  An important issue in functional time series analysis is whether an observed
series comes from a purely random process. We extend the BDS test, a
widely-used nonlinear independence test, to the functional time series. Like
the BDS test in the univariate case, the functional BDS test can act as the
model specification test to evaluate the adequacy of various prediction models
and as a nonlinearity test to detect the existence of nonlinear structures in a
functional time series after removing the linear structure exhibited. We show
that the test statistic from the functional BDS test has the same asymptotic
properties as those in the univariate case and provides the recommended range
of its hyperparameters. Additionally, empirical data analysis features its
applications in evaluating the adequacy of the fAR(1) and fGARCH(1,1) models in
fitting the daily curves of cumulative intraday returns (CIDR) of the VIX
index. We showed that the functional BDS test remedies the weakness of the
existing independence test in the literature, as the latter is restricted in
detecting linear structures, thus, can neglect nonlinear temporal structures.
