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.     

Asymmetric multivariate normal mixture GARCH

An asymmetric multivariate generalization of the recently proposed class of normal mixture GARCH models is developed. Issues of parametrization and estimation are discussed. Conditions for covariance stationarity and the existence of the fourth moment are derived, and expressions for the dynamic correlation structure of the process are provided. In an application to stock market returns, it is shown that the disaggregation of the conditional (co)variance process generated by the model provides substantial intuition. Moreover, the model exhibits a strong performance in calculating out-of-sample Value-at-Risk measures.     

Generalised long-memory GARCH models for intra-daily volatility

The class of fractionally integrated generalised autoregressive conditional heteroskedastic (FIGARCH) models is extended for modelling the periodic long-range dependence typically shown by volatility of most intra-daily financial returns. The proposed class of models introduces generalised periodic long-memory filters, based on Gegenbauer polynomials, into the equation describing the time-varying volatility of standard GARCH models. A fitting procedure is illustrated and its performance is evaluated by means of Monte Carlo simulations. The effectiveness of these models in describing periodic long-memory volatility patterns is shown through an empirical application to the Euro-Dollar intra-daily exchange rate.     

拖尾指数分布:代数拖尾特性及其参数估计

讨论了合成孔径雷达强度图像的拖尾指数分布的统计特性及其参数估计,提出了拖尾指数分布的仿真方法,并利用仿真数据验证了拖尾指数分布的代数拖尾特性.基于负数阶矩理论,提出了拖尾指数分布的对数矩参数估计方法,对数矩估计具有解析的表达式,并且不受矩阶次的影响.Monte Carlo仿真结果表明对数矩估计可以获得较高的估计精度,它是拖尾指数分布参数估计的高效方法.     

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.     

Surrogate modeling approximation using a mixture of experts based on EM joint estimation

An automatic method to combine several local surrogate models is presented. This method is intended to build accurate and smooth approximation of discontinuous functions that are to be used in structural optimization problems. It strongly relies on the Expectation−Maximization (EM) algorithm for Gaussian mixture models (GMM). To the end of regression, the inputs are clustered together with their output values by means of parameter estimation of the joint distribution. A local expert is then built (linear, quadratic, artificial neural network, moving least squares) on each cluster. Lastly, the local experts are combined using the Gaussian mixture model parameters found by the EM algorithm to obtain a global model. This method is tested over both mathematical test cases and an engineering optimization problem from aeronautics and is found to improve the accuracy of the approximation.     

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.     

Multivariate GARCH estimation via a Bregman-proximal trust-region method

The estimation of multivariate GARCH time series models is a difficult task mainly due to the excessive parametrization exhibited by the problem, usually referred to as the “curse of dimensionality”. For the VEC family, the number of parameters involved in the model grows as a polynomial of order four on the dimension of the problem and, additionally, these parameters are subjected to complex nonlinear constraints. So far, this problem has been addressed only in low dimensional cases with strong parsimony constraints for the diagonal three-dimensional VEC handled with ad-hoc techniques. A general formulation of the estimation problem in any dimension and a Bregman-proximal trust-region method for its solution is proposed. The Bregman-proximal approach allows to handle the constraints in a very efficient and natural way by staying in the primal space and the Trust-Region mechanism stabilizes and speeds up the scheme. Computational experiments confirm the very good performance of the proposed approach.     

Proportional data modeling via selection and estimation of a finite mixture of scaled Dirichlet distributions

This paper proposes an unsupervised algorithm for learning a finite mixture of scaled Dirichlet distributions. Parameters estimation is based on the maximum likelihood approach, and the minimum message length (MML) criterion is proposed for selecting the optimal number of components. This research work is motivated by the flexibility issues of the Dirichlet distribution, the widely used model for multivariate proportional data, which has prompted a number of scholars to search for generalizations of the Dirichlet. By introducing the extra parameters of the scaled Dirichlet, several useful statistical models could be obtained. Experimental results are presented using both synthetic and real datasets. Moreover, challenging real-world applications are empirically investigated to evaluate the efficiency of our proposed statistical framework.     

Bayesian testing for non-linearity in volatility modeling

Neural networks provide a tool for describing non-linearity in volatility processes of financial data and help to answer the question “how much” non-linearity is present in the data. Non-linearity is studied under three different specifications of the conditional distribution: Gaussian, Student-t and mixture of Gaussians. To rank the volatility models, a Bayesian framework is adopted to perform a Bayesian model selection within the different classes of models. In the empirical analysis, the return series of the Dow Jones Industrial Average index, FTSE 100 and NIKKEI 225 indices over a period of 16 years are studied. The results show different behavior across the three markets. In general, if a statistical model accounts for non-normality and explains most of the fat tails in the conditional distribution, then there is less need for complex non-linear specifications.     

Sea clutter modeling using an autoregressive generalized nonlinear-asymmetric GARCH model

The sea clutter modeling is critical to the radar design and assessment of relevant detection algorithms. In this paper, we investigate a family of generalized autoregressive conditional heteroscedastic (GARCH) processes to model the sea clutter as a time series, in which the current variance is dependent on historical information. The most general model (so-called the ALLGARCH model) provides more flexible variance structures to model non-Gaussian, asymmetry, and nonlinear properties of the clutter. However, after going through the usage of the ALLGARCH model, we find that it is not very suitable because the coefficients of the model, which are numerous, would be difficult to estimate in a real-time operating environment. Meanwhile, we find that some of the coefficients are negligible under almost all kinds of sea environments and weather conditions. Motivated by these observations, we propose a novel GARCH model for sea clutter modeling, which is a generalization of the nonlinear-asymmetric GARCH (NAGARCH) model. Considering the correlation between adjacent clutter returns, autoregressive terms are also introduced. By systematically analyzing practical sea clutter data under different sea environments, we demonstrate that the proposed model achieves comparable fitting effect to some commonly used statistical models. Also, we develop the corresponding generalized likelihood ratio test (GLRT) algorithm for the new model. Numerical simulations exhibit that the proposed detector achieves higher probability of detection, comparing with the AR-GARCH detector.     

Edgeworth-expanded gaussian mixture density modeling

Instead of increasing the order of the Edgeworth expansion of a single gaussian kernel, we suggest using mixtures of Edgeworth-expanded gaussian kernels of moderate order. We introduce a simple closed-form solution for estimating the kernel parameters based on weighted moment matching. Furthermore, we formulate the extension to the multivariate case, which is not always feasible with algebraic density approximation procedures.     

一种自组织混合模型在汇率波动性预测中的应用

汇率波动性的预测一直以来是研究金融市场者关注的焦点之一,本文拓展了一种基于自组织神经网络技术的,用于预测非平稳汇率波动性的自组织混合模型(SOMAR).SOMAR突破了传统模型对平稳性的假设,变全局建模为局部建模,使得全局非平稳数据变成局部平稳数据.同时,它也是一种基于神经元网络技术的非参数回归模型,结合传统回归模型的简易性和神经元网络算法的灵活性,拓展模型(ESOMAR)提高了对数据异构的适应性.在对汇率波动性的预测实验中,ESOMAR体现出优于传统回归模型和一些基于其它神经元网络模型的效果,并证明了它在预测金融数据方面所具有的价值.     

Characteristic function estimation of Ornstein-Uhlenbeck-based stochastic volatility models

Continuous-time stochastic volatility models are becoming increasingly popular in finance because of their flexibility in accommodating most stylized facts of financial time series. However, their estimation is difficult because the likelihood function does not have a closed-form expression. A characteristic function-based estimation method for non-Gaussian Ornstein-Uhlenbeck-based stochastic volatility models is proposed. Explicit expressions of the characteristic functions for various cases of interest are derived. The asymptotic properties of the estimators are analyzed and their small-sample performance is evaluated by means of a simulation experiment. Finally, two real-data applications show that the superposition of two Ornstein-Uhlenbeck processes gives a good approximation to the dependence structure of the process.     

GARCH,jumps and permanent and transitory components of volatility: the case of the Taiwan exchange rate

This paper investigates whether there are three distinctive features in financial asset prices, that is, time-varying conditional volatility, jumps and the component factors of volatility. It adopts a component-GARCH-Jump, which can efficiently capture the three features simultaneously. Our results demonstrate that the three features exist in the Taiwan exchange rate. Besides time-varying conditional volatility, our model identifies 172 jumps between 5 January 1988 and 21 March 2003. The empirical evidence shows that the permanent component of the conditional variance is a relatively smooth movement except for a fairly sharp shift which began in 1997. This means that the effect of the Asian crisis shock might very well have exerted not only a transitory jump effect, but also a permanent effect on Taiwan’s exchange rate.     

Continuous mixture modeling via goodness-of-fit ridges

We present a novel method for representing “extruded” distributions. An extruded distribution is an M-dimensional manifold in the parameter space of the component distribution. Representations of that manifold are “continuous mixture models”. We present a method for forming one-dimensional continuous Gaussian mixture models of sampled extruded Gaussian distributions via ridges of goodness-of-fit. Using Monte Carlo simulations and ROC analysis, we explore the utility of a variety of binning techniques and goodness-of-fit functions. We demonstrate that extruded Gaussian distributions are more accurately and consistently represented by continuous Gaussian mixture models than by finite Gaussian mixture models formed via maximum likelihood expectation maximization.     

Bayesian approaches to Gaussian mixture modeling

A Bayesian-based methodology is presented which automatically penalizes overcomplex models being fitted to unknown data. We show that, with a Gaussian mixture model, the approach is able to select an “optimal” number of components in the model and so partition data sets. The performance of the Bayesian method is compared to other methods of optimal model selection and found to give good results. The methods are tested on synthetic and real data sets     

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.     

Simulated minimum Hellinger distance estimation of stochastic volatility models

A simultaneously efficient and robust approach for distribution-free parametric inference, called the simulated minimum Hellinger distance (SMHD) estimator, is proposed. In the SMHD estimation, the Hellinger distance between the nonparametrically estimated density of the observed data and that of the simulated samples from the model is minimized. The method is applicable to the situation where the closed-form expression of the model density is intractable but simulating random variables from the model is possible. The robustness of the SMHD estimator is equivalent to the minimum Hellinger distance estimator. The finite sample efficiency of the proposed methodology is found to be comparable to the Bayesian Markov chain Monte Carlo and maximum likelihood Monte Carlo methods and outperform the efficient method of moments estimators. The robustness of the method to a stochastic volatility model is demonstrated by a simulation study. An empirical application to the weekly observations of foreign exchange rates is presented.