E. J. Hannan

Time series analysis

Course description: This course introduces statistical methods appropriate when sample observations are not independent, but rather, are logically ordered. Coverage will begin with the traditional ARIMA (Box-Jenkins) approach to time series analysis, and proceed through dynamic modeling and regression approaches to recent developments such as cointegration analysis, error correction models, and vector autoregression. Heavy emphasis will be given to fundamental concepts and applied work. Prerequisites for the course include a solid understanding of the fundamentals of statistical inference, regression analysis, matrix algebra, and the general linear model. Course requirements and evaluation: There will be four homework assignments and an exam. Grades will be based on the top 3 homework scores and the exam score, with each receiving equal weight. Homework assignments will include statistical problems and computer assignments requiring the use of statistical software. SAS, Stata, and E-Views will be used for various applications to give students a sense of the comparative strengths and weakness of different packages.

VECTOR LINEAR TIME SERIES MODELS

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;

THE ESTIMATION OF FREQUENCY

Very general forms of the strong law of large numbers and the central limit theorem are proved for estimates of the unknown parameters in a sinusoidal oscillation observed subject to error. In particular when the unknown frequency 0o, is in fact 0 or nt it is shown that the estimate, 0N, satisfies 0N = 0o for N ? No (w) where No (w) is an integer, determined by the realisation, w, of the process, that is almost surely finite.

On Limit Theorems for Quadratic Functions of Discrete Time Series

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).

Estimation of vector ARMAX models

The asymptotic properties of maximum likelihood estimates of a vector ARMAX system are considered under general conditions, relating to the nature of the exogenous variables and the innovation sequence and to the form of the parameterization of the rational transfer functions, from exogenous variables and innovations to the output vector. The exogenous variables are assumed to be such that the sample serial covariances converge to limits. The innovations are assumed to be martingale differences and to be nondeterministic in a fairly weak sense. Stronger conditions ensure that the asymptotic distribution of the estimates has the same covariance matrix as for Gaussian innovations but these stronger conditions are somewhat implausible. With each ARMAX structure may be associated an integer (the McMillan degree) and all structures for a given value of this integer may be topologised as an analytic manifold. Other parameterizations and topologisations of spaces of structures as analytic manifolds may also be considered and the presentation is sufficiently general to cover a wide range of these. Greater generality is also achieved by allowing for general forms of constraints.

The maximum of the periodogram

Let x(t), t = 1,..., T, be generated by a zero mean stationary process and let I([omega]) = [Sigma]x(t)expit[omega]2/T be the periodogram. Under general conditions, and in particular assuming only a finite 2nd moment, it is shown that max[omega]I([omega])/{2[pi]f([omega])logT}

Estimating the dimension of a linear system

The problem considered is that of estimating the integer or integers that prescribe the dimension of a linear system. These could be the Kronecker indices. Though attention is concentrated on the order or McMillan degree, which specifies the dimension of a minimal state vector, the same results are available for other cases. A fairly complete theorem is proved relating to conditions under which strong or weak convergence will hold for an estimate of the McMillan degree when the estimation is based on minimisation of a criterion of the form log det([Omega]n) + nC(T)/T, where [Omega]n, is the estimate of the prediction error covariance matrix and the McMillan degree is assumed to be n. The conditions relate to the prescribed sequence C(T).

The Estimation of the Prediction Error Variance

Abstract Spectral methods are used to construct an estimate of the variance of the prediction error for a normal, stationary process. The estimate obtained is shown to be strongly consistent and asymptotically normally distributed. Some aspects of the computations with respect to the fast Fourier transform are considered. The latter half of the article consists of a number of simulations, based on both generated and real data, which illustrate the results obtained. The relation between the estimate and that obtained from a high order autoregression is discussed.

Some properties of the parameterization of ARMA systems with unknown order

The first section of the paper introduces known theory relating to the description of the set of all ARMA structures via the concept of order, n, and the coordinatisation of all structures, M(n), of given order. The coordinates most easily used are related to the state space representation and in the first section these are related to coordinates obtained from the ARMA representation. In the second section geometric and topological properties of M(n) are considered. For example, the closure of M(n) is just the union of all M(j), j

The Estimation of Seasonal Variation in Economic Time Series

Abstract The problem of estimating the seasonal component in an economic time series is discussed and it is pointed out that the effects of any moving average operator on the seasonal component may be easily reversed so that one may use any suitable operator to remove the trend. The computational procedure is to estimate the seasonal index for the trend free series and to convert this index into a seasonal index for the original series by taking 12 term moving averages of this series (continued periodically) with weights depending on the operator used to remove trend. Weights for some commonly used operators are tabulated. The problem of estimating a slowly evolving seasonal is considered.