Bayesian time series and panel models : unit roots, dynamics and random effects

Abstract: This thesis consists of four papers and the main theme present is dependence, through time as in serial correlation, and across individuals, as in random effects. The individual papers may be grouped in many different ways. As is, the first two are concerned with autoregressive dynamics in a single time series and then a panel context, while the subject of the final two papers is parametric covariance modeling. Though set in a panel context the results in the latter are generally applicable. The approach taken is Bayesian. This choice is prompted by the coherent framework that the Bayesian principle offers for quantifying uncertainty and subsequently making inference in the presence of it. Recent advances in numerical methods have also made the Bayesian choice simpler.In the first paper an existing model to conduct inference directly on the roots of the autoregressive polynomial is extended to include seasonal components and to allow for a polynomial trend of arbitrary degree. The resulting highly flexible model robustifies against misspecification by implicitly averaging over different lag lengths, number of unit roots and specifications for the deterministic trend. An application to the Swedish real GDP illustrates the rich set of information about the dynamics of a time series that can be extracted using this modeling framework.The second paper offers an extension to a panel of time series. Limiting the scope, but at the same time simplifying matters considerably, the mean model is dropped restricting the applicability to non-trending panels. The main motivation of the extension is the construction of a flexible panel unit root test. The proposed approach circumvents the classical confusing problem of stating a relevant null hypothesis. It offers the possibility of more distinct inference with respect to unit root composition in the collection of time series. It also addresses the two important issues of model uncertainty and cross-section correlation. The model is illustrated using a panel of real exchange rates to investigate the purchasing power parity hypothesis.Many interesting panel models imply a structure on the covariance matrix in terms of a small number of parameters. In the third paper, exploiting this structure it is demonstrated how common panel data models lend themselves to direct sampling of the variance parameters. Not necessarily always practical, the implementation can be described by a simple and generally applicable template. For the method to be practical, simple to program and quick to execute, it is essential that the inverse of the covariance matrix can be written as a reasonably simple function of the parameters of interest. Also preferable but in no way necessary is the availability of a computationally convenient expression for the determinant of the covariance matrix as well as a bounded support for the parameters. Using the template, the computations involved in direct sampling and effect selection are illustrated in the context of a one- and two-way random effects model respectively.Having established direct sampling as a viable alternative in the previous paper, the generic template is applied to panel models with serial correlation in the fourth paper. In the case of pure serial correlation, with no random effects present, applying the template and using a Jeffreys type prior leads to very simple computations. In the very general setting of a mixed effects model with autocorrelated errors direct sampling of all variance parameters does not appear to be possible or at least not obviously practical. One important special case is identified in the model with the random individual effects model with autocorrelation.