<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.     

Extremes of Volterra series expansions with heavy-tailed innovations

In this paper, we look at the extremal behavior of Volterra series expansions generated by heavy-tailed innovations, via a point process formulation. Volterra series expansions are known to be the most general nonlinear representation for any stationary sequence. The so called complete convergence theorem on point processes we prove enable us to give in detail, the weak limiting behavior of various functionals of the underlying process including the asymptotic distribution of upper and lower order statistics. In particular, we investigate the limiting distribution of the sample maxima and the corresponding extremal index. The study of the extremal properties of finite order Volterra series expansions would be highly valuable in understanding the extremal behavior of nonlinear processes as well as understanding of order identification and adequacy of Volterra series when used as models in signal processing. In fact, such extremal properties may suggest a way of finding the order of a finite Volterra expansions which is consistent with the nonlinearities of the observed process.     

Testing the Cointegrating Rank with Uncorrelated but Dependent Errors

Abstract

We study the asymptotic behavior of the reduced rank estimator of the cointegrating space and adjustment space for vector error correction time series models with nonindependent innovations. It is shown that the distribution of the adjustment space can be quite different for models with iid innovations and models with nonindependent innovations. It is also shown that the likelihood ratio test remains valid when the assumption of iid Gaussian errors is relaxed. Monte Carlo experiments illustrate the finite sample performance of the likelihood ratio test using various kinds of weak error processes.     

基于ARIMA与神经网络集成的GDP时间序列预测研究

本文深入分析了单整自回归移动平均(ARIMA)模型与神经网络(NN)模型的预测特性和优劣,并在此基础上建立了由ARIMA模型和NN模型集成的GDP时间序列预测模型与算法。其基本思想是充分发挥两种模型在线性空间和非线性空间的预测优势,据此将GDP时间序列的数据结构分解为线性自相关主体和非线性残差两部分,首先用ARIMA模型预测序列的线性主体,然后用NN模型对其非线性残差进行估计,最终集成为整个序列的预测结果。仿真实验表明:集成模型的预测准确率显著高于单一模型的预测准确率,从而证实了集成模型用于GDP预测的有效性。     

基于人工神经网络和随机游走模型的汇率预测

由于金融数据具有随机性特征,使得建模和预测变得极其困难.提出一种组合预测方法,即假定任何金融时序数据由线性和非线性两部分组成,将其中线性部分的数据通过随机游走(RW)模型进行模拟,剩余的非线性残差部分由前馈神经网络(FANN)和诶尔曼神经网络(EANN)协同处理.从实证结果可知,该组合方法相比单独使用RW、FANN或EANN模型有更高的预测精度.     

A nonlinear time series model and estimation of missing observations

This paper formulates a nonlinear time series model which encompasses several standard nonlinear models for time series as special cases. It also offers two methods for estimating missing observations, one using prediction and fixed point smoothing algorithms and the other using optimal estimating equation theory. Recursive estimation of missing observations in an autoregressive conditionally heteroscedastic (ARCH) model and the estimation of missing observations in a linear time series model are shown to be special cases. Construction of optimal estimates of missing observations using estimating equation theory is discussed and applied to some nonlinear models.Authors were supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.     

Tests portmanteau multivariés d'adéquation de modèles VARMA faibles

In this Note, we consider portmanteau tests for testing the adequacy of vector autoregressive moving-average (VARMA) models under the assumption that the errors are uncorrelated but not necessarily independent. We relax the standard independence assumption to extend the range of application of the VARMA models, allowing us to treat linear representations of general nonlinear processes. We first study the joint distribution of the quasi-maximum likelihood estimator (QMLE) and the noise empirical autocovariances. We thus obtain the asymptotic distribution of residual empirical autocovariances and autocorrelations under weak assumptions on the noise. We deduce the asymptotic distribution of the Ljung–Box (or Box–Pierce) portmanteau statistics for VARMA models with nonindependent innovations. We propose a method to adjust the critical values of the portmanteau tests.     

An asymptotic theory for sample covariances of Bernoulli shifts

Covariances play a fundamental role in the theory of stationary processes and they can naturally be estimated by sample covariances. There is a well-developed asymptotic theory for sample covariances of linear processes. For nonlinear processes, however, many important problems on their asymptotic behaviors are still unanswered. The paper presents a systematic asymptotic theory for sample covariances of nonlinear time series. Our results are applied to the test of correlations.     

Multivariate arma models with generalized autoregressive linear innovation

In this paper we develop, in a multivariate framework, an alternative approach to the classical non linear analysis of time series. The proposed class of stochastic processes, of which the bilinear model is a special case, is based on a generalized autoregressive modelling of linear innovations. The probability structure is analyzed under quite general conditions. Moreover an important subclass of bilinear processes is studied in greater details. Finally, the usefulness of the results is illustrated via a numerical study.     

Tail dependence of random variables from ARCH and heavy-tailed bilinear models

Discussed in this paper is the dependent structure in the tails of distributions of random variables from some heavy-tailed stationary nonlinear time series. One class of models discussed is the first-order autoregressive conditional heteroscedastic (ARCH) process introduced by Engle (1982). The other class is the simple first-order bilinear models driven by heavy-tailed innovations. We give some explicit formulas for the asymptotic values of conditional probabilities used for measuring the tail dependence between two random variables from these models. Our results have significant meanings in finance.     

Multivariate Local Polynomial Fitting for Martingale Nonlinear Regression Models

Local polynomial modelling is a useful tool for nonlinear time series analysis. For nonlinear regression models with martingale difference errors, this paper presents a simple proof of local linear and local quadratic fittings under apparently minimal short-range dependence condition. Explicit formulae for the asymptotic bias and asymptotic variance are given, which facilitate numerical evaluations of these important quantities. The general theory is applied to nonparametric partial derivative estimation in nonlinear time series. A bias-adjusted method for constructing confidence intervals for first-order partial derivatives is described. Two examples, including the sunspots data, are used to demonstrate the use of local quadratic fitting for modelling and characterizing nonlinearity in time series data.     

Kernel estimation for time series: An asymptotic theory

We consider kernel density and regression estimation for a wide class of nonlinear time series models. Asymptotic normality and uniform rates of convergence of kernel estimators are established under mild regularity conditions. Our theory is developed under the new framework of predictive dependence measures which are directly based on the data-generating mechanisms of the underlying processes. The imposed conditions are different from the classical strong mixing conditions and they are related to the sensitivity measure in the prediction theory of nonlinear time series.     

Nonparametric changepoint detection for time series

We consider time series which change their structure repeatedly between a finite number of states, and we discuss algorithms to detect the changepoints for two particular situations. In the first case, the observed time series is a nonparametric autoregression of order p and the autoregression function changes sometimes. Here, we use a system of neural networks to estimate the autoregression functions and to detect the changepoints. In the second case, the time series is a piecewise linear process with stable innovations, where we assume that the various processes represent different dominating local frequencies, and we use wavelet packet coefficients to detect the changepoints.     

Asymptotic theory for curve-crossing analysis

We consider asymptotic properties of curve-crossing counts of linear processes and nonlinear time series by curves. Central limit theorems are obtained for curve-crossing counts of short-range dependent processes. For the long-range dependence case, the asymptotic distributions are shown to be either multiple Wiener–Itô integrals or integrals with respect to stable Lévy processes, depending on the heaviness of tails of the underlying processes.     

A partially linearized sigma point filter for latent state estimation in nonlinear time series models

A new technique for the latent state estimation of a wide class of nonlinear time series models is proposed. In particular, we develop a partially linearized sigma point filter in which random samples of possible state values are generated at the prediction step using an exact moment-matching algorithm and then a linear programming based procedure is used in the update step of the state estimation. The effectiveness of the new filtering procedure is assessed via a simulation example that deals with a highly nonlinear, multivariate time series representing an interest rate process.     

Asymptotics for estimation of quantile regressions with truncated infinite-dimensional processes

Many processes can be represented in a simple form as infinite-order linear series. In such cases, an approximate model is often derived as a truncation of the infinite-order process, for estimation on the finite sample. The literature contains a number of asymptotic distributional results for least squares estimation of such finite truncations, but for quantile estimation, results are not available at a level of generality that accommodates time series models used as finite approximations to processes of potentially unbounded order. Here we establish consistency and asymptotic normality for conditional quantile estimation of truncations of such infinite-order linear models, with the truncation order increasing in sample size. We focus on estimation of the model at a given quantile. The proofs use the generalized functions approach and allow for a wide range of time series models as well as other forms of regression model. The results are illustrated with both analytical and simulation examples.     

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.     

Asymptotic Normality of Kernel Density Estimators under Dependence

In this paper, we study the kernel methods for density estimation of stationary samples under generalized conditions, which unify both the linear and -mixing processes discussed in the literature and also adapt to the non-linear or/and non--mixing processes. Under general, mild conditions, the kernel density estimators are shown to be asymptotically normal. Some specific theorems are derived within various contexts, and their applications and relationship with the relevant references are considered. It is interesting that the conditions on the bandwidth may be very simple, even in the generalized context. The stationary sequences discussed cover a large number of (linear or nonlinear) time series and econometric models (such as the ARMA processes with ARCH errors).     

非线性自回归序列L1估计的渐近分布

由于时间序列数据中经常出现的厚尾特征使得通常的估计方法不再具有渐近的正态分布,在误差项二阶矩有限的条件下考虑了非线性自回归序列的L_1估计.采用局部线性近似的方法得到了具有凸样本路径的随机过程,在此基础上利用凸样本路径随机过程弱收敛的性质证明了非线性自回归序列L_1估计的渐近正态性及无偏性.     

Forecasting nonlinear time series with a hybrid methodology

In recent years, artificial neural networks (ANNs) have been used for forecasting in time series in the literature. Although it is possible to model both linear and nonlinear structures in time series by using ANNs, they are not able to handle both structures equally well. Therefore, the hybrid methodology combining ARIMA and ANN models have been used in the literature. In this study, a new hybrid approach combining Elman’s Recurrent Neural Networks (ERNN) and ARIMA models is proposed. The proposed hybrid approach is applied to Canadian Lynx data and it is found that the proposed approach has the best forecasting accuracy.