Covariance Matrix Estimation Research Articles

The stock market reaction to COVID-19 vaccination in ASEAN.

Previous studies have shown that the confirmed cases drive investor sentiment, reflecting the stock's return. Based on this, the vaccination growth is also expected to drive the investor's sentiment, which can be reflected in the return of the stock market in ASEAN. Therefore, this study explores the vaccination impact on stock returns in ASEAN countries. This study contributes to the gap of taking the COVID-19 vaccination impact to the stock return into account by using the panel regression model with HC and Driscoll and Kraay robust covariance matrix estimator, which addresses the cross-dependency and heterogeneity problems. This study is one of the early studies of the topic, especially in ASEAN. The panel regression model with HC and Driscoll and Kraay robust covariance matrix estimator uses three variables: the daily stocks return, vaccine growth, and cases growth. It is a balanced panel data that includes six countries and 117 daily series data, making 702 observations used in the study. The results show conflicting results where daily vaccination growth negatively affects the stock return. This problem can arise for several reasons, such as the uncertainty in the financial market and cross-dependency and heterogeneity detected in the model. We can see that the investors still have a negative sentiment because COVID-19 has resulted in uncertainty on the financial market in ASEAN. This gives us practical implications that the ASEAN country members' government needs to push vaccination policy more aggressively.

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.

A first order continuous timeVARwith random coefficients

This article considers a first order continuous time vector autoregression with random coefficients. We discuss some difficulties that arise when the exact discrete analogue is used for estimating the continuous time parameters and provide an estimation method based on an approximate discrete model. Some expressions for the estimator of the drift parameter matrix, for its approximated bias and for the covariance matrix of the parameter estimates are derived. The finite sample performance of the proposed method is studied by a Monte Carlo experiment. We also illustrate the advantages of our model in an application on the expectations theory of the term structure of interest rates. Results show that the performance of the proposed methodology is good, and allowing for time variation on coefficients results in large reductions in the root mean square error of the parameter estimates.

A Best Linear Empirical Bayes Method for High-Dimensional Covariance Matrix Estimation

Covariance matrix estimation plays a significant role in both in the theory and practice of portfolio analysis and risk management. This paper deals with the available data prior to developing a factor model to enhance covariance matrix estimation. Our work has two main outcomes. First, for a general linear model with unknown prior parameters, a class of best linear empirical Bayes estimators is established through two kinds of architectures to improve the estimation accuracy by utilizing additional data prior. The theoretical results indicate two key points: the proposed estimators are equivalent to the linear minimum mean-square error estimator when complete or sufficient partial data prior are provided; and the proposed estimators perform better than the optimal weighted least squares method, which ignores the data prior in each situation. Second, the proposed estimators are used for calculating a high-dimensional covariance matrix through factor models. The numerical example and the simulation results verify the effectiveness of our methods.

Tyler’s and Maronna’s M-estimators: Non-asymptotic concentration results

Tyler’s and Maronna’s M-estimators, as well as their regularized variants, are popular robust methods to estimate the scatter or covariance matrix of a multivariate distribution. In this work, we study the non-asymptotic behavior of these estimators, for data sampled from a distribution that satisfies one of the following properties: (1) independent sub-Gaussian entries, up to a linear transformation; (2) log-concave distributions; (3) distributions satisfying a convex concentration property. Our main contribution is the derivation of tight non-asymptotic concentration bounds of these M-estimators around a suitably scaled version of the data sample covariance matrix. Prior to our work, non-asymptotic bounds were derived only for Elliptical and Gaussian distributions. Our proof uses a variety of tools from non asymptotic random matrix theory and high dimensional geometry. Finally, we illustrate the utility of our results on two examples of practical interest: sparse covariance and sparse precision matrix estimation.

A data-driven statistical-stochastic surrogate modeling strategy for complex nonlinear non-stationary dynamics

We propose a statistical-stochastic surrogate modeling approach to predict the response of the mean and variance statistics under various initial conditions and external forcing perturbations. The proposed modeling framework extends the purely statistical modeling approach that is practically limited to the homogeneous statistical regime for high-dimensional state variables. The new closure system allows one to overcome several practical issues that emerge in the non-homogeneous statistical regimes. First, the proposed ensemble modeling that couples the mean statistics and stochastic fluctuations naturally produces positive-definite covariance matrix estimation, which is a challenging issue that hampers the purely statistical modeling approaches. Second, the proposed closure model, which embeds a non-Markovian neural-network model for the unresolved fluxes such that the variance of the dynamics is consistent, overcomes the inherent instability of the stochastic fluctuation dynamics. Effectively, the proposed framework extends the classical stochastic parametric modeling paradigm for the unresolved dynamics to a semi-parametric parameterization with a residual Long-Short-Term-Memory neural network architecture. Third, based on empirical information metric, we provide an efficient and effective training procedure by fitting a loss function that measures the differences between response statistics. Supporting numerical examples are provided with the Lorenz-96 model, a system of ODEs that admits the characteristic of chaotic dynamics with both homogeneous and inhomogeneous statistical regimes. In the latter case, we will see the effectiveness of the statistical prediction even though the resolved Fourier modes corresponding to the leading mean energy and variance spectra do not coincide.

Unbiased estimation of the OLS covariance matrix when the errors are clustered

When data are clustered, common practice has become to do OLS and use an estimator of the covariance matrix of the OLS estimator that comes close to unbiasedness. In this paper, we derive an estimator that is unbiased when the random-effects model holds. We do the same for two more general structures. We study the usefulness of these estimators against others by simulation, the size of the t-test being the criterion. Our findings suggest that the choice of estimator hardly matters when the regressor has the same distribution over the clusters. But when the regressor is a cluster-specific treatment variable, the choice does matter and the unbiased estimator we propose for the random-effects model shows excellent performance, even when the clusters are highly unbalanced.

An automated robust algorithm for clustering multivariate data

Clustering analysis is widely used in various applications, such as marketing, biology, medical science, finance, data mining, image processing, data analysis and pattern recognition. For instance, clustering can be used to: characterize the customer groups based on their purchasing patterns by discovering the distinct groups in the customer base; derive plant and animal taxonomies, categorize genes with similar functionalities and gain insight into structures inherent to populations; identify the cancer cells; and detect credit card fraud. The k-means, Hierarchical and self-organizing (Kohonen) map are widely used clustering algorithms. The practice demonstrated that these clustering algorithms have some significant limitations and drawbacks. This manuscript gives an automated robust algorithm for clustering multivariate data without prior information about the number of clusters. Robust estimate of location and covariance matrix are used to define Mahalanobis distance and corresponding radius of clustering algorithm. The algorithm is designed in a way that it controls both masking and swamping effects. It automatically divides a given data set into a number of clusters. Some properties pertaining to the algorithm are demonstrated which helps in finding clusters that accommodates observations with large deviation. A method to avoid the use of a fixed cutoff for determining outlier is discussed. The performance of the proposed algorithm is compared with the existing clustering algorithms and robust multiple outlier detection methods.

Probabilistic Forecast Reconciliation under the Gaussian Framework

Forecast reconciliation of multivariate time series maps a set of incoherent forecasts into coherent forecasts to satisfy a given set of linear constraints. Available methods in the literature either follow a projection matrix-based approach or an empirical copula-based reordering approach to revise the incoherent future sample paths to obtain reconciled probabilistic forecasts. The projection matrices are estimated either by optimizing a scoring rule such as energy or variogram score or simply using a projection matrix derived for point forecast reconciliation. This article proves that (a) if the incoherent predictive distribution is jointly Gaussian, then MinT (minimum trace) minimizes the logarithmic scoring rule for the hierarchy; and (b) the logarithmic score of MinT for each marginal predictive density is smaller than that of OLS (ordinary least squares). We illustrate these theoretical results using a set of simulation studies and the Australian domestic tourism dataset. The estimation of MinT needs to estimate the covariance matrix of the base forecast errors. We have evaluated the performance using the sample covariance matrix and shrinkage estimator. It was observed that the theoretical properties noted above are greatly impacted by the covariance matrix used and highlighted the importance of estimating it reliably, especially with high dimensional data.

A novel TomoSAR imaging method with few observations based on nested array

AbstractSynthetic aperture radar tomography (TomoSAR) baseline optimization method is capable of reducing system complexity and improving the temporal coherence of data, which has become important research in the field of TomoSAR. A nested TomoSAR technique is proposed, which introduces the nested array into TomoSAR as the baseline configuration. This technique obtains uniform and continuous difference co‐array through the nested array to increase the degrees of freedom of the system and expands the virtual aperture along the elevation direction. In order to make full use of the difference co‐array, covariance matrix of the echo needs to be obtained. Therefore, a TomoSAR sparse reconstruction algorithm based on the nested array is proposed, which uses adaptive covariance matrix estimation to improve the estimation performance in complex scenes. The effectiveness of the proposed method through simulated and real data experiments is demonstrated. Compared with traditional TomoSAR and coprime TomoSAR, the imaging results of this proposed method have better anti‐noise performance and retain more image information.

Large volatility matrix analysis using global and national factor models

Several large volatility matrix inference procedures have been developed, based on the latent factor model. They often assumed that there are a few of common factors, which can account for volatility dynamics. However, several studies have demonstrated the presence of local factors. In particular, when analyzing the global stock market, we often observe that nation-specific factors explain their own country’s volatility dynamics. To account for this, we propose the Double Principal Orthogonal complEment Thresholding (Double-POET) method, based on multi-level factor models, and also establish its asymptotic properties. Furthermore, we demonstrate the drawback of using the regular principal orthogonal component thresholding (POET) when the local factor structure exists. We also describe the blessing of dimensionality using Double-POET for local covariance matrix estimation. Finally, we investigate the performance of the Double-POET estimator in an out-of-sample portfolio allocation study using international stocks from 20 financial markets.

The empirical performance of option implied volatility surface-driven optimal portfolios

We apply a two-step strategy to forecast the dynamics of the volatility surface implicit in option prices to all American-style options written on the stocks that have entered the Dow Jones Industrial Average Index between 2004 and 2016. We explore whether the implied volatilities extracted through the two-step approach help improve the out-of-sample performance of minimum-variance portfolios. We find that, by using option-implied volatilities in estimating the covariance matrix, the ex-post volatility of the minimum-variance portfolio is lower when compared with the equal-weighted portfolio and a minimum-variance portfolio simply derived from the historical, sample covariance matrix estimator. Moreover, over most of our 13-year sample, the realized Sharpe, Sortino and information ratios increase when the sample covariance matrix estimator is replaced with its implied counterpart. However, the benefits of using option-implied information are countered by an increase in portfolio turnover that may imply higher (implicit) transaction costs. We also apply shrinkage methods to both the sample covariance estimator and the implied covariance estimator and note that they often lead to significant improvements in portfolio performance.

Rotationally invariant estimators on portfolio optimization to unveil financial risk’s states

Rotationally Invariant Estimators (RIE) are a new family of covariance matrix estimators based on random matrix theory and free probability. The family RIE has been proposed to improve the performance of an investment portfolio in the Markowitz model’s framework. Here, we apply state-of-the-art RIE techniques to improve the estimation of financial states via the correlation matrix. The Synthesized Clustering (SYNCLUS) and a dynamic programming algorithm for optimal one-dimensional clustering were employed to that aim. We found that the RIE estimations of the minimum portfolio risk increase the Active Information Storage (AIS) in the American and European markets. AIS’s local dynamic also mimics financial states’ behavior when estimating under the one-dimensional clustering algorithm. Our results suggest that in times of financial turbulence, RIE estimates can be of great advantage in minimizing risk exposure.

Bias compensation Kalman filter for 3D angle-only measurements target traking

Some sensors can only measure the angle of arrival of the signal, but the real position of the signal source is usually interesting. To obtain an accurate target position estimate, this paper studies the target tracking problem based on angle-only measurements in 3-Dimensional (3D) space. Firstly, the target position estimate is obtained by using weighted least-squares (WLS) algorithm, and the covariance matrix of the position estimate is calculated. Although this algorithm has low computational complexity, there is a bias problem due to the transformation of angle measurement equation from nonlinear to linear. To counter the bias, a bias compensation step is considered by estimating the correlation between the measurement matrix and noise vector. The classical Kalman filter (KF) is applied to the target position estimates before and after compensation. In numerical results, the performance advantages of the proposed bias compensation KF over the uncompensated algorithm and extended Kalman filter are verified. In addition, the impacts of angle measurement accuracy and sensor self-positioning accuracy are analyzed, indicating that the bias compensation Kalman filter can achieve high positioning performance with low computational cost.

Regularized fingerprinting in detection and attribution of climate change with weight matrix optimizing the efficiency in scaling factor estimation

Detection and attribution analyses play a central role in establishing the causal effect of human activities on global warming. The most commonly used method in such analyses, optimal fingerprinting, is a multiple regression where each covariate has a measurement error whose covariance matrix is the same as that of the regression error up to a known scale. Inferences about the regression coefficients are critical not only for making statements about detection and attribution but also for quantifying the uncertainty in important outcomes derived from detection and attribution analyses. When there are no errors-in-variables (EIV), the optimal weight matrix in estimating the regression coefficients is the precision matrix of the regression error. This matrix, however, is never known and has to be estimated from climate model simulations with regularization. The consequence is that the optimal fingerprinting method is not optimal, as believed in practice. We construct a weight matrix by inverting a nonlinear shrinkage estimate of the error covariance matrix that minimizes loss functions directly targeting the uncertainty of the resulting regression coefficient estimator. The resulting estimator of the regression coefficients is asymptotically optimal as the sample size of the climate model simulations and the matrix dimension go to infinity together with a limiting ratio. When EIVs are present, the estimator of the regression coefficients, based on the proposed weight matrix, is asymptotically more efficient than that based on the inverse of the existing linear shrinkage estimator of the error covariance matrix. The performance of the method is confirmed in finite sample simulation studies mimicking realistic situations in terms of the length of the confidence intervals and empirical coverage rates for the regression coefficients. In an application to detection and attribution analyses of the mean temperature at different spatial scales, the method yielded shorter confidence intervals which are important for such analyses in practice.

Scalable Bayesian High-dimensional Local Dependence Learning

In this work, we propose a scalable Bayesian procedure for learning the local dependence structure in a high-dimensional model where the variables possess a natural ordering. The ordering of variables can be indexed by time, the vicinities of spatial locations, and so on, with the natural assumption that variables far apart tend to have weak correlations. Applications of such models abound in a variety of fields such as finance, genome associations analysis and spatial modeling. We adopt a flexible framework under which each variable is dependent on its neighbors or predecessors, and the neighborhood size can vary for each variable. It is of great interest to reveal this local dependence structure by estimating the covariance or precision matrix while yielding a consistent estimate of the varying neighborhood size for each variable. The existing literature on banded covariance matrix estimation, which assumes a fixed bandwidth cannot be adapted for this general setup. We employ the modified Cholesky decomposition for the precision matrix and design a flexible prior for this model through appropriate priors on the neighborhood sizes and Cholesky factors. The posterior contraction rates of the Cholesky factor are derived which are nearly or exactly minimax optimal, and our procedure leads to consistent estimates of the neighborhood size for all the variables. Another appealing feature of our procedure is its scalability to models with large numbers of variables due to efficient posterior inference without resorting to MCMC algorithms. Numerical comparisons are carried out with competitive methods, and applications are considered for some real datasets.

Kernel Estimation of Bivariate Time-Varying Coefficient Model for Longitudinal Data with Terminal Event

We propose a nonparametric bivariate time-varying coefficient model for longitudinal measurements with the occurrence of a terminal event that is subject to right censoring. The time-varying coefficients capture the longitudinal trajectories of covariate effects along with both the followup time and the residual lifetime. The proposed model extends the parametric conditional approach given terminal event time in recent literature, and thus avoids potential model misspecification. We consider a kernel smoothing method for estimating regression coefficients in our model and use cross-validation for bandwidth selection, applying undersmoothing in the final analysis to eliminate the asymptotic bias of the kernel estimator. We show that the kernel estimates follow a finite-dimensional normal distribution asymptotically under mild regularity conditions, and provide an easily computed sandwich covariance matrix estimator. We conduct extensive simulations that show desirable performance of the proposed approach, and apply the method to analyzing the medical cost data for patients with end-stage renal disease. Supplementary materials for this article are available online.

High-Dimensional Covariance Estimation via Constrained Lq-Type Regularization

High-dimensional covariance matrix estimation is one of the fundamental and important problems in multivariate analysis and has a wide range of applications in many fields. In practice, it is common that a covariance matrix is composed of a low-rank matrix and a sparse matrix. In this paper we estimate the covariance matrix by solving a constrained Lq-type regularized optimization problem. We establish the first-order optimality conditions for this problem by using proximal mapping and the subspace method. The proposed stationary point degenerates to the first-order stationary points of the unconstrained Lq regularized sparse or low-rank optimization problems. A smoothing alternating updating method is proposed to find an estimator for the covariance matrix. We establish the convergence of the proposed calculation method. The numerical simulation results show the effectiveness of the proposed approach for high-dimensional covariance estimation.

A Regularized K-means Autoregressive Shrinkage Method to Portfolio Optimization

Estimation is always an issue of portfolio optimization. To address this issue this paper started from addressing global minimum variance (GMV) portfolio, which gives the weights as the function of estimated covariance matrix of the asset net returns. The inverse operation of the matrix always leads to ill-conditioned matrix due to the noise in the return series. Therefore, regularized sample covariance matrix is often used to address this problem. In this work, a regularized autoregressive shrinkage method for covariance matrix estimation with the historical data is developed, which regards the future covariance matrix as the function of the historical covariance matrix. K-means clustering algorithm is also added to enhance the performance of the regularized autoregressive model. The results demonstrate that the performance of GMV portfolio can be enhanced with regularization under autoregressive shrinkage model.

Total Variation-Based Distributed Kalman Filtering for Resiliency Against Byzantines

This article proposes a distributed Kalman filter (DKF) with enhanced robustness against Byzantine adversaries. A Byzantine agent is a legitimate network agent that, unlike an honest agent, manipulates information before sharing it with neighbors to impair the overall system performance. In contrast to the literature, the DKF is modeled as a distributed optimization problem where resiliency against Byzantine agents is accomplished by employing a total variation (TV) penalty term. We utilize a distributed subgradient algorithm to compute the state estimate and error covariance matrix updates of the DKF. Additionally, we prove that the proposed suboptimal solution converges to a neighborhood of the optimal centralized solution of the KF with a bounded radius when Byzantine agents are present. Numerical simulations corroborate the theoretical findings and demonstrate the robustness of the proposed DKF against Byzantine attacks.