Bloomfield (1987), Hurvich (1985) and Robinson (1991). However, adaption of the multivariate approach of Robinson (1991) to the spectral density estimator $\widehat{f}_{X,\lambda}(\sigma_r,\tau_s)$, for $r,s \in \{1,2, \ldots, k\}$, faces problems due to the high dimensionality of the periodogram operator involved.

We propose a simple approach to select the bandwidth $b$ used in our testing procedure which is based on the idea to overcome the high-dimensionality of the problem by selecting a single bandwidth based on the “on average" behavior of the pooled estimator $ 
  \widehat{f}_{\lambda}(\sigma_r,\tau_s)$, that is, its behavior over all points $r,s \in \{1,2, \ldots, k\}$ for which the functional objects are observed. To elaborate, define first the following quantities. The averaged periodogram $$\widehat{I}_T(\lambda) = \frac{1}{k^2}\sum_{r=1}^k\sum_{s=1}^k\big\{
  \frac{1}{2}\widehat{p}_{X,\lambda}(\sigma_r,\tau_s) + \frac{1}{2}\widehat{p}_{Y,\lambda}(\sigma_r,\tau_s)\big\}$$ and the averaged spectral density estimator $$\widehat{g}_b(\lambda) =\frac{1}{k^2}\sum_{r=1}^k\sum_{s=1}^k\big\{
  \frac{1}{2}\widehat{f}_{X,\lambda}(\sigma_r,\tau_s) + \frac{1}{2}\widehat{f}_{Y,\lambda}(\sigma_r,\tau_s)\big\}.$$ Notice that $\widehat{I}_T(\lambda)$ can be interpreted as the periodogram at frequency $ \lambda$ of the pooled univariate process $ \{V_t= \frac{1}{2}\int_0^1 X_t(s)ds + \frac{1}{2}\int_0^1 Y_t(s)ds, t \in {\mathbb Z}\}$ while $\widehat{g}_b(\lambda)$ is an estimator of the spectral density $g$ of $ \{V_t,t\in\Z\}$. We then choose the bandwidth $b$ by minimizing the objective function $$CV(b) = \frac{1}{N} \sum_{t=1}^N \big\{ \log(\widehat{g}_{-t}(\lambda_t) + \widehat{I}_T(\lambda_t)/\widehat{g}_{-t}(\lambda_t)\big\},$$ over a grid of values of $b$, where 