Reply to Discussion

We thank the discussants for their kind comments and questions, and the interesting questions raised. We address the contributions by named discussants. DANIEL PE ~ NA Professor Pe~ na contributes some insightful discussion and provides useful additional references and connections to recent literature. We are particularly interested in following up his commentary on factor models and look forward to seeing his new work on the topic with Poncela. A first set of key questions relate to interpretation of time series decompositions and the use of high-order AR (and TVAR) models when lower order ARMA (and TVARMA) models might be appropriate. A key reason for using high-order TVAR models in cases when TVARMA forms may be suggested is that it is simply more difficult to fit and use the latter classes of models. In applied work, little is lost with high-order AR and TVAR models that adequately represent the data structure, though fitting higher-order models requires adequate sample sizes and/or structured and informative priors. In a context where an MA component (or other non-AR structure) exists – as in the simulations run by Professor Pe~ na – the decomposition methods applied to high-order AR models will naturally identify collections of high-frequency latent components that correspond to the MA components. This is theoretically predicted and we need to be aware of the implications: when such components arise as quasi-periodic ARMA(2,1) processes, as is typical, they do not have physical interpretation as meaningful " cycles " in the series, but simply represent model approximation and noise contributions. Such " noise " components will have very low moduli relative to those of physically meaningful and interpretable components. For example, we repeated Professor Pe~ na's simulations by generating data from the ARMA process 1 , :6By t = 1 + :2Ba t and then fitting AR(4) models to the synthetic data. Evaluated at approximate MLEs of the AR(4) parameters, most of cases lead to at least a pair of complex roots which do not correspond to any " real " quasi-periodic component in the data. In all such cases, however, the moduli of these complex roots are very low, less than 0.5 and typically in the range of 0.2 to 0.4. In applied work, in-terpretable quasi-periodic components are typically rather persistent, with moduli higher than 0.8. We discuss these and related issues in Prado and West (1997, section 2), where we explore the question of …