Forecasting energy futures volatility with threshold augmented heterogeneous autoregressive jump models

Abstract

This study forecasts the volatility of two energy futures markets (oil and gas), using high-frequency data. We, first, disentangle volatility into continuous volatility and jumps. Second, we apply wavelet analysis to study the relationship between volume and the volatility measures for different horizons. Third, we augment the heterogeneous autoregressive (HAR) model by nonlinearly including both jumps and volume. We then propose different empirical extensions of the HAR model. Our study shows that oil and gas volatilities nonlinearly depend on public information (jumps), private information (continuous volatility), and trading volume. Moreover, our threshold augmented HAR model with heterogeneous jumps and continuous volatility outperforms HAR model in forecasting volatility.     

Term structure forecasting in affine framework with time-varying volatility

This study extends the affine Nelson–Siegel model by introducing the time-varying volatility component in the observation equation of yield curve, modeled as a standard EGARCH process. The model is illustrated in state-space framework and empirically compared to the standard affine and dynamic Nelson–Siegel model in terms of in-sample fit and out-of-sample forecast accuracy. The affine based extended model that accounts for time-varying volatility outpaces the other models for fitting the yield curve and produces relatively more accurate 6- and 12-month ahead forecasts, while the standard affine model comes with more precise forecasts for the very short forecast horizons. The study concludes that the standard and affine Nelson–Siegel models have higher forecasting capability than their counterpart EGARCH based models for the short forecast horizons, i.e., 1 month. The EGARCH based extended models have excellent performance for the medium and longer forecast horizons.     

On the invertibility of EGARCH(p,q)

Of the two most widely estimated univariate asymmetric conditional volatility models, the exponential GARCH (or EGARCH) specification is said to be able to capture asymmetry, which refers to the different effects on conditional volatility of positive and negative effects of equal magnitude, and leverage, which refers to the negative correlation between the returns shocks and subsequent shocks to volatility. However, the statistical properties of the (quasi-)maximum likelihood estimator (QMLE) of the EGARCH(p, q) parameters are not available under general conditions, but only for special cases under highly restrictive and unverifiable sufficient conditions, such as EGARCH(1,0) or EGARCH(1,1), and possibly only under simulation. A limitation in the development of asymptotic properties of the QMLE for the EGARCH(p, q) model is the lack of an invertibility condition for the returns shocks underlying the model. It is shown in this article that the EGARCH(p, q) model can be derived from a stochastic process, for which sufficient invertibility conditions can be stated simply and explicitly when the parameters respect a simple condition.¹¹Using the notation introduced in part 2, this refers to the cases where α ≥ |γ| or α ≤ ? |γ|. The first inequality is generally assumed in the literature related to the invertibility of EGARCH. This article provides (in the Appendix) an argument for the possible lack of invertibility when these conditions are not met. This will be useful in reinterpreting the existing properties of the QMLE of the EGARCH(p, q) parameters.     

Adaptive quasi-maximum likelihood estimation of GARCH models with Student’s t likelihood

ABSTRACT

This paper proposes an adaptive quasi-maximum likelihood estimation (QMLE) when forecasting the volatility of financial data with the generalized autoregressive conditional heteroscedasticity (GARCH) model. When the distribution of volatility data is unspecified or heavy-tailed, we worked out adaptive QMLE based on data by using the scale parameter η_f to identify the discrepancy between wrongly specified innovation density and the true innovation density. With only a few assumptions, this adaptive approach is consistent and asymptotically normal. Moreover, it gains better efficiency under the condition that innovation error is heavy-tailed. Finally, simulation studies and an application show its advantage.     

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.     

Quantile forecasts for financial volatilities based on parametric and asymmetric models

For financial volatilities such as realized volatility and volatility index, a new parametric quantile forecast strategy is proposed, focusing on forecast interval and value at risk (VaR) forecast. This fully addresses asymmetries in 3 parts: mean, volatility and distribution. The asymmetries are addressed by the LHAR (leverage heterogeneous autoregressive) model of McAleer and Medeiros (2008) and Corsi and Reno (2009) for the mean part, by the EGARCH model for the volatility part, and by the skew-t distribution for the error distribution part. The method is applied to the realized volatilities and the volatility indexes of the US S&P 500 index, the US NASDAQ index, the Korea KOSPI index in which significant asymmetries are identified. Considerable out-of-sample forecast improvements of the forecast interval and VaR forecast are demonstrated for the volatilities: forecast intervals of volatilities have better coverages with shorter lengths and VaR forecasts of volatility indexes have better violations if asymmetries are properly addressed rather than ignored. The proposed parametric method reveals considerably better out-of-sample performance than the recently proposed semiparametric quantile regression approach of Zikes and Barunik (2016).     

Generalized value at risk forecasting

Abstract

In this paper, using estimating function approach, a new optimal volatility estimator is introduced and based on the recursive form of the estimator a data-driven generalized EWMA model for value at risk (VaR) forecast is proposed. An appropriate data-driven model for volatility is identified by the relationship between absolute deviation and standard deviation for symmetric distributions with finite variance. It is shown that the asymptotic variance of the proposed volatility estimator is smaller than that of conventional estimators and is more appropriate for financial data with larger kurtosis. For IBM, Microsoft, Apple stocks and SP 500 index the proposed method is used to identify the model, estimate the volatility, and obtain minimum mean square error(MMSE) forecasts of VaR.     

Forecast Evaluation in the Presence of Unobserved Volatility

Abstract

A number of volatility forecasting studies have led to the perception that the ARCH- and Stochastic Volatility-type models provide poor out-of-sample forecasts of volatility. This is primarily based on the use of traditional forecast evaluation criteria concerning the accuracy and the unbiasedness of forecasts. In this paper we provide an analytical assessment of volatility forecasting performance. We use the volatility and log volatility framework to prove how the inherent noise in the approximation of the true- and unobservable-volatility by the squared return, results in a misleading forecast evaluation, inflating the observed mean squared forecast error and invalidating the Diebold–Mariano statistic. We analytically characterize this noise and explicitly quantify its effects assuming normal errors. We extend our results using more general error structures such as the Compound Normal and the Gram–Charlier classes of distributions. We argue that evaluation problems are likely to be exacerbated by non-normality of the shocks and that non-linear and utility-based criteria can be more suitable for the evaluation of volatility forecasts.     

Markov switch smooth transition HYGARCH model: Stability and estimation

Abstract

HYGARCH model is basically used to model long-range dependence in volatility. We propose Markov switch smooth-transition HYGARCH model, where the volatility in each state is a time-dependent convex combination of GARCH and FIGARCH. This model provides a flexible structure to capture different levels of volatilities and also short and long memory effects. The necessary and sufficient condition for the asymptotic stability is derived. Forecast of conditional variance is studied by using all past information through a parsimonious way. Bayesian estimations based on Gibbs sampling are provided. A simulation study has been given to evaluate the estimations and model stability. The competitive performance of the proposed model is shown by comparing it with the HYGARCH and smooth-transition HYGARCH models for some period of the S&P500 and Dow Jones industrial average indices based on volatility and value-at-risk forecasts.     

Generalized EGARCH Random Effect Models Application to Financial Time Series

In this article, we propose a simple alternative model to analyze the volatility of the financial time series. In the applications, the performance of this model is compared with the performance of the GARCH type models. Using GARCH, EGARCH, and the proposed models, we analyze the time series of the Bovespa and Dow Jones Industrial Average indexes. In the applications we can see that the proposed models have good performance compared with the usual GARCH type model.     

Forecasting using locally stationary wavelet processes

Locally stationary wavelet (LSW) processes, built on non-decimated wavelets, can be used to analyse and forecast non-stationary time series. They have been proved useful in the analysis of financial data. In this paper, we first carry out a sensitivity analysis, then propose some practical guidelines for choosing the wavelet bases for these processes. The existing forecasting algorithm is found to be vulnerable to outliers, and a new algorithm is proposed to overcome the weakness. The new algorithm is shown to be stable and outperforms the existing algorithm when applied to real financial data. The volatility forecasting ability of LSW modelling based on our new algorithm is then discussed and shown to be competitive with traditional GARCH models.     

Forecasting interest rates: a comparative assessment of some second-generation nonlinear models

Modeling and forecasting of interest rates has traditionally proceeded in the framework of linear stationary methods such as ARMA and VAR, but only with moderate success. We examine here three methods, which account for several specific features of the real world asset prices such as nonstationarity and nonlinearity. Our three candidate methods are based, respectively, on a combined wavelet artificial neural network (WANN) analysis, a mixed spectrum (MS) analysis and nonlinear ARMA models with Fourier coefficients (FNLARMA). These models are applied to weekly data on interest rates in India and their forecasting performance is evaluated vis-à-vis three GARCH models [GARCH (1,1), GARCH-M (1,1) and EGARCH (1,1)] as well as the random walk model. Both the WANN and MS methods show marked improvement over other benchmark models, and may thus hold out several potentials for real world modeling and forecasting of financial data.     

股指期货信息内含股价变动信息的挖掘 ――小波框架与支持向量回归的金融建模应用

 内容提要：中国股指期货的推出指日可待，交易者多了一种投资工具的同时也带来了新的风险。建立准确的金融时间序列预测模型是逐利及避险的方法之一，一直是学者专家研究的热点。本研究结合小波转换与支持向量回归，提出一个二阶段时间序列预测模型。先以离散小波框架将预测变量分解成不同尺度的多个子序列，揭示隐藏在预测变量内的信息，再以支持向量回归为工具，以这些子序列为预测变量建构SVR模型。本研究以日经225指数开盘价为预测目标，以期货开盘价为预测变量对模型进行实证研究，结果显示，该模型的预测绩效比单纯SVR模型及随机漫步模型好。未来可尝试以不同的基底函数作进一步研究。     

Markov switching component GARCH model: Stability and forecasting

ABSTRACT

This paper introduces an extension of the Markov switching GARCH model where the volatility in each state is a convex combination of two different GARCH components with time varying weights. This model has the dynamic behavior to capture the variants of shocks. The asymptotic behavior of the second moment is investigated and an appropriate upper bound for it is evaluated. Using the Bayesian method via Gibbs sampling algorithm, a dynamic method for the estimation of the parameters is proposed. Finally, we illustrate the efficiency of the model by simulation and also by considering two different set of empirical financial data. We show that this model provides much better forecasts of the volatility than the Markov switching GARCH model.     

A Comparison of the Runs Test for Volatility Forecastability and the LM Test for GARCH Using Aggregated Returns

Christoffersen and Diebold (2000 Christoffersen , P. F. , Diebold , F. X. ( 2000 ). How relevant is volatility forecasting for financial risk management? Review of Economics and Statistics 82 : 12 – 22 .[Crossref] , [Google Scholar]) have introduced a runs test for forecastable volatility in aggregated returns. In this note, we compare the size and power of their runs test and the more conventional LM test for GARCH by Monte Carlo simulation. When the true daily process is GARCH, EGARCH, or stochastic volatility, the LM test has better power than the runs test for the moderate-horizon returns considered by Christoffersen and Diebold. For long-horizon returns, however, the tests have very similar power. We also consider a qualitative threshold GARCH model. For this process, we find that the runs test has greater power than the LM test. Theresults support the use of the runs test with aggregated returns.     

GARCH-type forecasting models for volatility of stock market and MCS test

The prediction of time-changing volatility is an important task in the modeling of financial data. In the paper, a comprehensive analysis of the mean return and conditional variance of SSE380 index is performed to use GARCH, EGARCH and TGARCH models with Normal innovation and Student's t innovation. Conducting a bootstrap simulation study which shows the Model Confidence Set (MCS) captures the superior models across a range of significance levels. The experimental results show that, under various loss functions, the GARCH using Student's t innovation model is the best model for volatility predictions of SSE380 among the six models.     

Estimating high-frequency foreign exchange rate volatility with nonparametric ARCH models

High-frequency foreign exchange rate (HFFX) series are analyzed on an operational time scale using models of the ARCH class. Comparison of the estimated conditional variances focuses on the asymmetry and persistence issue. Estimation results for parametric models confirm standard results for HFFX series, namely high persistence and no significance of the asymmetry coefficient in an EGARCH model. To find out whether these results are robust against alternative specifications, nonparametric models are estimated. Local linear estimation techniques are applied to a nonparametric ARCH model of order one (CHARN). The results show significant asymmetry of the volatility function. To allow for both flexibility and persistence, a higher-order multiplicative model is fitted. The results show important asymmetries in volatility. In contrast to the EGARCH specification, the news impact curves have different shapes for different lags and tend to increase slower at the boundaries.     

基于AR—EGARCH的空气温度预测模型

天气衍生产品定价极其复杂,其中与温度相关的产品是目前研究热点,其定价核心在于温度变量的精确预测。传统AR-GARCH温度预测模型难以描述温度变量波动率的非对称性。基于此,构建了AR—EGARCH温度预测模型,它能够描述波动率的非对称性,更好地反映温度变化过程。对中国东部南北线六个较发达的城市进行实证研究,结果表明：六个城市的温度变化具有明显的非对称性,AR—EGARCH模型无论是拟合还是预测效果都较传统的AR-GARCH模型更优。     

中国股指期货与股票市场波动性关系的实证分析

使用修正的EGARCH模型与VaR方法检验股指期货的推出对中国股票市场波动性所产生的影响。采用的数据为沪深300指数,样本数据分为股指期货推出前,股指期货推出后的短期、中期和长期与样本数据全体五个时间段。研究表明,从股指期货推出的短期与中期来看,市场对信息的反应比较混乱。从长期来看,股指期货的推出加速了信息的传递速度并且弱化了非对称效应,并没有加大股市的波动性。VaR方法检验表明,股指期货的推出有效降低了股市风险,使A股市场更加成熟和完善。     

Empirical Performance and Asset Pricing in Hidden Markov Models

Abstract

To improve the empirical performance of the Black-Scholes model, many alternative models have been proposed to address leptokurtic feature, volatility smile, and volatility clustering effects of the asset return distributions. However, analytical tractability remains a problem for most alternative models. In this article, we study a class of hidden Markov models including Markov switching models and stochastic volatility models, that can incorporate leptokurtic feature, volatility clustering effects, as well as provide analytical solutions to option pricing. We show that these models can generate long memory phenomena when the transition probabilities depend on the time scale. We also provide an explicit analytic formula for the arbitrage-free price of the European options under these models. The issues of statistical estimation and errors in option pricing are also discussed in the Markov switching models.