A procedure for the modeling of non-stationary time series

Summary A minimum AIC procedure for the fitting of a locally stationary autoregressive model is proposed. The least squares computation for the procedure is realized by using the Householder transformation which makes the procedure computationally more flexible and efficient than the one originally proposed by Ozaki and Tong. The Institute of Statistical Mathematics     

Testing for serial correlation of unknown form in cointegrated time series models

Portmanteau test statistics are useful for checking the adequacy of many time series models. Here we generalized the omnibus procedure proposed by Duchesne and Roy (2004,Journal of Multivariate Analysis,89, 148–180) for multivariate stationary autoregressive models with exogenous variables (VARX) to the case of cointegrated (or partially nonstationary) VARX models. We show that for cointegrated VARX time series, the test statistic obtained by comparing the spectral density of the errors under the null hypothesis of non-correlation with a kernel-based spectral density estimator, is asymptotically standard normal. The parameters of the model can be estimated by conditional maximum likelihood or by asymptotically equivalent estimation procedures. The procedure relies on a truncation point or a smoothing parameter. We state conditions under which the asymptotic distribution of the test statistic is unaffected by a data-dependent method. The finite sample properties of the test statistics are studied via a small simulation study.     

Efficient on-line estimation of autoregressive parameters

New procedures for estimating autoregressive parameters in AR(m) models are proposed. The proposed method allows for incorporation of auxiliary information into the estimation process and produces estimation procedures, which are consistent and asymptotically efficient under certain regularity conditions. Also, these procedures are naturally on-line and do not require storing all the data. Theoretical results are presented in the case when m = 1. Two important particular cases are considered in detail: linear procedures and likelihood procedures with the LS truncations. A specific example is also presented to briefly discuss some practical aspects of applications of the procedures of this type.     

Prediction of multivariate time series by autoregressive model fitting

Suppose the stationary r-dimensional multivariate time series {y_t} is generated by an infinite autoregression. For lead times h≥1, the linear prediction of y_t+h based on y_t, y_t−1,… is considered using an autoregressive model of finite order k fitted to a realization of length T. Assuming that k → ∞ (at some rate) as T → ∞, the consistency and asymptotic normality of the estimated autoregressive coefficients are derived, and an asymptotic approximation to the mean square prediction error based on this autoregressive model fitting approach is obtained. The asymptotic effect of estimating autoregressive parameters is found to inflate the minimum mean square prediction error by a factor of (1 + kr/T).     

A Combinatorial Approach to Piecewise Linear Time Series Analysis

Over recent years, several nonlinear time series models have been proposed in the literature. One model that has found a large number of successful applications is the threshold autoregressive model (TAR). The TAR model is a piecewise linear process whose central idea is to change the parameters of a linear autoregressive model according to the value of an observable variable, called the threshold variable. If this variable is a lagged value of the time series, the model is called a self-exciting threshold autoregressive (SETAR) model. In this article, we propose a heuristic to estimate a more general SETAR model, where the thresholds are multivariate. We formulate the task of finding multivariate thresholds as a combinatorial optimization problem. We develop an algorithm based on a greedy randomized adaptive search procedure (GRASP) to solve the problem. GRASP is an iterative randomized sampling technique that has been shown to quickly produce good quality solutions for a wide variety of optimization problems. The proposed model performs well on both simulated and real data.     

Asymptotic Analysis of a Multiple Frequency Estimation Method

A recently proposed method of multiple frequency estimation for mixed-spectrum time series is analyzed. The so-called PF method is a procedure that combines the autoregressive (AR) representation of superimposed sinusoids with the idea of parametric filtering. The gist of the method is to parametrize a linear filter in accord with a certain parametrization property, as suggested by the particular form of the bias encountered by Prony′s least-squares estimator for the AR model. It is shown that for any parametric filter with this property, the least-squares estimator obtained from the filtered data is almost surely contractive as a function of the filter parameter and has a unique multivariate fixed-point in the vicinity of the true AR parameter. The fixed-point, known as the PF estimator, is shown to be stronly consistent for estimating the AR model, and the chronic bias of Prony′s estimator is thus eliminated. The almost sure convergence of an iterative algorithm that calculates the fixed-point and the asymptotic normality of the PF estimator are also established. The all-pole filter is considered as an example and application of the developed theory.     

ESTIMATION OF THE MIXED AR AND HIDDEN PERIODIC MODEL

ThisresearchissupportedbytheNationalNaturalScienceFoundationofChina.1.IntroductionGeneralizedhiddenperiodicmodelhasthefollowingformwhereacisthesetofallpositiveintegers,('~{((t);tEac}isastationarysequencewithzeromeanandcontinuousspectraldensity,i=n,qisanonnegativeinteger,'f=0,X=(Al,Az,',A,)isarealvectorwith--T相似文献   

Generalized Rank Estimates For An Autoregressive Time Series: A U-Statistic Approach

A class of weighted rank-based estimates for estimating the parameter vector of an autoregressive time series is considered. This class of estimates is similar to, and contains, the class proposed by Terpstra et al. [54]. Asymptotic linearity properties are derived for the so called GR-estimates. Based on these properties, the GR-estimates are shown to be asymptotically normal at rate n ^1/2. The theory of U-statistics along with a characterization of weak dependence that is inherent in stationary AR(p) models are the primary tools used to obtain the results. The so called pair-wise slopes estimator, which is a special case of this class of estimates, is discussed in an AR(1) context. This revised version was published online in June 2006 with corrections to the Cover Date.     

Autoregressive process modeling via the Lasso procedure

The Lasso is a popular model selection and estimation procedure for linear models that enjoys nice theoretical properties. In this paper, we study the Lasso estimator for fitting autoregressive time series models. We adopt a double asymptotic framework where the maximal lag may increase with the sample size. We derive theoretical results establishing various types of consistency. In particular, we derive conditions under which the Lasso estimator for the autoregressive coefficients is model selection consistent, estimation consistent and prediction consistent. Simulation study results are reported.     

Autoregressive approximation in nonstandard situations: the fractionally integrated and non-invertible cases

Autoregressive models are commonly employed to analyze empirical time series. In practice, however, any autoregressive model will only be an approximation to reality and in order to achieve a reasonable approximation and allow for full generality the order of the autoregression, h say, must be allowed to go to infinity with T, the sample size. Although results are available on the estimation of autoregressive models when h increases indefinitely with T such results are usually predicated on assumptions that exclude (1) non-invertible processes and (2) fractionally integrated processes. In this paper we will investigate the consequences of fitting long autoregressions under regularity conditions that allow for these two situations and where an infinite autoregressive representation of the process need not exist. Uniform convergence rates for the sample autocovariances are derived and corresponding convergence rates for the estimates of AR(h) approximations are established. A central limit theorem for the coefficient estimates is also obtained. An extension of a result on the predictive optimality of AIC to fractional and non-invertible processes is obtained.     

Computationally efficient learning of multivariate <Emphasis Type="Italic">t</Emphasis> mixture models with missing information

A finite mixture model using the multivariate t distribution has been well recognized as a robust extension of Gaussian mixtures. This paper presents an efficient PX-EM algorithm for supervised learning of multivariate t mixture models in the presence of missing values. To simplify the development of new theoretic results and facilitate the implementation of the PX-EM algorithm, two auxiliary indicator matrices are incorporated into the model and shown to be effective. The proposed methodology is a flexible mixture analyzer that allows practitioners to handle real-world multivariate data sets with complex missing patterns in a more efficient manner. The performance of computational aspects is investigated through a simulation study and the procedure is also applied to the analysis of real data with varying proportions of synthetic missing values.     

Copula-Type Estimators for Flexible Multivariate Density Modeling Using Mixtures

Copulas are popular as models for multivariate dependence because they allow the marginal densities and the joint dependence to be modeled separately. However, they usually require that the transformation from uniform marginals to the marginals of the joint dependence structure is known. This can only be done for a restricted set of copulas, for example, a normal copula. Our article introduces copula-type estimators for flexible multivariate density estimation which also allow the marginal densities to be modeled separately from the joint dependence, as in copula modeling, but overcomes the lack of flexibility of most popular copula estimators. An iterative scheme is proposed for estimating copula-type estimators and its usefulness is demonstrated through simulation and real examples. The joint dependence is modeled by mixture of normals and mixture of normal factor analyzer models, and mixture of t and mixture of t-factor analyzer models. We develop efficient variational Bayes algorithms for fitting these in which model selection is performed automatically. Based on these mixture models, we construct four classes of copula-type densities which are far more flexible than current popular copula densities, and outperform them in a simulated dataset and several real datasets. Supplementary material for this article is available online.     

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.     

Influence asymptotique de la correction par la moyenne sur l'estimation d'un modèle AR(1) périodique

This Note studies asymptotic influence of mean-correction on the parameter least squares estimation for a periodic AR(1) model. Unlike the stationary ARMA case, we show that fitting a periodic ARMA model with intercepts to the observed series can provide substantial gains in terms of asymptotic accuracy for the parameter least squares estimators compared with fitting a periodic ARMA model without intercepts to the mean-corrected series. To cite this article: A. Gautier, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Tolerance intervals for autoregressive models,with an application to hospital waiting lists

Long waiting lists are a symbol of inefficiencies of hospital services. The dynamics of waiting lists are complex, especially when trying to understand how the lists grow due to the demand of a particular treatment relative to a hospital's capacity. Understanding the uncertainty of forecasting growth/decline of waiting lists could help hospital managers with capacity planning. We address this uncertainty through the use of statistical tolerance intervals, which are intervals that contain a specified proportion of the sampled population at a given confidence level. Tolerance intervals are available for numerous settings, however, the approaches for autoregressive models are far more limited. This article fills that gap and establishes tolerance intervals for general AR(p) models, which may also have a mean or trend component present. A rigorous development of tolerance intervals in this setting is presented. Extensive simulation studies identify that good coverage properties are achieved when the AR process is stationary and the parameters of the AR model are well within the stationarity constraints. Otherwise, a bootstrap‐based correction can be applied to improve the coverage probabilities. Finally, the method is applied to the monthly number of patients on hospital waiting lists in England.     

On a Class of Shrinkage Priors for Covariance Matrix Estimation

We propose a flexible class of models based on scale mixture of uniform distributions to construct shrinkage priors for covariance matrix estimation. This new class of priors enjoys a number of advantages over the traditional scale mixture of normal priors, including its simplicity and flexibility in characterizing the prior density. We also exhibit a simple, easy to implement Gibbs sampler for posterior simulation, which leads to efficient estimation in high-dimensional problems. We first discuss the theory and computational details of this new approach and then extend the basic model to a new class of multivariate conditional autoregressive models for analyzing multivariate areal data. The proposed spatial model flexibly characterizes both the spatial and the outcome correlation structures at an appealing computational cost. Examples consisting of both synthetic and real-world data show the utility of this new framework in terms of robust estimation as well as improved predictive performance. Supplementary materials are available online.     

Sparse Vector Autoregressive Modeling

The vector autoregressive (VAR) model has been widely used for modeling temporal dependence in a multivariate time series. For large (and even moderate) dimensions, the number of the AR coefficients can be prohibitively large, resulting in noisy estimates, unstable predictions, and difficult-to-interpret temporal dependence. To overcome such drawbacks, we propose a two-stage approach for fitting sparse VAR (sVAR) models in which many of the AR coefficients are zero. The first stage selects nonzero AR coefficients based on an estimate of the partial spectral coherence (PSC) together with the use of BIC. The PSC is useful for quantifying the conditional relationship between marginal series in a multivariate process. A refinement second stage is then applied to further reduce the number of parameters. The performance of this two-stage approach is illustrated with simulation and real data examples. Supplementary materials for this article are available online.     

A semiparametric method for estimating nonlinear autoregressive model with dependent errors

The first-order nonlinear autoregressive model is considered and a semiparametric method is proposed to estimate regression function. In the presented model, dependent errors are defined as first-order autoregressive AR(1). The conditional least squares method is used for parametric estimation and the nonparametric kernel approach is applied to estimate regression adjustment. In this case, some asymptotic behaviors and simulated results for the semiparametric method are presented. Furthermore, the method is applied for the financial data in Iran’s Tejarat-Bank.     

Quantile forecasting based on a bivariate hysteretic autoregressive model with GARCH errors and time ‐varying correlations

To understand and predict chronological dependence in the second‐order moments of asset returns, this paper considers a multivariate hysteretic autoregressive (HAR) model with generalized autoregressive conditional heteroskedasticity (GARCH) specification and time‐varying correlations, by providing a new method to describe a nonlinear dynamic structure of the target time series. The hysteresis variable governs the nonlinear dynamics of the proposed model in which the regime switch can be delayed if the hysteresis variable lies in a hysteresis zone. The proposed setup combines three useful model components for modeling economic and financial data: (1) the multivariate HAR model, (2) the multivariate hysteretic volatility models, and (3) a dynamic conditional correlation structure. This research further incorporates an adapted multivariate Student t innovation based on a scale mixture normal presentation in the HAR model to tolerate for dependence and different shaped innovation components. This study carries out bivariate volatilities, Value at Risk, and marginal expected shortfall based on a Bayesian sampling scheme through adaptive Markov chain Monte Carlo (MCMC) methods, thus allowing to statistically estimate all unknown model parameters and forecasts simultaneously. Lastly, the proposed methods herein employ both simulated and real examples that help to jointly measure for industry downside tail risk.     

�̶�ЧӦ������ݲ�������ģ�͵ļ�Ȩ����LSDV����

This paper concerns with the estimation of a fixed effects panel data partially linear regression model with the idiosyncratic errors being an autoregressive process. For fixed effects short time series panel data, the commonly used autoregressive error structure fitting method will not result in a consistent estimator of the autoregressive coefficients. Here we propose an alternative estimation and show that the resulting estimator of the autoregressive coefficients is consistent and this method is workable for any order autoregressive error structure. Moreover, combining the B-spline approximation, profile least squares dummy variable (PLSDV) technique and consistently estimated the autoregressive error structure, we develop a weighted PLSDV estimator for the parametric component and a weighted B-spline series (BS) estimator for the nonparametric component. The weighted PLSDV estimator is shown to be asymptotically normal and more asymptotically efficient than the one which ignores the error autoregressive structure. In addition, this paper derives the asymptotic bias of the weighted BS estimator and establish its asymptotic normality as well. Simulation studies and an example of application are conducted to illustrate the finite sample performance of the proposed procedures.