Multivariate Stochastic Volatility Models: Bayesian Estimation and Model Comparison

In this paper we show that fully likelihood-based estimation and comparison of multivariate stochastic volatility (SV) models can be easily performed via a freely available Bayesian software called WinBUGS. Moreover, we introduce to the literature several new specifications that are natural extensions to certain existing models, one of which allows for time-varying correlation coefficients. Ideas are illustrated by fitting, to a bivariate time series data of weekly exchange rates, nine multivariate SV models, including the specifications with Granger causality in volatility, time-varying correlations, heavy-tailed error distributions, additive factor structure, and multiplicative factor structure. Empirical results suggest that the best specifications are those that allow for time-varying correlation coefficients.     

Common-factor stochastic volatility modelling with observable proxy

Multi-asset modelling is of fundamental importance to financial applications such as risk management and portfolio selection. In this article, we propose a multivariate stochastic volatility modelling framework with a parsimonious and interpretable correlation structure. Building on well-established evidence of common volatility factors among individual assets, we consider a multivariate diffusion process with a common-factor structure in the volatility innovations. Upon substituting an observable market proxy for the common volatility factor, we markedly improve the estimation of several model parameters and latent volatilities. The model is applied to a portfolio of several important constituents of the S&P500 in the financial sector, with the VIX index as the common-factor proxy. We find that the prediction intervals for asset forecasts are comparable to those of more complex dependence models, but that option-pricing uncertainty can be greatly reduced by adopting a common-volatility structure. The Canadian Journal of Statistics 48: 36–61; 2020 © 2020 Statistical Society of Canada     

Factor Stochastic Volatility in Mean Models: A GMM Approach

This paper provides a semiparametric framework for modeling multivariate conditional heteroskedasticity. We put forward latent stochastic volatility (SV) factors as capturing the commonality in the joint conditional variance matrix of asset returns. This approach is in line with common features as studied by Engle and Kozicki (1993), and it allows us to focus on identication of factors and factor loadings through first- and second-order conditional moments only. We assume that the time-varying part of risk premiums is based on constant prices of factor risks, and we consider a factor SV in mean model. Additional specification of both expectations and volatility of future volatility of factors provides conditional moment restrictions, through which the parameters of the model are all identied. These conditional moment restrictions pave the way for instrumental variables estimation and GMM inference.     

Multivariate Stochastic Volatility: A Review

The literature on multivariate stochastic volatility (MSV) models has developed significantly over the last few years. This paper reviews the substantial literature on specification, estimation, and evaluation of MSV models. A wide range of MSV models is presented according to various categories, namely, (i) asymmetric models, (ii) factor models, (iii) time-varying correlation models, and (iv) alternative MSV specifications, including models based on the matrix exponential transformation, the Cholesky decomposition, and the Wishart autoregressive process. Alternative methods of estimation, including quasi-maximum likelihood, simulated maximum likelihood, and Markov chain Monte Carlo methods, are discussed and compared. Various methods of diagnostic checking and model comparison are also reviewed.     

Multivariate Stochastic Volatility: A Review

The literature on multivariate stochastic volatility (MSV) models has developed significantly over the last few years. This paper reviews the substantial literature on specification, estimation, and evaluation of MSV models. A wide range of MSV models is presented according to various categories, namely, (i) asymmetric models, (ii) factor models, (iii) time-varying correlation models, and (iv) alternative MSV specifications, including models based on the matrix exponential transformation, the Cholesky decomposition, and the Wishart autoregressive process. Alternative methods of estimation, including quasi-maximum likelihood, simulated maximum likelihood, and Markov chain Monte Carlo methods, are discussed and compared. Various methods of diagnostic checking and model comparison are also reviewed.     

Monte Carlo Likelihood Estimation for Three Multivariate Stochastic Volatility Models

Estimating parameters in a stochastic volatility (SV) model is a challenging task. Among other estimation methods and approaches, efficient simulation methods based on importance sampling have been developed for the Monte Carlo maximum likelihood estimation of univariate SV models. This paper shows that importance sampling methods can be used in a general multivariate SV setting. The sampling methods are computationally efficient. To illustrate the versatility of this approach, three different multivariate stochastic volatility models are estimated for a standard data set. The empirical results are compared to those from earlier studies in the literature. Monte Carlo simulation experiments, based on parameter estimates from the standard data set, are used to show the effectiveness of the importance sampling methods.     

Bayesian analysis of multivariate stochastic volatility with skew return distribution

Multivariate stochastic volatility models with skew distributions are proposed. Exploiting Cholesky stochastic volatility modeling, univariate stochastic volatility processes with leverage effect and generalized hyperbolic skew t-distributions are embedded to multivariate analysis with time-varying correlations. Bayesian modeling allows this approach to provide parsimonious skew structure and to easily scale up for high-dimensional problem. Analyses of daily stock returns are illustrated. Empirical results show that the time-varying correlations and the sparse skew structure contribute to improved prediction performance and Value-at-Risk forecasts.     

Monte Carlo Likelihood Estimation for Three Multivariate Stochastic Volatility Models

Estimating parameters in a stochastic volatility (SV) model is a challenging task. Among other estimation methods and approaches, efficient simulation methods based on importance sampling have been developed for the Monte Carlo maximum likelihood estimation of univariate SV models. This paper shows that importance sampling methods can be used in a general multivariate SV setting. The sampling methods are computationally efficient. To illustrate the versatility of this approach, three different multivariate stochastic volatility models are estimated for a standard data set. The empirical results are compared to those from earlier studies in the literature. Monte Carlo simulation experiments, based on parameter estimates from the standard data set, are used to show the effectiveness of the importance sampling methods.     

Asymmetric Multivariate Stochastic Volatility

This paper proposes and analyses two types of asymmetric multivariate stochastic volatility (SV) models, namely, (i) the SV with leverage (SV-L) model, which is based on the negative correlation between the innovations in the returns and volatility, and (ii) the SV with leverage and size effect (SV-LSE) model, which is based on the signs and magnitude of the returns. The paper derives the state space form for the logarithm of the squared returns, which follow the multivariate SV-L model, and develops estimation methods for the multivariate SV-L and SV-LSE models based on the Monte Carlo likelihood (MCL) approach. The empirical results show that the multivariate SV-LSE model fits the bivariate and trivariate returns of the S&P 500, the Nikkei 225, and the Hang Seng indexes with respect to AIC and BIC more accurately than does the multivariate SV-L model. Moreover, the empirical results suggest that the univariate models should be rejected in favor of their bivariate and trivariate counterparts.     

Asymmetric Multivariate Stochastic Volatility

This paper proposes and analyses two types of asymmetric multivariate stochastic volatility (SV) models, namely, (i) the SV with leverage (SV-L) model, which is based on the negative correlation between the innovations in the returns and volatility, and (ii) the SV with leverage and size effect (SV-LSE) model, which is based on the signs and magnitude of the returns. The paper derives the state space form for the logarithm of the squared returns, which follow the multivariate SV-L model, and develops estimation methods for the multivariate SV-L and SV-LSE models based on the Monte Carlo likelihood (MCL) approach. The empirical results show that the multivariate SV-LSE model fits the bivariate and trivariate returns of the S&P 500, the Nikkei 225, and the Hang Seng indexes with respect to AIC and BIC more accurately than does the multivariate SV-L model. Moreover, the empirical results suggest that the univariate models should be rejected in favor of their bivariate and trivariate counterparts.     

Testing explosive bubbles with time-varying volatility

This article considers the problem of testing for an explosive bubble in financial data in the presence of time-varying volatility. We propose a weighted least squares-based variant of the Phillips et al.) test for explosive autoregressive behavior. We find that such an approach has appealing asymptotic power properties, with the potential to deliver substantially greater power than the established OLS-based approach for many volatility and bubble settings. Given that the OLS-based test can outperform the weighted least squares-based test for other volatility and bubble specifications, we also suggest a union of rejections procedure that succeeds in capturing the better power available from the two constituent tests for a given alternative. Our approach involves a nonparametric kernel-based volatility function estimator for computation of the weighted least squares-based statistic, together with the use of a wild bootstrap procedure applied jointly to both individual tests, delivering a powerful testing procedure that is asymptotically size-robust to a wide range of time-varying volatility specifications.     

Wavelet semi-parametric inference for long memory in volatility in the presence of a trend

Risk of investing in a financial asset is quantified by functionals of squared returns. Discrete time stochastic volatility (SV) models impose a convenient and practically relevant time series dependence structure on the log-squared returns. Different long-term risk characteristics are postulated by short-memory SV and long-memory SV models. It is therefore important to test which of these two alternatives is suitable for a specific asset. Most standard tests are confounded by deterministic trends. This paper introduces a new, wavelet-based, test of the null hypothesis of short versus long memory in volatility which is robust to deterministic trends. In finite samples, the test performs better than currently available tests which are based on the Fourier transform.     

Modelling stochastic volatility using generalized t distribution

In modelling financial return time series and time-varying volatility, the Gaussian and the Student-t distributions are widely used in stochastic volatility (SV) models. However, other distributions such as the Laplace distribution and generalized error distribution (GED) are also common in SV modelling. Therefore, this paper proposes the use of the generalized t (GT) distribution whose special cases are the Gaussian distribution, Student-t distribution, Laplace distribution and GED. Since the GT distribution is a member of the scale mixture of uniform (SMU) family of distribution, we handle the GT distribution via its SMU representation. We show this SMU form can substantially simplify the Gibbs sampler for Bayesian simulation-based computation and can provide a mean of identifying outliers. In an empirical study, we adopt a GT–SV model to fit the daily return of the exchange rate of Australian dollar to three other currencies and use the exchange rate to US dollar as a covariate. Model implementation relies on Bayesian Markov chain Monte Carlo algorithms using the WinBUGS package.     

Multivariate Stochastic Volatility Model With Realized Volatilities and Pairwise Realized Correlations

Abstract

Although stochastic volatility and GARCH (generalized autoregressive conditional heteroscedasticity) models have successfully described the volatility dynamics of univariate asset returns, extending them to the multivariate models with dynamic correlations has been difficult due to several major problems. First, there are too many parameters to estimate if available data are only daily returns, which results in unstable estimates. One solution to this problem is to incorporate additional observations based on intraday asset returns, such as realized covariances. Second, since multivariate asset returns are not synchronously traded, we have to use the largest time intervals such that all asset returns are observed to compute the realized covariance matrices. However, in this study, we fail to make full use of the available intraday informations when there are less frequently traded assets. Third, it is not straightforward to guarantee that the estimated (and the realized) covariance matrices are positive definite.

Our contributions are the following: (1) we obtain the stable parameter estimates for the dynamic correlation models using the realized measures, (2) we make full use of intraday informations by using pairwise realized correlations, (3) the covariance matrices are guaranteed to be positive definite, (4) we avoid the arbitrariness of the ordering of asset returns, (5) we propose the flexible correlation structure model (e.g., such as setting some correlations to be zero if necessary), and (6) the parsimonious specification for the leverage effect is proposed. Our proposed models are applied to the daily returns of nine U.S. stocks with their realized volatilities and pairwise realized correlations and are shown to outperform the existing models with respect to portfolio performances.     

上海燃料油期货套期保值比率选取的实证研究

通过对上海燃料油期货和现货价格的实证分析,表明期货和现货价格之间存在协整关系,同时价格的波动具有时变性和集聚性特征。考虑这两种特征,建立四个模型计算套期保值比率。结果表明,按照考虑协整关系建立的VECM模型估计的最优套保比率进行套期保值,套期保值效果最好,能使决策者面临的价格风险最小。     

Common Drifting Volatility in Large Bayesian VARs

The general pattern of estimated volatilities of macroeconomic and financial variables is often broadly similar. We propose two models in which conditional volatilities feature comovement and study them using U.S. macroeconomic data. The first model specifies the conditional volatilities as driven by a single common unobserved factor, plus an idiosyncratic component. We label this model BVAR with general factor stochastic volatility (BVAR-GFSV) and we show that the loss in terms of marginal likelihood from assuming a common factor for volatility is moderate. The second model, which we label BVAR with common stochastic volatility (BVAR-CSV), is a special case of the BVAR-GFSV in which the idiosyncratic component is eliminated and the loadings to the factor are set to 1 for all the conditional volatilities. Such restrictions permit a convenient Kronecker structure for the posterior variance of the VAR coefficients, which in turn permits estimating the model even with large datasets. While perhaps misspecified, the BVAR-CSV model is strongly supported by the data when compared against standard homoscedastic BVARs, and it can produce relatively good point and density forecasts by taking advantage of the information contained in large datasets.     

Deciding between GARCH and stochastic volatility via strong decision rules

The GARCH and stochastic volatility (SV) models are two competing, well-known and often used models to explain the volatility of financial series. In this paper, we consider a closed form estimator for a stochastic volatility model and derive its asymptotic properties. We confirm our theoretical results by a simulation study. In addition, we propose a set of simple, strongly consistent decision rules to compare the ability of the GARCH and the SV model to fit the characteristic features observed in high frequency financial data such as high kurtosis and slowly decaying autocorrelation function of the squared observations. These rules are based on a number of moment conditions that is allowed to increase with sample size. We show that our selection procedure leads to choosing the model that fits best, or the simplest model under equivalence, with probability one as the sample size increases. The finite sample size behavior of our procedure is analyzed via simulations. Finally, we provide an application to stocks in the Dow Jones industrial average index.     

Bayesian Dynamic Factor Models and Portfolio Allocation

We discuss the development of dynamic factor models for multivariate financial time series, and the incorporation of stochastic volatility components for latent factor processes. Bayesian inference and computation is developed and explored in a study of the dynamic factor structure of daily spot exchange rates for a selection of international currencies. The models are direct generalizations of univariate stochastic volatility models and represent specific varieties of models recently discussed in the growing multivariate stochastic volatility literature. We discuss model fitting based on retrospective data and sequential analysis for forward filtering and short-term forecasting. Analyses are compared with results from the much simpler method of dynamic variance-matrix discounting that, for over a decade, has been a standard approach in applied financial econometrics. We study these models in analysis, forecasting, and sequential portfolio allocation for a selected set of international exchange-rate-return time series. Our goals are to understand a range of modeling questions arising in using these factor models and to explore empirical performance in portfolio construction relative to discount approaches. We report on our experiences and conclude with comments about the practical utility of structured factor models and on future potential model extensions.     

Stochastic volatility models for exchange rates and their estimation using quasi-maximum-likelihood methods: an application to the South African Rand

This paper is concerned with the volatility modeling of a set of South African Rand (ZAR) exchange rates. We investigate the quasi-maximum-likelihood (QML) estimator based on the Kalman filter and explore how well a choice of stochastic volatility (SV) models fits the data. We note that a data set from a developing country is used. The main results are: (1) the SV model parameter estimates are in line with those reported from the analysis of high-frequency data for developed countries; (2) the SV models we considered, along with their corresponding QML estimators, fit the data well; (3) using the range return instead of the absolute return as a volatility proxy produces QML estimates that are both less biased and less variable; (4) although the log range of the ZAR exchange rates has a distribution that is quite far from normal, the corresponding QML estimator has a superior performance when compared with the log absolute return.     

Continuous Time Wishart Process for Stochastic Risk

Risks are usually represented and measured by volatility–covolatility matrices. Wishart processes are models for a dynamic analysis of multivariate risk and describe the evolution of stochastic volatility–covolatility matrices, constrained to be symmetric positive definite. The autoregressive Wishart process (WAR) is the multivariate extension of the Cox, Ingersoll, Ross (CIR) process introduced for scalar stochastic volatility. As a CIR process it allows for closed-form solutions for a number of financial problems, such as term structure of T-bonds and corporate bonds, derivative pricing in a multivariate stochastic volatility model, and the structural model for credit risk. Moreover, the Wishart dynamics are very flexible and are serious competitors for less structural multivariate ARCH models.