Weidong Liu, Shiqing Ling, Qi-Man Shao
In this paper we study the limiting distributions of the least-squares estimators for the non-stationary first-order threshold autoregressive (TAR(1)) model. It is proved that the limiting behaviors of the TAR(1) process are very different from those of the classical unit root model and the explosive AR(1).
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Moderate deviations for the mildly stationary autoregressive models with dependent errors
本文针对带相依误差的轻度平稳AR(1)模型，用鞅方法建立了回归项与误差项最小二乘估计的中偏差，并将其应用于近单位根二阶自回归过程分析及Durbin-Watson统计量的中偏差推导。
This paper studies the threshold estimation of a TAR model when the underlying threshold parameter is a random variable. It is shown that the Bayesian estimator is consistent and its limit distribution is expressed in terms of a limit likelihood ratio. Furthermore, convergence of moments of the estimators is also established. The limit distribution can be computed via explicit simulations from which testing and inference for the threshold parameter can be conducted. The obtained results are illustrated with numerical simulations.
Large sample statistical analysis of threshold autoregressive (TAR) models is usually based on the assumption that the underlying driving noise is uncorrelated. In this paper, we consider a model, driven by Gaussian noise with geometric correlation tail and derive a complete characterization of the asymptotic distribution for the Bayes estimator of the threshold parameter.
In this paper, we consider a continuous-time autoregressive fractionally integrated moving average (CARFIMA) model, which is defined as the stationary solution of a stochastic differential equation driven by a standard fractional Brownian motion. Like the discrete-time ARFIMA model, the CARFIMA model is useful for studying time series with short memory, long memory and antipersistence. We investigate the stationarity of the model and derive its covariance structure. In addition, we derive the spectral density function of a stationary CARFIMA process.
Model Specification Test for Stationary Functional Time Series
本文提出一个基于自回归近似和加权L2距离的通用框架，用于平稳函数型时间序列的模型设定检验，涵盖阶数检验、ARMA设定检验和可分离性检验，并证明了渐近有效性。
The approximation of a stationary time-series by finite order autoregressive (AR) and moving averages (MA) is a problem that occurs in many applications. In this paper we study asymptotic behavior of the spectral density of finite order approximations of wide sense stationary time series. It is shown that when the on the spectral density is non-vanishing in $[-π,π]$ and the covariance is summable, the spectral density of the approximating autoregressive sequence converges at the origin. Under additional mild conditions on the coefficients of the Wold decomposition it is also shown that the spe…
We consider the estimation of the transition matrix in the high-dimensional time-varying vector autoregression (TV-VAR) models. Our model builds on a general class of locally stationary VAR processes that evolve smoothly in time. We propose a hybridized kernel smoothing and $\ell^1$-regularized method to directly estimate the sequence of time-varying transition matrices. Under the sparsity assumption on the transition matrix, we establish the rate of convergence of the proposed estimator and show that the convergence rate depends on the smoothness of the locally stationary VAR processes only t…
The family of semi-stable laws is shown to be semi-selfdecomposable. Thus they qualify to model stationary first order autoregressive schemes. A connection between these autoregressive schemes with semi-stable marginals and semi-selfsimilar processes is given.
We explore the limits of the autoregressive (AR) sieve bootstrap, and show that its applicability extends well beyond the realm of linear time series as has been previously thought. In particular, for appropriate statistics, the AR-sieve bootstrap is valid for stationary processes possessing a general Wold-type autoregressive representation with respect to a white noise; in essence, this includes all stationary, purely nondeterministic processes, whose spectral density is everywhere positive. Our main theorem provides a simple and effective tool in assessing whether the AR-sieve bootstrap is a…
We investigate shrinkage priors on power spectral densities for complex-valued circular-symmetric autoregressive processes. We construct shrinkage predictive power spectral densities, which asymptotically dominate (i) the Bayesian predictive power spectral density based on the Jeffreys prior and (ii) the estimative power spectral density with the maximal likelihood estimator, where the Kullback-Leibler divergence from the true power spectral density to a predictive power spectral density is adopted as a risk. Furthermore, we propose general constructions of objective priors for Kähler paramete…
We introduce and explore a new class of stationary time series models for variance matrices based on a constructive definition exploiting inverse Wishart distribution theory. The main class of models explored is a novel class of stationary, first-order autoregressive (AR) processes on the cone of positive semi-definite matrices. Aspects of the theory and structure of these new models for multivariate "volatility" processes are described in detail and exemplified. We then develop approaches to model fitting via Bayesian simulation-based computations, creating a custom filtering method that reli…
Identification by non-Gaussianity in structural threshold and smooth transition vector autoregressive models
通过假设冲击相互独立且至多一个为高斯分布，证明了结构平滑转换向量自回归模型的统计可识别性，并提出了估计方法和混合识别策略。
Theory and inference for multivariate autoregressive binary models with an application to absence-presence data in ecology
该研究提出带平稳外生协变量的多元二元自回归模型通用类，证明平稳路径存在性条件，结合伪似然与成对似然方法实现多元probit动态模型的统计推断，为生态学物种存在-缺失数据分析提供理论支撑。
This paper studies some temporal dependence properties and addresses the issue of parametric estimation for a class of state-dependent autoregressive models for nonlinear time series in which we assume a stochastic autoregressive coefficient depending on the first lagged value of the process itself. We call such a model state-dependent first-order autoregressive process, (SDAR). We introduce some assumptions under which this class of models is strictly stationary and uniformly ergodic and we establish consistency and asymptotic normality of the quasi-maximum likelihood estimator of the paramet…
This paper introduces a local-to-unity/small sigma process for a stationary time series with strong persistence and non-negligible long run risk. This process represents the stationary long run component in an unobserved short- and long-run components model involving different time scales. More specifically, the short run component evolves in the calendar time and the long run component evolves in an ultra long time scale. We develop the methods of estimation and long run prediction for the univariate and multivariate Structural VAR (SVAR) models with unobserved components and reveal the impos…
A model-free bootstrap procedure for a general class of stationary time series is introduced. The theoretical framework is established, showing asymptotic validity of bootstrap confidence intervals for many statistics of interest. In addition, asymptotic validity of one-step ahead bootstrap prediction intervals is also demonstrated. Finite-sample experiments are conducted to empirically confirm the performance of the new method, and to compare with popular methods such as the block bootstrap and the autoregressive (AR)-sieve bootstrap.
A new mixture autoregressive model based on Student's $t$-distribution is proposed. A key feature of our model is that the conditional $t$-distributions of the component models are based on autoregressions that have multivariate $t$-distributions as their (low-dimensional) stationary distributions. That autoregressions with such stationary distributions exist is not immediate. Our formulation implies that the conditional mean of each component model is a linear function of past observations and the conditional variance is also time varying. Compared to previous mixture autoregressive models ou…
Conditional expectations given past observations in stationary time series are usually estimated directly by kernel estimators, or by plugging in kernel estimators for transition densities. We show that, for linear and nonlinear autoregressive models driven by independent innovations, appropriate smoothed and weighted von Mises statistics of residuals estimate conditional expectations at better parametric rates and are asymptotically efficient. The proof is based on a uniform stochastic expansion for smoothed and weighted von Mises processes of residuals. We consider, in particular, estimation…
Identifiability of VAR(1) model in a stationary setting
研究从平稳分布中辨识VAR(1)模型的有向加权图结构，利用代数统计中的雅可比拟阵方法，给出不同图产生不同稳态分布的充分图条件。
We consider a time-varying first-order autoregressive model with irregular innovations, where we assume that the coefficient function is Hölder continuous. To estimate this function, we use a quasi-maximum likelihood based approach. A precise control of this method demands a delicate analysis of extremes of certain weakly dependent processes, our main result being a concentration inequality for such quantities. Based on our analysis, upper and matching minimax lower bounds are derived, showing the optimality of our estimators. Unlike the regular case, the information theoretic complexity depen…
In this article we develop a tractable procedure for testing strict stationarity in a double autoregressive model and formulate the problem as testing if the top Lyapunov exponent is negative. Without strict stationarity assumption, we construct a consistent estimator of the associated top Lyapunov exponent and employ a random weighting approach for its variance estimation, which in turn are used in a t-type test. We also propose a GLAD estimation for parameters of interest, relaxing key assumptions on the commonly used QMLE. All estimators, except for the intercept, are shown to be consistent…
A class of multivariate periodic autoregressive models is proposed where coupling between time series is achieved through linear mean functions. Various response distributions with quadratic mean-variance relationships fit into the framework, including the negative binomial, gamma and Gaussian distributions. We develop an iterative algorithm to obtain unconditional means, variances and auto-/cross-covariances for models with higher order lags. Analytical solutions are given for the univariate model with lag one and multivariate models with linear mean-variance relationship. A special case of t…
Consider an autoregressive model with measurement error: we observe $Z_i=X_i+ε_i$, where $X_i$ is a stationary solution of the equation $X_i=f_{θ^0}(X_{i-1})+ξ_i$. The regression function $f_{θ^0}$ is known up to a finite dimensional parameter $θ^0$. The distributions of $X_0$ and $ξ_1$ are unknown whereas the distribution of $ε_1$ is completely known. We want to estimate the parameter $θ^0$ by using the observations $Z_0,..,Z_n$. We propose an estimation procedure based on a modified least square criterion involving a weight function $w$, to be suitably chosen. We give upper bounds for the ri…
Our aim in this work is to give explicit formula of the linear processes solution of autoregressive time series AR(2) with hint of generating functions theory by using the Horadam numbers and polynomials.
Here we develop a first order autoregressive model {Xn} that is marginally stationary where Xn is the sum/ extreme of k i.i.d observations. We prove that stationary solutions to these models are either semi-selfdecomposable/ extreme-semi-selfdecomposable or, sum/ extreme stable with respect to Harris distribution.
We develop a new methodology for the fitting of nonstationary time series that exhibit nonlinearity, asymmetry, local persistence and changes in location scale and shape of the underlying distribution. In order to achieve this goal, we perform model selection in the class of piecewise stationary quantile autoregressive processes. The best model is defined in terms of minimizing a minimum description length criterion derived from an asymmetric Laplace likelihood. Its practical minimization is done with the use of genetic algorithms. If the data generating process follows indeed a piecewise quan…
We consider a time series model involving a fractional stochastic component, whose integration order can lie in the stationary/invertible or nonstationary regions and be unknown, and an additive deterministic component consisting of a generalized polynomial. The model can thus incorporate competing descriptions of trending behavior. The stationary input to the stochastic component has parametric autocorrelation, but innovation with distribution of unknown form. The model is thus semiparametric, and we develop estimates of the parametric component which are asymptotically normal and achieve an …
Multivariate Self-Exciting Threshold Autoregressive Models with eXogenous Input
本文提出了一种具有外生输入的多元自激发阈值自回归模型，并给出了参数估计方法，探讨了模型的平稳性条件及参数估计算法的效率。
A novel first-order autoregressive moving average model for analyzing discrete-time series observed at irregularly spaced times is introduced. Under Gaussianity, it is established that the model is strictly stationary and ergodic. In the general case, it is shown that the model is weakly stationary. The lowest dimension of the state-space representation is given along with the one-step linear predictors and their mean squared errors. The maximum likelihood estimation procedure is discussed, and their finite-sample behavior is assessed through Monte Carlo experiments. These experiments show tha…
Moving Aggregate Modified Autoregressive Copula-Based Time Series Models (MAGMAR-Copulas)
本文提出MAGMAR-皮克罗模型，通过引入移动聚合部分改进传统皮克罗时间序列模型，以处理非马尔可夫过程，通过D-vine框架推导分布特性，并在美通胀数据上验证模型性能。
This paper introduces a Threshold Asymmetric Conditional Autoregressive Range (TACARR) formulation for modeling the daily price ranges of financial assets. It is assumed that the process generating the conditional expected ranges at each time point switches between two regimes, labeled as upward market and downward market states. The disturbance term of the error process is also allowed to switch between two distributions depending on the regime. It is assumed that a self-adjusting threshold component that is driven by the past values of the time series determines the current market regime. Th…
The article contains an overview over locally stationary processes. At the beginning time varying autoregressive processes are discussed in detail - both as as a deep example and an important class of locally stationary processes. In the next section a general framework for time series with time varying finite dimensional parameters is discussed with special emphasis on nonlinear locally stationary processes. Then the paper focusses on linear processes where a more general theory is possible. First a general definition for linear processes is given and time varying spectral densities are discu…
Inferring Latent Graphs from Stationary Signals Using a Graphical Autoregressive Model
本文提出图自回归模型GAR，将逆协方差表示为潜在图归一化拉普拉斯的二阶多项式，并用惩罚最大似然三步法推断潜在图，在模拟和S&P 500数据中优于高斯图模型。
The notion of confidence distributions is applied to inference about the parameter in a simple autoregressive model, allowing the parameter to take the value one. This makes it possible to compare to asymptotic approximations in both the stationary and the non stationary cases at the same time. The main point, however, is to compare to a Bayesian analysis of the same problem. A non informative prior for a parameter, in the sense of Jeffreys, is given as the ratio of the confidence density and the likelihood. In this way, the similarity between the confidence and non-informative Bayesian framew…
Testing the goodness-of-fit of a functional autoregressive model
针对函数型时间序列中的线性自相关模型，提出基于经验过程的拟合优度检验，推导了经验过程收敛到时间变换的Wiener过程的函数中心极限定理，并验证了检验在简单和复合原假设下的大样本性质及一致性。
Two-way Matrix Autoregressive Model with Thresholds
针对矩阵时间序列，提出双向阈值矩阵自回归模型（2-MART），允许行和列使用不同阈值变量，实现更大降维、更好拟合、更准预测和更合理解释。
For multivariate stationary time series many important properties, such as partial correlation, graphical models and autoregressive representations are encoded in the inverse of its spectral density matrix. This is not true for nonstationary time series, where the pertinent information lies in the inverse infinite dimensional covariance matrix operator associated with the multivariate time series. This necessitates the study of the covariance of a multivariate nonstationary time series and its relationship to its inverse. We show that if the rows/columns of the infinite dimensional covariance …
In this paper, we focus on isotropic and stationary sphere-cross-time random fields. We first introduce the class of spherical functional autoregressive-moving average processes (SPHARMA), which extend in a natural way the spherical functional autoregressions (SPHAR) recently studied in [8, 7]; more importantly, we then show that SPHAR and SPHARMA processes of sufficiently large order can be exploited to approximate every isotropic and stationary sphere-cross-time random field, thus generalizing to this infinite-dimensional framework some classical results on real-valued stationary processes. …
Change-point analysis for binomial autoregressive model with application to price stability counts
针对非平稳有界计数时间序列，提出带多个变点的BAR(1)模型，结合CUSUM检验与MDL原则检测并估计变点，并应用于欧盟价格稳定计数分析。
Likelihood asymptotics of stationary Gaussian arrays
本文针对依赖多维未知参数的平稳高斯时间序列三角阵列发展渐近似然理论，给出局部渐近正态的充分条件以保证极大似然估计渐近有效，并通过高频采样高斯过程和近单位根自回归模型验证理论。
We develop an estimator for the high-dimensional covariance matrix of a locally stationary process with a smoothly varying trend and use this statistic to derive consistent predictors in non-stationary time series. In contrast to the currently available methods for this problem the predictor developed here does not rely on fitting an autoregressive model and does not require a vanishing trend. The finite sample properties of the new methodology are illustrated by means of a simulation study and a financial indices study.
SAMoSSA: Multivariate Singular Spectrum Analysis with Stochastic Autoregressive Noise
本文提出两阶段算法SAMoSSA，结合多变量奇异谱分析（mSSA）与自回归（AR）噪声建模，解决了同时学习确定性与平稳成分的理论空白，并在多个基准数据集上实现5%至37%的预测性能提升。
Hawkes autoregressive processes: a new model for multiscale and heterogeneous processes
本文提出霍克斯自回归（HAR）模型，融合连续时间与离散时间动力学，并证明了平稳性、聚类表示、稳定性及遍历性等概率性质。
This study aimed to find temporal clusters for several commodity prices using the threshold non-linear autoregressive model. It is expected that the process of determining the commodity groups that are time-dependent will advance the current knowledge about the dynamics of co-moving and coherent prices, and can serve as a basis for multivariate time series analyses. The clustering of commodity prices was examined using the proposed clustering approach based on time series models to incorporate the time varying properties of price series into the clustering scheme. Accordingly, the primary aim …
In this paper, we study nonparametric models allowing for locally stationary regressors and a regression function that changes smoothly over time. These models are a natural extension of time series models with time-varying coefficients. We introduce a kernel-based method to estimate the time-varying regression function and provide asymptotic theory for our estimates. Moreover, we show that the main conditions of the theory are satisfied for a large class of nonlinear autoregressive processes with a time-varying regression function. Finally, we examine structured models where the regression fu…
In this paper we are interested in the Maximum Likelihood Estimator (MLE) of the vector parameter of an autoregressive process of order $p$ with regular stationary Gaussian noise. We exhibit the large sample asymptotical properties of the MLE under very mild conditions. Simulations are done for fractional Gaussian noise (fGn), autoregressive noise (AR(1)) and moving average noise (MA(1)).
The tail index, indicating the degree of fatness of the tail distribution, is an important component of extreme value theory since it dominates the asymptotic distribution of extreme values such as the sample maximum. In this paper, we consider the problem of testing for a change in the tail index of time series data. As a test, we employ the cusum test and investigate its null limiting distribution. Further, we derive the null limiting distribution of the cusum test based on the residuals from autoregressive models. Simulation results are provided for illustration.
We study the nonparametric covariance estimation of a stationary Gaussian field X observed on a regular lattice. In the time series setting, some procedures like AIC are proved to achieve optimal model selection among autoregressive models. However, there exists no such equivalent results of adaptivity in a spatial setting. By considering collections of Gaussian Markov random fields (GMRF) as approximation sets for the distribution of X, we introduce a novel model selection procedure for spatial fields. For all neighborhoods m in a given collection M, this procedure first amounts to computing …
The first purpose of this article is to obtain a.s. asymptotic properties of the maximum likelihood estimator in the autoregressive process driven by a stationary Gaussian noise. The second purpose is to show the local asymptotic normality property of the likelihoods ratio in order to get a notion of asymptotic efficiency and to build an asymptotically uniformly invariant most powerful procedure for testing the significance of the autoregressive parameter.
Extremal properties of max-autoregressive moving average processes for modelling extreme river flows
本文研究平稳 Max-ARMA 过程的极值性质，首次提出推导其极值指标和滞后渐近依赖系数的方法，并基于重新参数化拟合模型以捕捉高阈值下的河流流量行为，用英国泰晤士河数据进行了验证。
We propose a family of CUSUM-based statistics to detect the presence of changepoints in the deterministic part of the autoregressive parameter in a Random Coefficient AutoRegressive (RCA) sequence. In order to ensure the ability to detect breaks at sample endpoints, we thoroughly study weighted CUSUM statistics, analysing the asymptotics for virtually all possible weighing schemes, including the standardised CUSUM process (for which we derive a Darling-Erdos theorem) and even heavier weights (studying the so-called Rényi statistics). Our results are valid irrespective of whether the sequence i…
Most of the stationary first-order autoregressive integer-valued (INAR(1)) models were developed for a given thinning operator using either the forward approach or the backward approach. In the forward approach the marginal distribution of the time series is specified and an appropriate distribution for the innovation sequence is sought. Whereas in the backward setting, the roles are reversed. The common distribution of the innovation sequence is specified and the distributional properties of the marginal distribution of the time series are studied. In this article we focus on the backward app…
In this study, we present a criterion based on the analysis of EEG signals through the mean of the conventional power spectral density (PSD) in the aim to localize and detect the epileptic area of the brain. Firstly, as the EEG signals are commonly non stationary in practice, we processed the data with technique of differentiation in order to have the stationary which is convenient to model with autoregressive model (AR). For this, we have used many techniques for to determine the order which model better the data in this work. Therefore, we can characterize normal and abnormal activity which …
Inference for a New Signed Integer Valued Autoregressive Model Based on Pegram's Operator
本研究提出基于扩展二项分布与Pegram算子的SINARS(1)模型，可刻画非平稳序列差分后的正负整数平稳序列，采用两种参数估计方法并证明CML法渐近正态性，仿真及巴巴多斯新增病例差分数据分析验证了其有效性。
In this work we deal with the problem of fitting an error density to the goodness-of-fit test of the errors in nonlinear autoregressive time series models with stationary $α$-mixing error terms. The test statistic is based on the integrated squared error of the nonparametric error density estimate and the null error density. By deriving the asymptotic normality of test statistics in these models, we extend the result of Cheng and Sun (Statist. Probab. Lett. \textbf{78}, 1(2008), 50-59) in the model with i.i.d error terms to the more general case.
Vector autoregressive (VAR) models are popularly adopted for modelling high-dimensional time series, and their piecewise extensions allow for structural changes in the data. In VAR modelling, the number of parameters grow quadratically with the dimensionality which necessitates the sparsity assumption in high dimensions. However, it is debatable whether such an assumption is adequate for handling datasets exhibiting strong serial and cross-sectional correlations. We propose a piecewise stationary time series model that simultaneously allows for strong correlations as well as structural changes…
The volatility modeling for autoregressive univariate time series is considered. A benchmark approach is the stationary ARCH model of Engle (1982). Motivated by real data evidence, processes with non constant unconditional variance and ARCH effects have been recently introduced. We take into account such possible non stationarity and propose simple testing procedures for ARCH effects. Adaptive McLeod and Li's portmanteau and ARCH-LM tests for checking for second order dynamics are provided. The standard versions of these tests, commonly used by practitioners, suppose constant unconditional var…
Joint Probability Distribution of Prediction Errors of ARIMA
本文提出了一种基于ARIMA模型计算多步预测误差联合概率分布的方法，适用于平稳过程和本质平稳过程的单变量和多变量场景。
Nonlinear Causality in Time Series Networks: With Application to Motor Imagery vs Execution
针对时间序列网络中非线性因果关系难以识别的问题，提出基于阈值自回归模型的两阶段推断方法TAR4C，并在多通道EEG运动想象与执行实验中验证了其有效性。
Data Augmentation of Multivariate Sensor Time Series using Autoregressive Models and Application to Failure Prognostics
本文提出基于时变自回归过程的非平稳多元传感器时间序列数据增强方法，利用 CMAPSS 数据集和 AutoML 预测方案验证其可在数据稀缺场景下显著提升故障预测性能。
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