**1. Introduction**

In recent years, multivariate time series analysis has become an in important research field due to the positive improvements in both methodological and analytical computations. Based on these developments, it has been possible to assess the estimations of parameters in the models of multidimensional and complex time series. Parallelly with these developments, it has been a necessity to model datasets that have simultaneous and frequently changing together. Besides the increase of dataset dimensions, multidimensional volatility models have gained importance in respect of both economic and econometric parameter estimations due to the temporal fluctuations and changes. The information provided by the correlation structures of multidimensional volatility models has contributed a lot especially in optimal portfolio management, risk management, asset allocation and financial decisions. Moreover; as the volatility between different assets and markets can move together, multivariate analysis contributes statistical efficiency [1].

the models, the error was scaled out by normal, Gamma, and Beta distributions; the first one is AFactor-MSVOL-NOR model with normal distribution, the second one is AFactor-MSVOL-St model with Student-t distribution, and the last one is AFactor-MSVOL-Sl robust model with Slash distribution. Estimated AFactor-MSVOL models are bivariate and one-factor structure. Usage of Student-t and Slash distributions, while handling skewness and kurtosis features of returns, enabled a

*Bayesian Analysis of Additive Factor Volatility Models with Heavy-Tailed Distributions…*

Latent factor models prove the notion that high-dimensioned systems are just led by some random resources. Some factors are controlled by these random resources and these factors explain the interaction among the observations. Moreover; latent factor models are an efficient way of estimation of a dynamic covariance matrix. These models enable a decrease in the number of unknown parameters [12].

This model has several attractive features, including parsimony of the parameter space and the ability to capture the common features in asset returns and volatilities. Basic idea of Factor-MSVOL models was taken from multivariate ARCH models. In these models, returns are divided into two additive components. The first component has few factors that capture information about the pricing of all assets, while the other component is the error term that captures asset-specific

Stochastic discount Factor-MSVOL, which is also called as multiplicative Factor-

*iid <sup>N</sup>*ð Þ 0, <sup>Σ</sup>*<sup>ε</sup>* (1)

*iid <sup>N</sup>*ð Þ 0, 1 (2)

MSVOL model, was offered by [13]. He offered Bayesian analysis of structured dynamic factor models. Returns are divided into two multiplicative components in one-factor multiplicative model. As shown below, the first of these components is

scalar common factor and the other one is idiosyncratic error vector:

among the returns derives from the dependency in *εt*.

multiplicative MSVOL model is expanded as k-factor.

**2.2 Additive Factor-MSVOL model**

**35**

*yt* ¼ exp ð Þ *ht=*2 *εt*, *ε<sup>t</sup>* �

*ht*þ<sup>1</sup> ¼ *μ* þ *ϕ*ð Þþ *ht* � *μ η<sup>t</sup> η<sup>t</sup>* �

The first one Σ*<sup>ε</sup>* is accepted as 1 for identification. Compared to the MSVOL model, this model involves lesser parameters and it eases calculation. Different from AFactor-MSVOL model, correlation does not change according to time. Additionally, correlation in log-volatility is always equal to 1. The cross dependence

In [14] developed the one-factor model as k-factor. In their studies, [14] researched both the persistence amount of daily stock returns and the factors affecting common persistence components in volatility. In this study, the one-factor

The Factor-MSVOL model is one of the MSVOL approaches allowing the change of implicitly conditioned correlation matrix in time and producing time-varying correlation. Factor models and factors follow a stochastic volatility process. A kind

flexible approach as an alternative of normal distribution.

*DOI: http://dx.doi.org/10.5772/intechopen.93685*

**2. Model**

information.

**2.1 Multiplicative Factor-MSVOL model**

GARCH and stochastic volatility (SVOL) models, which are widely used in the estimation of volatility, are developed, analysed, and applied within the frame of multivariate the analysis. While multivariate GARCH (MGARCH) models are widely used, MSVOL models are often used in recent years. In his study [2], juxtaposed the most important studies on analysis and development of these models by comparing univariate and multivariate GARCH and SVOL models.

MSVOL models vary in different structures. These structures can be sorted as alternative specifications such as asymmetric models, factor models, time-varying correlation models and matrix exponential transformation, Cholesky decomposition, Wishart autoregressive models [1]. The reason for the limited use of MSVOL is the problems faced in the method of estimation in these models. The most important one among these problems is the problem of high dimension in multivariate analysis and this problem has been eased by using latent factor structures.

Factor-MSVOL models are divided into two groups according to how the factors involved in the mean equation. The first of these structures is additive Factor-MSVOL (AFactor-MSVOL) in which the factors are added summatively and the second one is multiplicative Factor-MSVOL in which the factors are added multiplicatively [3].

AFactor-MSVOL models are firstly offered by Harvey et al. [4]. Afterward, it was developed by [5–9]. The basic idea is taken from factor multivariate ARCH models; additionally, it is a more general state of factor decomposition of covariance structures in multivariate analysis. Returns are divided into two additive components. The first component involves a limited number of factors. The factors capture the information related to the pricing of the whole assets. The other component is the term of an error on the model and it captures the specific information of the asset [1].

Factor-MSVOL models derive from the field of financial econometrics. These models are often preferred to define the terms uncertainty and risk correctly. Asset allocation and asset pricing can be given as an example here. Additionally, it is also used in the arbitrage pricing theory and financial asset pricing model [10]. In comparison with other multivariate stochastic volatility models, Factor-MSVOL models can be estimated with lesser parameters. In this respect, they are parsimonious models in terms of parameters [11]. Factor models both reduce the number of parameters and allow the changing variance structure, it considerably explains the correlation.

Factor-MSVOL models aim to combine a plain, flexible, and robust structure. Like classical factor models, these models are easier in respect of degrading highdimensioned observation area into low-dimensioned orthogonal latent factor area [10]. Moreover; in the long term data, it is assessed with lesser deviation thanks to its being robust in case of unusual observations.

This study aims to model parameter estimations concerning AFactor-MSVOL models with normal distribution, Student-t distribution, Slash distribution assigned on the error within based on the Bayesian approach. For this purpose; S&P500 (Standard & Poor's 500) and SSEC (Shanghai Compound Index) index daily return series, involving the period between 10.20.2014 and 10.17.2019, were used. Among *Bayesian Analysis of Additive Factor Volatility Models with Heavy-Tailed Distributions… DOI: http://dx.doi.org/10.5772/intechopen.93685*

the models, the error was scaled out by normal, Gamma, and Beta distributions; the first one is AFactor-MSVOL-NOR model with normal distribution, the second one is AFactor-MSVOL-St model with Student-t distribution, and the last one is AFactor-MSVOL-Sl robust model with Slash distribution. Estimated AFactor-MSVOL models are bivariate and one-factor structure. Usage of Student-t and Slash distributions, while handling skewness and kurtosis features of returns, enabled a flexible approach as an alternative of normal distribution.
