Large deviations of realized volatility

This paper studies large and moderate deviation properties of a realized volatility statistic of high frequency financial data. We establish a large deviation principle for the realized volatility when the number of high frequency observations in a fixed time interval increases to infinity. Our large deviation result can be used to evaluate tail probabilities of the realized volatility. We also derive a moderate deviation rate function for a standardized realized volatility statistic. The moderate deviation result is useful for assessing the validity of normal approximations based on the central limit theorem. In particular, it clarifies that there exists a trade-off between the accuracy of the normal approximations and the path regularity of an underlying volatility process. Our large and moderate deviation results complement the existing asymptotic theory on high frequency data. In addition, the paper contributes to the literature of large deviation theory in that the theory is extended to a high frequency data environment.     

Goodness of fit test for ergodic diffusion processes

A goodness of fit test for the drift coefficient of an ergodic diffusion process is presented. The test is based on the score marked empirical process. The weak convergence of the proposed test statistic is studied under the null hypothesis and it is proved that the limit process is a continuous Gaussian process. The structure of its covariance function allows to calculate the limit distribution and it turns out that it is a function of a standard Brownian motion and so exact rejection regions can be constructed. The proposed test is asymptotically distribution free and it is consistent under any simple fixed alternative.     

股票价格序列中随机跳跃存在性的一种检验方法

提出股票价格序列跳跃的一种检验方法.假设价格具有连续样本路径,建立一个关于股票价格样本观察的统计量,利用中心极限定理求得该统计量的极限分布为正态分布,这样,当该统计量超出基于极限分布算出的临界水平时,可以拒绝原假设,认为样本中存在跳跃.用此方法来应用于中国股市沪深股票指数,得到了中国股市存在随机跳跃的直接证据.提出的跳跃检验方法无需对连续部分的波动率形式作过多的假设,克服了波动率模型对检验准确性的影响.结果对金融资产的定价、投资和风险管理都具有积极的意义.     

非参数固定效应Panel Data模型的统计推断

本文主要讨论了非参数固定效应Panel data模型的估计与检验问题,首先我们利用Profile最小二乘方法得到了固定效应与非参数部分的估计;接着基于比较原假设与备择假设下模型拟合的残差平方和的思想针对固定效应的检验问题构造了检验统计量,并给出了计算检验p-值的F分布逼近法。     

Khmaladze transformation of integrated variance processes with applications to goodness-of-fit testing

In the common nonparametric regression model the problem of testing for a specific parametric form of the variance function is considered. Recently Dette and Hetzler [8] proposed a test statistic which is based on an empirical process of pseudo residuals. The process converges weakly to a Gaussian process with a complicated covariance kernel depending on the data generating process. In the present paper we consider a standardized version of this process and propose an application of the Khmaladze transformation to obtain asymptotically distribution-free tests for the corresponding Kolmogorov-Smirnov and Cramér-von Mises functionals. The finite-sample properties of the proposed tests are investigated by means of a simulation study.      

On the goodness-of-fit testing for a switching diffusion process

We study the asymptotic behavior of the Cramér–von Mises type statistic in the goodness-of-fit hypotheses testing problem for ergodic diffusion processes. The basic (simple) hypothesis is defined by the stochastic differential equation with sign-type trend coefficient and known diffusion coefficient. It is shown that the limit distribution of the proposed test statistic (under hypothesis) is defined by the integral type functional of continuous Gaussian process. We provide the Karhunen–Loève expansion of the corresponding limiting process and show that the eigenfunctions in this expansion are expressed in terms of Bessel functions. This representation for the limit statistic allows us to approximate the threshold.     

Option pricing and hedging in incomplete market driven by Normal Tempered Stable process with stochastic volatility

This paper develops a distribution class, termed Normal Tempered Stable, by subordinating a drifted Brownian motion through a strictly increasing Tempered Stable process that generalizes the Variance Gamma and the Normal Inverse Gaussian and is used to model the logarithm asset returns. The newly added parameter is to create subclasses for all the distributions discovered in financial market. The empirical test suggests that time series of Technology stock returns in US market reject both the Variance Gamma distribution and the Normal Inverse Gaussian distribution and admit instead another subclass of the Normal Tempered Stable distribution. Furthermore, we introduce stochastic volatilities into the Normal Tempered Stable process and derive explicit formulae for option pricing and hedging by means of the characteristic function based methods. To answer the question of how well different models work in practice, we investigate four models adopting data on daily equity option prices and obtain several findings from the numerical results. To sum up, the Normal Tempered Stable process with stochastic volatility is able to adequately capture implied volatility dynamics and seen as a superior model relative to the jump-diffusion stochastic volatility model, based on the construction methodology that incorporates more sophisticated and flexible jump structure and the systematic and realistic treatment of volatility dynamics. The Normal Tempered Stable model turns out to have the competitive performance in an efficient manner given that it only requires three parameters.     

A multivariate empirical characteristic function test of independence with normal marginals

This paper proposes a semi-parametric test of independence (or serial independence) between marginal vectors each of which is normally distributed but without assuming the joint normality of these marginal vectors. The test statistic is a Cramér–von Mises functional of a process defined from the empirical characteristic function. This process is defined similarly as the process of Ghoudi et al. [J. Multivariate Anal. 79 (2001) 191] built from the empirical distribution function and used to test for independence between univariate marginal variables. The test statistic can be represented as a V-statistic. It is consistent to detect any form of dependence. The weak convergence of the process is derived. The asymptotic distribution of the Cramér–von Mises functionals is approximated by the Cornish–Fisher expansion using a recursive formula for cumulants and inversion of the characteristic function with numerical evaluation of the eigenvalues. The test statistic is finally compared with Wilks statistic for testing the parametric hypothesis of independence in the one-way MANOVA model with random effects.     

Empirical likelihood based goodness-of-fit testing for generalized linear mixed models

In this paper, we propose a bias-corrected empirical likelihood （BCEL） ratio to construct a goodness- of-fit test for generalized linear mixed models. BCEL test maintains the advantage of empirical likelihood that is self scale invariant and then does not involve estimating limiting variance of the test statistic to avoid deteri- orating power of test. Furthermore, the bias correction makes the limit to be a process in which every variable is standard chi-squared. This simple structure of the process enables us to construct a Monte Carlo test proce- dure to approximate the null distribution. Thus, it overcomes a problem we encounter when classical empirical likelihood test is used, as it is asymptotically a functional of Gaussian process plus a normal shift function. The complicated covariance function makes it difficult to employ any approximation for the null distribution. The test is omnibus and power study shows that the test can detect local alternatives approaching the null at parametric rate. Simulations are carried out for illustration and for a comparison with existing method.     

EM algorithm and its application to testing hypotheses

The conventional method for testing hypotheses is to find an exact or asymptotic distributionof a test statistic But when the model is complex and the sample size is small,difficulty often arises. Thispaper aims to present a method for finding maximum probability with the help of EM algorithm. For any fixedsample size,this method can be used not only to obtain an accurate test but also to check the real level of     

随机波动率模型下的单位根检验及对我国股市的实证研究

判断股市收益率波动的持续性问题通常归结为检验波动过程中是否存在着单位根。针对传统的单位根检验方法功效偏低,且检验的真实显著性水平容易失真的问题,本文采用随机波动率模型并利用改进的PP检验、KPSS检验,以及SIMEX检验方法对中国上证综合指数进行了实证研究.三种检验方法的结果都表明,股权分置改革后的上证综指收益率波动过程有单位根,波动呈现出很强的持续性.     

Bootstrap testing for discontinuities under long-range dependence

We consider testing for discontinuities in a trend function when the residual process exhibits long memory. Using a wavelet decomposition of the estimated trend function into a low-resolution and a high-resolution component, a test statistic is proposed based on blockwise resampling of estimated residual variances. Asymptotic validity of the test is derived. A simulation study illustrates finite sample properties.     

Gaussian stochastic volatility models: Scaling regimes,large deviations,and moment explosions

In this paper, we establish sample path large and moderate deviation principles for log-price processes in Gaussian stochastic volatility models, and study the asymptotic behavior of exit probabilities, call pricing functions, and the implied volatility. In addition, we prove that if the volatility function in an uncorrelated Gaussian model grows faster than linearly, then, for the asset price process, all the moments of order greater than one are infinite. Similar moment explosion results are obtained for correlated models.     

Asymptotic properties of some goodness-of-fit tests based on the L 1-norm

Some goodness-of-fit tests based on the L ₁-norm are considered. The asymptotic distribution of each statistic under the null hypothesis is the distribution of the L ₁-norm of the standard Wiener process on [0,1]. The distribution function, the density function and a table of some percentage points of the distribution are given. A result for the asymptotic tail probability of the L ₁-norm of a Gaussian process is also obtained. The result is useful for giving the approximate Bahadur efficiency of the test statistics whose asymptotic distributions are represented as the L ₁-norms of Gaussian processes.     

基于修正的已实现阈值幂变差的股市跳跃波动行为研究

基于非参数统计方法,利用考虑金融资产价格跳跃和杠杆效应的时点波动估计方法修正已实现阈值幂变差,构造甄别跳跃的检验统计量,对金融资产价格中的随机波动、有限活跃跳跃和无限活跃跳跃等问题进行综合研究。为同时吸收波动率的异方差集聚效应和收益率的非对称效应,对原有的已实现波动率异质自回归预测模型进行拓展,将非对称的异质性自回归模型的误差项设定为GARCH模型,以考察跳跃波动序列与连续波动序列之间的复杂关系。利用沪深股指高频数据进行实证研究,包括进行跳跃识别,跳跃活动程度检验和波动率预测效果对比。研究结果表明,沪深股市同时存在布朗运动成分、有限活跃跳跃和无限活跃跳跃成分,其中连续路径方差占主体。同时,收益和波动间的杠杆效应显著,无论短期还是长期,连续波动和跳跃波动对波动率的预测均具有显著影响,同时考虑股价的跳跃、波动和杠杆效应因素有助于更准确地刻画资产价格动态过程。     

Statistical inference for right-censored data with nonignorable missing censoring indicators

We consider the statistical inference for right-censored data when censoring indicators are missing but nonignorable, and propose an adjusted imputation product-limit estimator. The proposed estimator is shown to be consistent and converges to a Gaussian process. Furthermore, we develop an empirical processbased testing method to check the MAR (missing at random) mechanism, and establish asymptotic properties for the proposed test statistic. To determine the critical value of the test, a consistent model-based bootstrap method is suggested. We conduct simulation studies to evaluate the numerical performance of the proposed method and compare it with existing methods. We also analyze a real data set from a breast cancer study for an illustration.     

A Quantile Goodness-of-Fit Test for Cauchy Distribution, Based on Extreme Order Statistics

A test statistic for testing goodness-of-fit of the Cauchy distribution is presented. It is a quadratic form of the first and of the last order statistic and its matrix is the inverse of the asymptotic covariance matrix of the quantile difference statistic. The distribution of the presented test statistic does not depend on the parameter of the sampled Cauchy distribution. The paper contains critical constants for this test statistic, obtained from 50,000 simulations for each sample size considered. Simulations show that the presented test statistic is for testing goodness-of-fit of the Cauchy distributions more powerful than the Anderson-Darling, Kolmogorov-Smirnov or the von Mises test statistic.     

面板数据交互固定效应模型的方差分量检验

研究了面板数据交互固定效应模型中方差分量的检验问题.首先依据模型中误差项的估计构造辅助回归模型,然后根据该辅助回归构造检验统计量,对模型中的异方差性进行检验.进一步,通过构造不同的辅助回归模型和检验统计量可以判别异方差的来源.在一定正则条件下,得到了检验统计量在原假设和备择假设下的渐近分布,并说明所提出的检验方法不依赖于误差分布.最后,通过模拟研究对本文的检验方法进行评价,说明所提检验方法是有效的.     

Testing for a constant coefficient of variation in nonparametric regression by empirical processes

In the common nonparametric regression model, we consider the problem of testing the hypothesis that the coefficient of the scale and location function is constant. The test is based on a comparison of the standardized (by a local linear estimate of the scale function) observations with their mean. We show weak convergence of a centered version of this process to a Gaussian process under the null hypothesis and the alternative and use this result to construct a test for the hypothesis of a constant coefficient of variation in the nonparametric regression model. A small simulation study is also presented to investigate the finite sample properties of the new test.     

A note on the asymptotic distribution of the maximum likelihood estimator in a non-regular case

We consider the asymptotic distribution of the maximum likelihood estimator (MLE), when the log-likelihood ratio statistic weakly converges to the non-degenerated Gaussian process. We provide a simple expression for the density function of the asymptotic distribution by fundamental stochastic results. This note is helpful to investigate asymptotic properties of the MLE in a certain non-regular case.