  Auto-regressive conditionally heteroskedastic (ARCH) family models are still
used, by practitioners in business and economic policy making, as a conditional
volatility forecasting models. Furthermore ARCH models still are attracting an
interest of the researchers. In this contribution we consider the well known
GARCH(1,1) process and its nonlinear modifications, reminiscent of NGARCH
model. We investigate the possibility to reproduce power law statistics,
probability density function and power spectral density, using ARCH family
models. For this purpose we derive stochastic differential equations from the
GARCH processes in consideration. We find the obtained equations to be similar
to a general class of stochastic differential equations known to reproduce
power law statistics. We show that linear GARCH(1,1) process has power law
distribution, but its power spectral density is Brownian noise-like. However,
the nonlinear modifications exhibit both power law distribution and power
spectral density of the power law form, including 1/f noise.
