Introduction

Consider the autoregressive moving-average (ARMA) model of orders p and q, (1.1)$$\phi \left( B \right){y_t} = \theta \left( B \right){v_t}$$ where $$\phi \left( B \right) = - {\phi _0} - {\phi _1}B - \cdots - {\phi _p}{B^p},\theta \left( B \right) = - {\theta _0} - {\theta _1}B - \cdots - {\theta _q}{B^q},{\theta _0} = {\theta _0} = - 1,{\phi _p} \ne 0,{\theta _q} \ne 0$$, B is the backshift operator, and {v _t } is a sequence of independent and identically distributed random variables with means 0 and variances σ2 (> 0). The sequence {v _t } is called either a white noise process or an innovation process. In some time series books, the white noise process is defined as a sequence of uncorrelated random variables instead of that of independent random variables. In practical time series analysis, there is not as much difference between the two definitions. We assume that the model is stationary and invertible, i.e., the equations ϕ(z) = 0 and θ(z) = 0 have all the roots outside the unit circle. We assume that the two equations have no common root. This assumption is sometimes called coprimal. The stationarity and the invertibility conditions have been discussed by several authors. Interested readers may consult the references in Section 1.6. In statistical literature, the white noise process is frequently assumed to be Gaussian, i.e., normally distributed. There are some references about non-Gaussian ARMA processes in Section 1.6. In this book, we assume that the coefficients ϕ₁…, ϕ _p , ϕ₁…, ϕ _q , and the white noise variance σ2 are constants, i.e., they do not depend on time.