Multivariate volatility models

Correlations between asset returns are important in many financial applications. In recent years, multivariate volatility models have been used to describe the time-varying feature of the correlations. However, the curse of dimensionality quickly becomes an issue as the number of correlations is $k(k-1)/2$ for $k$ assets. In this paper, we review some of the commonly used models for multivariate volatility and propose a simple approach that is parsimonious and satisfies the positive definite constraints of the time-varying correlation matrix. Real examples are used to demonstrate the proposed model.Comment: Published at http://dx.doi.org/10.1214/074921706000001058 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

High-dimensional Linear Regression for Dependent Data with Applications to Nowcasting

Recent research has focused on $\ell_1$ penalized least squares (Lasso) estimators for high-dimensional linear regressions in which the number of covariates $p$ is considerably larger than the sample size $n$. However, few studies have examined the properties of the estimators when the errors and/or the covariates are serially dependent. In this study, we investigate the theoretical properties of the Lasso estimator for a linear regression with a random design and weak sparsity under serially dependent and/or nonsubGaussian errors and covariates. In contrast to the traditional case, in which the errors are independent and identically distributed and have finite exponential moments, we show that $p$ can be at most a power of $n$ if the errors have only finite polynomial moments. In addition, the rate of convergence becomes slower owing to the serial dependence in the errors and the covariates. We also consider the sign consistency of the model selection using the Lasso estimator when there are serial correlations in the errors or the covariates, or both. Adopting the framework of a functional dependence measure, we describe how the rates of convergence and the selection consistency of the estimators depend on the dependence measures and moment conditions of the errors and the covariates. Simulation results show that a Lasso regression can be significantly more powerful than a mixed-frequency data sampling regression (MIDAS) and a Dantzig selector in the presence of irrelevant variables. We apply the results obtained for the Lasso method to nowcasting with mixed-frequency data, in which serially correlated errors and a large number of covariates are common. The empirical results show that the Lasso procedure outperforms the MIDAS regression and the autoregressive model with exogenous variables in terms of both forecasting and nowcasting

Discussion of "Feature Matching in Time Series Modeling" by Y. Xia and H. Tong

Discussion of "Feature Matching in Time Series Modeling" by Y. Xia and H. Tong [arXiv:1104.3073]Comment: Published in at http://dx.doi.org/10.1214/11-STS345B the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Outlier detection in multivariate time series via projection pursuit

This article uses Projection Pursuit methods to develop a procedure for detecting outliers in a multivariate time series. We show that testing for outliers in some projection directions could be more powerful than testing the multivariate series directly. The optimal directions for detecting outliers are found by numerical optimization of the kurtosis coefficient of the projected series. We propose an iterative procedure to detect and handle multiple outliers based on univariate search in these optimal directions. In contrast with the existing methods, the proposed procedure can identify outliers without pre-specifying a vector ARMA model for the data. The good performance of the proposed method is verified in a Monte Carlo study and in a real data analysis

OUTLIER DETECTION IN MULTIVARIATE TIME SERIES VIA PROJECTION PURSUIT

This article uses Projection Pursuit methods to develop a procedure for detecting outliers in a multivariate time series. We show that testing for outliers in some projection directions could be more powerful than testing the multivariate series directly. The optimal directions for detecting outliers are found by numerical optimization of the kurtosis coefficient of the projected series. We propose an iterative procedure to detect and handle multiple outliers based on univariate search in these optimal directions. In contrast with the existing methods, the proposed procedure can identify outliers without pre-specifying a vector ARMA model for the data. The good performance of the proposed method is verified in a Monte Carlo study and in a real data analysis.

Denoising and Multilinear Dimension-Reduction of High-Dimensional Matrix-Variate Time Series via a Factor Model

This paper proposes a new multilinear projection method for dimension-reduction in modeling high-dimensional matrix-variate time series. It assumes that a $p_1\times p_2$ matrix-variate time series consists of a dynamically dependent, lower-dimensional matrix-variate factor process and a $p_1\times p_2$ matrix white noise series. Covariance matrix of the vectorized white noises assumes a Kronecker structure such that the row and column covariances of the noise all have diverging/spiked eigenvalues to accommodate the case of low signal-to-noise ratio often encountered in applications, such as in finance and economics. We use an iterative projection procedure to {reduce the dimensions and noise effects in estimating} front and back loading matrices and {to} obtain faster convergence rates than those of the traditional methods available in the literature. Furthermore, we introduce a two-way projected Principal Component Analysis to mitigate the diverging noise effects, and implement a high-dimensional white-noise testing procedure to estimate the dimension of the factor matrix. Asymptotic properties of the proposed method are established as the dimensions and sample size go to infinity. Simulated and real examples are used to assess the performance of the proposed method. We also compared the proposed method with some existing ones in the literature concerning the forecasting ability of the identified factors and found that the proposed approach fares well in out-of-sample forecasting.Comment: 57 Pages, 7 figures, 7 tables. arXiv admin note: text overlap with arXiv:2011.0902