Stable Autoregressive Models Research Articles

Forecasting of symmetric [formula omitted]stable autoregressive models by time series approach supported by artificial neural networks

Recent research activities in forecasting suggest that artificial neural networks can be a promising alternative to the traditional linear models. However, no single model, either linear or nonlinear is capable of obtaining the forecasts accurately. In this paper, a hybrid methodology that combines symmetric α-stable autoregressive time series and artificial neural networks is proposed. The methodology is validated through Monte-Carlo simulations. Moreover, the new method is used to model real empirical data thus showing the usefulness of heavy-tailed models supported by artificial neural networks in statistical modeling.

Tempered stable autoregressive models

In this article, we introduce and study a one sided tempered stable first order autoregressive model called TAR(1). Under the assumption of stationarity of the model, the marginal probability density function of the error term is found. It is shown that the distribution of the error term is infinitely divisible. Parameter estimation of the introduced TAR(1) process is done by adopting the conditional least square and method of moments based approach and the performance of the proposed methods are evaluated on simulated data. Also, we study an autoregressive model of order one with tempered stable innovations. Using appropriate test statistic it is shown that the model fits very well on real and simulated data. Our models generalize the inverse Gaussian and one-sided stable autoregressive models existing in the literature.

Estimation and Inference in Autoregressive Models with Trending Innovation Variance

This paper develops the asymptotic theory for stable autoregressive models in which the noise grows in a polynomial-like fashion. It is shown that the asymptotic distribution of the OLS estimator of the coefficient vector is multivariate normal with a covariance matrix that depends on the order, k, of the growth. A consistent estimator of k is proposed, which delivers heteroscedasticity-robust test statistics. The opposite case of variance decline is studied as well. It is demonstrated that by means of a simple data transformation producing the time reversed image of the original series, the problem of variance decrease can be reformulated in terms of that of polynomial-like growth. Simulation evidence suggests that the new procedures work quite well in small samples.

Developing stable autoregressive models of terrain topology

It is desirable to evaluate vehicle models over a wide range of terrain types, but it is computationally impractical to simulate over measured terrain. A method to characterise terrain topology is developed in this work to group terrain into meaningful sets with similar physical characteristics. Specifically, measured terrain profiles are considered realisations of an underlying stochastic process; an autoregressive model provides the mathematical framework to describe this process. The autocorrelation of the spatial derivative of the terrain profile is examined to determine the form of the model. The required order for the model is determined from the partial autocorrelation of the spatial derivative of the profile. The stability of the model is evaluated and enforced by transforming the AR difference equation into an Infinite Impulse Response filter representation. Finally, the method is applied to a set of USA highway data and an optimal model order is determined for this application.

Adaptive estimation of autoregressive models with time-varying variances

Stable autoregressive models are considered with martingale differences errors scaled by an unknown nonparametric time-varying function generating heterogeneity. An important special case involves structural change in the error variance, but in most practical cases the pattern of variance change over time is unknown and may involve shifts at unknown discrete points in time, continuous evolution or combinations of the two. This paper develops kernel-based estimators of the residual variances and associated adaptive least squares (ALS) estimators of the autoregressive coefficients. Simulations show that efficiency gains are achieved by the adaptive procedure.

Miscellanea. Random coefficient autoregressive models for longitudinal data

SUMMARY Stable autoregressive models with exogenous covariates can be used to mimic nonstationary, stabilising growth profiles in longitudinal data. If different individuals have very different trajectories, the autoregressive parameters can be interpreted as random variables varying from one individual to the next. If all random effects are assumed Gaussian, the model will be quite easy to estimate although the marginal distributions of the observations will no longer be Gaussian. However, this kind of model can only be used when the observations are equally spaced.

Principes de déviations modérées pour des modèles autorégressifs d'ordre p

We establish Moderate deviations principles (MDP) for some stable autoregressive models of order p: first for continuous unbounded additive functionnals of this process, second for the estimation error of the regression function (least-squares estimator in the stable linear case, kernel estimator in the Lipschitz non-linear case).

Propriétés asymptotiques des estimateurs du gradient et du gradient moyennisé pour les modèles autorégressifs stables

We consider a stable autoregressive model of order p. We establish, for the gradient estimator, a central limit theorem, a law of the iterated logarithm and an almost sure central limit theorem; a quadratic strong law of large numbers is proved for the estimation and for the associated prediction. Then we give a new estimator, the averaged gradient estimator, which is computed more easily than the classical least squares estimator, and enjoys the same optimal asymptotic properties as the latter.

Uniqueness of a two-step predictor based spectral estimator that generalizes the maximum entropy concept

Given a finite set of autocorrelations, it is well known that maximization of the entropy functional subject to this data leads to a stable autoregressive model. Since maximization of the entropy functional is equivalent to maximization of the minimum mean square error associated with one-step predictors, the problem of obtaining admissible extensions that maximize the k-step minimum-mean-square prediction error subject to the given autocorrelations has been shown to result in stable autoregressive moving-average (ARMA) extensions. The uniqueness of this true generalization of the maximum-entropy extension is proved here by a constructive procedure in the case of two-step predictors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>