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.