Large-dimensional Covariance Research Articles

A Heteroscedasticity-Robust Overidentifying Restriction Test with High-Dimensional Covariates

This article proposes an overidentifying restriction test for high-dimensional linear instrumental variable models. The novelty of the proposed test is that it allows the number of covariates and instruments to be larger than the sample size. The test is scale-invariant and robust to heteroscedastic errors. To construct the final test statistic, we first introduce a test based on the maximum norm of multiple parameters that could be high-dimensional. The theoretical power based on the maximum norm is higher than that in the modified Cragg-Donald test, the only existing test allowing for large-dimensional covariates. Second, following the principle of power enhancement, we introduce the power-enhanced test, with an asymptotically zero component used to enhance the power to detect some extreme alternatives with many locally invalid instruments. Finally, an empirical example of the trade and economic growth nexus demonstrates the usefulness of the proposed test.

Factor-Mimicking Portfolios for Climate Risk

We propose and implement a procedure to optimally hedge climate change risk. First, we construct climate risk indices through textual analysis of newspapers. Second, we present a new approach to compute factor-mimicking portfolios to build climate risk hedge portfolios. The new mimicking portfolio approach is much more efficient than traditional sorting or maximum correlation approaches by taking into account new methodologies of estimating large-dimensional covariance matrices in short samples. In an extensive empirical out-of-sample performance test, we demonstrate the superior all-around performance delivering markedly higher and statistically significant alphas and betas with the climate risk indices.

An Improved DCC Model Based on Large-Dimensional Covariance Matrices Estimation and Its Applications

The covariance matrix estimation plays an important role in portfolio optimization and risk management. It is well-known that portfolio is essentially a convex quadratic programming problem, which is also a special case of symmetric cone optimization. Accurate covariance matrix estimation will lead to more reasonable asset weight allocation. However, some existing methods do not consider the influence of time-varying factor on the covariance matrix estimations. To remedy this, in this article, we propose an improved dynamic conditional correlation model (DCC) by using nonconvex optimization model under smoothly clipped absolute deviation and hard-threshold penalty functions. We first construct a nonconvex optimization model to obtain the optimal covariance matrix estimation, and then we use this covariance matrix estimation to replace the unconditional covariance matrix in the DCC model. The result shows that the loss of the proposed estimator is smaller than other variants of the DCC model in numerical experiments. Finally, we apply our proposed model to the classic Markowitz portfolio. The results show that the improved dynamic conditional correlation model performs better than the current DCC models.

REGULARIZED ESTIMATION OF DYNAMIC PANEL MODELS

In a dynamic panel data model, the number of moment conditions increases rapidly with the time dimension, resulting in a large dimensional covariance matrix of the instruments. As a consequence, the generalized method of moments (GMM) estimator exhibits a large bias in small samples, especially when the autoregressive parameter is close to unity. To address this issue, we propose a regularized version of the one-step GMM estimator using three regularization schemes based on three different ways of inverting the covariance matrix of the instruments. Under double asymptotics, we show that our regularized estimators are consistent and asymptotically normal. These regularization schemes involve a tuning or regularization parameter which needs to be chosen. We derive a data-driven selection of this regularization parameter based on an approximation of the higher-order mean square error and show its optimality. As an empirical application, we estimate a model of income dynamics.

Minority oversampling for imbalanced time series classification

Many vital real-world applications involve time-series data with skewed distribution. Compared to traditional imbalanced learning problems, the classification of imbalanced time-series data is more challenging due to the high dimensionality and high inter-variable correlation. This paper proposes a structure-preserving Oversampling method to resolve the High-dimensional Imbalanced Time-series classification (OHIT). OHIT leverages a density-ratio-based shared nearest neighbor clustering algorithm to capture the modes of minority class in high-dimensional space. It for each mode applies the shrinkage technique of large-dimensional covariance matrix to obtain an accurate and reliable covariance structure. The structure-preserving synthetic samples are eventually generated based on the multivariate Gaussian distribution with the estimated covariance matrix. In addition, to further promote the performance of classifying imbalanced time-series data, we integrate OHIT into boosting framework to obtain a new ensemble algorithm OHITBoost. Extensive experiments on several publicly available time-series datasets (including unimodal and multimodal) demonstrate the effectiveness of OHIT and OHITBoost in terms of F1, G-mean, and AUC.

Cleaning large-dimensional covariance matrices for correlated samples.

We elucidate the problem of estimating large-dimensional covariance matrices in the presence of correlations between samples. To this end, we generalize the Marčenko-Pastur equationand the Ledoit-Péché shrinkage estimator using methods of random matrix theory and free probability. We develop an efficient algorithm that implements the corresponding analytic formulas based on the Ledoit-Wolf kernel estimation technique. We also provide an associated open-source Python library, called shrinkage, with a user-friendly API to assist in practical tasks of estimation of large covariance matrices. We present an example of its usage for synthetic data generated according to exponentially decaying autocorrelations.

Limiting Spectral Distribution of Large-Dimensional Sample Covariance Matrices Generated by the Periodic Autoregressive Model

The explicit representation for the limiting spectral moments of sample covariance matrices generated by the periodic autoregressive model (PAR) is established. We propose to use the moment-constrained maximum entropy method to estimate the spectral density function. The experiments show that the maximum entropy spectral density function curve obtained based on the fourth-order limiting spectral moment can match histograms of the eigenvalues of the covariance matrices very well.

How can an economic scenario generation model cope with abrupt changes in financial markets?

PurposeIt is quite possible that financial institutions including life insurance companies would encounter turbulent situations such as the COVID-19 pandemic before policies mature. Constructing models that can generate scenarios for major assets to cover abrupt changes in financial markets is thus essential for the financial institution's risk management.Design/methodology/approachThe key issues in such modeling include how to manage the large number of risk factors involved, how to model the dynamics of chosen or derived factors and how to incorporate relations among these factors. The authors propose the orthogonal ARMA–GARCH (autoregressive moving-average–generalized autoregressive conditional heteroskedasticity) approach to tackle these issues. The constructed economic scenario generation (ESG) models pass the backtests covering the period from the beginning of 2018 to the end of May 2020, which includes the turbulent situations caused by COVID-19.FindingsThe backtesting covering the turbulent period of COVID-19, along with fan charts and comparisons on simulated and historical statistics, validates our approach.Originality/valueThis paper is the first one that attempts to generate complex long-term economic scenarios for a large-scale portfolio from its large dimensional covariance matrix estimated by the orthogonal ARMA–GARCH model.

Analytical nonlinear shrinkage of large-dimensional covariance matrices

This paper establishes the first analytical formula for nonlinear shrinkage estimation of large-dimensional covariance matrices. We achieve this by identifying and mathematically exploiting a deep connection between nonlinear shrinkage and nonparametric estimation of the Hilbert transform of the sample spectral density. Previous nonlinear shrinkage methods were of numerical nature: QuEST requires numerical inversion of a complex equation from random matrix theory whereas NERCOME is based on a sample-splitting scheme. The new analytical method is more elegant and also has more potential to accommodate future variations or extensions. Immediate benefits are (i) that it is typically 1000 times faster with basically the same accuracy as QuEST and (ii) that it accommodates covariance matrices of dimension up to 10,000 and more. The difficult case where the matrix dimension exceeds the sample size is also covered.

Improved Shrinkage Estimator of Large-Dimensional Covariance Matrix under the Complex Gaussian Distribution

Estimating the covariance matrix of a random vector is essential and challenging in large dimension and small sample size scenarios. The purpose of this paper is to produce an outperformed large-dimensional covariance matrix estimator in the complex domain via the linear shrinkage regularization. Firstly, we develop a necessary moment property of the complex Wishart distribution. Secondly, by minimizing the mean squared error between the real covariance matrix and its shrinkage estimator, we obtain the optimal shrinkage intensity in a closed form for the spherical target matrix under the complex Gaussian distribution. Thirdly, we propose a newly available shrinkage estimator by unbiasedly estimating the unknown scalars involved in the optimal shrinkage intensity. Both the numerical simulations and an example application to array signal processing reveal that the proposed covariance matrix estimator performs well in large dimension and small sample size scenarios.

Improvement of sphericity test for large-dimensional covariance matrix

This paper discusses the problem of testing the sphericity of population covariance matrix when the sample size n and the dimensionality p both tend to infinity with p / n → c ∈ (0,1). A new test statistic is presented by utilizing an inequality, and asymptotic properties of the proposed statistic are derived for generally distributed population under the null hypothesis. Numerical simulations demonstrate that the proposed statistic has a significant improvement on test powers under the alternative hypothesis by comparing with some congeneric statistics.

Nonparametric confidence intervals for conditional quantiles with large-dimensional covariates

The first part of the paper is dedicated to the construction of a nonparametric confidence interval for a conditional quantile with a level depending on the sample size. When this level tends to 0 or 1 as the sample size increases, the conditional quantile is said to be extreme and is located in the tail of the conditional distribution. The proposed confidence interval is constructed by approximating the distribution of the order statistics selected with a nearest neighbor approach by a Beta distribution. We show that its coverage probability converges to the preselected probability and its accuracy is illustrated on a simulation study. When the dimension of the covariate increases, the coverage probability of the confidence interval can be very different from the nominal probability. This is a well known consequence of the data sparsity especially in the tail of the distribution. In a second part, a dimension reduction procedure is proposed in order to select more appropriate nearest neighbors in the right tail of the distribution and in turn to obtain a better coverage probability for extreme conditional quantiles. This procedure is based on the Tail Conditional Independence assumption introduced in (Gardes, Extremes, pp. 57--95, 18(3), 2018).

Regression analysis for multivariate process data of counts using convolved Gaussian processes

Research on Poisson regression analysis for process data of counts has developed rapidly in the last decade. One of the difficult problems in the multivariate case is how to construct a cross-correlation structure whilst, at the same time, making sure that the covariance matrix is positive definite. To address this issue, we propose to use convolved Gaussian processes (CGP) in this paper. This approach provides a semi-parametric model and offers a natural framework for modelling the common mean structure and covariance structure simultaneously. The CGP enables the model to define a different covariance structure for each component of the response variables. This flexibility ensures the model can cope with data coming from different sources or having different structures, and thus to provide accurate estimation and prediction. In addition, the model is able to accommodate large-dimensional covariates. The definition of the model, the inference and the implementation, as well as its asymptotic properties, are discussed. Comprehensive numerical examples with both simulation studies and real data are presented.

A Cholesky-based estimation for large-dimensional covariance matrices

ABSTRACTThis paper develops a new method to estimate a large-dimensional covariance matrix when the variables have no natural ordering among themselves. The modified Cholesky decomposition technique is used to provide a set of estimates of the covariance matrix under multiple orderings of variables. The proposed estimator is in the form of a linear combination of these available estimates and the identity matrix. It is positive definite and applicable in large dimensions. The merits of the proposed estimator are demonstrated through the numerical study and a real data example by comparison with several existing methods.

Factor Models for Portfolio Selection in Large Dimensions: The Good, the Better and the Ugly

Abstract This paper injects factor structure into the estimation of time-varying, large-dimensional covariance matrices of stock returns. Existing factor models struggle to model the covariance matrix of residuals in the presence of time-varying conditional heteroskedasticity in large universes. Conversely, rotation-equivariant estimators of large-dimensional time-varying covariance matrices forsake directional information embedded in market-wide risk factors. We introduce a new covariance matrix estimator that blends factor structure with time-varying conditional heteroskedasticity of residuals in large dimensions up to 1000 stocks. It displays superior all-around performance on historical data against a variety of state-of-the-art competitors, including static factor models, exogenous factor models, sparsity-based models, and structure-free dynamic models. This new estimator can be used to deliver more efficient portfolio selection and detection of anomalies in the cross-section of stock returns.

Semiparametric Estimation and Inference of Variance Function with Large Dimensional Covariates

Semiparametric Estimation and Inference of Variance Function with Large Dimensional Covariates

A homogeneity test of large dimensional covariance matrices under non-normality

A test statistic for homogeneity of two or more covariance matrices is presented when the distributions may be non-normal and the dimension may exceed the sample size. Using the Frobenius norm of t ...

Factor Models for Portfolio Selection in Large Dimensions: The Good, the Better and the Ugly

This paper injects factor structure into the estimation of time-varying, large-dimensional covariance matrices of stock returns. Existing factor models struggle to model the covariance matrix of residuals in the presence of time-varying conditional heteroskedasticity in large universes. Conversely, rotation-equivariant estimators of large-dimensional time-varying covariance matrices forsake directional information embedded in market-wide risk factors. We introduce a new covariance matrix estimator that blends factor structure with time-varying conditional heteroskedasticity of residuals in large dimensions up to 1000 stocks. It displays superior all-around performance on historical data against a variety of state-of-the-art competitors, including static factor models, exogenous factor models, sparsity-based models, and structure-free dynamic models. This new estimator can be used to deliver more efficient portfolio selection and detection of anomalies in the cross-section of stock returns.

A robust and efficient estimation and variable selection method for partially linear models with large-dimensional covariates

In this paper, a new robust and efficient estimation approach based on local modal regression is proposed for partially linear models with large-dimensional covariates. We show that the resulting estimators for both parametric and nonparametric components are more efficient in the presence of outliers or heavy-tail error distribution, and as asymptotically efficient as the corresponding least squares estimators when there are no outliers and the error distribution is normal. We also establish the asymptotic properties of proposed estimators when the covariate dimension diverges at the rate of $$o\left( {\sqrt{n} } \right) \mathrm{{ }}$$ . To achieve sparsity and enhance interpretability, we develop a variable selection procedure based on SCAD penalty to select significant parametric covariates and show that the method enjoys the oracle property under mild regularity conditions. Moreover, we propose a practical modified MEM algorithm for the proposed procedures. Some Monte Carlo simulations and a real data are conducted to illustrate the finite sample performance of the proposed estimators. Finally, based on the idea of sure independence screening procedure proposed by Fan and Lv (J R Stat Soc 70:849–911, 2008), a robust two-step approach is introduced to deal with ultra-high dimensional data.

Estimation of Large Covariance Matrices by Shrinking to Structured Target in Normal and Non-Normal Distributions

This paper addresses the estimation of large-dimensional covariance matrices under both normal and nonnormal distributions. The shrinkage estimators are constructed by convexly combining the sample covariance matrix and a structured target matrix. The optimal oracle shrinkage intensity is obtained analytically for any prespecified target in a class of matrices which includes various structured matrices such as banding, thresholding, diagonal, and block diagonal matrices. After deriving the unbiased and consistent estimates of some quantities in the oracle intensity involving unknown population covariance matrix, two classes of available optimal intensities are proposed under normality and nonnormality, respectively, by plug-in technique. For the target matrix with unknown parameter such as bandwidth in banded target, an analytic estimate of unknown parameter is provided. Both the numerical simulations and applications to signal processing and discriminant analysis show the comparable performance of the proposed estimators for large-dimensional data.