On the matrix-variate generalized hyperbolic distribution and its Bayesian applications

In the first part of the paper, we introduce the matrix-variate generalized hyperbolic distribution by mixing the matrix normal distribution with the matrix generalized inverse Gaussian density. The p-dimensional generalized hyperbolic distribution of [Barndorff-Nielsen, O. (1978). Hyperbolic distributions and distributions on hyperbolae. Scand. J. Stat., 5, 151–157], the matrix-T distribution and many well-known distributions are shown to be special cases of the new distribution. Some properties of the distribution are also studied. The second part of the paper deals with the application of the distribution in the Bayesian analysis of the normal multivariate linear model.     

Multi-modal tempered stable distributions and prosses with applications to finance

Abstract

The assumption of underlying return distribution plays an important role in asset pricing models. While the return distribution used in the traditional theories of asset pricing is the unimodal distribution, numerous studies which have investigated the empirical behavior of asset returns in financial markets use multi-modal distribution. We introduce a new parsimonious multi-modal distribution, referred to as the multi-modal tempered stable (MMTS) distribution. In this article we also generate the exponential Lévy market models and derive the value-at-risk (VaR) induced from them. To demonstrate the advantages, we will present the results of the parameter estimation and the VaRs for financial data.     

The Distribution of Stock Returns When the Market Is Up

Abstract

For some investments, the relation between stock returns and the market proxy is conventionally described by a linear regression model with the normality assumption. This paper derives the distribution of stock returns for a security in an upgrade (or downgrade) market with the assumption that the log stock returns of the market proxy follow a mixture of normal distributions. We discuss MLE and the method of moment estimation for parameters involved in the model. An analysis of stock data in Johannesburg Stock Exchange is included to illustrate the model. This note explains the phenomenon in financial analysis regarding the shape of the distribution of long-run stock returns limited on an upgrade or downgrade market index.     

The exponentiated-log-logistic geometric distribution: Dual activation

ABSTRACT

The log-logistic distribution is commonly used to model lifetime data. We propose a wider distribution, named the exponentiated log-logistic geometric distribution, based on a double activation approach. We obtain the quantile function, ordinary moments, and generating function. The method of maximum likelihood is used to estimate the model parameters. We propose a new extended regression model based on the logarithm of the exponentiated log-logistic geometric distribution. This regression model can be very useful in the analysis of real data and could provide better fits than other special regression models. The potentiality of the new models is illustrated by means of two applications to real lifetime data sets.     

Influence diagnostics in the capital asset pricing model under elliptical distributions

In this paper we consider the Capital Asset Pricing Model under Elliptical (symmetric) Distributions. This class of distributions, which contains the normal distribution, t, contaminated normal and power exponential, among others, offers a more flexible framework for modelling asset prices or returns. In order to analyze the sensibility to possible outliers and/or atypical returns of the maximum likelihood estimators, the local influence method was implemented. The results are illustrated by using a set of shares from companies who trade in the Chilean Stock Market. Our main conclusion is that symmetric distributions having heavier tails than those of the normal distribution, especially the t distribution with small degrees of freedom, show a better fit and allow the reduction of the influence of atypical returns in the maximum likelihood estimators.     

Fractionally integrated GARCH model with tempered stable distribution: a simulation study

With the growing availability of high-frequency data, long memory has become a popular topic in finance research. Fractionally Integrated GARCH (FIGARCH) model is a standard approach to study the long memory of financial volatility. The original specification of FIGARCH model is developed using Normal distribution, which cannot accommodate fat-tailed properties commonly existing in financial time series. Traditionally, the Student-t distribution and General Error Distribution (GED) are used instead to solve that problem. However, a recent study points out that the Student-t lacks stability. Instead, the Stable distribution is introduced. The issue of this distribution is that its second moment does not exist. To overcome this new problem, the tempered stable distribution, which retains most attractive characteristics of the Stable distribution and has defined moments, is a natural candidate. In this paper, we describe the estimation procedure of the FIGARCH model with tempered stable distribution and conduct a series of simulation studies to demonstrate that it consistently outperforms FIGARCH models with the Normal, Student-t and GED distributions. An empirical evidence of the S&P 500 hourly return is also provided with robust results. Therefore, we argue that the tempered stable distribution could be a widely useful tool for modelling the high-frequency financial volatility in general contexts with a FIGARCH-type specification.     

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.     

Stochastic Volatility Models Based on OU-Gamma Time Change: Theory and Estimation

We consider stochastic volatility models that are defined by an Ornstein–Uhlenbeck (OU)-Gamma time change. These models are most suitable for modeling financial time series and follow the general framework of the popular non-Gaussian OU models of Barndorff-Nielsen and Shephard. One current problem of these otherwise attractive nontrivial models is, in general, the unavailability of a tractable likelihood-based statistical analysis for the returns of financial assets, which requires the ability to sample from a nontrivial joint distribution. We show that an OU process driven by an infinite activity Gamma process, which is an OU-Gamma process, exhibits unique features, which allows one to explicitly describe and exactly sample from relevant joint distributions. This is a consequence of the OU structure and the calculus of Gamma and Dirichlet processes. We develop a particle marginal Metropolis–Hastings algorithm for this type of continuous-time stochastic volatility models and check its performance using simulated data. For illustration we finally fit the model to S&P500 index data.     

Bayesian analysis for heteroscedastic normal-Pareto mixture model with application

ABSTRACT

In this article, we propose a new distribution by mixing normal and Pareto distributions, and the new distribution provides an unusual hazard function. We model the mean and the variance with covariates for heterogeneity. Estimation of the parameters is obtained by the Bayesian method using Markov Chain Monte Carlo (MCMC) algorithms. Proposal distribution in MCMC is proposed with a defined working variable related to the observations. Through the simulation, the method shows a dependable performance of the model. We demonstrate through establishing model under a real dataset that the proposed model and method can be more suitable than the previous report.     

Modeling stock markets’ volatility using GARCH models with Normal,Student’s <Emphasis Type="Italic">t</Emphasis> and stable Paretian distributions

As GARCH models and stable Paretian distributions have been revisited in the recent past with the papers of Hansen and Lunde (J Appl Econom 20: 873–889, 2005) and Bidarkota and McCulloch (Quant Finance 4: 256–265, 2004), respectively, in this paper we discuss alternative conditional distributional models for the daily returns of the US, German and Portuguese main stock market indexes, considering ARMA-GARCH models driven by Normal, Student’s t and stable Paretian distributed innovations. We find that a GARCH model with stable Paretian innovations fits returns clearly better than the more popular Normal distribution and slightly better than the Student’s t distribution. However, the Student’s t outperforms the Normal and stable Paretian distributions when the out-of-sample density forecasts are considered.     

Adjusted Likelihood Inference in an Elliptical Multivariate Errors-in-Variables Model

In this paper, we obtain an adjusted version of the likelihood ratio (LR) test for errors-in-variables multivariate linear regression models. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal distribution as a special case. We derive a modified LR statistic that follows a chi-squared distribution with a high degree of accuracy. Our results generalize those in Melo and Ferrari (Advances in Statistical Analysis, 2010, 94, pp. 75–87) by allowing the parameter of interest to be vector-valued in the multivariate errors-in-variables model. We report a simulation study which shows that the proposed test displays superior finite sample behavior relative to the standard LR test.     

The structural Sharpe model under t-distributions

In this paper we consider Sharpe's single-index model or Sharpe's model, by assuming that the returns obtained follow a multivariate t elliptical distribution. Also, given that the returns of the market are not observable, the statistical analysis was made in the context of an errors-in-variables model. In order to analyze the sensibility to possible outliers and/or atypical returns of the maximum likelihood estimators the local influence method [10 Cook, R. D. 1986. Assessment of local influence. J. Roy. Statist. Soc. B, 48: 133–169.  [Google Scholar]] was implemented. The results are illustrated by using a set of shares of companies belonging to the Chilean Stock Market. The main conclusion is that the t model with small degrees of freedom is able to incorporate possible outliers and influential returns in the data.     

Modeling asset returns with alternative stable distributions *

In the 1960's Benoit Mandelbrot and Eugene Fama argued strongly in favor of the stable Paretian distribution as a model for the unconditional distribution of asset returns. Although a substantial body of subsequent empirical studies supported this position, the stable Paretian model plays a minor role in current empirical work.

While in the economics and finance literature stable distributions are virtually exclusively associated with stable Paretian distributions, in this paper we adopt a more fundamental view and extend the concept of stability to a variety of probabilistic schemes. These schemes give rise to alternative stable distributions, which we compare empirically using S&P 500 stock return data. In this comparison the Weibull distribution, associated with both the nonrandom-minimum and geometric-random summation schemes dominates the other stable distributions considered-including the stable Paretian model.     

On the Normal Inverse Gaussian Stochastic Volatility Model

In this article, the normal inverse Gaussian stochastic volatility model of Barndorff-Nielsen is extended. The resulting model has a more flexible lag structure than the original one. In addition, the second-and fourth-order moments, important properties of a volatility model, are derived. The model can be considered either as a generalized autoregressive conditional heteroscedasticity model with nonnormal errors or as a stochastic volatility model with an inverse Gaussian distributed conditional variance. A simulation study is made to investigate the performance of the maximum likelihood estimator of the model. Finally, the model is applied to stock returns and exchange-rate movements. Its fit to two stylized facts and its forecasting performance is compared with two other volatility models.     

A semi-parametric method for estimating the beta coefficients of the hidden two-sided asset return jumps

ABSTRACT

We introduce a new methodology for estimating the parameters of a two-sided jump model, which aims at decomposing the daily stock return evolution into (unobservable) positive and negative jumps as well as Brownian noise. The parameters of interest are the jump beta coefficients which measure the influence of the market jumps on the stock returns, and are latent components. For this purpose, at first we use the Variance Gamma (VG) distribution which is frequently used in modeling financial time series and leads to the revelation of the hidden market jumps' distributions. Then, our method is based on the central moments of the stock returns for estimating the parameters of the model. It is proved that the proposed method provides always a solution in terms of the jump beta coefficients. We thus achieve a semi-parametric fit to the empirical data. The methodology itself serves as a criterion to test the fit of any sets of parameters to the empirical returns. The analysis is applied to NASDAQ and Google returns during the 2006–2008 period.     

Influence diagnostics for the structural sharpe model under normal/independent distributions

A method is proposed in this paper to assess the local influence of minor perturbations for the Sharpe model when the normal distribution is replaced by normal/independent (NI) distributions. The family of NI distributions is an attractive class of symmetric heavy-tailed densities that includes as special cases the normal, t-Student, slash, and the contaminated normal distributions. Since the returns of the market are not observable, the statistical analysis is carried out in the context of an errors-in-variables model. An influence analysis for detecting influential observations (atypical returns) is developed to investigate the sensitivity of the maximum likelihood estimators. Diagnostic measures are obtained based on the conditional expectation of the complete-data log-likelihood function. The results are illustrated by using a set of shares of companies traded in the Chilean stock market.     

Weighted similarity tests for location-scale families of stable distributions

ABSTRACT

The class of stable distributions plays a central role in the study of asymptotic behavior of normalized partial sums, the same role performed by normal distribution among those with finite second moment. In this note, by exploiting the connection between stable laws and regularly varying functions, we present weighted similarity tests for stable location-scale families. The proposed weight functions are based on the 2nd-order Mallows distance between the empirical distribution and the root stable distribution. And the resulting statistics converge weakly to functionals of Brownian bridge.     

Modelling cell generation times by using the tempered stable distribution

Summary.  We show that the family of tempered stable distributions has considerable potential for modelling cell generation time data. Several real examples illustrate how these distributions can improve on currently assumed models, including the gamma and inverse Gaussian distributions which arise as special cases. Our applications concentrate on the generation times of oligodendrocyte progenitor cells and the yeast Saccharomyces cerevisiae . Numerical inversion of the Laplace transform of the probability density function provides fast and accurate approximations to the tempered stable density, for which no closed form generally exists. We also show how the asymptotic population growth rate is easily calculated under a tempered stable model.     

Shared frailty models with baseline generalized Pareto distribution

Abstract

In this article, we have considered three different shared frailty models under the assumption of generalized Pareto Distribution as baseline distribution. Frailty models have been used in the survival analysis to account for the unobserved heterogeneity in an individual risks to disease and death. These three frailty models are with gamma frailty, inverse Gaussian frailty and positive stable frailty. Then we introduce the Bayesian estimation procedure using Markov chain Monte Carlo (MCMC) technique to estimate the parameters. We applied these three models to a kidney infection data and find the best fitted model for kidney infection data. We present a simulation study to compare true value of the parameters with the estimated values. Model comparison is made using Bayesian model selection criterion and a well-fitted model is suggested for the kidney infection data.     

Which Factors are Risk Factors in Asset Pricing? A Model Scan Framework

Abstract

A key question for understanding the cross-section of expected returns of equities is the following: which factors, from a given collection of factors, are risk factors, equivalently, which factors are in the stochastic discount factor (SDF)? Though the SDF is unobserved, assumptions about which factors (from the available set of factors) are in the SDF restricts the joint distribution of factors in specific ways, as a consequence of the economic theory of asset pricing. A different starting collection of factors that go into the SDF leads to a different set of restrictions on the joint distribution of factors. The conditional distribution of equity returns has the same restricted form, regardless of what is assumed about the factors in the SDF, as long as the factors are traded, and hence the distribution of asset returns is irrelevant for isolating the risk-factors. The restricted factors models are distinct (nonnested) and do not arise by omitting or including a variable from a full model, thus precluding analysis by standard statistical variable selection methods, such as those based on the lasso and its variants. Instead, we develop what we call a Bayesian model scan strategy in which each factor is allowed to enter or not enter the SDF and the resulting restricted models (of which there are 114,674 in our empirical study) are simultaneously confronted with the data. We use a Student-t distribution for the factors, and model-specific independent Student-t distribution for the location parameters, a training sample to fix prior locations, and a creative way to arrive at the joint distribution of several other model-specific parameters from a single prior distribution. This allows our method to be essentially a scaleable and tuned-black-box method that can be applied across our large model space with little to no user-intervention. The model marginal likelihoods, and implied posterior model probabilities, are compared with the prior probability of 1/114,674 of each model to find the best-supported model, and thus the factors most likely to be in the SDF. We provide detailed simulation evidence about the high finite-sample accuracy of the method. Our empirical study with 13 leading factors reveals that the highest marginal likelihood model is a Student-t distributed factor model with 5 degrees of freedom and 8 risk factors.