  Functional magnetic resonance imaging or functional MRI (fMRI) is a very
popular tool used for differing brain regions by measuring brain activity. It
is affected by physiological noise, such as head and brain movement in the
scanner from breathing, heart beats, or the subject fidgeting. The purpose of
this paper is to propose a novel approach to handling fMRI data for infants
with high volatility caused by sudden head movements. Another purpose is to
evaluate the volatility modelling performance of multiple dependent fMRI time
series data. The models examined in this paper are AR and GARCH and the
modelling performance is evaluated by several statistical performance measures.
The conclusions of this paper are that multiple dependent fMRI series data can
be fitted with AR + GARCH model if the multiple fMRI data have many sudden head
movements. The GARCH model can capture the shared volatility clustering caused
by head movements across brain regions. However, the multiple fMRI data without
many head movements have fitted AR + GARCH model with different performance.
The conclusions are supported by statistical tests and measures. This paper
highlights the difference between the proposed approach from traditional
approaches when estimating model parameters and modelling conditional variances
on multiple dependent time series. In the future, the proposed approach can be
applied to other research fields, such as financial economics, and signal
processing. Code is available at \url{https://github.com/13204942/STAT40710}.
