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

Abstract

In this paper we propose a recursive online 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 non-stationary context -- in particular, asymptotic normality and an expression for the bias due to non-stationarity are established. By running two recursive online algorithms in parallel with different step sizes and taking a linear combination of the estimators, the rate of convergence can be improved for parameter curves from H\"{o}lder classes of order between 1 and 2.