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11.5.2 Why Does This Model Volatility?
I personally find the above "formal" definition lacking in motivation as to how it introduces
volatility. However, you can see how it is introduced by squaring both sides of the previous
equation:
Var(ɛ t ) = E[ɛ 2 t ] − (E[ɛ t ]) 2 (11.8)
= E[ɛ 2 t ] (11.9)
= E[wt 2 ] E[α 0 + α 1 ɛ 2 t−1] (11.10)
= E[α 0 + α 1 ɛ 2 t−1] (11.11)
= α 0 + α 1 Var(ɛ t−1 ) (11.12)
Where I have used the definitions of the variance Var(x) = E[x 2 ] − (E[x]) 2 and the linearity
of the expectation operator E, along with the fact that {w t } has zero mean and unit variance.
Thus we can see that the variance of the series is simply a linear combination of the variance
of the prior element of the series. Simply put, the variance of an ARCH(1) process follows an
AR(1) process.
It is interesting to compare the ARCH(1) model with an AR(1) model. Recall that the latter
is given by:
x t = α 0 + α 1 x t−1 + w t (11.13)
You can see that the models are similar in form with the exception of the white noise term.
11.5.3 When Is It Appropriate To Apply ARCH(1)?
What approach can be taken in order to determine whether an ARCH(1) model is appropriate
to apply to a series?
Consider that when we were attempting to fit an AR(1) model we were concerned with the
decay of the first lag on a correlogram of the series. However if we apply the same logic to the
square of the residuals and see whether we can apply an AR(1) to these squared residuals then
we have an indication that an ARCH(1) process may be appropriate.
Note that ARCH(1) should only ever be applied to a series that has already had an appropriate
model fitted sufficient to leave the residuals looking like discrete white noise. Since we can only tell
whether ARCH is appropriate or not by squaring the residuals and examining the correlogram,
we also need to ensure that the mean of the residuals is zero.
Crucially ARCH should only ever be applied to series that do not have any trends or seasonal
effects, which are those that have no evident serial correlation. ARIMA is often applied to such
a series (or even Seasonal ARIMA) at which point ARCH may be a good fit.
11.5.4 ARCH(p) Models
It is straightforward to extend ARCH to higher order lags. An ARCH(p) process is given by: