Covariances Estimation for Long-Memory Processes

Wu, Wei Biao; Huang, Yinxiao; Zheng, Wei

doi:10.1239/aap/1269611147

Cited by 18 publications

(13 citation statements)

References 33 publications

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“…There is a huge literature on asymptotic properties of sample auto-covariances. For linear processes this problem has been studied in Priestley (1981), Brockwell and Davis (1991), Hannan (1970), Anderson (1971), Hall and Heyde (1980), Hannan (1976), Hosking (1996), Phillips and Solo (1992), Wu and Min (2005) and Wu, Huang and Zheng (2010). If the lag k is fixed and bounded, thenγ k is basically the sample average of the stationary process of lagged products (X i X i−|k| ) and one can apply the limit theory for strong mixing processes; see Ibragimov and Linnik (1971), Eberlein and Taqqu (1986), Doukhan (1994) and Bradley (2007).…”

Section: Asymptotics Of Sample Auto-covariancesmentioning

confidence: 99%

Covariance Matrix Estimation in Time Series

Xiao

2012

Time Series Analysis: Methods and Applications

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Covariances play a fundamental role in the theory of time series and they are critical quantities that are needed in both spectral and time domain analysis. Estimation of covariance matrices is needed in the construction of confidence regions for unknown parameters, hypothesis testing, principal component analysis, prediction, discriminant analysis among others. In this paper we consider both low-and high-dimensional covariance matrix estimation problems and present a review for asymptotic properties of sample covariances and covariance matrix estimates. In particular, we shall provide an asymptotic theory for estimates of high dimensional covariance matrices in time series, and a consistency result for covariance matrix estimates for estimated parameters.

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Section: Asymptotics Of Sample Auto-covariancesmentioning

confidence: 99%

Covariance Matrix Estimation in Time Series

Xiao

2012

Time Series Analysis: Methods and Applications

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“…For a one-dimensional context, Xiao and Wu (2014) study a central limit theorem (CLT), Portmanteau tests and simultaneous inference for a growing number of lags based on a Gumbel type extreme value theory, see also the review Wu and Xiao (2011) and Jirak (2011). In Wu et al (2010) the estimation of autocovariances for long memory linear processes has been discussed and studied in depth including the case of a finite number of lags starting at a large lag k n with k n /n = o(1). Kouritzin (1995), also working within a linear process framework, established large-sample distributional asymptotics, based on strong approximations, of the sample cross-covariance matrix for two time series.…”

Section: Introductionmentioning

confidence: 99%

Large-sample approximations for variance-covariance matrices of high-dimensional time series

Steland¹,

Sachs²

2017

Bernoulli

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Distributional approximations of (bi-) linear functions of sample variancecovariance matrices play a critical role to analyze vector time series, as they are needed for various purposes, especially to draw inference on the dependence structure in terms of second moments and to analyze projections onto lower dimensional spaces as those generated by principal components. This particularly applies to the high-dimensional case, where the dimension d is allowed to grow with the sample size n and may even be larger than n. We establish large-sample approximations for such bilinear forms related to the sample variance-covariance matrix of a high-dimensional vector time series in terms of strong approximations by Brownian motions. The results cover weakly dependent as well as many long-range dependent linear processes and are valid for uniformly ℓ 1 -bounded projection vectors, which arise, either naturally or by construction, in many statistical problems extensively studied for high-dimensional series. Among those problems are sparse financial portfolio selection, sparse principal components, the LASSO, shrinkage estimation and change-point analysis for high-dimensional time series, which matter for the analysis of big data and are discussed in greater detail.

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“…Hence without the perturbation consideration, there is the danger of underestimating the fluctuation magnitude of the sample variance, namely, taking the fluctuation to be of the order N −1/2 when H < 3/4 and N 2H−2 when H > 3/4, whereas they are of the order N H−1 . We also mention that similar considerations also apply to the study of the asymptotic behavior of sample autocovariance/correlation (see, e.g., Hosking [48], Wu et al [84] and Lévy-Leduc et al [56]).…”

Section: Sample Variancementioning

confidence: 83%

How the Instability of Ranks Under Long Memory Affects Large-Sample Inference

Bai¹,

Taqqu²

2018

Statist. Sci.

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Under long memory, the limit theorems for normalized sums of random variables typically involve a positive integer called "Hermite rank". There is a different limit for each Hermite rank. From a statistical point of view, however, we argue that a rank other than one is unstable, whereas, a rank equal to one is stable. We provide empirical evidence supporting this argument. This has important consequences. Assuming a higher-order rank when it is not really there usually results in underestimating the order of the fluctuations of the statistic of interest. We illustrate this through various examples involving the sample variance, the empirical processes and the Whittle estimator.

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Covariances Estimation for Long-Memory Processes

Cited by 18 publications

References 33 publications

Covariance Matrix Estimation in Time Series

Covariance Matrix Estimation in Time Series

Large-sample approximations for variance-covariance matrices of high-dimensional time series

How the Instability of Ranks Under Long Memory Affects Large-Sample Inference

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