Stationary ARMA Processes Research Articles

Testing the order of a model using locally conic parametrization: population mixtures and stationary ARMA processes

In this paper, we address the problem of testing hypotheses using the likelihood ratio test statistic in nonidentifiable models, with application to model selection in situations where the parametrization for the larger model leads to nonidentifiability in the smaller model. We give two major applications: the case where the number of populations has to be tested in a mixture and the case of stationary ARMA$(p, q)$ processes where the order $(p, q)$ has to be tested. We give the asymptotic distribution for the likelihood ratio test statistic when testing the order of the model. In the case of order selection for ARMAs, the asymptotic distribution is invariant with respect to the parameters generating the process. A locally conic parametrization is a key tool in deriving the limiting distributions; it allows one to discover the deep similarity between the two problems.

On Rissanen's predictive stochastic complexity for stationary ARMA processes

Abstract The aim of this paper is to show that for a wide class of stationary ARMA processes Rissanen's predictive stochastic complexity is asymptotically equal to the lower bound provided by the Rissanen-Shannon inequality almost surely. An analogous theorem is proved for a more easily computable version of the predictive stochastic complexity based on the recursive estimation of the ARMA parameters.

Simplified conditions for noncausality between vectors in multivariate ARMA models

This article derives necessary and sufficient conditions for noncausality between two vectors of variables in stationary invertible ARMA processes. Earlier conditions proposed by Boudjellaba, Dufour, and Roy (1992a) are shown to hold under weaker regularity assumptions and then generalized to cover the important case where the two vectors do not necessarily embody all the variables considered in the analysis. The conditions so obtained can be considerably simpler and easier to implement than earlier ones. Testing of the conditions derived is also discussed and the results are applied to a model of Canadian money, income, and interest rates.

Mixing properties of ARMA processes

In this paper, we show that stationary vector ARMA processes are geometrically completely regular, and hence geometrically strong mixing, provided the innovations have absolutely continuous distribution with respect to Lebesgue measure.

Unit Canonical Correlations between Future and Past

Stationary vector ARMA processes $x(t), t = 0, \pm 1, \pm 2, \cdots$, of $n$ components are considered that are of full rank, and the situation where there are linear functions of the future $x(t), t > 0$, and the past $x(t), t \leq 0$ (more properly the present and the past) that have unit correlation. It is shown that the number of linearly independent such pairs (i.e., the number of unit canonical correlations between future and past) is the number of zeros of the determinant of the transfer functions, from innovations to outputs, that lie on the unit circle, counting these with their multiplicities.

On Adaptive Estimation in Stationary ARMA Processes

We consider the estimation problem for the parameter $\vartheta_0$ of a stationary ARMA $(p, q)$ process, with independent and identically, but not necessary normally distributed errors. First we prove local asymptotic normality (LAN) for this model. Then we construct locally asymptotically minimax (LAM) estimators, which asymptotically achieve the smallest possible covariance matrix. Utilizing these, we finally obtain strongly adaptive estimators, by using usual kernel estimators for the score function $\dot{\varphi} = -f'/2 f$, where $f$ denotes the density of the error distribution. These estimates turn out to be asymptotically optimal in the LAM sense for a wide class of symmetric densities $f$.

Computation of the autocovariances of stationary arma processes

Autoregressive-moving average (ARMA) models are often used for the purpose of forecasting a time series. As an aide to chosing a model, use is made of the autocorrelation function which is estimated from the data. If the only interest in the model is for forecasting purposes, then it is not necessary to compute the autocorrelation function associated with the chosen model. For this reason, a method for computation of the autocorrelation function is not usually included in the software used for identifying ARMA models. However, there are applications of ARMA models where it is important to compute the autocovariance function. This paper contains an algorithm and a listing of a FORTRAN program which computes the autocovariance directly from the solution to the difference equations which govern its behavior.