Showing papers on "STAR model published in 1995"

Some Aspects of Asymptotic Theory with Applications to Time Series Models

Abstract: The primary purpose of this paper is to review a very few results on some basic elements of large sample theory in a restricted structural framework, as described in detail in the recent book by LeCam and Yang (1990, Asymptotics in Statistics: Some Basic Concepts. New York: Springer), and to illustrate how the asymptotic inference problems associated with a wide variety of time series regression models fit into such a structural framework. The models illustrated include many linear time series models, including cointegrated models and autoregressive models with unit roots that are of wide current interest. The general treatment also includes nonlinear models, including what have become known as ARCH models. The possibility of replacing the density of the error variables of such models by an estimate of it (adaptive estimation) based on the observations is also considered.Under the framework in which the asymptotic problems are treated, only the approximating structure of the likelihood ratios of the observations, together with auxiliary estimates of the parameters, will be required. Such approximating structures are available under quite general assumptions, such as that the Fisher information of the common density of the error variables is finite and nonsingular, and the more specific assumptions, such as Gaussianity, are not required. In addition, the construction and the form of inference procedures will not involve any additional complications in the non-Gaussian situations because the approximating quadratic structure actually will reduce the problems to the situations similar to those involved in the Gaussian cases.

Modeling Volatility Dynamics

Abstract: Good macroeconomic and financial theorists, like all good theorists, want to get the facts straight before theorizing; hence, the explosive growth in the methodology and application of time-series econometrics in the last twenty-five years. Many factors fueled that growth, ranging from important developments in related fields (see Box and Jenkins, 1970) to dissatisfaction with the “incredible identifying restrictions” associated with traditional macroeconometric models (Sims, 1980) and the associated recognition that many tasks of interest, such as forecasting, simply do not require a structural model (see Granger and Newbold, 1979). A short list of active subfields includes vector autoregressions, index and dynamic factor models, causality, integration and persistence, cointegration, seasonality, unobserved-components models, state-space representations and the Kalman filter, regime-switching models, nonlinear dynamics, and optimal nonlinear filtering. Any such list must also include models of volatility dynamics. Models of autoregressive conditional heteroskedasticity (ARCH), in particular, provide parsimonious approximations to volatility dynamics and have found wide use in macroeconomics and finance1. The family of ARCH models is the subject of this chapter.

Spatial Correlation: A Suggested Alternative to the Autoregressive Model

Abstract: Issues relating to spatially autocorrelated disturbance terms are often considered in regional econometric models.1 Although various models have been suggested to describe such spatial correlation, one of the most widely used models is a spatial autoregressive (AR) model which was originally suggested by Whittle (1954) and then extensively studied by Cliff and Ord (1973).2 In the model the regression disturbance vector is viewed as the sum of two parts. One of these parts involves the product of a spatial weighting matrix and a scalar parameter, say p; the other is a random vector whose elements are typically assumed to be independent and identically distributed (i.i.d.) with zero mean and finite variance. We will henceforth refer to this random vector as the innovation vector, so as to distinguish it from the disturbance vector.

Order Choice in Nonlinear Autoregressive Models

Abstract: An extensive literature has been devoted to the problem of order choice in autoregressive models. Most of alternative methods to hypothesis tests are based on the minimization of the Akaike Information Criterion (AIC) or on some of its variants. These methods have the main drawback to have to assume a parametric form for the autoregression function. The aim of this paper is to present a nonparametric approach that allows to estimate the autoregression order without limiting ourself to any restrictive parametric class of processes. Our technique is in the same spirit as AIC criterion, in the sense that it is based on the minimization of some prediction error. Both theoretical and computational aspects of this method are discussed in this paper.

Modelling nonlinearity in U.S. Gross national product 1889–1987

Abstract: This paper considers modelling the annual logarithmed per capita gross national product of the United States in 1889–1987. Some authors have suggested that the parameters of the process generating the data have changed over time but formal parameter constancy tests do not support this argument. The series turns out to be nonlinear and can be adequately characterized by an exponential smooth transition autoregressive model. For comparison, a detrended series is also considered, found nonlinear and modelled using a logistic smooth transition autoregressive model. The behaviour of the estimated models is discussed, and it is seen that nonlinearity is needed to describe the response of the process to exceptionally large exogenous shocks. The properties of the models are further investigated by forecasting several years ahead, and the forecasts are compared with those from other linear and nonlinear models.

Results on estimation and testing for a unit root in the nonstationary autoregressive moving‐average model

Abstract: . We review the limiting distribution theory for Gaussian estimation of the univariate autoregressive moving-average (ARMA) model in the presence of a unit root in the autoregressive (AR) operator, and present the asymptotic distribution of the associated likelihood ratio (LR) test statistic for testing for a unit root in the ARMA model. The finite sample properties of the LR statistic as well as other unit root test procedures for the ARMA model are examined through a limited simulation study. We conclude that, for practical empirical work that relies on standard computations, the LR test procedure generally performs better than other standard procedures in the presence of a substantial moving-average component in the ARMA model.

Learning chaotic dynamics by neural networks

Abstract: We show that neural networks can accurately learn the dynamical laws of chaotic time series from a limited number of iterates. Moreover, for short-term predictions they clearly outperform conventional methods, like, for instance, linear autoregressive models and a nonlinear simplex-like algorithm. We reconstruct the dynamics of computer-generated data corresponding to the logistic equation-which is known to have negligible autocorrelation-and the Lorenz map-which has significant autocorrelation-. Unlike previous claims in the literature, in both cases properly trained neural networks show better predictive skill than the autoregressive and simplex-like models. Finally, we discuss briefly applications of neural networks in the analysis of real-world time series.

Modelling of an Epidemiological Time Series by a Threshold Autoregressive Model

Abstract: In this paper we fit a self-exciting threshold autoregressive (SETAR) model, introduced by Tong, to a recent epidemiological time series of reported cases of Salmonella typhimurium in France. The procedure proposed by Tsay for fitting this class of model is briefly presented. The fitted 'full' model is compared with a simple autoregressive (AR) model. Finally, we compare the full model with the 'restricted' model discussed by Thanoon. Our results favour modelling by a SETAR process instead of an AR process. Thus, the time series of infections due to Salmonella typhimurium exhibits a type of non-linearity which can be accounted for by a threshold model. For parsimony and ease of interpretation of the model, the restricted SETAR model is finally preferred.

Dynamic spectral analysis based on an autoregressive model with time-varying coefficients

Abstract: A method of dynamic spectral analysis based on an autoregressive model with time-varying coefficients is presented. For parameter estimation a method simpler than the Kalman approach is introduced. The capability of the estimation procedure is where demonstrated by applying it to simulated data.

Autoregressive Modelling of Markov Chains

Abstract: The reduction of the number of parameters in high-order Markov chain already inspired several articles. In particular, Raftery (1985) proposed an autoregressive modelling which utilizes a same transition matrix for every lag. In this paper, we show that a model of the same type, but utilizing different matrices, gives best results and is not harder to estimate, even when the number of data is small.

Maximum likelihood estimation of a generalized star(p; 1 p ) model

Abstract: A STAR model is characterized by autoregressive terms lagged both in time and space. The model we call GSTAR presents also contemporaneous spatial correlation. Under the hypothesis of stationarity we derive conditional maximum likelihood estimators of the autoregressive parameters and a consistent estimator of their covariance matrix.

Nonlinear maximum likelihood estimation of autoregressive time series

Abstract: Describes an algorithm for finding the exact, nonlinear, maximum likelihood (ML) estimators for the parameters of an autoregressive time series The authors demonstrate that the ML normal equations can be written as an interdependent set of cubic and quadratic equations in the AR polynomial coefficients They present an algorithm that algebraically solves this set of nonlinear equations for low-order problems For high-order problems, the authors describe iterative algorithms for obtaining a ML solution

Three-Dimensional Surface Characterization by Two-Dimensional Autoregressive Models

Abstract: An attempt is made to characterize and synthesize engineered surfaces. The proposed method, based on two-dimensional difference equations and two-dimensional linear autoregressive models, is not only an analytical tool to characterize but also to generate/synthesize three-dimensional surfaces with desired properties. The developed method expresses important three-dimensional surface characteristics such as the autocorrelation or power spectrum density functions in terms of the two-dimensional autoregressive coefficients

Parameter estimation of time-varying autoregressive models using the Gibbs sampler

Abstract: A method is described for applying a Markov chain Monte Carlo method known as the Gibbs sampler to the problem of estimating the parameters of a flexible time-varying autoregressive (TVAR) model with time dependent coefficients that are stationary stochastic processes.

A law of large numbers for the bifurcating autoregressive process

Abstract: The bifurcating autoregressive model is of both practical interest, as in the analysis of cell lineage data in Huggins & Staudte [3], and of theoretical interest in that whilst each line of descent is a first order autoregressive process the added complications when considering all lines of descent in a tree of data require new analytical techniques. Here laws of large numbers are given for several quantities associated with the bifurcating autoregressive process which are required to derive the asymptotic properties of both robust and maximum likelihood estimators

Prediction and Non-Gaussian Autoregressive Stationary Sequences

Abstract: The object of this paper is to show that under certain auxiliary assumptions a stationary autoregressive sequence has a best predictor in mean square that is linear if and only if the sequence is minimum phase or is Gaussian when all moments are finite.

Bias Nonmonotonicity in Stochastic Difference Equations

Abstract: We show that the bias of estimated parameters in autoregressive models can increase as the sample size grows. This unusual result is due to the effect of the initial sample observations that are typically neglected in theoretical asymptotoc analysis, in spite of their empirical relevance. Implications for practical economic modelling are considered.

Testing for Nonlinearities in Economic and Financial Time Series

Abstract: Cyclical asymmetry has been recognised as a nonlinear phenomenon in numerous recent studies examining various economic and financial time series. If the nonlinear phenomena can be modelled by a nonlinear stochastic structure like the bilinear (BL), exponential autoregressive (EAR), smooth transition autoregressive (STAR), or self-exciting threshold autoregressive (SETAR) types, then we need tests to enable us to identify these various nonlinear models. In this paper we suggest modifications to the Tsay (1991) general test for identifying nonlinearities of the BL, EAR, and SETAR types as they occur in time series. Our testing procedure is simulated to determine its empirical properties.

3D autoregressive model with proper transformation property under rotation

Abstract: For pattern recognition and image understanding, it is important to take invariance and/or covariance of features extracted from given data under some transformation into consideration. This makes various problems in pattern recognition and image understanding clear and easy. In this article, we present two autoregressive models which have proper transformation properties under rotations: one is a 2D autoregressive model (2D AR model) which has invariance under any 2D rotations and the other is a 3D autoregressive model (3D AR model) which has covariance under any 3D rotations. Our 2D AR model is based on a matrix representation of complex number. It is shown that our 2D AR model is equivalent to Otsu's complex AR model. On the other hand, our 3D autoregressive model is based on the representation theory of rotation group such as the fundamental representation of Lie algebra of SU(2)(Special Unitary group in 2D which includes rotation group), which is called Pauli's matrices.

Multivariate autoregressive time semes modeling: one scalar autoregressive model at-A-time

Abstract: Multivariate (or interchangeably multichannel) autoregressive (MCAR) modeling of stationary and nonstationary time series data is achieved doing things one channel at-a-time using only scalar computations on instantaneous data. The one channel at-a-time modeling is achieved as an instantaneous response multichannel autoregressive model with orthogonal innovations variance. Conventional MCAR models are expressible as linear algebraic transformations of the instantaneous response orthogonal innovations models. By modeling multichannel time series one channel at-a-time, the problems of modeling multichannel time series are reduced to problems in the modeling of scalar autoregressive time series. The three longstanding time series modeling problems of achieving a relatively parsimonious MCAR representation, of multichannel stationary time series spectral estimation and of the modeling of nonstationary covariance time series are addressed using this paradigm.

Mathematical modelling of nonlinear behaviour of seismicity

Abstract: In this paper, the nonlinear behaviour of seismic activities has been studied by means of the threshold autoregressive model and the exponential autoregressive model. The contents are as follows: (1) The theories and modelling methods of this two models have been studied. (2) One kind of explanation for the seismic cycle and order structure are given by means of the threshold autoregressive model. (3) According to the exponential autoregressive model, an inherent structure of the magnitude series are discussed, the different relations between magnitude and frequency in active period and quiet period are also explained in this paper.