Testing for jumps in the EGARCH process

This paper considers testing for jumps in the exponential GARCH (EGARCH) models with Gaussian and Student-t innovations. The Wald and log likelihood ratio tests contain a nuisance parameter unidentified under the null hypothesis of no jumps, and hence are unavailable for this problem, because jump probability and variance of jumps in the test statistic cannot be estimated under the null hypothesis of no jumps. It is shown that the nuisance parameter is cancelled out in the Lagrange multiplier (LM) test statistic, and hence that the test is nuisance parameter-free. The one-sided test is also proposed using the nonnegative constraint on jump variance. The actual size and power of the tests are examined in a Monte Carlo experiment. The test is applied to daily returns of S&P 500 as an illustrative example.     

Leverage, heavy-tails and correlated jumps in stochastic volatility models

Efficient and fast Markov chain Monte Carlo estimation methods for the stochastic volatility model with leverage effects, heavy-tailed errors and jump components, and for the stochastic volatility model with correlated jumps are proposed. The methods are illustrated using simulated data and are applied to analyze daily stock returns data on S&P500 index and TOPIX. Model comparisons are conducted based on the marginal likelihood for various SV models including the superposition model.     

Testing the Power of a Generalization of the KPSS-Tests against Fractionally Integrated Hypotheses

We analyse in this article the size and the power properties of differentgeneralizations of the KPSS-tests proposed by Hobjin et al. (1998) for testingthe null hypothesis of stationarity in univariate time series when thealternatives are of a fractional form. We show that the test based on the useof the Quadratic Spectral kernel along with an automatic bandwidth selectionprocedure produces the best results and thus, it might be employed for testingI(0) against I(d>0) stationary or nonstationary processes. An empiricalapplication, showing the performance of the tests in finite samples is alsocarried out at the end of the article.     

On generalised asymmetric stochastic volatility models

Stochastic volatility (SV) models have been considered as a real alternative to time-varying volatility of the ARCH family. Existing asymmetric SV (ASV) models treat volatility asymmetry via the leverage effect hypothesis. Generalised ASV models that take account of both volatility asymmetry and normality violation expressed simultaneously by skewness and excess kurtosis are introduced. The new generalised ASV models are estimated using the Bayesian Markov Chain Monte Carlo approach for parametric and log-volatility estimation. By using simulated and real financial data series, the new models are compared to existing SV models for their statistical properties, and for their estimation performance in within and out-of-sample periods. Results show that there is much to gain from the introduction of the generalised ASV models.     

Testing volatility autocorrelation in the constant elasticity of variance stochastic volatility model

The sample autocovariance of the suitably scaled squared returns of a given stock is shown here to be a consistent and asymptotically normal estimator of the theoretical autocovariance of the mean variance, when the data is generated by the Constant Elasticity of Variance stochastic volatility (CEV SV) process. By computing explicitly the asymptotic variance of the estimator, confidence bands are obtained for the theoretical autocovariance. For each one of the stock indexes S&P500, CAC40, FTSE, DAX and SMI the estimated confidence bands are compared with the theoretical autocovariances computed for several values of the model parameters. The results suggest that the CEV SV model is able to capture the empirical autocovariance detected on the observed data. Analogous results are derived for the theoretical autocorrelation function.     

On the applicability of stochastic volatility models

The applicability of the stochastic volatility (SV) model and the SV model with jumps for US. Treasury Bill yields data is investigated. The transformation of the continuous time models into regression models is considered and their error terms are examined. The applicability of the continuous time models to the real data is assessed by comparing some atypical properties of such error terms with an application to the real data and the generated data from the models. The empirical results indicate that the SV model and the SV model with jumps are not applicable to modeling the daily/weekly released US T-Bill secondary market yields data. Some trends and correlation structure are detected to exist in the error terms of the transformed regression models for the daily/weekly released US T-Bill yields data, while the error terms of the continuous time models are supposed to be uncorrelated. These results suggest that alternative models are needed to model such T-Bill yields data.     

Testing for nonlinearity in mean and volatility for heteroskedastic models

A simple test for threshold nonlinearity in either the mean or volatility equation, or both, of a heteroskedastic time series model is proposed. The procedure extends current Bayesian Markov chain Monte Carlo methods and threshold modelling by employing a general double threshold GARCH model that allows for an explosive, non-stationary regime. Posterior credible intervals on model parameters are used to detect and specify threshold nonlinearity in the mean and/or volatility equations. Simulation experiments demonstrate that the method works favorably in identifying model specifications varying in complexity from the conventional GARCH up to the full double-threshold nonlinear GARCH model with an explosive regime, and is robust to over-specification in model orders.     

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.     

Pricing currency options with support vector regression and stochastic volatility model with jumps

This paper presents an efficient currency option pricing model based on support vector regression (SVR). This model focuses on selection of input variables of SVR. We apply stochastic volatility model with jumps to SVR in order to account for sudden big changes in exchange rate volatility. We use forward exchange rate as the input variable of SVR, since forward exchange rate takes interest rates of a basket of currencies into account. Therefore, the inputs of SVR will include moneyness (spot rate/strike price), forward exchange rate, volatility of the spot rate, domestic risk-free simple interest rate, and the time to maturity. Extensive experimental studies demonstrate the ability of new model to improve forecast accuracy.     

Characteristic function estimation of Ornstein-Uhlenbeck-based stochastic volatility models

Continuous-time stochastic volatility models are becoming increasingly popular in finance because of their flexibility in accommodating most stylized facts of financial time series. However, their estimation is difficult because the likelihood function does not have a closed-form expression. A characteristic function-based estimation method for non-Gaussian Ornstein-Uhlenbeck-based stochastic volatility models is proposed. Explicit expressions of the characteristic functions for various cases of interest are derived. The asymptotic properties of the estimators are analyzed and their small-sample performance is evaluated by means of a simulation experiment. Finally, two real-data applications show that the superposition of two Ornstein-Uhlenbeck processes gives a good approximation to the dependence structure of the process.     

The hierarchical-likelihood approach to autoregressive stochastic volatility models

Many volatility models used in financial research belong to a class of hierarchical generalized linear models with random effects in the dispersion. Therefore, the hierarchical-likelihood (h-likelihood) approach can be used. However, the dimension of the Hessian matrix is often large, so techniques of sparse matrix computation are useful to speed up the procedure of computing the inverse matrix. Using numerical studies we show that the h-likelihood approach gives better long-term prediction for volatility than the existing MCMC method, while the MCMC method gives better short-term prediction. We show that the h-likelihood approach gives comparable estimations of fixed parameters to those of existing methods.     

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.     

Estimating stochastic volatility models of stock returns in Chinese markets

Volatility plays a key role in microstructure issues in the study of financial markets. Stochastic volatility (SV) models have been applied to the study of the behavior of financial variables. Two stock markets exist in China: Shanghai Stock Exchange and Shenzhen Stock Exchange. As emerging stock markets, investors are increasingly concerned about the volatilities of these two stock markets. We briefly introduce how to estimate SV models using the Markov chain Monte Carlo (MCMC) method. In order to do full and comprehensive analyses of the volatilities of stock returns, we estimated SV models using most of the historical data and the different data frequencies of the two Chinese markets. We found that estimated values of volatility parameters are very high for all data frequencies. This suggests that stock returns are extremely volatile even at long-term intervals in Chinese markets.     

Bayesian analysis of stochastic volatility models with mixture-of-normal distributions

Stochastic volatility (SV) models usually assume that the distribution of asset returns conditional on the latent volatility is normal. This article analyzes SV models with a mixture-of-normal distributions in order to compare with other heavy-tailed distributions such as the Student-t distribution and generalized error distribution (GED). A Bayesian method via Markov-chain Monte Carlo (MCMC) techniques is used to estimate parameters and Bayes factors are calculated to compare the fit of distributions. The method is illustrated by analyzing daily data from the Yen/Dollar exchange rate and the Tokyo stock price index (TOPIX). According to Bayes factors, we find that while the t distribution fits the TOPIX better than the normal, the GED and the normal mixture, the mixture-of-normal distributions give a better fit to the Yen/Dollar exchange rate than other models. The effects of the specification of error distributions on the Bayesian confidence intervals of future returns are also examined. Comparison of SV with GARCH models shows that there are cases that the SV model with the normal distribution is less effective to capture leptokurtosis than the GARCH with heavy-tailed distributions.     

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.     

Simulation-based sequential analysis of Markov switching stochastic volatility models

We propose a simulation-based algorithm for inference in stochastic volatility models with possible regime switching in which the regime state is governed by a first-order Markov process. Using auxiliary particle filters we developed a strategy to sequentially learn about states and parameters of the model. The methodology is tested against a synthetic time series and validated with a real financial time series: the IBOVESPA stock index (São Paulo Stock Exchange).     

Simulated minimum Hellinger distance estimation of stochastic volatility models

A simultaneously efficient and robust approach for distribution-free parametric inference, called the simulated minimum Hellinger distance (SMHD) estimator, is proposed. In the SMHD estimation, the Hellinger distance between the nonparametrically estimated density of the observed data and that of the simulated samples from the model is minimized. The method is applicable to the situation where the closed-form expression of the model density is intractable but simulating random variables from the model is possible. The robustness of the SMHD estimator is equivalent to the minimum Hellinger distance estimator. The finite sample efficiency of the proposed methodology is found to be comparable to the Bayesian Markov chain Monte Carlo and maximum likelihood Monte Carlo methods and outperform the efficient method of moments estimators. The robustness of the method to a stochastic volatility model is demonstrated by a simulation study. An empirical application to the weekly observations of foreign exchange rates is presented.     

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.     

The role of “leads” in the dynamic OLS estimation of cointegrating regression models

In this paper, we consider the role of “leads” of the first difference of integrated variables in the dynamic OLS estimation of cointegrating regression models. Specifically, we investigate Stock and Watson’s [J.H. Stock, M.W. Watson’s, A simple estimator of cointegrating vectors in higher order integrated systems, Econometrica 61 (1993) 783–820] claim that the role of leads is related to the concept of Granger causality by a Monte Carlo simulation. From the simulation results, we find that the dynamic OLS estimator without leads substantially outperforms that with leads and lags; we therefore recommend testing for Granger non-causality before estimating models.     

Analysis of filtering and smoothing algorithms for Lévy-driven stochastic volatility models

Filtering and smoothing algorithms that estimate the integrated variance in Lévy-driven stochastic volatility models are analyzed. Particle filters are algorithms designed for nonlinear, non-Gaussian models while the Kalman filter remains the best linear predictor if the model is linear but non-Gaussian. Monte Carlo experiments are performed to compare these algorithms across different specifications of the model including different marginal distributions and degrees of persistence for the instantaneous variance. The use of realized variance as an observed variable in the state space model is also evaluated. Finally, the particle filter's ability to identify the timing and size of jumps is assessed relative to popular nonparametric estimators.