Modeling financial time series with the skew slash distribution

Financial returns often present moderate skewness and high kurtosis. As a consequence, it is natural to look for a model that is exible enough to capture these characteristics. The proposal is to undertake inference for a generalized autoregressive conditional heteroskedastic (GARCH) model, where the innovations are assumed to follow a skew slash distribution. Both classical and Bayesian inference are carried out. Simulations and a real data example illustrate the performance of the proposed methodology.We acknowledge financial support by MCI grants 2007/04438/001 and MTM2008-03010

Bayesian Generalized Linear Mixed Effects Models Using Normal-Independent Distributions: Formulation and Applications

A standard assumption is that the random effects of Generalized Linear Mixed Effects Models (GLMMs) follow the normal distribution. However, this assumption has been found to be quite unrealistic and sometimes too restrictive as revealed in many real-life situations. A common case of departures from normality includes the presence of outliers leading to heavy-tailed distributed random effects. This work, therefore, aims to develop a robust GLMM framework by replacing the normality assumption on the random effects by the distributions belonging to the Normal-Independent (NI) class. The resulting models are called the Normal-Independent GLMM (NI-GLMM). The four special cases of the NI class considered in these models’ formulations include the normal, Student-t, Slash and contaminated normal distributions. A full Bayesian technique was adopted for estimation and inference. A real-life data set on cotton bolls was used to demonstrate the performance of the proposed NI-GLMM methodology

Modelos de regressão univariados e bivariados baseados nas distribuições de mistura de escala normal assimétrica sob a parametrização centrada

Orientador: Caio Lucidius Naberezny AzevedoDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Situações em que a variável resposta é contínua ou binária são bastante comuns em diversas áreas do conhecimento. Apesar de existirem diversos modelos para essas situações, em muitos casos, características como assimetria e caudas pesadas, não são contempladas adequadamente. Além disso, conjuntos de respostas bivariadas, contendo uma variável contínua e uma discreta, são comuns em muitos problemas reais, as quais também podem apresentar assimetria e caudas pesadas. A abordagem mais comum, no caso bivariado, é modelar cada variável separadamente, ignorando a potencial correlação entre elas, ou decompor a distribuição conjunta na distribuição marginal para a variável binária e na distribuição condicional para a variável contínua, dada a variável binária. A decomposição na distribuição marginal da variável contínua e na distribuição condicional da variável binária, dada a variável contínua, também é possível. Neste projeto desenvolvemos: uma classe de modelos de regressão linear baseada nas distribuições de mistura de escala normal assimétrica sob a parametrização centrada (MENAC), uma classe de modelos de regressão para dados binários com função de ligação associada a alguma distribuição MENAC, e uma classe de modelos de regressão misto para dados bivariados contínuo e binário, em que tanto a resposta contínua, quanto a função de ligação para a resposta binária pertencem a classe MENAC. Para introduzir a estrutura de dependência entre as duas variáveis resposta, consideramos uma estrutura de efeitos aleatórios comuns, cujas distribuições também pertencem a classe MENAC. Desenvolvemos procedimentos de estimação sob o paradigma bayesiano, assim como ferramentas de diagnóstico, contemplando análise residual e medidas de influência, bem como medidas de comparação de modelos. Realizamos estudos de simulação, considerando diferentes cenários de interesse, com o intuito de avaliar o desempenho das estimativas e das medidas de diagnóstico. As metodologias propostas foram ilustradas tanto com dados provenientes de estudos de simulação, quanto com conjuntos de dados reaisAbstract: Situations where the response variable is either continuous or binary are quite common in several fields of knowledge. Although there are several models for these situations, in many cases, characteristics such as asymmetry and heavy tails, are not properly treated. In addition, bivariate responses, containing one continuous and one discrete variable, are common in many real problems, which may also exhibit asymmetry and heavy tails. The most common approach in the bivariate case is to model each variable separately, ignoring the potential correlation between them, or to decompose the joint distribution into the marginal distribution of the binary variable and the conditional distribution of the continuous variable, given the binary variable. The decomposition into the marginal distribution of the continuous variable and the conditional distribution of the binary variable, given the continuous variable, it is also possible. In this project we developed: a class of linear regression models based on the skew scale mixture of normal distributions under the centered parameterization (SSMNC), a class of regression models for binary data with link function associated with some SSMNC distribution, and a class of mixed regression models for bivariate continuous and binary data, in which both the continuous response and the link function for the binary response, belong to the SSMNC class. To introduce the dependency structure between the two response variables, we consider a common random effects structure, whose distributions also belong to the SSMNC class. We developed estimation procedures under the Bayesian paradigm, also, diagnostic tools, including residual analysis and influence measures, as well as model comparison measures. We performed simulation studies, considering different scenarios of interest, in order to evaluate the performance of estimates and diagnostic measures. The proposed methodologies were illustrated with both data from simulation studies and with real data setsMestradoEstatisticaMestra em Estatística2015/25867-2131979/2016-7FAPESPCNP

Bayesian estimation of a dynamic conditional correlation model with multivariate Skew-Slash innovations

Financial returns often present a complex relation with previous observations, along with a slight skewness and high kurtosis. As a consequence, we must pursue the use of flexible models that are able to seize these special features: a financial process that can expose the intertemporal relation between observations, together with a distribution that can capture asymmetry and heavy tails simultaneously. A multivariate extension of the GARCH such as the Dynamic Conditional Correlation model with Skew-Slashinnovations for financial time series in a Bayesian framework is proposed in the present document, and it is illustrated using an MCMC within Gibbs algorithm performed onsimulated data, as well as real data drawn from the daily closing prices of the DAX,CAC40, and Nikkei indicesWe acknowledge financial support by Spanish Ministry of Economy and Competition Grant ECO2012-38442

A Bayesian Approach to Unit Lindley Mixed Model

In the applied filed of many areas such as Business and Economics, Physical, Biological, Medical, Environmental, and Social Sciences, Psychology, and Educations outcome measurements like rates, proportions, and fractions are common. Researchers have proven that using normal linear regression to analyze the relationship between outcome measurements such as rates and proportions and a set of independent variables can violates key assumptions of the method. Transformation techniques are usually applied to correct the assumptions, but this practice can violate the probability property of bounded nature of outcome variables. Even though beta regression was one of the popular methods to analyze bounded data situations, the latest development of unit-Lindley distribution and its associate regression was found to be superior. Unit-Lindley regression requires uncorrelated response variables; but in real applied fields, researchers may encounter clustered or correlated response variables. Thus, mixed models that are capable to handle correlated and clustered response variables are required. Therefore, unit-Lindley mixed model, that is capable to analyze correlated and bounded response variable and a set of independent variables is one of the suitable model choices. It is widely accepted and proven by researchers that the Bayesian parameter estimation approach is more advantageous over a classical approach in the case of mixed models in several ways. Despite the several advantages of the Bayesian approach, it has not been applied to the unit-Lindley mixed model. Therefore, this dissertation aims to develop a Bayesian approach to unit-Lindley mixed model. Additionally, the assumption of normality for random effects may not be appropriate when dealing with skewed data. Thus, this dissertation also aims to apply three distributional assumptions for random effects: normal, skew-normal, and skew-t. To achieve the research aims outlined in this dissertation, a Bayesian unit-Lindley mixed model with distributional assumptions of normal, skew-normal, and skew-t for random effects was developed. To implement them, a STAN program code was also developed and presented in this dissertation. A variety of simulated data situations were used to test all models in R. Leave-One-Out Cross-Validation Information Criterion (LOOIC) and Watanabe-Akaike Information Criterion (WAIC) were used to compare the modes. In addition, bias and RMSE of intercept and its variance were also used to support the results of LOOIC and WAIC. The results confirmed that a Bayesian unit-Lindley model with normal assumption of random effects performed better as compered to the model with two other assumptions when the data situation was approximately normal with normal random effects. However, when the response variable is skewed with a skewed random effects, model with the normal assumption was less robust than the model with skew-normal and skew-t random effects. While comparing the model with skew-normal and skew-t random effects, the model with skew-normal produced slightly less biased parameter estimation and RMSE and smaller LOOIC and WAIC. Finally, all the models were applied to analyze the child mortality rates across the countries of South Asia, and their performance was compared. After testing the models by using the simulation method and applying to a real data situation, it was confirmed that a Bayesian unit-Lindley model is ready to apply in any fields of the applied areas where measurement of outcome variable is clustered and bounded within unit interval