Least‐squares estimation of GARCH(1,1) models with heavy‐tailed errors

GARCH(1,1) models are widely used for modelling processes with time‐varying volatility. These include financial time series, which can be particularly heavy tailed. In this paper, we propose a novel log‐transform‐based least‐squares approach to the estimation of GARCH(1,1) models. Within this approach, the scale of the estimated volatility is dependent on an unknown tuning constant. By means of a backtesting exercise on both real and simulated data, we show that knowledge of the tuning constant is not crucial for Value at Risk prediction. However, this does not apply to many other applications where correct identification of the volatility scale is required. In order to overcome this difficulty, we propose two alternative two‐stage least‐squares estimators and we derive their asymptotic properties under very mild moment conditions for the errors. In particular, we establish the consistency and asymptotic normality at the standard convergence rate of n for our estimators. Their finite sample properties are assessed by means of an extensive simulation study.