An Estimating Function Approach
We introduce the method of estimating functions to study the
class of autoregressive conditional heteroskedasticity (ARCH)
models. We derive the optimal estimating functions by combining
linear and quadratic estimating functions. The resultant estimators
are more efficient than the quasi-maximum likelihood estimator. If
the assumption of conditional normality is imposed, the estimator
obtained by using the theory of estimating functions is identical
to that obtained by using the maximum likelihood method in finite
samples. The relative efficiencies of the estimating function
approach in comparison with the quasi-maximum likelihood estimator
are developed. We illustrate the estimating function approach using
a univariate GARCH(1,1) model with conditional Normal, Student-t,
and Gamma distributions. The efficiency benefits of the estimating
function (EF) approach relative to the quasi-maximum likelihood
approach are substantial for the Gamma distribution with large
skewness. Simulation analysis shows that the finite sample
properties of the estimators from the estimating function approach
are attractive. EF estimators tend to display less bias and root
mean squared error than the quasi-maximum likelihood estimator. The
efficiency gains are substantial for highly nonnormal
distributions. An example demonstrates that implementation of the
method is straightforward.
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