$\widehat{g}_{-t}(\lambda_t) = (Tb)^{-1}\sum_{s\in N_t} W((\lambda_t-\lambda_s)/b) \widehat{I}_T(\lambda_s)$ and $N_t=\{s: -N \leq s \leq N \ \mbox{and} \ s \neq \pm t\}$. That is, $ \widehat{g}_{-t}(\lambda_t)$ is the leave-one-out kernel estimator of $ g(\lambda)$, i.e., the estimator obtained after deleting the $t$-th frequency; see also Robinson (1991).\
Due to the computational complexity of the simulation analysis studied in the next section, the use of this automatic choice of the bandwidth $b$ will only be illustrated in the real-life data example.

Monte-Carlo Simulations
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We generated functional time series stemming from the following functional linear processes $$X_t=\varepsilon_t + a_1 \varepsilon_{t-1}+ a_2 \varepsilon_{t-2}, 
\label{mod-1}$$ and $$\quad Y_t=\varepsilon_t + a_1 \varepsilon_{t-1},
\label{mod-2}$$ $ t=1,2,\ldots,T$, where $\varepsilon_t$’s are i.i.d. Brownian Bridges. All curves were approximated using 21 equidistant points in the unit interval and transformed into functional objects using the Fourier basis with 21 basis functions. Three sample sizes $T=50$, $T=100$ and $T=200$ were considered and the bootstrap test was applied using three nominal levels, $\alpha=0.01$, $\alpha=0.05$ and $\alpha=0.10$. All bootstrap calculations were based on $B=1,000$ bootstrap replicates and $R=500$ model repetitions. To investigate the empirical size and power behavior of the bootstrap test, we set $a_1=0.8$ and consider a selection of $a_2$ values, i.e., $a_2 