Inference for Time Series Regression Models With Weakly Dependent and Heteroscedastic Errors

Motivated by the need to assess the significance of the trend in some macroeconomic series, this article considers inference of a parameter in parametric trend functions when the errors exhibit certain degrees of nonstationarity with changing unconditional variances. We adopt the recently developed self-normalized approach to avoid the difficulty involved in the estimation of the asymptotic variance of the ordinary least-square estimator. The limiting distribution of the self-normalized quantity is nonpivotal but can be consistently approximated by using the wild bootstrap, which is not consistent in general without studentization. Numerical simulation demonstrates favorable coverage properties of the proposed method in comparison with alternative ones. The U.S. nominal wages series is analyzed to illustrate the finite sample performance. Some technical details are included in the online supplemental material.     

基于核密度估计的非线性时间序列聚类

本文研究的是时间序列的聚类问题。由于现实世界中时间序列多数是非线性的,而现有的时间序列聚类问题大都是基于线性时间序列模型进行聚类的,本文提出了可以用于非线性时间序列的聚类方法。以时间序列的二维核密度估计之间的相似性作为非线性时间序列的距离度量,该距离度量方式是一种非参数的距离度量方法,考虑到了时间序列自相关结构的差异,能够粗糙地识别时间序列形状和动态相关结构的相似性。与理论研究结果相一致,我们的模拟实验结果也验证了这种距离度量的有效性。     

Nonparametric Trend Estimation for Periodic Autoregressive Time Series

A periodically stationary time series has seasonal variances. A local linear trend estimation is proposed to accommodate unequal variances. A comparison of this proposed estimator with the estimator commonly used for a stationary time series is provided. The optimal bandwidth selection for this new trend estimator is discussed.     

Estimating and Testing Nonlinear Local Dependence Between Two Time Series

ABSTRACT

The most common measure of dependence between two time series is the cross-correlation function. This measure gives a complete characterization of dependence for two linear and jointly Gaussian time series, but it often fails for nonlinear and non-Gaussian time series models, such as the ARCH-type models used in finance. The cross-correlation function is a global measure of dependence. In this article, we apply to bivariate time series the nonlinear local measure of dependence called local Gaussian correlation. It generally works well also for nonlinear models, and it can distinguish between positive and negative local dependence. We construct confidence intervals for the local Gaussian correlation and develop a test based on this measure of dependence. Asymptotic properties are derived for the parameter estimates, for the test functional and for a block bootstrap procedure. For both simulated and financial index data, we construct confidence intervals and we compare the proposed test with one based on the ordinary correlation and with one based on the Brownian distance correlation. Financial indexes are examined over a long time period and their local joint behavior, including tail behavior, is analyzed prior to, during and after the financial crisis. Supplementary material for this article is available online.     

A Scale-space Approach for Detecting Non-stationarities in Time Series

Abstract.  The presented method called Significant Non-stationarities, represents an exploratory tool for identifying significant changes in the mean, the variance, and the first-lag autocorrelation coefficient of a time series. The changes are detected on different time scales. The statistical inference for each scale is based on accurate approximation of the probability distribution, using test statistics being ratios of quadratic forms. No assumptions concerning the autocovariance function of the time series are made as the dependence structure is estimated non-parametrically. The results of the analyses are summarized in significance maps showing at which time points and on which time scales significant changes in the parameters occur. The performance of the given method is thoroughly studied by simulations in terms of observed significance level and power. Several examples, including a real temperature data set, are studied. The examples illustrate that it is important to carry out the analysis on several time horizons.     

First and Second Derivatives in Time Series Classification Using DTW

In our previous work, we developed a new distance function based on a derivative and showed that our algorithm is effective. In contrast to well-known measures from the literature, our approach considers the general shape of a time series rather than standard distance of function (value) comparison. The new distance was used in classification with the nearest neighbor rule. Now we improve on our previous technique by adding the second derivative. In order to provide a comprehensive comparison, we conducted a set of experiments, testing effectiveness on 47 time series datasets from a wide variety of application domains. Our experiments show that this new method provides a significantly more accurate classification on the examined datasets.     

Smoothing Time Series with Local Polynomial Regression on Time

Time series smoothers estimate the level of a time series at time t as its conditional expectation given present, past and future observations, with the smoothed value depending on the estimated time series model. Alternatively, local polynomial regressions on time can be used to estimate the level, with the implied smoothed value depending on the weight function and the bandwidth in the local linear least squares fit. In this article we compare the two smoothing approaches and describe their similarities. Through simulations, we assess the increase in the mean square error that results when approximating the estimated optimal time series smoother with the local regression estimate of the level.     

Bootstrapping moving average models

Summary In recent years, the bootstrap method has been extended to time series analysis where the observations are serially correlated. Contributions have focused on the autoregressive model producing alternative resampling procedures. In contrast, apart from some empirical applications, very little attention has been paid to the possibility of extending the use of the bootstrap method to pure moving average (MA) or mixed ARMA models. In this paper, we present a new bootstrap procedure which can be applied to assess the distributional properties of the moving average parameters estimates obtained by a least square approach. We discuss the methodology and the limits of its usage. Finally, the performance of the bootstrap approach is compared with that of the competing alternative given by the Monte Carlo simulation. Research partially supported by CNR and MURST.     

Parametric Inference in Stationary Time Series Models with Dependent Errors

This article is concerned with inference for the parameter vector in stationary time series models based on the frequency domain maximum likelihood estimator. The traditional method consistently estimates the asymptotic covariance matrix of the parameter estimator and usually assumes the independence of the innovation process. For dependent innovations, the asymptotic covariance matrix of the estimator depends on the fourth‐order cumulants of the unobserved innovation process, a consistent estimation of which is a difficult task. In this article, we propose a novel self‐normalization‐based approach to constructing a confidence region for the parameter vector in such models. The proposed procedure involves no smoothing parameter, and is widely applicable to a large class of long/short memory time series models with weakly dependent innovations. In simulation studies, we demonstrate favourable finite sample performance of our method in comparison with the traditional method and a residual block bootstrap approach.     

也谈由一般平均数时间数列计算序时平均数——兼与“由一般平均数计算序时平均数的方法释疑”一文商榷

本文指出了由—般平均数时间数列计算序时平均数在教科书上存在错误和“由一般平均数计算序时平均数的方法释疑”一文中的不足之处提出了—般平均数时间数列的序时平均数可以按照相对数时间数列计算序时平均数的方法计算,也可以根据平均指标基本公式计算。     

Comment on “Issues Involved With the Seasonal Adjustment of Economic Time Series” by William R. Bell and Steven C. Hiilmer

This article investigates the existence of multiple regimes in the U.S. economy during the 1923—1991 period. A technique known as regression tree analysis is applied to search for splits in the data, if any exist, rather than choosing a splitting point a priori as has been done in previous work. Using this technique, strong evidence for the existence of nonlinear behavior of U.S. output is found over this period. Monte Carlo results are presented to assess the significance of the regime changes that are found.     

Semiparametric Bayesian Inference for Time Series with Mixed Spectra

A Bayesian analysis is presented of a time series which is the sum of a stationary component with a smooth spectral density and a deterministic component consisting of a linear combination of a trend and periodic terms. The periodic terms may have known or unknown frequencies. The advantage of our approach is that different features of the data—such as the regression parameters, the spectral density, unknown frequencies and missing observations—are combined in a hierarchical Bayesian framework and estimated simultaneously. A Bayesian test to detect deterministic components in the data is also constructed. By using an asymptotic approximation to the likelihood, the computation is carried out efficiently using the Markov chain Monte Carlo method in O ( Mn ) operations, where n is the sample size and M is the number of iterations. We show empirically that our approach works well on real and simulated samples.     

Curve of Correlation for Time Series

The curve of correlation is a measure of local correlation between two random variables X and Y at the point X = x of the support of this variable. This article studies this local measure using the theory of time series for bivariate and univariate stationary stochastic process. We suggest local polynomial estimators for time series observing their consistency both theoretically and through simulations. For this, different sizes of series, bandwidths, and kernels, besides lags and models’ configurations were used. Applications have also been made using the daily returns of two financial series.     

Exact simulation of Gaussian Time Series from Nonparametric Spectral Estimates with Application to Bootstrapping

The circulant embedding method for generating statistically exact simulations of time series from certain Gaussian distributed stationary processes is attractive because of its advantage in computational speed over a competitive method based upon the modified Cholesky decomposition. We demonstrate that the circulant embedding method can be used to generate simulations from stationary processes whose spectral density functions are dictated by a number of popular nonparametric estimators, including all direct spectral estimators (a special case being the periodogram), certain lag window spectral estimators, all forms of Welch's overlapped segment averaging spectral estimator and all basic multitaper spectral estimators. One application for this technique is to generate time series for bootstrapping various statistics. When used with bootstrapping, our proposed technique avoids some – but not all – of the pitfalls of previously proposed frequency domain methods for simulating time series.     

Resampling Methods for Time Series Level Crossings

We present an application of subsampling and bootstrap methods for time series to determine the distribution of the estimator of zero crossings. The zero crossings method provides an alternative estimator of the lag-1 autocorrelation coefficient that is reducing the data storage requirements and is more robust with respect to outliers when compared to the classical estimator. The main results here are showing the consistency of subsampling, the consistency of moving block bootstrap, the consistency of non overlapping block bootstrap and the consistency of stationary bootstrap for this estimator. Theorems are formulated for Gaussian processes, elliptically symmetric processes and processes which are transformed Gaussian processes. Theoretical results are illustrated by simulations and practical data analysis. We have also shown that in practice the MBB method behaves better than the subsampling method.     

时间序列分析在经济预测中的应用

社会消费品零售总额是一项重要、敏感的政府统计。定期发布的消费品零售统计资料,常常引起国内外的强烈关注,间或还会引发一些疑义和争议。文章拟通过运用EXCEL及SAS软件建立季节分解模型和季节哑变量、ARIMA模型,对我国的社会消费零售总额的情况进行预测分析,从初步确定几个不同的模型中,把拟合效果最好的模型保留,并对模型的实用性进行了探讨。     

多维时间序列的条件独立性检验及其在股市相依关系中的应用

提出多维时间序列中各分量之间直接联系存在性的信息论检验方法,构造了条件互信息统计量检验分量间的条件独立性,统计量的显著性用置换检验决定.将提出的方法应用到国际股票市场,研究收益率序列相依关系,结果表明,此方法能有效检验各分量之间的直接联系和间接联系.     

Time Series Classification Based on Spectral Analysis

For time series data with obvious periodicity (e.g., electric motor systems and cardiac monitor) or vague periodicity (e.g., earthquake and explosion, speech, and stock data), frequency-based techniques using the spectral analysis can usually capture the features of the series. By this approach, we are able not only to reduce the data dimensions into frequency domain but also utilize these frequencies by general classification methods such as linear discriminant analysis (LDA) and k-nearest-neighbor (KNN) to classify the time series. This is a combination of two classical approaches. However, there is a difficulty in using LDA and KNN in frequency domain due to excessive dimensions of data. We overcome the obstacle by using Singular Value Decomposition to select essential frequencies. Two data sets are used to illustrate our approach. The classification error rates of our simple approach are comparable to those of several more complicated methods.     

Probabilistic Properties of Parametric Dual and Inverse Time Series Models Generated by ARMA Models

For the class of autoregressive-moving average (ARMA) processes, we examine the relationship between the dual and the inverse processes. It is demonstrated that the inverse process generated by a causal and invertible ARMA (p, q) process is a causal and invertible ARMA (q, p) model. Moreover, it is established that this representation is strong if and only if the generating process is Gaussian. More precisely, it is derived that the linear innovation process of the inverse process is an all-pass model. Some examples and applications to time reversibility are given to illustrate the obtained results.     

Some Properties and Generalizations of Non-negative Bayesian Time Series Models

We study the most basic Bayesian forecasting model for exponential family time series, the power steady model (PSM) of Smith, in terms of observable properties of one-step forecast distributions and sample paths. The PSM implies a constraint between location and spread of the forecast distribution. Including a scale parameter in the models does not always give an exact solution free of this problem, but it does suggest how to define related models free of the constraint. We define such a class of models which contains the PSM. We concentrate on the case where observations are non-negative. Probability theory and simulation show that under very mild conditions almost all sample paths of these models converge to some constant, making them unsuitable for modelling in many situations. The results apply more generally to non-negative models defined in terms of exponentially weighted moving averages. We use these and related results to motivate, define and apply very simple models based on directly specifying the forecast distributions.