Adaptive bandwidth selection in the long run covariance estimator of functional time series

In the analysis of functional time series an object which has seen increased use is the long run covariance function. It arises in several situations, including inference and dimension reduction techniques for high dimensional data, and new applications are being developed routinely. Given its relationship to the spectral density of finite dimensional time series, the long run covariance is naturally estimated using a kernel based estimator. Infinite order “flat-top” kernels remain a popular choice for such estimators due to their well documented bias reduction properties, however it has been shown that the choice of the bandwidth or smoothing parameter can greatly affect finite sample performance. An adaptive bandwidth selection procedure for flat-top kernel estimators of the long run covariance of functional time series is proposed. This method is extensively investigated using a simulation study which both gives an assessment of the accuracy of kernel based estimators for the long run covariance function and provides a guide to practitioners on bandwidth selection in the context of functional data.