Time-varying higher-order conditional moments and forecasting intraday VaR and Expected Shortfall

We estimate several GARCH- and Extreme Value Theory (EVT)-based models to forecast intraday Value-at-Risk (VaR) and Expected Shortfall (ES) for S&P 500 stock index futures returns for both long and short positions. Among the GARCH-based models we consider is the so-called Autoregressive Conditional Density (ARCD) model, which allows time-variation in higher-order conditional moments. ARCD model with time-varying conditional skewness parameter has the best in-sample fit among the GARCH-based models. The EVT-based model and the GARCH-based models which take conditional skewness and kurtosis (time-varying or otherwise) into account provide accurate VaR forecasts. ARCD model with time-varying conditional skewness parameter seems to provide the most accurate ES forecasts.     

Autoregresive conditional volatility, skewness and kurtosis

This paper proposes a GARCH-type model allowing for time-varying volatility, skewness and kurtosis. The model is estimated assuming a Gram–Charlier (GC) series expansion of the normal density function for the error term, which is easier to estimate than the non-central t distribution proposed by [Harvey, C. R. & Siddique, A. (1999). Autorregresive Conditional Skewness. Journal of Financial and Quantitative Analysis 34, 465–487). Moreover, this approach accounts for time-varying skewness and kurtosis while the approach by Harvey and Siddique [Harvey, C. R. & Siddique, A. (1999). Autorregresive Conditional Skewness. Journal of Financial and Quantitative Analysis 34, 465–487] only accounts for non-normal skewness. We apply this method to daily returns of a variety of stock indices and exchange rates. Our results indicate a significant presence of conditional skewness and kurtosis. It is also found that specifications allowing for time-varying skewness and kurtosis outperform specifications with constant third and fourth moments.     

CONDITIONALLY HETEROSKEDASTIC FACTOR MODELS WITH SKEWNESS AND LEVERAGE EFFECTS

Conditional heteroskedasticity, skewness and leverage effects are well‐known features of financial returns. The literature on factor models has often made assumptions that preclude the three effects to occur simultaneously. In this paper I propose a conditionally heteroskedastic factor model that takes into account the presence of both the conditional skewness and leverage effects. This model is specified in terms of conditional moment restrictions and unconditional moment conditions are proposed allowing inference by the generalized method of moments (GMM). The model is also shown to be closed under temporal aggregation. An application to daily excess returns on sectorial indices from the UK stock market provides strong evidence for dynamic conditional skewness and leverage with a sharp efficiency gain resulting from accounting for both effects. The estimated volatilitypersistence from the proposed model is lower than that estimated from models that rule out such effects. I also find that the longer the returns' horizon, the fewer conditionally heteroskedastic factors may be required for suitable modeling and the less strong is the evidence for dynamic leverage. Some of these results are in line with the main findings of Harvey and Siddique ( 1999 ) and Jondeau and Rockinger ( 2003 ), namely that accounting for conditional skewness impacts the persistence in the conditional variance of the return process. Copyright © 2012 John Wiley & Sons, Ltd.     

金融时间序列高阶矩的时变性与波动持续性研究

波动持续性与非对称性是二阶矩方差的两个典型特征,类似于方差风险。高阶矩风险也具有自己的特征。重点讨论了三阶矩偏度与四阶矩峰度的时变性特性与波动持续性特征,指出不仅方差具有波动持续性,而且高阶矩序列同样存在持续性,并给出了高阶矩存在持续性的定义以及波动的持续性定理的相关证明。     

Forecasting in GARCH models with polynomially modified innovations

Orthogonal polynomials can be used to modify the moments of the distribution of a random variable. In this paper, polynomially adjusted distributions are employed to model the skewness and kurtosis of the conditional distributions of GARCH models. To flexibly capture the skewness and kurtosis of data, the distributions of the innovations that are polynomially reshaped include, besides the Gaussian, also leptokurtic laws such as the logistic and the hyperbolic secant. Modeling GARCH innovations with polynomially adjusted distributions can effectively improve the precision of the forecasts. This strategy is analyzed in GARCH models with different specifications for the conditional variance, such as the APARCH, the EGARCH, the Realized GARCH, and APARCH with time-varying skewness and kurtosis. An empirical application on different types of asset returns shows the good performance of these models in providing accurate forecasts according to several criteria based on density forecasting, downside risk, and volatility prediction.     

Choosing instrumental variables in conditional moment restriction models

Properties of GMM estimators are sensitive to the choice of instrument. Using many instruments leads to high asymptotic asymptotic efficiency but can cause high bias and/or variance in small samples. In this paper we develop and implement asymptotic mean square error (MSE) based criteria for instrument selection in estimation of conditional moment restriction models. The models we consider include various nonlinear simultaneous equations models with unknown heteroskedasticity. We develop moment selection criteria for the familiar two-step optimal GMM estimator (GMM), a bias corrected version, and generalized empirical likelihood estimators (GEL), that include the continuous updating estimator (CUE) as a special case. We also find that the CUE has lower higher-order variance than the bias-corrected GMM estimator, and that the higher-order efficiency of other GEL estimators depends on conditional kurtosis of the moments.     

高阶矩波动性建模及应用

为度量高阶矩风险的动态特征、考察时变高阶矩风险对金融投资决策的影响，本文提出了一个新的高阶矩波动模型：NAGARCHSK-M模型。讨论了该模型的包容性，给出了关于高阶矩波动性建模的一整套建模技术，基于正态密度的Gram-Charlier展开给出了模型的参数估计方法。利用该模型对我国股市的高阶矩风险进行了动态描述，并讨论了时变方差风险、时变偏度风险和时变峰度风险对资产收益的影响。     

An Extensive Comparison of Some Well‐Established Value at Risk Methods

In the last two decades, several methods for estimating Value at Risk have been proposed in the literature. Four of the most successful approaches are conditional autoregressive Value at Risk, extreme value theory, filtered historical simulation and time‐varying higher order conditional moments. In this paper, we compare their performances under both an empirical investigation using 80 assets and a large Monte Carlo simulation. From our analysis, we conclude that most of the methods seem not to imply huge numerical difficulties and, according to usual backtests and performance measurements, extreme value theory presents the best results most of the times, followed by filtered historical simulation.     

Maximum entropy autoregressive conditional heteroskedasticity model

In many applications, it has been found that the autoregressive conditional heteroskedasticity (ARCH) model under the conditional normal or Student’s t distributions are not general enough to account for the excess kurtosis in the data. Moreover, asymmetry in the financial data is rarely modeled in a systematic way. In this paper, we suggest a general density function based on the maximum entropy (ME) approach that takes account of asymmetry, excess kurtosis and also of high peakedness. The ME principle is based on the efficient use of available information, and as is well known, many of the standard family of distributions can be derived from the ME approach. We demonstrate how we can extract information functional from the data in the form of moment functions. We also propose a test procedure for selecting appropriate moment functions. Our procedure is illustrated with an application to the NYSE stock returns. The empirical results reveal that the ME approach with a fewer moment functions leads to a model that captures the stylized facts quite effectively.     

Forecasting cryptocurrency volatility

This paper studies the behavior of cryptocurrencies’ financial time series, of which Bitcoin is the most prominent example. The dynamics of these series are quite complex, displaying extreme observations, asymmetries, and several nonlinear characteristics that are difficult to model and forecast. We develop a new dynamic model that is able to account for long memory and asymmetries in the volatility process, as well as for the presence of time-varying skewness and kurtosis. The empirical application, carried out on 606 cryptocurrencies, indicates that a robust filter for the volatility of cryptocurrencies is strongly required. Forecasting results show that the inclusion of time-varying skewness systematically improves volatility, density, and quantile predictions at different horizons.     

高阶矩风险与金融投资决策

针对传统投资组合理论没有考虑高阶矩风险这一缺陷,总结近期金融领域中有关偏度和峰度的研究成果,基于"均值-方差"效用函数的Taylor展开,讨论了投资者对高阶矩风险(偏度风险和峰度风险)的偏好特征。     

A comparison of financial duration models via density forecasts

Using density forecast evaluation techniques, we compare the predictive performance of econometric specifications that have been developed for modeling duration processes in intra-day financial markets. The model portfolio encompasses various variants of the Autoregressive Conditional Duration (ACD) model and recently proposed dynamic factor models. The evaluation is conducted on time series of trade, price and volume durations computed from transaction data of NYSE listed stocks. The results show that simpler approaches perform at least as well as more complex methods. With respect to modeling trade duration processes, standard ACD models successfully account for duration dynamics while none of the models provides an acceptable specification for the conditional duration distribution. We find that the Logarithmic ACD, if based on a flexible innovation distribution, provides a quite robust and useful framework for the modeling of price and volume duration processes.     

Stability of characterizations of normal distributions based on the conditional expected values of the sample skewness and the sample kurtosis

Characterizations of normal distributions given by Nguyen and Dinh (1998) based on conditional expected values of the sample skewness and the sample kurtosis, given the sample mean and the sample variance, are shown to be stable. Received: September 1998     

Temporal aggregation of volatility models

In this paper, we consider temporal aggregation of volatility models. We introduce semiparametric volatility models, termed square-root stochastic autoregressive volatility (SR-SARV), which are characterized by autoregressive dynamics of the stochastic variance. Our class encompasses the usual GARCH models and various asymmetric GARCH models. Moreover, our stochastic volatility models are characterized by multiperiod conditional moment restrictions in terms of observables. The SR-SARV class is a natural extension of the class of weak GARCH models. This extension has four advantages: (i) we do not assume that fourth moments are finite; (ii) we allow for asymmetries (skewness, leverage effect) that are excluded from weak GARCH models; (iii) we derive conditional moment restrictions and (iv) our framework allows us to study temporal aggregation of IGARCH models.     

L‐moments skewness and kurtosis as measures of regional convergence and cohesion

The paper deals with the statistical modeling of convergence and cohesion over time with the use of kurtosis, skewness and L‐moments. Changes in the shape of the distribution related to the spatial allocation of socio‐economic phenomena are considered as an evidence of global shift, divergence or convergence. Cross‐sectional time‐series statistical modeling of variables of interest is to overpass the minors of econometric theoretical models of convergence and cohesion determinants. L‐moments perform much more stable and interpretable than classical measures. Empirical evidence of panel data proves that one pure pattern (global shift, polarization or cohesion) rarely exists and joint analysis is required.     

Fixed Effects and Random Effects Estimation of Higher-order Spatial Autoregressive Models with Spatial Autoregressive and Heteroscedastic Disturbances

Abstract

This paper develops a unified framework for fixed effects (FE) and random effects (RE) estimation of higher-order spatial autoregressive panel data models with spatial autoregressive disturbances and heteroscedasticity of unknown form in the idiosyncratic error component. We derive the moment conditions and optimal weighting matrix without distributional assumptions for a generalized moments (GM) estimation procedure of the spatial autoregressive parameters of the disturbance process and define both an RE and an FE spatial generalized two-stage least squares estimator for the regression parameters of the model. We prove consistency of the proposed estimators and derive their joint asymptotic distribution, which is robust to heteroscedasticity of unknown form in the idiosyncratic error component. Finally, we derive a robust Hausman test of the spatial random against the spatial FE model.     

A new marked point process model for the federal funds rate target: Methodology and forecast evaluation

Forecasts of key interest rates set by central banks are of paramount concern for investors and policy makers. Recently it has been shown that forecasts of the federal funds rate target, the most anticipated indicator of the Federal Reserve Bank's monetary policy stance, can be improved considerably when its evolution is modeled as a marked point process (MPP). This is due to the fact that target changes occur in discrete time with discrete increments, have an autoregressive nature and are usually in the same direction. We propose a model which is able to account for these dynamic features of the data. In particular, we combine Hamilton and Jordà's [2002. A model for the federal funds rate target. Journal of Political Economy 110(5), 1135–1167] autoregressive conditional hazard (ACH) and Russell and Engle's [2005. A discrete-state continuous-time model of financial transactions prices and times: the autoregressive conditional multinomial-autoregressive conditional duration model. Journal of Business and Economic Statistics 23(2), 166 – 180] autoregressive conditional multinomial (ACM) model. The paper also puts forth a methodology to evaluate probability function forecasts of MPP models. By improving goodness of fit and point forecasts of the target, the ACH–ACM qualifies as a sensible modeling framework. Furthermore, our results show that MPP models deliver useful probability function forecasts at short and medium term horizons.     

A multivariate GARCH in mean approach to testing uncovered interest parity: evidence from Asia-Pacific foreign exchange markets

The existence of time-varying risk premia in deviations from uncovered interest parity (UIP) is investigated based on a conditional capital asset pricing model (CAPM) using data from four Asia-Pacific foreign exchange markets. A parsimonious multivariate generalized autoregressive conditional heteroskedasticity in mean (GARCH-M) parameterization is employed to model the conditional covariance matrix of excess returns. The empirical results indicate that when each currency is estimated separately with an univariate GARCH-M parameterization, no evidence of time-varying risk premia is found except Malaysian ringgit. However, when all currencies are estimated simultaneously with the multivariate GARCH-M parameterization, strong evidence of time-varying risk premia is detected. As a result, the evidence supports the idea that deviations from UIP are due to a risk premium and not to irrationality among market participants. In addition, the empirical evidence found in this study points out that simply modeling the conditional second moments is not sufficient enough to explain the dynamics of the risk premia. A time-varying price of risk is still needed in addition to the conditional volatility. Finally, significant asymmetric world market volatility shocks are found in Asia-Pacific foreign exchange markets.     

Shock-dependent conditional skewness in international aggregate stock markets

This article uses the S_U-normal distribution to model the dynamic behavior of skewness in ten international aggregate stock indices—five indices each from developed and emerging markets. The conditional skewness process is specified as both autoregressive and dependent on lagged return shocks. Our primary result is that a negative return shock skews the time-varying distribution to the right for mature markets but to the left for emerging markets. In addition, we find that the asymmetry in volatility is noticeably larger in developed markets than in emerging markets. Finally, including the skewness process in modeling has no effect on the asymmetry and persistence in volatility obtained. These results are different from those of previous studies, which demonstrate the existence of both effects.     

Extreme risk transmission channels between the stock index futures and spot markets: Evidence from China

We develop a skewness-dependent multivariate conditional autoregressive value at risk model (SDMV-CAViaR) to detect the extreme risk transmission channels between the Chinese stock index futures and spot markets. The proposed SDMV-CAViaR model improves the forecast performance of extreme risk by introducing the high-frequency realized skewness. Specifically, the realized skewness has a significant impact on the spillovers, but the realized volatility and realized kurtosis do not, which implies that the jump component plays an important role in extreme risk spillovers. The empirical results indicate there are bidirectional extreme risk spillovers between the stock index futures and spot markets, the decline of one market has direct and indirect channels to exacerbate the extreme risk of the other market. Firstly, the market decline will directly increase the extreme risk of related markets by decreasing market returns. Besides, the decline will indirectly increase the extreme risk by increasing the negative realized skewness and extreme risk spillovers.