Comparison of unit root tests for time series with level shifts

Unit root tests are considered for time series which have a level shift at a known point in time. The shift can have a very general nonlinear form, and additional deterministic mean and trend terms are allowed for. Prior to the tests, the deterministic parts and other nuisance parameters of the data generation process are estimated in a first step. Then, the series are adjusted for these terms and unit root tests of the Dickey–Fuller type are applied to the adjusted series. The properties of previously suggested tests of this sort are analysed and modifications are proposed which take into account estimation errors in the nuisance parameters. An important result is that estimation under the null hypothesis is preferable to estimation under local alternatives. This contrasts with results obtained by other authors for time series without level shifts.     

Fault Prediction Based on Dynamic Model and Grey Time Series Model in Chemical Processes

This paper combines grey model with time series model and then dynamic model for rapid and in-depth fault prediction in chemical processes. Two combination methods are proposed. In one method, historical data is in-troduced into the grey time series model to predict future trend of measurement values in chemical process. These predicted measurements are then used in the dynamic model to retrieve the change of fault parameters by model based diagnosis algorithm. In another method, historical data is introduced directly into the dynamic model to re-trieve historical fault parameters by model based diagnosis algorithm. These parameters are then predicted by the grey time series model. The two methods are applied to a gravity tank example. The case study demonstrates that the first method is more accurate for fault prediction.     

Bootstrap Approximation to Prediction MSE for State–Space Models with Estimated Parameters

Abstract. We propose simple parametric and nonparametric bootstrap methods for estimating the prediction mean square error (PMSE) of state vector predictors that use estimated model parameters. As is well known, substituting the model parameters by their estimates in the theoretical PMSE expression that assumes known parameter values results in underestimation of the true PMSE. The parametric method consists of generating parametrically a large number of bootstrap series from the model fitted to the original series, re‐estimating the model parameters for each series using the same method as used for the original series and then estimating the separate components of the PMSE. The nonparametric method generates the series by bootstrapping the standardized innovations estimated for the original series. The bootstrap methods are compared with other methods considered in the literature in a simulation study that also examines the robustness of the various methods to non‐normality of the model error terms. Application of the bootstrap method to a model fitted to employment ratios in the USA that contains 18 unknown parameters, estimated by a three‐step procedure yields unbiased PMSE estimators.     

NONLINEAR TIME SERIES REGRESSION FOR A CLASS OF AMPLITUDE MODULATED CONSINUSOIDS

Abstract. A central limit theorem is proved for estimates of the unknown parameters in a time series which is a sum of amplitude modulated consinusoids observed subject to error. The amplitude function depends upon unknown parameters as well as the length of the series. The frequency for each cosinusoid is also assumed to be unknown. Estimates and standard errors are obtained through nonlinear least squares in the frequency domain.     

Bend strength variation of ceramics in serial fabrication

We have performed a statistical evaluation of more than 10 000 experimental strength values of test pieces from a serial production of alumina products where in each production series 12 samples were broken. The Weibull parameters were obtained by three different methods: linear regression, maximum-likelihood and moments’ method. The methods give different values of parameters even for this large number of data, but all the correlation coefficients are high and similar. Evaluating Weibull parameters for each testing group gives expected systematically different values of particular Weibull modulus as compared to the whole population of collected data. However, when the order of experimental data is mixed, the average Weibull modulus is lower than that corresponding to time ordered data. This difference can be served as an indication for variation of products’ quality from series to series.     

A TEST FOR LINEARITY OF STATIONARY TIME SERIES

Abstract. A standard assumption that is often made in time series analysis is that the series conforms to a linear model. The object of this paper is to describe statistical tests for testing this assumption. The tests are constructed from the bispectral density function, and depend on the application of Hotelling T ². These tests are illustrated with two real time series and four simulated time series. Some guidelines about the choice of the parameters are also included.     

On the limit theory of mixed to unity VARs: Panel setting with weakly dependent errors

In this article, we re-visit a recent idea of Phillips and Lee (2015. Econometric Reviews 34: 1035 - 1056). They examine an empirically relevant situation when two time series exhibit different degrees of non-stationarity and one need to learn whether their persistence properties are the same. By bridging the asymptotic theory of local to unity and mildly explosive processes, they construct a Wald test for the commonality of the long-run behavior of the series. However, inference is complicated by the fact that their statistic does not converge in distribution under the null and diverges under the alternative. This is true if the parameters of the data generating process are known and a re-normalizing function can be constructed. If the parameters are unknown, which will be the case in practice, the test statistic may be divergent even under the null. We solve this problem by converting the original setting of vector time series into a panel setting with N individual vector series. We show that the proposed panel Wald test statistics converge to chi-squared distribution which is free of nuisance parameters under the null hypothesis of common local to unity behavior. The result is an extreme example of simplified asymptotics brought about by panel data.     

Experimental study of helium leakage parameters in flexible composite

The helium leakage characterizations of flexible composites are experimentally studied in this article. Three data processing techniques (Defining method, Fourier series method, and Fast series method) are used to evaluate the helium leakage parameters including the diffusion coefficient D, the solubility S, and the permeant rate k. The chamber pressure variation for helium permeation in flexible composite is measured by the differential pressure method. The results indicate that Fast series method is an effective technology in extracting the helium leakage parameters for flexible composites. © 2010 Wiley Periodicals, Inc. J Appl Polym Sci, 2010     

Velocity Uniformity of Microchannels in a Laminated‐Sheet Structure Under Parallel and Series Methods

An integral microchannel device is generally formed by lamination of multiple microchannel sheets. Uniform velocity distribution among the microchannels in each sheet shows a large influence on the device's performance. The effects of microchannel and manifold structure on the velocity uniformity in each sheet with different structural parameters under parallel and series laminated structures are investigated. For the laminated‐sheet structure where each sheet has the same structural parameters, a more uniform velocity distribution exists as compared to where the structural parameters are different. There exists a direct correlation between the velocity values in each sheet and structural parameters, whereas an inverse correlation is found under series laminated structure.     

基于时间序列的多岛遗传算法在水泥窑中的应用

研究时间序列在水泥回转窑中的建模问题,水泥回转窑熟料的煅烧是一个涉及物理、化学反应的复杂多变量、多扰动非线性过程.针对最小二乘在多参数高阶时间序列模型中难以寻找到较好的系数组,从而不能建立较精确的时间序列模型的情况,采用传统遗传算法也很难寻找出最优参数组,当利用多岛遗传算法对其进行参数寻优时,只要适当地设置算法里的一些关键参数即能得出满意的组解.实验结果表明：经过多岛遗传算法寻优得出的参数建立的时间序列模型,其拟合度满足模型曲线的精度要求.     

PROFILE SUMMARIES FOR ARIMA TIME SERIES MODEL PARAMETERS

Abstract. Autoregressive intergrated moving average (ARIMA) times series models are nonlinear in the parameters and so summarizing the inferential results for such models can be difficult. A common approach is to present parameter joint and marginal inference regions based on the linear approximation, and although such approximate regions are easy to calculate, it is not known generally whether they approximate the true regions adequately. In this paper we present exact approaches for summarizing the inferential results for time series model parameters, called profile t and profile trace plots, which are based on the work of Bates and Watts. Calculations for the profile plots are simple and can be used to determine exact regions, and so can be used to assess the accuracy of linear approximation regions. In addition to developing the profile plots for time series models, the main finding of the paper is that, for ARIMA model parameters, linear approximation regions are very satisfactory except when a parameter estimate is within about two standard errors of the stationarity or invertibility region boundary.     

ESTIMATION OF PERIODICALLY VARYING MEANS AND STANDARD DEVIATIONS IN TIME SERIES DATA

Abstract. Models which account for seasonal changes in mean and standard deviation in time series which are serially correlated are discussed. Full Gaussian maximum likelihood estimation of the parameters specifying the mean function, of the parameters specifying the standard deviation function, and of the parameters specifying the stationary serial correlation structure is discussed and the asymptotic distribution derived without the assumption of Gaussian data. A commonly used method which estimates the three components separately, and uses Fourier series parameterisation, is also reviewed and the asymptotic distribution for these estimates is also derived. Both of these have asymptotic covariances which depend upon third and fourth cumulants of the underlying distribution. When these vanish (e.g., Gaussian case) then a simple revision of the inefficient estimates yields estimates which are asymptotically equivalent to the Gaussian estimates. An application to a series of monthly Atlantic Ocean Sea surface temperatures is given to illustrate this last method.     

Gaussian Semi-parametric Estimation of Fractional Cointegration

Abstract. We analyse consistent estimation of the memory parameters of a nonstationary fractionally cointegrated vector time series. Assuming that the cointegrating relationship has substantially less memory than the observed series, we show that a multi-variate Gaussian semi-parametric estimate, based on initial consistent estimates and possibly tapered observations, is asymptotically normal. The estimates of the memory parameters can rely either on original (for stationary errors) or on differenced residuals (for nonstationary errors) assuming only a convergence rate for a preliminary slope estimate. If this rate is fast enough, semi-parametric memory estimates are not affected by the use of residuals and retain the same asymptotic distribution as if the true cointegrating relationship were known. Only local conditions on the spectral densities around zero frequency for linear processes are assumed. We concentrate on a bivariate system but discuss multi-variate generalizations and show the performance of the estimates with simulated and real data.     

Transport properties of chain molecules and the theorem of corresponding states

Prigogine's corresponding states approach to equilibrium properties of pure r-mers is extended to include transport properties. For r-mer molecules, besides the alike element pair potential parameters, additional quantities are required to characterize the geometrical structure of the molecule and thereby describe more adequately pair potential interactions. As an example, the theory is applied to the homologous series of liquid n-alkanes. The characteristic parameters for each member of the homologous series are determined using pure component viscosity data at one atmosphere and pure component denstiy data at one temperature and atmospheric pressure. The technique correlates pure component r-mer transport properties (viscosity — 981 points, thermal conductivity — 216 points) over a wide range of temperature and pressure within experimental error. Using the present technique, available self-diffusion coefficients are also correlated.     

DEFINITIONS AND REPRESENTATIONS OF MULTIVARIATE LONG‐RANGE DEPENDENT TIME SERIES

The notion of multivariate long‐range dependence is reexamined here from the perspectives of time and spectral domains. The role of the so‐called phase parameters is clarified and stressed throughout. In particular, examples of causal (one‐sided) representations of multivariate long‐range dependent time series with general‐phase parameters are constructed. A multivariate extension of the autoregressive fractionally integrated moving‐average series is introduced with explicit formulas for its autocovariance function.     

Poisson QMLE of Count Time Series Models

Regularity conditions are given for the consistency of the Poisson quasi‐maximum likelihood estimator of the conditional mean parameter of a count time series model. The asymptotic distribution of the estimator is studied when the parameter belongs to the interior of the parameter space and when it lies at the boundary. Tests for the significance of the parameters and for constant conditional mean are deduced. Applications to specific integer‐valued autoregressive (INAR) and integer‐valued generalized autoregressive conditional heteroscedasticity (INGARCH) models are considered. Numerical illustrations, Monte Carlo simulations and real data series are provided.     

WALSH-FOURIER ANALYSIS OF DISCRETE-VALUED TIME SERIES

Abstract. An approach to the analyses of discrete-valued time series is discussed. The analyses are accomplished in the spectral domain using the Walsh-Fourier transform which is based on Walsh functions. This approach will enable an investigator of discrete systems to analyse the data in terms of square waveforms and sequency rather than sine waves and frequency.
We develop a general signal-plus-noise type model for discrete-valued time series in which Walsh-Fourier spectral analysis is of interest. We consider the problems of detecting whether a common signal exists in repeated measures on discrete-valued time series and in discrete-valued processes collected in an experimental design. We show that these models may depend on unknown regression parameters and we develop consistent estimates of these parameters based on the finite Walsh-Fourier transform. Applications to certain Markov models are given; however, the methods presented also apply to non-Markov cases.     

MALDI for the estimation of viscosity parameters. A modified method which applies also to polycondensates

Recently, a method based on MALDI was described for the estimation of the parameters which characterize the viscosity of a polymer, namely the Mark-Houwink-Sakurada (MHS) parameters. The method makes use of samples with a narrow molar mass distribution and therefore, for polycondensates, it may become unpractical. A modification is proposed here, based on the universal calibration concept and on the coupling of size exclusion chromatography and MALDI. The new method is applied to two copolymers with units of styrene and maleic-anhydride and to a series of polymers and copolymers obtained by condensation. The MHS parameters are measured and they are compared with the values predicted by the method of adding the contributions coming from each functional group present in the chain, which is able to predict MHS parameters from first principles.     

A KALMAN FILTER APPROACH TO THE FORECASTING OF MONTHLY TIME SERIES AFFECTED BY Morris Festivals

Abstract. Many economic time series are affected by the moving dates of festivals. In this paper a moving festival effect is defined and incorporated into a dynamic linear model which specifies how the parameters of several unobservable components of a time series evolve stochastically in time. The merits of this approach in comparison to other approaches are discussed and demonstrated using empirical data of three selected time series.     

Bayesian analysis of multivariate Gaussian hidden Markov models with an unknown number of regimes

Multivariate Gaussian hidden Markov models with an unknown number of regimes are introduced here in the Bayesian setting and new efficient reversible jump Markov chain Monte Carlo algorithms for estimating both the dimension and the unknown parameters of the model are presented. Hidden Markov models are an extension of mixture models that can be applied to time series so as to classify the observations in a small number of groups, to understand when change points occur in the dynamics of the series and to model data heterogeneity through the switching among subseries with different means and covariance matrices. These aims can be achieved by assuming that the observed phenomenon is driven by a latent, or hidden, Markov chain. The methodology is illustrated through two different examples of multivariate time series.