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An Analysis of Tax Revenue Forecast Errors - WP 07/02

4  Forecast error decompositions

A better understanding of the source and nature of the Treasury's tax revenue forecasting errors is an important prerequisite to building more accurate and robust tax forecasting models. To this end, and as recommended in Schoefisch (2005), we now develop decompositions of the Treasury's past tax revenue forecasting errors into suitable structural components.

The decompositions considered include:

  • the disaggregation of total tax revenue forecast errors into their component tax types (Section 4.2);
  • the decomposition of individual tax revenue forecast errors into a component due to forecasting the macroeconomic variables that are a proxy for the tax-base used, and a component due to forecasting the ratio of tax revenue to proxy tax-base, or tax ratio (Section 4.3);
  • a further decomposition (Section 4.4) of the tax ratio into a trend measuring an underlying mean tax rate and a random error component. The former provides a benchmark against which tax ratio forecasts can be benchmarked and the latter is typically non-informative noise that is predicted only by its mean.

As discussed in Section 2.1, the Treasury's tax revenue forecasts are typically variations of simple multiplicative models that project tax revenue forward at the same rate as the growth forecast of the macroeconomic aggregate that serves as a proxy tax-base. As might be expected, within each tax type no one method has been used consistently and, instead, the Treasury's methods have been refined and modified over time. In addition, any forecasts produced by the methodology described in Section 2.1 are further modified by judgemental factors, both at the individual tax level by the Treasury’s tax forecasting unit, and subsequently at an aggregate level by an internal review panel of senior Treasury staff. As a consequence, the Treasury's tax revenue forecasting models and processes cannot be replicated exactly.

These considerations have led us to consider a simple benchmark model for each tax type that facilitates the decompositions referred to above. Although based on similar tax-base proxies, these models are not the same as the Treasury models, but do have the virtues of transparency, since they have a simple structural interpretation, and consistency over time. The benchmark model provides a structural decomposition of the individual tax revenues against which the Treasury's forecasts can now be assessed.

4.1  Benchmark model

The general structure of taxation suggests the simple model


which is a special case of (6) with unit elasticity βt = 1. Here the macroeconomic variable Xt is to be regarded as a proxy for the relevant tax-base of the tax concerned, the tax ratio Rt = Yt/Xt is the observed ratio of tax revenue to proxy tax-base, and the multiplicative errors et have unit mean so that αt can be interpreted as an underlying mean tax rate. A simpler version of this model was used in O'Neill (2005) to decompose tax forecasting errors into suitable components with Xt set at Canada's nominal GDP.

The systematic component αt is assumed to evolve smoothly over time to accommodate minor policy changes whose effects are phased in gradually over time, and also any discrepancies between the proxy tax-base Xt and the underlying true tax base. To maintain this assumption, it is possible that any abrupt one-off changes will need to be accounted for by prior adjustments made to the data. See Section 5.

With these assumptions and caveats in mind, the multiplicative model (8) can now be transformed into the additive model


where log αt is a trend and the εt = log et will be assumed to be stationary white noise, independent of αt. This simple model for log Rt belongs to the general class of structural time series models discussed in Harvey (1989).

Decompositions of the Treasury's tax forecasting errors can now be undertaken using this simple structural model and its components as a benchmark.

4.2  Decomposition of total tax revenue by tax type

Consider the case where the total tax revenue is denoted by Y(t) and the component revenues by Yj(t) (j=1,…,m) so that


and the Yj(t) follow models of the form (8). Given forecasts Ŷj(t) of the individual components Yj(t), a forecast of the aggregate Y(t) is given by

yielding the forecast error decompositions


and, in terms of proportionate errors,

where  measures Yj(t) as a proportion of the total tax revenue Y(t).

Using the approximation (3), the latter decomposition can now be framed in terms of logarithms to yield




with the e(Yj(t)) defined similarly. As before, the quality of the approximation is such that these errors can be interpreted as simple proportionate errors. Note that the forecast errors defined have the opposite sign to those more commonly adopted (logY(t) – log Ÿ(t) for proportionate errors and Y(t)Ÿ(t) for actual errors). However definition (11) allows for more natural interpretations with positive errors implying over-forecasting and negative errors implying under-forecasting.

In Section 6.1, it is shown that the Pj(t) evolve slowly and vary little over time by comparison to the e(Yj(t)). In this case, the proportionate forecast error for the total tax revenue Y(t) has mean-squared error given by





with var[.], cov[.], E[.] denoting variance, covariance and expectation respectively.

The decompositions (10) and (12) provide an appropriate framework for evaluating the relative contributions of the various tax forecasting errors to both individual tax components and their aggregates. Note that these particular decompositions are not dependent on the benchmark model (8).

4.3  Separating out the macroeconomic forecast errors

Consider the benchmark model (8) where, as before, Yt denotes a particular tax revenue, the macroeconomic variable Xt is a proxy for the tax-base, and Rt is the associated tax ratio. Given forecasts Ÿt,  of Yt, Xt respectively, a natural forecast of Rt is given by


so that the three forecasts satisfy the simple relationship


If Rt, Xt are independent or, more generally, if they are conditionally uncorrelated given past data, then the best predictors of Yt, Rt and Xt will satisfy (14). These and other considerations lead us to assume that (14) holds for the forecasts considered in this report so that Rt can be forecast by the simple predictor (13).

From the multiplicative relationships (8) and (14) we obtain


or, using the notation introduced in (10),


where these quantities are the proportionate forecast errors for each component. Note that (15) additively decomposes the total tax revenue proportionate forecast error e(Yt) into two component proportionate errors, one due to forecasting the tax ratio Rt and the other the macroeconomic variable Xt used as the tax-base proxy.

Multiplying (15) by Yt and using the approximation (3) yields the actual forecast error decomposition


which shows the influence of the respective errors in absolute terms.

The mean-squared proportionate forecast error of Yt is given by





This decomposition and (15) provide a suitable framework for separating out the forecast errors for the tax ratio Rt from those of the macroeconomic tax-base proxy Xt. They can also be used in conjunction with decompositions (10) and (12) to examine the relative contributions of the various tax-ratio forecasting errors.

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