Multivariate stochastic volatility with Bayesian dynamic linear models

This paper develops a Bayesian procedure for estimation and forecasting of the volatility of multivariate time series. The foundation of this work is the matrix-variate dynamic linear model, for the volatility of which we adopt a multiplicative stochastic evolution, using Wishart and singular multivariate beta distributions. A diagonal matrix of discount factors is employed in order to discount the variances element by element and therefore allowing a flexible and pragmatic variance modelling approach. Diagnostic tests and sequential model monitoring are discussed in some detail. The proposed estimation theory is applied to a four-dimensional time series, comprising spot prices of aluminium, copper, lead and zinc of the London metal exchange. The empirical findings suggest that the proposed Bayesian procedure can be effectively applied to financial data, overcoming many of the disadvantages of existing volatility models.     

Specification and testing of multiplicative time-varying GARCH models with applications

In this article, we develop a specification technique for building multiplicative time-varying GARCH models of Amado and Teräsvirta (2008, 2013). The variance is decomposed into an unconditional and a conditional component such that the unconditional variance component is allowed to evolve smoothly over time. This nonstationary component is defined as a linear combination of logistic transition functions with time as the transition variable. The appropriate number of transition functions is determined by a sequence of specification tests. For that purpose, a coherent modelling strategy based on statistical inference is presented. It is heavily dependent on Lagrange multiplier type misspecification tests. The tests are easily implemented as they are entirely based on auxiliary regressions. Finite-sample properties of the strategy and tests are examined by simulation. The modelling strategy is illustrated in practice with two real examples: an empirical application to daily exchange rate returns and another one to daily coffee futures returns.     

Comparison of asymmetric stochastic volatility models under different correlation structures

This paper conducts simulation-based comparison of several stochastic volatility models with leverage effects. Two new variants of asymmetric stochastic volatility models, which are subject to a logarithmic transformation on the squared asset returns, are proposed. The leverage effect is introduced into the model through correlation either between the innovations of the observation equation and the latent process, or between the logarithm of squared asset returns and the latent process. Suitable Markov Chain Monte Carlo algorithms are developed for parameter estimation and model comparison. Simulation results show that our proposed formulation of the leverage effect and the accompanying inference methods give rise to reasonable parameter estimates. Applications to two data sets uncover a negative correlation (which can be interpreted as a leverage effect) between the observed returns and volatilities, and a negative correlation between the logarithm of squared returns and volatilities.     

The copula directional dependence by stochastic volatility models

This paper proposes a copula directional dependence by using a bivariate Gaussian copula beta regression with Stochastic Volatility (SV) models for marginal distributions. With the asymmetric copula generated by the composition of two Plackett copulas, we show that our SV copula directional dependence by the Gaussian copula beta regression model is superior to the Kim and Hwang (2016) copula directional dependence by an asymmetric GARCH model in terms of the percent relative efficiency of bias and mean squared error. To validate our proposed method with the real data, we use Brent Crude Daily Price (BRENT), West Texas Intermediate Daily Price (WTI), the Standard & Poor’s 500 (SP) and US 10-Year Treasury Constant Maturity Rate (TCM) so that our copula SV directional dependence is overall superior to the Kim and Hwang (2016) copula directional dependence by an asymmetric GARCH model in terms of precision by the percent relative efficiency of mean squared error. In terms of forecasting using the real financial data, we also show that the Bayesian SV model of the uniform transformed data by a copula conditional distribution yields an improvement on the volatility models such as GARCH and SV.     

Score tests for zero inflation in generalized linear models

The authors develop score tests of goodness of fit for discrete generalized linear models against zero inflation. The binomial and Poisson models are treated as examples, and in the latter case the proposed test reduces to that of Broek (1995). Some simulation results and an illustrative example are presented.     

中国的通货膨胀及其不确定性

采用1990年1月以来居民消费价格指数(CPI)的月度数据,运用随机域回归模型、系列随机域的非线性检验方法和贝叶斯估计方法,对中国通货膨胀率与通货膨胀不确定性的关系进行了实证分析。研究发现:通货膨胀率与通货膨胀之间具有双向关系。通货膨胀率引起了通货膨胀不确定性,两者呈现U型关系;较高的通货膨胀不确定性引起通货膨胀率先升后降,呈现倒U曲线关系。     

Value at risk estimation under stochastic volatility models using adaptive PMCMC methods

In this paper, we propose a value-at-risk (VaR) estimation technique based on a new stochastic volatility model with leverage effect, nonconstant conditional mean and jump. In order to estimate the model parameters and latent state variables, we integrate the particle filter and adaptive Markov Chain Monte Carlo (MCMC) algorithms to develop a novel adaptive particle MCMC (A-PMCMC) algorithm. Comprehensive simulation experiments based on three stock indices and two foreign exchange time series show effectiveness of the proposed A-PMCMC algorithm and the VaR estimation technique.     

A New Model of Trend Inflation

This article introduces a new model of trend inflation. In contrast to many earlier approaches, which allow for trend inflation to evolve according to a random walk, ours is a bounded model which ensures that trend inflation is constrained to lie in an interval. The bounds of this interval can either be fixed or estimated from the data. Our model also allows for a time-varying degree of persistence in the transitory component of inflation. In an empirical exercise with CPI inflation, we find the model to work well, yielding more sensible measures of trend inflation and forecasting better than popular alternatives such as the unobserved components stochastic volatility model. This article has supplementary materials online.     

On periodic autoregressive stochastic volatility models: structure and estimation

To capture both the volatility evolution and the periodicity feature in the autocorrelation structure exhibited by many nonlinear time series, a Periodic AutoRegressive Stochastic Volatility (PAR-SV ) model is proposed. Some probabilistic properties, namely the strict and second-order periodic stationarity, are provided. Furthermore, conditions for the existence of higher-order moments are established. The autocovariance structure of the squares and higher order powers of the PAR-SV process is studied. Its dynamic properties are shown to be consistent with financial time series empirical findings. Ways in which the model may be estimated are discussed. Finally, a simulation study of the performance of the proposed estimation methods is provided and the PAR-SV is applied to model the spot rates of the euro and US dollar both against the Algerian dinar. The empirical analysis shows that the proposed PAR-SV model can be considered as a viable alternative to the periodic generalized autoregressive conditionally heteroscedastic (PGARCH) model.     

An empirical Bayesian forecast in the threshold stochastic volatility models

In the area of finance, the stochastic volatility (SV) model is a useful tool for modelling stock market returns. However, there is evidence that asymmetric behaviour of stock returns exists. A threshold SV (THSV) model is provided to capture this behaviour. In this study, we introduce a robust model created through empirical Bayesian analysis to deal with the uncertainty between the SV and THSV models. A Markov chain Monte Carlo algorithm is applied to empirically select the hyperparameters of the prior distribution. Furthermore, the value at risk from the resulting predictive distribution is also given. Simulation studies show that the proposed empirical Bayes model not only clarifies the acceptability of prediction but also reduces the risk of model uncertainty.     

Variable dimension via stochastic volatility model using FX rates

In this paper, changepoint analysis is applied to stochastic volatility (SV) models which aim to understand the locations and movements of high frequency FX financial time series. Bayesian inference using the Markov Chain Monte Carlo method is performed using a process called variable dimension for SV parameters. Interesting results are that FX series have locations where one or more positions of the sequence correspond to systemic changes, and overall non-stationarity, in the returns process. Furthermore, we found that the changepoint locations provide an informative estimate for all FX series. Importantly in most cases, the detected changepoints can be identified with economic factors relevant to the country concerned. This helps support the fact that macroeconomics news and the movement in financial price are positively related.     

Bayesian analysis of stochastic volatility models with flexible tails

An alternative distributional assumption is proposed for the stochastic volatility model. This results in extremely flexible tail behaviour of the sampling distribution for the observables, as well as in the availability of a simple Markov Chain Monte Carlo strategy for posterior analysis. By allowing the tail behaviour to be determined by a separate parameter, we reserve the parameters of the volatility process to dictate the degree of volatility clustering. Treatment of a mean function is formally integrated in the analysis.

Some empirical examples on both stock prices and exchange rates clearly indicate the presence of fat tails, in combination with high levels of volatility clustering. In addition, predictive distributions indicate a good fit with these typical financial data sets.     

Bayesian analysis of stochastic volatility models with flexible tails

An alternative distributional assumption is proposed for the stochastic volatility model. This results in extremely flexible tail behaviour of the sampling distribution for the observables, as well as in the availability of a simple Markov Chain Monte Carlo strategy for posterior analysis. By allowing the tail behaviour to be determined by a separate parameter, we reserve the parameters of the volatility process to dictate the degree of volatility clustering. Treatment of a mean function is formally integrated in the analysis.

Some empirical examples on both stock prices and exchange rates clearly indicate the presence of fat tails, in combination with high levels of volatility clustering. In addition, predictive distributions indicate a good fit with these typical financial data sets.     

On selecting a prior for the precision parameter of Dirichlet process mixture models

In hierarchical mixture models the Dirichlet process is used to specify latent patterns of heterogeneity, particularly when the distribution of latent parameters is thought to be clustered (multimodal). The parameters of a Dirichlet process include a precision parameter α$\alpha$ and a base probability measure _G0$G_0$. In problems where α$\alpha$ is unknown and must be estimated, inferences about the level of clustering can be sensitive to the choice of prior assumed for α$\alpha$. In this paper an approach is developed for computing a prior for the precision parameter α$\alpha$ that can be used in the presence or absence of prior information about the level of clustering. This approach is illustrated in an analysis of counts of stream fishes. The results of this fully Bayesian analysis are compared with an empirical Bayes analysis of the same data and with a Bayesian analysis based on an alternative commonly used prior.     

On regression-based tests for persistence in logarithmic volatility models

Building on the work of Pantula (1986), this paper discusses how the hypothesis of conditional variance nonstationarity in the logarithmic family of generalized autoregressive conditional heteroskedasticity (GARCH) and stochastic volatility processes may be tested using regression-based tests. The latter are easy to implement, have well-defined large-sample distributions, and are less sensitive to structural changes than tests based on the quasimaximum likelihood estimator.     

Eliciting prior information to enhance the predictive performance of bayesian graphical models

Both knowledge-based systems and statistical models are typically concerned with making predictions about future observables. Here we focus on assessment of predictive performance and provide two techniques for improving the predictive performance of Bayesian graphical models. First, we present Bayesian model averaging, a technique for accounting for model uncertainty.

Second, we describe a technique for eliciting a prior distribution for competing models from domain experts. We explore the predictive performance of both techniques in the context of a urological diagnostic problem.     

Local composite quantile regression estimation of time-varying parameter vector for multidimensional time-inhomogeneous diffusion models

This paper is dedicated to the study of the composite quantile regression (CQR) estimations of time-varying parameter vectors for multidimensional diffusion models. Based on the local linear fitting for parameter vectors, we propose the local linear CQR estimations of the drift parameter vectors, and verify their asymptotic biases, asymptotic variances and asymptotic normality. Moreover, we discuss the asymptotic relative efficiency (ARE) of the local linear CQR estimations with respect to the local linear least-squares estimations. We obtain that the local estimations that we proposed are much more efficient than the local linear least-squares estimations. Simulation studies are constructed to show the performance of the estimations proposed.     

Option pricing under model and parameter uncertainty using predictive densities

The theoretical price of a financial option is given by the expectation of its discounted expiry time payoff. The computation of this expectation depends on the density of the value of the underlying instrument at expiry time. This density depends on both the parametric model assumed for the behaviour of the underlying, and the values of parameters within the model, such as volatility. However neither the model, nor the parameter values are known. Common practice when pricing options is to assume a specific model, such as geometric Brownian Motion, and to use point estimates of the model parameters, thereby precisely defining a density function.We explicitly acknowledge the uncertainty of model and parameters by constructing the predictive density of the underlying as an average of model predictive densities, weighted by each model's posterior probability. A model's predictive density is constructed by integrating its transition density function by the posterior distribution of its parameters. This is an extension to Bayesian model averaging. Sampling importance-resampling and Monte Carlo algorithms implement the computation. The advantage of this method is that rather than falsely assuming the model and parameter values are known, inherent ignorance is acknowledged and dealt with in a mathematically logical manner, which utilises all information from past and current observations to generate and update option prices. Moreover point estimates for parameters are unnecessary. We use this method to price a European Call option on a share index.