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

TL;DR

This paper introduces a recursive online algorithm for estimating time-varying ARCH model parameters, providing asymptotic properties and a method to improve convergence rates in non-stationary contexts.

Contribution

It presents a novel recursive online estimation method for time-varying ARCH parameters, including analysis of its sampling properties and a technique to enhance convergence rates.

Findings

Estimator exhibits asymptotic normality in non-stationary settings

Bias due to non-stationarity is explicitly characterized

Combining estimators improves convergence for smooth parameter curves

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.