Block sampler and posterior mode estimation for asymmetric stochastic volatility models

A new efficient simulation smoother and disturbance smoother are introduced for asymmetric stochastic volatility models where there exists a correlation between today's return and tomorrow's volatility. The state vector is divided into several blocks where each block consists of many state variables. For each block, corresponding disturbances are sampled simultaneously from their conditional posterior distribution. The algorithm is based on the multivariate normal approximation of the conditional posterior density and exploits a conventional simulation smoother for a linear and Gaussian state-space model. The performance of our method is illustrated using two examples: (1) simple asymmetric stochastic volatility model and (2) asymmetric stochastic volatility model with state-dependent variances. The popular single move sampler which samples a state variable at a time is also conducted for comparison in the first example. It is shown that our proposed sampler produces considerable improvement in the mixing property of the Markov chain Monte Carlo chain.     

Bayesian testing for non-linearity in volatility modeling

Neural networks provide a tool for describing non-linearity in volatility processes of financial data and help to answer the question “how much” non-linearity is present in the data. Non-linearity is studied under three different specifications of the conditional distribution: Gaussian, Student-t and mixture of Gaussians. To rank the volatility models, a Bayesian framework is adopted to perform a Bayesian model selection within the different classes of models. In the empirical analysis, the return series of the Dow Jones Industrial Average index, FTSE 100 and NIKKEI 225 indices over a period of 16 years are studied. The results show different behavior across the three markets. In general, if a statistical model accounts for non-normality and explains most of the fat tails in the conditional distribution, then there is less need for complex non-linear specifications.     

Forecasting conditional correlations in stock,bond and foreign exchange markets

The paper forecasts conditional correlations between three classes of international financial assets, namely stock, bond and foreign exchange. Two countries are considered, namely Australia and New Zealand. Forecasting will be conducted using three multivariate GARCH models, namely the CCC model [T. Bollerslev, Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model, Rev. Econ. Stat. 72 (1990) 498–505], VARMA-GARCH model [S. Ling, M. McAleer, Asymptotic theory for a vector ARMA-GARCH model, Econometric Theory 19 (2003) 280–310], and VARMA-AGARCH model [M. McAleer, S. Hoti, F. Chan, Structure and asymptotic theory for multivariate asymmetric volatility, Econometric Rev., in press]. A rolling window technique is used to forecast 1-day ahead conditional correlations. To evaluate the impact of model specification on conditional correlations forecasts, this paper calculates and compares the correlations between conditional correlations forecasts resulted from the three models. The paper finds the evidence of volatility spillovers and asymmetric effect of negative and positive shock on the conditional variance in most pairs of series. However, it suggests that incorporating volatility spillovers and asymmetric do not contribute to better conditional correlations forecasts.     

A multivariate threshold stochastic volatility model

We introduce in this paper a multivariate threshold stochastic volatility model for multiple financial return time series. This model allows the dynamic structure of return and volatility to change according to a threshold model while accounting for the interdependence of financial returns. Through the threshold volatility modeling, we can understand the impact of market news on volatility asymmetry. Estimation of unknown parameters are carried out using Markov chain Monte Carlo techniques. Simulations show that our estimators are reliable in moderately large sample sizes. We apply the model to three market indice data and estimate time-varying correlations among the indice returns.     

Volatility dynamics of the US business cycle: A multivariate asymmetric GARCH approach

Most empirical investigations of the business cycles in the United States have excluded the dimension of asymmetric conditional volatility. This paper analyses the volatility dynamics of the US business cycle by comparing the performance of various multivariate generalised autoregressive conditional heteroskedasticity (GARCH) models. In particular, we propose two bivariate GARCH models to examine the evidence of volatility asymmetry and time-varying correlations concurrently, and then apply the proposed models to five sectors of Industrial Production of the United States. Our findings provide strong evidence of asymmetric conditional volatility in all sectors, and some support of time-varying correlations in various sectoral pairs. This has important policy implications for government to consider the effective countercyclical measures during recessions.     

On asymmetric generalised t stochastic volatility models

In stochastic volatility (SV) models, asset returns conditional on the latent volatility are usually assumed to have a normal, Student-t or exponential power (EP) distribution. An earlier study uses a generalised t (GT) distribution for the conditional returns and the results indicate that the GT distribution provides a better model fit to the Australian Dollar/Japanese Yen daily exchange rate than the Student-t distribution. In fact, the GT family nests a number of well-known distributions including the commonly used normal, Student-t and EP distributions. This paper extends the SV model with a GT distribution by incorporating general volatility asymmetry. We compare the empirical performance of nested distributions of the GT distribution as well as different volatility asymmetry specifications. The new asymmetric GT SV models are estimated using the Bayesian Markov chain Monte Carlo (MCMC) method to obtain parameter and log-volatility estimates. By using daily returns from the Standard and Poors (S&P) 500 index, we investigate the effects of the specification of error distributions as well as volatility asymmetry on parameter and volatility estimates. Results show that the choice of error distributions has a major influence on volatility estimation only when volatility asymmetry is not accounted for.     

Multivariate mixed normal conditional heteroskedasticity

A new multivariate volatility model where the conditional distribution of a vector time series is given by a mixture of multivariate normal distributions is proposed. Each of these distributions is allowed to have a time-varying covariance matrix. The process can be globally covariance stationary even though some components are not covariance stationary. Some theoretical properties of the model such as the unconditional covariance matrix and autocorrelations of squared returns are derived. The complexity of the model requires a powerful estimation algorithm. A simulation study compares estimation by maximum likelihood with the EM algorithm. Finally, the model is applied to daily US stock returns.     

Conditional confidence intervals for classification error rate

An observation is to be classified into one of several multivariate normal populations with equal covariance matrix. When the parameters are unknown, independent training samples are taken from the populations. We consider the construction of confidence intervals for the conditional error rate. The cases of two populations and three populations are studied in detail. We propose the conditional jackknife confidence interval and the conditional bootstrap confidence intervals of the conditional error rate. A Monte Carlo study is conducted to compare the confidence intervals.     

Modelling conditional correlations in the volatility of Asian rubber spot and futures returns

Asia is presently the most important market for the production and consumption of natural rubber. World prices of rubber are subject to not only to changes in demand, but also speculation regarding future markets. Japan and Singapore are the major future markets for rubber, while Thailand is one of the world's largest producers of rubber. As rubber prices are influenced by external markets, it is important to analyse the relationship between the relevant markets in Thailand, Japan and Singapore. The analysis is conducted using several alternative multivariate GARCH models. The empirical results indicate that the constant conditional correlations arising from the CCC model lie in the low to medium range. The results from the VARMA-GARCH model and the VARMA-AGARCH model suggest the presence of volatility spillovers and asymmetric effects of positive and negative return shocks on conditional volatility. Finally, the DCC model suggests that the conditional correlations can vary dramatically over time. In general, the dynamic conditional correlations in rubber spot and futures returns shocks can be independent or interdependent.     

Estimating stochastic volatility models using daily returns and realized volatility simultaneously

Realized volatility, which is the sum of squared intraday returns over a certain interval such as a day, has recently attracted the attention of financial economists and econometricians as an accurate measure of the true volatility. In the real market, however, the presence of non-trading hours and market microstructure noise in transaction prices may cause bias in the realized volatility. On the other hand, daily returns are less subject to noise and therefore may provide additional information on the true volatility. From this point of view, modeling realized volatility and daily returns simultaneously based on the well-known stochastic volatility model is proposed. Empirical studies using intraday data of Tokyo stock price index show that this model can estimate realized volatility biases and parameters simultaneously. The Bayesian approach is taken and an efficient sampling algorithm is proposed to implement the Markov chain Monte Carlo method for our simultaneous model. The result of the model comparison between the simultaneous models using both naive and scaled realized volatilities indicates that the effect of non-trading hours is more essential than that of microstructure noise and that asymmetry is crucial in stochastic volatility models. The proposed Bayesian approach provides an estimate of the entire conditional predictive distribution of returns under consideration of the uncertainty in the estimation of both biases and parameters. Hence common risk measures, such as value-at-risk and expected shortfall, can be easily estimated.     

高炉炼铁过程多元铁水质量非线性子空间建模及应用

高炉炼铁是一个物理化学反应复杂、多相多场耦合的大滞后、非线性动态系统,其关键工艺指标——铁水质量参数的检测、建模和控制一直是冶金工程和自动控制领域的难题.本文提出一种面向控制的数据驱动高炉炼铁多元铁水质量非线性子空间建模方法.首先,为了提高建模效率和降低计算复杂度,采用数据驱动典型相关性分析与相关性分析相结合的方法提取与铁水质量相关性最强的关键可控变量作为建模的输入变量;同时,为了更好地反映高炉非线性动态特性,将相关输入输出变量的时序和时滞关系在建模过程进行考虑;最后,采用基于最小二乘支持向量机（Least square support vector machine,LS-SVM）的非线性Hammerstein系统子空间辨识方法建立数据驱动的多元铁水质量非线性状态空间模型.同时,将核函数表示的模型非线性特性用多项式函数拟合,在仅损失很小模型精度的前提下大大降低模型的计算复杂度.基于实际数据的工业试验验证了所提建模方法的准确性、有效性和先进性.     

Matching the moments: a test of three representative agent models

No arbitrage profit opportunities implies state-contingent shadow prices that value the state-contingent payoffs of any asset traded in perfect markets. The shadow price is an implicit stochastic discount factor. I decompose the implicit model free nominal discount factor into its observable conditional mean—the price of a default free discount bond—and an unobservable innovation. I infer the unconditional covariance matrix between innovations in the model free discount factor and returns. Dynamic behavioural economic models specify, or imply, a discount factor. The theoretical mean of the nominal model generated discount factor is the observable price of a default free discount bond. A necessary condition for any model to be consistent with securities market data is that the moments of the deviations of the model-generated discount factor from its theoretical conditional mean (call the deviations residuals) match the moments of innovations in the model free discount factor. This is an easy specification test for any model. I match the moments for three representative agent specifications—power utility, habit formation, and a consumption-leisure choice model. The moments from the more complicated specifications come closer to matching the model free moments, but none of the representative agent models generates a covariance (risk premium) large enough to match the model free covariance     

A generalized Dynamic Conditional Correlation model for portfolio risk evaluation

We propose a generalization of the Dynamic Conditional Correlation multivariate GARCH model of Engle [R.F. Engle, Dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional heteroskedasticity models, Journal of Business and Economic Statistics 20 (2002) 339–350] and of the Asymmetric Dynamic Conditional Correlation model of Cappiello et al.[L. Cappiello, R.F. Engle, K. Sheppard, Asymmetric dynamics in the correlations of global equity and bond returns, Journal of Financial Econometrics 25 (2006) 537–572]. The model we propose introduces a block structure in parameter matrices that allows for interdependence with a reduced number of parameters. Our model nests the Flexible Dynamic Conditional Correlation model of Billio et al. [M. Billio, M. Caporin, M. Gobbo, Flexible dynamic conditional correlation multivariate GARCH for asset allocation, Applied Financial Economics Letters 2 (2006) 123–130] and is named Quadratic Flexible Dynamic Conditional Correlation Multivariate GARCH. In the paper, we provide conditions for positive definiteness of the conditional correlations. We also present an empirical application to the Italian stock market comparing alternative correlation models for portfolio risk evaluation.     

Forecasting large scale conditional volatility and covariance using neural network on GPU

Forecasting volatility is an important issue in financial econometric analysis. This paper aims to seek a computationally feasible approach for predicting large scale conditional volatility and covariance of financial time series. In the case of multi-variant time series, the volatility is represented by a Conditional Covariance Matrix (CCM). Traditional models for predicting CCM such as GARCH models are incapable of dealing with high-dimensional cases as there are O(N ²) parameters to be estimated in the case of N-variant asset return, and it is difficult to accelerate the computation of estimating these parameters by utilizing modern multi-core architecture. These GARCH models also have difficulties in modeling non-linear properties. The widely used Restricted Boltzmann Machine (RBM) is an energy-based stochastic recurrent neural network and its extended model, Conditional RBM (CRBM), has shown its capability in modeling high-dimensional time series. In this paper, we first propose a CRBM-based approach to forecast CCM and show how to capture the long memory properties in volatility, and then we implement the proposed model on GPU by using CUDA and CUBLAS. Experiment results indicate that the proposed CRBM-based model obtains better forecasting accuracy for low-dimensional volatility and it also shows great potential in modeling for large-scale cases compared with traditional GARCH models.     

GARCH,jumps and permanent and transitory components of volatility: the case of the Taiwan exchange rate

This paper investigates whether there are three distinctive features in financial asset prices, that is, time-varying conditional volatility, jumps and the component factors of volatility. It adopts a component-GARCH-Jump, which can efficiently capture the three features simultaneously. Our results demonstrate that the three features exist in the Taiwan exchange rate. Besides time-varying conditional volatility, our model identifies 172 jumps between 5 January 1988 and 21 March 2003. The empirical evidence shows that the permanent component of the conditional variance is a relatively smooth movement except for a fairly sharp shift which began in 1997. This means that the effect of the Asian crisis shock might very well have exerted not only a transitory jump effect, but also a permanent effect on Taiwan’s exchange rate.     

Intraday dynamic relationships between CSI 300 index futures and spot markets: a high-frequency analysis

Based on intraday 5-min high-frequency dataset, this paper empirically analyzes the intraday dynamic relationships between China’s CSI 300 index futures and spot markets with vector autoregression (VAR) and multivariate GARCH (MGARCH) models. By comparing four VAR–MGARCH models (dynamic conditional correlation, constant conditional correlation, diagonal and BEKK), the VAR–DCC–MGARCH model is found to fit the data the best and be preferred over the other models. The results of this model show that although there are bidirectional price causal relationships between the CSI 300 index futures and spot markets, the index futures return shock affects the spot market more severely than the spot return shock affects the futures market, indicating that the index futures market dominates the price discovery process between the two markets. There are bidirectional volatility spillovers effects between the CSI 300 index futures and spot markets, and the spillovers effects from index futures to spot almost equal to that from index spot to futures. The time-varying conditional correlations between the CSI 300 index futures and spot markets change from 0.4787 to 0.9594 across time, showing there is a strong positive correlation and linkage effect between the two markets. These results indicate that after a period of time of development, the price discovery performance of the CSI 300 index futures market has begun to function well, and the impact of the CSI 300 index futures market on its underlying spot market has strengthened.     

Bayesian model determination for multivariate ordinal and binary data

Different conditional independence specifications for ordinal categorical data are compared by calculating a posterior distribution over classes of graphical models. The approach is based on the multivariate ordinal probit model where the data are considered to have arisen as truncated multivariate normal random vectors. By parameterising the precision matrix of the associated multivariate normal in Cholesky form, ordinal data models corresponding to directed acyclic conditional independence graphs for the latent variables can be specified and conveniently computed. Where one or more of the variables are binary this parameterisation is particularly compelling, as necessary constraints on the latent variable distribution can be imposed in such a way that a standard, fully normalised, prior can still be adopted. For comparing different directed graphical models a reversible jump Markov chain Monte Carlo (MCMC) approach is proposed. Where interest is focussed on undirected graphical models, this approach is augmented to allow switches in the orderings of variables of associated directed graphs, hence allowing the posterior distribution over decomposable undirected graphical models to be computed. The approach is illustrated with several examples, involving both binary and ordinal variables, and directed and undirected graphical model classes.     

Joint forecasts of Dow Jones stocks under general multivariate loss function

When forecasts are assessed by a general loss (cost-of-error) function, the optimal point forecast is, in general, not the conditional mean, and depends on the conditional volatility—which, for stock returns, is time-varying. In order to provide forecasts of daily returns of 30 DJIA stocks under a general multivariate loss function, the following issues are addressed. We discuss what conditions define a multivariate loss function, and a simple class of such functions is proposed. Based on suitable combinations of univariate losses, the suggested multivariate functions are convenient for practical applications with many variables. To keep the computational aspect tractable, a flexible multivariate GARCH model is employed in estimating the conditional forecast distributions. The model easily copes with large number of series while allowing for skewness, fat tails, non-ellipticity, and tail dependence. Based on Engle’s DCC GARCH, it uses multivariate affine generalized hyperbolic distributions as conditional probability law, and the number of parameters to be estimated simultaneously does not depend on the number of series. The model is fitted using daily data from 2002 to 2007 (keeping data from 2008 for out-of-sample forecasts), and a bootstrap procedure is used to derive point forecasts under several multivariate loss functions of the proposed type.     

Wavelet-based detection of outliers in financial time series

Outliers in financial data can lead to model parameter estimation biases, invalid inferences and poor volatility forecasts. Therefore, their detection and correction should be taken seriously when modeling financial data. The present paper focuses on these issues and proposes a general detection and correction method based on wavelets that can be applied to a large class of volatility models. The effectiveness of the new proposal is tested by an intensive Monte Carlo study for six well-known volatility models and compared to alternative proposals in the literature, before it is applied to three daily stock market indices. The Monte Carlo experiments show that the new method is both very effective in detecting isolated outliers and outlier patches and much more reliable than other alternatives, since it detects a significantly smaller number of false outliers. Correcting the data of outliers reduces the skewness and the excess kurtosis of the return series distributions and allows for more accurate return prediction intervals compared to those obtained when the existence of outliers is ignored.     

Intra-daily information of range-based volatility for MEM-GARCH

Conventional GARCH modeling formulates an additive-error mean equation for daily return and an autoregressive moving-average specification for its conditional variance, without much consideration on the effects of intra-daily data. Using Engle’s multiplicative-error model (MEM) formulation, range-based volatility is proposed as an intraday proxy for several GARCH frameworks. The performances of these different approaches for two 8-year market data sets: the S&P 500 and the NASDAQ composite index, are studied and compared. The impact of significant changes in intraday data has been found to reflect in the MEM-GARCH volatility. For some frameworks it is also possible to use lagged values of range-based volatility to delay the intraday effects in the conditional variance estimation.