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.     

Bivariate inverse Weibull distribution

ABSTRACT

Recently it is observed that the inverse Weibull (IW) distribution can be used quite effectively to analyse lifetime data in one dimension. The main aim of this paper is to define a bivariate inverse Weibull (BIW) distribution so that the marginals have IW distributions. It is observed that the joint probability density function and the joint cumulative distribution function can be expressed in compact forms. Several properties of this distribution such as marginals, conditional distributions and product moments have been discussed. We obtained the maximum likelihood estimates for the unknown parameters of this distribution and their approximate variance– covariance matrix. We perform some simulations to see the performances of the maximum likelihood estimators. One data set has been re-analysed and it is observed that the bivariate IW distribution provides a better fit than the bivariate exponential distribution.     

An extension of the multivariate Chebyshev's inequality to a random vector with a singular covariance matrix

ABSTRACT

We extend Chebyshev's inequality to a random vector with a singular covariance matrix. Then we consider the case of a multivariate normal distribution for this generalization.     

Bayesian Inference on Multivariate Normal Covariance and Precision Matrices in a Star-Shaped Model with Missing Data

In this article, we study Bayesian estimation for the covariance matrix Σ and the precision matrix Ω (the inverse of the covariance matrix) in the star-shaped model with missing data. Based on a Cholesky-type decomposition of the precision matrix Ω = Ψ′Ψ, where Ψ is a lower triangular matrix with positive diagonal elements, we develop the Jeffreys prior and a reference prior for Ψ. We then introduce a class of priors for Ψ, which includes the invariant Haar measures, Jeffreys prior, and reference prior. The posterior properties are discussed and the closed-form expressions for Bayesian estimators for the covariance matrix Σ and the precision matrix Ω are derived under the Stein loss, entropy loss, and symmetric loss. Some simulation results are given for illustration.     

Investigating the sensitivity of Gaussian processes to the choice of their correlation function and prior specifications

A Gaussian process (GP) can be thought of as an infinite collection of random variables with the property that any subset, say of dimension n, of these variables have a multivariate normal distribution of dimension n, mean vector β and covariance matrix Σ [O'Hagan, A., 1994, Kendall's Advanced Theory of Statistics, Vol. 2B, Bayesian Inference (John Wiley & Sons, Inc.)]. The elements of the covariance matrix are routinely specified through the multiplication of a common variance by a correlation function. It is important to use a correlation function that provides a valid covariance matrix (positive definite). Further, it is well known that the smoothness of a GP is directly related to the specification of its correlation function. Also, from a Bayesian point of view, a prior distribution must be assigned to the unknowns of the model. Therefore, when using a GP to model a phenomenon, the researcher faces two challenges: the need of specifying a correlation function and a prior distribution for its parameters. In the literature there are many classes of correlation functions which provide a valid covariance structure. Also, there are many suggestions of prior distributions to be used for the parameters involved in these functions. We aim to investigate how sensitive the GPs are to the (sometimes arbitrary) choices of their correlation functions. For this, we have simulated 25 sets of data each of size 64 over the square [0, 5]×[0, 5] with a specific correlation function and fixed values of the GP's parameters. We then fit different correlation structures to these data, with different prior specifications and check the performance of the adjusted models using different model comparison criteria.     

On the Distribution of Matrix Quadratic Forms

A characterization of the distribution of the multivariate quadratic form given by X A X′, where X is a p × n normally distributed matrix and A is an n × n symmetric real matrix, is presented. We show that the distribution of the quadratic form is the same as the distribution of a weighted sum of non central Wishart distributed matrices. This is applied to derive the distribution of the sample covariance between the rows of X when the expectation is the same for every column and is estimated with the regular mean.     

ASYMPTOTIC DISTRIBUTION OF THE LARGEST EIGENVALUE

ABSTRACT

This paper studies the asymptotic distribution of the largest eigenvalue of the sample covariance matrix. The multivariate distribution for the population is assumed to be elliptical with finite kurtosis 3κ. An expression as an expectation is obtained for the distribution function of the largest eigenvalue regardless of the multiplicity, m, of the population's largest eigenvalue. The asymptotic distribution function and density function are evaluated numerically for m = 2,3,4,5. The bootstrap of the average of the m largest eigenvalues is shown to be consistent for any underlying distribution with finite fourth-order cumulants.     

Bootstrapping analogs of the one way MANOVA test

Abstract

Analogs of the classical one way MANOVA model have recently been suggested that do not assume that population covariance matrices are equal or that the error vector distribution is known. These tests are based on the sample mean and sample covariance matrix corresponding to each of the p populations. We show how to extend these tests using other measures of location such as the trimmed mean or coordinatewise median. These new bootstrap tests can have some outlier resistance, and can perform better than the tests based on the sample mean if the error vector distribution is heavy tailed.     

Analyzing multiple vector autoregressions through matrix-variate normal distribution with two covariance matrices

Abstract

This article proposes a new approach to analyze multiple vector autoregressive (VAR) models that render us a newly constructed matrix autoregressive (MtAR) model based on a matrix-variate normal distribution with two covariance matrices. The MtAR is a generalization of VAR models where the two covariance matrices allow the extension of MtAR to a structural MtAR analysis. The proposed MtAR can also incorporate different lag orders across VAR systems that provide more flexibility to the model. The estimation results from a simulation study and an empirical study on macroeconomic application show favorable performance of our proposed models and method.     

A permutation-based Bayesian approach for inverse covariance estimation

Abstract

Covariance estimation and selection for multivariate datasets in a high-dimensional regime is a fundamental problem in modern statistics. Gaussian graphical models are a popular class of models used for this purpose. Current Bayesian methods for inverse covariance matrix estimation under Gaussian graphical models require the underlying graph and hence the ordering of variables to be known. However, in practice, such information on the true underlying model is often unavailable. We therefore propose a novel permutation-based Bayesian approach to tackle the unknown variable ordering issue. In particular, we utilize multiple maximum a posteriori estimates under the DAG-Wishart prior for each permutation, and subsequently construct the final estimate of the inverse covariance matrix. The proposed estimator has smaller variability and yields order-invariant property. We establish posterior convergence rates under mild assumptions and illustrate that our method outperforms existing approaches in estimating the inverse covariance matrices via simulation studies.     

Incremental modelling for compositional data streams

ABSTRACT

Incremental modelling of data streams is of great practical importance, as shown by its applications in advertising and financial data analysis. We propose two incremental covariance matrix decomposition methods for a compositional data type. The first method, exact incremental covariance decomposition of compositional data (C-EICD), gives an exact decomposition result. The second method, covariance-free incremental covariance decomposition of compositional data (C-CICD), is an approximate algorithm that can efficiently compute high-dimensional cases. Based on these two methods, many frequently used compositional statistical models can be incrementally calculated. We take multiple linear regression and principle component analysis as examples to illustrate the utility of the proposed methods via extensive simulation studies.     

Power-transformed linear regression on quantile residual life for censored competing risks data

ABSTRACT

This paper proposes a power-transformed linear quantile regression model for the residual lifetime of competing risks data. The proposed model can describe the association between any quantile of a time-to-event distribution among survivors beyond a specific time point and the covariates. Under covariate-dependent censoring, we develop an estimation procedure with two steps, including an unbiased monotone estimating equation for regression parameters and cumulative sum processes for the Box–Cox transformation parameter. The asymptotic properties of the estimators are also derived. We employ an efficient bootstrap method for the estimation of the variance–covariance matrix. The finite-sample performance of the proposed approaches are evaluated through simulation studies and a real example.     

Efficient Estimation of the Seemingly Unrelated Regression Cointegration Model and Testing for Purchasing Power Parity

Abstract

This paper studies the efficient estimation of seemingly unrelated linear models with integrated regressors and stationary errors. We consider two cases. The first one has no common regressor among the equations. In this case, we show that by adding leads and lags of the first differences of the regressors and estimating this augmented dynamic regression model by generalized least squares using the long-run covariance matrix, we obtain an efficient estimator of the cointegrating vector that has a limiting mixed normal distribution. In the second case we consider, there is a common regressor to all equations, and we discuss efficient minimum distance estimation in this context. Simulation results suggests that our new estimator compares favorably with others already proposed in the literature. We apply these new estimators to the testing of the proportionality and symmetry conditions implied by purchasing power parity (PPP) among the G-7 countries. The tests based on the efficient estimates easily reject the joint hypotheses of proportionality and symmetry for all countries with either the United States or Germany as numeraire. Based on individual tests, our results suggest that Canada and Germany are the most likely countries for which the proportionality condition holds, and that Italy and Japan for the symmetry condition relative to the United States.     

Characterization of discrete scale invariant Markov sequences

ABSTRACT

Some special sampling of discrete scale invariant (DSI) processes are presented to provide a multi-dimensional self-similar process in correspondence. By imposing Markov property we show that the covariance functions of such Markov DSI sequences are characterized by variance, and covariance of adjacent samples in the first scale interval. We also provide a theoretical method for estimating spectral density matrix of corresponding multi-dimensional self-similar Markov process. Some examples such as simple Brownian motion (sBm) with drift and scale invariant autoregressive model are presented and these properties are investigated. We present two new method to estimate Hurst parameter of DSI processes and apply them to some sBm and also to the SP500 indices for some period which has DSI property. We compare our estimates with the maximum-likelihood and rescaled range (R/S) method which are applied to the corresponding multi-dimensional self-similar processes.     

Double sampling |S| control chart with variable sample size and variable sampling interval

This study presents a control chart for monitoring shifts in the covariance matrix of a multivariate normally distributed process. This chart combines the double sampling, variable sample size and variable sampling interval features, and is called the DSVSSI |S| chart. A Markov chain approach is developed to design the DSVSSI |S| chart, by minimizing the average time to signal (ATS), for a specified shift size in the covariance matrix. The DSVSSI |S| chart has a better ATS performance compared to other existing charts. An example is given to illustrate the operation of the DSVSSI |S| chart.     

Singular Gaussian graphical models: Structure learning

ABSTRACT

The goal of this article is to introduce singular Gaussian graphical models and their conditional independence properties. In fact, we extend the concept of Gaussian Markov Random Field to the case of a multivariate normally distributed vector with a singular covariance matrix. We construct, then, the associated graph’s structure from the covariance matrix’s pseudo-inverse on the basis of a characterization of the pairwise conditional independence. The proposed approach can also be used when the covariance matrix is ill-conditioned, through projecting data on a smaller subspace. In this case, our method ensures numerical stability and consistency of the constructed graph and significantly reduces the inference problem’s complexity. These aspects are illustrated using numerical experiments.     

A heteroscedastic structural errors-in-variables model with equation error

It is not uncommon with astrophysical and epidemiological data sets that the variances of the observations are accessible from an analytical treatment of the data collection process. Moreover, in a regression model, heteroscedastic measurement errors and equation errors are common situations when modelling such data. This article deals with the limiting distribution of the maximum-likelihood and method-of-moments estimators for the line parameters of the regression model. We use the delta method to achieve it, making it possible to build joint confidence regions and hypothesis testing. This technique produces closed expressions for the asymptotic covariance matrix of those estimators. In the moment approach we do not assign any distribution for the unobservable covariate while with the maximum-likelihood approach, we assume a normal distribution. We also conduct simulation studies of rejection rates for Wald-type statistics in order to verify the test size and power. Practical applications are reported for a data set produced by the Chandra observatory and also from the WHO MONICA Project on cardiovascular disease.     

Cross-validated mixed-datatype bandwidth selection for nonparametric cumulative distribution/survivor functions

ABSTRACT

We propose a computationally efficient data-driven least square cross-validation method to optimally select smoothing parameters for the nonparametric estimation of cumulative distribution/survivor functions. We allow for general multivariate covariates that can be continuous, discrete/ordered categorical or a mix of either. We provide asymptotic analysis, examine finite-sample properties through Monte Carlo simulation, and consider an illustration involving nonparametric copula modeling. We also demonstrate how the approach can also be used to construct a smooth Kolmogorov–Smirnov test that has a slightly better power profile than its nonsmooth counterpart.     

Influence diagnostics in constrained general linear models

ABSTRACT

Constrained general linear models (CGLMs) have wide applications in practice. Similar to other data analysis, the identification of influential observations that may be potential outliers is an important step beyond in the CGLMs. We develop multiple case-deletion diagnostics for detecting influential observations in the CGLMs. The diagnostics are functions of basic building blocks: studentized residuals, error contrast matrix, and the inverse of the response variable covariance matrix. The basic building blocks are computed only once from the complete data analysis and provide information on the influence of the data on different aspects of the model fit. Computational formulas are given which make the procedures feasible. An illustrative example with a real data set is also reported.     

Central Limit Theorem approximations for the number of runs in Markov-dependent binary sequences

We consider Markov-dependent binary sequences and study various types of success runs (overlapping, non-overlapping, exact, etc.) by examining additive functionals based on state visits and transitions in an appropriate Markov chain. We establish a multivariate Central Limit Theorem for the number of these types of runs and obtain its covariance matrix by means of the recurrent potential matrix of the Markov chain. Explicit expressions for the covariance matrix are given in the Bernoulli and a simple Markov-dependent case by expressing the recurrent potential matrix in terms of the stationary distribution and the mean transition times in the chain. We also obtain a multivariate Central Limit Theorem for the joint number of non-overlapping runs of various sizes and give its covariance matrix in explicit form for Markov dependent trials.