Conditional Heteroscedasticity Models with Time-Varying Parameters: Estimation and Asymptotics

This paper proposes using Chebyshev polynomials to approximate time-varying parameters of a GARCH model, where polynomial coefficients are estimated via numerical optimization using the function gradient descent method. We investigate the asymptotic properties of the estimates of polynomial coefficients and the subsequent estimate of conditional variance. Monte Carlo studies are conducted to examine the performance of the proposed polynomial approximation. With empirical studies of modelling daily returns of the US 30-year T-bond daily closing price and daily returns of the gold futures closing price, we find that in terms of in-sample fitting and out-of-sample forecasting, our proposed time-varying model outperforms the constant-parameter counterpart and a benchmark time-varying model.