Analytic moments for GJR-GARCH (1, 1) processes

For a GJR-GARCH(1, 1) specification with a generic innovation distribution we derive analytic expressions for the first four conditional moments of the forward and aggregated returns and variances. Moments for the most commonly used GARCH models are stated as special cases. We also derive the limits of these moments as the time horizon increases, establishing regularity conditions for the moments of aggregated returns to converge to normal moments. A simulation study using these analytic moments produces approximate predictive distributions which are free from the bias affecting simulations. An empirical study using almost 30 years of daily equity index, exchange rate and interest rate data applies Johnson SU and Edgeworth expansion distribution fitting to our closed-form formulae for higher moments of returns.     

Volatility and error transmission spillover effects: Evidence from three European financial regions

This study uses the multivariate GARCH-BEKK modelling approach to examine the transmission of news (both volatility and error) between portfolios of cross-listed equities within three European financial regions, that is, the Scandinavian (Denmark, Sweden, Finland and Norway), the Germanic (Austria, Switzerland and Germany) and the French area (Brussels, France, Italy, Holland and Spain). We find that the Finnish and Danish portfolios of cross-listed equities are the main transmitters of volatility relative to the Swedish and Norwegian portfolios of cross-listed equities. On the other hand, the Swiss portfolio of cross-listed equities is the major exporter of volatility and error to the other portfolios of cross-listed equities in the Germanic stock market area. Finally, the Paris, Amsterdam and Brussels stock exchanges are the major exporters of volatility and error to the portfolios of cross-listed equities traded on the Milan and Madrid stock exchanges.     

BAYESIAN INFERENCE METHODS FOR UNIVARIATE AND MULTIVARIATE GARCH MODELS: A SURVEY

This survey reviews the existing literature on the most relevant Bayesian inference methods for univariate and multivariate GARCH models. The advantages and drawbacks of each procedure are outlined as well as the advantages of the Bayesian approach versus classical procedures. The paper makes emphasis on recent Bayesian non‐parametric approaches for GARCH models that avoid imposing arbitrary parametric distributional assumptions. These novel approaches implicitly assume infinite mixture of Gaussian distributions on the standardized returns which have been shown to be more flexible and describe better the uncertainty about future volatilities. Finally, the survey presents an illustration using real data to show the flexibility and usefulness of the non‐parametric approach.     

On measuring skewness and kurtosis

The paper considers some properties of measures of asymmetry and peakedness of one dimensional distributions. It points to some misconceptions of the first and the second Pearson coefficients, the measures of asymetry and shape, that frequently occur in introductory textbooks. Also it presents different ways for obtaining the estimated values for the coefficients of skewness and kurtosis and statistical tests which include them.     

Optimal prediction under LINLIN loss: Empirical evidence

I compare the forecasts of returns from the mean predictor (optimal under MSE), with the pseudo-optimal and optimal predictor for an asymmetric loss function under the assumption that agents have an asymmetric LINLIN loss function. The results strongly suggest not using the conditional mean predictor under conditions of asymmetry. In general, forecasts can be improved by the use of optimal predictor rather than the pseudo-optimal predictor, suggesting that the loss reduction from using the optimal predictor can actually be important for practitioners as well.     

股市收益波动的非对称性研究

随着我国股票市场的逐步发展及证券市场监管体制的逐步完善,股票市场收益波动所呈现的非对称性效应是否会发生逆转引人关注。本文运用不同的GARCH类条件异方差模型(EGARCH、TARCH、PARCH以及非对称CARCH模型),选取1998年1月5日至2006年11月6日上证综指的日收盘价格指数数据,分析和验证了股票收益波动的非对称性效应。结果表明,上海股市股票收益波动仍然具有非对称性,即利空消息对收益波动的冲击效应大于同等程度的利好消息,从而验证了非对称效应的不可逆转性。     

Fourth order pseudo maximum likelihood methods

We extend PML theory to account for information on the conditional moments up to order four, but without assuming a parametric model, to avoid a risk of misspecification of the conditional distribution. The key statistical tool is the quartic exponential family, which allows us to generalize the PML2 and QGPML1 methods proposed in Gourieroux et al. (1984) to PML4 and QGPML2 methods, respectively. An asymptotic theory is developed. The key numerical tool that we use is the Gauss-Freud integration scheme that solves a computational problem that has previously been raised in several fields. Simulation exercises demonstrate the feasibility and robustness of the methods.     

波动率预测:GARCH模型与隐含波动率

在预测未来波动率时,究竟是基于历史数据的时间序列模型还是基于期权价格的隐含波动率模型效率更高?本文对香港恒生指数期权市场所含信息的研究发现,在预测期限较短(一周)时,GARCH(1,1)模型所含信息较多,预测能力最强,但在预测较长期限(一个月)时,隐含波动率所含信息较多,预测能力较强。同时,期权市场交易越活跃,所反映的信息就越全面,隐含波动率的预测能力也就越强。     

The large-sample joint distribution of key circular statistics

In this paper we derive the large-sample asymptotic joint distribution of the statistics used as fundamental measures of central location, concentration, skewness and kurtosis in the analysis of circular data. The importance of the distributional result in relation to inference for the corresponding population measures is illustrated, with various new confidence set constructions being derived and applied in the analysis of data from an animal orientation experiment.     

Using neural network to forecast stock index option price: a new hybrid GARCH approach

This study aims to apply a new hybrid approach to estimate volatility in neural network option-pricing model. The analytical results also indicate that the new hybrid method can be used to forecast the prices of derivative securities. Owing to combines the grey forecasting model with the GARCH to improve the estimated ability, the empirical evidence shows that the new hybrid GARCH model outperforms the other approaches in the neural network option-pricing model.     

MGARCH models: Trade-off between feasibility and flexibility

Multivariate GARCH (MGARCH) models need to be restricted so that their estimation is feasible in large systems and so that the covariance stationarity and positive definiteness of conditional covariance matrices are guaranteed. This paper analyzes the limitations of some of the popular restricted parametric MGARCH models that are often used to represent the dynamics observed in real systems of financial returns. These limitations are illustrated using simulated data generated by general VECH models of different dimensions in which volatilities and correlations are interrelated. We show that the restrictions imposed by the BEKK model are very unrealistic, generating potentially misleading forecasts of conditional correlations. On the other hand, models based on the DCC specification provide appropriate forecasts. Alternative estimators of the parameters are important in order to simplify the computations, and do not have implications for the estimates of conditional correlations. The implications of the restrictions imposed by the different specifications of MGARCH models considered are illustrated by forecasting the volatilities and correlations of a five-dimensional system of exchange rate returns.     

A SCIENTIFIC CLASSIFICATION OF VOLATILITY MODELS

Abstract.  Modeling volatility, or predictable changes over time and space in a variable, is crucial in the natural and social sciences. Life can be volatile, and anything that matters, and which changes over time and space, involves volatility. Without volatility, many temporal and spatial variables would simply be constants. Our purpose is to propose a scientific classification of the alternative volatility models and approaches that are available in the literature, following the Linnaean taxonomy. This scientific classification is used because the literature has evolved as a living organism, with the birth of numerous new species of models.     

GARCH族模型的预测能力比较:一种半参数方法

半参数GARCH模型无须设定条件分布的具体形式。本文首先将一种效率较高、易于实施的半参数方法——估计函数方法应用于10类常见的GARCH结构,并给出证据,显示该方法能显著提高GARCH族模型的波动率预测绩效。然后,应用估计函数方法,较为全面地比较各类GARCH结构的预测能力。为给出统计意义下的结果,并减少数据窥察问题,研究中分别使用OLS和SPA检验法进行绩效评价。结果发现,与其他GARCH类结构相比,EGARCH和APARCH模型能够较好地描述股市收益率的波动过程。     

交易量适合作为股价波动信息的代理变量吗?——来自中国沪深股市的证据

本文利用中国沪深股市日交易数据,采用多元GARCH模型从信息传递的角度进行实证研究,结果表明:股价对交易量具有显著的波动溢出效应,但交易量对股价的波动溢出效应不明显。这种波动的单向溢出说明在应对信息的冲击上股价比交易量能更快地做出反应,其后才通过波动溢出在交易量上得到反映,股价波动对成交量波动具有先导作用。因此,从波动冲击传导和信息传递的角度看,单纯地将交易量视为股价变动信息的代理变量还缺乏稳健的统计证据。