  In data rich environments we may sometimes deal with time series that are
probability density-function valued, such as observations of cross-sectional
income distributions over time. To apply the methods of functional time series
analysis to such observations, we should first embed them in a linear space in
which the essential properties of densities are preserved under addition and
scalar multiplication. Bayes Hilbert spaces provide one way to achieve this
embedding. In this paper we investigate the use of Bayes Hilbert spaces to
model cointegrated density-valued linear processes. We develop an I(1)
representation theory for cointegrated linear processes in a Bayes Hilbert
space, and adapt existing statistical procedures for estimating the
corresponding attractor space to a Bayes Hilbert space setting. We revisit
empirical applications involving earnings and wage densities to illustrate the
utility of our approach.
