Basic Data Analysis for Time Series with R

6 THE MOVING AVERAGE MODELS MA(1) AND MA(2)

6.1 THE MOVING AVERAGE MODEL

The second commonly used model for temporal patterns in the errors is the moving average structure. The structure MA(l) is ϵ_j = −b_lw_{j − l}… − b₂w_{j − 2} − b₁w_{j − 1} + w_j, where w_k are all white noise. In this chapter the focus is on the special cases MA(1) and MA(2). As with AR(m) models, the general MA(l) cases are dealt with in Chapter 14.

6.2 THE AUTOCORRELATION FOR MA(1) MODELS

The basic model for MA(1) is ϵ_j = −b₁w_{j − 1} + w_j. Using the relationship(s) ϵ_j = −b₁w_{j − 1} + w_j,    ϵ_{j − 1} = −b₁w_{j − 2} + w_{j − 1},    and    ϵ_{j − 2} = −b₁w_{j − 3} + w_{j − 2}, it is easy to derive (Exercise 1) the autocorrelation function, R_k C₀ = σ²_MA(1) = (1 + b²₁)σ_w², C₁ = −b₁σ²_w, and C_j = 0 for j ⩾ 2.

From this it follows that: R_o = 1, R₁ = −b₁/(1 + b²₁), and R_k = 0 for k ⩾ 2.

There is an invertibility condition |b₁| < 1. The reason for this condition will be more apparent in Chapter 14. Notice that, for any real b₁, all R_k, k = 0, 1, 2, … are valid, so the invertibility condition is obviously not functioning in the same way as the stability conditions for AR(m) models.

6.3 A DUALITY BETWEEN MA(L) AND AR(M) MODELS

The following argument is not particularly rigorous but hints at some more general results found in Chapter 14. Recall, for the AR(1) model, ϵ_n = a₁ϵ_{n − 1} + w_n, so ϵ_n = a²₁ϵ_{n − 2} + a₁w_{n − 1} + w_n and in general ϵ_n = a^k₁ϵ_{n − k} + ∑^k_{j = 1}a^{j − 1}₁w_{n − j + 1}. More generally, let k → ∞. Because |a₁| < 1, then ϵ_n ≈ ∑^∞_{j = 1}a^{j − ...}