Whittle Estimation of ARCH Models

For a class of parametric ARCH models, Whittle estimation based on squared
observations is shown to be fi-consistent and asymptotically normal. Our conditions
require the squares to h a ~ seh ort memory autocorrelation, by comparison
with the work of Zaffaroni (1999, "Gaussian Inference on Certain Long-Range
Dependent Volatility Models," Preprint), who established the same propertie!, on
the basis of an alternative class of models with martingale difference levels and
long memory autocorrelated squares.

1, INTRODUCTION
Conditional heteroskedasticity arises in much analysis of economic and financial
time series data. Even series that appear not to be autocorrelated may exhibit
dependence in their squares, a notable example being daily asset returns.
For a covariance stationary process, x,, t = 0,t1,...,suppose that, almost surely,
where
and 3; is the n-field of events generated by x,, s 5 t. The requirement
@, > O,$, r 0,j 4 1, ensures positivity of the conditional variance lz,, whereas
convergence conditions on the 4,will be imposed in the sequel. The .u, are
observable in some applications, whereas in others they could be innovations
in a time series model or regression errors.