Akademisk avhandling

som för avläggande av filosofie doktorsexamen

vid Stockholms universitet

offentligen försvaras i

hörsal 1, hus A, Frescati

onsdagen den 24 maj 2000 kl 10.00

av

Mattias Villani

fil lic

Stockholms universitet

**Abstract**

Cointegration concerns at least two time series which
individually drift extensively, but which cannot wander too far away from
each other. This simple concept has caused what is perhaps best described
as a revolution in empirical macroeconomics.

This thesis is devoted to a Bayesian analysis of
cointegration in vector autoregressive processes. Four included papers
treat various important aspects of cointegration, e. g. estimation, testing
and prediction.

In the first paper, the posterior distribution of
the lag length in the vector autoregressive process is derived using the
newly introduced fractional Bayes approach. Several properties of the posterior
distribution are explored by analytical and simulation methods.

The second paper introduces a Bayesian reference
analysis of the parameters in the cointegrated vector autoregression with
a given number of long run relations. The novelty is the focus on the well-known
fact that only the space spanned by the cointegration vectors, the cointegration
space, is uniquely determined. The uniform distribution over the space
of cointegration spaces is proposed as a reference prior for the cointegration
vectors and the prior on the adjustment coefficients is chosen to make
the overall prior consistent. A simple numerical evaluation of the posterior
distribution based on Gibbs sampling is derived.

The third paper derives the posterior probability
of restrictions on the cointegration space. Such posterior probabilities
have several important advantages over traditional test of hypotheses,
e. g. they have a clear interpretation and are internally consistent in
the sense that the conclusions are not dependent on the particular sequence
of tests chosen. In a small simulation study, the performance of the Bayesian
method was nearly uniformly better than the likelihood ratio test and always
superior to the two most widely used information criteria.

The fourth and last paper presents a procedure for
calculating the complete predictive distribution for a sequence of future
values of the process. The uncertainty in the choice of long run relations
is appropriately handled via an easily specified prior distribution. The
procedure is applied to a seven-variable macroeconomic system with the
purpose of forecasting inflation.

**Key words:** Bayesian inference, Cointegration, Estimation, Lag
length, Prediction, Restrictions.

ISBN 91-7265-098-2