4.4 A model-based decomposition of tax forecast errors
Here we consider the tax ratio Rt given by the benchmark model (8). Since αt has been assumed to be independent of εt and et = exp(εt) has unit mean, the best forecast of Rt will always be the same as the best forecast of αt. In essence, the benchmark model decomposes Rt into a structural forecastable component αt and a non-informative noise component et. These considerations lead us to assume, in addition to (14), that the forecasts considered in this report satisfy
(17)
where 
are the forecasts of Rt and αt respectively.
Now (6) and (17) yield the decomposition
or
(18)
where these proportionate error components are defined in the same way as before and nt = –εt is non-systematic white noise error. Here the mean-squared proportionate forecast error of Rt is given by
(19)
where, to a good approximation,
provided E(εt) is close to zero and the forecasts 
are closely correlated to the optimal forecasts of Rt.
These decompositions can now be used to determine the relative contributions, within tax revenue types, of the proportionate forecast errors of the tax ratio trend αt and, just as importantly, the nature and size of the non-informative noise components εt. They can also be used in conjunction with the decompositions given in the previous sections to better understand the inter-relationships between the various tax types and their error components.
