Bayesian Analysis of Additive Factor Volatility Models with Heavy-Tailed Distributions with Specific Reference to S&P 500 and SSEC Indices

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:

The first one Σεis 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:

yitrefers to the observation values in t period of i serial. εt=ε1t…εNt′is the error vector which shows normal distribution with Σεcovariance matrix and 0 mean. Diagonal elements of Σεcovariance matrix are unity and off-diagonal elements are defined as ρij. Variance of this model is produced by AR(1) process:

Here, ηt=η1t…ηNt′with 0 mean and multivariate of ∑ηmatrix is normal. This model, Eq. (4), N×1ht, can be generalised as multivariate AR(1) and even ARMA process. If we handle the multivariate random walk model of h_t, which is its special case:

wtand ξtelements are N×1_{vectors in case} wit=logyit2and ξt=logεit2+1.27i=1,…,N. iis N×1vector 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:

_As θk≤N, N×kparameter matrix, htand ηtk×1vectors, Σηk×kpositively defined matrix, h¯is an N×1vector in which the first kelements are zeros and the last N−kelements are unbounded, Harvey et al. [4] estimated this model with QML method. Common factors are transformed as θ∗=θR′and ht∗=Rhtto evaluate the factor loading [4].

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

and ht=h1t…hpthp+1,t,…,hp+q,t.

Bis a p×qmatrix of factor loadings. For i<j,i≤jbij=0, for i≤qbii=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 Vtdoes 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 tperiod are expressed as yt=y1ty2t′. Additionally, when it is showed as εt=ε1tε2t′, ηt=η1tη2t′, μt=μ1tμ2t′and ht=h1,th2,t′, two varieties one-Factor MSVOL models are such as below:

and h0=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. htrepresents the log-volatility of the common factor (ft) which takes place in A Factor-MSVOL model. The conditional correlation between y1tand y2tis as below:

σε12=σε22=0is 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:

h0=μ, v=v1v2′. In this model, heavy-tailed Studenttdistribution is used for return shocks. The conditional correlation between y1tand y2tis as below:

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 εtυt∼slash0∑ευwith σεt2∼betaαβprior distribution.

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.

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:

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.

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.

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 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 λvariable 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];

Here μis a mean vector, eis error vector and have normal distribution. λvariable 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, λvariable is 1 and the error term shows normal distribution. As λvariate shows Gamma υ2υ2distribution, it converges Student-t; and as λvariate shows Betaυ1distribution 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 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 ϕare all nearby unity and seem to propose random walk behaviour for ht. The mean of ϕis 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 value of Gelman-Rubin statistics and convergence occurs.

DIC allows comparison between the models by taking into consideration the complexity of the model [27, 28]. p_d is expressed as efficient parameter number. p_d 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.