Innovation algorithm in ARMA process

Most of the works in Time Series Analysis are based on the Auto Regressive Integrated Moving Average (ARIMA) models presented by Box and Jenkins(1976). If the data exhibits no apparent deviation from stationarity and if it has rapidly decreasing autocorrelation function then a suitable ARMA(p,q) model is fit to the given data. Selection of the orders of p and q is one of the crucial steps in Time Series Analysis. Most of the methods to determine p and q are based on the autocorrelation function and partial autocorrelation function as suggested by Box and Jenkins (1976). Many new techniques have emerged in the literature and it is found that most of them are of very little use in determining the orders of p and q when both of them are non-zero. The Durbin-Levinson algorithm and Innovation algorithm (Brockwell and Davis, 1987) are used as recursive methods for computing best linear predictors in an ARMA(p,q) model. These algorithms are modified to yield an effective method for ARMA model identification so that the values of order p and q can be determined from them. The new method is developed and its validity and usefulness is illustrated by many theoretical examples. This method can also be applied to any real world data.     

左删失数据下ARMA(p,q)模型的估计

在观测数据左删失情形下由K—M估计方法得到,严平稳遍历序列{Xt}的均值和自协方差函数的估计,从而获得ARMA（p,q）模型的参数估计,且所给估计量是强相合估计．     

基于ARMA(p,q)利息力生存年金精算现值模型

企业年金是养老保险体系的重要组成部分,其定价的合理性正受到越来越多的关注.主要是基于一般的ARM A(p,q)模型得到了随机利率下生存年金的精算现值模型,分别给出了年金给付的一阶矩和二阶矩,这对年金保险的合理收费和避免收不抵支情况的出现具有重要的指导意义.     

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．     

Whittle estimation in a heavy-tailed GARCH(1,1) model

The squares of a GARCH(p,q) process satisfy an ARMA equation with white noise innovations and parameters which are derived from the GARCH model. Moreover, the noise sequence of this ARMA process constitutes a strongly mixing stationary process with geometric rate. These properties suggest to apply classical estimation theory for stationary ARMA processes. We focus on the Whittle estimator for the parameters of the resulting ARMA model. Giraitis and Robinson (2000) show in this context that the Whittle estimator is strongly consistent and asymptotically normal provided the process has finite 8th moment marginal distribution.

We focus on the GARCH(1,1) case when the 8th moment is infinite. This case corresponds to various real-life log-return series of financial data. We show that the Whittle estimator is consistent as long as the 4th moment is finite and inconsistent when the 4th moment is infinite. Moreover, in the finite 4th moment case rates of convergence of the Whittle estimator to the true parameter are the slower, the fatter the tail of the distribution.

These findings are in contrast to ARMA processes with iid innovations. Indeed, in the latter case it was shown by Mikosch et al. (1995) that the rate of convergence of the Whittle estimator to the true parameter is the faster, the fatter the tails of the innovations distribution. Thus the analogy between a squared GARCH process and an ARMA process is misleading insofar that one of the classical estimation techniques, Whittle estimation, does not yield the expected analogy of the asymptotic behavior of the estimators.     

ARMA identification

In view of recent results on the asymptotic behavior of the prediction error covariance for a state variable system (see Ref. 1), an identification scheme for autoregressive moving average (ARMA) processes is proposed. The coefficients of thed-step predictor determine asymptotically the system momentsU ₀,...,U _d–1. These moments are also nonlinear functions of the coefficients of the successive 1-step predictors. Here, we estimate the state variable parameters by the following scheme. First, we use the Burg technique (see Ref. 2) to find the estimates of the coefficients of the successive 1-step predictors. Second, we compute the moments by substitution of the estimates provided by the Burg technique for the coefficients in the nonlinear functions relating the moments with the 1-step predictor coefficients. Finally, the Hankel matrix of moment estimates is used to determine the coefficients of the characteristic polynomial of the state transition matrix (see Refs. 3 and 4).A number of examples for the state variable systems corresponding to ARMA(2, 1) processes are given which show the efficiency of this technique when the zeros and poles are separated. Some of these examples are also studied with an alternative technique (see Ref. 5) which exploits the linear dependence between successive 1-step predictors and the coefficients of the transfer function numerator and denominator polynomials.In this paper, the problems of order determination are not considered; we assumed the order of the underlying system. We remark that the Burg algorithm is a robust statistical procedure. With the notable exception of Ref. 6 that uses canonical correlation methods, most identification procedures in control are based on a deterministic analysis and consequently are quite sensitive to errors. In general, spectral identification based on the windowing of data lacks the resolving power of the Burg technique, which is a super resolution method.This work was supported by NATO Research Grant No. 585/83, by University of Nice, by Thomson CSF-DTAS, by Instituto Nacional de Investigação Científica, and by CIRIT (Comissió Interdepartmental de Recerca i Innovació Technológica de Catalunya). The work of the third author was also partially supported by Army Research Office Contract DAAG-29-84-k-005.Simple ARMA(2, 1) Basic language analysis programs to construct random data were written by the second author and Dr. K. D. Senne, MIT Lincoln Laboratory. Lack of stability of the direct estimation was observed at TRW with the help of Dr. G. Butler. Analysis programs in FORTRAN for ARMA(p, q) were written and debugged at CAPS by the fifth author. The research was helped by access to VAXs at Thomson CSF-DTAS, Valbonne, France, at CAPS, Instituto Superior Técnico, and at Universitat Politécnica de Catalunya. In particular, the authors explicitly acknowledge Thomson CSF-DTAS and Dr. H. Gautier for extending the use of their facilities to the authors from September 1983 until June 1984, when the examples presented were simulated.on leave from University of Southern California, Los Angeles, California.formerly visiting at Massachusetts Institute of Technology and Laboratory for Information and Decision Systems, while on leave from Instituto Superior Técnico, Lisbon, Portugal.     

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.     

低碳经济背景下中国铅精炼企业产销存量预测的ARMA模型构建及其应用研究

分析了ARMA模型构建中的数学方法,构建了低碳经济背景下中国铅精炼企业产销存量预测的ARMA模型,检验结果显示,ARMA(4,0)模型是对低碳经济背景下中国铅精炼企业产销存量进行短期预测的较优模型,同时最后采用该模型对中国铅精炼企业未来5年的产销存量进行了预测,并就预测结果提出了促进我国精炼铅市场低碳可持续发展的三种政策建议。     

Sequential and non-sequential acceptance sampling plans for autocorrelated processes using ARMA(p,q) models

This paper presents variable acceptance sampling plans based on the assumption that consecutive observations on a quality characteristic(X) are autocorrelated and are governed by a stationary autoregressive moving average (ARMA) process. The sampling plans are obtained under the assumption that an adequate ARMA model can be identified based on historical data from the process. Two types of acceptance sampling plans are presented: (1) Non-sequential acceptance sampling: In this case historical data is available based on which an ARMA model is identified. Parameter estimates are used to determine the action limit (k) and the sample size(n). A decision regarding acceptance of a process is made after a complete sample of size n is selected. (2) Sequential acceptance sampling: Here too historical data is available based on which an ARMA model is identified. A decision regarding whether or not to accept a process is made after each individual sample observation becomes available. The concept of Sequential Probability Ratio Test (SPRT) is used to derive the sampling plans. Simulation studies are used to assess the effect of uncertainties in parameter estimates and the effect of model misidentification (based on historical data) on sample size for the sampling plans. Macros for computing the required sample size using both methods based on several ARMA models can be found on the author’s web page .     

ARMA model order and parameter estimation using genetic algorithms

A new method for simultaneously determining the order and the parameters of autoregressive moving average (ARMA) models is presented in this article. Given an ARMA (p, q) model in the absence of any information for the order, the correct order of the model (p, q) as well as the correct parameters will be simultaneously determined using genetic algorithms (GAs). These algorithms simply search the order and the parameter spaces to detect their correct values using the GA operators. The proposed method works on the principle of maximizing the GA fitness value relying on the deviation between the actual plant output, with or without an additive noise, and the estimated plant output. Simulation results show in detail the efficiency of the proposed approach. In addition to that, a practical model identification and parameter estimation is conducted in this article with results obtained as desired. The new method is compared with other well-known methods for ARMA model order and parameter estimation.     

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.     

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.     

基于蒙特卡洛-马尔科夫链(MCMC)的ARMA模型选择

AIC与SIC等准则函数方法是ARMA模型选择过程中经常使用的方法。但是,当模型的阶数很高时,无法计算并比较每一个备选模型的准则函数值。本文提出了一个基于蒙特卡洛-马尔科夫链方法的随机模型生成方法,以产生准则函数值最小的备选模型。实际应用表明本文的方法在处理拥有大量备选模型的ARMA模型选择问题时有很好的效果。     

Recursive method for ARMA model estimation (II)

In this paper a new recursive method for ARMA model estimation is given. Same as in [1], the order's estimator is strongly consistent, and the pearameter's estimaters defer to CLT and LIL under a natural condition. Compared with the previous methods suggested by Hannan & Kavalieris (1984), Wang Shouren & Chen Zhaoguo (1985) and Franke (1985), this method has some advantages: the amount of calculat on work is smaller, the minimum-phase property of coefficient estimators can be guaranteed, the BAN estimators for MA or AR model can be obtained directly, and the simulation shows that this method is more accurate in estimating the order and parameters.     

On the recursive fitting of subset autoregressive-moving average process

For fitting data with an ARMA model, the major topic we come cross is how to determine the order (p, q). More over we may wish to search for a subset ARMA model. This paper offers a recursive procedure for this purpose, it is computationally very cheap. If the data are generated truly by a subset ARMA model, then the procedure can be proved to give a consistent identification.Institute of Applied Mathematics, Academia Sinica     

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.     

Properties of batch means from stationary ARMA time series

The batch-means process arising from an arbitary autoregressive moving-average (ARMA) process time series is derived. As side results, the variance and correlation structures of the batch-means process as functions of the batch size and parameters of the original process are obtained. Except for the first-order ARMA process, for which a closed-form expression is obtained, the parameters of the batch-means process are determined numerically.     

Identification of Refined ARMA Echelon Form Models for Multivariate Time Series

In the present article, we are interested in the identification of canonical ARMA echelon form models represented in a “refined” form. An identification procedure for such models is given by Tsay (J. Time Ser. Anal.10(1989), 357-372). This procedure is based on the theory of canonical analysis. We propose an alternative procedure which does not rely on this theory. We show initially that an examination of the linear dependency structure of the rows of the Hankel matrix of correlations, with originkin (i.e., with correlation at lagkin position (1, 1)), allows us not only to identify the Kronecker indicesn₁, …, n_d, whenk=1, but also to determine the autoregressive ordersp₁, …, p_d, as well as the moving average ordersq₁, …, q_dof the ARMA echelon form model by settingk>1 andk<1, respectively. Successive test procedures for the identification of the structural parametersn_i,p_i, andq_iare then presented. We show, under the corresponding null hypotheses, that the test statistics employed asymptotically follow chi-square distributions. Furthermore, under the alternative hypothesis, these statistics are unbounded in probability and are of the formNδ{1+o_p(1)}, whereδis a positive constant andNdenotes the number of observations. Finally, the behaviour of the proposed identification procedure is illustrated with a simulated series from a given ARMA model.     

指数回归-ARMA模型在我国人均生活电力消费量预测中的应用

随着我国经济快速增长、居民收入水平的显著提高,生活电力消费量快速增长。本文以1983-2006年我国人均生活电力消费量的历史数据为基础,根据趋势图拟合出与之相似的指数回归曲线,然后对其残差序列进行分析和识别,找出适合我国人均生活电力消费量的指数回归-ARMA模型,并根据此模型进行预测分析。     

The Application of Kernel Smoothing to Time Series Data

There are already a lot of models to fit a set of stationary time series, such as AR, MA, and ARMA models. For the non-stationary data, an ARIMA or seasonal ARIMA models can be used to fit the given data. Moreover, there are also many statistical softwares that can be used to build a stationary or non-stationary time series model for a given set of time series data, such as SAS, SPLUS, etc. However, some statistical softwares wouldn＇t work well for small samples with or without missing data, especially for small time series data with seasonal trend. A nonparametric smoothing technique to build a forecasting model for a given small seasonal time series data is carried out in this paper. And then, both the method provided in this paper and that in SAS package are applied to the modeling of international airline passengers data respectively, the comparisons between the two methods are done afterwards. The results of the comparison show us the method provided in this paper has superiority over SAS＇s method.