**3. Empirical analysis**

#### **3.1 Dataset**

This study aims to model parameter estimations concerning AFactor-MSVOL models with Student-t, Slash and normal distributions assigned to the error. For this purpose; S&P500 and SSEC index daily return series, involving the period between 10.20.2014 and 10.17.2019, were used. Among the models the error was scaled by normal, Gamma, and Beta distributions; the first one is AFactor-MSV-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. Analyses of data were carried out with R and WinBugs programmes. Daily mean logarithmic return series were determined by:

*Bayesian Analysis of Additive Factor Volatility Models with Heavy-Tailed Distributions… DOI: http://dx.doi.org/10.5772/intechopen.93685*

$$Y\_t = \mathbf{100} \times (\log P\_{t^-} \log P\_{t^-1}) \tag{29}$$

$$y\_t = Y\_t - \frac{1}{T} \sum\_{t=1}^{T} Y\_t \tag{30}$$

S&P500 index is composed of stocks of the most valuable 500 companies in USA. On the other hand, SSEC has the most important and the biggest companies of China. Commercial and financial relations between the USA and China not only affect themselves but also global economy. Commercial and financial tensions between them and the anxieties on currency wars can negatively affect Asia and Europe stock markets. Therefore; index values of two grand economies such as China and USA are preferred for analyses. In **Figure 1**, time series plots for S&P500 and SSEC return series are given.

Descriptive statistic values of S&P500 and SSEC series are given in **Table 1**. S&P500 and SSEC series have negative mean returns. It seems that SSEC return series have more volatility. Moreover, both of the series are negatively skew. Kurtosis level is higher for both S&P500 and SSEC. Jarque-Bera normality test results show that series do not have a normal distribution.

In **Table 2**, Ljung-Box and ARCH-LM test results are illustrated in some lags. As Q statistics of Ljung-Box test are examined, null hypothesis that there is not autocorrelation is rejected for both of the series in 20th and 50th lags. It refers that autocorrelation exists in series. According to the ARCH test results, ARCH effect is seen in the whole series. It shows the necessity of preferring the models allowing heteroscedastic structures in the analyses of volatility in return series.

#### **Figure 1.**

*Time series plots for S&P500 and SSEC returns.*


#### **Table 1.**

*Descriptive statistics of S&P500 and SSEC return series.*


method used in case posterior distribution has a closed-form and it is a kind of iterative method reproducing random values from these values. The full conditional density function is obtained by Gibbs sampling as all the unknown parameters are

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

In this study, parameter estimations are obtained by modelling three different prior distributions assigned on the error term. In modelling, error term is scaled with *λ* variable and normal/independent (or scaled mixture) defined distributions are used. As y variable, which shows normal/independent distribution, is expressed

*y* ¼ *μ* þ

*e* ffiffi *λ*

Here *μ* is a mean vector, *e* is error vector and have normal distribution. *λ* variable that takes place in the model shows different distributions according to the degrees of freedom of *υ*, and it is defined as random variable with positive valence. As degrees of freedom goes infinite, *λ* variable is 1 and the error term shows normal

> <sup>2</sup> , *<sup>υ</sup>* 2

As an addition to the AFactor-MSVOL offered by [3] and heavy-tailed AFactor-MSVOL models, bivariate one-factor AFactor-MSVOL model in which the error

In **Table 3**, posterior mean values of the parameters, standard errors and 95% credible intervals are shown. Using different initial values for each model, two chains are formed. Total iteration number in each chain is determined as 500,000

> Sd 0.2899 0.292 0.614 %95 CI [�2.208, �1.054] [�2.231, �1.072] [�5.531, �3.102]

Sd 0,005198 0.006 0.055 %95 CI [0.9788,0.9988] [0.978, 0.999] [0.708, 0.926]

Sd 36.89 36.339 0.076 %95 CI [42.81, 183.0] [39.950,177.002] [0.496, 0.793]

Sd 0.02329 0.018 0.052 %95 CI [0.1335,0.2248] [0.123,0.194] [0.074, 0.278]

Sd 0.006022 0.007 0.042 %95 CI [3.984E-8, 0.0216] [0.000,0.023] [0.260, 0.426]

Sd 0.01054 0.011 0.037 %95 CI [0.1583, 0.1996] [0.131, 0.174] [0.242, 0.387]

**μ** Mean �1.59 �1.586 �3.930

**Ø** Mean 0.9910 0.991 0.830

**<sup>2</sup>** Mean 95.68 87.230 0.653

**d** Mean 0.178 0.158 0.174

**<sup>ε</sup><sup>1</sup>** Mean 0.003672 0.001 0.345

**<sup>ε</sup><sup>2</sup>** Mean 0.1781 0.151 0.309

**AFactor-MSVOL-NOR AFactor-MSVOL-St AFactor-MSVOL-Sl**

term is scaled with Slash distribution is estimated in the analysis.

and as *λ* variate shows Betað Þ *υ*, 1 distribution in [0,1] closed interval, it converges

p (31)

� � distribution, it converges Student-t;

given and parameters are estimated with this method.

in longitudinal model given below [25];

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

distribution. As *λ* variate shows Gamma *<sup>υ</sup>*

Slash distribution.

**3.3 Findings**

**ση**

**σ2**

**σ2**

**41**

#### **Table 2.**

*Ljung-box and ARCH-LM test results.*

#### **3.2 Bayesian estimation**

The most important factor, which limits the usage of Factor-MSVOL models, is difficulty in estimating the statistics, whereupon some methods were offered for estimation. In these methods, quasi maximum likelihood, simulated maximum likelihood, and Bayesian MCMC are offered as the most efficient methods. Bayesian MCMC method is very efficient against high dimension problems of the dataset [8, 9, 17].

In this study, parameter estimations are obtained by the Bayesian approach. As it is known, in parameter estimation it is supposed that the error term shows the normal distribution, but this assumption is not valid in case unusual points exist, therefore error term has a heterogeneous variance. This case is often faced in longitudinal datasets. In case unusual points exist in datasets, researchers generally prefer some strategies such as keeping the outliers, removing outliers, and recoding outliers. If keeping the outliers is chosen, the heavy-tailed distribution must be preferred rather than normal distribution. Otherwise, it causes statistical inferences.

In recent years, multidimensional analytical operations in computational science have become easier thanks to the advances in computer technology. In parallel with these advances and usage of the Bayesian approach, using more robust models in analyses has increased in the observation of unusual points. In the Bayesian approach, model parameters are random variables and it is supposed that it shows a known distribution. The Bayesian approach relies on the combination of subjective experiences of the researcher, the prior information obtained from the former studies, and the likelihood obtained from data. Posterior information is achieved from the combination with prior information. This information is defined with a known distribution function and parameter estimations are achieved from the posterior distribution.

#### Posterior ∝Prior X Likelihood

In the Bayesian approach, in obtained of the posterior distribution of parameters requires multidimensional integral computations in multidimensional and longitudinal datasets. This difficulty is overcome by the development of iterative methods such as MCMC. MCMC methods are based on the randomly generate parameter values from posterior distribution; thus, some analytically difficult problems are easily solved by simulation techniques. In this study, parameter estimations are obtained by Gibbs sampling which is also a MCMC method. Gibbs sampling is a

*Bayesian Analysis of Additive Factor Volatility Models with Heavy-Tailed Distributions… DOI: http://dx.doi.org/10.5772/intechopen.93685*

method used in case posterior distribution has a closed-form and it is a kind of iterative method reproducing random values from these values. The full conditional density function is obtained by Gibbs sampling as all the unknown parameters are given and parameters are estimated with this method.

In this study, parameter estimations are obtained by modelling three different prior distributions assigned on the error term. In modelling, error term is scaled with *λ* variable and normal/independent (or scaled mixture) defined distributions are used. As y variable, which shows normal/independent distribution, is expressed in longitudinal model given below [25];

$$y = \mu + \frac{e}{\sqrt{\lambda}}\tag{31}$$

Here *μ* is a mean vector, *e* is error vector and have normal distribution. *λ* variable that takes place in the model shows different distributions according to the degrees of freedom of *υ*, and it is defined as random variable with positive valence. As degrees of freedom goes infinite, *λ* variable is 1 and the error term shows normal distribution. As *λ* variate shows Gamma *<sup>υ</sup>* <sup>2</sup> , *<sup>υ</sup>* 2 � � distribution, it converges Student-t; and as *λ* variate shows Betað Þ *υ*, 1 distribution in [0,1] closed interval, it converges Slash distribution.

#### **3.3 Findings**

As an addition to the AFactor-MSVOL offered by [3] and heavy-tailed AFactor-MSVOL models, bivariate one-factor AFactor-MSVOL model in which the error term is scaled with Slash distribution is estimated in the analysis.

In **Table 3**, posterior mean values of the parameters, standard errors and 95% credible intervals are shown. Using different initial values for each model, two chains are formed. Total iteration number in each chain is determined as 500,000


#### *Linear and Non-Linear Financial Econometrics - Theory and Practice*


#### **Table 3.**

*Posterior mean values of the parameters in the AFactor-MSVOL models.*

and the iteration number that must be omitted in the burn-in is 250,000. Thus, when the first burn-in period of 250,000 is omitted, a Gibbs chain of 250,000 is obtained for each parameter by means of saving each iteration value.

It is seen that for AFactor-MSVOL and AFactor-MSVOL-St models Ø parameter of posterior mean value is so close to the unit value. It refers that latent volatility had random walk behaviour. On the other hand, factor process for all the models was highly obtained. It is seen that standard deviation of posterior mean value of Ø parameter is too low. According to this, logarithmic volatility of time-varying latent components shows persistent features. Posterior mean value of Ø parameter is lower in AFactor-MSVOL-Sl model in comparison to the other models, while the posterior means of *ϕ* are all nearby unity and seem to propose random walk behaviour for *ht*. The mean of *ϕ* is close to unity with a low standard deviation under all specifications, offering persistent time-varying log-volatility for latent components. Factor loading for the estimated models are determined as 0.178, 0.158, and 0.17, respectively. The overall variance-covariance is decomposed into a component which is due to the variation in the common factor and a component reflecting the variation in the idiosyncratic errors. Diebold and Nerlov [26] suggest the common factor reflects the flow of new information relevant to the pricing of all assets, upon which asset-specific shocks represented by the idiosyncratic errors are superimposed (**Figures 2**, **3** and **4**).

Gelman-Rubin statistics is an approach to determining convergence. According to it, convergence takes place in case means of variance within the chain and the variance values between the chains are equal. In this case, Gelman-Rubin statistics is about 1. In **Table 4**, Gelman-Rubin statistics of estimated models for parameter estimation take place. According to this, it is seen that all the parameters take 1

**μ** 1.00 1.00 1.01 **Ø** 1.00 1.00 1.00

**<sup>2</sup>** 1.02 1.00 1.00 **d** 1.00 1.00 1.00

**<sup>ε</sup><sup>1</sup>** 1.01 1.04 1.00

**AFactor-MSVOL-NOR AFactor-MSVOL-St AFactor-MSVOL-Sl**

value of Gelman-Rubin statistics and convergence occurs.

*Kernel density of AFactor-MSVOL-SL model μ and Ø parameters.*

*Kernel density of AFactor-MSVOL-St model v1 and v2 parameters.*

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

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

**Figure 3.**

**Figure 4.**

**ση**

**σ2**

**43**

**Figure 2.** *Kernel density estimation of AFactor-MSVOL-NOR model μ and Ø parameters.*

*Bayesian Analysis of Additive Factor Volatility Models with Heavy-Tailed Distributions… DOI: http://dx.doi.org/10.5772/intechopen.93685*

**Figure 3.** *Kernel density of AFactor-MSVOL-St model v1 and v2 parameters.*

**Figure 4.** *Kernel density of AFactor-MSVOL-SL model μ and Ø parameters.*

Gelman-Rubin statistics is an approach to determining convergence. According to it, convergence takes place in case means of variance within the chain and the variance values between the chains are equal. In this case, Gelman-Rubin statistics is about 1. In **Table 4**, Gelman-Rubin statistics of estimated models for parameter estimation take place. According to this, it is seen that all the parameters take 1 value of Gelman-Rubin statistics and convergence occurs.



points. Therefore; it is seen that Student-t and Slash distributions are applicable as an alternative of normal distribution in the analysis of financial return series. Moreover, it is possible to say that heavy-tailed distributions can substitute normal

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

This study was presented at the 20th International Symposium on Econometrics, Operations Research and Statistics, 12–14 February 2016, Ankara, Turkey. (Oral

distribution in case deviated observation values are not present.

\* and Burcu Mestav<sup>2</sup>

Sciences, Çanakkale Onsekiz Mart University, Çanakkale, Turkey

\*Address all correspondence to: verdaatmaca@comu.edu.tr

1 Department of Econometrics, Biga Faculty of Economics and Administrative

2 Department of Statistics, Faculty of Arts and Sciences, Çanakkale Onsekiz Mart

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

**Acknowledgements**

**Additional classification**

**JEL codes:** C11, C46, G19, C59, C15.

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

presentation).

**Author details**

**45**

Verda Davasligil Atmaca<sup>1</sup>

University, Çanakkale, Turkey

provided the original work is properly cited.

#### **Table 4.**

*Gelman-Rubin diagnostic test.*

DIC allows comparison between the models by taking into consideration the complexity of the model [27, 28]. pd is expressed as efficient parameter number. pd model gives the approximate value of parameter number and measures the complexity of the model. DIC can take both negative and positive values. It causes negative valorisation of both deviation and DIC. In conclusion, the model with the lowest DIC value must be chosen from alternative models [29]. In **Table 5**, DIC values of each three values are given; according to this, the model with the lowest DIC values should be chosen.


**Table 5.** *DIC values.*
