A local spectral approach for assessing time series model misspecification

We consider band-limited frequency-domain goodness-of-fit testing for stationary time series, without smoothing or tapering the periodogram, while taking into account the effects of parameter uncertainty (from maximum-likelihood estimation). We are principally interested in modeling short econometric time series, typically with 100 to 150 observations, for which data-driven bandwidth selection procedures for kernel-smoothed spectral density estimates are unlikely to have adequate levels. Our mathematical results take parameter uncertainty directly into account, allowing us to obtain adequate level properties at small sample sizes. The main theorems provide very general results involving joint normality for linear functionals of powers of the periodogram, while accounting for parameter uncertainty, which can be used to determine the level and power of a wide array of statistics. We discuss several applications, such as spectral peak testing and testing for the inclusion of an Unobserved Component, and illustrate our methods on a time series from the Energy Information Administration.     

Gauss inequalities on ordered linear spaces

Markov inequalities on ordered linear spaces are tightened through the α-unimodality of the corresponding measures. Modality indices are studied for various induced measures, including the singular values of a random matrix and the periodogram of a time series. These tools support a detailed study of linear inference and the ordering of random matrices, to include fixed and random designs and probability bounds on their comparative efficiencies. Other applications include probability bounds on quadratic forms and of order statistics on Rⁿ, on periodograms in the analysis of time series, and on run-length distributions in multivariate statistical process control. Connections to other topics in applied probability and statistics are noted.     

On multiplicative seasonal modelling for vector time series

Many time series encountered in real applications display seasonal behavior. In this paper, we consider multiplicative seasonal vectorial autoregressive moving average (SVARMA) models to describe seasonal vector time series. We discuss conditional maximum likelihood estimation of the model parameters, allowing them to satisfy general linear constraints. Having fitted a model, residual autocovariances (or autocorrelations) have been found useful in checking time series models. Consequently, we obtain the asymptotic distributions of the residual autocovariance matrices. As applications of these results, Portmanteau test statistics are proposed and their asymptotic distributions are studied. The finite-sample properties of the test statistics are evaluated using Monte Carlo experiments.     

Rank order statistics for time series models

Some invariance principles are obtained for the one-sample rank order statistics of a -mixing or strong mixing type time series. The estimation of the center of symmetry of the time series and tests for serial dependence are considered as applications.     

基于Bootstrap的厚尾相依序列持久性变点检验

基于修正方差比率函数给出一种检验厚尾序列持久性变点的统计量.在无变点的假设下得到了统计量的渐近分布.为避免检验渐近分布中的厚尾指数,构造Bootstrap抽样方法来确定渐近分布的经验临界值.数值模拟研究结果说明修正方差比率统计量及Bootstrap抽样方法的有效性.     

Testing for cointegration using induced-order statistics

In this paper we explore the usefulness of induced-order statistics in the characterization of integrated series and of cointegration relationships. We propose a non-parametric test statistic for testing the null hypothesis of two independent random walks against wide cointegrating alternatives including monotonic nonlinearities and certain types of level shifts in the cointegration relationship. We call our testing device the induced-order Kolmogorov–Smirnov cointegration test (KS), since it is constructed from the induced-order statistics of the series, and we derive its limiting distribution. This non-parametric statistic endows the test with a number of desirable properties: invariance to monotonic transformations of the series, and robustness for the presence of important parameter shifts. By Monte Carlo simulations we analyze the small sample properties of this test. Our simulation results show the robustness of the induced order cointegration test against departures from linear and constant parameter models. This paper is an extension of the work of Aparicio and Granger (1995) and Aparicio and Escribano (1998).     

Diagnostic checking of multivariate nonlinear time series models with martingale difference errors

In this article, we derive the asymptotic distribution of residual autocovariance and autocorrelation matrices for a general class of multivariate nonlinear time series models by assuming only that the error term is a martingale difference sequence. Two types of applications are developed: global test statistics of the portmanteau type and one-lag test statistics, which describe the residual correlation at individual lags. To illustrate the proposed methodology, simulation results are reported for diagnosing multivariate threshold time series models. The following test statistics are compared: the classical test statistics presuming independent errors and the proposed methodology which supposes only martingale difference errors.     

An Exponential Inequality for U-Statistics Under Mixing Conditions

The family of U-statistics plays a fundamental role in statistics. This paper proves a novel exponential inequality for U-statistics under the time series setting. Explicit mixing conditions are given for guaranteeing fast convergence, the bound proves to be analogous to the one under independence, and extension to non-stationary time series is straightforward. The proof relies on a novel decomposition of U-statistics via exploiting the temporal correlatedness structure. Such results are of interest in many fields where high-dimensional time series data are present. In particular, applications to high-dimensional time series inference are discussed.     

Hypotheses Testing for Error-in-Variables Models

In this paper, hypotheses testing based on a corrected score function are considered. Five different testing statistics are proposed and their asymptotic distributions are investigated. It is shown that the statistics are asymptotically distributed according to the chisquare distribution or can be written as a linear combination of chisquare random variables with one degree of freedom. A small scale numerical Monte Carlo study is presented in order to compare the empirical size and power of the proposed tests. A comparative calibration example is used to illustrate the results obtained.     

On the calculation of cumulants of estimators arising from a linear time series regression model

Bounds for higher-order cumulants of statistics arising from a linear time series regression model are investigated. A result given in Brillinger is proved and extended. The bounds permit derivation of asymptotic moments and asymptotic normality for estimators of parameters in the model. Two examples are given as illustrations.     

Semi-parametric estimation of partially linear single-index models

One of the most difficult problems in applications of semi-parametric partially linear single-index models (PLSIM) is the choice of pilot estimators and complexity parameters which may result in radically different estimators. Pilot estimators are often assumed to be root-n consistent, although they are not given in a constructible way. Complexity parameters, such as a smoothing bandwidth are constrained to a certain speed, which is rarely determinable in practical situations.In this paper, efficient, constructible and practicable estimators of PLSIMs are designed with applications to time series. The proposed technique answers two questions from Carroll et al. [Generalized partially linear single-index models, J. Amer. Statist. Assoc. 92 (1997) 477-489]: no root-n pilot estimator for the single-index part of the model is needed and complexity parameters can be selected at the optimal smoothing rate. The asymptotic distribution is derived and the corresponding algorithm is easily implemented. Examples from real data sets (credit-scoring and environmental statistics) illustrate the technique and the proposed methodology of minimum average variance estimation (MAVE).     

带季节线性趋势的季节时间序列的单位根检验

本文为检验带季节线性趋势的季节时间序列的单位根，提出了调整季节线性趋势的统计量，导出这些统计量的极限分布，并用Monte Carlo方法比较研究这些统计量的检验势．     

Asymptotic behavior of central order statistics from stationary processes

In this paper, we show that central order statistics from strictly stationary and ergodic sequences are strongly consistent estimators of population quantiles provided that the quantiles are unique. We generalize this result to strictly stationary but not necessarily ergodic sequences. We also describe three types of possible asymptotic behavior of central order statistics in the case when the corresponding population quantile is not unique. We give applications of the presented results to linear processes with both absolutely continuous and discrete innovations.     

厚尾金融时间序列的贝叶斯单位根检验

在金融时间序列分析中,单位根检验是一个相当重要的研究问题。对于数据生成过程为自回归的厚尾金融时序数据,古典的ADF单位根检验统计量应用非常困难。本文则在贝叶斯框架下,发展了检验带有未知自由度厚尾t分布的自回归金融时间序列单位根的贝叶斯方法。蒙特卡罗模拟结果显示本文发展的方法能够取得好的检验功效。最后,用万科地产月度历史收益数据来演示了本文发展的方法。     

线性模型中自相关的经验似然比检验

研究线性回归模型中的自相关检验问题,用经验似然的方法构造检验统计量,得到了零假设下检验统计量的渐近分布,我们的检验方法不但可以检验一阶自相关,也可以检验高阶自相关,数值模拟表明检验方法具有良好的检验功效.     

Testing strategies for model specification

A crucial element in the development of econometric methodology during the past decade has been the concern with testing as opposed to estimating econometric models. In this paper we discuss—especially for the econometric analysis of time series—the main types of test procedures, and we also investigate the opportunities to uphold the Neyman-Pearson theory in the context of thorough model specification testing. In applied work it is quite usual to carry out several tests on the same set of sample data. We consider an extension of the Neyman-Pearson framework to the case of such repeated testing, and examine situations where the various hypotheses under test have a particular nesting structure. For the case where a sequence of superposed alternatives is tested by so-called marginal tests, we prove that the various test statistics are asymptotically independent under a common null hypothesis if the statistics are based on either the likelihood-ratio, or the Wald, or the Lagrange-multiplier approach. Testing a particular null hypothesis against a series of juxtaposed alternatives appears to lead to independent test statistics only in specific circumstances. It is shown how independence of test statistics enables the control over the overall Type I error probability, which is an essential element in the Neyman-Pearson theory. Using the notions of constructive hypotheses and auxiliary hypotheses, we can draw a clear distinction between specification tests and misspecification tests. Next an overview is given of approaches to and examples of specification and misspecification testing. With respect to the former, attention is paid to the problem determining the order of dynamics and discriminating between system dynamics and error dynamics. The misspecification testing is reviewed for specification error, nonconstancy of coefficients, heteroscedasticity, serial dependence, and nonnormality of disturbances. Also the problem of testing for several misspecifications jointly or sequentially is considered. Finally we discuss the options and associated difficulties in implementing the various tests in an overall testing strategy.     

Asymptotic theory of some tests for a possible change in the regression slope occurring at an unknown time point

Summary Based on least squares estimators and aligned rank order statistics, some testing procedures for a possible change in the regression slope occurring at an unknown time point are considered. The asymptotic theory of the proposed tests rests on certain invariance principles relating to least squares estimators and aligned rank order statistics, and these are developed here.Work supported by the National Heart, Lung and Blood Institute, Contract NIH-NHLBI-71-2243 from the (U.S.) National Institutes of Health     

Nonparametric tests for multiple regression under progressive censoring

For continuous observations from time-sequential studies, suitable Cramér-von Mises and Kolmogorov-Smirnov types of (nonparametric) statistics (based on linear rank statistics) for testing hypotheses on some multiple-regression models are proposed and studied. The asymptotic theory of these tests is provided for both the null and (local) alternative hypotheses situations and is based on the weak convergence of suitable rank order processes (on the D[0, 1] space) to certain functions of Brownian motions. Bahadur efficiency results are also presented. Empirical values of the percentile points of the null distributions of the proposed test statistics, obtained through simulation studies, are also provided.     

基于灰色理论的线性回归模型的参数理论及应用

线性回归分析是数理统计学的基本内容之一,但传统数理统计学中的线性回归分析,是建立在非模糊的随机数据上的线性回归估计和回归系数检验。而现实经济社会中大量存在含有模糊或灰色等不分明性的数据,面对这类不分明性数据,简单地使用传统的统计分析方法显然是不足取的。要想较为科学合理地分析与决策,需要利用灰色系统的相关理论,应用于随机系统信息,从而建立灰色线性回归估计、预测和灰色回归系数检验的基本理论方法,并把该方法应用于金融分析实例中,与经典线性回归分析方法进行对比,足见灰色线性回归方法能够提供比经典线性回归较多的有效信息,从而提出处理不分明数据新的方法。     

股票收益率的次指数分布拟合

股票收益率等金融时间序列具有重尾特征,因而不适于用正态分布来描述,次指数分布族S是一类重尾分布族,能够很好的处理具有偏态、重尾特征的金融时间序列,本文对上证指数的收益率进行了次指数分布拟合,并给出了在险价值(VaR)的估计。