Parametric Inference and Dynamic State Recovery From Option Panels

We develop a new parametric estimation procedure for option panels observed with error. We exploit asymptotic approximations assuming an ever increasing set of option prices in the moneyness (cross‐sectional) dimension, but with a fixed time span. We develop consistent estimators for the parameters and the dynamic realization of the state vector governing the option price dynamics. The estimators converge stably to a mixed‐Gaussian law and we develop feasible estimators for the limiting variance. We also provide semiparametric tests for the option price dynamics based on the distance between the spot volatility extracted from the options and one constructed nonparametrically from high‐frequency data on the underlying asset. Furthermore, we develop new tests for the day‐by‐day model fit over specific regions of the volatility surface and for the stability of the risk‐neutral dynamics over time. A comprehensive Monte Carlo study indicates that the inference procedures work well in empirically realistic settings. In an empirical application to S&P 500 index options, guided by the new diagnostic tests, we extend existing asset pricing models by allowing for a flexible dynamic relation between volatility and priced jump tail risk. Importantly, we document that the priced jump tail risk typically responds in a more pronounced and persistent manner than volatility to large negative market shocks.     

考虑微观结构噪声与测量误差的波动率预测

以测量误差的分布理论为基础，本文将微观结构噪声的影响引入到测量误差的方差中，构建了包含微观结构噪声影响的HARQ-N模型。使用蒙特卡洛模拟与中国股市的高频数据对HAR、HARQ、HARQ-N模型与HAR-RV-N-CJ模型的估计和预测进行了比较，研究发现，HARQ模型和HARQ-N模型的测量误差修正项对波动率的影响系数统计显著为负，HARQ-N模型的测量误差项影响系数远大于HARQ模型，更大程度地减弱当期微观结构噪声和测量误差的影响。并且，考虑微观结构噪声和测量误差的HARQ-N模型样本内和样本外预测效果在统计上显著优于HAR模型、HARQ模型与HAR-RV-N-CJ模型。     

Correcting the Errors: Volatility Forecast Evaluation Using High‐Frequency Data and Realized Volatilities

We develop general model‐free adjustment procedures for the calculation of unbiased volatility loss functions based on practically feasible realized volatility benchmarks. The procedures, which exploit recent nonparametric asymptotic distributional results, are both easy‐to‐implement and highly accurate in empirically realistic situations. We also illustrate that properly accounting for the measurement errors in the volatility forecast evaluations reported in the existing literature can result in markedly higher estimates for the true degree of return volatility predictability.     

双因子非对称已实现SV模型及其实证研究

在基本的SV模型中引入包含丰富日内高频信息的已实现测度,同时考虑其偏差修正以及波动率非对称性与长记忆性,构建了双因子非对称已实现SV（2FARSV）模型.进一步基于连续粒子滤波算法,给出了2FARSV模型参数的极大似然估计方法.蒙特卡罗模拟实验表明,给出的估计方法是有效的.采用上证综合指数和深证成份指数日内高频数据计算已实现波动率（RV）和已实现极差波动率（RRV）,对2FARSV模型进行了实证研究.结果表明：RV和RRV都是真实日度波动率的有偏估计（下偏）,但RRV相比RV是更有效的波动率估计量;沪深股市具有强的波动率持续性以及显著的波动率非对称性（杠杆效应与规模效应）;2FARSV模型相比其它已实现波动率模型具有更好的数据拟合效果,该模型能够充分地捕获沪深股市波动率的动态特征（时变性、聚集性、非对称性与长记忆性）.     

金融波动的赋权“已实现”双幂次变差及其应用

金融波动是金融研究中的热点问题。金融高频数据比低频数据包含了更丰富的日内收益波动信息,因此对金融高频时间序列的研究成为金融领域中备受关注的焦点。"已实现"波动是利用高频数据计算金融波动率的全新方法,目前在金融高频数据的研究中应用十分广泛,但它具有误差较大和不稳健的缺点,因此各种改进方法应运而生,其中"已实现"双幂次变差克服了"已实现"波动的不稳健的缺点。本文提出赋权"已实现"双幂次变差的概念,不但继承了"已实现"双幂次变差的稳健性,而且满足无偏性和最小方差性,通过理论证明和实证研究都表明其能够更准确的度量金融波动率。     

中国ETF基金价格“已实现”波动率、跟踪误差之间的Granger关系研究

在高频数据条件下,中国ETF基金价格"已实现"波动率与跟踪误差之间是否存在着因果关系并存在着信息的先导效应？基于"已实现"波动、跟踪误差计算方法及Granger因果检验过程、VAR模型等,本文对此进行了深入研究。研究结果认为:中国ETF基金价格"已实现"波动率与两种跟踪误差分别具有Granger因果关系,后者是前者的Granger原因;中国ETF基金价格"已实现"波动率序列与两种跟踪误差序列的同期及一、二阶滞后相关性较高,而跟踪误差滞后于"已实现"波动率;当ETF基金的跟踪误差受外部市场条件的某一冲击后,将给ETF基金价格"已实现"波动率带来同向的冲击,这一冲击具有一定的持续性和滞后性。     

基于局部相关系数和截尾扭曲混合Copula的杠杆效应识别和度量

作为风险资产收益和波动之间关系的度量，杠杆效应是金融市场数据三大分布特征之一，在波动预测、资产定价和风险管理中起着重要作用。日内高频数据计算的已实现波动作为波动的代理变量，解决了波动不能观测的问题，实现了用波动和收益直接建模捕捉杠杆效应。深入了解收益和已实现波动的相关模式并以此构建二者的联合分布是正确度量杠杆效应的关键。本文以局部相关系数为工具研究收益和波动在不同取值范围内的相关性变化，实证研究结果表明，与负收益冲击引起波动增加一样，正收益冲击也会引起波动增加，这与传统杠杆效应理论并不一致，与Chen和Ghysels（2011）对美国股票市场的实证结果一致。为正确捕捉和度量实证结果反映出的杠杆效应，在扭曲混合Copula构造方法基础上，本文用截尾扭曲函数构造扭曲混合Copula，以此作为收益和已实现波动的联合分布，再现收益和已实现波动的局部相关性特征。以上证综指2013.1.29日至2017.4.30区间内日内1分钟高频数据为样本进行实证分析表明，本文构造的Copula函数具有和实际数据一致的局部相关特征，能够正确刻画市场表现出的杠杆效应。Copula拟合优度的非参数检验表明，实际数据不拒绝本文构造的Copula函数，而现有文献采用的单成分Copula函数和两成分混合Copula函数均被拒绝。本文为收益和已实现波动的联合建模提供参考，具有基础重要性。     

Asymptotic Efficiency in Parametric Structural Models with Parameter‐Dependent Support

In certain auction, search, and related models, the boundary of the support of the observed data depends on some of the parameters of interest. For such nonregular models, standard asymptotic distribution theory does not apply. Previous work has focused on characterizing the nonstandard limiting distributions of particular estimators in these models. In contrast, we study the problem of constructing efficient point estimators. We show that the maximum likelihood estimator is generally inefficient, but that the Bayes estimator is efficient according to the local asymptotic minmax criterion for conventional loss functions. We provide intuition for this result using Le Cam's limits of experiments framework.     

Designing Realized Kernels to Measure the ex post Variation of Equity Prices in the Presence of Noise

This paper shows how to use realized kernels to carry out efficient feasible inference on the ex post variation of underlying equity prices in the presence of simple models of market frictions. The weights can be chosen to achieve the best possible rate of convergence and to have an asymptotic variance which equals that of the maximum likelihood estimator in the parametric version of this problem. Realized kernels can also be selected to (i) be analyzed using endogenously spaced data such as that in data bases on transactions, (ii) allow for market frictions which are endogenous, and (iii) allow for temporally dependent noise. The finite sample performance of our estimators is studied using simulation, while empirical work illustrates their use in practice.     

Impossibility Results for Nondifferentiable Functionals

We examine challenges to estimation and inference when the objects of interest are nondifferentiable functionals of the underlying data distribution. This situation arises in a number of applications of bounds analysis and moment inequality models, and in recent work on estimating optimal dynamic treatment regimes. Drawing on earlier work relating differentiability to the existence of unbiased and regular estimators, we show that if the target object is not differentiable in the parameters of the data distribution, there exist no estimator sequences that are locally asymptotically unbiased or α‐quantile unbiased. This places strong limits on estimators, bias correction methods, and inference procedures, and provides motivation for considering other criteria for evaluating estimators and inference procedures, such as local asymptotic minimaxity and one‐sided quantile unbiasedness.     

基于四次幂差修正HAR模型的股指期货波动率预测

股指期货波动率建模与预测是揭示其波动运行规律和市场风险是重要途径。本文基于跳跃、好坏波动率与符号跳跃建立四组HAR模型,提出单级纠偏HARQ类模型和多级纠偏HARQF类模型,实证研究揭示股指期货波动运行规律,并采用MCS检验来评估模型优劣。HAR建模考察连续与跳跃波动、好与坏波动率的两种已实现波动分解。为了降低波动率估计偏差,基于最小化MSE准则确定最优抽样频率,利用已实现核修正的ADS检测法识别跳跃,采用已实现核估计修正好坏波动率与符号跳跃。基于沪深300股指期货的实证研究表明：连续波动比跳跃波动对未来已实现波动贡献更大;好坏波动率具有不对称波动冲击,而符号跳跃对未来波动具有负向冲击;好坏波动率分解优于连续与跳跃波动分解;中位数已实现四次幂差能够显著提升HAR类模型的样本内外预测能力;与样本内预测相反,样本外预测中单级纠偏HARQ类模型优于多级纠偏HARQF类模型;MCS检验得出HARQ-RV-SJ模型表现最佳。研究结论与启示对认识股指期货波动规律和市场风险具有意义。     

Assessment of Uncertainty in High Frequency Data: The Observed Asymptotic Variance

The availability of high frequency financial data has generated a series of estimators based on intra‐day data, improving the quality of large areas of financial econometrics. However, estimating the standard error of these estimators is often challenging. The root of the problem is that traditionally, standard errors rely on estimating a theoretically derived asymptotic variance, and often this asymptotic variance involves substantially more complex quantities than the original parameter to be estimated. Standard errors are important: they are used to assess the precision of estimators in the form of confidence intervals, to create “feasible statistics” for testing, to build forecasting models based on, say, daily estimates, and also to optimize the tuning parameters. The contribution of this paper is to provide an alternative and general solution to this problem, which we call Observed Asymptotic Variance. It is a general nonparametric method for assessing asymptotic variance (AVAR). It provides consistent estimators of AVAR for a broad class of integrated parameters Θ = ∫ θ_t dt, where the spot parameter process θ can be a general semimartingale, with continuous and jump components. The observed AVAR is implemented with the help of a two‐scales method. Its construction works well in the presence of microstructure noise, and when the observation times are irregular or asynchronous in the multivariate case. The methodology is valid for a wide variety of estimators, including the standard ones for variance and covariance, and also for more complex estimators, such as, of leverage effects, high frequency betas, and semivariance.     

已实现波动GAS-HEAVY模型及其实证研究

引入日内高频数据计算的已实现波动，能够提高波动模型预测能力。本文将日收益和已实现波动联合建模，提出一种新的波动模型。选取尺度调整t分布和F分布作为日收益和已实现波动的分布，更为充分和灵活地捕捉厚尾性，采用得分驱动方法设定波动模型更新项，得出广义自回归得分（GAS）波动模型，提高对实际模型的逼近效率。本文对模型遍历性和平稳性进行证明，并与同类模型进行比较。蒙特卡罗模拟实验显示，在数据生成过程误设的情况下本文提出的GAS-HEAVY模型比同类模型具有更好的数据拟合效果。基于沪综指、深成指和沪深300指数2013.1至2017.4日内1分钟高频数据实证分析表明，不同损失函数的SPA检验下GAS-HEAVY模型的波动预测能力显著强于其它同类模型。本文给出的GAS-HEAVY模型为有关理论研究和市场应用提供了新的波动计量工具。     

Bootstrapping Realized Volatility

We propose bootstrap methods for a general class of nonlinear transformations of realized volatility which includes the raw version of realized volatility and its logarithmic transformation as special cases. We consider the independent and identically distributed (i.i.d.) bootstrap and the wild bootstrap (WB), and prove their first‐order asymptotic validity under general assumptions on the log‐price process that allow for drift and leverage effects. We derive Edgeworth expansions in a simpler model that rules out these effects. The i.i.d. bootstrap provides a second‐order asymptotic refinement when volatility is constant, but not otherwise. The WB yields a second‐order asymptotic refinement under stochastic volatility provided we choose the external random variable used to construct the WB data appropriately. None of these methods provides third‐order asymptotic refinements. Both methods improve upon the first‐order asymptotic theory in finite samples.     

The Realized Laplace Transform of Volatility

We introduce and derive the asymptotic behavior of a new measure constructed from high‐frequency data which we call the realized Laplace transform of volatility. The statistic provides a nonparametric estimate for the empirical Laplace transform function of the latent stochastic volatility process over a given interval of time and is robust to the presence of jumps in the price process. With a long span of data, that is, under joint long‐span and infill asymptotics, the statistic can be used to construct a nonparametric estimate of the volatility Laplace transform as well as of the integrated joint Laplace transform of volatility over different points of time. We derive feasible functional limit theorems for our statistic both under fixed‐span and infill asymptotics as well as under joint long‐span and infill asymptotics which allow us to quantify the precision in estimation under both sampling schemes.     

Higher Order Properties of Gmm and Generalized Empirical Likelihood Estimators

In an effort to improve the small sample properties of generalized method of moments (GMM) estimators, a number of alternative estimators have been suggested. These include empirical likelihood (EL), continuous updating, and exponential tilting estimators. We show that these estimators share a common structure, being members of a class of generalized empirical likelihood (GEL) estimators. We use this structure to compare their higher order asymptotic properties. We find that GEL has no asymptotic bias due to correlation of the moment functions with their Jacobian, eliminating an important source of bias for GMM in models with endogeneity. We also find that EL has no asymptotic bias from estimating the optimal weight matrix, eliminating a further important source of bias for GMM in panel data models. We give bias corrected GMM and GEL estimators. We also show that bias corrected EL inherits the higher order property of maximum likelihood, that it is higher order asymptotically efficient relative to the other bias corrected estimators.     

Forecasting With Model Uncertainty: Representations and Risk Reduction

We consider forecasting with uncertainty about the choice of predictor variables. The researcher wants to select a model, estimate the parameters, and use the parameter estimates for forecasting. We investigate the distributional properties of a number of different schemes for model choice and parameter estimation, including: in‐sample model selection using the Akaike information criterion; out‐of‐sample model selection; and splitting the data into subsamples for model selection and parameter estimation. Using a weak‐predictor local asymptotic scheme, we provide a representation result that facilitates comparison of the distributional properties of the procedures and their associated forecast risks. This representation isolates the source of inefficiency in some of these procedures. We develop a simulation procedure that improves the accuracy of the out‐of‐sample and split‐sample methods uniformly over the local parameter space. We also examine how bootstrap aggregation (bagging) affects the local asymptotic risk of the estimators and their associated forecasts. Numerically, we find that for many values of the local parameter, the out‐of‐sample and split‐sample schemes perform poorly if implemented in the conventional way. But they perform well, if implemented in conjunction with our risk‐reduction method or bagging.     

Estimating Long Memory in Volatility

We consider semiparametric estimation of the memory parameter in a model that includes as special cases both long‐memory stochastic volatility and fractionally integrated exponential GARCH (FIEGARCH) models. Under our general model the logarithms of the squared returns can be decomposed into the sum of a long‐memory signal and a white noise. We consider periodogram‐based estimators using a local Whittle criterion function. We allow the optional inclusion of an additional term to account for possible correlation between the signal and noise processes, as would occur in the FIEGARCH model. We also allow for potential nonstationarity in volatility by allowing the signal process to have a memory parameter d^*1/2. We show that the local Whittle estimator is consistent for d^*∈(0,1). We also show that the local Whittle estimator is asymptotically normal for d^*∈(0,3/4) and essentially recovers the optimal semiparametric rate of convergence for this problem. In particular, if the spectral density of the short‐memory component of the signal is sufficiently smooth, a convergence rate of n^2/5−δ for d^*∈(0,3/4) can be attained, where n is the sample size and δ>0 is arbitrarily small. This represents a strong improvement over the performance of existing semiparametric estimators of persistence in volatility. We also prove that the standard Gaussian semiparametric estimator is asymptotically normal if d^*=0. This yields a test for long memory in volatility.     

Econometric Analysis of Realized Covariation: High Frequency Based Covariance,Regression, and Correlation in Financial Economics

This paper analyses multivariate high frequency financial data using realized covariation. We provide a new asymptotic distribution theory for standard methods such as regression, correlation analysis, and covariance. It will be based on a fixed interval of time (e.g., a day or week), allowing the number of high frequency returns during this period to go to infinity. Our analysis allows us to study how high frequency correlations, regressions, and covariances change through time. In particular we provide confidence intervals for each of these quantities.     

Robust Estimation and Inference for Jumps in Noisy High Frequency Data: A Local‐to‐Continuity Theory for the Pre‐Averaging Method

We develop an asymptotic theory for the pre‐averaging estimator when asset price jumps are weakly identified, here modeled as local to zero. The theory unifies the conventional asymptotic theory for continuous and discontinuous semimartingales as two polar cases with a continuum of local asymptotics, and explains the breakdown of the conventional procedures under weak identification. We propose simple bias‐corrected estimators for jump power variations, and construct robust confidence sets with valid asymptotic size in a uniform sense. The method is also robust to certain forms of microstructure noise.