A Simple Test for White Noise in Functional Time Series

Abstract

We propose a new procedure for white noise testing of a functional time series. Our approach is based on an explicit representation of the L 2 -distance between the spectral density operator and its best (L 2 -)approximation by a spectral density operator corresponding to a white noise process. The estimation of this distance can be easily accomplished by sums of periodogram kernels, and it is shown that an appropriately standardized version of the estimator is asymptotically normal distributed under the null hypothesis (of functional white noise) and under the alternative. As a consequence, we obtain a very simple test (using the quantiles of the normal distribution) for the hypothesis of a white noise functional process. In particular, the test does not require either the estimation of a long-run variance (including a fourth order cumulant) or resampling procedures to calculate critical values. Moreover, in contrast to all other methods proposed in the literature, our approach also allows testing for 'relevant' deviations from white noise and constructing confidence intervals for a measure that measures the discrepancy of the underlying process from a functional white noise process.

A general white noise test based on kernel lag-window estimates of the spectral density operator

“…, u n from {u t } t∈Z , we are then interested in testing the hypothesis H 0 : Q = 0 versus H 1 : Q > 0. Zhang (2016) and Bagchi et al (2018) develop general tests of H 0 versus H 1 . In this paper, we focus on testing H 1 versus the stronger but still relevant hypothesis H 0,iid : the sequence {u t } t∈Z is independent and identically distributed in H.…”

“…We provide some further discussion on this in Section 5. By contrast the tests of Zhang (2016) and Bagchi et al (2018) are built for such sequences, although the test of Bagchi et al (2018) seems to be undersized.…”

“…Each of these statistics are indelibly connected in that they reduce to weighted sums of semi-norms of empirical autocovariance operators, and hence it is natural to consider alternate ways of choosing the weights in order to increase power; see Escanciano and Lobato (2009) for a nice discussion of these connections for scalar portmanteau tests. The statistics developed in Zhang (2016) and Bagchi et al (2018), which are based on the periodogram operator, effectively give equal weights to the autocovariance operators at all lags. One might expect then that reweighting the periodogram so that it is a more efficient estimate of the spectral density operator might lead to more powerful tests.…”

“…In practice, we recommend the use of the statistic T β * 1 n with the data driven bandwidth for general white noise testing as long as conditional heteroscedasticity is not thought to be an issue. In the case when conditional heteroscedasiticy is of concern, the statistic of Zhang (2016) or Bagchi et al (2018) is expected to give more reliable results.…”

Novel Financial Capital Flow Forecast Framework Using Time Series Theory and Deep Learning: A Case Study Analysis of Yu’e Bao Transaction Data

“…In the model evaluation, the residual between the ideal modeling result and the real data [29] should consist of white noise, which means that no correlations occur between variables, the mean value is 0, and the residual satisfies a Gaussian distribution.…”