Forecasting high-dimensional realized volatility matrices using a factor model

Summary The primary purpose of this paper is to propose empirical measures of the structural differences between two communities of plants or animals composed of the same species. Structure is defined to consist of; 1) the species in the community, 2) the pattern of interactions as represented by the covariance or correlation matrix of successive observations on each species, and 3) the mean abundances of each species in each of the two communities. Statistical tests are proposed for testing whether the covariance matrices and the vectors of mean densities for each community are equal and empirical measures of the differences between the covariance matrices and mean vectors are proposed. Given unequal covariance or correlation matrices the factor analysis model is used to derive empirical measures of the degree to which each variable of the ecosystem is responsible for the observed defferences in the pattern of interactions in each community. These tests and measures were applied to data gathered by Hunter (1966) on the abundances of six species of Drosophila censused monthly over a period of approximately two and a half years in two adjacent, but different habitats near Bogota, colombia. The two covariance matrices were significantly different indicating different patterns of interactions in the two Drosophila communities.

High dimensional covariance matrices have attracted much attention of statisticians and econometricians during the past decades. Vast literature is devoted to the research in high dimensional covariance matrices. However, most of them are for constant covariance matrices. In many applications, constant covariance matrices are not appropriate, e.g. in portfolio allocation, dynamic covariance matrices would make much more sense. Simply assuming each entry of a covariance matrix is a function of time to introduce a dynamic structure would not work. In this paper, we are going to introduce a class of high dimensional dynamic covariance matrices in which a kind of additive structure is embedded. We will show the proposed high dimensional dynamic covariance matrices have many advantages in applications. An estimation procedure is also proposed to estimate the proposed high dimensional dynamic covariance matrices. Asymptotic properties are built to justify the proposed estimation procedure. Intensive simulation studies show the proposed estimation procedure works very well when sample size is finite. Finally, we apply the proposed high dimensional dynamic covariance matrices, together with the proposed estimation procedure, to portfolio allocation. The results look very interesting.

Estimating time-varying conditional covariance matrices of financial returns play important role in portfolio analysis, risk management, and financial econometrics research. The availability of high-frequency financial data can provide an additional data source for dynamic covariance modeling. In this paper, we propose to use the information of asset return vector and realized covariance measures simultaneously to develop a new conditional covariance matrix model. We derive the stationary condition of the new model. We use the normal and Wishart distributions to construct the quasi-log-likelihood function. We also consider the variance targeting (VT) method, which plugs in the weighted average of the sample covariance matrix of returns and the sample mean of realized covariance measure for the unconditional covariance matrix, in order to maximize the quasi-log-likelihood function. We show the consistency and asymptotic normality of the quasi-maximum likelihood (QML) and VT estimators. We investigate the finite sample property of these estimators via Monte Carlo experiments. The empirical example for the bivariate data of the Nikkei 225 index and its futures indicates that the first-step VT estimation could have non-negligible effects on the standard errors of the second-step VT estimates.

Radar data suffers from the curse of dimensionality, and with the deluge of data, due to advances in sensors, and technology, it has becomes increasingly challenging to interpret, visualize and process data. In this paper, we assume that the dimensions of the data matrix is large. That is, we assume that the number of variables as well as the number of samples is large, however, their ratio is a finite constant. We further assume a structured covariance matrix, comprised of low rank matrices added to a scaled identity matrix. The behavior of their principal components (PCs) and eigenvalues of the estimated covariance matrix is analyzed using well known tools in random matrix theory (RMT) and almost sure limits are studied. Simulations reveal that the analysis is valid for many distributions, not necessarily Gaussian, provided these samples are statistically independent. The theory is applied to the covariance matrices generated from the KASSPER database.

Estimating large covariance and precision (inverse covariance) matrices has become increasingly important in high dimensional statistics because of its wide applications. The estimation problem is challenging not only theoretically due to the constraint of its positive definiteness, but also computationally because of the curse of dimensionality. Many types of estimators have been proposed such as thresholding under the sparsity assumption of the target matrix, banding and tapering the sample covariance matrix. However, these estimators are not always guaranteed to be positive-definite, especially, for finite samples, and the sparsity assumption is rather restrictive. We propose a novel two-stage adaptive method based on the Cholesky decomposition of a general covariance matrix. By banding the precision matrix in the first stage and adapting the estimates to the second stage estimation, we develop a computationally efficient and statistically accurate method for estimating high dimensional precision matrices. We demonstrate the finite-sample performance of the proposed method by simulations from autoregressive, moving average, and long-range dependent processes. We illustrate its wide applicability by analyzing financial data such S&P 500 index and IBM stock returns, and electric power consumption of individual households. The theoretical properties of the proposed method are also investigated within a large class of covariance matrices.

High Dimensional Covariance Matrix Estimation in Approximate Factor Models

Estimation of the high-dimensional banded covariance matrix is widely used in multivariate statistical analysis. To ensure the validity of estimation, we aim to test the hypothesis that the covariance matrix is banded with a certain bandwidth under the high-dimensional framework. Though several testing methods have been proposed in the literature, the existing tests are only powerful for some alternatives with certain sparsity levels, whereas they may not be powerful for alternatives with other sparsity structures. The goal of this paper is to propose a new test for the bandedness of high-dimensional covariance matrix, which is powerful for alternatives with various sparsity levels. The proposed new test also be used for testing the banded structure of covariance matrices of error vectors in high-dimensional factor models. Based on these statistics, a consistent bandwidth estimator is also introduced for a banded high dimensional covariance matrix. Extensive simulation studies and an application to a prostate cancer dataset from protein mass spectroscopy are conducted for evaluating the effectiveness of the proposed adaptive tests blue and bandwidth estimator for the banded covariance matrix.

Realized Covariance Estimation in Dynamic Portfolio Optimization

A nonparametric eigenvalue-regularized integrated covariance matrix estimator for asset return data

Upcoming and existing large-scale surveys of galaxies require accurate theoretical predictions of the dark matter clustering statistics for thousands of mock galaxy catalogs. We demonstrate that this goal can be achieve with our new Parallel Particle-Mesh (PM) Nbody code (PPM-GLAM) at a very low computational cost. We run about 15,000 simulations with ~2 billion particles that provide ~1% accuracy of the dark matter power spectra P(k) for wave-numbers up to k~ 1h/Mpc. Using this large data-set we study the power spectrum covariance matrix, the stepping stone for producing mock catalogs. In contrast to many previous analytical and numerical results, we find that the covariance matrix normalised to the power spectrum C(k,k')/P(k)P(k') has a complex structure of non-diagonal components. It has an upturn at small k, followed by a minimum at k=0.1-0.2h/Mpc. It also has a maximum at k=0.5-0.6h/Mpc. The normalised covariance matrix strongly evolves with redshift: C(k,k')~delta(t)^alpha P(k)P(k'), where delta is the linear growth factor and alpha ~ 1-1.25, which indicates that the covariance matrix depends on cosmological parameters. We also show that waves longer than 1Gpc have very little impact on the power spectrum and covariance matrix. This significantly reduces the computational costs and complexity of theoretical predictions: relatively small volume ~ (1Gpc)^3 simulations capture the necessary properties of dark matter clustering statistics. All the power spectra obtained from many thousands of our simulations are publicly available.

This paper develops an improved covariance matrix estimator in the mean-variance optimisation setting. Well-known problems with the sample covariance matrix are that it is singular when the number of observations is less than the number of assets, and can be nearly singular when the number of observations exceeds the number of assets. Therefore, using the sample covariance matrix as an input in mean-variance optimisation can result in unreasonable optimal portfolios and badly biased estimates of Sharpe ratios. We address this problem by imposing structure on the estimated covariance matrix by putting constraints on the Sharpe ratio, asset return variances, and the variance of the global minimum variance portfolio. We show that the constrained maximum likelihood estimator (CMLE) performs better than the sample covariance matrix. Moreover, when the shrinkage approach is applied to the CMLE and single index covariance matrix, it performs better than the shrinkage estimator of Ledoit and Wolf (2004).

One of the major challenges in multivariate analysis is the estimation of population covariance matrix from the sample covariance matrix (SCM). Most recent covariance matrix estimators use either shrinkage transformations or asymptotic results from Random Matrix Theory (RMT). Both of these techniques try to achieve a similar goal which is to remove noisy correlations and add structure to SCM to overcome the bias-variance trade-off. Both methods have their respective pros and cons. In this paper, we propose an improved estimator which exploits the advantages of these techniques by taking optimally weighted convex combination of covariance matrices estimated by shrinkage transformation and a filter based on RMT. It is a generalized estimator which can adapt to changing sampling noise conditions by performing hyperparameter optimization. Using data from six of the world's biggest stock exchanges, we show that the proposed estimator outperforms the existing estimators in minimizing the out-of-sample risk of the portfolio and hence predicts population statistics more precisely. The proposed estimator can be useful in a wide range of machine learning and signal processing applications. © 1994-2012 IEEE.

Support vector regression is a promising method for time-series prediction, as it has good generalisability and an overall stable behaviour. Recent studies have shown that it can describe the dynamic characteristics of financial processes and make more accurate forecasts than other machine learning techniques. The first main contribution of this paper is to propose a methodology for dynamic modelling and forecasting covariance matrices based on support vector regression using the Cholesky decomposition. The procedure is applied to range-based covariance matrices of returns, which are estimated on the basis of low and high prices. Such prices are most often available with closing prices for many financial series and contain more information about volatility and relationships between returns. The methodology guarantees the positive definiteness of the forecasted covariance matrices and is flexible, as it can be applied to different dependence patterns. The second contribution of the paper is to show with an example of the exchange rates from the forex market that the covariance matrix forecasts calculated using the proposed approach are more accurate than the forecasts from the benchmark dynamic conditional correlation model. The advantage of the suggested procedure is higher during turbulent periods, i.e., when forecasting is the most difficult and accurate forecasts matter most.

Summary Ensemble optimization (referred to throughout the remainder of the paper as EnOpt) is a rapidly emerging method for reservoir-model-based production optimization. EnOpt uses an ensemble of controls to approximate the gradient of the objective function with respect to the controls. Current implementations of EnOpt use a Gaussian ensemble of control perturbations with a constant covariance matrix, and thus a constant perturbation size, during the entire optimization process. The covariance-matrix-adaptation evolutionary strategy is a gradient-free optimization method developed in the “machine learning” community, which also uses an ensemble of controls, but with a covariance matrix that is continually updated during the optimization process. It was shown to be an efficient method for several difficult but small-dimensional optimization problems and was recently applied in the petroleum industry for well location and production optimization. In this study, we investigate the scope to improve the computational efficiency of EnOpt through the use of covariance-matrix adaptation (referred to throughout the remainder of the paper as CMA-EnOpt). The resulting method is applied to the waterflooding optimization of a small multilayer test model and a modified version of the Brugge benchmark model. The controls used are inflow-control-valve settings at predefined time intervals for injectors and producers with undiscounted net present value as the objective function. We compare EnOpt and CMA-EnOpt starting from identical covariance matrices. For the small model, we achieve only slightly higher (0.7 to 1.8%) objective-function values and modest speedups with CMA-EnOpt compared with EnOpt. Significantly higher objective-function values (10%) are obtained for the modified Brugge model. The possibility to adapt the covariance matrix, and thus the perturbation size, during the optimization allows for the use of relatively large perturbations initially, for fast exploration of the control space, and small perturbations later, for more-precise gradients near the optimum. Moreover, the results demonstrate that a major benefit of CMA-EnOpt is its robustness with respect to the initial choice of the covariance matrix. A poor choice of the initial matrix can be detrimental to EnOpt, whereas the CMA-EnOpt performance is near-independent of the initial choice and produces higher objective-function values at no additional computational cost.

We consider the estimation of a Gaussian random vector x observed through a linear transformation H and corrupted by additive Gaussian noise with a known covariance matrix, where the covariance matrix of x is known to lie in a given region of uncertainty that is described using bounds on the eigenvalues and on the elements of the covariance matrix. Recently, two criteria for minimax estimation called difference regret (DR) and ratio regret (RR) were proposed and their closed form solutions were presented assuming that the eigenvalues of the covariance matrix of x are known to lie in a given region of uncertainty, and assuming that the matrices H T Cw -1 H and C x are jointly diagonalizable, where C w and C x denote the covariance matrices of the additive noise and of x respectively. In this work we present a new criterion for the minimax estimation problem which we call the generalized difference regret (GDR), and derive a new minimax estimator which is based on the GDR criterion where the region of uncertainty is defined not only using upper and lower bounds on the eigenvalues of the parameter's covariance matrix, but also using upper and lower bounds on the individual elements of the covariance matrix itself. Furthermore, the new estimator does not require the assumption of joint diagonalizability and it can be obtained efficiently using semidefinite programming. We also show that when the joint diagonalizability assumption holds and when there are only eigenvalue uncertainties, then the new estimator is identical to the difference regret estimator. The experimental results show that we can obtain improved mean squared error (MSE) results compared to the MMSE, DR, and RR estimators.