基于拟似然方法的股票收益与波动率关系及其应用研究

股票市场中收益与波动率的关系研究在金融证券领域起着很重要的作用,而随机波动率模型能够很好地拟合这种关系。本文将拟似然方法和渐近拟似然方法运用在随机波动率模型的参数估计方面,渐近拟似然方法可以避免因为人为的结构错误指定而造成的偏差,比较稳健。本文采用拟似然和渐近拟似然方法对随机波动率模型的参数估计进行了模拟探索,并和两种已有估计方法进行了对比,结果表明拟似然和渐近拟似然方法在模型的参数估计方面有着很好的估计结果。实证研究中,选取2000-2015年标普500指数作为研究对象,结果显示所选数据具有金融时间序列的常见特征。本文为金融证券领域中股票收益与波动率关系及其应用研究提供了一定的启示。     

Non-stationary quasi-likelihood and asymptotic optimality

This article is concerned with non-stationary time series which does not require the full knowledge of the likelihood function. Consequently, a quasi-likelihood is employed for estimating parameters instead of the maximum (exact) likelihood. For stationary cases, Wefelmeyer (1996) and Hwang and Basawa (2011a,b), among others, discussed the issue of asymptotic optimality of the quasi-likelihood within a restricted class of estimators. For non-stationary cases, however, the asymptotic optimality property of the quasi-likelihood has not yet been adequately addressed in the literature. This article presents the asymptotic optimal property of the non-stationary quasi-likelihood within certain estimating functions. We use a random norm instead of a constant norm to get limit distributions of estimates. To illustrate main results, the non-stationary ARCH model, branching Markov process, and non-stationary random-coefficient AR process are discussed.     

Criteria for estimating the variance function used in the asymptotic quasi-likelihood approach

The estimation of the variance function of a linear regression model used in the asymptotic quasi-likelihood approach is considered. It is shown that the variance function used in the determination of the asymptotic quasi-likelihood estimates encompasses the variance functions commonly found in the literature. Selection criteria of the most appropriate estimate of the variance function for given data are established. These criteria are based on a graphical technique and a chi-squared test.     

Asymptotic properties of maximum quasi-likelihood estimator in quasi-likelihood non linear models with stochastic regression

In this paper, we establish the asymptotic properties of maximum quasi-likelihood estimator (MQLE) in quasi-likelihood non linear models (QLNMs) with stochastic regression under some mild regular conditions. We also investigate the existence, strong consistency, and asymptotic normality of MQLE in QLNMs with stochastic regression.     

ON THE EFFICIENCY OF THE QUASI-LIKELIHOOD ESTIMATORS FOR EXPONENTIAL FAMILIES WITH EXTRA VARIATION

Bounds are obtained for the asymptotic efficiency of the quasi-likelihood estimator relative to the likelihood estimators for exponential families with random variables in the exponent. We conclude that the quasi-likelihood is very efficient in the Poisson case for small means or small extra variation.     

One-step local quasi-likelihood estimation

Local quasi-likelihood estimation is a useful extension of local least squares methods, but its computational cost and algorithmic convergence problems make the procedure less appealing, particularly when it is iteratively used in methods such as the back-fitting algorithm, cross-validation and bootstrapping. A one-step local quasi-likelihood estimator is introduced to overcome the computational drawbacks of the local quasi-likelihood method. We demonstrate that as long as the initial estimators are reasonably good, the one-step estimator has the same asymptotic behaviour as the local quasi-likelihood method. Our simulation shows that the one-step estimator performs at least as well as the local quasi-likelihood method for a wide range of choices of bandwidths. A data-driven bandwidth selector is proposed for the one-step estimator based on the pre-asymptotic substitution method of Fan and Gijbels. It is then demonstrated that the data-driven one-step local quasi-likelihood estimator performs as well as the maximum local quasi-likelihood estimator by using the ideal optimal bandwidth.     

Asymptotics for penalised splines in generalised additive models

This paper discusses asymptotic theory for penalised spline estimators in generalised additive models. The purpose of this paper is to establish the asymptotic bias and variance as well as the asymptotic normality of the ridge-corrected penalised spline estimator. Furthermore, the asymptotics for the penalised quasi-likelihood fit in mixed models are also discussed.     

Analysis of experiments using the asymptotic quasi-likelihood approach

Comparison of treatment effects in an experiment is usually done through analysis of variance under the assumption that the errors are normally and independently distributed with zero mean and constant variance. The traditional approach in dealing with non-constant variance is to apply a variance stabilizing transformation and then run the analysis on the transformed data. In this approach, the conclusions of analysis of variance apply only to the transformed population. In this paper, the asymptotic quasi-likelihood method is introduced to the analysis of experimental designs. The weak assumptions of the asymptotic quasi-likelihood method make it possible to draw conclusions on heterogeneous populations without transforming them. This paper demonstrates how to apply the asymptotic quasi-likelihood technique to three commonly used models. This gives a possible way to analyse data given a complex experimental design.     

On Bias Reduction in Robust Inference for Generalized Linear Models

Abstract.  It is well known that one or more outlying points in the data may adversely affect the consistency of the quasi-likelihood or the likelihood estimators for the regression effects. Similar to the quasi-likelihood approach, the existing outliers-resistant Mallow's type quasi-likelihood (MQL) estimation approach may also produce biased regression estimators. As a remedy, by using a fully standardized score function in the MQL estimating equation, in this paper, we demonstrate that the fully standardized MQL estimators are almost unbiased ensuring its higher consistency performance. Both count and binary responses subject to one or more outliers are used in the study. The small sample as well as asymptotic results for the competitive estimators are discussed.     

A comparison of efficiencies between quasi-likelihood and pseudo-likelihood estimates in non-separable spatial–temporal binary data

The goal of this paper is to compare the performance of two estimation approaches, the quasi-likelihood estimating equation and the pseudo-likelihood equation, against model mis-specification for non-separable binary data. This comparison, to the authors’ knowledge, has not been done yet. In this paper, we first extend the quasi-likelihood work on spatial data to non-separable binary data. Some asymptotic properties of the quasi-likelihood estimate are also briefly discussed. We then use the techniques of a truncated Gaussian random field with a quasi-likelihood type model and a Gibbs sampler with a conditional model in the Markov random field to generate spatial–temporal binary data, respectively. For each simulated data set, both of the estimation methods are used to estimate parameters. Some discussion about the simulation results are also included.     

Asymptotic properties of approximate maximum quasi-likelihood estimator in quasi-likelihood nonlinear models with random effects

Quasi-likelihood nonlinear models with random effects (QLNMWRE) include generalized linear models with random effects and quasi-likelihood nonlinear models as special cases. In this paper, some regularity conditions analogous to those given by Breslow and Clatyton (1993) are proposed. On the basis of the proposed regularity conditions and Laplace approximation, the existence, the strong consistency and asymptotic normality of the approximate maximum quasi-likelihood estimation of the fixed effects are proved in QLNMWRE.     

Nonparametric quasi-likelihood for right censored data

Quasi-likelihood was extended to right censored data to handle heteroscedasticity in the frame of the accelerated failure time (AFT) model. However, the assumption of known variance function in the quasi-likelihood for right censored data is usually unrealistic. In this paper, we propose a nonparametric quasi-likelihood by replacing the specified variance function with a nonparametric variance function estimator. This nonparametric variance function estimator is obtained by smoothing a function of squared residuals via local polynomial regression. The rate of convergence of the nonparametric variance function estimator and the asymptotic limiting distributions of the regression coefficient estimators are derived. It is demonstrated in simulations that for finite samples the proposed nonparametric quasi-likelihood method performs well. The new method is illustrated with one real dataset.     

A New Kind of Asymptotic Quasi-Score Estimating Function

A new definition of asymptotic quasi-score sequence of estimating functions is given and studied. The relationship between asymptotic quasi-likelihood and quasi-likelihood estimates is investigated. A new practical approach for obtaining a good estimate of θ in the model y _t = f_t (θ) + m_t without any prior knowledge on the nature of E ( m ² _t |F _{t −1}) is suggested, where f_t is a predictable process and m_t is a martingale difference process. Two examples are used to show that the approach is practicable.     

Asymptotic properties of maximum quasi-likelihood estimator in quasi-likelihood nonlinear models with misspecified variance function

The quasi-likelihood function proposed by Wedderburn [Quasi-likelihood functions, generalized linear models, and the Gauss–Newton method. Biometrika. 1974;61:439–447] broadened the application scope of generalized linear models (GLM) by specifying the mean and variance function instead of the entire distribution. However, in many situations, complete specification of variance function in the quasi-likelihood approach may not be realistic. Following Fahrmeir's [Maximum likelihood estimation in misspecified generalized linear models. Statistics. 1990;21:487–502] treating with misspecified GLM, we define a quasi-likelihood nonlinear models (QLNM) with misspecified variance function by replacing the unknown variance function with a known function. In this paper, we propose some mild regularity conditions, under which the existence and the asymptotic normality of the maximum quasi-likelihood estimator (MQLE) are obtained in QLNM with misspecified variance function. We suggest computing MQLE of unknown parameter in QLNM with misspecified variance function by the Gauss–Newton iteration procedure and show it to work well in a simulation study.     

First-order random coefficient integer-valued autoregressive processes

A first-order random coefficient integer-valued autoregressive (RCINAR(1)) model is introduced. Ergodicity of the process is established. Moments and autocovariance functions are obtained. Conditional least squares and quasi-likelihood estimators of the model parameters are derived and their asymptotic properties are established. The performance of these estimators is compared with the maximum likelihood estimator via simulation.     

Asymptotic Properties of the Maximum Quasi-Likelihood Estimator in Quasi-Likelihood Nonlinear Models

Quasi-likelihood nonlinear models (QLNM) are a further extension of generalized linear models by only specifying the expectation and variance functions of the response variable. In this article, some mild regularity conditions are proposed. These regularity conditions, respectively, assure the existence, strong consistency, and the asymptotic normality of the maximum quasi-likelihood estimator (MQLE) in QLNM.     

On confidence intervals associated with quasi-likelihood

Starting from an unbiased estimating function and the corresponding quasi-likelihood, we consider a large class of test statistics. An explicit and readily applicable higher order asymptotic formula is derived for comparing the statistics in the class with respect to the expected lengths of the confidence intervals given by their inversion.     

Estimation in the generalized linear empirical bayes model using the extended quasi-likelihood

A generalized linear empirical Bayes model is developed for empirical Bayes analysis of several means in natural exponential families. A unified approach is presented for all natural exponential families with quadratic variance functions (the Normal, Poisson, Binomial, Gamma, and two others.) The hyperparameters are estimated using the extended quasi-likelihood of Nelder and Pregibon (1987), which is easily implemented via the GLIM package. The accuracy of these estimates is developed by asymptotic approximation of the variance. Two data examples are illustrated.     

Rate of Strong Consistency of Maximum Quasi-Likelihood Estimator in Multivariate Generalized Linear Models

In this article, we consider a multivariate generalized linear model with adaptive design matrix and general link function. The asymptotic singularity of design matrix of adaptive design presents challenge when establishing the large sample properties of the maximum quasi-likelihood estimate (MQLE). Under some mild conditions, we obtain a strong convergence rate of the MQLE.     

Local polynomial quasi-likelihood regression with truncated and dependent data

The local polynomial quasi-likelihood estimation has several good statistical properties such as high minimax efficiency and adaptation of edge effects. In this paper, we construct a local quasi-likelihood regression estimator for a left truncated model, and establish the asymptotic normality of the proposed estimator when the observations form a stationary and α-mixing sequence, such that the corresponding result of Fan et al. [Local polynomial kernel regression for generalized linear models and quasilikelihood functions, J. Amer. Statist. Assoc. 90 (1995), pp. 141–150] is extended from the independent and complete data to the dependent and truncated one. Finite sample behaviour of the estimator is investigated via simulations too.