Chapter 3: Autoregressive and moving average processes

2 Moving average models Definition. The moving average model of order q, or MA(q), is defined to be Xt = t + θ1 t−1 + θ2 t−2 + · · ·+ θq t−q, where t i.i.d. ∼ N(0, σ). Remarks: 1. Without loss of generality, we assume the mean of the process to be zero. 2. Here θ1, . . . , θq (θq 6= 0) are the parameters of the model. 3. Sometimes it suffices to assume that t ∼WN(0, σ). Here we assume normality mainly to simplify our discussion. 4. By defining the moving average operator as Θ(B) = 1 + θ1B + θ2B + · · ·+ θqB We may also write the MA(q) process in the equivalent form Xt = Θ(B) t. Θ(z) is also known as the MA polynomial for z ∈ C. Proposition. Let {Xt} follow the MA(q) model. Then 1. EXt = 0, 2. varXt = (1 + θ 1 + · · ·+ θ q) σ, ∗Please send any comments and corrections to [email protected]. †Updated on 28 Jan.