ARCH-M (in mean)

The ARCH model can be extended to allow the mean of the series to be a function of
its own variance. This parameterization is referred to as the ARCH-in-Mean (or
ARCH-M) model and is formed by adding a risk-related component to the return
equation, in other words, the conditional variance ht . Hence, equation (2.4) can be
rewritten as:
Ytómtdòjhtòet (2.13)
Therefore the ARCH-M model allows the conditional variance to explain directly the
dependent variable in the mean equation of (2.13). The estimation process consists of solving equations (2.13) and (2.5) recursively. The term jht has a time-varying
impact on the conditional mean of the series. A positive j implies that the conditional
mean of Y increases as the conditional variance increases.
The (G)ARCH-M model specification is ideal for equity returns since it provides a
unified framework to estimate jointly the volatility and the time-varying expected
return (mean) of the series by the inclusion of the conditional variance in the mean
equation. Unbiased estimates of assets risk and return are crucial to the mean
variance utility approach and other related asset pricing theories. Finance theory
states that rational investors should expect a higher return for riskier assets. The
parameter j in equation (2.13) can be interpreted as the coefficient of relative risk
aversion of a representative investor and when recursively estimated, the same
coefficient jht can be seen as the time-varying risk premium. Since, in the presence
of ARCH, the variance of returns might increase over time, the agents will ask for
greater compensation in order to hold the asset. A positive j implies that the agent
is compensated for any additional risk.
Thus, the introduction of ht into the mean is another non-linear function of past
information. Since the next period’s variance, htò1, is known with certainty the next
period’s return forecast, E(Ytò1), can be obtained recursively. Assuming that mt is
known we can rearrange equation (2.13) as
E[Ytò1DIt ]•mtò1ó'tò1dò#htò1 (2.14)
Thus, the series’ expectation at tò1 (i.e. one period ahead into the future) is equal
to the series’ conditional mean, mtò1, at the same period. Unlike the unconditional
mean, kóE(Y), which is not a random variable, the conditional mean is a function
of past volatility, and because it uses information for the period up to t, can generally
be forecasted more accurately.
In contrast to the linear GARCH model, consistent estimation of the parameter
estimates of an ARCH-M model are sensitive to the model specification. A model is
said to be misspecified in the presence of simultaneity bias in the conditional mean
equation as defined in equation (2.13). This arises because the estimates for the
parameters in the conditional mean equation are not independent of the estimates
of the parameters in the conditional variance. Therefore, it has been argued that a
misspecification in the variance equation will lead to biased and inconsistent estimates
for the conditional mean equation.