  HYGARCH process is the commonly used long memory process in modeling the
long-rang dependence in volatility.
  Financial time series are characterized by transition between phases of
different volatility levels. The smooth transition HYGARCH (ST-HYGARCH) model
is proposed to model time-varying structure with long memory property. The
asymptotic behavior of the second moment is studied and an upper bound for it
is derived. A score test is developed to check the smooth transition property.
The asymptotic behavior of the proposed model and the score test is examined by
simulation. The proposed model is applied to the \textit{S}\&\textit{P}500
indices for some period which show evidence of smooth transition property and
demonstrates out-performance of the ST-HYGARCH than HYGARCH in forecasting.
