Normal mixture quasi maximum likelihood estimation for non-stationary TGARCH(1,1) models

Although quasi maximum likelihood estimator based on Gaussian density (G-QMLE) is widely used to estimate GARCH-type models, it does not perform successfully when error distribution is either skewed or leptokurtic. This paper proposes normal mixture quasi-maximum likelihood estimator (NM-QMLE) for non-stationary TGARCH(1,1) models. We show that, under mild regular conditions, there is no consistent estimator for the intercept, and the proposed estimator for any other parameter is consistent.     

Inference for 2-D GARCH models

The purpose of this paper is, in the first step, to consider a class of GMM estimators with interesting asymptotic properties and a reasonable number of computations for two dimensionally indexed Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model. In the second step, we use the central limit theorem of Huang (1992) for spatial martingale differences to establish the LAN property for general two-dimensional discrete models on a regular grid with Gaussian errors. We then apply this result to the spatial GARCH model and derive the limit distribution of the maximum likelihood estimators of the parameters. Results of numerical simulations are presented.     

Bayesian tail‐risk forecasting using realized GARCH

A realized generalized autoregressive conditional heteroskedastic (GARCH) model is developed within a Bayesian framework for the purpose of forecasting value at risk and conditional value at risk. Student‐t and skewed‐t return distributions are combined with Gaussian and student‐t distributions in the measurement equation to forecast tail risk in eight international equity index markets over a 4‐year period. Three realized measures are considered within this framework. A Bayesian estimator is developed that compares favourably, in simulations, with maximum likelihood, both in estimation and forecasting. The realized GARCH models show a marked improvement compared with ordinary GARCH for both value‐at‐risk and conditional value‐at‐risk forecasting. This improvement is consistent across a variety of data and choice of distributions. Realized GARCH models incorporating a skewed student‐t distribution for returns are favoured overall, with the choice of measurement equation error distribution and realized measure being of lesser importance. Copyright © 2017 John Wiley & Sons, Ltd.     

Convergence results for maximum likelihood type estimators in multivariable ARMA models II

The consistency proof for the (Gaussian quasi) maximum likelihood estimator in multivariable ARMA models as given in Dunsmuir and Hannan (1976, Adv, in Appl. Probab. 8, 339–364) rests on a certain property of the underlying parameter space, called B6 in their paper. It is not known whether the usual parameter spaces like the manifold M(n) or the parameter spaces corresponding to echelon forms satisfy condition B6, since the argument given by Dunsmuir and Hannan to establish this fact is inconclusive. In Pötscher (1987, J. Multivariate Anal. 21 29–52) it was shown how consistency can be proved without relying on B6 if the data generating process is Gaussian. In this note we show that the Gaussianity assumption can be replaced by ergodicity thus restoring Dunsmuir and Hannan's consistency proof to its full generality and extending it to parameter spaces which do not satisfy condition B6.     

On the comparison of the pre-test and shrinkage phi-divergence test estimators for the symmetry model of categorical data

The estimation problem of the parameters in a symmetry model for categorical data has been considered for many authors in the statistical literature (for example, Bowker (1948) [1], Ireland et al. (1969) [2], Quade and Salama (1975) [3], Cressie and Read (1988) [4], Menéndez et al. (2005) [5]) without using uncertain prior information. It is well known that many new and interesting estimators, using uncertain prior information, have been studied by a host of researchers in different statistical models, and many papers have been published on this topic (see Saleh (2006) [9] and references therein). In this paper, we consider the symmetry model of categorical data and we study, for the first time, some new estimators when non-sample information about the symmetry of the probabilities is considered. The decision to use a “restricted” estimator or an “unrestricted” estimator is based on the outcome of a preliminary test, and then a shrinkage technique is used. It is interesting to note that we present a unified study in the sense that we consider not only the maximum likelihood estimator and likelihood ratio test or chi-square test statistic but we consider minimum phi-divergence estimators and phi-divergence test statistics. Families of minimum phi-divergence estimators and phi-divergence test statistics are wide classes of estimators and test statistics that contain as a particular case the maximum likelihood estimator, likelihood ratio test and chi-square test statistic. In an asymptotic set-up, the biases and the risk under the squared loss function for the proposed estimators are derived and compared. A numerical example clarifies the content of the paper.     

正态条件下带AR(1)-型方差结构GMANOVA-MANOVA模型极大似然估计的小样本特征

研究了一类带一阶自回归(AR(1))-型方差结构的广义多元方差分析-多元方差分析(GMANO VA-MANOVA)模型参数极大似然估计的小样本特征.对带AR(1)-型方差结构GMANOVA-MANOVA模型,文章在正态条件下给出了参数极大似然估计存在的一个充分必要条件,讨论了极大似然估计唯一的充分条件.在该充分条件下,文章证明了相关系数极大似然估计的精确分布只与相关系数有关,并依此给出了自相关系数简单假设H0:ρ=0v.s.H1:ρ≠0的一个不需要叠代计算估计的检验,同时模拟表明该检验为无偏检验且势函数与似然比检验势函数无太大差异.     

Quasi-maximum likelihood estimation in GARCH processes when some coefficients are equal to zero

The asymptotic distribution of the quasi-maximum likelihood (QML) estimator is established for generalized autoregressive conditional heteroskedastic (GARCH) processes, when the true parameter may have zero coefficients. This asymptotic distribution is the projection of a normal vector distribution onto a convex cone. The results are derived under mild conditions. For an important subclass of models, no moment condition is imposed on the GARCH process. The main practical implication of these results concerns the estimation of overidentified GARCH models.     

On asymptotic theory for multivariate GARCH models

The paper investigates the asymptotic theory for a multivariate GARCH model in its general vector specification proposed by Bollerslev, Engle and Wooldridge (1988) [4], known as the VEC model. This model includes as important special cases the so-called BEKK model and many versions of factor GARCH models, which are often used in practice. We provide sufficient conditions for strict stationarity and geometric ergodicity. The strong consistency of the quasi-maximum likelihood estimator (QMLE) is proved under mild regularity conditions which allow the process to be integrated. In order to obtain asymptotic normality, the existence of sixth-order moments of the process is assumed.     

Some closed form robust moment‐based estimators for the MEM(1,1)

In this paper, we extend the closed form moment estimator (ordinary MCFE) for the autoregressive conditional duration model given by Lu et al (2016) and propose some closed form robust moment‐based estimators for the multiplicative error model to deal with the additive and innovational outliers. The robustification of the closed form estimator is done by replacing the sample mean and sample autocorrelation with some robust estimators. These estimators are more robust than the quasi‐maximum likelihood estimator (QMLE) often used to estimate this model, and they are easy to implement and do not require the use of any numerical optimization procedure and the choice of initial value. The performance of our proposal in estimating the parameters and forecasting conditional mean μ_t of the MEM(1,1) process is compared with the proposals existing in the literature via Monte Carlo experiments, and the results of these experiments show that our proposal outperforms the ordinary MCFE, QMLE, and least absolute deviation estimator in the presence of outliers in general. Finally, we fit the price durations of IBM stock with the robust closed form estimators and the benchmarks and analyze their performances in estimating model parameters and forecasting the irregularly spaced intraday Value at Risk.     

Semiparametric Maximum Likelihood for Missing Covariates in Parametric Regression

We consider parameter estimation in parametric regression models with covariates missing at random. This problem admits a semiparametric maximum likelihood approach which requires no parametric specification of the selection mechanism or the covariate distribution. The semiparametric maximum likelihood estimator (MLE) has been found to be consistent. We show here, for some specific models, that the semiparametric MLE converges weakly to a zero-mean Gaussian process in a suitable space. The regression parameter estimate, in particular, achieves the semiparametric information bound, which can be consistently estimated by perturbing the profile log-likelihood. Furthermore, the profile likelihood ratio statistic is asymptotically chi-squared. The techniques used here extend to other models.