Multivariate Stochastic Volatility Research Articles

Multivariate Stochastic Volatility with Co-Heteroscedasticity

Abstract A new methodology that decomposes shocks into homoscedastic and heteroscedastic components is developed. This specification implies there exist linear combinations of heteroscedastic variables that eliminate heteroscedasticity; a property known as co-heteroscedasticity. The heteroscedastic part of the model uses a multivariate stochastic volatility inverse Wishart process. The resulting model is invariant to the ordering of the variables, which is shown to be important for volatility estimation. By incorporating testable co-heteroscedasticity restrictions, the specification allows estimation in moderately high-dimensions. The computational strategy uses a novel particle filter algorithm, a reparameterization that substantially improves algorithmic convergence and an alternating-order particle Gibbs that reduces the amount of particles needed for accurate estimation. An empirical application to a large Vector Autoregression (VAR) is provided, finding strong evidence for co-heteroscedasticity and that the new method outperforms some previously proposed methods in terms of forecasting at all horizons. It is also found that the structural monetary shock is 98.8 % homoscedastic, and that investment and the SP 500 index are nearly 100 % determined by fat tail heteroscedastic shocks. A Monte Carlo experiment illustrates that the new method estimates well the characteristics of approximate factor models with heteroscedastic errors.

Retraction Note: A Bayesian analysis based on multivariate stochastic volatility model: evidence from green stocks

Retraction Note: A Bayesian analysis based on multivariate stochastic volatility model: evidence from green stocks

Global uncertainty and potential shelters: gold, bitcoin, and currencies as weak and strong safe havens for main world stock markets

This article investigates five safe-haven asset responses from 2014 to 2022, including the unprecedented COVID-19 crisis, Russian invasion of Ukraine, and sharp US interest rate increases of 2015 and 2022. We apply the unique approach of the multivariate factor stochastic volatility (MSV) model, which is extremely efficient for financial market analysis and allows us to conduct dynamic factor analysis of safe-haven relationships that cannot be observed directly. The research sample consists of five prospective safe-haven assets—gold, bitcoin, the euro, the Japanese yen, and the Swiss franc—and five primary world stock market indices—the S&P 500, Financial Times Stock Exchange (FTSE) 100, DAX, STOXX Europe 600, and Nikkei 225. Our findings are useful for investors searching for the best safe-haven assets among gold, bitcoin, and currencies to hedge against financial turmoil in global stock markets. Our unique findings suggest that safe-haven effects work differently for gold and the yen; that is, the Japanese yen acts as the strongest safe haven across all stock indices. Bitcoin is not a strong safe-haven currency since it has zero days of negative correlations with the considered stock indices, but it is a weak safe-haven during times of financial distress. Consequently, we state that strong and weak safe-haven properties vary across time and place. The novelty of our study lies in the methodological complexity of the MSV model (used for the first time to find the best safe-haven asset properties), dynamic factor analysis, a long-term research sample covering the Russian invasion of Ukraine in 2022, and an international investor perspective focusing on the world’s leading stock markets. We extend earlier studies by analyzing the interrelations of the world’s leading stock market indices with five potential safe-haven assets during the long period of 2014–2022 and using a unique dynamic factor analysis to show the differentiated behaviors of the Japanese yen and gold. Additionally, the main innovative contribution is a new framework of weak and strong safe-haven asset classifications not previously applied in the literature.

Multivariate Stochastic Volatility Modeling via Integrated Nested Laplace Approximations: A Multifactor Extension

This study introduces a multivariate extension to the class of stochastic volatility models, employing integrated nested Laplace approximations (INLA) for estimation. Bayesian methods for estimating stochastic volatility models through Markov Chain Monte Carlo (MCMC) can become computationally burdensome or inefficient as the dataset size and problem complexity increase. Furthermore, issues related to chain convergence can also arise. In light of these challenges, this research aims to establish a computationally efficient approach for estimating multivariate stochastic volatility models. We propose a multifactor formulation estimated using the INLA methodology, enabling an approach that leverages sparse linear algebra and parallelization techniques. To evaluate the effectiveness of our proposed model, we conduct in-sample and out-of-sample empirical analyses of stock market index return series. Furthermore, we provide a comparative analysis with models estimated using MCMC, demonstrating the computational efficiency and goodness of fit improvements achieved with our approach.

Co-volatility dynamics in global cryptocurrency and conventional asset classes: a multivariate stochastic factor volatility approach

PurposeThis study aims to investigate the co-volatility patterns between cryptocurrencies and conventional asset classes across global markets, encompassing 26 global indices ranging from equities, commodities, real estate, currencies and bonds.Design/methodology/approachIt used a multivariate factor stochastic volatility model to capture the dynamic changes in covariance and volatility correlation, thus offering empirical insights into the co-volatility dynamics. Unlike conventional research on price or return transmission, this study directly models the time-varying covariance and volatility correlation.FindingsThe study uncovers pronounced co-volatility movements between cryptocurrencies and specific indices such as GSCI Energy, GSCI Commodity, Dow Jones 1 month forward and U.S. 10-year TIPS. Notably, these movements surpass those observed with precious metals, industrial metals and global equity indices across various regions. Interestingly, except for Japan, equity indices in the USA, Canada, Australia, France, Germany, India and China exhibit a co-volatility movement. These findings challenge the existing literature on cryptocurrencies and provide intriguing evidence regarding their co-volatility dynamics.OriginalityThis study significantly contributes to applying asset pricing models in cryptocurrency markets by explicitly addressing price and volatility dynamics aspects. Using the stochastic volatility model, the research adding methodological contribution effectively captures cryptocurrency volatility's inherent fluctuations and time-varying nature. While previous literature has primarily focused on bitcoin and a few other cryptocurrencies, this study examines the stochastic volatility properties of a wide range of cryptocurrency indices. Furthermore, the study expands its scope by examining global asset markets, allowing for a comprehensive analysis considering the broader context in which cryptocurrencies operate. It bridges the gap between traditional asset pricing models and the unique characteristics of cryptocurrencies.

On an asymmetric multivariate stochastic difference volatility: structure and estimation

<abstract><p>In this study, we explored an asymmetric multivariate stochastic difference volatility model that extends various probabilistic and statistical properties previously discussed in the literature. We rigorously established that the model exhibits periodic stationarity and periodic ergodicity. Additionally, we delved into the robust consistency and asymptotic normality of the Quasi-Maximum Likelihood Estimator (QMLE), providing a comprehensive analysis of its theoretical underpinnings. Finally, we demonstrated the practical applicability of our major findings through a series of pertinent applications. This work not only contributes to the existing body of knowledge on stochastic volatility modeling, but also opens new avenues for further research in this domain.</p></abstract>

Large Order-Invariant Bayesian VARs with Stochastic Volatility

Many popular specifications for Vector Autoregressions (VARs) with multivariate stochastic volatility are not invariant to the way the variables are ordered due to the use of a lower triangular parameterization of the error covariance matrix. We show that the order invariance problem in existing approaches is likely to become more serious in large VARs. We propose the use of a specification which avoids the use of this lower triangular parameterization. We show that the presence of multivariate stochastic volatility allows for identification of the proposed model and prove that it is invariant to ordering. We develop a Markov chain Monte Carlo algorithm which allows for Bayesian estimation and prediction. In exercises involving artificial and real macroeconomic data, we demonstrate that the choice of variable ordering can have non-negligible effects on empirical results when using the nonorder invariant approach. In a macroeconomic forecasting exercise involving VARs with 20 variables we find that our order-invariant approach leads to the best forecasts and that some choices of variable ordering can lead to poor forecasts using a conventional, non-order invariant, approach.

Do stochastic risks flow between industrial and precious metals, Islamic stocks, green bonds, green stocks, clean investments, major foreign exchange rates, and Bitcoin?

This study examines the interactions between Bitcoin (BTC) and other major financial markets (forex, equity and bond, and industrial and precious metals). This interaction is viewed from the spectrum of spillover analysis, hedging, and portfolio analyses. Due to the high volatility and volatility clustering nature of most financial series, particularly BTC, we rely on a Multivariate Factor Stochastic Volatility (mvFSV) Model of Kastner et al. (2017). In order to capture the influence of extreme market conditions, we use a quantile-based mvFSV Model. Among other things, results confirm the importance of quantile analysis as results are sensitive to market conditions. For instance, currencies and industrial and precious metals are net transmitters of shocks at lower and middle quantiles, while these assets change their status to the net receiver of shocks at the upper quantile. Results of the total connectedness show that there is a high degree of interconnectedness among the assets, as the total connectedness index ranges between 86% and 100%. Besides, Islamic equities perform such as the most currencies and conventional assets in terms of the transmission of shocks and diversification benefits. The hedging potentials of Bitcoin over other assets were also examined, and results confirm that the hedging effectiveness is commodity-specific. Policy implications of these results are discussed.

Information overflow between monetary policy transparency and inflation expectations using multivariate stochastic volatility models

The forward-looking policy is useful for joint decision-making between public and monetary authorities. The study calculates the monetary policy transparency index, inflation expectations, and their volatility spillover effects at a data-driven angle. Optimization mechanisms of monetary policy transparency influencing inflation and inflation expectations volatility, and transparency index based on market participants’ optimization decisions are analysed. Multivariate stochastic volatility models for measuring the volatility of inflation and inflation expectations are analysed. Additionally, the relationships between inflation and inflation expectations volatility and transparency are tested empirically. Tests show that: first, the most suitable models for measuring volatility of inflation and inflation expectations are obtained. Second, monetary policy transparency has a negative effect on the volatility of inflation and inflation expectations. Third, there is a unidirectional Granger relationship from the volatility of inflation expectations to the volatility of inflation, and it indicates that the volatility of inflation expectations has an important guiding effect on that of inflation, and it is necessary to stabilize inflation expectations for stabilizing inflation. Fourth, improving monetary policy transparency can not only reduce inflation, inflation expectations, inflation volatility, inflation expectations volatility, but also reduce the sacrifice rate, that is, to stabilize the price level with less anti-inflation costs.

Gaussian Variational Approximations for High-dimensional State Space Models

We consider a Gaussian variational approximation of the posterior density in high-dimensional state space models. The number of parameters in the covariance matrix of the variational approximation grows as the square of the number of model parameters, so it is necessary to find simple yet effective parametrisations of the covariance structure when the number of model parameters is large. We approximate the joint posterior density of the state vectors by a dynamic factor model, having Markovian time dependence and a factor covariance structure for the states. This gives a reduced description of the dependence structure for the states, as well as a temporal conditional independence structure similar to that in the true posterior. We illustrate the methodology on two examples. The first is a spatio-temporal model for the spread of the Eurasian collared-dove across North America. Our approach compares favorably to a recently proposed ensemble Kalman filter method for approximate inference in high-dimensional hierarchical spatio-temporal models. Our second example is a Wishart-based multivariate stochastic volatility model for financial returns, which is outside the class of models the ensemble Kalman filter method can handle.

Hybrid SV‐GARCH, t‐GARCH and Markov‐switching covariance structures in VEC models—Which is better from a predictive perspective?

SummaryWe compare predictive performance of a multitude of alternative Bayesian vector autoregression (VAR) models allowing for cointegration and time‐varying conditional covariances, described by different multivariate stochastic volatility (MSV) models, including their hybrids with multivariate GARCH processes (MSV‐MGARCH), as well as t‐GARCH and Markov‐switching structures. The forecast accuracy is evaluated mainly through predictive Bayes factors, but energy scores and the probability integral transform are also used. Two empirical studies, for the US and Polish economies, are based on a small model of monetary policy comprising inflation, unemployment and interest rate. The results indicate that capturing conditional heteroskedasticity by some MSV‐MGARCH specifications contributes the most to the forecasting power of the VAR/VEC model.

Dynamic factor, leverage and realized covariances in multivariate stochastic volatility

In the stochastic volatility models for multivariate daily stock returns, it has been found that the estimates of parameters become unstable as the dimension of returns increases. To solve this problem, we focus on the factor structure of multiple returns and consider two additional sources of information: first, the stock index associated with the market factor and, second, the realized covariance matrix calculated from high-frequency data. The proposed dynamic factor model with the leverage effect and realized measures is applied to 10 top stocks composing the exchange traded fund linked with the investment return of the S&P 500 index and the model is shown to have a stable advantage in portfolio performance.

Modeling the Dynamic Correlations among Cryptocurrencies: New Evidence from Multivariate Factor Stochastic Volatility Model

Modeling the Dynamic Correlations among Cryptocurrencies: New Evidence from Multivariate Factor Stochastic Volatility Model

Multivariate continuous-time autoregressive moving-average processes on cones

In this article we study multivariate continuous-time autoregressive moving-average (MCARMA) processes with values in convex cones. More specifically, we introduce matrix-valued MCARMA processes with Lévy noise and present necessary and sufficient conditions for processes from this class to be cone valued. We derive specific hands-on conditions in the following two cases: First, for classical MCARMA on Rd with values in the positive orthant Rd+. Second, for MCARMA processes on real square matrices assuming values in the cone of symmetric and positive semi-definite matrices. Both cases are relevant for applications and we give several examples of positivity ensuring parameter specifications. In addition to the above, we discuss the capability of positive semi-definite MCARMA processes to model the spot covariance process in multivariate stochastic volatility models. We justify the relevance of MCARMA based stochastic volatility models by an exemplary analysis of the second order moment structure of positive semi-definite well-balanced Ornstein–Uhlenbeck based models.

Measuring and Quantifying Uncertainty in Volatility Spillovers: A Bayesian Approach

Volatility spillover measures are crucial for studying connectivity of financial time series. Understanding how financial time series are interconnected can help, for example, portfolio managers and policymakers in their decision process. Besides estimating the spillover effects themselves, it is important to estimate the corresponding uncertainty which current approaches lack. We propose a fully Bayesian approach based on a multivariate stochastic volatility model, which allows us to estimate the distribution of the volatility spillovers and naturally leads to uncertainty quantification.

Quantile-based spillover connectedness among stochastic volatilities of ESG equities, Islamic and conventional stocks with implications for portfolio management

PurposeThis study aims to investigate the relationship between stock markets, environmental, social and governance (ESG) factors and Shariah-compliant in an integrated framework.Design/methodology/approachThe authors employ the multivariate factor stochastic volatility (mvFSV) framework to extract the volatility of the different sectoral indices. Based on this evidence, the authors employ the quantile vector autoregressive (QVAR) approach to examine the dynamic spillover connectedness among the aforementioned indices.FindingsThe study emphasizes the following major findings: (1) significant time-varying spillover connectedness across quantiles, (2) bidirectional and asymmetric spillover effect among the ESG index and the other sectoral indices, (3) the strength of spillover connectedness is time-varying across quantiles, (4) based on the perspective of portfolio optimization, ESG market is a significant strong forecasting contributor to conventional and Shariah-compliant markets, (5) overall, the findings point out serious quantile pass-through effect among ESG index and the other sectoral indices during the COVID-19 health crisis.Originality/valueThis study extends the previous literature in the following ways. First, to the best of the researchers’ knowledge, none of the existing studies have investigated the relationship between stock markets, ESG factors and Shariah-compliant in an integrated framework. Second, this study extends the previous scholarships by applying the mvFSV. Third, the authors propose a new rolling version to estimate dynamic spillovers, namely the rolling-window quantile VAR method. This approach provides a great advantage in computing the dynamics of return and variance spillover between variables in terms not only of the overall factor but also of the net (pairwise) aspect.

Optimal contracting with asymmetric belief and complementarity

PurposeIn this study, a dynamic contracting model is developed between a venture capitalist (VC) and an entrepreneur (EN) to explore the influence of asymmetric beliefs regarding output-relevant parameters, agency conflicts and complementarity on the VC's posterior beliefs through the EN's unobservable effort choices to influence the optimal dynamic contract.Design/methodology/approachThe authors construct the contracting model by incorporating the VC's effort, which is ignored in most studies. Using backward induction and a discrete-time approximation approach, the authors solve the continuous-time contract design problem, which evolves into a nonlinear ordinary differential equation (ODE).FindingsThe optimal equity share that the VC provides to the EN decreases over time. In accordance with the empirical evidence, the EN's optimistic beliefs regarding the project's profitability positively affect its equity share. However, the interactions between the optimal equity share, project risk and both partners' degrees of risk aversion are not monotonic. Moreover, the authors find that the optimal equity share increases with the degree of complementarity, which indicates that the EN is willing to cooperate with the VC. This study’s results also show that the optimal equity shares at each time are interdependent if the VC is risk-averse and independent if the VC is risk-neutral.Research limitations/implicationsIn conclusion, the authors highlight two potential directions for future research. First, the authors only considered a single VC, whereas in practice, a risk project may be carried out by multiple VCs, and it is interesting to discuss how the degree of complementarity affects the number of VCs that ENs contract. Second, the authors may introduce jumps and consider more general multivariate stochastic volatility models for output dynamics and analyze the characteristics of the optimal contracts. Third, further research can deal with other forms of discretionary output functions concerning complementarity, such as Cobb–Douglas and constant elasticity of substitution (See Varian, 1992).Social implicationsThe results of this study have several implications. First, it offers a novel approach to designing dynamic contracts that are specific and easy to operate. To improve the complicated venture investment situation and abate conflict between contractual parties, this study plays a good reference role. Second, the synergy effect proposed in this study provides a theoretical explanation for the executive compensation puzzle in economics, in which managers are often “rewarded for luck” (Bertrand and Mullainathan, 2001; Wu et al., 2018). This result indicates a realistic perspective on financing and establishing cooperative relationships, which enhances the efficiency of venture investment. Third, from an empirical standpoint, one can apply this framework to study research and development (R&D) problems.Originality/valueFirst, the authors introduce asymmetric beliefs and Bayesian learning to study the dynamic contract design problem and discuss their effects on equity share. Second, the authors incorporate the VC's effort into the contracting problem, and analyze the synergistic effect of effort complementarity on the optimal dynamic contract.

Estimation of the Cholesky Multivariate Stochastic Volatility Model Using Iterated Filtering

Aim: The paper aims to propose a new estimation method for the Cholesky Multivariate Stochastic Volatility Model based on the iterated filtering algorithm (Ionides et al., 2006, 2015). Methodology: The iterated filtering method is a frequentist-based technique that through multiple repetitions of the filtering process, provides a sequence of iteratively updated parameter estimates that converge towards the maximum likelihood estimate. Results: The effectiveness of the proposed estimation method was shown in an empirical example in which the Cholesky Multivariate Stochastic Volatility Model was used in a study on safe-haven assets of one market index: Standard and Poor’s 500 and three safe-haven candidates: gold, Bitcoin and Ethereum. Implications and recommendations: In further research, the iterating filtering method may be used for more advanced multivariate stochastic volatility models that take into account, for example, the leverage effect (as in Ishihara et al., 2016) and heavy-tailed errors (as in Ishihara and Omori, 2012). Originality/Value: The main contribution of the paper is the proposition of a new estimation method for the Cholesky Multivariate Stochastic Volatility Model based on iterated filtering algorithm This is one of the few frequentist-based statistical inference methods for multivariate stochastic volatility models.

A Bayesian analysis based on multivariate stochastic volatility model: evidence from green stocks.

Green stocks are companies environmental protective and friendly. We test Green stock index in Shanghai Stock Exchange and China Securities Index as safe-havens for global investors. Suitable multivariate-SV model and Bayesian method are used to estimate the spillover effect between different assets among local and global markets. We choose multivariate volatility model because it can efficiently simulate the spillover effect by using machine learning MCMC method. The results show that the Environmental Protection Index (EPI) of Shanghai Stock Exchange (SSE) and China Securities Index (CSI) have no significant volatility spillover from Shanghai Stock index, S &P index, gold price, oil future prices of USA and China. During COVID-19 pandemic, we find Green stock index is a suitable safe-haven with low volatility spillover. Green stock indexes has a strongly one-way spillover to the crude oil future price. Environmentally friendly investor can use diversity green assets to provide a low risk investment portfolio in EPI stock market. The DCGCt-MSV model using machine learning of MCMC method is accurate and outperform others in Bayes parameter estimation.

Macroeconomic forecasting and variable ordering in multivariate stochastic volatility models

We document five novel empirical findings on the well-known potential ordering drawback associated with the time-varying parameter vector autoregression with stochastic volatility developed by Cogley and Sargent (2005) and Primiceri (2005). First, the ordering does not affect point prediction. Second, the standard deviation of the predictive densities implied by different orderings can differ substantially. Third, the average length of the prediction intervals is also sensitive to the ordering. Fourth, the best ordering for one variable in terms of log-predictive scores does not necessarily imply the best ordering for another variable under the same metric. Fifth, the ordering problem becomes exacerbated in conditional forecasting exercises. Then, we consider three alternative ordering invariant models: a canonical discounted Wishart stochastic volatility model and two dynamic stochastic correlation models. When the forecasting performance of these ordering invariant models is compared to Cogley, Primiceri, and Sargent’s ordering variant model, the former underperforms relative to all orderings and the latter two have an out-of-sample forecasting performance comparable with the median outcomes across orderings.