STATIONARITY AND CENTRAL LIMIT THEOREM ASSOCIATED WITH BILINEAR TIME SERIES MODELS

Abstract. Consider the general bilinear times series model where {X_t; t= 0, L1, …} is a p-variate process, C (p x (s+ 1)), A (p x p). B _t(p x p) (1 ≤j≤q) are arbitrary matrices of constants, εT=[εt,…εt-q+1] and {εt; t=0, ±1, …} is a strictly stationary ergodic sequence of random variables. We investigate a set of minimal regularity conditions (on C, A, B _j and {ε_t}) under which we can establish the existence and causality of X _t and the asymptotic normality of the sample mean derived from { X _t}.     

A UNIFIED APPROACH TO ARMA MODEL IDENTIFICATION AND PRELIMINARY ESTIMATION

Abstract. This paper reviews several different methods for identifying the orders of autoregressive-moving average models for time series data. The case is made that these have a common basis, and that a unified approach may be found in the analysis of a matrix G, defined to be the covariance matrix of forecast values.
The estimation of this matrix is considered, emphasis being placed on the use of high order autoregression to approximate the predictor coefficients. Statistical procedures are proposed for analysing G, and identifying the model orders.
A simulation example and three sets of real data are used to illustrate the procedure, which appears to be a very useful tool for order identification and preliminary model estimation.     

ESTIMATION OF AUTOREGRESSIVE MOVING-AVERAGE ORDER GIVEN AN INFINITE NUMBER OF MODELS AND APPROXIMATION OF SPECTRAL DENSITIES

Abstract. A modification of the minimum Akaike information criterion (AIC) procedure (and of related procedures like the Bayesian information criterion (BIC)) for order estimation in autoregressive moving-average (ARMA) models is introduced. This procedure has the advantage that consistency for the order estimators obtained via this procedure can be established without restricting attention to only a finite number of models. The behaviour of these newly introduced order estimators is also analysed for the case when the data-generating process is not an ARMA process (transfer function/spectral density approximation). Furthermore, the behaviour of the order estimators obtained via minimization of BIC (or of related criteria) is investigated for a non-ARMA data-generating process.