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

6.1  Total tax revenue decomposition

Figure 2 – Forecast errors for total tax revenue and major tax types
Percentage forecast errors (top row), weighted percentage forecast errors (middle row) and actual forecast errors (bottom row) for total tax revenue (black), PAYE (red), GST (green), corporate tax (blue), net other persons tax (cyan) and other taxes (magenta). Time series plots are given on the left and boxplots on the right.
Figure 2: Forecast errors for total tax revenue and major tax types.
Source: The Treasury

The results of applying decomposition (20) are given in Figure 2. For each tax revenue as well as total tax revenue, the percentage forecast errors are plotted as well as the weighted percentage forecast errors and the actual forecast errors. Note that, in this case, the weighted percentage forecast errors are just the actual forecast errors expressed as a percentage of the total tax revenue Y(t). This simple interpretation follows from the relation

    

where Ÿj(t) is the forecast of Yj(t) and the tax share .

The boxplots show the marginal distributions of the various forecast errors with the medians (middle bars of the boxes) located centrally between the quartiles (ends of the boxes) for some, but not all, of the tax revenues indicating symmetric distributions. In particular, corporate tax and other taxes would appear to have negatively skewed distributions. The notches give an approximate 95% confidence interval for the true median of the distribution of forecast errors concerned and so give an indication of whether the forecasts are biased. Adopting this criterion suggests that all tax revenue forecasts are biased downwards (negatively biased forecast errors) with the exception of corporate tax and net other persons tax, although the evidence is marginal in some cases. The whiskers (bars extending from the boxes) indicate the range of the data.

Although individual percentage forecast errors can be quite large (corporate tax and net other persons tax forecast errors are good examples), their effect on the total tax revenue error is moderated by their tax shares Pj(t). The weighted percentage forecast errors reflect the errors that do directly impact on the total tax revenue error. These present quite a different picture and show that, while corporate tax is still clearly the largest source of errors, it is followed by PAYE and GST with the other taxes (net other persons tax in particular) now playing a more minor role. This illustrates the importance of considering the weighted percentage forecast errors.

Apart from scale, the actual forecast errors displayed in both the boxplots and the time series plots differ very little from those of the weighted percentage forecast errors. This suggests that little is lost by focussing on just the weighted percentage forecast errors and the percentage forecast errors. Within any individual tax revenues, it will be sufficient to consider just the percentage forecast errors since the Pj(t) are approximately constant.

The time series plots of the various tax forecast errors indicate that some of them may be serially correlated. The lag one autocorrelations were calculated in each case for the weighted percentage errors for each tax type and these are given in Table 2 together with summary statistics such as the mean forecast error (bias), the forecast error standard deviation and the root mean squared forecast error (RMSE). The square of the latter is the mean squared error (MSE) which is the sum of the forecast error variance and the squared bias. The bias, standard deviation and RMSE values reflect what has already been seen and commented on in the boxplots. The Durbin-Watson test statistics indicated that the weighted percentage forecast errors for total tax revenues, PAYE and corporate tax have significant lag one autocorrelations indicating serial correlation. However the limited number of observations available makes these results marginal.

Table 2– Summary statistics for the weighted percentage forecast errors– Summary statistics for the weighted percentage forecast errors
  Total tax PAYE GST Corporate Net OP Other
Bias -1.31 -0.62 -0.46 -0.25 0.22 -0.30
Standard deviation 3.16 0.74 0.60 1.79 0.43 0.68
RMSE 3.28 0.94 0.73 1.72 0.47 0.71
Lag one autocorrelation 0.52 0.50 0.15 0.52 -0.12 0.26

Source: The Treasury

The contemporaneous correlations between the weighted percentage forecast errors for the individual tax revenues are given in Table 3. These are, on the whole, not significant with the exception of the correlation between the weighted percentage forecast errors for PAYE and other taxes, and possibly PAYE and corporate tax. These associations, if present, may be to do with the macroeconomic variables used as tax-base proxies in each case, or may be related to other causes. These issues should become clearer when the percentage forecast errors for the individual tax revenues are decomposed further.

Table 3 – Contemporaneous correlations for the weighted percentage forecast errors

– Contemporaneous correlations for the weighted percentage forecast errors
  PAYE GST Corporate Net OP Other
PAYE 1 0.35 0.56 0.07 0.82
GST 0.35 1 0.45 -0.11 0.08
Corporate 0.56 0.45 1 0.31 0.32
Net other persons 0.07 -0.11 0.31 1 0.27
Other 0.82 0.08 0.32 0.27 1

Source: The Treasury

The lag one cross-correlations of the weighted percentage forecast errors were also calculated and indicated that the forecast errors for GST led those of both PAYE and corporate tax. Again, however, these results are marginal and may have more to do with the macroeconomic variables used.

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