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This article proposes a unifying theory, or the Golden Rule, of forecasting. The Golden Rule of Forecasting is to be conservative. A conservative forecast is consistent with cumulative knowledge about the present and the past. To be conservative, forecasters must seek out and use all knowledge relevant to the problem, including knowledge of methods validated for the situation. Twenty-eight guidelines are logically deduced from the Golden Rule. A review of evidence identified 105 papers with experimental comparisons; 102 support the guidelines. Ignoring a single guideline increased forecast error by more than two-fifths on average. Ignoring the Golden Rule is likely to harm accuracy most when the situation is uncertain and complex, and when bias is likely. Non-experts who use the Golden Rule can identify dubious forecasts quickly and inexpensively. To date, ignorance of research findings, bias, sophisticated statistical procedures, and the proliferation of big data, have led forecasters to violate the Golden Rule. As a result, despite major advances in evidence-based forecasting methods, forecasting practice in many fields has failed to improve over the past half-century.
Golden Rule of Forecasting: Be Conservative
J. Scott Armstrong
The Wharton School, University of Pennsylvania, 700 Huntsman Hall,
3730 Walnut Street, Philadelphia, PA 19104, U.S.A.
and Ehrenberg-Bass Institute, Adelaide, Australia
Kesten C. Green
University of South Australia School of Business
and Ehrenberg-Bass Institute, Adelaide, Australia
Andreas Graefe
Department of Communication Science and Media Research
LMU Munich, Germany
21 January 2015
Acknowledgments: Kay A. Armstrong, Fred Collopy, Jason Dana, Peter Fader, Robert Fildes,
Everette Gardner, Paul Goodwin, David A. Griffith, Nigel Harvey, Robin Hogarth, Michael
Lawrence, Barbara Mellers, Mike Metcalf, Don Peters, Fotios Petropoulos, Nada R. Sanders,
Steven P. Schnaars, and Eric Stellwagen provided reviews. This does not imply that all
reviewers were in agreement with all of our conclusions. Geoff Allen, Hal Arkes, Bill Ascher,
Bob Clemen, Shantayanan Devarajan, Magne Jørgensen, Geoffrey Kabat, Peter Pronovost,
Lisa Shu, Jean Whitmore, and Clifford Winston suggested improvements. Kesten Green
presented a version of the paper at the University of South Australia in May 2013, and at the
International Symposiums on Forecasting in Seoul in June 2013 and in Rotterdam in June
2014. Thanks also to the many authors who provided suggestions on our summaries of their
research. Hester Green, Emma Hong, Jennifer Kwok, and Lynn Selhat edited the paper.
Responsibility for any errors remains with the authors.
This article proposes a unifying theory, or Golden Rule, of forecasting. The Golden Rule of
Forecasting is to be conservative. A conservative forecast is consistent with cumulative
knowledge about the present and the past. To be conservative, forecasters must seek out and
use all knowledge relevant to the problem, including knowledge of methods validated for the
situation. Twenty-eight guidelines are logically deduced from the Golden Rule. A review of
evidence identified 105 papers with experimental comparisons; 102 support the guidelines.
Ignoring a single guideline increased forecast error by more that two-fifths on average.
Ignoring the Golden Rule is likely to harm accuracy most when the situation is uncertain and
complex, and when bias is likely. Non-experts who use the Golden Rule can identify dubious
forecasts quickly and inexpensively. To date, ignorance of research findings, bias, sophisticated
statistical procedures, and the proliferation of big data, have led forecasters to violate the
Golden Rule. As a result, despite major advances in evidence-based forecasting methods,
forecasting practice in many fields has failed to improve over the past half-century.
Keywords: analytics, bias, big data, causality, checklists, combining, elections, index method,
judgmental bootstrapping, structured analogies, uncertainty.
Imagine that you are a manager who hires a consultant to predict profitable locations for
stores. The consultant applies the latest statistical techniques to large databases to develop a
forecasting model. You do not understand the consultant’s procedures, but the implications of
the forecasts are clear: invest in new outlets. The consultant’s model is based on statistically
significant associations in the data. Your colleagues are impressed by the consultant’s report,
and support acting on it. Should you?
To answer that question, and the general question of how best to go about forecasting, this
paper proposes a general rule: a Golden Rule of Forecasting. The short form of the Golden
Rule is to be conservative. The long form is to be conservative by adhering to cumulative
knowledge about the situation and about forecasting methods. Conservatism requires a valid
and reliable assessment of the forecasting problem in order to make effective use of cumulative
knowledge about the situation, and about evidence-based forecasting procedures.
The Golden Rule applies to all forecasting problems, but is especially important when bias is
likely and when the situation is uncertain and complex. Such situations are common in physical
and biological systemsas with climate, groundwater, mine yield, and species success
businessas with investment returnsand public policyas with the effects of government
projects, laws, and regulations.
Work on this paper started with a narrow conception of the application of conservatism to
forecasting: reduce the amount of change that is forecast in the presence of uncertainty. That
philosophy is the basis of regression analysis, which regresses toward the mean. The narrow
conception created its own contradictions, however, because reducing the amount of change
predicted is not conservative when a larger change is more consistent with cumulative
knowledge. Consider, for example, that it would not be conservative to reduce growth forecasts
for a less-developed nation that has made big reductions in barriers to trade and investment, and
in the regulation of business. Deliberations on this point led to the definition of conservatism
proposed for the Golden Rule. To the authors’ knowledge, the foregoing definition of
conservatism has not been used in the forecasting literature, but it is consistent with Zellner’s
description of a “sophisticatedly simple model” being one that “takes account of the techniques
and knowledge in a field and is logically sound” (Zellner, 2001, p. 259).
The Golden Rule Checklist
The checklist of 28 operational guidelines provided in this article follows logically from the
definition of conservatism. The checklist can help forecasters to be conservative by applying
the Golden Rule.
Subsequent searches for papers with comparative evidence relevant to the 28 guidelines
involved Internet literature searches, investigating references in important papers, asking key
researchers, and posting requests on the Internet. Email messages were then sent to the lead
authors of articles cited in substantive ways in order to check whether any relevant evidence
had been overlooked and to ensure the evidence is properly summarized. Reminder messages
were sent to authors who did not respond and to some co-authors. Eighty-four percent of
authors for whom valid email addresses were found responded.
The unit of analysis for assessing evidence on the guidelines is the paper or chapter. While
findings from individual studies are sometimes mentioned in this article, where a paper
includes more than one relevant comparison these are averaged before calculating any
summary figures. Averages are geometric means of error reductions. Where they are available,
error reductions are based on appropriate evidence-based and intuitive measures of error
(Armstrong, 2001c); often median-absolute-percentage-errors.
Table 1 shows the improvements in accuracy achieved by following a guideline relative to
using a less-conservative approach. The guidelines are each denoted by a check box. The total
number of papers, the number of papers providing evidence included in the calculation of the
average percentage error reduction, and the error reduction accompanies each guideline. For
example, for the first checklist item, seven comparisons were identified. Of those, three studies
provided evidence on the relative accuracy of forecasts from evidence-based methods validated
for the situation. The average error reduction is 18 percent. Almost all of the evidence identified
in the searches supports the guidelines, and each of the 21 guidelines for which evidence was
identified is supported by the overwhelming balance of that evidence. The last row of the Table
shows that the weighted average error reduction per guideline was 31 percent. The balance of
this section describes each of the guidelines and the evidence.
Problem formulation (1)
Forecasters should first formulate the forecasting problem. Proper formulation calls for use of
cumulative knowledge about the situation and the selection of relevant evidence-based
forecasting methods.
Use all important knowledge and information (1.1)
Use all relevant, reliable, and important information, and no more. It is important to exclude
unimportant and dubious variables because their use will harm predictive validity. That is one
of the major objections to the application of complex statistical techniques, or analytics, to “big
data”see, e.g. Sanders, 2014, pp. 195196 and 204, for illustrations of the problems with this
approach. To identify important information, ask a heterogeneous group of experts to
independently list data sources, relevant variables, directions and strengths of the variables’
effects. In addition, ask experts to justify their judgments. Search the literature for evidence
about causal relationships. Especially useful are meta-analyses, where structured procedures are
used to summarize the findings of experimental studies. Non-experimental data might be useful
in situations where experimental data are lacking, but should be used with great cautionsee
Kabat 2008 on health risk studies for illustrations of problems with analysis of non-
experimental data.
Conservative forecasting requires knowing the current situation, and so forecasters should
seek out the most recent data. For example, to forecast demand for ice cream in Sydney in the
coming week, it would be important to know that a big cruise ship was due to arrive and that a
week of perfect beach weather was expected.
The need to weight recent history more heavily should not, however, lead one to conclude that
things are so different now that historical data and knowledge should be ignored. Such claims
should be met with demands for evidence. The mantra that the world in general or a particular
situation is outside of previous experience is popular among CEOs and political leaders. U.S.
President Dwight Eisenhower, for example, stated that, “Things are more like they are now
than they ever were before.” The belief that things are different now has led to disastrous
forecasts by governments, businesses, and investors. The many and varied speculative bubbles
from Dutch tulip bulbs to Dot-com stocks provide examples of the failed forecasts of investors
who believed the situation was different from previous experience. Schnaars (1989) provides
many further examples. If need be, conduct experiments to assess the effects of recent changes,
or identify and analyze the outcomes of analogous situations.
Use all important knowledge and information by selecting evidence-based methods validated
for the situation (1.1.1)
Forecasters should use only procedures that have been empirically validated under conditions
similar to those of the situation being forecast. Fortunately, there is much evidence on which
forecasting methods provide the most accurate forecasts under which conditions. Evidence,
derived from empirical comparisons of the out-of-sample accuracy of forecasts from alternative
methods, is summarized in Principles of Forecasting (Armstrong, 2001d). The handbook is a
collaborative effort by 40 forecasting researchers and 123 expert reviewers. Updates since the
book’s publication are provided at
Do not assume that published forecasting methods have been validated. Many statistical
forecasting procedures have been proposed without adequate validation studies, simply on the
basis of experts’ opinions. An example is a published model for forecasting sales that was
tested on only six holdout observations from three different products. A reanalysis of the
model’s performance using a more extensive dataset, consisting of 14 products and 55 holdout
observations, found no evidence that the complex utility-based model yields more accurate
forecasts than a much simpler evidence-based extrapolation model (Goodwin and Meeran,
Statisticians have generally shown little interest in how well their proposed methods perform
in empirical validation tests. A check of the Social Science and the Science Citation Indices
(SSCI and SCI) found that four key comparative validation studies on time-series
forecasting were cited on average only three times per year between 1974 and 1991 in all
the statistics journals indexed (Fildes and Makridakis, 1995). Many thousands of empirical
time-series studies were published over that period. In other words, most researchers
ignored cumulative knowledge about forecasting methods.
Forecasters should validate any method they propose against evidence-based methods. Clients
should ask about independent validation testing rather than assume that it was done. For
example, independent evaluations of popular commercial programs sold by Focus Forecasting
concluded that these forecasts were substantially less accurate than forecasts from exponential
smoothing (Flores and Whybark, 1986; Gardner and Anderson, 1997) and damped smoothing
(Gardner, Anderson-Fletcher, and Wickes, 2001).
One validated approach, Rule-based Forecasting (RBF), embodies existing knowledge about
which methods work best under what conditions in the form of rules. RBF involves 99 rules for
how to forecast given up to 28 conditions of time series data. For example, the method varies
the weights on alternative extrapolation forecasts depending on the forecast horizon, causal
forces, and variability of the historical data. The conditions also allow for the incorporation of
experts’ domain knowledge. RBF provided the most accurate forecasts for annual data in the
M-Competition. There was a reduction in the Median Average Percentage Errors (MdAPE) of
18 percent for one-year-ahead forecasts compared to that for the equal-weights combined
forecastthe next most accurate method. For six-year ahead forecasts, the error reduction
versus equal-weights combining was 42 percent (Collopy and Armstrong 1992). Vokurka,
Flores and Pearce (1996) provide additional support for differential weights RBF. They used
automated procedures for rule selection and found that errors for 6-year-ahead forecasts of M-
Competiton data were 15 percent less than those for the equal-weights combined forecasts.
Fildes and Petropoulos (this issue) also provide evidence that forecasters can reduce
forecast error by using basic knowledge about the characteristics of series being forecast
along with simple evidence-based rules for selecting extrapolation methods and weighting
them. That approach led to a five percent error reduction in their study.
Despite the extensive evidence on forecasting methods, many forecasters overlook that
knowledge. Consider the U.N. Intergovernmental Panel on Climate Change’s (IPCC’s)
forecasts of dangerous manmade global warming (Randall et al., 2007). An audit of the
procedures used to generate these forecasts found that they violated 72 of the 89 relevant
forecasting principles such as “compare track records of various forecasting methods”
(Green and Armstrong, 2007a). As a consequence of overlooking evidence on what forecasting
procedures should be used in the situation, the IPCC used invalid forecasting methods to
generate the forecasts that have been used as the basis for costly government policies.
Use all important knowledge and information by decomposing to best use knowledge,
information, judgment (1.1.2)
Decomposition allows forecasters to better match forecasting methods to the situation, for
example by using causal models to forecast market size, using data from analogous
geographical regions to extrapolate market-share, and using information about recent changes
in causal factors to help forecast trends. While decomposition is often applicable, paucity of
knowledge or data may rule its use out for some problems.
There are two types of decomposition: additive and multiplicative.
Additive decomposition involves making forecasts for segments separately and then adding
them, a procedure that is also known as segmentation, tree analysis, or bottom-up forecasting.
Segments might be a firm’s sales for different products, geographical regions, or demographic
Another additive decomposition procedure is to estimate the current status or initial value of a
time seriesa process that is sometimes referred to as nowcastingand to then add a forecast
of the trend. The repeated revisions of official economic data suggest much uncertainty about
initial levels. For example, Runkle (1998) found that the difference between initial and revised
estimates of quarterly GDP growth from 1961 to 1996 varied from 7.5 percentage points
upward to 6.2 percentage points downward. Zarnowitz (1967) found that about 20 percent of
the total error in predicting one-year-ahead GNP in the U.S. arose from errors in
estimating the current GNP.
Armstrong (1985, pp. 286287) reports on nine studies on additive decomposition, all of which
showed gains in forecast accuracy. Only one of the studies (Kinney Jr., 1971) included an
effect size. That study, on company earnings, found that the mean absolute percentage error
(MAPE) was reduced by 17 percent in one comparison and 3.4 percent in another.
Dangerfield and Morris (1992) used exponential smoothing models to forecast all 15,753
unique series derived by aggregating pairs of the 178 monthly time-series used in the M-
Competition (Makridakis et al., 1982) that included at least 48 observations in the specification
set. The additive decomposition forecasts derived by combining forecasts from exponential
smoothing models of the individual series were more accurate for 74 percent of two-item
series. The MAPE of the bottom-up forecasts was 26 percent smaller than for the top-down
forecasts. Similarly, Jørgensen (2004) finds that when seven teams of experts forecast project
completion times, the errors of bottom-up forecasts were 49 percent smaller than the errors of
direct forecasts.
Carson, Cenesizoglu, and Parker (2011) forecast total monthly U.S. commercial air travel
passengers for 2003 and 2004. They estimated an econometric model using data from 1990 to
2002 in order to directly forecast aggregate passenger numbers. They used a similar approach
to estimate models for forecasting passenger numbers for each of the 179 busiest airports using
regional data, and then added across airports to get an aggregate forecast. The mean absolute
error (MAE) from the recomposed forecasts was about half that from the aggregate forecasts,
and was consistently lower over horizons from one-month-ahead to 12-months-ahead.
Additive decomposition enables forecasters to include information on many important
variables when there are large databases. For example, Armstrong and Andress (1970) used
data from 2,717 gas stations to derive a segmentation model that used 11 of an initial 19
variables selected based on domain knowledgee.g. building age, and open 24 hours. They
used the same data to estimate a stepwise regression model that included all 19 variables. The
two models were used to forecast sales for 3,000 holdout gas stations. The segmentation model
forecasts had a MAPE of 41 percent and provided an error reduction of 29 percent compared to
the 58 percent MAPE of the regression model’s forecasts. The finding is consistent with the
fact that segmentations can properly incorporate more information than regression analysis.
Because data on the current level are often unreliable, forecasters should seek alternative
estimates. Consider combining the latest survey data with estimates from exponential
smoothing—with a correction for lag—or with a regression model’s estimate of the level
at t=0. Armstrong (1970), for example, estimated a cross-sectional regression model using
annual sales of photographic equipment in each of 17 countries for 1960 to 1965. Backcasts
were made for annual sales from 1955 to 1953. One approach started with the survey data and
added the trend over time by using an econometric model. Another approach used a
combination of survey data and econometric estimates of the starting values, and then added
the trend. No matter what the weights, forecasts based on the combined estimates of the starting
values were more accurate than forecasts based on survey data estimates of the starting values
alone. The a priori weights reduced the backcast errors for 14 of the 17 countries. On average
across the countries, the mean absolute percentage error (MAPE) was reduced from 30 percent
to 23 percent, an error reduction of 23 percent.
Another study, on forecasting U.S. lodging market sales, examined the effect of estimating the
current level and trend separately. An econometric model provided 28 forecasts from 1965
through 1971 using successive updating. The MAPE was reduced by 28 percent when the
current level was based on a combination of survey data and the econometric forecast. Another
test, done with forecasts from an extrapolation model, found the MAPE was reduced by 44
percent (Tessier and Armstrong, this issue).
Multiplicative decomposition involves dividing the problem into elements that can be forecast
and then multiplied. For example, multiplicative decomposition is often used to forecast a
company’s sales by multiplying forecasts of total market sales by forecasts of market share. As
with additive decomposition, the procedure is likely to be most useful when the decomposition
allows the use of more information in the forecasting process, and when there is much
uncertainty. If there is little uncertainty, then little gain is expected.
Perhaps the most widely used application of multiplicative decomposition is to obtain separate
estimates for seasonal factors for time-series forecasting. For forecasts over 18-month
horizons for 68 monthly economic series from the M-competition, Makridakis et al.
(1982) showed that seasonal factors reduced the MAPE by 23 percent.
MacGregor (2001) tested the effects of multiplicative decomposition in three experimental
studies of judgmental prediction including 31 problems that involved high uncertainty. For
example, how many pieces of mail were handled by the U.S. Postal Service last year? The
subjects made judgmental predictions for each component. The averages of the predictions for
each component were then multiplied. Relative to directly forecasting the aggregate figure,
decomposition reduced median error ratios by 36 percent in one study, 50 percent in another,
and 67 percent in the third (MacGregor’s Exhibit 2).
Avoid bias (1.2)
Forecasters sometimes depart from prior knowledge due to unconscious biases such as
optimism. Financial and other incentives, deference to authority, and confusing forecasting
with planning can also cause forecasters to ignore prior knowledge or to choose methods that
have not been validated.
Bias might be deliberate if the purpose of the forecasts is to further an organizational or a
political objective, such as with profit forecasts to help raise capital for a risky venture, or cost-
benefit forecasts for large-scale public works projects. For example, one study analyzed more
than 10,000 judgmental adjustments of quantitative model forecasts for one-step-ahead
pharmaceutical sales forecasts. In 57 percent of 8,411 forecasts, the experts adjusted the
forecast upwards, whereas downward adjustments occurred only 42 percent of the time.
Optimism remained even after experts were informed about their bias, as the feedback
decreased the rate of upward adjustments only slightly to 54 percent of 1,941 cases (Legerstee
and Franses, 2013). Another study found that first-year demand forecasts for 62 large rail
transportation projects were consistently optimistic, with a median overestimate of demand of
96 percent (Flyvbjerg, 2013).
Avoid bias by concealing the purpose of the forecast (1.2.1)
By ensuring that forecasters are unaware of the purpose of the forecast, one can eliminate
intentional biases. To implement this guideline, give the forecasting task to independent
forecasters who are not privy to the purpose of the forecast.
Avoid bias by specifying multiple hypotheses and methods (1.2.2)
Obtaining experimental evidence on multiple reasonable hypotheses is an ideal way to avoid
bias. Doing so should help to overcome even unconscious bias, such as confirmation bias, by
encouraging the forecaster to test reasonable alternatives to the favorite. The approach has a
long tradition in science as Chamberlin (1890, 1965) described. For example, to assess the
effects of a pharmaceutical product, use different methods and measures to test how it performs
relative to alternative treatments, including no treatment. Prasad et al. (2013, p.1) summarized
findings from the testing of a variety of medical procedures and found that “of the 363 articles
testing standard of care, 146 (40.2%) reversed that practice, whereas 138 (38.0%) reaffirmed
Forecasters should consider using an appropriate no-change model as a benchmark
hypothesis. The no-change model is a reasonable conservative approach for many complex and
highly uncertain problems. The no-change model is, however, not always conservative: There
are cases where cumulative knowledge calls for change. For example, consider that you sell
baked beans and have a small market share. You reduce your price by 20 percent. A no-change
model for forecasting unit sales would not be conservative. You should rely instead on
knowledge about the price elasticity of similar products. In other words, forecasters should test
alternative hypotheses, methods, and models to the extent that a skeptical critic would not be
able to point to a plausible and important alternative that was not tested.
Given the power of the no-change model in many situations, the Relative Absolute Error
(RAE) was developed to compare the accuracy of forecasts from alternative models
(Armstrong and Collopy, 1992). It is the error of a forecast from a proposed model relative to
that of a forecast from a credible no-change or other benchmark model. Thus, a RAE less than
1.0 means the forecasts are more accurate than the benchmark forecasts, and a RAE greater
than 1.0 means the forecasts are worse than the benchmark forecasts.
Avoid bias by obtaining signed ethics statements before and after forecasting (1.2.3)
To reduce deliberate bias, obtain signed ethics statements from the forecasters before they
start, and again at the completion of the forecasting project. Ideally, the statement would
declare that the forecaster understands and will follow evidence-based forecasting procedures,
and would include declarations of any actual or potential conflicts of interest. Laboratory
studies have shown that when people reflect on their ethical standards, they behave more
ethically (Armstrong 2010, pp. 8994, reviews studies on this issue; also see Shu, Mazar, Gino,
Ariely, and Bazerman, 2012).
Provide full disclosure for independent audits, replications, extensions (1.3)
Replications are fundamental to scientific progress. Audits are good practice in government
and business, and might provide valuable evidence in a legal damages case. Even the
possibility that a forecasting procedure might be audited or replicated is likely to encourage the
forecaster to take more care to follow evidence-based procedures. To facilitate these benefits,
forecasters should fully disclose the data and methods used for forecasting, and describe how
they were selected.
Cumulative knowledge, and hence full disclosure, is vital to the Golden Rule. Failures to
disclose are often due to oversights, but are sometimes intentional. For example, in preparation
for a presentation to a U.S. Senate Science Committee hearing, the first author requested the
data used by U.S. Fish and Wildlife Service researchers as the basis of their forecasts that polar
bears are endangered. The researchers refused to provide the data on the grounds that they were
“using them” (Armstrong, Green, and Soon 2008).
Replications are important for detecting mistakes. Gardner (1984) found 23 books and
articles, most of which were peer-reviewed, that included mistakes in the formula for the trend
component of exponential smoothing model formulations. Gardner (1985) also found mistakes
in the exponential smoothing programs used in two companies.
Finally, Weimann (1990) finds a correlation of 0.51 between comprehensive reporting of
methodologyas measured by the number of methodological deficiencies reportedand the
accuracy of election polls. The finding is consistent with the notion that those who report more
fully on the limitations of their methodology are less biased, and thus their forecasts are more
Judgmental methods (2)
Judgmental forecasts are often used for important decisions such as whether to start a war,
launch a new product, acquire a company, buy a house, select a CEO, get married, or stimulate
the economy.
Avoid unaided judgment (2.1)
Use structured, validated procedures to make effective use of knowledge that is available in
the form of judgment. Unaided judgment is not conservative because it is a product of faulty
memories, inadequate mental models, and unreliable mental processing, to mention only a few
of the shortcomings that prevent good use of judgment. As a result, when the situation is
complex and uncertain, forecasts by experts using their unaided judgment are no more accurate
than those of non-experts (Armstrong, 1980). Green (2005) finds that forecasts of the decisions
that would be made in eight conflict situations that were obtained by using simulated
interactiona form of role-playing that involves structuring judgmentreduced error relative
to unaided judgment forecasts by 45 percent.
Moreover, when experts use their unaided judgment, they tend to more easily remember
recent, extreme, and vivid events. Thus, they overemphasize the importance of recent events, as
was shown in a study of 27,000 political and economic forecasts made over a 20-year period by
284 experts from different fields (Tetlock 2005).
Unaided judges tend to see patterns in the past and predict their persistence, despite lacking
reasons for the patterns. Even forecasting experts are tempted to depart from conservatism in
this way. For example, when attendees at the 2012 International Symposium on Forecasting
were asked to forecast the annual global average temperature for the following 25 years on two
50-year charts, about half of the respondents drew zigzag lines (Green and Armstrong, 2014).
They likely drew the zigzags to resemble the noise or pattern in the historical series (Harvey,
1995)a procedure that is almost certain to increase forecast error relative to a straight line.
Use alternative wording and pretest questions (2.2)
The way a question is framed can have a large effect on the answer. Hauser (1975, Chapter
15) provides examples of how wording affects responses. One example was the proportion of
people who answered “yes” to alternatively worded questions about free speech in 1940. The
questions and the percentage of affirmative responses are: (1) “Do you believe in freedom of
speech?” 96 percent; (2) “Do you believe in freedom of speech to the extent of allowing
radicals to hold meetings and express their views to the community?” 39 percent. To reduce
response errors, pose the question in multiple ways, pre-test the different wordings to ensure
they are understood as intended, and combine the responses to the alternative questions.
Ask judges to write reasons against the forecast (2.3)
Asking judges to explain their forecasts in writing is conservative because it encourages them
to consider more information and contributes to full disclosure.
Koriat, Lichtenstein, and Fischhoff (1980) asked 73 subjects to pick the correct answer to each
of ten general knowledge questions and then to judge the probability that their choice was
correct. For ten additional questions, the subjects were asked to make their picks and write
down as many reasons for and against each pick that they could think of. Their errors were 11
percent less when they provided reasons. In their second experiment, subjects predicted the
correct answers to general knowledge questions and provided one reason to support their
prediction (n=66), to contradict their prediction (n=55), or both (n=68). Providing a single
contradictory reason reduced error by 4 percent compared to providing no reason. Providing
supporting reasons had only a small effect on accuracy.
Additional evidence was provided in an experiment by Hoch (1985). Students predicted the
timing of their first job offer, the number of job offers, and starting salaries. Those who wrote
reasons why their desired outcome might not occur made more accurate forecasts.
Use judgmental bootstrapping (2.4)
People are often inconsistent in applying their knowledge. For example, they might suffer
from information overload, boredom, fatigue, distraction, or forgetfulness. Judgmental
bootstrapping protects against these problems by applying forecasters’ implicit rules in a
consistent way. In addition, the bootstrapping regression model is conservative in that it gives
less weight to variables when uncertainty is high.
To use judgmental bootstrapping, develop a quantitative model to infer how an expert or
group of experts makes forecasts. To do so, ask an expert to make forecasts for artificial cases
in which the values of the causal factors vary independently of one another. Then, estimate a
regression model of the expert’s forecasts against the variables. A key condition is that the final
model must exclude any variable that affects the forecast in a way that is opposite to what is
known about causality from prior knowledge, especially experimental evidence.
A review found eleven studies using cross-sectional data from various fields, including
personnel selection, psychology, education, and finance (Armstrong, 2001a). Forecasts from
judgmental bootstrapping models were more accurate than those from unaided judgment in
eight studies, there was no difference in two studies, and they were less accurate in one study in
which an incorrect belief on causality was applied more consistently. Most of these studies
reported accuracy in terms of correlations. One of them, however, reported an error reduction
of 6.4 percent.
Use structured analogies (2.5)
A situation of interest, or target situation, is likely to turn out like analogous situations. Using
evidence on behavior from analogous situations is conservative because doing so increases the
knowledge applied to the problem.
To forecast using structured analogies, ask five to 20 independent experts to identify
analogous situations from the past, describe similarities and differences, rate each analogy’s
similarity to the target situation, and then report the outcome of each. An administrator
calculates a modal outcome for a set of experts by using each expert’s top-rated analogy. The
modal outcome serves as the forecast for the target situation.
Research on structured analogies is in its infancy, but the findings of substantial improvements
in accuracy for complex uncertain situations are encouraging. In one study, eight conflict
situations, including union-management disputes, corporate takeover battles, and threats of war
were described to experts. Unaided expert predictions of the decisions made in these situations
were little more accurate than randomly selecting from a list of feasible decisions. In contrast,
by using structured analogies to obtain 97 forecasts, errors were reduced by 25 percent relative
to guessing. Furthermore, the error reduction was as much as 39 percent for the 44 forecasts
derived from data provided by experts who identified two or more analogies (Green and
Armstrong, 2007b).
Structured analogies can provide easily understood forecasts for complex projects. For
example, to forecast whether the California High Speed Rail (HSR) would cover its costs, a
forecaster could ask experts to identify similar HSR systems worldwide and obtain information
on their profitability. The Congressional Research Service did that and found that “few if any
HSR lines anywhere in the world have earned enough revenue to cover both their construction
and operating costs, even where population density is far greater than anywhere in the United
States” (Ryan and Sessions, 2013).
In Jørgensen’s (2004) study on forecasting the software development costs of two projects, the
errors of the forecasts from two teams of experts who recalled the details of analogous projects
are 82 percent smaller than the errors of top-down forecasts from five other teams of experts
who did not recall the details of any analogous situation. In addition, the errors in the forecasts
informed by analogies are 54 percent smaller than the errors of seven bottom-up forecasts from
seven teams of experts.
Nikolopoulos, Litsa, Petropoulos, Bougioukos, and Khammash (this issue) test a variation of
the structured analogies method: structured analogies from an interacting group. Their
approach reduced the average percentage error relative to unaided judgment by 54 percent.
Combine independent forecasts from judges (2.6)
To increase the amount of information considered and to reduce the effects of biases, combine
anonymous independent forecasts from judges. For example, experts can make useful
predictions about how others would behave in some situations. Avoid using traditional group
meetings to combine experts’ forecasts. The risk of bias is high in face-to-face meetings
because people can be reluctant to share their opinions in order to avoid conflict or ridicule.
Managers often rely needlessly on the unaided judgments that emerge from group meetings as
forecasts for important decisions. Experimental evidence demonstrates that it is easy to find
structured combining methods that produce forecasts from expert judgments that are more
accurate than those from traditional group meetings (Armstrong, 2006b).
The Delphi technique is one established and validated structured judgmental forecasting
method for combining experts’ forecasts. Delphi is a multi-round survey that elicits
independent and anonymous forecasts and reasons for them from a panel of experts. After each
round, a summary of the forecasts and reasons is provided to the experts. The experts can revise
their forecasts, free from group pressures, in later rounds. A review of the literature concluded
that Delphi provided forecasts that were more accurate than forecasts from traditional face-to-
face meetings in five studies and less accurate in one; two studies showed no difference (Rowe
and Wright, 2001). A laboratory experiment involving estimation tasks found that Delphi is
easier to understand than prediction markets (Graefe and Armstrong, 2011).
Armstrong (2001b) presents evidence from seven studies that involved combining forecasts
from four to as many as 79 experts. Combining the forecasts reduced error by an average of 12
percent compared to the typical expert forecast. Another study analyzes the accuracy of expert
forecasts on the outcomes of the three U.S. presidential elections from 2004 to 2012. The error
of the combined forecasts from 12 to 15 experts was 12 percent less than that of the forecast by
the typical expert (Graefe, Armstrong, Jones, and Cuzán, 2014).
Good results can be achieved by combining forecasts from eight to twelve experts with
diverse knowledge of the problem and biases that are likely to differ. Surprisingly, expertise
does not have to be high, and often has little impact on forecast accuracy (Armstrong, 1980;
Tetlock, 2005). Graefe (2014) finds that voters’ combined expectations of who will win
provided forecasts that were more accurate than the expectations of the typical individual
expert, with errors 32 percent smaller across six U.S. presidential elections. Combined voter
expectations were also more accurate than the single-expert complex statistical forecasts for the
2012 U.S. presidential election at for all of the 100-day period leading up
to Election Day. Combined voter expectations reduced MAE by an average of 38 percent.
Nikolopoulos, Litsa, Petropoulos, Bougioukos, and Khammash (this issue) obtained five
forecasts about the outcomes of two government programs from a group of 20 experts using
their unaided judgment, and from groups of experts using either semi-structured analogies or
the Delphi method. The two structured approaches to combining judgmental forecasts reduced
average percentage error relative to unaided judgment by eight and 27 percent.
In some situations, people are experts about their own behavior. The standard method for
combining judgments of one’s likely behavior is an intentions survey. There is a close
relationship between intentions and behavior as shown in the meta-analysis by Kim and Hunter
(1993), especially for high-involvement decisions (Morwitz, 2001). Here, again, it is harmful to
make judgmental revisions (Wright and McRae, 2007).
Forecasts from intentions surveys are more accurate when they are very short-term predictions
about important events. For example, while polls that ask people who they intend to vote for
have no predictive value for long-term forecasts, they are highly accurate shortly before
Election Day (Erikson and Wlezien, 2012).
Extrapolation methods (3)
Extrapolation for forecasting is in part conservative because it is based on data about past
behavior. Extrapolation can be used with time-series data or cross-sectional data. For an
example of the latter, behavioral responses to gun law changes in some states can be used to
predict responses in other states.
Extrapolation ceases to be conservative when knowledge about the situation that is not
contained in the time-series or cross-sectional data is at odds with the extrapolation. Thus, there
have been attempts to incorporate judgments into extrapolation. This section examines
approaches to incorporating more knowledge into extrapolations.
Use the longest time-series of valid and relevant data (3.1)
This guideline is based on the logic of the Golden Rule. The alternative of selecting a
particular starting point for estimating a time-series forecasting model, or of selecting a specific
subset of cross-sectional data, allows the forecaster considerable influence over the forecast that
will result. For example, McNown, Rogers, and Little (1995) showed that an extrapolation
model predicted increases in fertility when based on five years of historical data, but declines in
fertility when based on 30 years of data. Similar findings had been published earlier. For
example, Dorn’s (1950) review of research on population forecasts led him to conclude that
they were insufficiently conservative due to an over emphasis on recent trends.
By using the longest obtainable series, or all obtainable cross-sectional data, one reduces the
risk of biasing forecasts, whether intentionally or unintentionally.
Decompose by causal forces (3.2)
Causal forces that may affect a time series can be classified as growing, decaying, supporting,
opposing, regressing, and unknown (Armstrong and Collopy 1993). Growth, for example,
means that the causal forces will lead the series to increase, irrespective of the historical trend.
Ask domain expertspeople with expert knowledge about the situationto identify the effects
of causal forces on the trend of the series to be forecast.
When forecasting a time-series that is the product of opposing causal forces such as growth
and decay, decompose the series into the components affected by those forces and extrapolate
each component separately. By doing so, the forecaster is being conservative by using
knowledge about the expected trend in each component. Consider the problem of forecasting
highway deaths. The number of deaths tends to increase with the number of miles driven, but to
decrease as the safety of vehicles and roads improve. Because of the conflicting forces, the
direction of the trend in the fatality rate is uncertain. By decomposing the problem into miles-
driven-per-year and deaths-per-mile-driven, the analyst can use knowledge about the individual
trends to extrapolate each component. The forecast for the total number of deaths per year is
calculated as the product of the two components.
Armstrong, Collopy, and Yokum (2005) test the value of decomposition by causal forces for
twelve annual time-series of airline and automobile accidents, airline revenues, computer sales,
and cigarette production. Decomposition was hypothesized to provide more accurate forecasts
than those from extrapolations of the global series if each of the components could be forecast
over a simulation period with less error than could the aggregate, or if the coefficient of
variation about the trend line of each of the components was less than that for the global series.
Successive updating produced 575 forecasts, some for forecast horizons of one-to-five-years
and some for horizons of one-to-10-years. For the nine series that met one or both of the two
conditions, forecasting the decomposed series separately reduced the Median Relative Absolute
Error (MdRAE) of the combined forecasts by a geometric mean average of 64 percent relative
to forecasts from extrapolating the global series. (The original text of that paper states the error
reduction was 56 percent, but that is a typographical error.)
Modify trends to incorporate more knowledge (3.3)
Extrapolate conservatively by relying on cumulative knowledge about the trend. In many
situations, conservatism calls for a reduction in the magnitude of the trend, which is commonly
referred to as damping. Damping keeps the forecasts closer to the estimate of the current
situation. However, damping might not be conservative if it were to lead to a substantial
departure from a consistent long-term trend arising from well-supported and persistent causal
forces. For example, Moore’s Law, which states that computer performance doubles roughly
every two years, has held up for more than half a century, and there is reason to expect that the
causal forces will persist (Mollick 2006). Thus, modifying to incorporate more knowledge can
also involve moving a short-term trend toward a long-term trend. Without strong evidence that
the causal factors had changed, forecasts derived assuming a weakening of Moore’s Law
would not be conservative.
Damping is also not conservative for situations in which an important change in causal forces
is expected to increase a trend, as might be caused by a substantial reduction in corporate taxes,
elimination of a tariff, or introduction of a substantially improved product. The following
guidelines help to identify situations where modifying trends is conservative.
Modify trends… if the series is variable or unstable (3.3.1)
Variability and stability can be assessed by statistical measures or judgmentallyor both.
Most of the research to date uses statistical measures.
In a review of ten papers, Armstrong (2006a) concludes that damping the trend by using only
statistical rules on the variability in the historical data yielded an average error reduction of
about 4.6 percent. A reanalysis of the papers using the procedures of this review finds eight of
the papers (Gardner and McKenzie, 1985; Makridakis et al., 1982; Makridakis and Hibon,
2000; Gardner, 1990; Schnaars, 1986; Gardner and Anderson, 1997; Miller and Liberatore,
1993; Fildes, Hibon, Makridakis, and Meade, 1998) include relevant evidence on error
reduction from damping trends when forecasting by extrapolation. The average error reduction
across the eight papers is 12 percent. In all but one of the papers, accuracy was improved by
In his review of research on exponential smoothing, Gardner (2006) concludes that “ is still
difficult to beat the application of a damped trend to every time series” (p. 637). Since the gains
can be achieved easily and without any intervention, the adoption of the damped-trend
exponential smoothing method would lead to substantial savings for production and inventory
control systems worldwide. Further gains in accuracy can be achieved by modifying trends to
incorporate knowledge about the situation and expert judgment in structured ways as the
following guidelines describe.
Modify trends… if the historical trend conflicts with causal forces (3.3.2)
If the causal forces acting on a time-series conflict with the observed trend in a time-
series, a condition called a contrary series, damp the trend heavily toward the no-change
forecast. To identify casual forces, ask a small group of experts (three or more) for their
assessment and adopt the majority judgment. Experts typically need only a minute or so
to assess the causal forces for a given series, or for a group of related series.
Causal forces may be sufficiently strong as to reverse a long-term trend, such as when a
government regulates an industry. In that case, one would expect the iron law of
regulation to prevail (Armstrong and Green, 2013) with consequent losses of consumer
welfare as Winston (2006) finds.
Research findings to date suggest a simple guideline that works well for contrary series:
ignore trends. Armstrong and Collopy (1993) apply this contrary-series guideline to forecasts
from Holt’s exponential smoothingwhich takes no account of causal forces. Twenty annual
time-series from the M-Competition were rated as contrary. By removing the trend term from
Holt’s exponential smoothing, the median average percentage error (MdAPE) was reduced by
18 percent for one-year-ahead forecasts, and by 40 percent for six-year-ahead forecasts.
Additional testing used contrary series from four other data sets: annual data on Chinese
epidemics, unit product sales, economic and demographic variables, and quarterly data on U.S.
Navy personnel numbers. On average, the MdAPE for the no-trend forecasts was 17 percent
less than Holt’s for 943 one-step-ahead forecasts. For 723 long-range forecasts, which were
six-ahead for annual and 18-ahead for quarterly data, the error reduction averaged 43 percent
over the four data sets, a geometric mean of 31.4 percent across all 10 comparisons.
Modify trends… if the forecast horizon is longer than the historical series (3.3.3)
Uncertainty is higher when the forecast horizon is longer than the length of the historical time-
series. If making forecasts in such a situation cannot be avoided, consider (1) damping the trend
toward zero as the forecast horizon increases, and (2) averaging the trend with trends from
analogous series. U.S. Fish and Wildlife Service scientists violated this guideline and
overlooked the need for damping when they used only five years of historical data to forecast
an immediate, strong, and long-term reversal in the trend of polar bear population numbers
(Armstrong, Green, and Soon, 2008).
Wright and Stern (this issue) found that using an average of analogous products’ sales growth
trends for forecasting sales of new pharmaceutical products over their first year reduced the
MAPE by 43 percent compared to forecasts from a standard marketing model, exponential-
gamma, when 13 weeks of sales data were used for calibration.
Modify trends… if the short- and long-term trend directions are inconsistent (3.3.4)
If the direction of the short-term trend is inconsistent with that of the long-term trend, the
short-term trend should be damped towards the long-term trend as the forecast horizon
lengthens. Assuming no major change in causal forces, a long-term trend represents more
knowledge about the behavior of the series than does a short-term trend.
Modify seasonal factors to reflect uncertainty (3.4)
When the situation is uncertain, seasonal adjustment can harm accuracy as was shown long
ago by, for example, Groff (1973), and Nelson (1972). Having only a few years of data, large
variations in the estimates of seasonal factors from one year to the next, and ignorance about
what might cause seasonality are all sources of uncertainty.
One conservative response to uncertainty about seasonality is to damp seasonal factors toward
1.0. That approach has been the most successful one to date. Other approaches to consider are
to combine the estimate of a seasonal factor with those for the time period before and the period
after; and to combine the seasonal factors estimated for the target series with those estimated
for analogous series. The two latter approaches incorporate more information, and might
therefore improve upon a damping approach based only on statistical relationships.
Modify seasonal factors… if estimates vary substantially across years (3.4.1)
If estimates of the size of seasonal factors vary substantially from one year to the next, this
suggests uncertainty. Variations might be due to shifting dates of major holidays, strikes,
natural disasters, irregular marketing actions such as advertising or price reductions, and so on.
To deal with variations in seasonal factor estimates, damp the estimates or use the average of
each seasonal factor with those from the time periods immediately before and after.
Miller and Williams (2004) damped the seasonal factors for the 1,428 monthly series of the
M3-Competition based on the degree of variability. Forecasts based on damped seasonal
factors were more accurate for 59 to 65 percent of the series, depending on the horizon. For
series where the tests of variability called for damping, MAPEs were reduced by about four
Chen and Boylan (2008) test seasonal factor damping procedures on 218 monthly series of
lightbulb sales. They found that two damping procedures on average reduced the error
symmetrical MAPEof forecasts for all but one of 12 combinations of estimation period
two, three, and four yearand forecast horizonone, three, six, and nine month. Calculating
from Chen and Boylan’s Table 6, average error reduction was 3.1 percent.
Modify seasonal factors… if few years of data are available (3.4.2)
Damp seasonal factors stronglyor perhaps avoid using themunless there are sufficient
years of historical data from which to estimate them. Chen and Boylan (2008) find that
seasonal factors harmed accuracy when they were estimated from fewer than three years of
To compensate for a lack of information, consider estimating seasonal factors from analogous
series. For example, for a recently developed ski field, one could combine seasonal factors
from time-series on analogous fields with those from the new field. Withycombe’s (1989)
study finds reduced forecast errors in a test using 29 products from six analogous product lines
from three different companies. Combining seasonal factors across the products in each
product line provided forecasts that were more accurate than those based on estimates of
seasonality for the individual product in 56 percent of 289 one-month-ahead forecasts.
Combining seasonal factors from analogous series reduced the mean-squared-error of the
forecasts for each of the product lines from two to 21 percent.
In an analysis of 44 series of retail sales data from a large U.K. department store chain, Bunn
and Vassilopoulos (1999) find that forecasts from models that used seasonal factors estimated
from analogous series were consistently more accurate than forecasts from models that used
seasonal factors estimated from the target series data alone. When analogies were from the
same business class as the target series, the reductions in the Mean Absolute Deviation errors
(MADs) compared to forecasts from standard seasonal adjustment were between eight and 25
percent, depending on the model used.
Gorr, Olligschlaeger, and Thompson (2003) combine seasonal crime rates from six precincts
in Pittsburgh. The combined-seasonality forecast errors were about eight percent smaller than
the individual seasonality forecast errors.
Modify seasonal factors… if causal knowledge is weak (3.4.3)
Without prior knowledge on the causes of seasonality in the series to be forecast, seasonal
factors are likely to increase forecasting error. To the extent that the causal knowledge is weak,
damp the factors toward cumulative knowledge on seasonality. If there is no established causal
basis for seasonality, do not use seasonal factors.
Combine forecasts from alternative extrapolation methods and alternative data (3.5)
Armstrong (2001b, p. 428) finds error reductions from combining forecasts from different
extrapolation methods in five studies. The error reductions ranged from 4.3 to 24.2 percent,
with an average of 16 percent.
Analogous time-series can provide useful information for extrapolation models. The
information is relevant for levelsor base rates for cross-sectional dataand for trends. For
example, consider that one wishes to forecast sales of the Hyundai Genesis automobile. Rather
than relying only on the Genesis sales trend data, use the data for all luxury cars to forecast the
trend, and then combine the two forecasts.
Causal methods (4)
Regression analysis is currently the most common approach for developing and estimating
causal models. The method is conservative in that it regresses to the mean value of the series in
response to unattributed variability in the data. However, regression analysis has characteristics
that limit its usefulness for forecasting.
Regression is not sufficiently conservative because it does not reflect uncertainty regarding
causal effects arising from omitted variables, predicting the causal variables, changing causal
relationships, and inferred causality if variables in the model correlate with important excluded
variables over the estimation period. In addition, using statistical significance tests and
sophisticated statistical methods to help select predictor variables is problematic when large
databases are used. That is because sophisticated statistical techniques and an abundance of
observations tend to seduce forecasters and their clients away from using cumulative
knowledge and evidence-based forecasting procedures. In other words, they lead forecasters to
ignore the Golden Rule. For a more detailed discussion of problems with using regression
analysis for forecasting, see Armstrong (2012a), and Soyer and Hogarth (2012).
Use prior knowledge to specify variables, relationships, and effects (4.1)
Scientific discoveries about causality were made in the absence of sophisticated statistical
analyses. For example, John Snow identified the cause of cholera in London in 1854 as a result
of “the clarity of the prior reasoning, the bringing together of many different lines of evidence,
and the amount of shoe leather Snow was willing to use to get the data” (Freedman 1991, p.
Only variables that are known to be related to the variable to be forecast should be included in
a model. Ideally, variables should be identified from well established theorye.g., price
elasticities for normal goodsobvious relationshipse.g., rainfall and crop productionor
experimental evidence. For simple problems, one might use statistical analysis of non-
experimental data, but valid causal relationships cannot be discovered in this way for complex
A priori analyses to obtain knowledge, and to specify variables, relationships, and effects, can
be time consuming, expensive, and difficult. Finding and understanding the relevant research is
necessary. Perhaps unsurprisingly, then, since the middle of the Twentieth Century, forecasters
have turned to sophisticated statistical procedures such as stepwise regression and data mining,
along with large databases and high-speed computers, in the hope that these would replace the
need for a priori analyses. Ziliak and McCloskey (2004) provide evidence for this trend with
their analysis of papers that were published in the American Economic Review in the 1980s and
then in the 1990s. While 32 percent chose variables solely on the basis of statistical significance
in the 1980s, 74 percent did so in the 1990s.
There is little reason to believe that statistical analyses will lead to better forecasting models.
Consider the case of data mining. Data mining involves searching for relationships in data
without a priori analysis. Academic literature on data mining goes back many decades. A
Google Scholar Search for “data mining” and “predict or forecast” at the end of December
2014 produced about 175,000 hits. Two of the leading books on data mining have each
been cited more than 23,000 times. Despite the efforts in support of data mining,
comparative studies that show data mining provides substantive and consistent
improvements in forecast accuracy are lacking.
Keogh and Kasetty (2003) conduct a comprehensive search for empirical studies on data
mining. They criticize the failure of data mining researchers to test alternative methods. To
address the lack of testing, they tested procedures from more than 25 papers on data mining on
50 diverse empirical data sets. In a personal correspondence with the first author of this article,
Keogh wrote:
“[Professor X] claimed to be able to do 68% accuracy. I sent them some ‘stock’ data and
asked them to do prediction on it, they got 68% accuracy. However, the ‘stock’ data I sent them
was actually random walk! When I pointed this out, they did not seem to think it important.
The same authors have another paper in [the same journal], doing prediction of respiration data.
When I pointed out that they were training and testing on the same data and therefore their
experiments are worthless, they agreed (but did not withdraw the paper). The bottom line is that
although I read every paper on time-series data mining, I have never seen a paper that
convinced me that they were doing anything better than random guessing for prediction.
Maybe there is such a paper out there, but I doubt it.”
More than ten years later, the authors of this article asked Keogh for an update. He responded
on 15 January 2015, “I have never seen a paper that convinces me that the data mining (big
data) community are making a contribution to forecasting (although I have seen papers that
make that claim).
Statistical analyses of non-experimental data are unlikely ever to successfully replace a priori
analyses of experimental data.
Armstrong (1970) tests the value of a priori analysis in his study on forecasting international
camera sales. A fully specified model was developed from prior knowledge about causal
relationships before analyzing data. Data from 1960 to 1965 for 17 countries were then used to
estimate regression model coefficients. The final model coefficients were calculated as an
average of the a priori estimates and regression coefficients, a process later referred to as a poor
man’s Bayesian regression analysis. To test the predictive value of the approach, the model was
used to forecast backwards in time (backcast) 1954’s camera sales. Compared to forecasts from
a benchmark regression model with statistically estimated coefficients, forecasts from the
model with coefficient estimates that included the a priori knowledge reduced MAPE by 23
percent. Another test estimated models using 1960 to 1965 data for 19 countries. The models
were then used to predict market size in 11 holdout countries. The models that used a priori
knowledge in estimating coefficients reduced the MAPE of forecasts by 40 percent.
Economists and other social scientists concerned with specifying relationships use elasticities
to summarize prior knowledge. Elasticities are unit-free and easy to understand. They represent
the percentage change that occurs in the variable to be forecast in response to a one-percent
change in the causal variable. For example, a price elasticity of demand of -1.5 would mean
that if the price were increased by 1 percent, all else being equal, one would expect unit sales to
go down by 1.5 percent. Forecasters can examine prior research in order to estimate elasticities
and their plausible lower and upper bounds for the situation they are concerned with. For
example, in forecasting sales, one can find income, price, and advertising elasticities for various
product types in published meta-analyses. If little prior research exists, obtain estimates by
surveying domain experts.
Modify effect estimates to reflect uncertainty (4.2)
Causal variable coefficients should be modified in the direction of having no effect when
uncertainty about the effect that variables have on the dependent variable is high and when the
forecaster is uncertain about how much the causal variables will change. Modification of that
kind is referred to as damping, or shrinkage. In general, the greater the uncertainty, the greater
should be the damping.
Another strategy for addressing uncertainty over relationships is to adjust the weights of the
causal variables so that they are more equal with one another, in other words to adjust the
variable coefficients towards equality. Equalizing requires expressing the variables as
differences from their mean divided by their standard deviationi.e., as standardized
variablesestimating the model coefficients using regression analysis, and then adjusting the
estimated coefficients toward equality. When uncertainty about relative effect sizes is high,
consider assigning equal-weights to the standardized variables, which is the most extreme case
of equalizing. Dana and Dawes (2004) analyze the relative predictive performance of
regression and equal-weights models for five real non-experimental social science datasets and
a large number of synthetic datasets. The regression weights models failed to yield forecasts
that were more accurate than those from equal-weights models, except for sample sizes larger
than one hundred observations per predictor and situations in which prediction error was likely
to be very smalli.e., adjusted-R2 > .9. The optimal approach most likely lies in between these
two methods, statistically optimal and equal, and so averaging the forecasts from an equal-
weights model and a regression model is a sensible strategy.
As Graefe (this issue) summarizes, much evidence since the 1970s shows that equal-weights
models often provide more accurate ex ante forecasts than do regression models. Graefe’s
article also provides evidence from U.S. presidential election forecasting. Equal-weights
variants of nine established regression models yielded forecasts that were more accurate for six
of the nine models. On average, the equal-weights models’ forecasts reduced the MAE
compared to the original regression models’ forecasts by five percent.
Use all important variables (4.3)
When estimating relationships using non-experimental data, regression models can properly
include only a subset of variablestypically about threeno matter the sample size. However,
important practical problems often involve more than three important variables and a lack of
experimental data. For example, the long-run economic growth rates of nations are likely
affected by many important variables. In addition, causal variables may not vary over periods
for which data are available, and so regression models cannot provide estimates of the causal
relationships of these variables.
Index models, on the other hand, allow for the inclusion of all knowledge about causal
relationships that is important into a single model. The index method draws on an insight from
Benjamin Franklin’s “method for deciding doubtful matters” (Sparks, 1844). Franklin
suggested listing all relevant variables, identifying their directional effects, and weighting them
by importance. Index models might also be called knowledge models, because they can
represent all knowledge about factors affecting the thing being forecast.
To develop an index model, use prior knowledge to identify all relevant variables and their
expected directional influence on whatever is being forecaste.g., a candidate’s performance
in a job. Ideally one should develop an index model using knowledge gained by reviewing
experimental studies. In fields where experimental studies are scarce, survey independent
experts who, among them, have diverse knowledge. Calculate an index score by determining
the values of variables for a situation of interest and then adding the values. Consider using
different weights for the variables only if there is strong prior evidence that the variables have
differential effects. The index score is then used to calculate the forecast. For selection
problems, the option with the highest score is favored. For numerical forecasts, use a simple
linear regression model to estimate the relationship between the index score and the variable to
be predictede.g., box-office sales of a new movie.
Consider the problem of predicting judges’ decisions in court casesa situation for which the
causal variables are determined by the relevant law. Kort (1957) uses the index method to test
the predictability of U.S. Supreme Court decisions on right-to-counsel cases. Kort selected
right-to-counsel cases because experts considered the Court’s decisions on these cases to be
unpredictable. Kort’s review of the law led to the identification of 26 key variables, for
example the youth and the literacy of the offender. He assigned the variables importance
valuesweightsbased on an analysis of 14 cases decided between 1932 and 1947. Kort then
tested the resulting model by forecasting the decisions made in 14 out-of-sample cases decided
between 1947 and 1956. The model’s index scores accurately forecast 12 decisions. Two
decisions were too close to call based on the index scores.
The index method has also been used to forecast U.S. presidential elections, a situation with
knowledge about a large number of causal variables. An index model based on 59 biographical
variables correctly predicted the winners in 28 of 30 U.S. presidential elections up through
2012 (Armstrong and Graefe, 2011). For the four elections from 1996, the many-variable
biographical model provided forecasts that reduced MAE by 37 percent compared to the
average econometric model; all of which included few causal variables.
Another index model was based on surveys of how voters expected U.S. presidential
candidates to handle up to 47 important issues. The model correctly predicted the election
winner in 10 of the 11 elections up to 2012 (Graefe and Armstrong, 2013). For the three
elections from 2000, issues-index model forecast errors (MAEs) were 50 percent smaller than
the average econometric model forecast error.
Graefe (this issue) creates an index model by adding the standardized values of all 29
variables that were used by nine established models for forecasting the results of U.S.
presidential election. Across the 10 elections from 1976 to 2012, the errors of the forecasts
from that index model were 48 percent smaller than the errors of the typical individual
regression model forecasts, and were 29 percent smaller than the errors of the forecasts from
the most accurate individual model.
Another index model was developed to predict the effectiveness of advertisements based on
the use of evidence-based persuasion principles. Advertising novices were asked to rate how
effectively each relevant principle was applied for each ad in 96 pairs of print ads. The ad with
the higher index score was predicted to be the more effective ad of the pair. The index-score
predictions were compared to advertising experts’ unaided judgments, the typical approach for
such forecasts. The experts were correct for 55 percent of the pairs whereas the index scores
were correct for 75 percent, an error reduction of 43 percent (Armstrong, Du, Green, and
Graefe 2014).
Combine forecasts from dissimilar models (4.4)
One way to deal with the limitations of regression analysis is to develop different models with
different variables and data, and to then combine the forecasts from each model. In a study on
10-year-ahead forecasts of population in 100 counties of North Carolina, the average MAPE
for a set of econometric models was 9.5 percent. In contrast, the MAPE for the combined
forecasts was only 5.8 percent, an error reduction of 39 percent (Namboodiri and Lalu, 1971).
Armstrong (2001b, p. 428) found error reductions from combining forecasts from different
causal models in three studies. The error reductions were 3.4 percent for Gross National
Product forecasts, 9.4 percent for rainfall runoff forecasts, and 21 percent for plant and
equipment forecasts.
Another test involved forecasting U.S. presidential election results. Most of the well-known
regression models for this task are based on a measure of the incumbent’s performance in
handling the economy and one or two other variables. The models differ in the variables and in
the data used. Across the six elections from 1992 to 2012, the combined forecasts from all of
the published models in each yearthe number of which increased from 6 to 22 across the six
electionshad a MAE that was 30 percent less than that of the typical model (Graefe,
Armstrong, Jones Jr., and Cuzán, 2014).
Combine forecasts from diverse evidence-based methods (5)
Combining forecasts from evidence-based methods is conservative in that more knowledge is
used, and the effects of biases and mistakes such as data errors, computational errors, and poor
model specification are likely to offset one another. Consequently, combining forecasts reduces
the likelihood of large errors. Equally weighting component forecasts is conservative in the
absence of strong evidence of large differences in out-of-sample forecast accuracy from
different methods.
Interestingly, the benefits of combining are not intuitively obvious. In a series of experiments
with highly qualified MBA students, a majority of participants thought that averaging estimates
would deliver only average performance (Larrick and Soll, 2006).
A meta-analysis by Armstrong (2001b, p. 428) finds 11 studies on the effect of averaging
forecasts from different methods. On average, the errors of the combined forecasts were 11.5
percent lower than the average error of the component forecasts. More recent research on U.S.
presidential election forecasting (Graefe, Armstrong, Jones, and Cuzán, 2014) finds much
larger gains when forecasts are combined from different evidence-based methods that draw
upon different data. Averaging forecasts within and across four established election-forecasting
methods (polls, prediction markets, expert judgments, and regression models) yielded forecasts
that were more accurate than those from each of the component methods. Across six elections,
the average error reduction compared to the typical component method forecast error was 47
Many scholars have proposed methods for how to best weight the component forecasts.
However, Clemen’s (1989) review of over 200 published papers from the fields of forecasting,
psychology, statistics, and management science concluded that simple averagesi.e., using
equal-weightsusually provides the most accurate forecasts.
Graefe, Küchenhoff, Stierle, and Riedl (2014) find that simple averages provide forecasts that
are more accurate than those from Bayesian combining methods in four of five studies on
economic forecasting, with an average error reduction of five percent. Their study also provides
new evidence from U.S. presidential election forecasting, where the error of the simple average
forecasts were 25 percent less than the error of the Bayesian Model Averaging forecasts. A
study that tested the range of theoretically possible combinations finds that easily understood
and implemented heuristics, such as take-the-average, will, in most situations, perform as well
as the rather complex Bayesian approach (Goodwin, this issue).
Avoid unstructured judgmental adjustments to forecasts (6)
Judgmental adjustments tend to reduce objectivity and to introduce biases and random errors.
For example, a survey of 45 managers in a large conglomerate found that 64 percent of them
believed that “forecasts are frequently politically motivated(Fildes and Hastings, 1994).
In psychology, extensive research on cross-sectional data found that one should not make
unstructured subjective adjustments to forecasts from a quantitative model. A summary of
research on personnel selection revealed that employers should rely on forecasts from validated
statistical models. For example, those who will make the decision should not meet job
candidates, because doing so leads them to adjust forecasts to the detriment of accuracy (Meehl
Unfortunately, forecasters and managers are often tempted to make unstructured adjustments
to forecasts from quantitative methods. One study of forecasting in four companies finds that
91 percent of more than 60,000 statistical forecasts were judgmentally adjusted (Fildes,
Goodwin, Lawrence, and Nikolopoulos, 2009). Consistent with this finding, a survey of
forecasters at 96 U.S. corporations found that about 45 percent of the respondents claimed that
they always made judgmental adjustments to statistical forecasts, while only nine percent said
that they never did (Sanders and Manrodt 1994). Legerstee and Franses (2014) find that 21
mangers in 21 countries adjusted 99.7 percent of the 8,411 one-step-ahead sales forecasts for
pharmaceutical products. Providing experts with feedback on the harmful effects of their
adjustments had little effectthe rate of adjustments was reduced only to 98.4 percent.
Most forecasting practitioners expect that judgmental adjustments will lead to error reductions
of between five and 10 percent (Fildes and Goodwin, 2007). Yet little evidence supports that
belief. For example, Franses and Legerstee (2010) analyze the relative accuracy of forecasts
from models and forecasts that experts had subsequently adjusted for 194 combinations of one-
step-ahead forecasts in 35 countries and across seven pharmaceutical product categories. On
average, the adjusted forecasts were less accurate than the original model forecasts in 57
percent of the 194 country-category combinations.
Judgmental adjustments that are the product of structured procedures are less harmful. In an
experiment by Goodwin (2000), 48 subjects reviewed one-period ahead statistical sales
forecasts. When no specific instructions were provided, the subjects adjusted 85 percent of the
statistical forecasts; the revised forecasts had a median absolute percentage error (MdAPE) of
10 percent. In comparison, when subjects were asked to justify any adjustments by picking a
reason from a pre-specified list, they adjusted only 35 percent of the forecasts. The MdAPE
was 3.6 percent and thus 64 percent less than the error of the unstructured adjustment. In both
cases, however, the judgmental adjustments yielded forecasts that were 2.8 percent less
accurate than the original statistical forecasts.
Judgmental adjustments should only be considered when the conditions for successful
adjustment are met and when bias can be avoided (Goodwin and Fildes, 1999; Fildes,
Goodwin, Lawrence, and Nikolopoulos, 2009). In particular, accuracy-enhancing judgmental
adjustments may be possible when experts have good knowledge of important influences not
included in the forecasting model such as special events and changes in causal forces (Fildes
and Goodwin, 2007). Estimates of the effects should be made in ignorance of the model
forecasts, but with knowledge of what method and information the model is based upon
(Armstrong and Collopy, 1998; Armstrong, Adya, and Collopy, 2001). The experts’ estimates
should be derived in a structured way (Armstrong and Collopy, 1998), and the rationale and
process documented and disclosed (Goodwin, 2000). In practice, documentation of the reasons
for adjustments is uncommon (Fildes and Goodwin, 2007). The final forecasts should be
composed from the model forecasts and the experts’ adjustments. Judgmental adjustment under
these conditions is conservative in that more knowledge and information is used in the
forecasting process. Sanders and Ritzman (2001) found that subjective adjustments helped in
six of the eight studies in which the adjustments were made by those with domain knowledge,
but in only one of the seven studies that involved judges who lacked domain knowledge.
Checklists are useful when dealing with complex problems. Unaided judgment is inadequate
for analyzing the multifarious aspects of complex problems. Checklists are of enormous value
as a tool to help decision-makers working in complex fields. Think of skilled workers involved
with, for example, manufacturing and healthcare (Gawande, 2010).
In their review of 15 studies on the use of checklists in healthcare, Hales and Provonost (2006)
find substantial improvements in outcomes in all studies. For example, an experiment on
avoiding infection in intensive care units of 103 Michigan hospitals required physicians to
follow five rules when inserting catheters: (1) wash hands, (2) clean the patient’s skin, (3) use
full-barrier precautions when inserting central venous catheters, (4) avoid the femoral site, and
(5) remove unnecessary catheters. Adhering to this simple checklist reduced the median
infection rate from 2.7 per 1,000 patients to zero after three months. Benefits persisted 16 to 18
months after the checklist was introduced, and infection rates decreased by 66 percent.
Another study reports on the application of a 19-item checklist to surgical procedures on
thousands of patients in eight hospitals in cities around the world. Following the introduction of
the checklist, death rates declined by almost half, from 1.5 to 0.8 percent, and complications
declined by over one-third, from 11 to seven percent (Haynes, Weiser, Berry, Lipsitz, Breizat,
and Dellinger, 2009).
Given the effects of mistakes on human welfare, making decisions about complex problems
without the aid of a checklist when one is available is foolish. In fact, organizations and
regulators often require the use of checklists and penalize those who fail to follow them. The
completion of an aviation checklist by memory, for example, is considered a violation of
proper procedures.
Checklists should follow evidence. For example, evidence is identified for 21 of the 28
guidelines in the Golden Rule checklist. The other seven guidelines are logical consequences of
the Golden Rule’s unifying theory of conservatism in forecasting. Checklists based on faulty
evidence or faulty logic might cause harm by encouraging users to do the wrong thing and to
do so more consistently. Checklists that omit critical items risk doing more harm than good in
the hands of trusting users.
Even a comprehensive evidence-based checklist might be misapplied. To reduce the effects of
biases, omissions, and misinterpretations, ask two or more people to apply the checklist to the
problem independently. Select people who are likely to be unbiased and ask them to sign a
statement declaring that they have no biases pertaining to the problem at hand.
Computer-aided checklists are especially effective. Boorman (2001) finds that they decreased
errors by an additional 46 percent as compared to paper checklists. With that in mind, a
computer-aided Golden Rule checklist is available at no cost from
The Golden Rule has face validity in that forecasting experts tend to agree with the guidelines
of the Golden Rule Checklist. In a survey of forecasting experts conducted while this article
was being written, most respondents stated that they typically follow or would consider
following all but three of the guidelines. The guidelines that most experts disagreed with were
1.2.1/1.2.2which were originally formulated as one guideline: “specify multiple hypotheses
or conceal the purpose of the forecast”and 2.6“use structured analogies”. The survey
questionnaire and responses are available at
Table 2 summarizes the evidence on conservatism by type of forecasting method. There
are at least 12 papers providing evidence for each method, 105 papers in total for all
methods including combining, but excluding the guideline on unstructured judgmental
adjustments (6). Conservatism is found to improve or not harm forecast accuracy in 102
or 97 percent of the 105 papers. Rejecting conservative procedures increased error for the
type of method by between 25 percent, for extrapolation methods, and roughly 45 percent
for both problem formulation and causal methods. In other words, no matter what type of
forecasting method is appropriate for the forecasting problem, formulating the problem
and implementing forecasting methods in accordance with the relevant conservative
guidelines will avoid substantial error.
Table 3 summarizes the evidence to date on the Checklist guidelines. All the evidence is
consistent with the guidelines provided in the Checklist, and the gains in accuracy are
large on average. Details on how these improvements were assessed are provided in a
spreadsheet available from
There are, however, gaps in the evidence. For example, no evidence was found for seven of
the guidelines, and five guidelines were supported by single comparisons only. Research on
those guidelines would likely improve knowledge on how to most effectively implement
conservatism in forecasting.
Tracking down relevant studies is difficult, so there are likely to be more than the 109
papers with experimental comparisons identified in this article. Surely, then, new or
improved ways of being conservative can be found and improvements can be made in
how and when to apply the guidelines.
Current forecasting practice
Pop management books on forecasting appear over the years, and usually claim that
forecasting is predestined to fail. The books are often popular, but are the claims that
forecasting is impossible true?
No, they are not.
Substantial advances have been made in the development and validation of forecasting
procedures over the past century. That is evident, for example, in the astonishing improvements
summarized in Table 3. Improvements in forecasting knowledge have, however, had little
effect on practice in some areas. For example, in his review of forecasting for population,
economics, energy, transportation, and technology, Ascher (1978) concludes that forecast
accuracy had not improved over time. Similar findings have been obtained in agriculture (Allen
1994); population (Booth 2006, and Keilman 2008); sales (McCarthy, Davis, Golicic, and
Mentzer, 2006); and public transportation (Flyvbjerg, Skamris, Holm, and Buhl, 2005).
In other areas, forecasting practice has improved. Weather, sports, and election forecasting are
examples. Why does progress occur in some areas and not others?
The answer appears to be that practitioners in many areas fail to use evidence-based
forecasting procedures. That neglect might be due to ignorance of proper forecasting
procedures, or to the desire to satisfy a client with a forecast that supports a pre-determined
Bias can be introduced by forecasters as well as by clients. For example, once a forecasting
method is established, those who benefit from the status quo will fight against change. This
occurred when Billy Beane of the Oakland Athletics baseball team adopted evidence-based
forecasting methods for selecting and playing baseball players. The baseball scouts had been
making forecasts about player performance using their unaided judgment, and they were
incensed by Beane’s changes. Given the won-lost records, it soon became obvious that
Oakland won more games after the team had implemented evidence-based selection. The
change is described in Michael Lewis’s book, Moneyball, and depicted in the movie of the
same name. Most sports teams have now learned that they can either adopt evidence-based
forecasting procedures for selecting players, or lose more games (Armstrong, 2012b).
Statisticians may have a biased influence on forecasting research and practice in that their
skills lead them to prefer complex statistical methods and large databases. That bias would tend
to lead them to depart from cumulative knowledge about forecasting methods and domain
knowledge about the problem. The authors of this paper have between them about 75 years of
experience with forecasting, and they have done many literature reviews, yet they have not
found evidence that complex statistical procedures can produce consistent and reliable
improvements in the forecast accuracy relative to conservative forecasts from simple validated
procedures (Green and Armstrong, this issue).
Forecasters are more motivated to adopt evidence-based methods when they work in a field in
which there is competition, in which the forecasting is repetitive rather than one-off, and in
which forecast errors are salient to forecast users. Such fields include sports betting,
engineering, agriculture, and weather forecasting. Weather forecasters, for example, are well
calibrated in their short-term forecasts (Murphy and Winkler, 1984). In another example,
independently prepared forecasts of U.S. presidential election vote shares are unbiased and
extremely accurate (Graefe et al., 2014).
How to use the Golden Rule Checklist to improve forecasting practice
The Golden Rule Checklist provides evidence-based standards for forecasting procedures.
Using the Checklist requires little trainingintelligent people with no background in
forecasting can use it. Clients can require that forecasters use the checklist in order to fulfill
their contract. Clients can also rate the forecasting procedures used by forecasters against the
checklist. As an additional safeguard against bias, clients can ask independent raters to rate the
forecasting procedures used by forecasters against the checklist. Taking that extra step helps to
guard against violations of the Golden Rule by the client, as well as by the forecaster.
If the client is unable to assess whether the forecaster followed the guidelines in the Checklist,
the client should reject the forecasts on the basis that the forecaster provided inadequate
information on the forecasting process. If guidelines were violated, clients should insist that the
forecaster corrects the violations and resubmits forecasts.
The accuracy of forecasts should be judged relative to those from the next best method or
other evidence-based methodswith errors measured in ways that are relevant to decision
makersand not by reference to a graphical display. The latter can easily be used to suggest
that the forecasts and outcomes are somewhat similar.
Software providers could help their clients avoid violations of the Golden Rule by
implementing the Checklist guidelines as defaults in forecasting software. For example, it
would be a simple and inexpensive matter to include the contrary-series rule (3.3.2), and to
avoid using seasonal factors if there are fewer than three years of data (3.4.2).
The checklist can be applied quickly and at little expense. With about two hours of
preparation, analysts who understand the forecasting procedure should be able to guard against
forecasts that are unconservative. The goal of the Checklist is to ensure that there are no
violations. Remember that even a single violation can have a substantial effect on
accuracy. On average, the violation of a typical guideline increases the forecast error by
44 percent.
When bad outcomes occur in medicine, doctors are often sued if they failed to follow proper
evidence-based procedures. In engineering, aviation, and mining a failure to follow proper
procedures can lead to lawsuits even when damages have not occurred. The interests of both
clients and forecasters would be better served if clients insisted that forecasters use the
evidence-based Golden Rule Checklist, and that they sign a document to certify that they did
The first paragraph of this paper asked how a decision maker should evaluate a forecast. This
article proposes following the Golden Rule. The Golden Rule provides a unifying theory of
forecasting: Be conservative by adhering to cumulative knowledge about the situation and
about forecasting methods. The theory is easy to understand and provides the basis for a
checklist that forecasters and decision-makers can use to improve the accuracy of forecasts and
to reject forecasts that are likely to be biased and dangerously inaccurate.
The Golden Rule Checklist provides easily understood guidance on how to make forecasts for
any situation. The 28 guidelines in the Checklist are simple, using the definition of simplicity
provided by Green and Armstrong (this issue).
Use of the Golden Rule Guidelines improves accuracy substantially and consistently no
matter what is being forecast, what type of forecasting method is used, how long the forecast
horizon, how much data are available, how good the data are, or what criteria are used for
accuracy. The Golden Rule is especially useful for situations in which decision makers are
likely to be intimidated by forecasting experts.
The error reduction from following a single guidelinebased on experimental comparisons
from 70 papersranged from four to 64 percent and averaged 31 percent. In other words,
violating a single guideline typically increased forecast error by 44 percent. Imagine the effect
of violating more than one guideline.
The Golden Rule makes scientific forecasting comprehensible and accessible to all: Analysts,
clients, critics, and lawyers should use the checklist to ensure that there are no violations of the
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Table 1: Golden Rule Checklist with evidence on error reduction
Problem formulation
Use all important knowledge and information by…
selecting evidence-based methods validated for the situation
decomposing to best use knowledge, information, judgment
Avoid bias by…
concealing the purpose of the forecast
specifying multiple hypotheses and methods
obtaining signed ethics statements before and after forecasting
Provide full disclosure for independent audits, replications, extensions
Judgmental methods
Avoid unaided judgment
Use alternative wording and pretest questions
Ask judges to write reasons against the forecasts
Use judgmental bootstrapping
Use structured analogies
Combine independent forecasts from judges
Extrapolation methods
Use the longest time-series of valid and relevant data
Decompose by causal forces
Modify trends to incorporate more knowledge if the…
series is variable or unstable
historical trend conflicts with causal forces
forecast horizon is longer than the historical series
short and long-term trend directions are inconsistent
Modify seasonal factors to reflect uncertainty if…
estimates vary substantially across years
few years of data are available
causal knowledge is weak
Combine forecasts from alternative extrapolation methods, data
Causal methods
Use prior knowledge to specify variables, relationships, and effects
Modify effect estimates to reflect uncertainty
Use all important variables
Combine forecasts from dissimilar models
Combine forecasts from diverse evidence-based methods
Avoid unstructured judgmental adjustments to forecasts
Totals and Unweighted Average
* N: Number of papers with findings on effect direction.
n: Number of papers with findings on effect size. %: Average effect size (geometric mean)
Table 2: Evidence on accuracy of forecasts from conservative procedures by method type
----- Number of Comparisons -----
Method type
better or
Error increase
Problem formulation
All method types
Weighted average*
*Weighted by total papers
Table 3: Evidence on the 28 Golden Rule Guidelines
Evidence available on 21
Effect size reported for 20
More than one paper with effect size comparison 15
Range of error reductions -17 71%
Average error reduction per guideline 31%
... The rule for accurate forecast proposed by Armstrong et al. (2015) determines that the data must be extensive and valid. Although it is now possible to collect continuous long-term POS data, the longer the term span of the data to be analyzed, the lower the possibility of the timeseries structure being uniform across the entire time span. ...
... The proposed method theoretically improves the accuracy and the reliability of the recent linear trend estimated by non-piecewise linear approximation in the sense of AIC. Furthermore, it is considered that the optimal time span is used for automatically estimating the recent linear trend, which is required by the Armstrong et al. (2015)'s rule, please see section "Introduction." We are interested in the recent trend, regardless of the existence of the structural breaks. ...
... careless decisions about data time span may return misleading results and lead us to erroneously strategize marketing. The case study supports the assumed dependence of the forecast result on time span, claimed by Dekimpe and Hanssens (1995), McNown et al. (1995), and Armstrong et al. (2015). ...
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Due to the structural breaks in time series, estimated current trend of product sales differs between the case where data for the entire span are used and the case where only the most recent data are used. The purpose of this study is to establish the piecewise linear approximation (PLA) as a trend analysis method that accounts for the structural breaks. PLA uses the complete data to simultaneously estimate the breakpoints and the continuously connected trends, immediately before and after the break. Thus, PLA not only ensures the ease of interpretation of the results, but also eliminates the probability of using discretion by uniquely determining the current trend, making the estimated result reliable. The case study demonstrates the proposition that, several products’ sales trends and the necessity of determining an appropriate time span for data analysis, underwent changes at least once. The method’s validity is demonstrated by showing the changes of the sales trends immediately after analyzed store’s renovation. Data collection from the time period immediately prior to a structural break can help identify the factors changing the product sales.
... Further, HFC guidelines required our system to make predictions for new datasets and sources on the fly, often with only a couple hours notice before users could access the IFPs. Therefore, simple time-series models tended to outperform more complex, topic-specific models-a result supported by the general success of conservative forecasting approaches (Armstrong, Green, and Graefe 2015;Makridakis, Spiliotis, and Assimakopoulos 2018). Initially, all forecasts were based on the Auto ARIMA model (Hyndman and Athanasopoulos 2018), but later, this was supplanted by an ensemble (labeled "PHE2" below) of Auto ARIMA and an exponential smoothing state space model (Hyndman and Athanasopoulos 2018), which emerged from an overall pool of 28 candidate models. ...
... In complex settings, simpler, traditional statistical methods often outperform novel, complex methods (Makridakis, Spiliotis, and Assimakopoulos 2018). Research on forecast combinations supports the success of simple conservative methods (Armstrong, Green, and Graefe 2015). ...
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Sound decision‐making relies on accurate prediction for tangible outcomes ranging from military conflict to disease outbreaks. To improve crowdsourced forecasting accuracy, we developed SAGE, a hybrid forecasting system that combines human and machine generated forecasts. The system provides a platform where users can interact with machine models and thus anchor their judgments on an objective benchmark. The system also aggregates human and machine forecasts weighting both for propinquity and based on assessed skill while adjusting for overconfidence. We present results from the Hybrid Forecasting Competition (HFC)—larger than comparable forecasting tournaments—including 1085 users forecasting 398 real‐world forecasting problems over 8 months. Our main result is that the hybrid system generated more accurate forecasts compared to a human‐only baseline, which had no machine generated predictions. We found that skilled forecasters who had access to machine‐generated forecasts outperformed those who only viewed historical data. We also demonstrated the inclusion of machine‐generated forecasts in our aggregation algorithms improved performance, both in terms of accuracy and scalability. This suggests that hybrid forecasting systems, which potentially require fewer human resources, can be a viable approach for maintaining a competitive level of accuracy over a larger number of forecasting questions.
... Thus, we are a bit conservative, in the sense that we use standard errors of the estimates that are a little high. This is appropriate, given the often-observed underestimation of uncertainty in the prediction of densities (Armstrong et al. 2015, Makridakis et al. 2019. ...
... Hence, we do not attempt to reduce the residual variance, on purpose. Again, this is appropriate, as it reflects a cautious and conservative attitude towards density forecasting (Armstrong et al. 2015, Makridakis et al. 2019. ...
Technical Report
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We present a probabilistic forecast for the immigrant population of Norway and their Norwegian-born children ("second generation") broken down by age, sex, and three types of country background: 1. West European countries plus the United States, Canada, Australia, and New Zealand; 2. East European countries that are members of the European Union; 3. other countries. First, we compute a probabilistic forecast of the population of Norway by age and sex, but irrespective of migration background. The future development of the population is simulated 3 000 times by stochastically varying parameters for mortality, fertility and international migration to 2060. We add migrant group detail using stochastically varying random shares to split up each result from the previous step into six subgroups with immigration background, and one for the non-immigrants. The probabilistic forecast is calibrated against the Medium Variant of Statistics Norway's official population projection. The results show that a few population trends that were predicted for Norway to 2060 are quite certain: strong increases in the size of the immigrant population (more specifically those who belong to group 3) and of Norwegian-born children of immigrants. As to the age structures of immigrants and their children, the prediction intervals around the forecasts of these persons in one-year age groups are so wide that there is little information in these forecasts. For the population as a whole (irrespective of migrant background), forecasts for the age structure in one-year age groups are reliable up to around 2040, except for children born after 2022. For later years, the intervals become very wide for all ages. But ageing is certain.
... The prediction model is directly related to the decision model. Armstrong et al. (2015) define "being conservative" as the golden rule of forecasting, which is to "reduce the amount of change that is expected in the presence of uncertainty" (Armstrong et al., , p. 1718. Green and Armstrong (2015) suggest employing evidence-based methods to make predictions, adopting the most effective model when tested against the facts (Armstrong et al., , p. 1729. ...
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Purpose This paper examines the relationship between marketing automation emergence and the marketers' use of heuristics in their decision-making processes. Heuristics play a role for the integration of human decision-making models and automation in augmentation processes, particularly in marketing where automation is widespread. Design/methodology/approach This study analyzes qualitative data about the impact of marketing automation on the scope of heuristics in decision-making models, and it is based on evidence collected from interviews with twenty-two experienced marketers. Findings Marketers make extensive use of heuristics to manage their tasks. While the adoption of new automatic marketing tools modify the task environment and field of use of traditional decision-making models, the adoption of heuristics rules with a different scope is essential to defining inputs, interpreting/evaluating outputs and control the marketing automation system. Originality/value The paper makes a contribution to research on the relationship between marketing automation and decision-making models. In particular, it proposes the results of in-depth interviews with senior decision makers to assess the impact of marketing automation on the scope of heuristics as decision-making models adopted by marketers.
... From the beginning, SF has leveraged the insights of experts. In particular, the Delphi method, developed by the RAND Corporation, works with multiple rounds of expert surveys to anticipate future developments, to establish plausible time frames for their emergence and to identify the underlying factors that drive developments (Armstrong, Green, & Graefe, 2015;von der Gracht, 2012). In our context, we avoid placing constraints on the process of identifying SF methods by defining the method too narrowly. ...
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Many organizations use design thinking (DT) to develop future products and services. DT is often used for its ability to serve as a common “language” and platform to enable market-facing departments and technology-oriented units to cocreate innovations. DT has been shown to be a powerful tool for helping to identify and connect the needs of average customers (personas) with technical solutions that form the basis for winning products. In this paper, we investigate the extent to which DT professionals already use strategic foresight (SF) methods that anticipate future customer needs and highlight emerging technologies to expand classical DT and anchor their projects in the future. Using survey data on 302 DT projects, we report on the extent to which SF methods are used in DT projects, the overriding types of SF methods in DT projects, and their impact on project success.
... It is likely that significant linear trends would probably "damp" outside of the range of the data, particularly in situations where there are many non-respondents. This is a reason why damped trend forecasting methods that give conservative forecasts are often successful (e.g., Armstrong et al. 2015;Gardner 2015). In the use of wave analysis in the experimental section, this lack of conservatism is an advantage, as linear extrapolation is used to create worst case bounds for correlations. ...
This study proposes a method of nonresponse assessment based on meta-analytical file-drawer techniques, also known as worst-case resistance testing (WCRT), and suitable for a wide range of data collection scenarios. A general method is devised to estimate the number of significantly different nonrespondents it would take to significantly alter the results of an analysis. Estimates of nonrespondents can be plotted against effect sizes using "n-curves", with similar interpretation to p-curves or power curves. Variants of the general method are derived for tests of means and correlations. A sample using a well-established survey instrument from previous behavioral research is used to test the method. The results suggest that employing worst-case resistance testing can be used on its own or in conjunction with wave analysis to precisely flag nonresponse risks.
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For C-V2X systems, the selection of the best beam in a real-time mode becomes an increasingly critical and yet open topic. Most of the existing approaches adopt either conventional ARIMA or ANN. Recently, there has been research on adopting sequence-to-sequence (Seq2Seq) predictors with attentions to extract time series features and emphasis on critical information to achieve data prediction. In this paper, a Seq2Seq predictor integrating with a Transitional Matrix based Hard attention is presented and validated through an artificial test dataset with predefined transitional states. At first, the transition probability matrix is generated from previous time series data and fed into the “hard” attention module of Seq2Seq predictor to determine the weights during the training phase. Secondly, the presented Seq2Seq predictor was implemented and adopted to predict the best beams of a C-V2X beamforming selector built up by the authors. Experiments were conducted and captured data were used to validate the performance of the predictor. When compared with baseline models, the presented predictor can achieve an enhanced prediction accuracy in a gain of 10-12%.
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People often use analogies when forecasting, but in an unstructured manner. We propose a structured judgmental procedure whereby experts list analogies, rate similarity to the target, and match outcomes with possible target outcomes. An administrator would then derive a forecast from the information. When predicting decisions made in eight conflict situations, unaided experts’ forecasts were little better than chance at 32% accurate. In contrast, 46% of structured-analogies forecasts were accurate. Among experts who were able to think of two or more analogies and who had direct experience with their closest analogy, 60% of forecasts were accurate. Collaboration did not help. Key words: availability, case-based reasoning, comparison, decision, method.
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This article presents results from the first statistically significant study of traffic forecasts in transportation infrastructure projects. The sample used is the largest of its kind, covering 210 projects in 14 nations worth US$59 billion. The study shows with very high statistical significance that forecasters generally do a poor job of estimating the demand for transportation infrastructure projects. The result is substantial downside financial and economic risks. Such risks are typically ignored or downplayed by planners and decision makers, to the detriment of social and economic welfare. For nine out of ten rail projects passenger forecasts are overestimated; average overestimation is 106 percent. This results in large benefit shortfalls for rail projects. For half of all road projects the difference between actual and forecasted traffic is more than plus/minus 20 percent. Forecasts have not become more accurate over the 30-year period studied. If techniques and skills for arriving at accurate demand forecasts have improved over time, as often claimed by forecasters, this does not show in the data. The causes of inaccuracy in forecasts are different for rail and road projects, with political causes playing a larger role for rail than for road. The cure is transparency, accountability, and new forecasting methods. The challenge is to change the governance structures for forecasting and project development. The article shows how planners may help achieve this.
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Judgmental and statistical forecasts can each bring advantages to the forecasting process. One way forecasters can integrate these methods is to adjust statistical forecasts based on judgment. However, judgmental adjustments can bias forecasts and harm accuracy. Forecasters should consider six principles in deciding when and how to use judgment in adjusting statistical forecasts: (1) Adjust statistical forecasts if there is important domain knowledge; (2) adjust statistical forecasts in situations with a high degree of uncertainty; (3) adjust statistical forecasts when there are known changes in the environment; (4) structure the judgmental adjustment process; (5) document all judgmental adjustments made and periodically relate to forecast accuracy; (6) consider mechanically integrating judgmental and statistical forecasts over adjusting.
Principles of Forecasting: A Handbook for Researchers and Practitioners summarizes knowledge from experts and from empirical studies. It provides guidelines that can be applied in fields such as economics, sociology, and psychology. It applies to problems such as those in finance (How much is this company worth?), marketing (Will a new product be successful?), personnel (How can we identify the best job candidates?), and production (What level of inventories should be kept?). The book is edited by Professor J. Scott Armstrong of the Wharton School, University of Pennsylvania. Contributions were written by 40 leading experts in forecasting, and the 30 chapters cover all types of forecasting methods. There are judgmental methods such as Delphi, role-playing, and intentions studies. Quantitative methods include econometric methods, expert systems, and extrapolation. Some methods, such as conjoint analysis, analogies, and rule-based forecasting, integrate quantitative and judgmental procedures. In each area, the authors identify what is known in the form of `if-then principles', and they summarize evidence on these principles. The project, developed over a four-year period, represents the first book to summarize all that is known about forecasting and to present it so that it can be used by researchers and practitioners. To ensure that the principles are correct, the authors reviewed one another's papers. In addition, external reviews were provided by more than 120 experts, some of whom reviewed many of the papers. The book includes the first comprehensive forecasting dictionary.
We develop a simple method for forecasting and benchmarking consumer trial of new products. First, we extend the exponential trial growth models used in controlled test markets to the context of national product launches. This provides a marketing science benchmark against which our new approach can be compared.
When deciding for whom to vote, voters should select the candidate they expect to best handle issues, all other things equal. A simple heuristic predicted that the candidate who is rated more favorably on a larger number of issues would win the popular vote. This was correct for nine out of ten U.S. presidential elections from 1972 to 2008. We then used simple linear regression to relate the incumbent’s relative issue ratings to the actual two-party popular vote shares. The resulting model yielded out-of-sample forecasts that were competitive with those from the Iowa Electronic Markets and other established quantitative models. This model has implications for political decision-makers, as it can help to track campaigns and to decide which issues to focus on.
Today we find ourselves in possession of stupendous know-how, which we willingly place in the hands of the most highly skilled people. But avoidable failures are common, and the reason is simple: the volume and complexity of our knowledge has exceeded our ability to consistently deliver it - correctly, safely or efficiently. In this groundbreaking book, Atul Gawande makes a compelling argument for the checklist, which he believes to be the most promising method available in surmounting failure. Whether you're following a recipe, investing millions of dollars in a company or building a skyscraper, the checklist is an essential tool in virtually every area of our lives, and Gawande explains how breaking down complex, high pressure tasks into small steps can radically improve everything from airline safety to heart surgery survival rates. Fascinating and enlightening, The Checklist Manifesto shows how the simplest of ideas could transform how we operate in almost any field.