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model, as well as the conditional heteroskedastic models of the ARCH and GARCH families.
These models will provide us with our first realistic attempts at forecasting asset prices.
In this section, however, we are going to introduce the Moving Average of order q model,
known as MA(q). This is a component of the more general ARMA model and as such we need
to understand it before moving further.
10.5.1 Rationale
A Moving Average model is similar to an Autoregressive model, except that instead of being a
linear combination of past time series values, it is a linear combination of the past white noise
terms.
Intuitively, this means that the MA model sees such random white noise "shocks" directly at
each current value of the model. This is in contrast to an AR(p) model, where the white noise
"shocks" are only seen indirectly, via regression onto previous terms of the series.
A key difference is that the MA model will only ever see the last q shocks for any particular
MA(q) model, whereas the AR(p) model will take all prior shocks into account, albeit in a
decreasingly weak manner.
10.5.2 Definition
Mathematically the MA(q) is a linear regression model and is similarly structured to AR(p):
Definition 10.5.1. Moving Average Model of order q. A time series model, {x t }, is a moving
average model of order q, MA(q), if:
x t = w t + β 1 w t−1 + . . . + β q w t−q (10.11)
of B:
Where {w t } is white noise with E(w t ) = 0 and variance σ 2 .
If we consider the Backward Shift Operator, B then we can rewrite the above as a function φ
x t = (1 + β 1 B + β 2 B 2 + . . . + β q B q )w t = φ q (B)w t (10.12)
We will make use of the φ function in subsequent chapters.
10.5.3 Second Order Properties
As with AR(p) the mean of a MA(q) process is zero. This is easy to see as the mean is simply a
sum of means of white noise terms, which are all themselves zero.
q∑
Mean: µ x = E(x t ) = E(w i ) = 0 (10.13)
i=0
Var: σ 2 w(1 + β 2 1 + . . . + β 2 q ) (10.14)