j − μ) + εi − q∑
θkεi−k, (10.1)
where the innovations εi are independent and identically distributed, with mean 0 and variance σ2ε . It is assumed that εi is independent of Yi−1 = (Y1, . . . , Yi−1). Often, the innovations are assumed to be Gaussian. One can rewrite the relationship (10.1) more concisely as
φ(B)(Yi − μ) = θ(B)εi,
where φ(z) = 1−∑pj=1 φjzj, θ(z) = 1− ∑q
k=1 θkz k and B is the lag operator,
i.e., BYi = Yi−1. In order for the process to admit a stationary distribution, it is necessary and sufficient that the roots of φ(z) are all outside the unit circle in the complex plane. Also, to obtain a unique representation, one imposes that the roots of θ(z) are all outside the unit circle in the complex plane.
