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.     

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.     

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.     

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.     

Classical and Bayesian Analysis of Univariate and Multivariate Stochastic Volatility Models

In this paper, efficient importance sampling (EIS) is used to perform a classical and Bayesian analysis of univariate and multivariate stochastic volatility (SV) models for financial return series. EIS provides a highly generic and very accurate procedure for the Monte Carlo (MC) evaluation of high-dimensional interdependent integrals. It can be used to carry out ML-estimation of SV models as well as simulation smoothing where the latent volatilities are sampled at once. Based on this EIS simulation smoother, a Bayesian Markov chain Monte Carlo (MCMC) posterior analysis of the parameters of SV models can be performed.     

Classical and Bayesian Analysis of Univariate and Multivariate Stochastic Volatility Models

In this paper, efficient importance sampling (EIS) is used to perform a classical and Bayesian analysis of univariate and multivariate stochastic volatility (SV) models for financial return series. EIS provides a highly generic and very accurate procedure for the Monte Carlo (MC) evaluation of high-dimensional interdependent integrals. It can be used to carry out ML-estimation of SV models as well as simulation smoothing where the latent volatilities are sampled at once. Based on this EIS simulation smoother, a Bayesian Markov chain Monte Carlo (MCMC) posterior analysis of the parameters of SV models can be performed.     

Inference for Lévy‐Driven Stochastic Volatility Models via Adaptive Sequential Monte Carlo

Abstract. We investigate simulation methodology for Bayesian inference in Lévy‐driven stochastic volatility (SV) models. Typically, Bayesian inference from such models is performed using Markov chain Monte Carlo (MCMC); this is often a challenging task. Sequential Monte Carlo (SMC) samplers are methods that can improve over MCMC; however, there are many user‐set parameters to specify. We develop a fully automated SMC algorithm, which substantially improves over the standard MCMC methods in the literature. To illustrate our methodology, we look at a model comprised of a Heston model with an independent, additive, variance gamma process in the returns equation. The driving gamma process can capture the stylized behaviour of many financial time series and a discretized version, fit in a Bayesian manner, has been found to be very useful for modelling equity data. We demonstrate that it is possible to draw exact inference, in the sense of no time‐discretization error, from the Bayesian SV model.     

Selection of Multivariate Stochastic Volatility Models via Bayesian Stochastic Search

We propose a Bayesian stochastic search approach to selecting restrictions on multivariate regression models where the errors exhibit deterministic or stochastic conditional volatilities. We develop a Markov chain Monte Carlo (MCMC) algorithm that generates posterior restrictions on the regression coefficients and Cholesky decompositions of the covariance matrix of the errors. Numerical simulations with artificially generated data show that the proposed method is effective in selecting the data-generating model restrictions and improving the forecasting performance of the model. Applying the method to daily foreign exchange rate data, we conduct stochastic search on a VAR model with stochastic conditional volatilities.     

Multivariate Stochastic Volatility Models with Correlated Errors

We develop a Bayesian approach for parsimoniously estimating the correlation structure of the errors in a multivariate stochastic volatility model. Since the number of parameters in the joint correlation matrix of the return and volatility errors is potentially very large, we impose a prior that allows the off-diagonal elements of the inverse of the correlation matrix to be identically zero. The model is estimated using a Markov chain simulation method that samples from the posterior distribution of the volatilities and parameters. We illustrate the approach using both simulated and real examples. In the real examples, the method is applied to equities at three levels of aggregation: returns for firms within the same industry, returns for different industries, and returns aggregated at the index level. We find pronounced correlation effects only at the highest level of aggregation.     

Multivariate Stochastic Volatility Models with Correlated Errors

We develop a Bayesian approach for parsimoniously estimating the correlation structure of the errors in a multivariate stochastic volatility model. Since the number of parameters in the joint correlation matrix of the return and volatility errors is potentially very large, we impose a prior that allows the off-diagonal elements of the inverse of the correlation matrix to be identically zero. The model is estimated using a Markov chain simulation method that samples from the posterior distribution of the volatilities and parameters. We illustrate the approach using both simulated and real examples. In the real examples, the method is applied to equities at three levels of aggregation: returns for firms within the same industry, returns for different industries, and returns aggregated at the index level. We find pronounced correlation effects only at the highest level of aggregation.     

A Stochastic Volatility Model With Markov Switching

This article presents a new way of modeling time-varying volatility. We generalize the usual stochastic volatility models to encompass regime-switching properties. The unobserved state variables are governed by a first-order Markov process. Bayesian estimators are constructed by Gibbs sampling. High-, medium- and low-volatility states are identified for the Standard and Poor's 500 weekly return data. Persistence in volatility is explained by the persistence in the low- and the medium-volatility states. The high-volatility regime is able to capture the 1987 crash and overlap considerably with four U.S. economic recession periods.     

A Range-Based Multivariate Stochastic Volatility Model for Exchange Rates

In this paper we present a parsimonious multivariate model for exchange rate volatilities based on logarithmic high–low ranges of daily exchange rates. The multivariate stochastic volatility model decomposes the log range of each exchange rate into two independent latent factors, which could be interpreted as the underlying currency specific components. Owing to the empirical normality of the logarithmic range measure the model can be estimated conveniently with the standard Kalman filter methodology. Our results show that our model fits the exchange rate data quite well. Exchange rate news seems to be currency specific and allows identification of currency contributions to both exchange rate levels and exchange rate volatilities.     

A Range-Based Multivariate Stochastic Volatility Model for Exchange Rates

In this paper we present a parsimonious multivariate model for exchange rate volatilities based on logarithmic high-low ranges of daily exchange rates. The multivariate stochastic volatility model decomposes the log range of each exchange rate into two independent latent factors, which could be interpreted as the underlying currency specific components. Owing to the empirical normality of the logarithmic range measure the model can be estimated conveniently with the standard Kalman filter methodology. Our results show that our model fits the exchange rate data quite well. Exchange rate news seems to be currency specific and allows identification of currency contributions to both exchange rate levels and exchange rate volatilities.     

A Stochastic Volatility Model With Realized Measures for Option Pricing

Abstract

Based on the fact that realized measures of volatility are affected by measurement errors, we introduce a new family of discrete-time stochastic volatility models having two measurement equations relating both observed returns and realized measures to the latent conditional variance. A semi-analytical option pricing framework is developed for this class of models. In addition, we provide analytical filtering and smoothing recursions for the basic specification of the model, and an effective MCMC algorithm for its richer variants. The empirical analysis shows the effectiveness of filtering and smoothing realized measures in inflating the latent volatility persistence—the crucial parameter in pricing Standard and Poor’s 500 Index options.     

Multivariate Stochastic Volatility via Wishart Processes: A Comment

This comment refers to an error in the methodology for estimating the parameters of the model developed by Philipov and Glickman for modeling multivariate stochastic volatility via Wishart processes. For estimation they used Bayesian techniques. The derived expressions for the full conditionals of the model parameters as well as the expression for the acceptance ratio of the covariance matrix are erroneous. In this erratum all necessary formulae are given to guarantee an appropriate implementation and application of the model.     

Foreign Exchange Intervention by the Bank of Japan: Bayesian Analysis Using a Bivariate Stochastic Volatility Model

A bivariate stochastic volatility model is employed to measure the effect of intervention by the Bank of Japan (BOJ) on daily returns and volume in the USD/YEN foreign exchange market. Missing observations are accounted for, and a data-based Wishart prior for the precision matrix of the errors to the transition equation that is in line with the likelihood is suggested. Empirical results suggest there is strong conditional heteroskedasticity in the mean-corrected volume measure, as well as contemporaneous correlation in the errors to both the observation and transition equations. A threshold model is used for the BOJ reaction function, which is estimated jointly with the bivariate stochastic volatility model via Markov chain Monte Carlo. This accounts for endogeneity between volatility in the market and the BOJ reaction function, something that has hindered much previous empirical analysis in the literature on central bank intervention. The empirical results suggest there was a shift in behavior by the BOJ, with a movement away from a policy of market stabilization and toward a role of support for domestic monetary policy objectives. Throughout, we observe “leaning against the wind” behavior, something that is a feature of most previous empirical analysis of central bank intervention. A comparison with a bivariate EGARCH model suggests that the bivariate stochastic volatility model produces estimates that better capture spikes in in-sample volatility. This is important in improving estimates of a central bank reaction function because it is at these periods of high daily volatility that central banks more frequently intervene.     

Foreign Exchange Intervention by the Bank of Japan: Bayesian Analysis Using a Bivariate Stochastic Volatility Model

A bivariate stochastic volatility model is employed to measure the effect of intervention by the Bank of Japan (BOJ) on daily returns and volume in the USD/YEN foreign exchange market. Missing observations are accounted for, and a data-based Wishart prior for the precision matrix of the errors to the transition equation that is in line with the likelihood is suggested. Empirical results suggest there is strong conditional heteroskedasticity in the mean-corrected volume measure, as well as contemporaneous correlation in the errors to both the observation and transition equations. A threshold model is used for the BOJ reaction function, which is estimated jointly with the bivariate stochastic volatility model via Markov chain Monte Carlo. This accounts for endogeneity between volatility in the market and the BOJ reaction function, something that has hindered much previous empirical analysis in the literature on central bank intervention. The empirical results suggest there was a shift in behavior by the BOJ, with a movement away from a policy of market stabilization and toward a role of support for domestic monetary policy objectives. Throughout, we observe “leaning against the wind” behavior, something that is a feature of most previous empirical analysis of central bank intervention. A comparison with a bivariate EGARCH model suggests that the bivariate stochastic volatility model produces estimates that better capture spikes in in-sample volatility. This is important in improving estimates of a central bank reaction function because it is at these periods of high daily volatility that central banks more frequently intervene.     

Equilibrium dynamics of ice streams: a Bayesian statistical analysis

Studies of the behaviors of glaciers, ice sheets, and ice streams rely heavily on both observations and physical models. Data acquired via remote sensing provide critical information on geometry and movement of ice over large sections of Antarctica and Greenland. However, uncertainties are present in both the observations and the models. Hence, there is a need for combining these information sources in a fashion that incorporates uncertainty and quantifies its impact on conclusions. We present a hierarchical Bayesian approach to modeling ice-stream velocities incorporating physical models and observations regarding velocity, ice thickness, and surface elevation from the North East Ice Stream in Greenland. The Bayesian model leads to interesting issues in model assessment and computation.     

Bayesian Analysis of Stochastic Volatility Models

New techniques for the analysis of stochastic volatility models in which the logarithm of conditional variance follows an autoregressive model are developed. A cyclic Metropolis algorithm is used to construct a Markov-chain simulation tool. Simulations from this Markov chain converge in distribution to draws from the posterior distribution enabling exact finite-sample inference. The exact solution to the filtering/smoothing problem of inferring about the unobserved variance states is a by-product of our Markov-chain method. In addition, multistep-ahead predictive densities can be constructed that reflect both inherent model variability and parameter uncertainty. We illustrate our method by analyzing both daily and weekly data on stock returns and exchange rates. Sampling experiments are conducted to compare the performance of Bayes estimators to method of moments and quasi-maximum likelihood estimators proposed in the literature. In both parameter estimation and filtering, the Bayes estimators outperform these other approaches.     

Bayesian Estimation of Stochastic Frontier Models with Multivariate Skew t Error Terms

The study of proportions is a common topic in many fields of study. The standard beta distribution or the inflated beta distribution may be a reasonable choice to fit a proportion in most situations. However, they do not fit well variables that do not assume values in the open interval (0, c), 0 < c < 1. For these variables, the authors introduce the truncated inflated beta distribution (TBEINF). This proposed distribution is a mixture of the beta distribution bounded in the open interval (c, 1) and the trinomial distribution. The authors present the moments of the distribution, its scoring vector, and Fisher information matrix, and discuss estimation of its parameters. The properties of the suggested estimators are studied using Monte Carlo simulation. In addition, the authors present an application of the TBEINF distribution for unemployment insurance data.