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

6.2  Individual tax revenue decompositions

In the following subsections, we apply decompositions (21) and (22) to the percentage forecast errors of each tax revenue including the total tax revenue. Within any individual tax revenue, it is sufficient to consider just the percentage forecast errors, rather than the weighted percentage forecast errors, since the tax shares Pj(t) are approximately constant over time t.

6.2.1  Total tax revenue

Although total tax revenue is not forecast directly, but is constructed indirectly by summing the forecasts of each tax type, it may still be useful to disaggregate the total tax revenue forecast errors with respect to nominal GDP, since the tax-to-GDP ratio is something that is typically focussed on at each forecasting round. Such a model is also discussed in Schoefisch (2005) and O'Neill (2005). This simple model provides a direct forecast against which to benchmark the indirect forecast obtained by aggregation and, ideally, the two forecasts should be combined to give a better forecast overall. See Granger (1989) for a review of the advantages of combining forecasts.


Figure 3 – Total tax revenue and nominal GDP
The top plots show total tax revenue (solid black) and its forecast (dashed black), scaled GDP (solid red) and its forecast (dashed red), the associated tax ratio (solid blue) and its forecast (dashed blue), and the tax ratio trend (solid green). The remaining time series plots and boxplots show the percentage forecast errors due to forecasting total tax revenue (black), GDP (red), tax ratio (blue), tax ratio trend (green) and residual error (cyan).
Figure 3: Total tax revenue and nominal GDP.
Source: The Treasury

The results of applying decompositions (21) and (22) are shown in Figure 3 together with total tax revenue and its forecast, GDP and its forecast, the associated tax-to-GDP ratio and its forecast, and the tax ratio trend. Evidently, the forecasts of total tax revenue and GDP have both underestimated the actual outcomes consistently from 2000. Moreover, the tax ratio forecasts appear to be overestimating actual outcomes when the tax ratio trends downwards, and underestimating actual outcomes when it trends upwards.

The boxplots in Figure 3 show that, for decomposition (21), GDP forecast errors are significantly biased downwards and it is these that are contributing to the predominantly negative percentage forecast errors for total tax revenue. The tax ratio percentage forecast errors do not appear to be biased, but have higher volatility (standard deviation) than the GDP percentage forecast errors. For decomposition (22), the boxplots indicate that the percentage forecast errors due to forecasting the tax ratio trend may be positively biased. The statistical measures of bias, standard deviation and RMSE given in Table 4 also support these observations. In particular, the relative sizes of the RMSE values for the tax ratio percentage forecast errors and those of the non-systematic error component suggest that there are forecast gains to be had, even with this simple model.

Total tax revenue Yt and its components: GDP Xt, associated tax ratio Rt, tax ratio trend αt and residual et

Table 4 – Summary statistics for total tax revenue percentage forecast errors
  Yt Xt Rt αt et
Bias -1.31 -1.58 0.26 0.35 -0.09
Standard deviation 3.16 1.97 2.32 1.65 1.36
RMSE 3.28 2.46 2.23 1.62 1.30
Lag one autocorrelation 0.52 0.12 0.00 0.59 -0.61

Source: The Treasury

The plots of the time series in Figure 3 suggest that some may be serially correlated. The results of applying decompositions (21) and (22) are shown in Figure 3 together with total tax revenue and its forecast, GDP and its forecast, the associated tax-to-GDP ratio and its forecast, and the tax ratio trend. Evidently, the forecasts of total tax revenue and GDP have both underestimated the actual outcomes consistently from 2000. Moreover, the tax ratio forecasts appear to be overestimating actual outcomes when the tax ratio trends downwards, and underestimating actual outcomes when it trends upwards.

The boxplots in Figure 3 show that, for decomposition (21), GDP forecast errors are significantly biased downwards and it is these that are contributing to the predominantly negative percentage forecast errors for total tax revenue. The tax ratio percentage forecast errors do not appear to be biased, but have higher volatility (standard deviation) than the GDP percentage forecast errors. For decomposition (22), the boxplots indicate that the percentage forecast errors due to forecasting the tax ratio trend may be positively biased. The statistical measures of bias, standard deviation and RMSE given in Table 4 also support these observations. In particular, the relative sizes of the RMSE values for the tax ratio percentage forecast errors and those of the non-systematic error component suggest that there are forecast gains to be had, even with this simple model.

Table 4 gives the lag one autocorrelations of the various percentage forecast errors together with other summary statistics. There appears to be significant lag one autocorrelation in the percentage forecast errors due to forecasting the tax ratio trend, possibly an artefact of the effect mentioned earlier, and a suggestion of significant lag one autocorrelation in the percentage forecast errors for total tax revenue and the residual.

Each forecast error decomposition (21) and (22) led to two additive components whose cross-correlations were not significantly different from zero. A marginal exception was GDP and the tax ratio where the percentage forecast errors of the former appeared to lead the latter by one year, perhaps indicating that lagged GDP might be a better tax-base proxy. The tax ratio trend provides a good fit to the data suggesting that the benchmark model is reasonable.

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