Publisher Summary The chapter discusses the development of a rather complete inferential theory for ARMAX models. The first problem in the development is the coordinatization of spaces of such structures. Coordinates are needed both for computations and because a central limit theorem must be expressed in terms of them. One such coordinatization would follow from the use of the coefficient matrices in g, h, j in the scalar m.f.d., but there are many others. The chapter highlights the algebraic and topological description of ARMAX systems. In this connection, control engineers have played a premier part. The chapter focuses on the asymptotic properties of maximum likelihood (ML) estimators. The ML estimator is obtained by optimizing this likelihood. The chapter explains the asymptotic properties of these estimators without assuming the data to be Gaussian and also discusses the basis of the assumptions that appear to be minimal.

The statistical theory of linear systems

Under general conditions strong consistency of certain estimates of the maximum lags of an autoregressive moving average process is established. A theorem on weak consistency is also proved and in certain cases where consistency does not hold the probability of over-estimation of a maximum lag is evaluated.

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The Estimation of the Order of an ARMA Process

A linear time-series model is considered to be one for which a stationary time series, which is purely non-deterministic, has the best linear predictor equal to the best predictor. A general inferential theory is constructed for such models and various estimation procedures are shown to be equivalent. The treatment is considerably more general than previous treatments. The case where the series has mean which is a linear function of very general kinds of regressor variables is also discussed and a rather general form of central limit theorem for regression is proved. The central limit results depend upon forms of the central limit theorem for martingales.

The asymptotic theory of linear time series models

Let Yt satisfy the stochastic difference equation for t = 1,2,…, where et are independent and identically distributed random variables with mean zero and variance σ2 and the initial conditions (Y−p+1,…, Y0) are fixed constants. It is assumed that the process is invertible and that the true, but unknown, roots m1,m2,…,mp of satisfy the hypothesis Hd: m1 = … = md = 1 and |mj| < 1 for j = d + 1,…,p. We present a reparameterization of the model for Yt that is convenient for testing the hypothesis Hd. We consider the asymptotic properties of (i) a likelihood ratio type “F-statistic” for testing the hypothesis Hd, (ii) a likelihood ratio type t-statistic for testing the hypothesis Hd against the alternative Hd−1. Using these asymptotic results, we obtain two sequential testing procedures that are asymptotically consistent.

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Testing for Unit Roots in Time Series Data

This paper presents proofs of the strong law of large numbers and the central limit theorem for estimators of the parameters in quite general finite-parameter linear models for vector time series. The estimators are derived from a Gaussian likelihood (although Gaussianity is not assumed) and certain spectral approximations to this. An important example of finite-parameter models for multiple time series is the class of autoregressive moving-average (ARMA) models and a general treatment is given for this case. This includes a discussion of the problems associated with identification in such models. LINEAR PROCESSES; VECTOR ARMA MODELS; IDENTIFICATION; LIMIT THEOREMS;

Vector linear time series models

In this paper it is shown how martingale theorems can be used to appreciably widen the scope of classical inferential results concerning autocorrelations in time series analysis. The object of study is a process which is basically the second-order stationary purely non-deterministic process and contains, in particular, the mixed autoregressive and moving average process. We obtain a strong law and a central limit theorem for the autocorrelations of this process under very general conditions. These results show in particular that, subject to mild regularity conditions, the classical theory of inference for the process in question goes through if the best linear predictor is the best predictor (both in the least squares sense).

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E. J. Hannan

Papers

On Limit Theorems for Quadratic Functions of Discrete Time Series