Weak invariance principles for sums of dependent random functions

Berkes, István; Horváth, Lajos; Rice, Gregory

doi:10.1016/j.spa.2012.10.003

Cited by 52 publications

(77 citation statements)

References 15 publications

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“…The arbitrary constants δ and κ in are needed to guarantee that a weak approximation theorem of Berkes et al. (), which we use in our proofs, holds for every segment.…”

Section: Functional Factor Modelmentioning

confidence: 99%

Change point tests in functional factor models with application to yield curves

Bardsley

Horváth

Kokoszka

et al. 2017

The Econometrics Journal

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Summary Motivated by the problem of the detection of a change point in the mean structure of yield curves, we introduce several methods to test the null hypothesis that the mean structure of a time series of curves does not change. The mean structure does not refer merely to the level of the curves, but also to their range and other aspects of their shape, most prominently concavity. The performance of the tests depends on whether possible break points in the error structure, which refers to the random variability in the aspects of the curves listed above, are taken into account or not. If they are not taken into account, then an existing change point in the mean structure may fail to be detected with a large probability. The paper contains a complete asymptotic theory, a simulation study and illustrative data examples, as well as details of the numerical implementation of the testing procedures.

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“…The arbitrary constants δ and κ in are needed to guarantee that a weak approximation theorem of Berkes et al. (), which we use in our proofs, holds for every segment.…”

Section: Functional Factor Modelmentioning

confidence: 99%

Change point tests in functional factor models with application to yield curves

Bardsley

Horváth

Kokoszka

et al. 2017

The Econometrics Journal

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“…It follows directly from that

(E ‖ ξ_{i} - ξ_{i, m} ‖_{2}^{q} true)^{1 / q} = O (m^{- γ}),

where γ > 1, hence for some p < q ,

(E ‖ ξ_{i} - ξ_{i, m} ‖_{2}^{q} true)^{1 / p} = O (m^{- ζ}),

where ζ > 1. This gives that

{falsefalse}_{\sum}^{m = 1} (E ‖ ξ_{i} - ξ_{i, m} ‖_{2}^{q} true)^{1 / p} < \infty .

Therefore the sequence { ξ i } satisfies the conditions of Theorem 1.1 of Berkes et al (), and hence with

Ξ_{T} (t, s, x) : = \sqrt{T} ({\tilde{C}}_{X Y} (t, s, x) - \frac{⌊ T x ⌋}{T} C_{X Y} (t, s)) = \frac{1}{\sqrt{T}} {falsefalse}_{\sum}^{i = 1} ξ_{i} (t, s),

…”

Section: Appendix a Proofs Of Technical Resultsmentioning

confidence: 85%

“…Theorem 1.1 of Berkes et al ( ) in L 2 [0,1] 2 Consider a mean zero strictly stationary sequence of stochastic processes

false{ξ_{i} false(t, s false), t, s \in false[0, 1 false], i \in double-struckZ false}

. Suppose there exists a function f : S ∞ → L 2 [0,1] 2 , and i.i.d.…”

Section: Appendix a Proofs Of Technical Resultsmentioning

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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“…In this case assumption (1.2) follows from the joint measurability of the ϵ i (t, ω)'s. Assumption (1.3) is stronger than the requirement  ∞ ℓ=1 (E∥η j − η j,ℓ ∥ 2 ) 1/2 < ∞ used by Hörmann and Kokoszka (2010), Berkes et al (2013) and Jirak (2013) to establish the central limit theorem for sums of Bernoulli shifts. Since we need the central limit theorem for sample correlations, higher moment conditions and a faster rate in the approximability with ℓ-dependent sequences are needed.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Testing for independence between functional time series

Horváth

Rice

2015

Journal of Econometrics

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Article history:Available online xxxx JEL classification: C12 C32Keywords: Functional observations Tests for independence Weak dependence Long run covariance function Central limit theorem a b s t r a c t Frequently econometricians are interested in verifying a relationship between two or more time series. Such analysis is typically carried out by causality and/or independence tests which have been well studied when the data is univariate or multivariate. Modern data though is increasingly of a high dimensional or functional nature for which finite dimensional methods are not suitable. In the present paper we develop methodology to check the assumption that data obtained from two functional time series are independent. Our procedure is based on the norms of empirical cross covariance operators and is asymptotically validated when the underlying populations are assumed to be in a class of weakly dependent random functions which include the functional ARMA, ARCH and GARCH processes.

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Weak invariance principles for sums of dependent random functions

Cited by 52 publications

References 15 publications

Change point tests in functional factor models with application to yield curves

Change point tests in functional factor models with application to yield curves

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

Testing for independence between functional time series

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