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An Introduction to State and Local Public Finance
Thomas A. Garrett and John C. Leatherman
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PART 2 - SELECTED APPLICATIONS IN PUBLIC FINANCE
Revenue forecasting involves the use of analytical techniques to project the amount of financial resources available in the future. In the public sector, revenues come from taxes, fees, license sales or intergovernmental transfers. Forecasting attempts to identify the relationship between the factors that drive revenues (tax rates, building permits issued, retail sales) and the revenues government collects (property taxes, user fees, sales taxes). The ability to accurately project future resources is critical to avoiding budgetary shortfalls or collecting excess taxes or fees. For the federal government, even small errors in projecting revenue can result in serious budget problems such as large surpluses or deficits. Thus, revenue forecasting is fundamental to both state and federal governments, as well as many larger municipalities. As local governments continue to shift reliance from the property tax to user fee-based revenues, forecasting will be increasingly important to smaller units of government and department administrators.
Revenue forecasts can apply to aggregate total revenue or to single revenue sources such as sales tax revenues or property tax revenues. There is no single method for projecting revenues. Rather, different methods tend to work better depending on the type of revenue. Similarly, there is no standard time-frame over which to attempt a forecast. State government might look ahead to the next year’s budget, while managers of a city water system may be concerned about a twenty year time horizon. Finally, revenue forecasting is intimately tied to the public policy process and is thus subject to considerable scrutiny and even political pressure.
Government fiscal policy is affected by the context in which it is formed. It deals with not only economic but also political concerns. It is essential to establish assumptions and procedures that concerned parties agree upon, as well as a mechanism for evaluating the validity of revenue forecasts. Thus, a disciplined process is needed. Guajardo and Miranda (2000) suggest a seven step process. The following steps are applied to each type of revenue to be forecast.
There are a wide range of forecasting techniques available (Frank, 1993; Makridakis and Wheelwright, 1987, 1989; Guajardo and Miranda, 2000). They range from relatively informal qualitative techniques to highly sophisticated quantitative techniques. In revenue forecasting, more sophisticated does not necessarily mean more accurate. In fact, an experienced finance officer can often "guess" what is likely to happen with a great deal of accuracy. In general, forecasters use a variety of techniques, recognizing that some perform better than others depending on the nature of the revenue source.
i. Qualitative Forecasting Methods
Qualitative forecasting methods rely on judgements about future revenue collection. These techniques are often referred to as judgmental or nonextrapolative approaches. In addition to their relatively small dependence on numbers, these techniques frequently do not provide a rigorous specification of underlying assumptions.
Among the most commonly used methods of forecasting is judgmental forecasting. This technique involves having an individual or small group of people make assessments of likely future conditions. While sounding ad hoc, the technique can produce very good estimates, especially when experienced persons are involved. The forecaster will utilize experience in conjunction with consideration of historical trends, current economic conditions, and other factors relevant to the revenue source.
Judgmental approaches tend to work best when background conditions are changing rapidly. When economic, political or administrative conditions are in flux, quantitative methods may not capture important information about factors that are likely to alter historical patterns.
A variation of the judgmental approach is consensus forecasting. Here, experts familiar with factors affecting a particular type of revenue meet to discuss near-term conditions in order to reach agreement about what is likely to happen to revenue collections. For example, municipal public administrators might meet with persons familiar with the local real estate market, economists monitoring local, state and national conditions, and representatives of local financial institutions to come up with a consensus forecast of future building permit applications. Consensus forecasting tends to work best when there is little historical information to draw upon that might be used with a quantitative forecasting method.
Judgmental forecasting approaches certainly have their place among forecasting methods. To some extent, a judgmental perspective needs to supplement any forecasting technique, even the most quantitatively rigorous methods. As might be suspected, however, judgmental approaches can be subject to bias and other sources of error. Guajardo and Miranda (2000) provide the following list of the major weaknesses of qualitative forecasting methods:
ii. Quantitative Forecasting Methods
Quantitative methods relay on numerical data relevant to the revenue source. Quantitative methods also make explicit the assumptions and procedures used to generate forecasts. Finally, quantitative methods will also generally assign a margin of error to forecasts, providing a indication of the degree of uncertainty associated with the estimates.
There are two general types of quantitative forecasting methods. The first is a time series approach that consists of a large number of techniques that generally use past trends to project future revenues. The second general approach, while still incorporating time series data, constructs causal models that use the variables assumed to influence the level of a particular revenue.
In general, quantitative methods do a better job of predicting future revenues than do qualitative methods (Cirincione, et al., 1999; Makridakis and Wheelwright, 1989). Simpler quantitative methods also generally perform as well as more complex methods (Makridakis, et al., 1984). Finally, the time series approach typically outperforms the causal modeling approaches, at least in the near-term, given the uncertainty associated with capturing all the relevant economic factors that influence revenue generation (Frank, 1993).
Time series approaches are the "bread and butter" of forecasting. They have been used extensively in the private sector and have been subject to substantial evaluation. Today, computer software exists that automatically applies the appropriate technique given the characteristics of the data entered. The underlying assumption of time series techniques is that patterns associated with past values in a data series can be used to project future values.
In using time series techniques, Frank (1993) identifies several essential concepts that need consideration prior to the selection of technique. The first is what constitutes a trend. This fundamentally questions how long a data series is required for the technique to be able to identify any underlying pattern in the data. There are no definitive guidelines as to the number of data points required in constructing a data set. Generally, the data should cover a period of at least several years and, depending on the technique used, should include a minimum of 24 observations and perhaps as many as 50 or more observations.
Cyclicality in time series refers to the extent to which the revenue source is influenced by general business cycles. Again, with local governments moving away from the relatively stable and predictable property tax to sales taxes and user fees, the need to take into consideration the effects of business cycles becomes relatively more important.
Similarly, seasonality is another cyclic phenomenon that needs consideration. This is typically the case when the observations are monthly or quarterly. The mathematical formulas employed can be adjusted to determine both the degree of seasonality that may exist as well as whether seasonality is increasing or decreasing over time.
Randomness is another factor that affects time series data. Randomness refers to unexpected events that may distort trends that otherwise exist over the long-term. Events such as natural disasters, political crisis, and the outbreak of war can result in temporary distortions in trends. Randomness can also result from natural variations around average or typical behavior. When the data series have a constant mean and variance over time, this is known a stationarity. Stationarity exists if the data series were divided into several parts and the independent averages of the means and variances of each part were about equal. If the average of each mean or variance were substantially different, nonstationarity would be suggested. When randomness tends to characterize a data series time series techniques do not perform very well, as performing econometric analyses on nonstationary data can often result in biased estimates.
b. Descriptions of Time Series Forecasting Models
There are a large number of time series approaches that are used in forecasting. Cirincione, et al., (1999) discuss a number of issues in their use and provide a nice summary of a variety of techniques in an appendix to their article. This presentation builds on the technical description found there.
A naive forecasting model simply assumes the revenue available at time t is the same amount available at time t-1. This is also known as the random walk approach.
where Ft is the forecast at time t, and At-1 is the actual value at time t-1.
A variation of this involves averaging the two prior periods to generate the estimate. Yet another variation involves adjusting for any seasonality that may be present. Naive forecasting is often used when the data series is unpredictable. It is also used in expert forecasting as the starting point for estimates that are then adjusted mentally.
Moving average models are probably the most commonly used time series approach among local governments. As implied by the name, the future value to be forecast is based on the average of N previous periods. It is a moving average because the oldest data points are dropped off as new ones are added.
where F is the forecast at time t, At-i is the actual value at time t-i, and N is the number of time periods averaged.
The length of time to include in the average depends on the degree of variation present in the data series. To the extent there appears a high degree of randomness in the data, a longer period is used. Similarly, to the extent cyclicality or seasonality is present in the data, longer time periods are required. An amount of trial and error will be needed to find the best fitting model, although new software can very rapidly identify the time period producing the minimum forecast error. While more complex time series techniques can perform better than the moving average, it does a reasonably good job and is often used as the benchmark against which other methods are compared.
3. Exponential Smoothing Models
The single exponential smoothing model is one of the common forecasting techniques used in the private sector. The model is a moving average of forecasts that have been corrected for the error observed in preceding forecasts. In the first smoothing model, there is assumed no trend or seasonal pattern.
where Ft is the forecast at time t, A t-i is the actual value at time t-i, and N is the number of time periods averaged.
The parameter 
is the
smoothing coefficient and has an estimated value between zero and one. It is
referred to as an exponential smoothing model because the value of tends to
affect past values exponentially. As
approaches one, the forecast
resembles a short-term moving average, while an
closer to zero tends to
resemble long-term moving averages. Regardless of the value of
, however, exponential
smoothing tends to give more recent values higher implicit weights. Again,
is typically estimated using
trial and error to secure the best fitting model, but software today can
rapidly find the model that minimizes forecast error.
The single parameter smoothing model presented above can be adapted to take into account trends that may be present in the data. The form presented here is called the Holt Model. In addition to the smoothing parameter estimated in the exponential smoothing model, a parameter representing the trend is also estimated.
Following the exposition found in Cirincione, et al. (1999), the
forecast at time t for k periods into the future equals the level
of the series at t plus the product of k and the trend at time
t. The level of series is estimated as a function of the actual value of
the series at time t, the level of the series at a previous time, and
the estimated trend at a previous time. The parameter is a smoothing
coefficient. The trend at time t is estimated to be a function of the
smoothed value of the change in level between the two time periods and the
estimated trend for the previous time period. The values for the smoothing
parameters, 
and
, are between zero and one.
where Ft+k is the forecast at time k
periods in the future, At is the actual value at
time t, St is the level of the series at
time t, Tt is the trend at time t,
and 
and
are smoothing parameters.
5. Damped Trend Exponential Smoothing
While the Holt Model takes into consideration the trend that may be
inherent in the data series, it somewhat unrealistically assumes the trend
continues in perpetuity. This means it can overshoot estimates several time
periods in the future. A variation known as damped trend exponential smoothing
has the effect of dampening the trend as time continues into subsequent
periods. It includes a third parameter,
, with a value between zero and
one that specifies a rate of decay in the trend.
where 
is the forecast
at time k periods in the future, At is the
actual value at time t, St is the level of
the series at time t, Tt is the trend at
time t, and , ,
are smoothing parameters.
6. Holt-Winter’s Linear Seasonal Smoothing
This model adapts Holt’s method to include a seasonal component in addition to a smoothing coefficient and a trend parameter. The first variant of the model is additive. This assumes the seasonality is constant over the series being forecast.
where Ft+k is the forecast at time k
periods in the future, At is the actual value at
time t, St is the level of the series at
time t, Tt is the trend at time t,
It is the seasonal index at time t,
s is the seasonal index counter, and
,
, and
are smoothing parameters.
The multiplicative variant of this model assumes that the seasonality is changing over the length of the series.
Incorporating seasonality, of course, increases the data requirements – typically three to four years of monthly data. The model is also quite complex, estimating smoothing, trend and seasonal parameters simultaneously. Because of these difficulties, many communities use simpler methods such as single or double exponential smoothing methods.
ARIMA is an acronym for autoregressive integrated moving average. Autoregressive and moving average refer to two of the components of the model, while integrated refers to the process of translating the calculations into a metric that can be interpreted.
ARIMA modeling has three components (Frank, 1993). In the model identification stage, the forecaster must decide whether the time series is autoregressive, moving average, or both. This is usually done by visually inspecting diagrams of the data or employing various statistical techniques. In the second stage, model estimation and diagnostic checks, the forecaster verifies the original model identification is correct. This requires subjecting the model to a variety of diagnostics. If the model checks out, the forecaster then proceeds to the third stage, forecasting.
The principle advantage of using the ARIMA approach is that the method can generate confidence intervals around the forecasts. This actually serves as another check of the validity of the model. If it predicts a high degree of confidence about a dubious forecast, the modeler may have to respecify the form of the model.
In order to achieve best results using the Box-Jenkins ARIMA approach, three assumptions need be met. The first is the generally accepted threshold of 50 data points. This tends to be a significant obstacle for many local governments who may collect data only annually for some types of revenue.
The second assumption is that the data series is stationary, i.e. that the data series varies around a constant mean and variance. Running a regression on two non-stationary variables can result in spurious results. If the data is non-stationary, the data series needs differencing and/or the addition of a time trend. If the data is trend non-stationary only, then adding a linear time trend to the model will render the series stationary. Trend non-stationary data have a mean and variance that change over time by a constant amount. If the data is first-difference non-stationary, then first differencing of the data will render the series stationary. Differencing involves subtracting the observation in time t by the observation in time t-1 for all observations. Whether the data requires these types of treatment should become apparent at the identification stage, and is generally easily accomplished with econometric software programs.
The third assumption of ARIMA models is that the series be homoscedastic, i.e. has a constant variance. If the amplitude of the variance around the mean is great even after differencing, the series is considered heteroscedastic. The remedy for this problem may be simple or complex and involves measures such as using the natural logarithm of the data, using square or cubed roots, or truncating the data series (cutting out certain values).
The first component of the ARIMA process is the autoregressive component. The autoregressive component predicts future values based on a linear combination of prior values. An autoregressive process of order p can be shown as:
where Ft is the predicted value at time
t, At-p is the actual value at time
t, and p’s are the estimated
parameters.
The moving average component provides forecasts based on prior forecasting errors. The moving average component of a model for a q-order process can be shown as:
where Ft is the predicted value at time
t, 
t-q is the forecast error at time
t, and the p
‘s are the estimated parameters.
These two components together form autoregressive moving average (ARMA) models. ARMA models assume a stationary data series before first differencing or the inclusion of a time trend. If a series has been rendered trend or difference stationary, the above models form the Box-Jenkins ARIMA model (Box and Jenkins, 1976). The number of autoregressive and moving average lags in an ARIMA model is represented as ARIMA(p,d,q), where d is the degree of difference, i.e. d=1 if the data is first differenced, d=2 if the data is second differenced, etc. If d=0, the ARIMA model is an ARMA(p,q). Further derivations can also take into account seasonality by considering autoregressive or moving average trends that occur at certain points in time. In the case of seasonality the ARIMA model is expressed as ARIMA(p,d,q)(P,Q) where P is the number of seasonal autoregressive lags and Q is the number of seasonal moving average lags. Seasonality is a consideration with relatively frequent data, such as weekly, monthly, or possibly quarterly data.
Causal forecasting models generally tend to be among the more complex techniques, having large data requirements and requiring a high degree of statistical skill. These approaches tend to work best for revenues that are heavily influenced by economic factors, such as business license fees, income taxes, and retail sales taxes. Thus, external data representing relevant economic performance indicators are used to predict the level of revenue expected. Some of the common economic information incorporated into these models include local population, income, and price information (Wong ,1995).
The complexity of causal models varies. The simplest type would be a simple linear regression model that might attempt to project revenue as a function of time, for example. Multiple regression models incorporate any number of relevant explanatory variables, including important policy variables as binary dummy variables. Binary variables take the value of one if a specific time period is represented and a value of zero otherwise. To illustrate, following Cirincione et al. (1999), four common regression models employing ordinary least squares can be show as:
where Ft is the predicted value at time
t, Tt is the value of the time at time
t, Tt2 is the squared value of
time at time t, 
is the
linear trend parameter associated with time,
is the quadratic trend parameter associated with time squared,
Ds is a binary dummy variable for each of the
s seasons, and 
is the
parameter value associated with each season. The estimated values on the dummy
variables reveal the average level of the dependent variable during the
designated time period. Testing the equality of the dummy coefficients can
reveal whether there are significant differences in the average level of the
dependent variable across seasonal periods.
Econometric forecasts are structured similar to regression equations, but can include estimates of change across multiple dimensions. Thus, complex events and relationships can be modeled where the output from one equation is fed into another equation as they are solved simultaneously. The types of revenue for which econometric forecasts are most useful include corporate tax, personal income tax, real estate tax, sales tax, and user charges and fees, such as building and construction permits.
This brief overview of revenue forecasting belies the fact that forecasting is a major field of economics. The intricacies and variations can not be represented thoroughly in a brief section. Yet, for those concerned with public finance, the topic is one of growing importance, especially at the municipal level. While some of these techniques are likely beyond the capability of local government managers, improvements in computer software and assistance available from universities and outreach providers increases the plausibility of using these tools even in smaller units of government.