On the estimation and application of max-stable processes

The theory of max-stable processes generalizes traditional univariate and multivariate extreme value theory by allowing for processes indexed by a time or space variable. We consider a particular class of max-stable processes, known as M4 processes, that are particularly well adapted to modeling the extreme behavior of multiple time series. We develop procedures for determining the order of an M4 process and for estimating the parameters. To illustrate the methods, some examples are given for modeling jumps in returns in multivariate financial time series. We introduce a new measure to quantify and predict the extreme co-movements in price returns.     

Multivariate Stochastic Volatility Models with Correlated Errors

We develop a Bayesian approach for parsimoniously estimating the correlation structure of the errors in a multivariate stochastic volatility model. Since the number of parameters in the joint correlation matrix of the return and volatility errors is potentially very large, we impose a prior that allows the off-diagonal elements of the inverse of the correlation matrix to be identically zero. The model is estimated using a Markov chain simulation method that samples from the posterior distribution of the volatilities and parameters. We illustrate the approach using both simulated and real examples. In the real examples, the method is applied to equities at three levels of aggregation: returns for firms within the same industry, returns for different industries, and returns aggregated at the index level. We find pronounced correlation effects only at the highest level of aggregation.     

Multivariate Stochastic Volatility Models with Correlated Errors

We develop a Bayesian approach for parsimoniously estimating the correlation structure of the errors in a multivariate stochastic volatility model. Since the number of parameters in the joint correlation matrix of the return and volatility errors is potentially very large, we impose a prior that allows the off-diagonal elements of the inverse of the correlation matrix to be identically zero. The model is estimated using a Markov chain simulation method that samples from the posterior distribution of the volatilities and parameters. We illustrate the approach using both simulated and real examples. In the real examples, the method is applied to equities at three levels of aggregation: returns for firms within the same industry, returns for different industries, and returns aggregated at the index level. We find pronounced correlation effects only at the highest level of aggregation.     

HIGH-DIMENSIONAL PARAMETRIC MODELLING OF MULTIVARIATE EXTREME EVENTS

Multivariate extreme events are typically modelled using multivariate extreme value distributions. Unfortunately, there exists no finite parametrization for the class of multivariate extreme value distributions. One common approach is to model extreme events using some flexible parametric subclass. This approach has been limited to only two or three dimensions, primarily because suitably flexible high-dimensional parametric models have prohibitively complex density functions. We present an approach that allows a number of popular flexible models to be used in arbitrarily high dimensions. The approach easily handles missing and censored data, and can be employed when modelling componentwise maxima and multivariate threshold exceedances. The approach is based on a representation using conditionally independent marginal components, conditioning on positive stable random variables. We use Bayesian inference, where the conditioning variables are treated as auxiliary variables within Markov chain Monte Carlo simulations. We demonstrate these methods with an application to sea-levels, using data collected at 10 sites on the east coast of England.     

Solving Noisy ICA Using Multivariate Wavelet Denoising with an Application to Noisy Latent Variables Regression

A novel approach to solve the independent component analysis (ICA) model in the presence of noise is proposed. We use wavelets as natural denoising tools to solve the noisy ICA model. To do this, we use a multivariate wavelet denoising algorithm allowing spatial and temporal dependency. We propose also using a statistical approach, named nested design of experiments, to select the parameters such as wavelet family and thresholding type. This technique helps us to select more convenient combination of the parameters. This approach could be extended to many other problems in which one needs to choose parameters between many choices. The performance of the proposed method is illustrated on the simulated data and promising results are obtained. Also, the suggested method applied in latent variables regression in the presence of noise on real data. The good results confirm the ability of multivariate wavelet denoising to solving noisy ICA.     

Computation aspects of the parameter estimates of linear mixed effects model in multivariate repeated measures set-up

The number of parameters mushrooms in a linear mixed effects (LME) model in the case of multivariate repeated measures data. Computation of these parameters is a real problem with the increase in the number of response variables or with the increase in the number of time points. The problem becomes more intricate and involved with the addition of additional random effects. A multivariate analysis is not possible in a small sample setting. We propose a method to estimate these many parameters in bits and pieces from baby models, by taking a subset of response variables at a time, and finally using these bits and pieces at the end to get the parameter estimates for the mother model, with all variables taken together. Applying this method one can calculate the fixed effects, the best linear unbiased predictions (BLUPs) for the random effects in the model, and also the BLUPs at each time of observation for each response variable, to monitor the effectiveness of the treatment for each subject. The proposed method is illustrated with an example of multiple response variables measured over multiple time points arising from a clinical trial in osteoporosis.     

Statistical Inference for Curved Fibrous Objects in 3D – Based on Multiple Short Observations of Multivariate Autoregressive Processes

This paper deals with statistical inference on the parameters of a stochastic model, describing curved fibrous objects in three dimensions, that is based on multivariate autoregressive processes. The model is fitted to experimental data consisting of a large number of short independently sampled trajectories of multivariate autoregressive processes. We discuss relevant statistical properties (e.g. asymptotic behaviour as the number of trajectories tends to infinity) of the maximum likelihood (ML) estimators for such processes. Numerical studies are also performed to analyse some of the more intractable properties of the ML estimators. Finally the whole methodology, i.e., the fibre model and its statistical inference, is applied to appropriately describe the tracking of fibres in real materials.     

Maximum likelihood estimation of skew-t copulas with its applications to stock returns

The multivariate Student-t copula family is used in statistical finance and other areas when there is tail dependence in the data. It often is a good-fitting copula but can be improved on when there is tail asymmetry. Multivariate skew-t copula families can be considered when there is tail dependence and tail asymmetry, and we show how a fast numerical implementation for maximum likelihood estimation is possible. For the copula implicit in a multivariate skew-t distribution, the fast implementation makes use of (i) monotone interpolation of the univariate marginal quantile function and (ii) a re-parametrization of the correlation matrix. Our numerical approach is tested with simulated data with data-driven parameters. A real data example involves the daily returns of three stock indices: the Nikkei225, S&P500 and DAX. With both unfiltered returns and GARCH/EGARCH filtered returns, we compare the fits of the Azzalini–Capitanio skew-t, generalized hyperbolic skew-t, Student-t, skew-Normal and Normal copulas.     

Lattice-oriented percolation system applied to volatility behavior of stock market

In this paper, a discrete time series of stock price process is modeled by the two-dimensional lattice-oriented bond percolation system. Percolation theory, as one of statistical physics systems, has brought new understanding and techniques to a broad range of topics in nature and society. According to this financial model, we studied the statistical behaviors of the stock price from the model and the real stock prices by comparison. We also investigated the probability distributions, the long memory and the long-range correlations of price returns for the actual data and the simulative data. The empirical research exhibits that for proper parameters, the simulative data of the financial model can fit the real markets to a certain extent.     

Regression for doubly inflated multivariate Poisson distributions

Dependent multivariate count data occur in several research studies. These data can be modelled by a multivariate Poisson or Negative binomial distribution constructed using copulas. However, when some of the counts are inflated, that is, the number of observations in some cells are much larger than other cells, then the copula-based multivariate Poisson (or Negative binomial) distribution may not fit well and it is not an appropriate statistical model for the data. There is a need to modify or adjust the multivariate distribution to account for the inflated frequencies. In this article, we consider the situation where the frequencies of two cells are higher compared to the other cells and develop a doubly inflated multivariate Poisson distribution function using multivariate Gaussian copula. We also discuss procedures for regression on covariates for the doubly inflated multivariate count data. For illustrating the proposed methodologies, we present real data containing bivariate count observations with inflations in two cells. Several models and linear predictors with log link functions are considered, and we discuss maximum likelihood estimation to estimate unknown parameters of the models.     

Bayesian modeling of dynamic extreme values: extension of generalized extreme value distributions with latent stochastic processes

This paper develops Bayesian inference of extreme value models with a flexible time-dependent latent structure. The generalized extreme value distribution is utilized to incorporate state variables that follow an autoregressive moving average (ARMA) process with Gumbel-distributed innovations. The time-dependent extreme value distribution is combined with heavy-tailed error terms. An efficient Markov chain Monte Carlo algorithm is proposed using a state-space representation with a finite mixture of normal distributions to approximate the Gumbel distribution. The methodology is illustrated by simulated data and two different sets of real data. Monthly minima of daily returns of stock price index, and monthly maxima of hourly electricity demand are fit to the proposed model and used for model comparison. Estimation results show the usefulness of the proposed model and methodology, and provide evidence that the latent autoregressive process and heavy-tailed errors play an important role to describe the monthly series of minimum stock returns and maximum electricity demand.     

Initializing EM using the properties of its trajectories in Gaussian mixtures

A strategy is proposed to initialize the EM algorithm in the multivariate Gaussian mixture context. It consists in randomly drawing, with a low computational cost in many situations, initial mixture parameters in an appropriate space including all possible EM trajectories. This space is simply defined by two relations between the two first empirical moments and the mixture parameters satisfied by any EM iteration. An experimental study on simulated and real data sets clearly shows that this strategy outperforms classical methods, since it has the nice property to widely explore local maxima of the likelihood function.     

Cholesky-GARCH models with applications to finance

Instantaneous dependence among several asset returns is the main reason for the computational and statistical complexities in working with full multivariate GARCH models. Using the Cholesky decomposition of the covariance matrix of such returns, we introduce a broad class of multivariate models where univariate GARCH models are used for variances of individual assets and parsimonious models for the time-varying unit lower triangular matrices. This approach, while reducing the number of parameters and severity of the positive-definiteness constraint, has several advantages compared to the traditional orthogonal and related GARCH models. Its major drawback is the potential need for an a priori ordering or grouping of the stocks in a portfolio, which through a case study we show can be taken advantage of so far as reducing the forecast error of the volatilities and the dimension of the parameter space are concerned. Moreover, the Cholesky decomposition, unlike its competitors, decompose the normal likelihood function as a product of univariate normal likelihoods with independent parameters, resulting in fast estimation algorithms. Gaussian maximum likelihood methods of estimation of the parameters are developed. The methodology is implemented for a real financial dataset with seven assets, and its forecasting power is compared with other existing models.     

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.     

Variable selection for multivariate generalized linear models

Generalized linear models (GLMs) are widely studied to deal with complex response variables. For the analysis of categorical dependent variables with more than two response categories, multivariate GLMs are presented to build the relationship between this polytomous response and a set of regressors. Traditional variable selection approaches have been proposed for the multivariate GLM with a canonical link function when the number of parameters is fixed in the literature. However, in many model selection problems, the number of parameters may be large and grow with the sample size. In this paper, we present a new selection criterion to the model with a diverging number of parameters. Under suitable conditions, the criterion is shown to be model selection consistent. A simulation study and a real data analysis are conducted to support theoretical findings.     

Bayesian inference of multiple structural change models with asymmetric GARCH errors

Structural change in any time series is practically unavoidable, and thus correctly detecting breakpoints plays a pivotal role in statistical modelling. This research considers segmented autoregressive models with exogenous variables and asymmetric GARCH errors, GJR-GARCH and exponential-GARCH specifications, which utilize the leverage phenomenon to demonstrate asymmetry in response to positive and negative shocks. The proposed models incorporate skew Student-t distribution and prove the advantages of the fat-tailed skew Student-t distribution versus other distributions when structural changes appear in financial time series. We employ Bayesian Markov Chain Monte Carlo methods in order to make inferences about the locations of structural change points and model parameters and utilize deviance information criterion to determine the optimal number of breakpoints via a sequential approach. Our models can accurately detect the number and locations of structural change points in simulation studies. For real data analysis, we examine the impacts of daily gold returns and VIX on S&P 500 returns during 2007–2019. The proposed methods are able to integrate structural changes through the model parameters and to capture the variability of a financial market more efficiently.

     

Modeling with a large class of unimodal multivariate distributions

In this paper we introduce a new class of multivariate unimodal distributions, motivated by Khintchine's representation for unimodal densities on the real line. We start by introducing a new class of unimodal distributions which can then be naturally extended to higher dimensions, using the multivariate Gaussian copula. Under both univariate and multivariate settings, we provide MCMC algorithms to perform inference about the model parameters and predictive densities. The methodology is illustrated with univariate and bivariate examples, and with variables taken from a real data set.     

Finite mixtures of multivariate Poisson distributions with application

In the present paper we examine finite mixtures of multivariate Poisson distributions as an alternative class of models for multivariate count data. The proposed models allow for both overdispersion in the marginal distributions and negative correlation, while they are computationally tractable using standard ideas from finite mixture modelling. An EM type algorithm for maximum likelihood (ML) estimation of the parameters is developed. The identifiability of this class of mixtures is proved. Properties of ML estimators are derived. A real data application concerning model based clustering for multivariate count data related to different types of crime is presented to illustrate the practical potential of the proposed class of models.     

Cross-validation approximation in functional linear regression

Cross-validation has been widely used in the context of statistical linear models and multivariate data analysis. Recently, technological advancements give possibility of collecting new types of data that are in the form of curves. Statistical procedures for analysing these data, which are of infinite dimension, have been provided by functional data analysis. In functional linear regression, using statistical smoothing, estimation of slope and intercept parameters is generally based on functional principal components analysis (FPCA), that allows for finite-dimensional analysis of the problem. The estimators of the slope and intercept parameters in this context, proposed by Hall and Hosseini-Nasab [On properties of functional principal components analysis, J. R. Stat. Soc. Ser. B: Stat. Methodol. 68 (2006), pp. 109–126], are based on FPCA, and depend on a smoothing parameter that can be chosen by cross-validation. The cross-validation criterion, given there, is time-consuming and hard to compute. In this work, we approximate this cross-validation criterion by such another criterion so that we can turn to a multivariate data analysis tool in some sense. Then, we evaluate its performance numerically. We also treat a real dataset, consisting of two variables; temperature and the amount of precipitation, and estimate the regression coefficients for the former variable in a model predicting the latter one.     

Multivariate Tests of Mean-Variance Efficiency and Spanning With a Large Number of Assets and Time-Varying Covariances

We develop a finite-sample procedure to test the mean-variance efficiency and spanning hypotheses, without imposing any parametric assumptions on the distribution of model disturbances. In so doing, we provide an exact distribution-free method to test uniform linear restrictions in multivariate linear regression models. The framework allows for unknown forms of nonnormalities as well as time-varying conditional variances and covariances among the model disturbances. We derive exact bounds on the null distribution of joint F statistics to deal with the presence of nuisance parameters, and we show how to implement the resulting generalized nonparametric bounds tests with Monte Carlo resampling techniques. In sharp contrast to the usual tests that are not even computable when the number of test assets is too large, the power of the proposed test procedure potentially increases along both the time and cross-sectional dimensions.