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Basic Data Analysis for Time Series with R by DeWayne Derryberry

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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 ϵj = −blwjl… − b2wj − 2b1wj − 1 + wj, where wk 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 ϵj = −b1wj − 1 + wj. Using the relationship(s) ϵj = −b1wj − 1 + wj,    ϵj − 1 = −b1wj − 2 + wj − 1,    and    ϵj − 2 = −b1wj − 3 + wj − 2, it is easy to derive (Exercise 1) the autocorrelation function, Rk C0 = σ2MA(1) = (1 + b21w2, C1 = −b1σ2w, and Cj = 0 for j ⩾ 2.

From this it follows that: Ro = 1, R1 = −b1/(1 + b21), and Rk = 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 Rkk = 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, ϵn = a1ϵn − 1 + wn, so ϵn = a21ϵn − 2 + a1wn − 1 + wn and in general ϵn = ak1ϵnk + ∑kj = 1aj − 11wnj + 1. More generally, let k → ∞. Because |a1| < 1, then ϵn ≈ ∑j = 1aj − ...

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