The HGARCH model allows long‐memory dependence in volatilities. A new HGARCH model with time‐varying amplitude is presented in this paper. Moment properties of the model are discussed. A score test is derived to check the time‐varying behaviour of the amplitude. Value‐at‐risk testings are done to evaluate the forecasting capability. Simulation and empirical results provide further support to the proposed model.

HYGARCH process is commonly used for modeling long memory volatility. Many financial time series are characterized by transition between different levels of volatilities. Smooth transition HYGARCH (ST-HYGARCH) model is proposed to model smooth transition between components of HYGARCH process. The behavior of the conditional variance in the ST-HYGARCH are allowed to change smoothly over time. The asymptotic finiteness of the second moment is studied. A score test is developed to check the smooth transition property. The performance of the new proposed model and the score test are examined by some simulations. Applying the log returns of some part of S&P500 and Dow Jones industrial average indexes, we show the competing performance of the ST-HYGARCH model in comparison to HYGARCH and ST-GARCH models in forecasting volatility and value-at-risk.

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