Modeling Heteroscedasticity in the Presence of Serial Correlations in Discrete-time Stochastic Series: A GARCH-in-Mean Approach

Imoh Udo Moffat, Emmanuel Alphonsus Akpan

Asian Journal of Probability and Statistics, Page 1-16
DOI: 10.9734/ajpas/2019/v4i230109

Background: In modeling heteroscedasticity of returns, it is often assumed that the series are uncorrelated. In practice, such series with small time periods between observations can be observed to contain significant serial correlations, hence the motivation for this research.

Aim: The aim of this research is to assess the existence of serial correlations in the return series of Zenith Bank Plc, which is targeted at identifying their effects on the parameter estimates of heteroscedastic models.

Materials and Methods: The data were obtained from the Nigerian Stock Exchange spanning from January 3, 2006, to November 24, 2016, having 2690 observations. The hybridized Autoregressive Integrated Moving Average-Generalized Autoregressive Conditional Heteroscedasticity (ARIMA-GARCH-type) models such as Autoregressive Integrated Moving Average-Generalized Autoregressive Conditional Heteroscedasticity (ARIMA-GARCH), Autoregressive Integrated Moving Average-Exponential Generalized Autoregressive Conditional Heteroscedasticity (ARIMA-EGARCH) and the Autoregressive Integrated Moving Average-Glosten, Jagannathan and Runkle Generalized Autoregressive Conditional Heteroscedastic (ARIMA-GJRGARCH) under normal and student-t distributions were employed to model the conditional variance while the GARCH-in-Mean-GARCH-type model corresponding to the selected ARIMA-GARCH-type model was applied to appraise the possible existence of serial correlations.

Results: The findings of this study showed that heteroscedasticity exists and appeared to be adequately captured by ARIMA(2,1,1)-EGARCH(1,1) model under student-t distribution but failed to account for the presence of serial correlations in the series. Meanwhile, its counterpart, GARCH-in-Mean-EGARCH(1,1) model under student-t distribution sufficiently appraised the existence of serial correlations.

Conclusion: One remarkable implication is that the estimates of the parameters of ARIMA-GARCH-type model are likely to be biased when the presence of serial correlations is ignored. Also, the application of GARCH-in-Mean-GARCH-type model possibly provides the feedback mechanism or interaction between the variance and mean equations.

Estimation of Malaria Symptom Data Set using Hidden Markov Model

Drinold Mbete, Kennedy Nyongesa, Joseph Rotich

Asian Journal of Probability and Statistics, Page 1-29
DOI: 10.9734/ajpas/2019/v4i230110

Clinical study of malaria presents a modeling challenge as patients disease status and progress is partially observed and assessed at discrete clinic visit times. Since patients initiate visits based on symptoms, intense research has focused on identication of reliable prediction for exposure, susceptibility to infection and development of severe malaria complications. Despite detailed literature on malaria infection and transmission, very little has been documented in the existing literature on malaria symptoms modeling, yet these symptoms are common. Furthermore, imperfect diagnostic tests may yield misclassication of observed symptoms. Place and Duration of Study: The main objective of this study is to develop a Bayesian Hidden Markov Model of Malaria symptoms in Masinde Muliro University of Science and Technology student population. An expression of Hidden Markov Model is developed and the parameters estimated through the forward-backward algorithm.

Bayesian Estimation of Normal Linear Regression Model with Heteroscedasticity Error Structures

Asian Journal of Probability and Statistics, Page 1-13
DOI: 10.9734/ajpas/2019/v4i230111

Non-constant error variance in Normal Linear Regression Model (NLRM) is an econometric problem generally referred to as heteroscedasticity. Its presence renders statistical inference invalid. Classical approach to its detection, estimation and remediation are widely discussed in the econometric literature. However, estimation of a NLRM using the Bayesian approach when heteroscedasticity problem is present is a major gap in the existing stock of knowledge on this subject. This approach has grown widely in recent times because it combines out-of-sample information with observed data. The study derived Bayesian estimators of the NLRM in the presence of functional forms of heteroscedasticity. Variance was treated as a linear function and as an exponential function of exogenous variables. The estimators are found to be unbiased and consistent and the precision values tend to zero. The estimates obtained from the estimators approximately 95% draws fall within each of the corresponding credible interval. Therefore, the results obtained for the derived Bayesian estimators for different functional forms of heteroscedasticity considered are similar, thus, providing a credible alternative to the existing classical methods which depend solely on the sample information.

Comparing Zero-inflated Poisson, Zero-inflated Negative Binomial and Zero-inflated Geometric in Count Data with Excess Zero

M. I. Adarabioyo, R. A. Ipinyomi

Asian Journal of Probability and Statistics, Page 1-10
DOI: 10.9734/ajpas/2019/v4i230113

Count data often violate the assumptions of a normal distribution due to the fact that they are bounded by their lowest value which is zero. The Poison distribution is sometimes suggested but when the assumption of equal mean and variance is violated due to over-dispersion and presence of zeros we tend to look in the direction of other models. Zero-inflated data falls in this category. The zero-inflated and hurdle models have been found to fit this scenario. The proportions of zero in the data often affect the choice of the models. Our study used the Monte Carlo design to sample 1000 cases from positively skewed distribution with 1.25 as mean vector and 0.10 as zero-inflation parameter. The data was analysed using the method of the maximum likelihood estimation. The Zero-Inflated Poisson, Zero-Inflated Negative Binomial and Zero-Inflated Geometric were fitted; the standard error and Akaike Information Criterion were obtained as measures of model validation with ZIP outperformed ZINB and ZIG.

Review of Bayesian Analysis in Additive Hazards Model

Enrique Ernesto, Alvarez, Maximiliano Luis, Riddick

Asian Journal of Probability and Statistics, Page 1-14
DOI: 10.9734/ajpas/2019/v4i230112

In Survival Analysis, the focus of interest is a time T* until the occurrence of some event. A set of explanatory variables (denoted by a vector Z) is considered to analyze if there is a relationship between any of them and T*. Accordingly, the hazard function'' is defined: $\lambda(t,z) := \lim_{\Delta \downarrow 0} \frac{P[T\leq t+ \Delta \vert T >t,Z=z]}{\Delta} .$ Several models are defined based on this, as is the case of the additive model (among others). Bayesian techniques allow to incorporate previous knowledge or presumption information about the parameters into the model. This area grows extensively since the computationally techniques increase, giving rise to powerful Markov Chain Monte Carlo (MCMC) methods, which allow to generate random samples from the desired distributions. The purpose of this article is to offer a summary of the research developed in Bayesian techniques to approach the additive hazard models.