Autocovariance Structure of Markov Regime Switching Models and Model Selection

We show that the covariance function of a second-order stationary vector Markov regime switching time series has a vector ARMA( p , q ) representation, where upper bounds for p and q are elementary functions of the number of regimes. These bounds apply to vector Markov regime switching processes with both mean–variance and autoregressive switching. This result yields an easily computed method for setting a lower bound on the number of underlying Markov regimes from an estimated autocovariance function.     

A Joint Regression Variable and Autoregressive Order Selection Criterion

Abstract.  In linear regression models with autocorrelated errors, we apply the residual likelihood approach to obtain a residual information criterion (RIC), which can jointly select regression variables and autoregressive orders. We show that RIC is a consistent criterion. In addition, our simulation studies indicate that it outperforms heuristic selection criteria – the Akaike information criterion and the Bayesian information criterion – when the signal-to-noise ratio is not weak.     

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.     

REGRESSION, AUTOREGRESSION MODELS

Abstract. The accuracy of least squares fitted regression autoregression models as approximations to more general stochastic structures is considered, attention being paid to the accuracy of the estimates of coefficients, of the innovations sequence and to the behaviour of the order (i.e., maximum lag) as determined by methods such as IAC, BIC. A key part is played by an accurate evaluation of the quantity where ετ(t) is the estimated innovation sequehce from an h_Tth- order regression, autoregression. Attempts are made to attain results near to the best possible and to establish almost sure convergence rates.     

ON THE UNBIASEDNESS PROPERTY OF AIC FOR EXACT OR APPROXIMATING LINEAR STOCHASTIC TIME SERIES MODELS

Abstract. A rigorous analysis is given of the asymptotic bias of the log maximum likelihood as an estimate of the expected log likelihood of the maximum likelihood model, when a linear model, such as an invertible, gaussian ARMA ( p, q ) model, with or without parameter constraints, is fit to stationary, possibly non-gaussian observations. It is assumed that these data arise from a model whose spectral density function either (i) coincides with that of a member of the class of models being fit, or, that failing, (ii) can be well-approximated by invertible ARMA ( p, q ) model spectral density functions in the class, whose ARMA coefficients are parameterized separately from the innovations variance. Our analysis shows that, for the purpose of comparing maximum likelihood models from different model classes, Akaike's AIC is asymptotically unbiased, in case (i), under gaussian or separate parametrization assumptions, but is not necessarily unbiased otherwise. In case (ii), its asymptotic bias is shown to be of the order of a number less than unity raised to the power max { p, q } and so is negligible if max { p, q } is not too small. These results extend and complete the somewhat heuristic analysis given by Ogata (1980) for exact or approximating autoregressive models.     

Book Review

Abraham Boyarsky and Pawel Gora, Laws of Chaos: Invariant Measures and Dynamical Systems in one Dimension, Probability and Its Applications     

Book Review

The effect of humid ageing on the bond strength of steel-steel cyanoacrylate bonds has been investigated. Such systems rapidly lose most of their bondstrength while under similar conditions polycarbonate-polycarbonate bonds remain virtually unaffected. This loss of strength can be attributed to the formation of an oxide layer at the substrate - adhesive interface and the simultaneous surface hydrolysis of the polymer. It nas also been shown that the adhesive itself is relatively impermeable to water and that the most likely method of ingress into the bond would be via a “wicking” mechanism along the substrate/polymer interface.