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

Rice, Gregory; Shang, Han Lin

doi:10.1111/jtsa.12229

Cited by 52 publications

(72 citation statements)

References 35 publications

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“…An approach that balances these two concerns is to take h = Mn 1/(1+2 τ ) , where the constant M is estimated from the data and increases with the level of serial correlation. It can be shown (see Rice and Shang ()) that the optimal constant M in terms of asymptotically minimizing the mean‐squared normed error of

{\overset{false^}{C}}_{ε}

is of the form

M = false(2 italicτ false‖ frakturq C_{ε}^{(τ)} {{false‖}^{2})}^{1 / (1 + 2 τ)} {()(, [‖ C_{ε} {false‖}^{2} + {\{\int_{0}^{1} C_{ε} (u, u) normald u\}}^{2}], \int_{- \infty}^{\infty}, w_{τ}^{2}, false(x false), d, x)}^{- 1 / (1 + 2 τ)},

where

C_{italicε}^{false(italicτ false)}

is related to the τ th derivative of a spectral density operator evaluated at frequency 0. The unknown quantities in M can be estimated by using pilot estimates of C ɛ and

C_{italicε}^{false(italicτ false)}

to produce an estimated bandwidth

h = \overset{false^}{M} n^{1 / (1 + 2 τ)}

.…”

Section: Implementation Detailsmentioning

confidence: 99%

“…A comparison of the accuracy in terms of mean‐squared normed error of

{\overset{false^}{C}}_{ε}

for a multitude of bandwidth and weight function combinations is provided in Rice and Shang (), but a comparative study of how these estimators perform in problems of inference has not been conducted, to the best of our knowledge. With results that are reported in the on‐line supplement, the proposed break point detection method was compared for all combinations of the Bartlett, Parzen () and a version of the flat top (Politis and Romano, ) weight functions with the four bandwidth choices of h = n 1/3 , h = n 1/4 , h = n 1/5 and

h = \overset{false^}{M} n^{1 / (1 + 2 τ)}

for the data‐generating processes that are considered in the simulation study that is presented below, as well as some additional processes exhibiting stronger temporal dependence.…”

Section: Implementation Detailsmentioning

confidence: 99%

See 1 more Smart Citation

Detecting and Dating Structural Breaks in Functional Data Without Dimension Reduction

Aue

Rice

Sönmez

2017

Journal of the Royal Statistical Society Series B: Statistical Methodology

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158

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Summary Methodology is proposed to uncover structural breaks in functional data that is ‘fully functional’ in the sense that it does not rely on dimension reduction techniques. A thorough asymptotic theory is developed for a fully functional break detection procedure as well as for a break date estimator, assuming a fixed break size and a shrinking break size. The latter result is utilized to derive confidence intervals for the unknown break date. The main results highlight that the fully functional procedures perform best under conditions when analogous estimators based on functional principal component analysis are at their worst, namely when the feature of interest is orthogonal to the leading principal components of the data. The theoretical findings are confirmed by means of a Monte Carlo simulation study in finite samples. An application to annual temperature curves illustrates the practical relevance of the procedures proposed.

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{\overset{false^}{C}}_{ε}

is of the form

M = false(2 italicτ false‖ frakturq C_{ε}^{(τ)} {{false‖}^{2})}^{1 / (1 + 2 τ)} {()(, [‖ C_{ε} {false‖}^{2} + {\{\int_{0}^{1} C_{ε} (u, u) normald u\}}^{2}], \int_{- \infty}^{\infty}, w_{τ}^{2}, false(x false), d, x)}^{- 1 / (1 + 2 τ)},

where

C_{italicε}^{false(italicτ false)}

is related to the τ th derivative of a spectral density operator evaluated at frequency 0. The unknown quantities in M can be estimated by using pilot estimates of C ɛ and

C_{italicε}^{false(italicτ false)}

to produce an estimated bandwidth

h = \overset{false^}{M} n^{1 / (1 + 2 τ)}

.…”

Section: Implementation Detailsmentioning

confidence: 99%

“…A comparison of the accuracy in terms of mean‐squared normed error of

{\overset{false^}{C}}_{ε}

h = \overset{false^}{M} n^{1 / (1 + 2 τ)}

for the data‐generating processes that are considered in the simulation study that is presented below, as well as some additional processes exhibiting stronger temporal dependence.…”

Section: Implementation Detailsmentioning

confidence: 99%

Detecting and Dating Structural Breaks in Functional Data Without Dimension Reduction

Aue

Rice

Sönmez

2017

Journal of the Royal Statistical Society Series B: Statistical Methodology

Self Cite

158

View full text Add to dashboard Cite

show abstract

“…We also repeated all simulations using the Parzen weight function, which is of order two, and found that changing the weight function had relatively limited effect relative to the differences in changing the bandwidth; see Andrews () for the definition of the Parzen weight function. To select the bandwidth B , one could employ any of a number of methods, for example using the criteria discussed in Hörmann and Kokoszka () or by adapting the data driven procedure in Rice and Shang (). We study below the choice of B = T 1/(2 b + 1) where b = 1 and b = 2, which diverge at an optimal rate in terms of minimizing the asymptotic mean squared normed error of the estimator

\overset{D}{true}

when the weight function is of order b .…”

Section: Methodsmentioning

confidence: 99%

Inference for the Lagged Cross‐Covariance Operator Between Functional Time Series

Rice

Shum

2019

Journal Time Series Analysis

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When considering two or more time series of functional data objects, for instance those derived from densely observed intraday stock price data of several companies, the empirical cross‐covariance operator is of fundamental importance due to its role in functional lagged regression and exploratory data analysis. Despite its relevance, statistical procedures for measuring the significance of such estimators are currently undeveloped. We present methodology based on a functional central limit theorem for conducting statistical inference for the cross‐covariance operator estimated between two stationary, weakly dependent, functional time series. Specifically, we consider testing the null hypothesis that the two series possess a specified cross‐covariance structure at a given lag. Since this test assumes that the series are jointly stationary, we also develop a change‐point detection procedure to validate this assumption of independent interest. The most imposing technical hurdle in implementing the proposed tests involves estimating the spectrum of a high dimensional spectral density operator at frequency zero. We propose a simple dimension reduction procedure based on functional principal component analysis to achieve this, which is shown to perform well in a simulation study. We illustrate the proposed methodology with an application to densely observed intraday price data of stocks listed on the New York stock exchange‐20.40

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“…Viewing the choice of b as the selection of the truncation lag in the aforementioned lag window type estimator, allows for the use of some results available in the literature in order to select b. To elaborate, the choice of the truncation lag in the functional set-up has been discussed in Horváth et al (2016) and Rice and Shang (2017), where different procedures to select this parameter have been investigated. In our context, we found the simple rule proposed by Rice and Shang (2017) quite effective according to which the block length b is set equal to the smallest integer larger or equal to n 1/3 .…”

Section: Simulationsmentioning

confidence: 99%

“…To elaborate, the choice of the truncation lag in the functional set-up has been discussed in Horváth et al (2016) and Rice and Shang (2017), where different procedures to select this parameter have been investigated. In our context, we found the simple rule proposed by Rice and Shang (2017) quite effective according to which the block length b is set equal to the smallest integer larger or equal to n 1/3 . In the case of n 1 = n 2 = 200 observations, this rule leads to the value b = 6.…”

Section: Simulationsmentioning

confidence: 99%

Testing equality of autocovariance operators for functional time series

Pilavakis

Paparoditis

Sapatinas

2020

Journal Time Series Analysis

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We consider strictly stationary stochastic processes of Hilbert space-valued random variables and focus on tests of the equality of the lag-zero autocovariance operators of several independent functional time series. A moving block bootstrap-based testing procedure is proposed which generates pseudo random elements that satisfy the null hypothesis of interest. It is based on directly bootstrapping the time series of tensor products which overcomes some common difficulties associated with applications of the bootstrap to related testing problems. The suggested methodology can be potentially applied to a broad range of test statistics of the hypotheses of interest. As an example, we establish validity for approximating the distribution under the null of a fully functional test statistic based on the Hilbert-Schmidt distance of the corresponding sample lag-zero autocovariance operators, and show consistency under the alternative. As a prerequisite, we prove a central limit theorem for the moving block bootstrap procedure applied to the sample autocovariance operator which is of interest on its own. The finite sample size and power performance of the suggested moving block bootstrap-based testing procedure is illustrated through simulations and an application to a real-life dataset is discussed.

show abstract

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

Cited by 52 publications

References 35 publications

Detecting and Dating Structural Breaks in Functional Data Without Dimension Reduction

Detecting and Dating Structural Breaks in Functional Data Without Dimension Reduction

Inference for the Lagged Cross‐Covariance Operator Between Functional Time Series

Testing equality of autocovariance operators for functional time series

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