Adaptive likelihood estimator of conditional variance function

Modelling volatility in the form of conditional variance function has been a popular method mainly due to its application in financial risk management. Among others, we distinguish the parametric GARCH models and the nonparametric local polynomial approximation using weighted least squares or gaussian likelihood function. We introduce an alternative likelihood estimate of conditional variance and we show that substitution of the error density with its estimate yields similar asymptotic properties, that is, the proposed estimate is adaptive to the error distribution. Theoretical comparison with existing estimates reveals substantial gains in efficiency, especially if error distribution has fatter tails than Gaussian distribution. Simulated data confirm the theoretical findings while an empirical example demonstrates the gains of the proposed estimate.