Multivariate Hyper-Rotated GARCH-BEKK

For large multivariate models of generalized autoregressive conditional heteroskedasticity (GARCH), it is important to reduce the number of parameters to cope with the ‘curse of dimensionality’. Recently, Laurent, Rombouts and Violante (2014 “Multivariate Rotated ARCH Models” Journal of Econometrics 179: 16–30) developed the rotated multivariate GARCH model, which focuses on the parameters for standardized variables. This paper extends the rotated multivariate GARCH model by considering a hyper-rotation, which uses a more flexible structure for the rotation matrix. The paper shows an alternative representation based on a random coefficient vector autoregressive and moving-average (VARMA) process, and provides the regularity conditions for the consistency and asymptotic normality of the quasi-maximum likelihood (QML) estimator for VARMA with hyper-rotated multivariate GARCH. The paper investigates the finite sample properties of the QML estimator for the new model. Empirical results for four exchange rate returns show the new specifications works satisfactory for reducing the number of parameters.

Confidence Regions for Parameters in Stationary Time Series Models With Gaussian Noise

This article develops two new empirical likelihood methods for long-memory time series models based on adjusted empirical likelihood and mean empirical likelihood. By application of Whittle likelihood, one obtains a score function that can be viewed as the estimating equation of the parameters of the long-memory time series model. An empirical likelihood ratio is obtained which is shown to be asymptotically chi-square distributed. It can be used to construct confidence regions. By adding pseudo samples, we simultaneously eliminate the non-definition of the original empirical likelihood and enhance the coverage probability. Finite sample properties of the empirical likelihood confidence regions are explored through Monte Carlo simulation, and some real data applications are carried out.

Manifold-adaptive dimension estimation revisited

Data dimensionality informs us about data complexity and sets limit on the structure of successful signal processing pipelines. In this work we revisit and improve the manifold adaptive Farahmand-Szepesvári-Audibert (FSA) dimension estimator, making it one of the best nearest neighbor-based dimension estimators available. We compute the probability density function of local FSA estimates, if the local manifold density is uniform. Based on the probability density function, we propose to use the median of local estimates as a basic global measure of intrinsic dimensionality, and we demonstrate the advantages of this asymptotically unbiased estimator over the previously proposed statistics: the mode and the mean. Additionally, from the probability density function, we derive the maximum likelihood formula for global intrinsic dimensionality, if i.i.d. holds. We tackle edge and finite-sample effects with an exponential correction formula, calibrated on hypercube datasets. We compare the performance of the corrected median-FSA estimator with kNN estimators: maximum likelihood (Levina-Bickel), the 2NN and two implementations of DANCo (R and MATLAB). We show that corrected median-FSA estimator beats the maximum likelihood estimator and it is on equal footing with DANCo for standard synthetic benchmarks according to mean percentage error and error rate metrics. With the median-FSA algorithm, we reveal diverse changes in the neural dynamics while resting state and during epileptic seizures. We identify brain areas with lower-dimensional dynamics that are possible causal sources and candidates for being seizure onset zones.

Multiscale regression on unknown manifolds

<abstract><p>We consider the regression problem of estimating functions on $ \mathbb{R}^D $ but supported on a $ d $-dimensional manifold $ \mathcal{M} ~~\subset \mathbb{R}^D $ with $ d \ll D $. Drawing ideas from multi-resolution analysis and nonlinear approximation, we construct low-dimensional coordinates on $ \mathcal{M} $ at multiple scales, and perform multiscale regression by local polynomial fitting. We propose a data-driven wavelet thresholding scheme that automatically adapts to the unknown regularity of the function, allowing for efficient estimation of functions exhibiting nonuniform regularity at different locations and scales. We analyze the generalization error of our method by proving finite sample bounds in high probability on rich classes of priors. Our estimator attains optimal learning rates (up to logarithmic factors) as if the function was defined on a known Euclidean domain of dimension $ d $, instead of an unknown manifold embedded in $ \mathbb{R}^D $. The implemented algorithm has quasilinear complexity in the sample size, with constants linear in $ D $ and exponential in $ d $. Our work therefore establishes a new framework for regression on low-dimensional sets embedded in high dimensions, with fast implementation and strong theoretical guarantees.</p></abstract>