On strict positive definiteness of product and product-sum covariance models

Although positive definiteness is a sufficient condition for a function to be a covariance, the stronger strict positive definiteness is important for many applications, especially in spatial statistics, since it ensures that the kriging equations have a unique solution. In particular, spatial-temporal prediction has received a lot of attention, hence strictly positive definite spatial-temporal covariance models (or equivalently strictly conditionally negative definite variogram models) are needed.In this paper the necessary and sufficient condition for the product and the product-sum space-time covariance models to be strictly positive definite (or the variogram function to be strictly conditionally negative definite) is given. In addition it is shown that an example appeared in the recent literature which purports to show that product-sum covariance functions may be only semi-definite is itself invalid. Strict positive definiteness of the sum of products model is also discussed.     

Space–time correlation analysis: a comparative study

Space–time correlation modelling is one of the crucial steps of traditional structural analysis, since space–time models are used for prediction purposes. A comparative study among some classes of space–time covariance functions is proposed. The relevance of choosing a suitable model by taking into account the characteristic behaviour of the models is proved by using a space–time data set of ozone daily averages and the flexibility of the product-sum model is also highlighted through simulated data sets.     

Bayesian tests on components of the compound symmetry covariance matrix

Complex dependency structures are often conditionally modeled, where random effects parameters are used to specify the natural heterogeneity in the population. When interest is focused on the dependency structure, inferences can be made from a complex covariance matrix using a marginal modeling approach. In this marginal modeling framework, testing covariance parameters is not a boundary problem. Bayesian tests on covariance parameter(s) of the compound symmetry structure are proposed assuming multivariate normally distributed observations. Innovative proper prior distributions are introduced for the covariance components such that the positive definiteness of the (compound symmetry) covariance matrix is ensured. Furthermore, it is shown that the proposed priors on the covariance parameters lead to a balanced Bayes factor, in case of testing an inequality constrained hypothesis. As an illustration, the proposed Bayes factor is used for testing (non-)invariant intra-class correlations across different group types (public and Catholic schools), using the 1982 High School and Beyond survey data.     

Positive and negative non-separability for space–time covariance models

Separable spatio-temporal covariance models, defined as the product of purely spatial and purely temporal covariance functions, are often used in practice, but frequently they only represent a convenient assumption. On the other hand, non-separable models are receiving a lot of attention, since they are more flexible to handle empirical covariances showed up in applications. Different forms of non-separability for space–time covariance functions have been recently defined in the literature. In this paper, the notion of positive and negative non-separability is further formalized in order to distinguish between pointwise and uniform non-separability. Various well-known non-separable space–time stationary covariance models are analyzed and classified by using the new definition of non-separability. In particular, wide classes of non-separable spatio-temporal covariance functions, able to capture positive and negative non-separability, are proposed and some examples of these classes are given. General results concerning the non-separability of spatial–temporal covariance functions obtained by a linear combination of spatial–temporal covariance functions and some stability properties are also presented. These results can be helpful to generate as well as to select appropriate covariance models for describing space–time data.     

Modeling Conditional Covariances With Economic Information Instruments

We propose a new model for conditional covariances based on predetermined idiosyncratic shocks as well as macroeconomic and own information instruments. The specification ensures positive definiteness by construction, is unique within the class of linear functions for our covariance decomposition, and yields a simple yet rich model of covariances. We introduce a property, invariance to variate order, that assures estimation is not impacted by a simple reordering of the variates in the system. Simulation results using realized covariances show smaller mean absolute errors (MAE) and root mean square errors (RMSE) for every element of the covariance matrix relative to a comparably specified BEKK model with own information instruments. We also find a smaller mean absolute percentage error (MAPE) and root mean square percentage error (RMSPE) for the entire covariance matrix. Supplementary materials for practitioners as well as all Matlab code used in the article are available online.     

Positive-definite modification of a covariance matrix by minimizing the matrix $$\ell_{\infty}$$ ℓ ∞ norm with applications to portfolio optimization

AStA Advances in Statistical Analysis - The covariance matrix, which should be estimated from the data, plays an important role in many multivariate procedures, and its positive definiteness...     

New HEAVY Models for Fat-Tailed Realized Covariances and Returns

ABSTRACT

We develop a new score-driven model for the joint dynamics of fat-tailed realized covariance matrix observations and daily returns. The score dynamics for the unobserved true covariance matrix are robust to outliers and incidental large observations in both types of data by assuming a matrix-F distribution for the realized covariance measures and a multivariate Student's t distribution for the daily returns. The filter for the unknown covariance matrix has a computationally efficient matrix formulation, which proves beneficial for estimation and simulation purposes. We formulate parameter restrictions for stationarity and positive definiteness. Our simulation study shows that the new model is able to deal with high-dimensional settings (50 or more) and captures unobserved volatility dynamics even if the model is misspecified. We provide an empirical application to daily equity returns and realized covariance matrices up to 30 dimensions. The model statistically and economically outperforms competing multivariate volatility models out-of-sample. Supplementary materials for this article are available online.     

Simulation of stationary Gaussian vector fields

In earlier work we described a circulant embedding approach for simulating scalar-valued stationary Gaussian random fields on a finite rectangular grid, with the covariance function prescribed. Here, we explain how the circulant embedding approach can be used to simulate Gaussian vector fields. As in the scalar case, the simulation procedure is theoretically exact if a certain non-negativity condition is satisfied. In the vector setting, this exactness condition takes the form of a nonnegative definiteness condition on a certain set of Hermitian matrices. The main computational tool used is the Fast Fourier Transform. Consequently, when implemented appropriately, the procedure is highly efficient, in terms of both CPU time and storage.     

The Generalized Conditional Autoregressive Wishart Model for Multivariate Realized Volatility

It is well known that in finance variances and covariances of asset returns move together over time. Recently, much interest has been aroused by an approach involving the use of the realized covariance (RCOV) matrix constructed from high-frequency returns as the ex-post realization of the covariance matrix of low-frequency returns. For the analysis of dynamics of RCOV matrices, we propose the generalized conditional autoregressive Wishart (GCAW) model. Both the noncentrality matrix and scale matrix of the Wishart distribution are driven by the lagged values of RCOV matrices, and represent two different sources of dynamics, respectively. The GCAW is a generalization of the existing models, and accounts for symmetry and positive definiteness of RCOV matrices without imposing any parametric restriction. Some important properties such as conditional moments, unconditional moments, and stationarity are discussed. Empirical examples including sequences of daily RCOV matrices from the New York Stock Exchange illustrate that our model outperforms the existing models in terms of model fitting and forecasting.     

Superiority comparisons of homogeneous linear estimators

In the paper homogeneous linear estimators of the parameter vector of the general linear model are compared in terms of their MSE matrices. A necessary and sufficient condition for the difference of two MSE matrices to be positive definite is obtained and its practical existence discussed. The non-negative definiteness of the difference also receives attention, and conditions for this case are discussed. The absence of any conditions of the above type is taken into consideration as well.     

Asymmetric matrix-valued covariances for multivariate random fields on spheres

ABSTRACT

Matrix-valued covariance functions are crucial to geostatistical modelling of multivariate spatial data. The classical assumption of symmetry of a multivariate covariance function is overly restrictive and has been considered as unrealistic for most of the real data applications. Despite of that, the literature on asymmetric covariance functions has been very sparse. In particular, there is some work related to asymmetric covariances on Euclidean spaces, depending on the Euclidean distance. However, for data collected over large portions of planet Earth, the most natural spatial domain is a sphere, with the corresponding geodesic distance being the natural metric. In this work, we propose a strategy based on spatial rotations to generate asymmetric covariances for multivariate random fields on the d-dimensional unit sphere. We illustrate through simulations as well as real data analysis that our proposal allows to achieve improvements in the predictive performance in comparison to the symmetric counterpart.     

Multivariate Positive Dependence on the Simplex

The Liouville and Generalized Liouville families have been proposed as parametric models for data constrained to the simplex. These families have generated practical interest owing primarily to inadequacies, such as a completely negative covariance structure, that are inherent in the better-known Dirichlet class. Although there is some numerical evidence suggesting that the Liouville and Generalized Liouville families can produce completely positive and mixed covariance structures, no general paradigms have been developed. Research toward this end might naturally be focused on the many classical "positive dependence" concepts available in the literature, all of which imply a nonnegative covariance structure. However, in this article it is shown that no strictly positive distribution on the simplex can possess any of these classical dependence properties. The same result holds for Liouville and generalized Liouville distributions even if the condition of strict positivity is relaxed.     

A Class of Convolution‐Based Models for Spatio‐Temporal Processes with Non‐Separable Covariance Structure

Abstract. In this article, we propose a new parametric family of models for real‐valued spatio‐temporal stochastic processes S ( x , t ) and show how low‐rank approximations can be used to overcome the computational problems that arise in fitting the proposed class of models to large datasets. Separable covariance models, in which the spatio‐temporal covariance function of S ( x , t ) factorizes into a product of purely spatial and purely temporal functions, are often used as a convenient working assumption but are too inflexible to cover the range of covariance structures encountered in applications. We define positive and negative non‐separability and show that in our proposed family we can capture positive, zero and negative non‐separability by varying the value of a single parameter.     

A Note on Sufficient Conditions for Valid Unmodified t Testing in Correlation Analysis with Autocorrelated and Heteroscedastic Sample Data

Traditionally, sphericity (i.e., independence and homoscedasticity for raw data) is put forward as the condition to be satisfied by the variance–covariance matrix of at least one of the two observation vectors analyzed for correlation, for the unmodified t test of significance to be valid under the Gaussian and constant population mean assumptions. In this article, the author proves that the sphericity condition is too strong and a weaker (i.e., more general) sufficient condition for valid unmodified t testing in correlation analysis is circularity (i.e., independence and homoscedasticity after linear transformation by orthonormal contrasts), to be satisfied by the variance–covariance matrix of one of the two observation vectors. Two other conditions (i.e., compound symmetry for one of the two observation vectors; absence of correlation between the components of one observation vector, combined with a particular pattern of joint heteroscedasticity in the two observation vectors) are also considered and discussed. When both observation vectors possess the same variance–covariance matrix up to a positive multiplicative constant, the circularity condition is shown to be necessary and sufficient. “Observation vectors” may designate partial realizations of temporal or spatial stochastic processes as well as profile vectors of repeated measures. From the proof, it follows that an effective sample size appropriately defined can measure the discrepancy from the more general sufficient condition for valid unmodified t testing in correlation analysis with autocorrelated and heteroscedastic sample data. The proof is complemented by a simulation study. Finally, the differences between the role of the circularity condition in the correlation analysis and its role in the repeated measures ANOVA (i.e., where it was first introduced) are scrutinized, and the link between the circular variance–covariance structure and the centering of observations with respect to the sample mean is emphasized.     

A multivariate volatility vine copula model

This article proposes a dynamic framework for modeling and forecasting of realized covariance matrices using vine copulas to allow for more flexible dependencies between assets. Our model automatically guarantees positive definiteness of the forecast through the use of a Cholesky decomposition of the realized covariance matrix. We explicitly account for long-memory behavior by using fractionally integrated autoregressive moving average (ARFIMA) and heterogeneous autoregressive (HAR) models for the individual elements of the decomposition. Furthermore, our model incorporates non-Gaussian innovations and GARCH effects, accounting for volatility clustering and unconditional kurtosis. The dependence structure between assets is studied using vine copula constructions, which allow for nonlinearity and asymmetry without suffering from an inflexible tail behavior or symmetry restrictions as in conventional multivariate models. Further, the copulas have a direct impact on the point forecasts of the realized covariances matrices, due to being computed as a nonlinear transformation of the forecasts for the Cholesky matrix. Beside studying in-sample properties, we assess the usefulness of our method in a one-day-ahead forecasting framework, comparing recent types of models for the realized covariance matrix based on a model confidence set approach. Additionally, we find that in Value-at-Risk (VaR) forecasting, vine models require less capital requirements due to smoother and more accurate forecasts.     

Unconstrained parametrizations for variance-covariance matrices

The estimation of variance-covariance matrices through optimization of an objective function, such as a log-likelihood function, is usually a difficult numerical problem. Since the estimates should be positive semi-definite matrices, we must use constrained optimization, or employ a parametrization that enforces this condition. We describe here five different parametrizations for variance-covariance matrices that ensure positive definiteness, thus leaving the estimation problem unconstrained. We compare the parametrizations based on their computational efficiency and statistical interpretability. The results described here are particularly useful in maximum likelihood and restricted maximum likelihood estimation in linear and non-linear mixed-effects models, but are also applicable to other areas of statistics.     

Alternative expectation formulas for real-valued random vectors

Abstract

When the elements of a random vector take any real values, formulas of product moments are obtained for continuous and discrete random variables using distribution/survival functions. The random product can be that of strictly increasing functions of random variables. For continuous cases, the derivation based on iterated integrals is employed. It is shown that Hoeffding’s covariance lemma is algebraically equal to a special case of this result. For discrete cases, the elements of a random vector can be non-integers and/or unequally spaced. A discrete version of Hoeffding’s covariance lemma is derived for real-valued random variables.     

Blur-generated non-separable space–time models

Statistical space–time modelling has traditionally been concerned with separable covariance functions, meaning that the covariance function is a product of a purely temporal function and a purely spatial function. We draw attention to a physical dispersion model which could model phenomena such as the spread of an air pollutant. We show that this model has a non-separable covariance function. The model is well suited to a wide range of realistic problems which will be poorly fitted by separable models. The model operates successively in time: the spatial field at time t +1 is obtained by 'blurring' the field at time t and adding a spatial random field. The model is first introduced at discrete time steps, and the limit is taken as the length of the time steps goes to 0. This gives a consistent continuous model with parameters that are interpretable in continuous space and independent of sampling intervals. Under certain conditions the blurring must be a Gaussian smoothing kernel. We also show that the model is generated by a stochastic differential equation which has been studied by several researchers previously.     

PARTIAL CORRELATION AND CONDITIONAL CORRELATION AS MEASURES OF CONDITIONAL INDEPENDENCE

This paper investigates the roles of partial correlation and conditional correlation as measures of the conditional independence of two random variables. It first establishes a sufficient condition for the coincidence of the partial correlation with the conditional correlation. The condition is satisfied not only for multivariate normal but also for elliptical, multivariate hypergeometric, multivariate negative hypergeometric, multinomial and Dirichlet distributions. Such families of distributions are characterized by a semigroup property as a parametric family of distributions. A necessary and sufficient condition for the coincidence of the partial covariance with the conditional covariance is also derived. However, a known family of multivariate distributions which satisfies this condition cannot be found, except for the multivariate normal. The paper also shows that conditional independence has no close ties with zero partial correlation except in the case of the multivariate normal distribution; it has rather close ties to the zero conditional correlation. It shows that the equivalence between zero conditional covariance and conditional independence for normal variables is retained by any monotone transformation of each variable. The results suggest that care must be taken when using such correlations as measures of conditional independence unless the joint distribution is known to be normal. Otherwise a new concept of conditional independence may need to be introduced in place of conditional independence through zero conditional correlation or other statistics.     

Parameter expansion for sampling a correlation matrix: an efficient GPX-RPMH algorithm

Sampling the correlation matrix (R) plays an important role in statistical inference for correlated models. There are two main constraints on a correlation matrix: positive definiteness and fixed diagonal elements. These constraints make sampling R difficult. In this paper, an efficient generalized parameter expanded re-parametrization and Metropolis-Hastings (GPX-RPMH) algorithm for sampling a correlation matrix is proposed. Drawing all components of R simultaneously from its full conditional distribution is realized by first drawing a covariance matrix from the derived parameter expanded candidate density (PXCD), and then translating it back to a correlation matrix and accepting it according to a Metropolis-Hastings (M-H) acceptance rate. The mixing rate in the M-H step can be adjusted through a class of tuning parameters embedded in the generalized candidate prior (GCP), which is chosen for R to derive the PXCD. This algorithm is illustrated using multivariate regression (MVR) models and a simulation study shows that the performance of the GPX-RPMH algorithm is more efficient than that of other methods.