On the Identifiability of Minimal VARMA Representations

By studying the geometry of relevant Hilbert spaces, we give a characterization of the identifiable standard representations of multivariate ARMA models in terms of the autocovariance function.     

A multiple window scan statistic for time series models

In this article we extend the results derived for scan statistics in Wang and Glaz (2014) for independent normal observations. We investigate the performance of two approximations for the distribution of fixed window scan statistics for time series models. An R algorithm for computing multivariate normal probabilities established in Genz and Bretz (2009) can be used along with proposed approximations to implement fixed window scan statistics for ARMA models. The accuracy of these approximations is investigated via simulation. Moreover, a multiple window scan statistic is defined for detecting a local change in the mean of a Gaussian white noise component in ARMA models, when the appropriate length of the scanning window is unknown. Based on the numerical results, for power comparisons of the scan statistics, we can conclude that when the window size of a local change is unknown, the multiple window scan statistic outperforms the fixed window scan statistics.     

Consistent and asymptotically normal estimators for cyclically time-dependent linear models

We consider a general class of time series linear models where parameters switch according to a known fixed calendar. These parameters are estimated by means of quasi-generalized least squares estimators. conditions for strong consistency and asymptotic normality are given. Applications to cyclical ARMA models with non constant periods are considered.     

误差服从多元t分布的一类线性模型下参数估计的若干注记

研究一类线性模型下参数估计的若干问题．这类模型包含了多个因变量线性模型、增长曲线模型、扩充的增长曲线模型、似乎不相关回归方程组、方差分量模型等常用模型．在这类线性模型下,证明了当误差服从多元t分布时与误差服从多元正态分布时,具有相同的完全统计量和无偏估计,且在后一种情况下的充分统计量必为前一种情况下的充分统计量．对于带有多种协方差结构的前述几种模型,把在误差服从多元正态分布下,相应的协方差阵及有关参数的一致最小风险无偏(UMRU)估计存在性的结论推广到了相应的误差服从多元t分布情形．此外,对于误差服从多元t分布的这类统一的线性模型,给出了回归系数的线性可估函数的无偏估计的协方差阵的C-R下界．     

Performance bound for myopic order-up-to inventory policies under stationary demand processes

We consider a single-item infinite-horizon inventory system operating in discrete time and study the performance of the popular myopic order-up-to policy when demand is driven by a general autoregressive moving average (ARMA) stationary process. We derive a suboptimality bound for a system that operates under full demand backlogging, linear holding and backordering costs, and a constant replenishment lead time. We illustrate our results for the case in which demand follows an ARMA(1,1) process, which includes two commonly used demand models, MA(1) and AR(1), as special cases.     

IDENTIFICATION OF MULTIVARIATE ARMA MODELS

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The functional central limit theorem for linear processes with strong near-epoch dependent innovations

This paper discusses linear processes with innovations exhibiting asymptotic weak dependence by being strong near-epoch dependent functions of mixing processes. The functional central limit theorem for the normalized partial sum process is established. The conditions given essentially improve on existing results in the literature in terms of the “size” requirement for the amount of dependence. It is also shown that two important econometric models, ARMA and GARCH models, are strong near-epoch dependent sequences.     

郑州期货糖08年9月合约价格的分析与预测

本文对郑州期货糖0809主力合约的价格首先进行多元线性回归,探求其与纽约期货糖价和郑州现货糖价的关系,进而在发现回归残差具有周期效应的基础上进行时间序列的频域分析,并同时考虑各种突发事件的影响,在模型中加入示性变量进行适当修正,经过ADF单位根检验确定此时的残差已为平稳序列之后建立ARMA模型,并接受最终残差为白噪声。将上述分解过程进行整合,估计模型系数并剔除其中的不显著变量便得到最终的拟合方程,在此基础上对后续三天的郑州期货糖价进行动态预测,结果显示真实价格均落在所给95%置信区间内。     

Multivariate Dispersion Models

We introduce a class of multivariate dispersion models suitable as error distributions for generalized linear models with multivariate non-normal responses. The models preserve some of the main properties of the multivariate normal distribution, and include the elliptically contoured distributions and certain other known distributions as special cases. We give explicit methods for constructing multivariate proper dispersion models. This is exemplified by constructing multivariate gamma, Laplace, hyperbola, and von Mises distributions.     

Using ARMA models to forecast workpiece roundness error in a turning operation

The paper describes the methodology for developing autoregressive moving average (ARMA) models to represent the workpiece roundness error in the machine taper turning process. The method employs a two stage approach in the determination of the AR and MA parameters of the ARMA model. It first calculates the parameters of the equivalent autoregressive model of the process, and then derives the AR and MA parameters of the ARMA model. Akaike's Information Criterion (AIC) is used to find the appropriate orders m and n of the AR and MA polynomials respectively. Recursive algorithms are developed for the on-line implementation on a laboratory turning machine. Evaluation of the effectiveness of using ARMA models in error forecasting is made using three time series obtained from the experimental machine. Analysis shows that ARMA(3,2) with forgetting factor of 0.95 gives acceptable results for this lathe turning machine.     

Normal Linear Regression Models With Recursive Graphical Markov Structure

A multivariate normal statistical model defined by the Markov properties determined by an acyclic digraph admits a recursive factorization of its likelihood function (LF) into the product of conditional LFs, each factor having the form of a classical multivariate linear regression model (≡WMANOVA model). Here these models are extended in a natural way to normal linear regression models whose LFs continue to admit such recursive factorizations, from which maximum likelihood estimators and likelihood ratio (LR) test statistics can be derived by classical linear methods. The central distribution of the LR test statistic for testing one such multivariate normal linear regression model against another is derived, and the relation of these regression models to block-recursive normal linear systems is established. It is shown how a collection of nonnested dependent normal linear regression models (≡Wseemingly unrelated regressions) can be combined into a single multivariate normal linear regression model by imposing a parsimonious set of graphical Markov (≡Wconditional independence) restrictions.     

The effect of noise reduction in measuring the linear and nonlinear dependency of financial markets

The daily closing prices of several stock market indices are examined to analyse whether noise reduction matters in measuring dependencies of the financial series. We consider the effect of noise reduction on the linear and nonlinear measure of dependencies. We also use singular spectrum analysis as a powerful method for filtering financial series. We compare the results with those obtained by ARMA and GARCH models as linear and nonlinear methods for filtering the series. We also examine the findings on an artificial data set namely the Hénon map.     

Robust Simulation-Based Estimation of ARMA Models

This article proposes a new approach to the robust estimation of a mixed autoregressive and moving average (ARMA) model. It is based on the indirect inference method that originally was proposed for models with an intractable likelihood function. The estimation algorithm proposed is based on an auxiliary autoregressive representation whose parameters are first estimated on the observed time series and then on data simulated from the ARMA model. To simulate data the parameters of the ARMA model have to be set. By varying these we can minimize a distance between the simulation-based and the observation-based auxiliary estimate. The argument of the minimum yields then an estimator for the parameterization of the ARMA model. This simulation-based estimation procedure inherits the properties of the auxiliary model estimator. For instance, robustness is achieved with GM estimators. An essential feature of the introduced estimator, compared to existing robust estimators for ARMA models, is its theoretical tractability that allows us to show consistency and asymptotic normality. Moreover, it is possible to characterize the influence function and the breakdown point of the estimator. In a small sample Monte Carlo study it is found that the new estimator performs fairly well when compared with existing procedures. Furthermore, with two real examples, we also compare the proposed inferential method with two different approaches based on outliers detection.     

On Hsu's theorem in multivariate regression

The paper deals with optimal quadratic unbiased estimation of the unknown dispersion matrix in multivariate regression models without assuming normality of the errors. We show that Hsu's theorem for univariate regression models continues to multivariate models with no additional assumptions. Furthermore optimal quadratic plus linear estimating functions for regression coefficients are considered, and we investigate whether the ordinary linear estimates are the best. This leads to a new theorem which is similar to that of Hsu.     

Self-weighted generalized empirical likelihood methods for hypothesis testing in infinite variance ARMA models

This paper develops the generalized empirical likelihood (GEL) method for infinite variance ARMA models, and constructs a robust testing procedure for general linear hypotheses. In particular, we use the GEL method based on the least absolute deviations and self-weighting, and construct a natural class of statistics including the empirical likelihood and the continuous updating-generalized method of moments for infinite variance ARMA models. The self-weighted GEL test statistic is shown to converge to a \(\chi ^2\)-distribution, although the model may have infinite variance. Therefore, we can make inference without estimating any unknown quantity of the model such as the tail index or the density function of unobserved innovation processes. We also compare the finite sample performance of the proposed test with the Wald-type test by Pan et al. (Econom Theory 23:852–879, 2007) via some simulation experiments.     

基于Gibbs抽样的贝叶斯稳健ARMA模型研究

针对ARMA模型建模过程中模型识别和参数估计易受观测值异常点影响问题,构建了同时考虑加性异常点和更新性异常点的ARMA模型．运用基于Gibbs抽样的Markov Chain Monte Carlo贝叶斯方法,估计稳健ARMA模型参数,同步确定观测值中异常点的位置,辨别异常点类型．并利用我国人口自然增长数据进行仿真分析,研究结果表明：贝叶斯方法能够有效地识别ARMA序列的异常点．     

The exact likelihood for a multivariate ARMA model

A number of algorithms are presented for calculating the exact likelihood of a multivariate ARMA model. There are two aspects to the algorithms. Firstly, the parameterization is in terms of AR parameters and autocovariances. This obviates difficulties with initial MA estimates. Secondly, the algorithms explicitly account for specification of the lag structure of the multivariate time series. Additionally, an algorithm is presented to deal with missing data. The algorithms are, of themselves, not new but they have not been applied to likelihood construction in the manner discussed here.     

Strictly stationary solutions of multivariate ARMA equations with i.i.d. noise

We obtain necessary and sufficient conditions for the existence of strictly stationary solutions of multivariate ARMA equations with independent and identically distributed driving noise. For general ARMA(p, q) equations these conditions are expressed in terms of the coefficient polynomials of the defining equations and moments of the driving noise sequence, while for p =?1 an additional characterization is obtained in terms of the Jordan canonical decomposition of the autoregressive matrix, the moving average coefficient matrices and the noise sequence. No a priori assumptions are made on either the driving noise sequence or the coefficient matrices.     

Admissibilities of linear estimator in a class of linear models with a multivariate t error variable

This paper discusses admissibilities of estimators in a class of linear models,which include the following common models:the univariate and multivariate linear models,the growth curve model,the extended growth curve model,the seemingly unrelated regression equations,the variance components model,and so on.It is proved that admissible estimators of functions of the regression coefficient β in the class of linear models with multivariate t error terms,called as Model II,are also ones in the case that error terms have multivariate normal distribution under a strictly convex loss function or a matrix loss function.It is also proved under Model II that the usual estimators of β are admissible for p 2 with a quadratic loss function,and are admissible for any p with a matrix loss function,where p is the dimension of β.     

<Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> linear interpolator for missing values in time series

We propose a minimum mean absolute error linear interpolator (MMAELI), based on theL ₁ approach. A linear functional of the observed time series due to non-normal innovations is derived. The solution equation for the coefficients of this linear functional is established in terms of the innovation series. It is found that information implied in the innovation series is useful for the interpolation of missing values. The MMAELIs of the AR(1) model with innovations following mixed normal andt distributions are studied in detail. The MMAELI also approximates the minimum mean squared error linear interpolator (MMSELI) well in mean squared error but outperforms the MMSELI in mean absolute error. An application to a real series is presented. Extensions to the general ARMA model and other time series models are discussed. This research was supported by a CityU Research Grant and Natural Science Foundation of China.