Parameter changes in GARCH model

A new method for detecting the parameter changes in generalized autoregressive heteroskedasticity GARCH (1,1) model is proposed. In the proposed method, time series observations are divided into several segments and a GARCH (1,1) model is fitted to each segment. The goodness-of-fit of the global model composed of these local GARCH (1,1) models is evaluated using the corresponding information criterion (IC). The division that minimizes IC defines the best model. Furthermore, since the simultaneous estimation of all possible models requires huge computational time, a new time-saving algorithm is proposed. Simulation results and empirical results both indicate that the proposed method is useful in analysing financial data.     

基于EGARCH-M模型的沪深300股指期货跨期套利研究——一种修正的协整关系

现有的基于协整股指期货跨期套利策略主要利用GARCH模型进行建模。该模型忽略了杠杆效应的存在,也未考虑条件方差可能影响协整方程。在引入EGARCH-M模型进行套利研究的基础上,提出一种新的协整关系——修正的协整;利用沪深300股指期货合约的每分钟收盘价进行实证分析,结果表明:正态EGARCH-M模型对数据的拟合效果优于传统的GARCH模型,通过设定合理的交易机制可以获得良好的套利结果;非正态的EGARCH-M模型在拟合效果和捕捉套利机会方面都比正态模型具有更好的表现,且套利效果也有显著提高。     

Value-at-risk estimation with new skew extension of generalized normal distribution

In this paper, we introduce a new distribution, called the alpha-skew generalized normal (ASGN), for GARCH models in modeling daily Value-at-Risk (VaR). Basic structural properties of the proposed distribution are derived including probability and cumulative density functions, moments and stochastic representation. The real data application based on ISE-100 index is given to show the performance of GARCH model specified under ASGN innovation distribution with respect to normal, Student’s-t, skew normal and generalized normal models in terms of the VaR accuracy. The empirical results show that GARCH model with ASGN innovation distribution generates the most accurate VaR forecasts for all confidence levels.     

On double hysteretic heteroskedastic model

ABSTRACT

This paper proposes a hysteretic autoregressive model with GARCH specification and a skew Student's t-error distribution for financial time series. With an integrated hysteresis zone, this model allows both the conditional mean and conditional volatility switching in a regime to be delayed when the hysteresis variable lies in a hysteresis zone. We perform Bayesian estimation via an adaptive Markov Chain Monte Carlo sampling scheme. The proposed Bayesian method allows simultaneous inferences for all unknown parameters, including threshold values and a delay parameter. To implement model selection, we propose a numerical approximation of the marginal likelihoods to posterior odds. The proposed methodology is illustrated using simulation studies and two major Asia stock basis series. We conduct a model comparison for variant hysteresis and threshold GARCH models based on the posterior odds ratios, finding strong evidence of the hysteretic effect and some asymmetric heavy-tailness. Versus multi-regime threshold GARCH models, this new collection of models is more suitable to describe real data sets. Finally, we employ Bayesian forecasting methods in a Value-at-Risk study of the return series.     

A new algorithm for maximum likelihood estimation in normal scale-mixture generalized autoregressive conditional heteroskedastic models

In this paper, we propose a new generalized autoregressive conditional heteroskedastic (GARCH) model using infinite normal scale-mixtures which can suitably avoid order selection problems in the application of finite normal scale-mixtures. We discuss its theoretical properties and develop a two-stage algorithm for the maximum likelihood estimator to estimate the mixing distribution non-parametric maximum likelihood estimator (NPMLE) as well as GARCH parameters (two-stage MLE). For the estimation of a mixing distribution, we employ a fast computational algorithm proposed by Wang [On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. J R Stat Soc Ser B. 2007;69:185–198] under the gradient characterization of the non-parametric mixture likelihood. The GARCH parameters are then estimated either using the expectation-mazimization algorithm or general optimization scheme. In addition, we propose a new forecasting algorithm of value-at-risk (VaR) using the two-stage MLE and the NPMLE. Through a simulation study and real data analysis, we compare the performance of the two-stage MLE with the existing ones including quasi-maximum likelihood estimator based on the standard normal density and the finite normal mixture quasi maximum estimated-likelihood estimator (cf. Lee S, Lee T. Inference for Box–Cox transformed threshold GARCH models with nuisance parameters. Scand J Stat. 2012;39:568–589) in terms of the relative efficiency and accuracy of VaR forecasting.     

The generalized Gudermannian distribution: inference and volatility modelling

In this paper, we introduce a new distribution, called generalized Gudermannian (GG) distribution, and its skew extension for GARCH models in modelling daily Value-at-Risk (VaR). Basic structural properties of the proposed distribution are obtained including probability density and cumulative distribution functions, moments, and stochastic representation. The maximum likelihood method is used to estimate unknown parameters of the proposed model and finite sample performance of maximum likelihood estimates are evaluated by means of Monte-Carlo simulation study. The real data application on Nikkei 225 index is given to demonstrate the performance of GARCH model specified under skew extension of GG innovation distribution against normal, Student's-t, skew normal and generalized error and skew generalized error distributions in terms of the accuracy of VaR forecasts. The empirical results show that the GARCH model with GG innovation distribution produces the most accurate VaR forecasts for all confidence levels.     

Consistency of maximum likelihood estimators for the regime-switching GARCH model

The regime-switching GARCH (generalized autoregressive conditionally heteroscedastic) model incorporates the idea of Markov switching into the more restrictive GARCH model, which significantly extends the GARCH model. However, the statistical inference for such an extended model is rather difficult because observations at any time point then depend on the whole regime path and the likelihood becomes intractable quickly as the length of observations increases. In this paper, by transforming it into an infinite order ARCH model, we obtain the possibility of writing a likelihood which can be handled directly and the consistency of the maximum likelihood estimators is proved. Simulation studies to illustrate the consistency and asymptotic normality of the estimators (for both Gaussian and non-Gaussian innovations) and a model specification problem are presented.     

Finite nonparametric grach model for foreign exchange volatility

GARCH model has been commonly used to describe the volatility of foreign exchange returns, which typically depends on returns many lags before, While the GARCH model provides a simple geometric decaying structure for persistence in time, it restricts tiie impact of variables to Quadratic functions. A finite nonparametric GARCH model is proposed that allows the variables' impact to be a smooth function of any form. A direct local polynomial estimation method for this finite GARCH model is proposed based on results on proportional additive model, and is applied to the German Mark (DEM)/US Dollar (USD) daily returns data. Estimators uf both the decaying rate and the impact function are obtained. Diagnostics show satisfactory out-of-sampie prediction based on the proposed model, which helps to better understand the dynamics of foreign exchange volatility.     

Value-at-risk forecasting based on Gaussian mixture ARMA–GARCH model

In this paper, we develop a new forecasting algorithm for value-at-risk (VaR) based on ARMA–GARCH (autoregressive moving average–generalized autoregressive conditional heteroskedastic) models whose innovations follow a Gaussian mixture distribution. For the parameter estimation, we employ the conditional least squares and quasi-maximum-likelihood estimator (QMLE) for ARMA and GARCH parameters, respectively. In particular, Gaussian mixture parameters are estimated based on the residuals obtained from the QMLE of GARCH parameters. Our algorithm provides a handy methodology, spending much less time in calculation than the existing resampling and bias-correction method developed in Hartz et al. [Accurate value-at-risk forecasting based on the normal-GARCH model, Comput. Stat. Data Anal. 50 (2006), pp. 3032–3052]. Through a simulation study and a real-data analysis, it is shown that our method provides an accurate VaR prediction.     

ASSESSING AND TESTING FOR THRESHOLD NONLINEARITY IN STOCK RETURNS

This paper proposes a test for threshold nonlinearity in a time series with generalized autore‐gressive conditional heteroscedasticity (GARCH) volatility dynamics. This test is used to examine whether financial returns on market indices exhibit asymmetric mean and volatility around a threshold value, using a double‐threshold GARCH model. The test adopts the reversible‐jump Markov chain Monte Carlo idea of Green, proposed in 1995, to calculate the posterior probabilities for a conventional GARCH model and a double‐threshold GARCH model. Posterior evidence favouring the threshold GARCH model indicates threshold nonlinearity with asymmetric behaviour of the mean and volatility. Simulation experiments demonstrate that the test works very well in distinguishing between the conventional GARCH and the double‐threshold GARCH models. In an application to eight international financial market indices, including the G‐7 countries, clear evidence supporting the hypothesis of threshold nonlinearity is discovered, simultaneously indicating an uneven mean‐reverting pattern and volatility asymmetry around a threshold return value.     

基于混频已实现GARCH模型的波动预测与VaR度量

本文将Hansen等（2012）的Realized GARCH模型扩展为包含日内收益率、日收益率以及已实现波动率的混频已实现GARCH模型（M-Realized GARCH模型）。该模型将日内交易分为前后两段,引入了混频均值方程,并对混频均值方程的残差分别建立条件波动率方程和已实现日波动率方程。本文采用2013-2016年沪深300指数混频数据,分别在扰动项服从正态分布、t分布和广义误差分布的假设下,采用损失函数、SPA检验、kupiec检验和动态分位数检验法,对GARCH、Realized GARCH和M-Realized GARCH模型的波动率预测和VaR度量效果对比研究,得出M-Realized GARCH模型能提高预测精度,且VaR实际失败率与理论失败率一致,失败发生之间不相关。最后,本文利用Block bootstrap方法抽样得到混频数据,模拟证明了M-Realized GARCH模型比Realized GARCH模型具有更高的预测精度。     

A General Multivariate Threshold GARCH Model With Dynamic Conditional Correlations

We introduce a new multivariate GARCH model with multivariate thresholds in conditional correlations and develop a two-step estimation procedure that is feasible in large dimensional applications. Optimal threshold functions are estimated endogenously from the data and the model conditional covariance matrix is ensured to be positive definite. We study the empirical performance of our model in two applications using U.S. stock and bond market data. In both applications our model has, in terms of statistical and economic significance, higher forecasting power than several other multivariate GARCH models for conditional correlations.     

Kalman filter-based modelling and forecasting of stochastic volatility with threshold

We propose a parametric nonlinear time-series model, namely the Autoregressive-Stochastic volatility with threshold (AR-SVT) model with mean equation for forecasting level and volatility. Methodology for estimation of parameters of this model is developed by first obtaining recursive Kalman filter time-update equation and then employing the unrestricted quasi-maximum likelihood method. Furthermore, optimal one-step and two-step-ahead out-of-sample forecasts formulae along with forecast error variances are derived analytically by recursive use of conditional expectation and variance. As an illustration, volatile all-India monthly spices export during the period January 2006 to January 2012 is considered. Entire data analysis is carried out using EViews and matrix laboratory (MATLAB) software packages. The AR-SVT model is fitted and interval forecasts for 10 hold-out data points are obtained. Superiority of this model for describing and forecasting over other competing models for volatility, namely AR-Generalized autoregressive conditional heteroscedastic, AR-Exponential GARCH, AR-Threshold GARCH, and AR-Stochastic volatility models is shown for the data under consideration. Finally, for the AR-SVT model, optimal out-of-sample forecasts along with forecasts of one-step-ahead variances are obtained.     

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.     

A Bayesian Analysis of Autoregressive Models with Exogenous Variables and Power-Transformed and Threshold GARCH Errors

Consider a class of autoregressive models with exogenous variables and power transformed and threshold GARCH (ARX-PTTGARCH) errors, which is a natural generalization of the standard and special GARCH model. We propose a Bayesian method to show that combining Gibbs sampler and Metropolis-Hastings algorithm to give a Bayesian analysis can be applied to estimate parameters of ARX-PTTGARCH models with success.     

Stepwise local influence in generalized autoregressive conditional heteroskedasticity models

Detection of outliers or influential observations is an important work in statistical modeling, especially for the correlated time series data. In this paper we propose a new procedure to detect patch of influential observations in the generalized autoregressive conditional heteroskedasticity (GARCH) model. Firstly we compare the performance of innovative perturbation scheme, additive perturbation scheme and data perturbation scheme in local influence analysis. We find that the innovative perturbation scheme give better result than other two schemes although this perturbation scheme may suffer from masking effects. Then we use the stepwise local influence method under innovative perturbation scheme to detect patch of influential observations and uncover the masking effects. The simulated studies show that the new technique can successfully detect a patch of influential observations or outliers under innovative perturbation scheme. The analysis based on simulation studies and two real data sets show that the stepwise local influence method under innovative perturbation scheme is efficient for detecting multiple influential observations and dealing with masking effects in the GARCH model.     

Semiparametric Time Series Models with Log‐concave Innovations: Maximum Likelihood Estimation and its Consistency

We study semiparametric time series models with innovations following a log‐concave distribution. We propose a general maximum likelihood framework that allows us to estimate simultaneously the parameters of the model and the density of the innovations. This framework can be easily adapted to many well‐known models, including autoregressive moving average (ARMA), generalized autoregressive conditionally heteroscedastic (GARCH), and ARMA‐GARCH models. Furthermore, we show that the estimator under our new framework is consistent in both ARMA and ARMA‐GARCH settings. We demonstrate its finite sample performance via a thorough simulation study and apply it to model the daily log‐return of the FTSE 100 index.     

Characteristic function-based inference for GARCH models with heavy-tailed innovations

We consider estimation and goodness-of-fit tests in GARCH models with innovations following a heavy-tailed and possibly asymmetric distribution. Although the method is fairly general and applies to GARCH models with arbitrary innovation distribution, we consider as special instances the stable Paretian, the variance gamma, and the normal inverse Gaussian distribution. Exploiting the simple structure of the characteristic function of these distributions, we propose minimum distance estimation based on the empirical characteristic function of properly standardized GARCH-residuals. The finite-sample results presented facilitate comparison with existing methods, while the new procedures are also applied to real data from the financial market.     

A Stochastic Volatility Model With Conditional Skewness

We develop a discrete-time affine stochastic volatility model with time-varying conditional skewness (SVS). Importantly, we disentangle the dynamics of conditional volatility and conditional skewness in a coherent way. Our approach allows current asset returns to be asymmetric conditional on current factors and past information, which we term contemporaneous asymmetry. Conditional skewness is an explicit combination of the conditional leverage effect and contemporaneous asymmetry. We derive analytical formulas for various return moments that are used for generalized method of moments (GMM) estimation. Applying our approach to S&P500 index daily returns and option data, we show that one- and two-factor SVS models provide a better fit for both the historical and the risk-neutral distribution of returns, compared to existing affine generalized autoregressive conditional heteroscedasticity (GARCH), and stochastic volatility with jumps (SVJ) models. Our results are not due to an overparameterization of the model: the one-factor SVS models have the same number of parameters as their one-factor GARCH competitors and less than the SVJ benchmark.     

A new generalization of skew-T distribution with volatility models

In this paper, we propose a new generalized alpha-skew-T (GAST) distribution for generalized autoregressive conditional heteroskedasticity (GARCH) models in modelling daily Value-at-Risk (VaR). Some mathematical properties of the proposed distribution are derived including density function, moments and stochastic representation. The maximum likelihood estimation method is discussed to estimate parameters via a simulation study. Then, the real data application on S&P-500 index is performed to investigate the performance of GARCH models specified under GAST innovation distribution with respect to normal, Student's-t and Skew-T models in terms of the VaR accuracy. Backtesting methodology is used to compare the out-of-sample performance of the VaR models. The results show that GARCH models with GAST innovation distribution outperforms among others and generates the most conservative VaR forecasts for all confidence levels and for both long and short positions.