Moving Average Models

3.2 The traditional approach

A moving average model of order q, denoted by MA(q):

q

yt = a0 + X akzt-k + £t, ^ ~ NI(0, a2), t Є T, (28.18)

k—1

is traditionally viewed as a DGM with a normal white noise process {et, t Є T} (see (28.4)) as the input and {yt, t Є T}, as the output process.

The question which naturally arises at this stage is "how does the DGM (28.18) fit into the orthogonal decomposition given in (28.9)?" A naive answer will be yt = E(yt | o(et-1, et-2,…, et-q)) + et, t Є T. However, such an answer is misleading because operational conditioning cannot be defined in terms of an unobserved stochastic process {et, t Є T}. In view of this, the next question is "how is the formulation (28.18) justified as a statistical Generating Mechanism?" The answer lies with the following theorem.

Wold decomposition theorem. Let {yt, t Є T}, be a normal stationary process and define the unobservable derived process {e t, t Є T}, by:

et = yt – E(yt | o(Y0-i)), with E(e) = 0, E(e2) = о2 > 0. (28.19)

Then, the process {yt, t Є T}, can be expressed in the form:

Подпись:yt – p = Wt + f ak£t-k, t Є T.

k=0

Подпись: (i) (ii) et ~ NI(0, о2), t =1, 2,…,

f a2 < -, for ak = COv(yt, £t-k), k = 0, 1, 2, … k=0 var(e t_k)

(iii) for wt = f ykwt-k, E(wtes) = 0, for all t, s = 1, 2,

k=1

It is important to note that the process {wt, t Є T}, is deterministic in the sense that it’s perfectly predictable from its own past; since o(wt-1, wt-2,…) = 17=-rxo(Y°_1), the right-hand side being the remote past of the process {yt, t Є T} (see Wold, 1938). In view of this, the MA(^) decomposition often excludes the remote past:

yt – P = f ak£t-k + et, t Є T. (28.21)

k=1

As it stands, the MA(^) formulation is non-operational because it involves an infinite number of unknown parameters. The question arises whether one can truncate the MA(ro) at some finite value q < T, in order to get an operational model. As the Wold decomposition theorem stands, no such truncation is justifi­able. For such a truncation to be formally justifiable we need to impose certain temporal dependence restrictions on the covariances o|T| = cov( yt, yt-T). The most natural dependence restriction in the case of stationary processes is that of ergodicity; see Hamilton (1994) and Phillips (1987).

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