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

Abstract

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 "flattop" 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 flattop 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.

A Plug‐in Bandwidth Selection Procedure for Long‐Run Covariance Estimation with Stationary Functional Time Series

“…Via the same simulation procedure as described earlier, summary statistics of the normed estimation errors are reported in Table 1 in the Supporting Information of this article (Rice and Shang, ), for settings 3, 4 and 5 using final Bartlett weight functions, for the flat‐top pilot with adaptive bandwidth, which we call ‘adaptive flat‐top pilot’, and for the ‘oracle bandwidth’, which uses a flat‐top weight function with the optimal bandwidth that minimized the normed estimation error computed over all positive integer bandwidths with a maximum of 50. We summarize the results of this simulation as follows: The adaptive bandwidth estimation procedure of Horváth et al () performed the best for the FMA 1 (0) and FMA 0.5 (8) DGPs. For other DGPs, including the FAR(1) processes considered, setting 4 was generally shown to be the most accurate choice in our proposed plug‐in method. In majority of these cases, the proposed methods of settings 3, 4 and 5 beat the adaptive pilot estimator. The oracle estimator was typically five times as accurate if not more compared with the methods we considered, despite the fact that our empirical bandwidth is not restricted to the integers.…”

“…Horváth et al () and Panaretos and Tavakoli () define analogous smoothed periodogram type estimates of the long‐run covariance and spectral density operators for functional time series; however, the problem of bandwidth selection has been only lightly investigated in this setting. Horváth et al () propose an adaptive bandwidth selection algorithm that is designed for the infinite‐order ‘flat‐top’ weight function of Politis and Romano (). Bandwidth selection methodology for finite‐order weight functions have not yet been considered.…”

On the asymptotic normality of kernel estimators of the long run covariance of functional time series

“…Another approach is to minimize (2.15) with respect to h after pilot estimators are chosen for E∥Γ 1 ∥ 2 and ∥F∥ 2 ; see [10] for a development in the univariate case. A data driven estimator is discussed in [28,41] for the ''flat top'' kernel, i.e. when q = ∞.…”

“…The estimator in (1.1) was introduced in [26], where it is shown to be consistent under mild conditions, and its applications are developed in [28,30] in the context of inference for the mean and stationarity testing with functional time series. Hörmann et al [21] develops an analog of dynamic principal component analysis based on the spectral density operator of functional time series, which is directly related to the long run covariance operator.…”

Detecting and Localizing Differences in Functional Time Series Dynamics: A Case Study in Molecular Biophysics

“…Zhang et al (2011) and Aston and Kirch (2012a,b) considered the change point detection of the mean function. Concerning inference on the second-order structure, Horváth, Kokoszka, and Reeder (2013) and Horváth, Rice, and Whipple (2014) proposed a consistent estimator for the longrun covariance operator, Kokoszka and Reimherr (2013) established the asymptotic normality of the sample covariance operator and its eigenfunctions, while Zhang and Shao (2015) considered the comparison of the covariance operators of two functional time series, a problem that has attracted considerable attention in the case of two collections of iid functional data (Panaretos, Kraus, and Maddocks 2010;Boente, Rodriguez, and Sued 2011;Kraus and Panaretos 2012;Horváth and Kokoszka 2012;Fremdt et al 2013;Paparoditis and Sapatinas 2014;Pigoli et al 2014;Boente, Rodriguez, and Sued 2014). The covariance operator, however, fails to capture any of the dynamics of a functional time series; and the long-run covariance operator captures only crude aspects of the time dynamics (being the sum of the autocovariance operators over lags).…”