The ARCH(1)

The ARCH model is based on the principal that speculative price changes contain
volatility clusters. Suppose that a security’s returns Yt can be modeled as:
Ytómtdòet (2.4)
where mt is a vector of variables with impact on the conditional mean of Yt , and et is the
residual return with zero mean, Etñ1(et )ó0, and variance Etñ1(e2
t )óht . The conditional
mean are expected returns that changes in response to current information. The
square of the residual return, often referred to as the squared error term, e2
t , can be
modeled as an autoregressive process. It is this that forms the basis of the Autoregressive
Conditional Heteroskedastic (ARCH) model. Hence the first-order ARCH
can be written:
htóuòae2
tñ1 (2.5)
where u[0 and aP0, and ht denotes the time-varying conditional variance of Yt .
This is described as a first-order ARCH process because the squared error term e2
tñ1
is lagged one period back. Thus, the conditional distribution of et is normal but its
conditional variance is a linear function of past squared errors.
ARCH models can validate scientifically a key characteristic of time series data
that ‘large changes tend to be followed by large changes – of either sign – and small
changes tend to be followed by small changes’. This is often referred to as the
clustering effect and, as discussed earlier, is one of the major explanations behind
the violation of the i.i.d. assumptions. The usefulness of ARCH models relates to its
ability to deal with this effect by using squared past forecast errors e2
tñ1 to predict
future variances. Hence, in the ARCH methodology the variance of Yt is expressed
as a (non-linear) function of past information, it validates earlier concerns about
heteroskedastic stock returns and meets a necessary condition for modeling volatility
as conditional on past information and as time varying.