The first component of the sum represents the part of the historical time series attributable
to innovations since T. The second component is termed as a ‘base projection’ of XTþj and is formed solely from information available at time T. The historical decomposition assigns
responsibility for the difference between the base projection and the actual series among
the innovations of the variables in the VAR. The second equation makes it clear that the
innovations since T in all variables yield the actual series.
The importance of any variable, or set of variables, can be determined by the extent to
which the introduction of the innovations since T in that variable, or set of variables, closes
the gap between the base projection and the actual series.
The VAR model is used to forecast the behavior of the interest rate during the cycles,
using the information up to the month immediately before the cycle begins. The deviations
of the forecasts from the actual series are considered the historical errors. The latter are due
to the fact that the model is not able to predict the shocks that have occurred on different
variables. Introducing the actual behavior of the variables during the cycles, the model is
able to eliminate the errors. The importance of the different variables to explain the cycles
is measured by their contribution to reduce the historical errors.