CONSISTENT UNDERESTIMATION BIAS, THE
ASYMMETRICAL LOSS FUNCTION, AND HOMOGENEOUS
SOURCES OF BIAS IN STATE REVENUE FORECASTS
William R. Voorhees*
ABSTRACT. One component of revenue forecast error has been attributed to
the phenomena of consistent underestimation bias due asymmetrical loss.
Because underestimation of revenue forecast results in less loss to forecasters
than overestimations, there appears to be a bias for forecasters to underestimate
revenue forecasts. This paper confirms this hypothesis. Additionally, with the
greater usage of national forecasting organizations that provide economic
forecasts on which revenue forecasts are based, a secondary source of forecaster
bias may be present in many state level forecasts. This hypothesis is supported
by the increase in number of states using such organizations and a decrease in
the standard deviation of the annual mean percentage state forecast error.
INTRODUCTION
Revenue forecast error often is attributed to consistent
underestimation bias due to an asymmetrical loss function. Because
forecasters are subject to a greater loss when they overestimate revenue
than when they underestimate revenue, there is an incentive for
forecasters to under forecast revenues and thus avoid losses they may
encounter with overestimated revenue. Forecaster loss is manifested in
many forms including loss of potential salary increases, loss of
reputation as a forecaster and loss of job responsibilities. Research to
date has considered the revenue forecaster as the source of
underestimation bias, but the recent usage of external economic forecasts
may also be introducing bias into forecasts. Many states utilize a
conditional forecasting process where a forecast is made for economic
-------------------* William Voorhees, Ph.D., is an Assistant Professor, School of Public Affairs,
Arizona State University. His publications and research interests include topics
in revenue forecasting, governmental accounting, and public finance.
conditions and then the revenues are forecast from the economic
forecast. If the economic forecast is underestimated, then an accurate
revenue-forecasting model will underestimate revenue. Recent trends
indicate that states are increasing their reliance on external forecasts
generated by a very limited number of national forecasting consultants.
In October of 2002, the two primary firms, Data Resources, Inc., (DRI)
and Wharton Econometric Forecasting Associates (WEFA) merged into
a single company called Global Insights, further reducing sources of
economic forecasts. Companies providing economic forecasts on a fee
basis may well be operating under a similar set of asymmetrical risk
factors as are revenue forecasters. Renewal of contracts for economic
forecasts depends on both the accuracy and the impact that the error of
the forecast has on governmental disruption. Because an under forecast
will always result in less disruption than an over forecast, third party
economic forecasters are incented to underestimation bias. As more
states utilize a limited number of economic forecasting services, the error
for the economic forecast should become homogenized as indicated by a
decreasing variance of error across state revenue forecasts.
This paper first considers the literature on underestimation bias and
the effects on forecasts attributed to fiscal stress. Next state forecasts are
examined for a consistent underestimation bias for the years 1979 thru
2002. Finally, aggregate state forecast error variances are examined for
consistency across years, which would indicate the introduction of a
homogenous error source.
SOURCES OF BIAS IN REVENUE FORECASTS
In addition to random error, bias also creates an opportunity for the
forecast to be in error. Although all forecasts have error, an unbiased,
strongly rational forecast has error that is attributable only to
randomness. A forecast is said to be strongly rational if the forecast and
the actual revenues, conditional upon the influences of a set of full
information, are equal (Feenberg, Gentry, Gilroy & Rosen, 1989). In
other words, given a set of full information available and taking into
account the effects of full information, the difference between the
forecast and the actual revenue should be zero. Weak rationality exists
when the set of information is incomplete yet the forecasters achieve the
correct answer. When bias is introduced into a strongly rational forecast,
the error takes on an additional systemic element in addition to the
element of randomness.
Forecast error at the federal level has raised many concerns in recent
years. During the 1980s, actual revenues often fell short of the
projections leaving the government with larger than anticipated deficits,
while in the later part of the 1990s, actual revenues have been exceeding
the forecasts. Overly optimistic assumptions were the primary cause of
the shortfalls of the 1980s (Shumavon, 1981). David Stockman, Director
of the Office of Management and Budget during much of that period,
claims that optimistic assumptions were utilized intentionally to justify
the Reagan administration’s tax reduction package (Stockman, 1985).
Howard (1987), who argues there is little difference between
Congressional Budget Office (CBO) and Office of Management and
Budget (OMB) projections, presents a different picture. The similarity of
OMB and CBO projections would suggest that political motivations are
not the foundation of a consistent bias.
At the local level, several studies have indicated that there appears to
be a conservative or underestimation bias present in their forecasts
(Larkey & Smith, 1984). In a 1988 study, Frank found that Florida cities
tended to underestimate revenues by approximately 8% (Frank, 1988). A
1992 study found that in Pennsylvania, both cities and counties
underestimate total revenues consistently, but that property tax and
intergovernmental
revenues
were
consistently
overestimated
(Bretschneider, Bunch and Gorr, 1992).
At the state level, mixed results on the presence of bias have been
reported. One study of general sales tax forecasts in twenty-eight states
for the years 1981 through 1986 found overestimates with a mean
percentage error of 0.06. A 1989 study by Cassidy, Kamlet, and Nagin,
consisting of twenty-three states between 1978 and 1987 found that 59%
of their forecasts were underestimations with a mean percent error of 0.51. According to the researchers, the results cast substantial doubt on
the prevailing literature that there is an underestimation bias at the state
level. They base this on a t-test, finding that the results were only
marginally significantly different from zero at p = 0.046. Because no
corrections were utilized for serial correlation within states, they argue
that the standard error would exceed the “conventional standards”
(Cassidy, Kamlet & Nagin, 1989).
Contradicting these findings are other studies that have found that
there is an underestimation bias in revenue forecasting generally
attributed to an asymmetrical loss function of the forecasters. Feenberg et
al. (1989) studied three states and found they all had a bias towards
underestimation. Another study that investigated forecasting in New
Jersey found that forecasts favored underestimation bias with the means
of the forecast errors being significantly different from zero (Gentry,
1989). A study on forecasting in Illinois also found a conservative bias
(Albritton & Dran, 1987). Finally, a 1992 study found that the use of
outside consultants resulted in an underestimation bias except when there
political competition existed (Bretschneider and Gorr, 1992). Taken in
total, these results suggest that the evidence for forecast bias must be
considered inconclusive.
Political Aspects of Forecast Error Bias
The political environment is clearly one possible source of
asymmetrical loss for forecasters. However, the political environment
may favor either underestimation or overestimation depending on policy
objectives of the political actors and their influence over the forecasters.
In discussing this relationship, Wachs argues that public administrators
must consider both psychological and sociological dimensions of
forecasting and that forecasters often are faced with ethical dilemmas.
The organizational locus of the forecaster may cause the forecaster to
produce forecast outcomes that optimize the policy preferences of the
organization’s goals (Wachs, 1982). In interviews of state forecasters, it
was found that the low-end forecast was usually accepted. As one
interviewee stated, “I am a hero when there is more money than I
predicted and a villain when there is less. Let me tell you, it is better to
be a hero than a villain” (Rodgers & Joyce, 1996).
In a study on the forecast accuracy between the Congressional
Budget Office (CBO) and the Office of Management and Budget
(OMB), Howard (1987) found that the OMB produced consistently
optimistic forecasts of assumptions. Howard attributes this to an
optimistic bias of the executive budget. In a similar study, Shumavon
found that the CBO consistently produced more accurate forecasts than
the OMB. He attributed the increased accuracy to organizational
differences that insulated the CBO staff from partisan political pressures
(Shumavon, 1981).
Rubin (1987) has suggested that conservative revenue estimates may
be a result of not only the lack of knowledge about the economy and
overly simple forecasting techniques but also a conscious political
decision to distort the estimates.
The effort to buffer against uncertainty may combine with the fiscal
conservativeness of the finance officials to create a “normal” bias toward
underestimation. This tendency may be exaggerated if there is
conservative political leadership trying to reduce the level of taxation and
services: revenue underestimates may hold down service levels and
cause cuts in services. The eventual revenue surpluses resulting from
underestimates of revenue encourage reduction in tax levels. Low
revenue estimates may discourage expansionary departmental requests,
or failing that may enhance the power of the city administrator to cut
departmental budget requests (Rubin, 1987).
As suggested by Rubin, the political argument generally is based on
the claim that conservative politicians are more likely to cause the
forecast to be minimized in order to reduce the level of services. Cassidy,
Kamlet, and Nagin (1989) make the argument that the direction of
forecast error is difficult if not impossible to predict by either party or
ideological position. While one may argue that conservative politicians
attempt to minimize revenue forecasts in order to reduce spending and
liberal politicians attempt to maximize revenue forecasts to increase
spending and service levels, the opposite can be shown. For example, at
the federal level, David Stockman (1985) claims that forecasts were
overestimated intentionally to pave the way for President Reagan’s tax
cuts. Thus while ideology may indeed influence estimation, the direction
of the influence is ambiguous.
While the direction of the forecast may not be logically attributable
to partisan/ideological influences, the accuracy of the forecast is another
question. Accuracy is measured as the absolute value of the deviation
from the actual and does not account for direction. Partisan or ideological
influences may arise from competition between parties, which challenge
the assumptions of the revenue forecast. The lack of a dominant party
may result in reduced competition and lead to an imbalance in political
power, thus forecasts may go unchallenged (Bretschneider, Gorr, Grizzle
& Klay, 1989). In their 1989 study, Bretschneider et al. found evidence
that party dominance was influential in state revenue forecasts. This
result is supported partially by Gentry (1989) who, in an extension of
Feenberg, Gentry, Gilroy, and Rosen’s (1989) study on forecasting
rationality, found that party dominance has significant influence on New
Jersey inheritance tax forecasts. This result is further supported by the
case study of Ohio forecasting by Shkurti and Winefordner. They
suggest, “forecast accuracy can be achieved even in a highly partisan
political environment, provided that the officials involved perceive the
advantages of submitting unbiased forecasts” (Shkurti & Winefordner,
1989). Another study found that unified governments produce more
accurate forecasts than do split governments, attributing the accuracy to
the efforts of the party in power to avoid a loss of political capital
(Voorhees, 2000; Voorhees, 2004). However, contrary to the above
results, Cassidy, Kamlet, and Nagin (1989) found that party dominance
does not influence forecast accuracy.
Fiscal Stress, Underestimation Bias, and the Asymmetrical Loss
Function
Several studies have considered the effects of stress on forecast
accuracy. In California, stress induced by Proposition 13 was found to
cause underestimates resulting in greater accuracy (Chapman, 1982). A
study of 133 Illinois cities found that cities with growing revenue, low
property taxes, and no fiscal stress tended to underestimate revenues. On
the other hand, cities with stagnant revenue growth, high property taxes,
and fiscal stress tended to overestimate revenues (Rubin, 1987).
These results might be explained by the natural tendency of
forecasters towards an underestimation bias. If forecasters perceive a
greater loss when the forecast is higher than actual revenues than when it
is less than actual revenues, they might be incented towards
underestimation. Shortfalls can dramatically affect the operations of
government requiring cutbacks and tax increases. Both of these options
are distasteful to politicians and the public. On the other hand, a surplus
from underestimated revenue is not seen as negatively as the shortfall. In
some cases, the public may even see a surplus as a positive indicator of
government performing better than expected. This creates an
asymmetrical loss curve where the loss to a forecaster is less when
revenues are underestimated than when they are overestimated.
If the asymmetrical loss function is at work during periods of nonstress, then what maladies might occur during periods of fiscal stress?
Fiscal stress may cause an asymmetrical loss curve to shift thus making
overestimation less costly than normal and underestimation more costly.
The effect of this shift is to offset some of the underestimation bias. The
shift in the asymmetrical loss curve results from additional pressures on
forecasters to predict desirable forecasts as opposed to accurate forecasts
resulting in less underestimation bias than normal. At the same time,
revenues are likely to be falling in real dollars (or failing to grow at their
historical rates). These two effects result in a convergence of the forecast
and actual revenue in times of stress. Thus, the shift in the asymmetrical
loss curve, coupled to declining revenues, actually improves forecast
accuracy.
One interesting topic that has yet to be addressed in the literature is
whether overestimation risk results in an additive or multiplicative
influence. If we assume a linear function, this would be manifested by
either an increase in the Y-intercept or by the slope of the line. In the
former case, the risk of overestimation as compared to underestimation,
is raised equally across all levels of overestimation, while in the later, the
excess risk due to overestimation increases with the overestimation.
National Economic Forecasts: A Homogenous Source of Bias
So far, research has assumed that the asymmetrical loss function is a
phenomenon of state and local forecasters. However, research on sales
tax forecasting shows that 75% of the states first develop an economic
forecast before the revenue forecast (Klay & Grizzle, 1992). In recent
years, states have been relying more on external or national forecasting
firms such as DRI and WEFA (now Global Insights) to provide them
with the economic forecast on which they base their revenue forecast.
Supposedly, the volume of forecasts made by these firms would allow
them to utilize the best techniques and hire the most knowledgeable
forecasters, something many states may be constrained from doing
(Bretschneider & Schroeder, 1988). In addition to utilizing national
forecasting firms, 44% of the states utilize a council of economic
advisers to help arrive at economic forecasts (Voorhees, 2002). These
conditional economic forecasts need to be considered as possible sources
of forecast bias
A survey taken by the author in the spring of 1999 showed that 39
states out of 46 or 85% of the states responding utilized national
forecasting firms in 1997. This is compared to a 1990 study that found 29
states out of 44 or 66% of the responding states utilized a national
economic forecasts (Federation of Tax Administrators, 1993). This
represents a 19% increase in the use of national forecasting firms over a
period of just eight years and indicates a significant change in forecasting
policies within state governments, the effects of which have not been
adequately measured. If one were to consider that most state forecasters
share information on forecasts, the effective saturation of the national
forecasting firms might actually have an impact close to 100% of the
states.
A single forecaster performing both economic and revenue forecasts
might result in bias in both the economic and revenue forecasts, but the
threat of a highly inaccurate forecast would constrain the total bias. On
the other hand, if the tasks of forecasting the economy and revenue are
separated and assigned to two independent forecasters, the constraint on
the error bias would be expected to decrease. If so, then this would result
in an increase in error if the bias of both the economic and revenue
forecasts were biased in the same direction and there is an absence of
contravening factors. Were state forecasters prescient as to the economic
forecast bias, the economic forecast bias could be incorporated into the
revenue forecast bias increasing the overall accuracy
It is reasonable to assume that an asymmetrical loss function would
be in existence for national forecasting firms just as it is believed to be
for state forecasters. During 1997–1998, it was found that both state
economic forecasters and private forecasters tended to underestimate
economic forecasts (Davis & Boyd, 1999). In theory, the asymmetric
loss function might be more applicable to the national forecasting firms
than to state forecasters. An economic forecast that results in a shortfall
would surely raise the attention of state officials and possibly jeopardize
future contracts between the government and the forecasting firm.
Additionally, individual forecasters for the national firms would be
subject to the same asymmetrical risks as a forecaster for the state or
local government. In contrast, an economic forecast that resulted in a
revenue windfall would not be nearly as serious to state officials and the
contract would not be placed in as great of jeopardy.
Testing for Consistent Underestimation Bias
Using secondary data from the Fiscal Survey of States (National
Governors Association, 1978–2002), tests for underestimation bias,
consistency and homogeneous sources are performed. The current
literature list several methods for measuring forecast error including
mean absolute percentage error (MAPE), mean absolute deviation
(MAD) and root mean squared error (RMSE) (Chase, 1995; Makridakis
& Wheelwright, 1989). This study utilizes the mean percentage error
(MPE) because it preserves the directionality of the error and identifies
bias. Because the other methods utilized either an absolute error or a
squared error, directionality is lost and with it the ability to determine
bias. The MPE is calculated by subtracting the actual revenue (A) from
the forecast revenue (F) for a given each state (i) for a each year (y),
dividing the result by the actual forecast for the respective state and year,
summing across all forecasts and then dividing by the number of
forecasts (n). This average will typically be close to zero since the
positive random errors will tend to offset the negative random errors. If
this is not close to zero, then bias should be suspected.
MPE 

Fyi  Ayi
Ayi
n
(1)
For a given year, this formula will result in a single percentage
representing aggregate forecast error for all states and will have a
positive or negative sign. A positive number indicates overestimation
bias, a negative number indicates underestimation bias and a zero
indicates no bias. The test for bias consistency is performed by
calculating the MPE for the population (fifty states) by year and then
comparing over forecast years to under forecast years. If the MPEs for
each year are substantially positive or negative then it can be said that the
bias is consistent. On the other hand, if no bias exists, then one would
expect approximately half of the errors to exceed zero and half to fall
below zero.
Finally, the issue of homogenous bias source is considered. Is it
possible that a substantial portion of forecast error is being introduced
based on the economic forecasts of a few economic forecasting firms?
While this question cannot be answered without considering the actual
economic forecast error, something that is not readily available for most
states, one is able to gain some insight as to the influence of these
homogenous sources.
One may consider the total variance of the revenue forecast as
consisting of two parts, a variance attributable to the economic forecasts
and a variance attributable to the revenue forecasts. As states make
greater use of economic forecasts produced by national forecasting firms,
the economic forecast portion of the variance will decrease due to
increased homogeneity of data sources and assumptions. However, how
the greater use of national firms would affect the accuracy of the
economic forecast is ambiguous. Initially it might be presumed that
better forecast techniques would be utilized to vastly improve the
forecast. However the use of a national firm may be due to personnel
resource constraints rather than any attempt to improve accuracy.
Because most states utilize a conditional forecast, the portion of
forecast variance attributable to the revenue forecast would be a function
of the revenue forecasting techniques including organizational processes.
Improvements in revenue forecasting techniques should result in both a
decreasing variance and improved accuracy.
RESULTS
When observations for the 24-year period are aggregated, we find a
mean percentage forecast error of -1.45. This seems reasonable, as many
states will plan for a two percent error in their budgetary process. The
standard deviation for this distribution was 8.6. Table 1 lists the
percentage error for the states by year. From this table one finds that for
20 of the 24 years the aggregate forecast is a negative amount, adding
strength to the hypothesis that the bias is consistent. These results
confirm the studies by Feenberg, Gentry et al. (1989) and dissuade us
from accepting the previous studies that deny the existence of an
underestimation bias.
Additionally, in the periods 1979 through 2002 there were only 339
(28%) occurrences of positive forecasts out of 1194 observations. If only
random error were influencing the forecasts one would expect that the
mix of overestimations and underestimations would approximate 50%.
This further indicates that a consistent underestimation bias is present.
Plotting a regression line through the annual mean percentage error, we
find that on the average underestimation error has been increasing by
approximately 0.05% every year (see Chart 1).
CHART 1
State Forecast Mean Percentage Error: 1979-2002
12.0
10.0
Percentage Mean Error
8.0
Linear (Percentage Mean
Error)
Mean Percentage Error
6.0
4.0
2.0
0.0
-2.0
-4.0
y = -0.0485x + 95.234
R2 = 0.0087
-6.0
-8.0
1975
1980
1985
1990
Year
1995
2000
2005
Next, the data is tested to see if a homogeneous bias source, such as
a national forecasting firm, might be present. This is tested by looking at
the standard deviation of the annual mean percentage errors. Chart 2
illustrates a substantial decrease in the standard deviation over the years
indicating that forecast errors are becoming more homogeneous across
states regardless of whether the forecast is accurate or not. In other
words, the standard deviation from the aggregate state percentage
forecast error in any given year has been decreasing over time. This
correlates negatively to the increase in the use of national economic
forecasting firms, which might well explain the decreases in the standard
deviation over the years. Consider first that the revenue forecast is a
function of the economic forecast and that error in the economic forecast
is reflected in the revenue forecast and ultimately in the revenue forecast
error. If each state were to produce its own economic forecast, the
distribution of the economic forecast error would presumably be greater
than if the states utilized a single economic forecast. If, however, states
migrate to a single economic forecast, one would expect to see the
variance of the revenue forecast error across states to diminish
somewhat. These results support the suggestion that state forecasts are
being influenced by a shift to national forecasting firms. However, the
reader needs to be cautioned that these are only preliminary indicators
and that other factors might influence a reduction of standard deviations.
TABLE 1
Aggregate State Forecast Error by Year
Year
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
Mean Percentage Error
N
-1.9062
-0.8210
-3.4124
5.2109
9.6861
-5.6386
-4.3825
-0.1945
-2.4946
-4.0832
-5.8261
-1.0458
1.3551
-0.7212
-2.3000
-4.0591
-3.4985
-1.9350
-3.5516
-3.3154
-1.5193
-4.2467
-2.0722
6.1046
48
50
50
50
50
50
50
50
49
50
50
50
50
50
50
50
50
49
50
50
49
49
50
49
Std. Deviation
10.0463
9.8720
7.8213
15.0738
13.3564
11.6660
5.9091
5.0431
6.9557
6.5679
8.0596
7.1554
6.6474
5.0665
6.5791
8.2115
6.4665
4.5458
4.1748
4.1026
5.2154
6.1827
6.4397
5.3425
CHART 2
Standard Deviation of Mean State Forecast Error, 1979-2003
Standard Deviation of Mean State Forecast Error
16.0
14.0
Standard
Deviation of Mean
Error
12.0
Linear (Standard
Deviation of Mean
Error)
10.0
8.0
6.0
4.0
y = -0.2999x + 604.17
R2 = 0.4975
2.0
0.0
1975
1980
1985
1990
1995
2000
2005
Year
CONCLUSION
This paper has considered the problem of underestimation bias and
found evidence of it during a twenty-four year period between 1979 and
2002. The results show that states averaged a 1.45% underestimation
error. Consistency of the underestimation was also confirmed by testing
the forecast error for each of the years individually where all but four of
the twenty-four years were found to be underestimates. Finally, in the
belief that national forecast firms are influencing the revenue forecast
error, the standard deviations of the annual mean percentage error
forecasts were examined. It was found that the standard errors of the
annual mean percentage forecasts have been decreasing over time.
Although other variables may also be influencing this trend, it is believed
that both the increased usage and conglomeration of national economic
forecasting may be introducing a homogenous source of bias into the
state revenue forecast equation. In terms of underestimation bias, this
poses a troubling situation for state forecasters in that both state revenue
forecasters and national economic forecasters are introducing
underestimation bias into the revenue forecast. The allocation of the
revenue and economic forecast to different parties, each with incentives
to underestimate, also prevents the revenue forecaster from
understanding the degree of bias that may be present in the economic
forecast.
Correcting a situation of double bias is not an easy task. One
approach might be to incent forecasters, both revenue and economic,
towards accuracy and at the same time reduce their loss exposure when
negative forecasts do occur. Naturally, this is easier said than done.
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