Gaussian Approximation for High Dimensional Time Series

By Danna Zhang, Wei Biao Wu. 2015

140 citations recorded by the source corpus.

We consider the problem of approximating sums of high-dimensional stationary time series by Gaussian vectors, using the framework of functional dependence measure. The validity of the Gaussian approximation depends on the sample size $n$, the dimension $p$, the moment condition and the dependence of the underlying processes. We also consider an estimator for long-run covariance matrices and study its convergence properties. Our results allow constructing simultaneous confidence intervals for mean vectors of high-dimensional time series with asymptotically correct coverage probabilities. A Gaussian multiplier bootstrap method is proposed. A simulation study indicates the quality of Gaussian approximation with different $n$, $p$ under different moment and dependence conditions.

Subfields: Applied Mathematics, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e1cb6a4d1cf2c6e6b6c8de0d63e505150e28a9f6

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Citations in the corpus

High-dimensional econometrics and regularized GMM (2018, background): “[123, 46, 122] develop time series extensions of the high-dimensional CLT.”
Network quantile autoregression (2019, methodology): “As one possible solution, we discuss the problem under the framework of Gaussian approximation theory, which is formulated by Zhang and Cheng (2014) and Zhang and Wu (2015) for time series analysis.”
Nearly optimal central limit theorem and bootstrap approximations in high dimensions (2020, result): “lecting penalty parameters for regularized estimators of high-dimensional models; see [6] for details on these and other examples. Various bounds on and on closely related quantities were derived in [12, 13, 15, 42, 20, 41, 34, 30, 17, 23, 32, 31, 19, 18] but in our discussion, we only focus on the ”
Finite sample change point inference and identification for high‐dimensional mean vectors (2017, background): “Bootstrap approximations have been extended to partial sums of weakly dependent time series in [50, 52] and more general nonlinear statistics in [14].”
Estimation and Inference about Heterogeneous Treatment Effects in High-Dimensional Dynamic Panels. (2017, background): “…Gaussian approximation (Chernozhukov et al. (2013), Chernozhukov et al. (2014), Chernozhukov et al. (2015), Chernozhukov et al. (2017b), Zhang and Wu (2017), Chernozhukov et al. (2019a), Chernozhukov et al. (2019b)) by allowing for a panel setting with observations that are weakly…”
Beyond Gaussian approximation: Bootstrap for maxima of sums of independent random vectors (2017, methodology): “It can be also used in time series settings (Zhang and Wu, 2015) and high-dimensional regression (Zhang and Zhang, 2014; Belloni et al.”
Improved central limit theorem and bootstrap approximations in high dimensions (2019, unknown): “ian approximation of the form (2) allows us to develop powerful inference methods for high-dimensional data in applications discussed above and has stimulated further developments into dependent data [36, 35, 15], U-statistics [17, 9, 10], Malliavin calculus [18], and homogeneous sums [25]. Despite ”
High-dimensional central limit theorems by Stein’s method (2020, background): “s and improvements of the error bound, Chen (2018); Chen and Kato (2019); Song, Chen and Kato (2019) for extensions to U-statistics, Chernozhukov, Chetverikov and Kato (2019); Zhang and Cheng (2018); Zhang and Wu (2017) for sums of dependent random vectors, and Belloni et al. (2018) for a general su”