BAYESIAN ASSESSMENT OF DIMENSIONALITY IN REDUCED RANK REGRESSION JUKKA CORANDER AND MATTIAS VILLANI Abstract. We consider Bayesian inference about the dimensionality in the multivariate reduced rank regression framework, which encompasses several models such as MANOVA, factor analysis and cointegration models for mul- tiple time series. The fractional Bayes approach is used to derive a closed form approximation to the posterior distribution of the dimensionality and some as- ymptotic properties of the approximation are proved. Finite sample properties are studied by simulation and the method is applied to growth curve data and cointegrated multivariate time series. 1. Introduction A common situation in multivariate analysis involves exploration of the rela- tionships between sets of variables, either by explicit parametric models or by descriptive methods such as principal components and canonical correlations. Al- though it was early understood that these instances may be jointly represented in terms of multivariate regression obeying a so called reduced rank structure for certain parameters (see, e.g., the pioneering work by Anderson, 1951), such an approach has only recently been fully appreciated by the general statistical community. An essential strength of the reduced rank regression (RRR) framework is its generality, as it encompasses several well-known models such that MANOVA, fac- tor analysis, linear simultaneous equations models and many other models for multiple time series. For a thorough treatment concerning the time series models, see Ahn and Reinsel (1990), Geweke (1996), Johansen (1995), Velu, Reinsel and Wichern (1986) and for the others, see Anderson (1984, 1994), and the references therein. An excellent review of various issues may also be found in Reinsel and Velu (1998). The typical model uncertainty in regular full rank multivariate regression is about the choice of relevant predictor variables. Several reasonable solutions are available for this latter model choice problem, see, e.g., Brown et al. (1999) or George and Foster (2000). It has been more of a challenge to produce sensible Key words and phrases. Cointegration, Fractional Bayes, Growth curves, Multivariate Regression. We thank an associate editor and two referees for helpful comments. The first author is grateful for financial support from the Academy of Finland, grant no. 50203. The second author gratefully acknowledges financial support from the Swedish Council of Research in Humanities and Social Sciences (HSFR), grant no. F0582/1999 and the Swedish Research Council (Veten- skapsrådet) grant no. 412-2002-1007. 1