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would further depress the price of equities within these portfolios. Since the larger portfolios are
generally highly correlated anyway, this could trigger significant downward volatility.
These "sell-off" periods, as well as many other forms of volatility that occur in finance, lead
to heteroskedasticity that is serially correlated and hence conditional on periods of increased
variance. Thus we say that such series are conditional heteroskedastic.
One of the challenging aspects of conditional heteroskedastic series is that if we were to plot
the correlogram of a series with volatility we might still see what appears to be a realisation
of stationary discrete white noise. That is, the volatility itself is hard to detect purely from
the correlogram. This is despite the fact that the series is most definitely non-stationary as its
variance is not constant in time.
We are going to describe a mechanism for detecting conditional heteroskedastic series in this
chapter and then use the ARCH and GARCH models to account for it, ultimately leading to
more realistic forecasting performance, and thus more profitable trading strategies.
11.5 Autoregressive Conditional Heteroskedastic Models
We have now discussed conditional heteroskedasticity (CH) and its importance within financial
series. We want a class of models that can incorporate CH in a natural way. We know that the
ARIMA model does not account for CH, so how can we proceed?
Well, how about a model that utilises an autoregressive process for the variance itself ? That
is, a model that actually accounts for the changes in the variance over time using past values of
the variance.
This is the basis of the Autoregressive Conditional Heteroskedastic (ARCH) model. We
will begin with the simplest possible case, namely an ARCH model that depends solely on the
previous variance value in the series.
11.5.1 ARCH Definition
Definition 11.5.1. Autoregressive Conditional Heteroskedastic Model of Order Unity.
A time series {ɛ t } is given at each instance by:
ɛ t = σ t w t (11.5)
Where {w t } is discrete white noise, with zero mean and unit variance, and σ 2 t is given by:
σ 2 t = α 0 + α 1 ɛ 2 t−1 (11.6)
Where α 0 and α 1 are parameters of the model.
We say that {ɛ t } is an autoregressive conditional heteroskedastic model of order unity, denoted
by ARCH(1). Substituting for σ 2 t , we receive:
ɛ t = w t
√α 0 + α 1 ɛ 2 t−1 (11.7)