**2. Model**

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 information.

#### **2.1 Multiplicative Factor-MSVOL model**

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

$$y\_t = \exp\left(h\_t/2\right)\varepsilon\_t,\qquad\qquad\varepsilon\_t \stackrel{iid}{\sim} N(\mathbf{0}, \Sigma\_t)\tag{1}$$

$$h\_{t+1} = \mu + \phi(h\_t - \mu) + \eta\_t \qquad \eta\_t \stackrel{iid}{\sim} N(\mathbf{0}, \mathbf{1}) \tag{2}$$

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

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 multiplicative MSVOL model is expanded as k-factor.

#### **2.2 Additive Factor-MSVOL model**

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

of Factor-SVOL model that does not allow time-varying correlations was offered by [13]. On the other hand, Harvey et al. [4] introduced a common factor in the linearized state-space version of the basic MSVOL model. In this context, the most basic MSVOL model specification is by:

$$\mathcal{Y}\_{it} = \varepsilon\_{it} \left( \exp \left\{ h\_{it} \right\} \right)^{1/2} i = 1, \dots, N \, t = 1, \dots, T \tag{3}$$

Φ ¼ *diag φ*1, …, *φ<sup>p</sup>*þ*<sup>q</sup>*

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

Φ ¼ *diag φ*1, …, *φ<sup>p</sup>*þ*<sup>q</sup>*

*B* is a *p* � *q* matrix of factor loadings. For *i* <*j*, *i* ≤*j bij* ¼ 0, for *i* ≤*qbii* ¼ 1, and all remaining elements are unconstrained. Therefore, each of the factors and errors in this model develops according to SVOL models. Similar to this model, except the fact that *Vt* does not change in time under restriction, another model was handled in [6] and [16]. In [6] estimated their models with Markov Chain Monte Carlo (MCMC) method. On the other hand, [16] showed how to assess MLE with the Efficient Importance Sampling method. Presented by [17], a more generalised version of these models allows spikes in observation equations and the errors are

In [3] showed that additive factor models are by both time-varying volatility and correlations. In this context, they offered two varieties one-Factor SVOL model and they showed that the correlation between two return series is related to the volatility of the factor. According to this, logarithmic returns observed in t period are

. Additionally, when it is showed as **ε<sup>t</sup>** ¼ *ε*1*<sup>t</sup>* ð Þ , *ε*2*<sup>t</sup>* <sup>0</sup>

*<sup>ε</sup>*1, *σ*<sup>2</sup> *ε*2

*ε*2

*<sup>ε</sup>*<sup>2</sup> *exp* ð Þ �*ht*

*iid* N **0**, diag *σ*<sup>2</sup>

and *h0* ¼ 0. This model is offered by [5, 6]. The first component that takes place in return equation involves a small number of factors which includes the information related to the pricing of the whole assets. The second term is error term peculiar to equation; it involves specific information of the asset. A Factor-MSVOL model allows high kurtosis and volatility cluster. It also enables cross dependency in both returns and volatility. ht represents the log-volatility of the common factor (ft) which takes place in A Factor-MSVOL model. The conditional correlation between

*d exp h*ð Þ*<sup>t</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q � �

<sup>¼</sup> *<sup>d</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q � �

*<sup>ε</sup>*<sup>2</sup> ¼ 0 is not, so correlation coefficient changes in time. Correlation

Offered by [3], specification of two varieties one-factor AFactor-MSVOL model,

*<sup>ε</sup>*<sup>1</sup> *exp* ð Þ �*ht <sup>d</sup>*<sup>2</sup> <sup>þ</sup> *<sup>σ</sup>*<sup>2</sup>

dynamics is dependent on the dynamics of ht; likewise, the correlation is an increasing function of ht. It refers that the correlation will be high as much as the

*ε*1 � � *<sup>d</sup>*<sup>2</sup> *exp h*ð Þþ*<sup>t</sup> <sup>σ</sup>*<sup>2</sup>

*exp h*ð Þþ*<sup>t</sup> <sup>σ</sup>*<sup>2</sup>

1 þ *σ*<sup>2</sup>

which allows heavy-tailed distribution, is as below:

, two varieties one-Factor MSVOL

� � � � (18)

*iid* N 0, 1 ð Þ (19)

, *η<sup>t</sup>* ¼

(20)

(21)

and *ht* ¼ *h*1*<sup>t</sup>*, …, *hpt*, *hp*þ1,*t*,…,*hp*þ*q*,*<sup>t</sup>*

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

distributed by heavy-tailed-t [18].

, *y*2*<sup>t</sup>* � �<sup>0</sup>

, *μ<sup>t</sup>* ¼ *μ*1*<sup>t</sup>*, *μ*2*<sup>t</sup>* ð Þ<sup>0</sup> and *h<sup>t</sup>* ¼ *h*1,*<sup>t</sup>* ð Þ , *h*2,*<sup>t</sup>* <sup>0</sup>

**y***<sup>t</sup>* ¼ *D ft* þ **ε***t***ε***<sup>t</sup>* �

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

expressed as **y***<sup>t</sup>* ¼ *y*1*<sup>t</sup>*

*y*1*<sup>t</sup>* and *y*2*<sup>t</sup>* is as below:

common factor volatility is high.

*σ*2 *<sup>ε</sup>*<sup>1</sup> <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup>

**37**

models are such as below:

*η*1*<sup>t</sup>*, *η*2*<sup>t</sup>* ð Þ<sup>0</sup>

� �*:*

� � (14)

� � (16)

<sup>Σ</sup>*ηη* <sup>¼</sup> *diag <sup>σ</sup>*1,*ηη*, …, *<sup>σ</sup><sup>p</sup>*þ*q*,*ηη* � � (15)

<sup>Σ</sup>*ηη* <sup>¼</sup> *diag <sup>σ</sup>*1,*ηη*, …, *<sup>σ</sup><sup>p</sup>*þ*q*,*ηη* � � (17)

*yit* refers to the observation values in t period of i serial. *ε<sup>t</sup>* ¼ *ε*1*<sup>t</sup>* ð Þ , …, *εNt* <sup>0</sup> is the error vector which shows normal distribution with Σε covariance matrix and 0 mean. Diagonal elements of Σε covariance matrix are unity and off-diagonal elements are defined as *ρ*ij. Variance of this model is produced by AR(1) process:

$$h\_{\rm it} = \gamma\_i + qh\_{\rm it-1} + \eta\_{\rm it}i = \mathbf{1}, \ldots, N \tag{4}$$

Here, *<sup>η</sup><sup>t</sup>* <sup>¼</sup> *<sup>η</sup>*1*<sup>t</sup>*, …, *<sup>η</sup>Nt* ð Þ<sup>0</sup> with 0 mean and multivariate of <sup>P</sup> *<sup>η</sup>* matrix is normal. This model, Eq. (4), *N* � 1 *ht*, can be generalised as multivariate AR(1) and even ARMA process. If we handle the multivariate random walk model of *ht*, which is its special case:

$$w\_t = -\mathbf{1}.\mathbf{27}i + h\_t + \xi\_t \tag{5}$$

$$h\_t = h\_{t-1} + \eta\_t \tag{6}$$

*wt* and *<sup>ξ</sup><sup>t</sup>* elements are *<sup>N</sup>* � <sup>1</sup> vectors in case *wit* <sup>¼</sup> log *<sup>y</sup>*<sup>2</sup> *it* and *<sup>ξ</sup>***<sup>t</sup>** <sup>¼</sup> log *<sup>ε</sup>*<sup>2</sup> *it* þ 1*:*27 *i* ¼ 1, …, *N*. *i* is *N* � 1 vector which is composed of unit values.

Common factors can be included in multivariate stochastic variance models; they are unobservable components of time series models. In [4] modelled with a multivariate random walk by considering the persistence in volatility. According to this, Eq. (4) is by:

$$
\omega\_t = -\mathbf{1}.27i + \theta h\_t + \overline{h} + \xi\_t,\tag{7}
$$

$$\begin{aligned} h\_t &= h\_{t-1} + \eta\_t \\ Var(\eta\_t) &= \sum\_{\eta} \end{aligned} \tag{8}$$

As *θ* k≤ N, *N* � *k* parameter matrix, *ht* and *η***<sup>t</sup>** *k* � 1 vectors, *Ση k* � *k* positively defined matrix, *h* is an *N* � 1 vector in which the first *k* elements are zeros and the last *N* � *k* elements are unbounded, Harvey et al. [4] estimated this model with QML method. Common factors are transformed as *<sup>θ</sup>* **<sup>∗</sup>** <sup>¼</sup> *<sup>θ</sup>R*<sup>0</sup> and *<sup>h</sup>* **<sup>∗</sup>** *<sup>t</sup>* ¼ *Rh***<sup>t</sup>** to evaluate the factor loading [4].

Following the model offered by [15], another kind of MSVOL factor model was handled by [8] as below:

$$\mathbf{y}\_t = \mathbf{B}\mathbf{f}\_t + \mathbf{V}\_t^{1/2}\boldsymbol{\varepsilon}\_t\mathbf{e}\_t \sim \mathbf{N}\_\mathbf{p}(\mathbf{0}, \mathbf{I})\tag{9}$$

$$f\_t = D\_t^{1/2} \gamma\_t \gamma\_t \sim N\_q(0, I) \tag{10}$$

$$h\_{t+1} = \mu + \Phi(h\_t - \mu) + \eta\_t \eta\_t \sim N\_{p+q}\left(0, \sum\_{\eta \eta} \right) \tag{11}$$

$$\mathbf{V\_{t}} = \operatorname{diag}\left(\exp\left(h\_{1t}\right), \dots, \exp\left(h\_{pt}\right)\right) \tag{12}$$

$$\mathbf{D}\_{\mathbf{t}} = \operatorname{diag} \left( \exp \left( h\_{p+1\downarrow} \right), \dots, \exp \left( h\_{p+q\downarrow} \right) \right) \tag{13}$$

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

$$\Phi = \text{diag}\left(\rho\_1, \dots, \rho\_{p+q}\right) \tag{14}$$

$$\Sigma\_{\eta\eta} = \text{diag}\left(\sigma\_{1,\eta\eta}, \dots, \sigma\_{p+q,\eta\eta}\right) \tag{15}$$

$$\Phi = \text{diag}\left(\rho\_1, \dots, \rho\_{p+q}\right) \tag{16}$$

$$\Sigma\_{\eta\eta} = \operatorname{diag} \left( \sigma\_{1,\eta\eta}, \dots, \sigma\_{p+q,\eta\eta} \right) \tag{17}$$

and *ht* ¼ *h*1*<sup>t</sup>*, …, *hpt*, *hp*þ1,*t*,…,*hp*þ*q*,*<sup>t</sup>* � �*:*

*B* is a *p* � *q* matrix of factor loadings. For *i* <*j*, *i* ≤*j bij* ¼ 0, for *i* ≤*qbii* ¼ 1, and all remaining elements are unconstrained. Therefore, each of the factors and errors in this model develops according to SVOL models. Similar to this model, except the fact that *Vt* does not change in time under restriction, another model was handled in [6] and [16]. In [6] estimated their models with Markov Chain Monte Carlo (MCMC) method. On the other hand, [16] showed how to assess MLE with the Efficient Importance Sampling method. Presented by [17], a more generalised version of these models allows spikes in observation equations and the errors are distributed by heavy-tailed-t [18].

In [3] showed that additive factor models are by both time-varying volatility and correlations. In this context, they offered two varieties one-Factor SVOL model and they showed that the correlation between two return series is related to the volatility of the factor. According to this, logarithmic returns observed in t period are expressed as **y***<sup>t</sup>* ¼ *y*1*<sup>t</sup>* , *y*2*<sup>t</sup>* � �<sup>0</sup> . Additionally, when it is showed as **ε<sup>t</sup>** ¼ *ε*1*<sup>t</sup>* ð Þ , *ε*2*<sup>t</sup>* <sup>0</sup> , *η<sup>t</sup>* ¼ *η*1*<sup>t</sup>*, *η*2*<sup>t</sup>* ð Þ<sup>0</sup> , *μ<sup>t</sup>* ¼ *μ*1*<sup>t</sup>*, *μ*2*<sup>t</sup>* ð Þ<sup>0</sup> and *h<sup>t</sup>* ¼ *h*1,*<sup>t</sup>* ð Þ , *h*2,*<sup>t</sup>* <sup>0</sup> , two varieties one-Factor MSVOL models are such as below:

$$\mathbf{y}\_t = Df\_t + \mathbf{e}\_t \mathbf{e}\_t \overset{iid}{\sim} \mathbf{N}\left(\mathbf{0}, \text{diag}\left(\sigma\_{\epsilon1}^2, \sigma\_{\epsilon2}^2\right)\right) \tag{18}$$

$$h\_{t+1} = \mu + \varrho(h\_t \cdot \mu) + \sigma\_\eta \eta\_t, \eta\_t \stackrel{iid}{\sim} \mathbf{N}(\mathbf{0}, \mathbf{1}) \tag{19}$$

and *h0* ¼ 0. This model is offered by [5, 6]. The first component that takes place in return equation involves a small number of factors which includes the information related to the pricing of the whole assets. The second term is error term peculiar to equation; it involves specific information of the asset. A Factor-MSVOL model allows high kurtosis and volatility cluster. It also enables cross dependency in both returns and volatility. ht represents the log-volatility of the common factor (ft) which takes place in A Factor-MSVOL model. The conditional correlation between *y*1*<sup>t</sup>* and *y*2*<sup>t</sup>* is as below:

$$\frac{d\exp\left(h\_t\right)}{\sqrt{\left(\exp\left(h\_t\right) + \sigma\_{\epsilon1}^2\right)\left(d^2\exp\left(h\_t\right) + \sigma\_{\epsilon2}^2\right)}}\tag{20}$$

$$=\frac{d}{\sqrt{1+\sigma\_{\epsilon1}^2 \exp\left(-h\_t\right)\left(d^2+\sigma\_{\epsilon2}^2 \exp\left(-h\_t\right)\right)}}\tag{21}$$

*σ*2 *<sup>ε</sup>*<sup>1</sup> <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup> *<sup>ε</sup>*<sup>2</sup> ¼ 0 is not, so correlation coefficient changes in time. Correlation dynamics is dependent on the dynamics of ht; likewise, the correlation is an increasing function of ht. It refers that the correlation will be high as much as the common factor volatility is high.

Offered by [3], specification of two varieties one-factor AFactor-MSVOL model, which allows heavy-tailed distribution, is as below:

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

$$\mathbf{y}\_t = Df\_t + \mathbf{e}\_t \mathbf{e}\_t \stackrel{iid}{\sim} \mathbf{t}\left(\mathbf{0}, \text{diag}\left(\sigma\_{\varepsilon1}^2, \sigma\_{\varepsilon2}^2\right), \mathbf{v}\right) \tag{22}$$

$$f\_t = \exp\left(h\_t/2\right)u\_t, u\_t \stackrel{\text{iid}}{\sim} \mathbf{t}(\mathbf{0}, \mathbf{1}, \boldsymbol{\alpha}),\tag{23}$$

*Yt* <sup>¼</sup> <sup>100</sup> � ð Þ log *Pt*‐ log *Pt*‐<sup>1</sup> (29)

*Yt* (30)

**S&P500 SSEC**

*yt* <sup>¼</sup> *Yt* � <sup>1</sup>

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

and SSEC return series are given.

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

**Figure 1.**

**Table 1.**

**39**

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

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

*T* X *T*

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

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

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

results show that series do not have a normal distribution.

heteroscedastic structures in the analyses of volatility in return series.

**Sample size** 1177 1177 **Mean** �0.2784 �0.2862 **Maximum** 2.1777 2.3282 **Minimum** �2.112 �4.1485 **Standard deviation** 0.39 0.6802 **Skewness** �0.1463 �0.9852 **Kurtosis** 4.7226 6.0157 **Jarque-Bera (possibility)** 1098.0 (3.7495e-239) 1965.2 (0.0000)

*t*¼1

$$h\_{t+1} = \mu + \varrho(h\_t \cdot \mu) + \sigma\_\eta \eta\_t, \eta\_t \stackrel{iid}{\sim} \mathbf{N}(\mathbf{0}, \mathbf{1}) \tag{24}$$

h0 ¼ μ, **v** ¼ ð Þ v1, v2 <sup>0</sup> . In this model, heavy-tailed Studentt distribution is used for return shocks. The conditional correlation between *y*1*<sup>t</sup>* and *y*2*<sup>t</sup>* is as below:

$$\frac{d}{\sqrt{\mathbf{1} + \frac{a\mathbf{e}}{v\_1}\sigma\_{\epsilon1}^2 \exp\left(-h\_t\right) \left(d^2 + \frac{a\mathbf{e}}{v\_2}\sigma\_{\epsilon2}^2 \exp\left(-h\_t\right)\right)}}\tag{25}$$

In addition to these models, AFactor-MSVOL-Sl model in which error distribution is scaled with Slash distribution is defined as:

The error for the AFactor-MSVOL-Sl model is shown as Slash distribution *<sup>ε</sup><sup>t</sup> <sup>υ</sup><sup>t</sup>* ð Þ� <sup>j</sup> *slash* 0,<sup>P</sup> *<sup>ε</sup>*, *<sup>υ</sup>* � � with *<sup>σ</sup>*<sup>2</sup> *<sup>ε</sup><sup>t</sup>* � *beta*ð Þ *α*, *β* prior distribution.

$$\mathbf{y}\_t = Df\_t + \mathbf{e}\_t, \mathbf{e}\_t \sim \text{slash}\left(\mathbf{0}, \text{diag}\left(\sigma\_{\varepsilon1}^2, \sigma\_{\varepsilon2}^2\right), \mathbf{v}\right) \tag{26}$$

$$f\_t = \exp\left(h\_t/2\right)u\_t, u\_t \sim \text{slash}(0, \mathbf{1}, o) \tag{27}$$

$$h\_{t+1} = \mu + \varrho(h\_t \cdot \mu) + \sigma\_\eta \eta\_t, \eta\_t \stackrel{iid}{\sim} \mathbf{N}(\mathbf{0}, \mathbf{1}) \tag{28}$$

Philipov and Glickman [19] offered high-dimensioned additive factor-MSVOL models in their studies. In this study, factor covariance matrix is led by Wishart random process. On the other hand, it is known that daily return series are leptokurtic. In context of stochastic volatility, [20] and [21] presented empirical proofs on the usage of heavy-tailed distribution in conditioned mean equation. Moreover, [22] analysed SVOL models with Student-t distribution and GED. Daily data analysis of JPY/Dollar and TOPIX were carried out by the method of MCMC. Comparison of distributions, in respect of accordance, was calculated with Bayesian factor values. It is determined that SVOL-t model assorts with both of the data compared to SVOL-normal and SVOL-GED models.

In [23] analysed new-class linear factor models. In these models, factors are latent and covariance matrix is followed with MSVOL process. Wu et al. [24] proposed dynamic correlated latent factor SVOL model structure in his studies. According to the results of analysis led by MCMC method, statistically comprehensible results were obtained for financial and economic data.
