Asymmetric ARCH (AARCH)

An ARCH model with properties similar to those of EGARCH is the asymmetric ARCH
(AARCH). In its simplest form the conditional variance ht can be written as
htóuòa(etñ1òc)2òbhtñ1 (2.18)
The conditional variance parameterization in equation (2.18) is a quadratic function
of one-period-past error (etñ1òc)2. Since the model of equation (2.18) and higherorder
versions of this model formulation still lie within the parametric ARCH, it can
therefore, be interpreted as the quadratic projection of the squared series on the
information set.
The (G)AARCH has similar properties to the GARCH but unlike the latter, which explores only the magnitude of past errors, the (G)AARCH allows past errors to have
an asymmetric effect on ht . That is, because c can take any value, a dynamic
asymmetric effect of positive and negative lagged values of et on ht is permitted. If c
is negative the conditional variance will be higher when etñ1 is negative than when it
is positive. If có0 the (G)AARCH reduces to a (G)ARCH model. Therefore, the
(G)AARCH, like the EGARCH, can capture the leverage effect present in the stock
market data. When the sum of ai and bj (where ió1, . . . , p and jó1, . . . , q are the
orders of the (G)AARCH(p, q) process) is unity then analogously to the GARCH, the
model is referred to as Integrated (G)AARCH.
As with the GARCH process, the autocorrelogram and partial autocorrelogram of
the squares of e, as obtained by an AR(1), can be used to identify the {p, q} orders. In
the case of the (G)AARCH(1, 1), the unconditional variance of the process is given by
p2ó(uòc2a)/(1ñañb) (2.19)
Other asymmetric specifications
GJR or threshold:
htóuòbhtñ1òae2
tñ1òcS¯ tñ1e2
tñ1 (2.20)
where S¯ tó1 if et\0, S¯ tó0 otherwise.
Non-linear asymmetric GARCH:
htóuòbhtñ1òa(etñ1òc htñ1)2 (2.21)