  Point forecast reconciliation of collection of time series with linear
aggregation constraints has evolved substantially over the last decade. A few
commonly used methods are GLS (generalized least squares), OLS (ordinary least
squares), WLS (weighted least squares), and MinT (minimum trace). GLS and MinT
have similar mathematical expressions, but they differ by the covariance matrix
used. OLS and WLS can be considered as special cases of MinT where they differ
by the assumptions made about the structure of the covariance matrix. All these
methods ensure that the reconciled forecasts are unbiased, provided that the
base forecasts are unbiased. The ERM (empirical risk minimizer) approach was
proposed to relax the assumption of unbiasedness.
  This paper proves that (a) GLS and MinT reduce to the same solution; (b) on
average, a method similar to ERM (which we refer to as MinT-U) can produce
better forecasts than MinT (lowest total mean squared error) which is then
followed by OLS and then by base; and (c) the mean squared error of each series
in the structure for MinT-U is smaller than that for MinT which is then
followed by that for either OLS or base forecasts. We show these theoretical
results using a set of simulation studies. We also evaluate them using the
Australian domestic tourism data set.
