  Forecast reconciliation of multivariate time series is the process of mapping
a set of incoherent forecasts into coherent forecasts to satisfy a given set of
linear constraints. Commonly used projection matrix based approaches for point
forecast reconciliation are OLS (ordinary least squares), WLS (weighted least
squares), and MinT (minimum trace). Even though point forecast reconciliation
is a well-established field of research, the literature on generating
probabilistic forecasts subject to linear constraints is somewhat limited.
Available methods follow a two-step procedure. Firstly, it draws future sample
paths from the univariate models fitted to each series in the collection (which
are incoherent). Secondly, it uses a projection matrix based approach or
empirical copula based reordering approach to account for contemporaneous
correlations and linear constraints. The projection matrices are estimated
either by optimizing a scoring rule such as energy or variogram score, or
simply using a projection matrix derived for point forecast reconciliation.
  This paper proves that (a) if the incoherent predictive distribution is
Gaussian then MinT minimizes the logarithmic scoring rule; and (b) the
logarithmic score of MinT for each marginal predictive density is smaller than
that of OLS. We show these theoretical results using a set of simulation
studies. We also evaluate them using the Australian domestic tourism data set.
