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# 3  Towards a model framework

The tax revenue forecasting procedures used by the Treasury suggest a multiplicative model for the monthly, quarterly or annual levels of taxation revenue considered. A simple example is

(1)

where Yt denotes a particular tax revenue, Xt denotes a macroeconomic predictor such as GDP, and et denotes multiplicative error which varies about a mean of unity. If the transformed macroeconomic predictor  can be thought of as a proxy for the relevant tax-base, then α can be interpreted as a mean tax rate. Other multiplicative variables can be included and parameters such as α and β may also be time-dependent. Monthly or quarterly variables may have seasonal variation and all are likely to be affected, to some degree, by longer-term economic cycles.

In terms of continuously-compounding growth rates, (1) becomes

(2)

where Δ denotes the difference operator (), the parameter β can be interpreted as an elasticity, and the εt now correspond to additive errors (possibly stationary) with zero mean. Note that the approximation

(3)

yields

to a good degree of approximation provided the right-hand side of the above (a simple growth rate) is small. Throughout this report, we mainly consider the more-commonly used compound growth rates of the form ΔlogZt rather than their simple growth rate equivalents. The reasons for this are largely technical convenience and a direct link to continuous time growth models, but little is lost in adopting either definition since they differ very little in practice.

Consider (2) and forecasting the tax revenue growth rate ΔlogYt. If the predictor ΔlogXt is known and the εt are independent, in addition to having zero mean, then the best predictor of ΔlogYt is given by

(4)

or, using the above approximations and simple growth rates,

(5)

provided the various growth rates are relatively small. Forecast functions such as these lie at the heart of the Treasury's current tax forecasting methods described in Section 2.1. This suggests that the Treasury's tax forecasting methods could be regarded as optimal predictors for simple models that are suitable variants of (1).

This linkage between a model, such as (1), and its forecast function is not unique since other models can be found that will yield the same forecast function (5). However the simplicity of (1) and its growth rate model (2) make it a suitable starting point for a model framework within which the Treasury’s tax forecasting methods can be embedded. This is the strategy that has been adopted here.

## 3.1  Alternative models

Schoefisch (2005) notes that a number of the Treasury’s tax revenue forecasting methods assume that the elasticity β in (1) is identically unity. He questions this assumption, noting that β may well depend on the phase of the economic cycle which could impact differentially on the various components of GDP. In addition, the parameter α is also likely to change slowly over time to accommodate structural changes in New Zealand's economy.

Such considerations suggest a more general model of the form

(6)

where Yt, Xt and et are as in (1), but now the mean rate αt and elasticity βt are assumed to vary over time. In this case, models for the evolution of αt and βt are needed in order to use (6) for forecasting. It is also possible that seasonal factors will need to be included in (6) in the case of monthly or quarterly data. The analysis of such a model would directly address many of the recommendations made in Schoefisch (2005).

In terms of growth rates, (6) becomes

(7)

where, as in (2), the εt correspond to additive errors with zero mean. In practice, the mean rate αt and elasticity βt are likely to evolve smoothly over time and so Δlogαt, Δβt will typically be small, or have small variance, relative to the other sources of variation. Such considerations lead to modelling log αt as a stochastic trend within a suitable structural time series framework. See Harvey (1989) for a full discussion of this general class of models.

Many other variations of (6) are possible using different tax-base proxies where Xt is replaced by geometric combinations of one or more macroeconomic regressors and their lagged values. An example is

which allows the tax-base proxy to be a moving geometric combination of current and past values of the macroeconomic driver Xt. This model can also be framed in terms of growth rates and, in this case, a long-run co-integrating relationship between log Yt and log Xt may be needed. Alternatively, these macroeconomic drivers could be replaced by lagged values of Yt or some other tax revenue series. A simple example is

where the αt are assumed to be evolving smoothly over time and the log et are white noise errors. If |β| < 1, then log Yt reduces to a conventional time series trend plus additive error where the latter is a first-order autoregression. If β = 1, then the tax revenue growth rates ΔlogYt follow a trend plus error model. This simple model is readily generalised to include other more complex time series models.

In short, the general model (6) provides a flexible modelling framework for forecasting tax revenues and their growth rates, either in terms of suitable macroeconomic drivers and their lags that are proxies for the tax-base, or time series models involving just the tax revenue alone, or a combination of both.

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