Modelling nonlinearities and heavy tails via threshold normal mixture GARCH models

A new class of flexible threshold normal mixture GARCH models is proposed for the analysis and modelling of the stylized facts appeared in many financial time series. A Bayesian stochastic method is developed and presented for the analysis of the proposed model allowing for automatic model determination and estimation of the thresholds and their unknown number. A computationally feasible algorithm that explores the posterior distribution of the threshold models is designed using Markov chain Monte Carlo stochastic search methods. A simulation study is conducted to assess the performance of the proposed method, and an empirical application of the proposed model is illustrated using real data.     

Robust M-estimation of multivariate GARCH models

The Gaussian quasi-maximum likelihood estimator of Multivariate GARCH models is shown to be very sensitive to outliers in the data. A class of robust M-estimators for MGARCH models is developed. To increase the robustness of the estimators, the use of volatility models with the property of bounded innovation propagation is recommended. The Monte Carlo study and an empirical application to stock returns document the good robustness properties of the M-estimator with a fat-tailed Student t loss function.     

Currency hedging strategies using dynamic multivariate GARCH

This paper examines the effectiveness of using futures contracts as hedging instruments of: (1) alternative models of volatility for estimating conditional variances and covariances; (2) alternative currencies; and (3) alternative maturities of futures contracts. For this purpose, daily data of futures and spot exchange rates of three major international currencies, Euro, British pound and Japanese yen, against the American dollar, are used to analyze hedge ratios and hedging effectiveness resulting from using two different maturity currency contracts, near-month and next-to-near-month contract. We estimate four multivariate volatility models (namely CCC, VARMA-AGARCH, DCC and BEKK), and calculate optimal portfolio weights and optimal hedge ratios to identify appropriate currency hedging strategies. The hedging effectiveness index suggests that the best results in terms of reducing the variance of the portfolio are for the USD/GBP exchange rate. The empirical results show that futures hedging strategies are slightly more effective when the near-month future contract is used for the USD/GBP and USD/JPY currencies. Moreover, the CCC and AGARCH models provide similar hedging effectiveness, which suggests that dynamic asymmetry may not be crucial empirically, although some differences appear when the DCC and BEKK models are used.     

Heavy-tailed mixture GARCH volatility modeling and Value-at-Risk estimation

This paper presents a heavy-tailed mixture model for describing time-varying conditional distributions in time series of returns on prices. Student-t component distributions are taken to capture the heavy tails typically encountered in such financial data. We design a mixture MT(m)-GARCH(p, q) volatility model for returns, and develop an EM algorithm for maximum likelihood estimation of its parameters. This includes formulation of proper temporal derivatives for the volatility parameters. The experiments with a low order MT(2)-GARCH(1, 1) show that it yields results with improved statistical characteristics and economic performance compared to linear and nonlinear heavy-tail GARCH, as well as normal mixture GARCH. We demonstrate that our model leads to reliable Value-at-Risk performance in short and long trading positions across different confidence levels.     

Bayesian estimation of the Gaussian mixture GARCH model

Bayesian inference and prediction for a generalized autoregressive conditional heteroskedastic (GARCH) model where the innovations are assumed to follow a mixture of two Gaussian distributions is performed. The mixture GARCH model can capture the patterns usually exhibited by many financial time series such as volatility clustering, large kurtosis and extreme observations. A Griddy-Gibbs sampler implementation is proposed for parameter estimation and volatility prediction. Bayesian prediction of the Value at Risk is also addressed providing point estimates and predictive intervals. The method is illustrated using the Swiss Market Index.     

Real-time anomaly detection in hyperspectral images using multivariate normal mixture models and GPU processing

Hyperspectral imaging, which records a detailed spectrum of light arriving in each pixel, has many potential uses in remote sensing as well as other application areas. Practical applications will typically require real-time processing of large data volumes recorded by a hyperspectral imager. This paper investigates the use of graphics processing units (GPU) for such real-time processing. In particular, the paper studies a hyperspectral anomaly detection algorithm based on normal mixture modelling of the background spectral distribution, a computationally demanding task relevant to military target detection and numerous other applications. The algorithm parts are analysed with respect to complexity and potential for parallellization. The computationally dominating parts are implemented on an Nvidia GeForce 8800 GPU using the Compute Unified Device Architecture programming interface. GPU computing performance is compared to a multi-core central processing unit implementation. Overall, the GPU implementation runs significantly faster, particularly for highly data-parallelizable and arithmetically intensive algorithm parts. For the parts related to covariance computation, the speed gain is less pronounced, probably due to a smaller ratio of arithmetic to memory access. Detection results on an actual data set demonstrate that the total speedup provided by the GPU is sufficient to enable real-time anomaly detection with normal mixture models even for an airborne hyperspectral imager with high spatial and spectral resolution.     

Fast clustering of GARCH processes via Gaussian mixture models

The financial econometrics literature includes several Multivariate GARCH models where the model parameter matrices depend on a clustering of financial assets. Those classes might be defined a priori or data-driven. When the latter approach is followed, one method for deriving asset groups is given by the use of clustering methods. In this paper, we analyze in detail one of those clustering approaches, the Gaussian mixture GARCH. This method is designed to identify groups based on the conditional variance dynamic parameters. The clustering algorithm, based on a Gaussian mixture model, has been recently proposed and is here generalized with the introduction of a correction for the presence of correlation across assets. Finally, we introduce a benchmark estimator used to assess the performances of simpler and faster estimators. Simulation experiments show evidence of the improvements given by the correction for asset correlation.     

A Fuzzy Asymmetric GARCH model applied to stock markets

In this paper, we derive a new class of flexible threshold asymmetric Generalized Autoregression Conditional Heteroskedasticity (GARCH) models. We use this tool for analysis and modeling of the properties that are apparent in many financial time series. In general, the transmission of volatility in the stock market is time-varying, nonlinear, and asymmetric with respect to both positive and negative results. Given this fact, we adopt the method of fuzzy logic systems to modify the threshold values for an asymmetric GARCH model. Our simulations use stock market data from the Taiwan weighted index (Taiwan), the Nikkei 225 index (Japan), and the Hang Seng index (Hong Kong) to illustrate the performance of our proposed method. From the simulation results, we have determined that the forecasting of volatility performance is significantly improved if the leverage effect of clustering is considered along with the use of expert knowledge enabled by the GARCH model.     

Cluster analysis using multivariate normal mixture models to detect differential gene expression with microarray data

DNA microarrays make it possible to study simultaneously the expression of thousands of genes in a biological sample. Univariate clustering techniques have been used to discover target genes with differential expression between two experimental conditions. Because of possible loss of information due to use of univariate summary statistics, it may be more effective to use multivariate statistics. We present multivariate normal mixture model based clustering analyses to detect differential gene expression between two conditions.Deviating from the general mixture model and model-based clustering, we propose mixture models with specific mean and covariance structures that account for special features of two-condition microarray experiments. Explicit updating formulas in the EM algorithm for three such models are derived. The methods are applied to a real dataset to compare the expression levels of 1176 genes of rats with and without pneumococcal middle-ear infection to illustrate the performance and usefulness of this approach. About 10 genes and 20 genes are found to be differentially expressed in a six-dimensional modeling and a bivariate modeling, respectively. Two simulation studies are conducted to compare the performance of univariate and multivariate methods. Depending on data, neither method can always dominate the other. The results suggest that multivariate normal mixture models can be useful alternatives to univariate methods to detect differential gene expression in exploratory data analysis.     

An analysis of the flexibility of Asymmetric Power GARCH models

The Asymmetric Power GARCH (APGARCH) model allows a wider class of power transformations than simply taking the absolute value or squaring the data as in classical heteroscedastic models. A dynamic estimation is used to compare the three GARCH families and examine their forecasting performances in a value-at-risk setting. The results suggest that the optimal power transformation obtained with the APGARCH model is virtually never statistically different from 1 or 2. Moreover, although some indices switch between these two values over the time, the measures of accuracy and efficiency used to assess the performance of VaR forecasts indicate that the additional flexibility brought by the APGARCH model provides little, if any, improvements for risk management.     

Volatility dynamics of the US business cycle: A multivariate asymmetric GARCH approach

Most empirical investigations of the business cycles in the United States have excluded the dimension of asymmetric conditional volatility. This paper analyses the volatility dynamics of the US business cycle by comparing the performance of various multivariate generalised autoregressive conditional heteroskedasticity (GARCH) models. In particular, we propose two bivariate GARCH models to examine the evidence of volatility asymmetry and time-varying correlations concurrently, and then apply the proposed models to five sectors of Industrial Production of the United States. Our findings provide strong evidence of asymmetric conditional volatility in all sectors, and some support of time-varying correlations in various sectoral pairs. This has important policy implications for government to consider the effective countercyclical measures during recessions.     

Model and distribution uncertainty in multivariate GARCH estimation: A Monte Carlo analysis

Multivariate GARCH models are in principle able to accommodate the features of the dynamic conditional covariances; nonetheless the interaction between model parametrization of the second conditional moment and the conditional density of asset returns adopted in the estimation determines the fitting of such models to the observed dynamics of the data. Alternative MGARCH specifications and probability distributions are compared on the basis of forecasting performances by means of Monte Carlo simulations, using both statistical and financial forecasting loss functions.     

Bounded multivariate generalized Gaussian mixture model using ICA and IVA

Pattern Analysis and Applications - A bounded multivariate generalized Gaussian mixture model with a full covariance matrix is proposed for modeling data in a bounded support region. For model...     

Constrained monotone EM algorithms for finite mixture of multivariate Gaussians

The likelihood function for normal multivariate mixtures may present both local spurious maxima and also singularities and the latter may cause the failure of the optimization algorithms. Theoretical results assure that imposing some constraints on the eigenvalues of the covariance matrices of the multivariate normal components leads to a constrained parameter space with no singularities and at least a smaller number of local maxima of the likelihood function. Conditions assuring that an EM algorithm implementing such constraints maintains the monotonicity property of the usual EM algorithm are provided. Different approaches are presented and their performances are evaluated and compared using numerical experiments.     

Fast computation of high-dimensional multivariate normal probabilities

A new efficient method is proposed to compute multivariate normal probabilities over rectangles in high dimensions. The method exploits four variance reduction techniques: conditional Monte Carlo, importance sampling, splitting and control variates. Simulation results are presented that evaluate the performance of the new proposed method. The new method is designed for computing small exceedance probabilities.     

Generalized F-tests for the multivariate normal mean

Based on Läuter’s [Läuter, J., 1996. Exact t and F tests for analyzing studies with multiple endpoints. Biometrics 52, 964-970] exact t test for biometrical studies related to the multivariate normal mean, we develop a generalized F-test for the multivariate normal mean and extend it to multiple comparison. The proposed generalized F-tests have simple approximate null distributions. A Monte Carlo study and two real examples show that the generalized F-test is at least as good as the optional individual Läuter’s test and can improve its performance in some situations where the projection directions for the Läuter’s test may not be suitably chosen. The generalized F-test could be superior to individual Läuter’s tests and the classical Hotelling T²-test for the general purpose of testing the multivariate normal mean. It is shown by Monte Carlo studies that the extended generalized F-test outperforms the commonly-used classical test for multiple comparison of normal means in the case of high dimension with small sample sizes.     

Improved Stein-type shrinkage estimators for the high-dimensional multivariate normal covariance matrix

Many applications require an estimate for the covariance matrix that is non-singular and well-conditioned. As the dimensionality increases, the sample covariance matrix becomes ill-conditioned or even singular. A common approach to estimating the covariance matrix when the dimensionality is large is that of Stein-type shrinkage estimation. A convex combination of the sample covariance matrix and a well-conditioned target matrix is used to estimate the covariance matrix. Recent work in the literature has shown that an optimal combination exists under mean-squared loss, however it must be estimated from the data. In this paper, we introduce a new set of estimators for the optimal convex combination for three commonly used target matrices. A simulation study shows an improvement over those in the literature in cases of extreme high-dimensionality of the data. A data analysis shows the estimators are effective in a discriminant and classification analysis.     

Extension of the mixture of factor analyzers model to incorporate the multivariate t-distribution

Mixtures of factor analyzers enable model-based density estimation to be undertaken for high-dimensional data, where the number of observations n is small relative to their dimension p. However, this approach is sensitive to outliers as it is based on a mixture model in which the multivariate normal family of distributions is assumed for the component error and factor distributions. An extension to mixtures of t-factor analyzers is considered, whereby the multivariate t-family is adopted for the component error and factor distributions. An EM-based algorithm is developed for the fitting of mixtures of t-factor analyzers. Its application is demonstrated in the clustering of some microarray gene-expression data.     

Semi‐supervised classification of facial expression using a mixture of multivariate t distributions

Abstract: This paper addresses the semi‐supervised classification of facial expression images using a mixture of multivariate t distributions. The facial expression features are first extracted into labelled graph vectors using the Gabor wavelet transformation. We then learn a mixture of multivariate t distributions by using the labelled graph vectors, and set correspondence between the component distributions and the basic facial emotions. According to this correspondence, the classification of a given testing image is implemented in a probabilistic way according to its fitted posterior probabilities of component memberships. Specifically, we perform hard classification of the testing image by assigning it into an emotional class that the corresponding mixture component has the highest posterior probability, or softly use the posterior probabilities as the estimates of the semantic ratings of expressions. The experimental results on the Japanese female facial expression database, Ekman's Pictures of Facial Affect database and the AR database demonstrate the effectiveness of the proposed method.     

On fast supervised learning for normal mixture models with missing information

It is an important research issue to deal with mixture models when missing values occur in the data. In this paper, computational strategies using auxiliary indicator matrices are introduced for efficiently handling mixtures of multivariate normal distributions when the data are missing at random and have an arbitrary missing data pattern, meaning that missing data can occur anywhere. We develop a novel EM algorithm that can dramatically save computation time and be exploited in many applications, such as density estimation, supervised clustering and prediction of missing values. In the aspect of multiple imputations for missing data, we also offer a data augmentation scheme using the Gibbs sampler. Our proposed methodologies are illustrated through some real data sets with varying proportions of missing values.