Bayesian Reduced Rank Regression with Fixed Dimensionality

 by Mattias Villani

 Research Report 1998:6

 Department of Statistics, Stockholm University, S-106 91 Stockholm, Sweden


A Bayesian analysis of the reduced rank regression with fixed dimensionality is presented. The relevance of the model is shown by discussing two of its many encompassed models: MANOVA and cointegration models.
A normalization is necessary in order to identify the model and it is shown that different normalizations are needed for the encompassed models.
The consequences of neglecting the reduced rank condition in the construction of a diffuse prior are shown to be severe and a new diffuse prior is derived.
A fully operational and easily implemented procedure based on Gibbs sampling and Metropolis-Hastings methods is developed to handle the diffuse analysis based on the new prior.
The diffuse analysis is exemplified on several simulated and real-world data sets and it seems that the new diffuse prior overcomes the problem of local non-identification that appeared in earlier analyses based on priors which did not take the reduced rank restriction into account.
It is further argued that the special nature of the model that results from the reduced rank restriction and the specific applications in which the model is used calls for a discontinous prior which we model by a finite mixture distribution where some of the components are Dirac delta functions. The numerical evaluation of the posterior for the mixture prior is treated and it is shown that parts of the posterior are analytically tractable.

 Key words: Bayesian, Cointegration, MANOVA, Matrix Cauchy, Metropolis-Hastings, Mixture prior, Reduced rank regression. 

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Last update: 1998-09-28 / KH