  This paper develops on-line inference for the multivariate local level model,
with the focus being placed on covariance estimation of the innovations. We
assess the application of the inverse Wishart prior distribution in this
context and find it too restrictive since the serial correlation structure of
the observation and state innovations is forced to be the same. We generalize
the inverse Wishart distribution to allow for a more convenient correlation
structure, but still retaining approximate conjugacy. We prove some relevant
results for the new distribution and we develop approximate Bayesian inference,
which allows simultaneous forecasting of time series data and estimation of the
covariance of the innovations of the model. We provide results on the steady
state of the level of the time series, which are deployed to achieve
computational savings. Using Monte Carlo experiments, we compare the proposed
methodology with existing estimation procedures. An example with real data
consisting of production data from an industrial process is given.
