## A recursive online
algorithm for the estimation of time-varying ARCH parameters

In this paper we propose an online recursive algorithm for
estimating the parameters of a time-varying ARCH process. The
estimation is done by updating the estimator at time point
(t-1) with observations about the time point t to yield an
estimator of the parameter at time point t. The sampling
properties of this estimator are studied in a nonstationary context,
in particular asymptotic normality
and an expression for the bias due to nonstationarity are established.
The minimax risk for the tvARCH process is evaluated and compared with
the mean squared error of the online recursive estimator.
It is shown, if the time-varying parameters belong to a Hoelder class
of order less
than or equal to one, then, with a suitable choice of step-size,
the recursive online estimator attains the local
minimax bound. On the other hand, if the order of the Hoelder class is greater
than one, the minimax bound for the recursive algorithm
cannot be attained. However,
by running two recursive online algorithms
in parallel with different step-sizes and
taking a linear combination of the estimators the
minimax bound can be attained for
Hoelder classes of order between one and two.

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