Asymptotic properties of $M$-estimator for $GARCH(1,1)$ model parameters

$GARCH(1,1)$ model is used for analysis and forecasting of financial and economic time series. In the classical version, the maximum likelihood method is used to estimate the model parameters. However, this method is not convenient for analysis of models with residuals distribution different from normal. In this paper, we consider $M$-estimator for the $GARCH(1,1)$ model parameters, which is a generalization of the maximum likelihood method. An algorithm for constructing an $M$-estimator is described and its asymptotic properties are studied. A set of conditions is formulated under which the estimator is strictly consistent and has an asymptotically normal distribution. This method allows to analyze models with different residuals distributions; in particular, models with stable and tempered stable distributions that allow to take into account the features of real financial data: volatility clustering, heavy tails, asymmetry