Autoregressive Model Coefficients Research Articles

BIAS IN THE ESTIMATION OF MULTIVARIATE AUTOREGRESSIONS

summaryA derivation of the bias of the least squares estimate of the coefficients of a multivariate autoregressive model is presented. The explicit expression obtained is of use when, in the case of small samples, one needs to make a bias correction prior to bootstrapping.

Conditional Heteroscedastic Time Series Models

Abstract Under the traditional linear time series or regression setting, the conditional variance of one-step-ahead prediction is time invariant. Experience in conjunction with data analysis, however, suggests that the variability of a process might well depend on the available information. This reality has motivated extensive research to relax the constant variance assumption imposed by the traditional linear time series model, and several classes of generalized parametric models designed specifically for handling nonhomogeneity of a process have been proposed recently. In particular, the random coefficient autoregressive (RCA) models were widely investigated by time series analysts and the autoregressive conditional heteroscedastic (ARCH) models were investigated by econometricians. The interesting fact is that the ARCH processes can be regarded as special cases of the RCA model. In this article, I first give the relationship between these two types of models and show that the special feature of these t...

The mixing property of bilinear and generalised random coefficient autoregressive models

The paper gives sufficient conditions for the absolute regularity of bilinear models. Our approach is based on their Markovian representation. The above property is a direct consequence of the geometric ergodicity of the Markovian process in this representation. The latter process belongs to what we call the generalised random coefficients autoregressive models. Conditions for the geometric ergodicity and also for the existence of moments for this model are given. Our results generalise that of Feigin and Tweedie.

The Box-Jenkins approach to random coefficient autoregressive modelling

Recent time series research has been directed towards the relaxation of the assumption that time series models have constant coefficients. One class of models to emerge as a result of this has been that of random coefficient autoregressive models. This paper demonstrates how the Box-Jenkins three-step approach of model specification, estimation and diagnostic checking may be applied to this class of models.

The Box-Jenkins approach to random coefficient autoregressive modelling

Recent time series research has been directed towards the relaxation of the assumption that time series models have constant coefficients. One class of models to emerge as a result of this has been that of random coefficient autoregressive models. This paper demonstrates how the Box-Jenkins three-step approach of model specification, estimation and diagnostic checking may be applied to this class of models.

Book Reviews

Book reviewed in this article: Random Coefficient Autoregressive Models: An Introduction.: By Des F. Nicholls & Barry G. Quinn. Neyman‐From Life. By Constance Reid. Tests for Preference. By J. J. Dik. Higher Order Asymptotics for Simple Linear Rank Statistics. Mathematical Centre Tracts 151. Reliability in the Acquisitions Processes. By Douglas J. De Priest and Robert L. Launer (Eds).

RANDOM COEFFICIENT AUTOREGRESSIVE PROCESSES:A MARKOV CHAIN ANALYSIS OF STATIONARITY AND FINITENESS OF MOMENTS

Abstract. Simple yet practically efficient conditions for the ergodicity of a Markov chain on a general state space have recently been developed. We illustrate their application to non‐linear time series models and, in particular, to random coefficient autoregressive models.As well as ensuring the existence of a unique stationary distribution, geometric rates of convergence to stationarity are ensured. Moreover, sufficient conditions for the existence and convergence of moments can be determined by a closely related method. The latter conditions, in particular, are new.

Testing for Unit Roots in Seasonal Time Series

Abstract Regression estimators of coefficients in seasonal autoregressive models are described. The percentiles of the distributions for time series that have unit roots at the seasonal lag are computed by Monte Carlo integration for finite samples and by analytic techniques and Monte Carlo integration for the limit case. The tabled distributions may be used to test the hypothesis that a time series has a seasonal unit root.

Book Reviews

Book Reviews

Comparison of least-squares and stochastic gradient lattice predictor algorithms using two performance criteria

The least-squares (LS) and stochastic gradient (SG) lattice prediction algorithms are compared using two different performance criteria. These are a) output mean squared error and b) the accuracy of the autoregressive, estimate obtained from the mean values of the lattice coefficients, assuming a stationayinput. It is found that the second performance criterion is more sensitive than the first. This spectral performance criterion is a measure of the accuracy of the estimatcd autoregressive model coefficients. Bias in the LS and SG coefficient estimates can cause significant deviation of the asymptotic estimates from the actual input spectrum: The similarly between the LS and SG lattice algorithms enables comparative simulations with analogous initial conditions and exponential weighting constants. For both types of comparisons, the LS algorithm offers a modest performance improvement over the SG algorithms simulated. This improvement is more noticeable when the input is highly correlated. It is also found that slight changes in the SG lattice algorithm may significantly affect its performance.

The estimation of multivariate random coefficient autoregressive models

Using a two stage regression procedure estimates of the unknown parameters of a class of multivariate random coefficient autoregressive models are obtained. The estimates are shown, under fairly general conditions, to be strongly consistent and to have a distribution which converges to that of a normally distributed random vector.

THE ESTIMATION OF RANDOM COEFFICIENT AUTOGRESSIVE MODELS. II

Abstract. In Nicholls and Quinn (1980) a procedure was proposed for the determination of strongly consistent estimates of random coefficient autoregressive models. These estimates are used here as starting values in a Newton‐Raphson algorithm which is employed to obtain the maximum likelihood estimates of a class of random coefficient autoregressions. The maximum likelihood estimates are shown to be strongly consistent and to satisfy a central limit theorem. The problem of testing for the randomness of the coefficients is also briefly discussed. The results of a number of simulations are reported which illustrate the theoretical results obtained.

THE ESTIMATION OF RANDOM COEFFICIENT AUTOREGRESSIVE MODELS. I

Abstract. This paper is concerned with autoregressive models in which the coefficients are assumed to be not constant but subject to random perturbations so that we are considering a class of random coefficient autoregressive models. By means of a two stage regression procedure estimates of the unknown parameters of these models are obtained. The estimates are shown to be strongly consistent and to satisfy a central limit theorem. A number of Monte Carlo experiments was carried out to illustrate the estimation procedure and their results are reported.

Relative efficiencies of some simple Bayes estimators of coefficients in dynamic models — I

In this paper we consider some approximations to Bayes estimators of coefficients in simple autoregressive models and give an example of a Monte Carlo experiment where these approximate Bayes estimators yield a substantial improvement over the usual sampling theory or quasi-Bayesian estimators. The practical situation is represented by the case where the coefficient vector is known to lie in or on a hypersphere of radius r with center at 0. We show that arbitrariness in the choice of the value of r is often not catastrophic if r is sufficiently large, but finite.

Linear system identification by autoregressive model decomposition

Fast, simple and parallel identification algorithms are highly desirable in many applications. However, the assumptions which make the algorithm fast and simple also make the algorithm sensitive to deviations from these assumptions. In view of these considerations, this paper develops a sequential algorithm for the identification of discrete-time linear systems. The identification algorithm is based on decomposition of the autoregressive model. This decomposition approach identifies autoregressive model coefficients by minimizing the squared error performance index by use of the multilevel hierarchical decomposition procedure and by use of the stochastic approximation algorithm. Computer simulations illustrate the performance of the identification algorithm and comparison of the results with those obtained from the undecomposed stochastic approximation algorithm.