## 6.2 THE AUTOCORRELATION FOR MA(1) MODELS

The basic model for MA(1) is ϵ_{j} = −*b*_{1}*w*_{j − 1} + *w*_{j}. Using the relationship(s)
ϵ_{j} = −*b*_{1}*w*_{j − 1} + *w*_{j}, ϵ_{j − 1} = −*b*_{1}*w*_{j − 2} + *w*_{j − 1}, and ϵ_{j − 2} = −*b*_{1}*w*_{j − 3} + *w*_{j − 2}, it is easy to derive (Exercise 1) the autocorrelation function, *R*_{k}
*C*_{0} = σ^{2}_{MA(1)} = (1 + *b*^{2}_{1})σ_{w}^{2}, *C*_{1} = −*b*_{1}σ^{2}_{w}, and *C*_{j} = 0 for *j* ⩾ 2.

From this it follows that: *R*_{o} = 1, *R*_{1} = −*b*_{1}/(1 + *b*^{2}_{1}), and *R*_{k} = 0 for *k* ⩾ 2.

There is an invertibility condition |*b*_{1}| < 1. The reason for this condition will be more apparent in Chapter 14. Notice that, for any real *b*_{1}, 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} = *a*_{1}ϵ_{n − 1} + *w*_{n}, so ϵ_{n} = *a*^{2}_{1}ϵ_{n − 2} + *a*_{1}*w*_{n − 1} + *w*_{n} and in general ϵ_{n} = *a*^{k}_{1}ϵ_{n − k} + ∑^{k}_{j = 1}*a*^{j − 1}_{1}*w*_{n − j + 1}. More generally, let *k* → ∞. Because |*a*_{1}| < 1, then ϵ_{n} ≈ ∑^{∞}_{j = 1}*a*^{j − ...}