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Abstract

Problem How to help practitioners, academics, and decision makers use experimental research findings to substantially reduce forecast errors for all types of forecasting problems. Methods Findings from our review of forecasting experiments were used to identify methods and principles that lead to accurate forecasts. Cited authors were contacted to verify that summaries of their research were correct. Checklists to help forecasters and their clients undertake and commission studies that adhere to principles and use valid methods were developed. Leading researchers were asked to identify errors of omission or commission in the analyses and summaries of research findings. Findings Forecast accuracy can be improved by using one of 15 relatively simple evidence-based forecasting methods. One of those methods, knowledge models, provides substantial improvements in accuracy when causal knowledge is good. On the other hand, data models – developed using multiple regression, data mining, neural nets, and “big data analytics” – are unsuited for forecasting. Originality Three new checklists for choosing validated methods, developing knowledge models, and assessing uncertainty are presented. A fourth checklist, based on the Golden Rule of Forecasting, was improved. Usefulness Combining forecasts within individual methods and across different methods can reduce forecast errors by as much as 50%. Forecasts errors from currently used methods can be reduced by increasing their compliance with the principles of conservatism (Golden Rule of Forecasting) and simplicity (Occam’s Razor). Clients and other interested parties can use the checklists to determine whether forecasts were derived using evidence-based procedures and can, therefore, be trusted for making decisions. Scientists can use the checklists to devise tests of the predictive validity of their findings.
Forecasting Methods and Principles: Evidence-Based Checklists
J. Scott Armstrong1 and Kesten C. Green2
ABSTRACT
Problem: How to help practitioners, academics, and decision makers use experimental research findings to
substantially reduce forecast errors for all types of forecasting problems.
Methods: Findings from our review of forecasting experiments were used to identify methods and principles
that lead to accurate forecasts. Cited authors were contacted to verify that summaries of their research were
correct. Checklists to help forecasters and their clients practice and commission studies that adhere to principles
and use valid methods were developed. Leading researchers were asked to identify errors of omission or
commission in the analyses and summaries of research findings.
Findings: Forecast accuracy can be improved by using one of 15 relatively simple evidence-based
forecasting methods. One of those methods, knowledge models, provides substantial improvements in accuracy
when causal knowledge is good. On the other hand, data modelsdeveloped using multiple regression, data
mining, neural nets, and “big data analytics”—are unsuited for forecasting.
Originality: Three new checklists for choosing validated methods, developing knowledge models, and
assessing uncertainty are presented. A fourth checklist, based on the Golden Rule of Forecasting, was improved.
Usefulness: Combining forecasts within individual methods and across different methods can reduce
forecast errors by as much as 50%. Forecasts errors from currently used methods can be reduced by increasing
their compliance with the principles of conservatism (Golden Rule of Forecasting) and simplicity (Occam’s
Razor). Clients and other interested parties can use the checklists to determine whether forecasts were derived
using evidence-based procedures and can, therefore, be trusted for making decisions. Scientists can use the
checklists to devise tests of the predictive validity of their findings.
Key words: combining forecasts, data models, decomposition, equalizing, expectations, extrapolation, knowledge
models, intentions, Occam’s razor, prediction intervals, predictive validity, regression analysis, uncertainty
Authors’ notes:
1. This paper will be published in the Journal of Global Scholars of Marketing Science. We were pleased to
do so because of the interest by their new editor, Arch Woodside, in papers with useful findings, and the
journal’s promise of fast decisions and publication, offer of OpenAccess publication, and policy of
publishing in both English and Mandarin. The journal has also supported our use of a structured abstract
and provision of links to cited papers to the benefit of readers.
2. We received no funding for this paper and have no commercial interests in any method.
3. Most readers should be able to read this paper in less than one hour.
4. We endeavored to conform with the Criteria for Science Checklist at GuidelinesforScience.com.
Acknowledgments: We thank our reviewers, Hal Arkes, Kay A. Armstrong, Roy Batchelor, David Corkindale, Alfred G.
Cuzán, John Dawes, Robert Fildes, Paul Goodwin, Andreas Graefe, Rob Hyndman, Randall Jones, Magne Jorgensen,
Spyros Makridakis, Kostas Nikolopoulos, Keith Ord, Don Peters, and Malcolm Wright. Thanks also to those who made
useful suggestions: Raymond Hubbard, Frank Schmidt, Phil Stern, and Firoozeh Zarkesh. And to our editors: Harrison
Beard, Amy Dai, Simone Liao, Brian Moore, Maya Mudambi, Esther Park, Scheherbano Rafay, and Lynn Selhat. Finally, we
thank the authors of the papers that we cited for their substantive findings for their prompt confirmation and useful
suggestions on how to best summarize their work.
1 The Wharton School, University of Pennsylvania, Philadelphia, PA 19104, U.S.A. and Ehrenberg-Bass Institute,
University of South Australia Business School: +1 610 622 6480; armstrong@wharton.upenn.edu
2 School of Commerce and Ehrenberg-Bass Institute, University of South Australia Business School, University of
South Australia, City West Campus, North Terrace, Adelaide, SA 5000; kesten.green@unisa.edu.au.
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INTRODUCTION
Forecasts are important for decision-making in businesses and other organizations, and for governments.
A survey of practitioners, educators, and decision-makers found that they rated accuracyas the most important
of 13 criteria for judging forecasts (Yokum and Armstrong, 1995). Researchers were especially concerned with
accuracy. Consistent with that finding, improving forecast accuracy is the primary concern of this paper.
Since the 1930s, researchers have responded to the need for accurate forecasts by conducting
experiments testing multiple reasonable methods. The findings from those ground-breaking experiments have
greatly improved forecasting knowledge. In the late-1990s, 39 forecasting researchers from a variety of
disciplines summarized scientific knowledge on forecasting. They were assisted by 123 expert reviewers
(Armstrong 2001). The findings were used to develop 139 principles (condition-action statements), for
forecasting in various situations. In 2015, two papers further condensed forecasting knowledge as two
overarching principles: simplicity and conservatism (Green and Armstrong 2015, and Armstrong, Green, and
Graefe 2015, respectively).
While the advances in forecasting knowledge allow for substantial improvements in forecast accuracy,
that knowledge is largely ignored in academic journal articles and, we expect, also by practitioners. At the time
that the original 139 forecasting principles were published in 2001, a review of 17 forecasting textbooks found
that the typical book mentioned only 19% of the principles (Cox and Loomis 2001). Moreover, forecasting
software packages, which could help to ensure that the principles are used, were found to ignore about half of the
forecasting principles (Tashman and Hoover 2001).
CHECKLISTS TO IMPROVE FORECASTING
The use of evidence-based checklists avoids the need for memorizing and simplifies complex tasks. In
fields such as medicine, aeronautics, and engineering, a failure to follow an appropriate checklist can be grounds
for a lawsuit.
The use of checklists is supported by much research (e.g., Hales and Pronovost 2006). One experiment
assessed the effects of using a 19-item checklist for a hospital procedure. The study compared thousands of
patient outcomes in hospitals in eight cities around the world before and after the checklist was used. Use of the
checklist reduced deaths from 1.5% to 0.8% in the month after the medical procedures (Haynes et al. 2009).
Importantly, checklists improve decision-making even when the knowledge incorporated in them is well-known
to practitioners, and is known to be important (Hales and Pronovost 2006). To ensure that they include the latest
evidence, checklists should be revised routinely.
Convincing people to use checklists is easy. When engineers and medical doctors are told they must use
the checklist as a condition of their employment, and when use of the checklist is monitored, they use the
checklists. When we have paid people modest sums to complete tasks by using checklists, almost all of those
who accepted the task did so effectively. For example, to assess the persuasiveness of print advertisements, raters
hired through Amazon’s Mechanical Turk used a 195-item checklist to evaluate advertisementsconformance to
persuasion principles. The inter-rater reliability was high (Armstrong, Du, Green, and Graefe 2016).
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RESEARCH METHODS
We reviewed prior experimental research on which forecasting methods and principles lead to improved
forecast accuracy. To do so, we first identified relevant research by:
1) searching the Internet, mostly using Google Scholar;
2) contacting leading researchers for suggestions of important experimental findings;
3) checking key papers referred to in experimental studies and meta-analyses;
4) putting our working paper online with requests for evidence that we might have overlooked;
5) providing links to all papers in an OpenAccess version of this paper in order to allow readers to check
our interpretations of the original findings.
Given the enormous number of papers with promising titles, we screened papers by assessing whether
the “Abstract” or “Conclusions” sections provided evidence on the comparative value of alternative methods, and
full disclosure. Only a small percentage of the papers with promising titles met those criteria.
Only studies that examine many out-of-sample (ex ante) forecasts are considered as evidence in this
paper. For cross-sectional data, the jack-knifeprocedure allows for many forecasts by using all but one data
point to estimate the model, making a forecast for the excluded observation, then replacing that observation and
excluding another, and so on until forecasts have been made for all data points. Successive updating can be used
to increase the number of out-of-sample forecasts for time-series data. For example, to test the predictive validity
of alternative models for forecasting the next 100 years of global mean temperatures, annual forecasts were made
for horizons from one to 100 years-ahead starting in 1851. The forecasts were updated as if in 1852, then 1853,
and so on, thus providing errors for 157 one-year-ahead forecasts… and 58 one-hundred-year-ahead forecasts
(Green, Armstrong, and Soon 2009).
We attempted to contact the authors of all papers that we cited regarding substantive findings. We did so
on the basis of evidence that findings cited in papers in leading scientific journals are often described incorrectly
(Wright and Armstrong 2008). We asked the authors if our summary of their findings was correct and whether
our description could be improved. We also asked them to suggest relevant papers that we had overlooked
especially papers describing experiments with findings that conflicted with our conclusions. That practice was
shown to contribute to a substantially more comprehensive search for evidence than was achieved by computer
searches (Armstrong and Pagell 2003). In the case of six papers, we could not agree with the authors on the
interpretation of findings. We discarded our citations of those papers, as they were not essential to the purpose of
this paper.
Of the 90 papers with substantive findings that were not our own, we were able to contact the authors of
73 and received substantive, and often helpful, replies from 69. We coded the papers in the references section of
this paper, including the results of our efforts to contact authors.
Our review led to the development of five checklists. They provide evidence-based guidance on
forecasting methods, knowledge models, the Golden Rule of Forecasting, simplicity, and uncertainty.
VALID FORECASTING METHODS: CHECKLIST AND EVIDENCE
The predictive validity of a forecasting method is assessed by comparing the accuracy of forecasts from
the method with the accuracy of forecasts from currently used methods, or from simple benchmark methods such
as the naïve no-trend model, or from other evidence-based methods. Such testing of multiple reasonable
hypotheses is a requirement of the scientific method as described by Chamberlin (1890).
For categorical forecastssuch as whether a, b, or c will happen, or which of them would be better
accuracy is typically measured as a variation of percent correct. For quantitative forecasts, accuracy is assessed
by differences between ex ante forecasts and data on what actually transpired. The benchmark error measure for
evaluating forecasting methods is the Relative Absolute Error, or “RAE.It has been shown to be more reliable
than the Root Mean Square Error (Armstrong and Collopy 1992). Tests of a new methoda development of the
RAEcalled the Unscaled Mean Bounded Relative Absolute Error (UMBRAE)suggest that it is superior to
the RAE and other proposed alternatives (Chen, Twycross, and Garibaldi 2017). We suggest using both the RAE
4
and UMBRAE until additional testing has been done to provide a definitive conclusion on which is the better
measure.
Exhibit 1 lists 15 individual evidence-based forecasting methods. They are consistent with forecasting
principles and have been shown to provide out-of-sample forecasts with superior accuracy. The Exhibit also
identifies the knowledge needed to use each method. Combining within and across methods is recommended
(Checklist items 16 and 17.)
Exhibit 1: Forecasting Methods Application Checklist
Name of forecasting problem: ________________________________________________________________
Forecaster: ____________________________________________________ Date: ______________________
Method
Knowledge needed
Usable
method
Forecaster*
Respondents/Experts
(ý)
Judgmental methods
1.
Prediction markets
Survey/market design
Domain; Problem
o
2.
Multiplicative decomposition
Domain; Structural relationships
Domain
o
3.
Intentions surveys
Survey design
Own plans/behavior
o
4.
Expectations surveys
Survey design
Others’ behavior
o
5.
Expert surveys (Delphi, etc.)
Survey design
Domain
o
6.
Simulated interaction
Survey/experimental design
Normal human responses
o
7.
Structured analogies
Survey design
Analogous events
o
8.
Experimentation
Experimental design
Normal human responses
o
9.
Expert systems
Survey design
Domain
o
Quantitative methods (Judgmental inputs sometimes required)
10.
Extrapolation
Time-series methods; Data
n/a
o
11.
Rule-based forecasting
Causality; Time-series methods
Domain
o
12.
Judgmental bootstrapping
Survey/Experimental design
Domain
o
13.
Segmentation
Causality; Data
Domain
o
14.
Simple regression
Causality; Data
Domain
o
15.
Knowledge models
Cumulative causal knowledge
Domain
o
16.
Combining forecasts from a single method… o
SUM of VARIATIONS
[ ]
17.
Combining forecasts from several methods…o
COUNT of METHODS
[ ]
*Forecasters must always know about the forecasting problem, which may require consulting with the forecast client and domain
experts, and consulting the research literature.
†Experts who are consulted by the forecaster about their domain knowledge should be aware of relevant findings from
experiments. Failing that, the forecaster is responsible for obtaining that knowledge.
For most forecasting problems, several of the methods will be usable, and should be used, as we describe
below. An electronic version of the Exhibit 1 checklist is provided at ForecastingPrinciples.com in the top menu
bar under “Methods Checklist.”
Because we are concerned with methods that have been shown to improve forecast accuracy relative to
methods that are commonly used in practice, we do not discuss all methods that have been used for forecasting.
For example, multiple regression analysis is apparently one of the most widely used methods for developing
forecasting models. Given the evidence summarized in this paper, however, we recommend against the use of
multiple regression analysis and other data modeling approaches.
5
Clients should ask forecasters what methods they will use and why. If they mention a method that is not
listed in Exhibit 1, they should be asked to produce evidence that their method provides forecasts with smaller
errors than the relevant methods listed in the Exhibit.
Judgmental Methods
Expertise based on experience in similar situations can be useful for forecasting. Experience can lead to
simple “rules of thumb,” or heuristics, that provide quick forecasts for rapid decision-making. For example, the
emergency landing of US Airways Flight 1549the “Miracle on the Hudson”was a success because the pilot
used the gaze heuristic to forecast that landing on the Hudson River was a viable option, whereas returning to La
Guardia Airport was not (Hafenbrädl, Waeger, Marewski, and Gigerenzer 2016). Extensive research conducted
by Gerd Gigerenzer and the ABC group of the Max Planck Institute for Human Development in Berlin has found
that simple heuristics are superior to more complex and information intensive methods for many practical
problems.
For situations in which there are two or more important causal factors and where experts do not receive
frequent well-summarized feedback on the accuracy of their predictions, however, expertise and experience are,
on their own, of no apparent value. Such situations are common in business and government decision making.
Even leading experts’ unaided judgmental forecasts often turn out to be disastrously wrong, sometimes to the
delight of the media (e.g., see Cerf and Navasky 1998; Perry 2017).
Research on the accuracy of experts’ unaided judgmental forecasts about complex situations dates from
the early 1900s. An early review of the research led to the Seer-Sucker Theory: “No matter how much evidence
exists that seers do not exist, suckers will pay for the existence of seers” (Armstrong 1980). The Seer-Sucker
Theory has held up well over the years; in particular, a 20-year study comparing the accuracy of many forecasts
from experts with those of forecasts from novices and from naïve rules provided support (Tetlock 2005).
While unaided expert judgments should be avoided, topic experts can play a vital role in forecasting
when their judgments are incorporated using evidence-based methods. The next section describes nine structured
methods for forecasting using expert judgment.
1. Prediction markets
Prediction marketsalso known as betting markets, information markets, and futures marketshave
been used for forecasting since the 16th century (Rhode and Strumpf 2004). Monetary rewards attract people who
believe they have knowledge or information that enables them to make accurate predictions about the situation
they are betting on.
Prediction markets are especially useful when knowledge is dispersed and many participants are
motivated to trade repeatedly. Markets can rapidly revise forecasts when new information becomes available.
Forecasters using prediction markets need to be familiar with designing prediction markets and surveys.
The accuracy of forecasts from prediction markets was tested in eight published comparisons in the field
of business forecasting (Graefe 2011). The results were mixed. For example, prediction markets’ out-of-sample
forecast errors were 28% smaller than those from no-change models in one comparison. On the other hand,
averaging people’s judgments outperformed market forecasts in two of three comparisons. In another
comparison, forecasts from the Iowa Electronic Market (IEM) prediction market across the 100 days before each
U.S. presidential election from 2004 though 2016 were, on average, less accurate than forecasts from the
RealClearPolitics poll average, a survey of experts, and citizen forecasts (Graefe 2017a). The IEM prediction
market limits the bets to no more than $500, which likely reduces the number and motivation of participants.
Comparative accuracy tests based on 44 elections in eight countries other than the U.S., however, found that
forecasts from betting markets were more accurate than forecasts by experts, econometric models, and polls
(Graefe 2017b).
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2. Multiplicative decomposition
Decomposition has long been a key element of forecasting. A Google search for “decomposition” and
either “forecast” or “predict” found over 45 million results in December 2017.
Multiplicative decomposition involves dividing a forecasting problem into parts, forecasting each part
separately, and multiplying the forecasts of the parts to forecast the whole. For example, to forecast sales for a
brand, a firm might separately forecast total market sales and market share, and then multiply those components.
Decomposition is expected to be most effective at reducing forecast errors when suitable forecasting methods,
data availability, and directional effects of causal factors vary among the parts.
To assess the effect of decomposition on forecast accuracy, subjects in an experiment were presented
with five problems from an almanac, such as “How many packs (rolls) of Polaroid color films do you think were
used in the United States in 1970?” Some subjects were asked to make estimates of the total figure, while others
were asked to estimate each of the decomposed elements (Armstrong, Denniston, and Gordon 1975). Across that
study and two similar studies, forecast error was reduced by an average of 42% (MacGregor 2001).
Another study used graphical software to display the 68 monthly series from the M-Competition
(Makridakis et al. 1982) in ways that were designed to help users identify and forecast seasonality and trend
independently using their judgment. The study found that three postgraduate students with knowledge of time
series analysis and the software produced forecasts for one to 12 months into the future that had errors that were
7% less than those from the leading M-Competition method of deseasonalized single exponential smoothing. The
error reduction from software-assisted judgmental decomposition by 35 novices forecasting five time-series each
was 5% (Table II, Edmundson 1990).
3. Intentions surveys
Intentions surveys ask people how they plan to behave in specified situations. They can be used, for
example, to predict how people would respond to major changes in the design of a product. One meta-analysis
included 47 comparisons with over 10,000 subjects, and another provided a meta-analysis of 10 meta-analyses
involving over 83,000 subjects. Both found a strong relationship between people’s intentions and their future
behavior (Kim and Hunter 1993; Sheeran 2002).
Intentions surveys are especially useful when historical data are not available. They are most likely to
provide useful forecasts for short forecast time-horizons, and for important decisions (Morwitz 2001; Morwitz,
Steckel, and Gupta 2007).
To assess people’s intentions, the forecaster should prepare brief unbiased descriptions of the situation
(Armstrong and Overton 1971). Intentions should be expressed as probabilities such as 0 = ‘No chance, or almost
no chance (1 in 100)’, to 10 = ‘Certain, or practically certain (99 in 100).’ Responses can be used to calculate a
forecast of how people will behave, such as “3.2% of the population will buy the product in the next three
months” (Morwitz 2001).
The way a question is asked can have a large effect on responses. Two ways to reduce response error are
to: (1) pretest the questions to ensure that the respondents understand them in the way the forecaster intends, and
(2) use alternative ways to state a question, then average responses across questions. For more advice, see
Bradburn, Sudman, and Wansink (2004).
Including a monetary incentive to respond along with the questionnaire reduces non-response error
(Armstrong and Yokum 1994). The forecasters should resend the questionnaire to non-responders in follow-up
waves. Doing so allows one to estimate the effect of non-response by extrapolating across waves (Armstrong and
Overton 1977). Additional evidence-based procedures for selecting samples and obtaining high response rates are
described in Dillman, Smyth, and Christian (2014).
4. Expectations surveys
Expectations surveys ask people how they expect they or others will behave. Expectations differ from
intentions because people realize that the situation can change. For example, if you were asked whether you
intend to purchase a vehicle over the next year, you might say that you have no intention of doing so. However,
you realize that it is possible that your vehicle will develop a major problem. As a consequence, you might expect
7
that there is a chance that you will purchase a new car. As with intentions surveys, expectations surveys should
use probability scales, follow evidence-based procedures for survey design, use representative samples, obtain
high response rates, and correct for non-response bias by extrapolating across waves.
Following the U.S. government’s 1932 prohibition of prediction markets for political elections,
expectation surveyswhich poll a representative sample of potential voters on how others would votewere
introduced (Hayes 1936). Those “citizen expectations” surveys correctly predicted the popular vote winners of
the U.S. Presidential elections in 89% of the 217 surveys from 1932 to 2012. Furthermore, citizens’ expectations
provided more accurate out-of-sample forecasts of the national vote share than polls, prediction markets, models,
and experts across the seven U.S. Presidential elections from 1988 to 2012 (Graefe 2014), and again in 2016.
Over the 100 days before the 2016 election, the error of citizens’ expectations forecasts of the popular vote in
seven U.S. Presidential elections from 1992 through 2016 averaged 1.2 percentage points. In comparison, the
error of a typical poll aggregator was, at 2.6 percentage points, more than twice as high. (Graefe, Armstrong,
Jones, and Cuzán, 2017).
5. Expert surveys
Use written questions and instructions for self-completion surveys to ensure that each expert is
questioned in the same way. Apply the same procedures for developing questions as those described for
expectations surveys above.
Forecasters should obtain forecasts from at least five experts, and up to 20 for important forecasts
(Hogarth 1978). That advice was followed in forecasting the popular vote for U.S. Presidential elections from
2004 to 2016, when surveys of about fifteen experts led to an average error of 1.6 percentage points, compared to
1.7 percentage points for combined polls (Graefe, Armstrong, Jones, and Cuzán 2017, and personal
correspondence with Graefe). Additional advice on the design of expert surveys is provided in Armstrong (1985,
pp.108-116).
Delphi is an extension of the expert survey approach whereby the survey is conducted over two or more
rounds. After each round, anonymous summaries of the experts’ forecasts and reasons are provided to the experts.
The process is repeated until forecasts change little between roundsusually two or three rounds are sufficient.
Use the median or mode of the experts’ final-round forecasts as the Delphi forecast. Delphi is expected to be most
useful when the different experts each have different information relevant to the problem (Jones, Armstrong, and
Cuzán 2007).
Forecasts from Delphi were more accurate than forecasts from traditional meetings in five studies, about
the same accuracy in two, and less accurate in one. Delphi forecasts were more accurate than forecasts from
traditional surveys of expert opinion for 12 of 16 studies, with two ties and two cases in which Delphi was less
accurate. Among those 24 comparisons, Delphi improved accuracy in 71% and harmed it in 12% (Rowe and
Wright 2001).
Delphi is attractive to managers because judgments from dispersed experts can be obtained without the
expense of arranging meetings. It has an advantage over prediction markets in that the participants provide
reasons for their forecasts (Green, Armstrong, and Graefe 2007). Software for the procedure is freely available at
ForecastingPrinciples.com.
6. Simulated interaction
Simulated interaction uses role-playing to forecast decisions by two or more parties with conflicting
interests. Situations that have been used for testing the method include an attempt to secure an exclusive
distribution arrangement with a major supplier, a union-management dispute over pay and conditions, and artists
demanding that the government provide them with financial support.
The forecaster provides each role-player with a description of one the main protagonists’ roles, and a
brief description of the situation including a list of possible decisions. The role-players are asked to engage in
realistic interactions with one another, staying in their roles until a decision is reached. The simulations typically
last less than an hour.
Relative to unaided expert judgmentthe most common method—simulated interaction reduced
forecast errors by 57% on average for eight conflict situations, including those described above and an attempted
8
hostile takeover of a corporation, and a military standoff between two countries over access to water (Green
2005). The method seems to work best when naïve role players do not know each other, have no prior opinions
about the situation, and no agenda beyond that indicated by their role.
The alternative approach of “putting oneself in the other person’s shoes” has been proposed. U.S.
Secretary of Defense Robert McNamara suggested that if he had done this during the Vietnam War, he would
have made better decisions.3 A test of the “role-thinking” approach, however, found no improvement in forecast
accuracy relative to that of unaided judgment. It is too difficult to think through the interactions in a complex
situationactive role-playing between parties is necessary to provide sufficient realism (Green and Armstrong
2011).
7. Structured analogies
The structured analogies method involves asking ten or so experts to suggest situations that were similar
to the one for which a forecast is required, the target situation. The experts are given a description of the target
situation and are asked to identify analogous situations, rate their similarity to the target, and match the outcomes
of their analogies with possible outcomes of the target situation. An administrator takes the target situation
outcome implied by each expert’s top-rated analogy and calculates the modal outcome as the forecast. The
method should not be confused with the common use of analogies to justify a decision that is preferred by the
forecaster or client.
Structured analogies forecasts were 41% more accurate than unaided judgment forecasts in forecasting
decisions in the eight real conflicts used in research on the simulated interaction method described above (Green
and Armstrong 2007a). Structured analogies were also used to forecast the effects of incentives to promote laptop
purchases by university students, and a program offering certification on Internet safety to parents of high school
students. The error of those structured analogies forecasts was 8% lower than the error of forecasts from unaided
judgment (Nikolopoulos, Petropoulos, Bougioukos, and Khammash 2015). A procedure akin to structured
analogies was used to forecast box office revenue for 19 unreleased movies, in which raters identified analogous
movies from a database and rated them for similarity. The revenue forecasts from the analogies were adjusted for
advertising expenditure and whether the movie was a sequel. Errors from the structured analogies forecasts were
less than half those of forecasts from simple and complex regression models (Lovallo, Clarke and Camerer
2012). Across the ten comparative tests from the three studies described above, the error reductions from using
structured analogies averaged about 40%.
8. Experimentation
Experimentation is widely used and is the most valid and reliable method for determining cause-and-
effect relationships. Knowledge of the direction of effects and estimates of the strength of effects can then be used
to make forecasts. Experiments can be conducted in laboratories. An analysis of organizational behavior
experiments found that laboratory experiments yielded similar findings to field experiments (Locke 1986).
Alternatively, forecasters can analyze natural experiments to identify causal relationships and make
forecasts. For example, the regulation and deregulation of industries provided natural experiments on the effect of
regulation on consumer welfare. Winston (1993) found that regulation harmed customers in eight of the nine
markets for which such experimental data were available, and was of no net benefit in the case of the ninth
market.
9. Expert systems
Expert systems are developed by asking experts to describe the steps they take while they make
forecasts, then describing that process using software. The resulting expert system should be complete, simple,
and clearly described.
A review of 15 comparisons found that expert system forecasts were more accurate than forecasts from
unaided judgments (Collopy, Adya and Armstrong 2001). Two of the studieson gas, and on mail order
catalogue salesfound that the expert systemsforecast errors were 10% and 5% smaller, respectively, than
3 From the 2003 documentary film, “Fog of War.”
9
those of unaided judgment. While the evidence available on predictive validity is scant, the method appears
promising.
Quantitative Methods
Quantitative methods require numerical data on or related to the forecasting problem. Quantitative
methods can also draw upon judgmental methods, such as decomposition, in order to make the best use of
knowledge and data. These models also enable the explicit use of causal relationships.
This section describes six evidence-based quantitative forecasting methods. Other than the first of the
methods (extrapolation), the methods rely heavily on causal knowledge to forecast the effects of changes in
causal variables. Such forecasts can be used for policy making, and for developing contingency plans.
Forecasting what will happen when the causal variables are out of the decision makerscontrol, however,
requires that the causal variables are accurately forecast.
10. Extrapolation
While extrapolation methods can be used for any problem requiring forecasts of a time series, they are
especially useful when little is known about the factors affecting the forecast variable, causal variables are not
expected to change much, or causal variables cannot be forecast with much accuracy.
Exponential smoothing, which dates back to Brown (1959 and 1962), is easy to understand. It is a
sensible approach, because it uses all historical data in a moving average that puts more weight on the most recent
data. For a review of exponential smoothing, see Gardner (2006).
One should not assume that a trend will continue at the same rate, even in the short-term. It could
increase or decrease in response to changes in the causal forces that drive the trend. The greater the uncertainty
about the situation, the greater is the need to damp the trend toward zerothe no change forecast. A review of 10
experimental comparisons found that, on average, damping the trend toward zero reduced forecast errors by
almost 5% and reduced the risk of large errors compared to forecasts that assumed a constant trend (Armstrong
2006). Gardner’s software for damped-trend extrapolation can be found at ForcastingPrinciples.com. When there
is a long-term trend and the causal factors are expected to continuesuch as with the real prices of resources
(Simon 1996)damping toward the long-term trend is appropriate.
When extrapolating for time periods less than a year, estimate the effects of seasonal influences and
remove them from the data. Forecast the seasonally-adjusted series, then seasonalize the forecasts. In forecasts
for 68 monthly economic series over 18-month horizons from the M-Competition, seasonal adjustment reduced
forecast errors by 23% (Makridakis, Andersen, Carbone, et al. 1984, Table 14).
Forecasters should damp statistical estimates of seasonal influences. Such estimates are uncertain and
standard seasonal adjustment procedures tend to overfitthe data. Miller and Williams (2003, 2004) provide
procedures for damping seasonal factors. When they damped the seasonal adjustments for the 1,428 monthly
time-series from the M3-Competition, the accuracy of the forecasts improved for 59% to 65% of the time series,
depending on the horizon. The broad findings were replicated by Boylan, Goodwin, Mohammadipour, and
Syntetos (2015). Software for the Miller-Williams procedures and the M3-Competition data are freely available
at .
Damping by averaging seasonal factors across analogous series also improves forecast accuracy. In one
study, combining seasonal factors from related products, such as snow blowers and snow shovels, reduced the
average forecast error by about 20% (Bunn and Vassilopoulos 1999). In another study, pooling monthly seasonal
factors for crime rates for six city precincts reduced the error of exponential smoothing forecasts by about 7%
compared to using seasonal factors that were estimated individually for each precinct (Gorr, Oligschlager, and
Thompson 2003, Figure 4).
Multiplicative decomposition can be used to incorporate causal knowledge into extrapolation forecasts.
For example, when forecasting time-series data, it often happens that the series is affected by causal forces
characterized as growth, decay, opposing, regressing, supporting, or unknown. In such a case, one can
decompose the time series by causal forces that have different directional effects, extrapolate each component,
and then recombine. Doing so is likely to improve accuracy under two conditions: (1) domain knowledge can be
used to structure the problem so that causal forces differ for two or more of the component series, and (2) it is
10
possible to obtain relatively accurate forecasts for each component. For example, to forecast motor vehicle
deaths, one study forecast the number of miles driven, a series that would be expected to grow, and the death rate
per million passenger miles, a series that would be expected to decrease due to better roads and safer cars. The
two extrapolation forecasts were then multiplied to get total deaths. When tested on five time series that clearly
met the two conditions, decomposition by causal forces reduced out-of-sample forecast errors by two-thirds. For
the four series that partially met the conditions, decomposition by causal forces reduced error by one-half. There
was no gain or loss in forecast accuracy when the conditions did not apply (Armstrong, Collopy, and Yokum
2005).
Additive decomposition can also be considered for extrapolation problems. One approach that is useful
when the most recent data are uncertain or liable to subsequent revision is to forecast the starting level and trend
separately, and then add thema procedure called “nowcasting.” Three comparative studies found that, on
average, nowcasting reduced errors for short-range forecasts by 37% (Tessier and Armstrong 2015).
11. Rule-based forecasting
Rule-based forecasting (RBF) uses knowledge about evidence-based extrapolation along with causal
knowledge to forecast time-series data. To use RBF, first identify which of 28 features best characterize the
series to be forecast. Features include forecast horizons, the amount of data available, and the existence of
outliers. Then use the 99 RBF rules to weight the alternative extrapolation models and combine the models’
forecasts (Armstrong, Adya and Collopy 2001).
For one-year-ahead ex ante forecasts of 90 annual series from the M-Competition (available on
ForecastingPrinciples.com), the Median Absolute Percentage Error of RBF forecasts was 13% smaller than that
of equally weighted combined forecasts. For six-year-ahead ex ante forecasts, the RBF forecast errors were 42%
smaller, likely due to the increasing importance of causal effects over longer horizons. RBF forecasts were also
more accurate than equally weighted combinations of forecasts in situations involving strong trends, low
uncertainty, stability, and good domain expertise. RBF forecasts had little or no accuracy advantage over
unweighted combinations of forecasts for other situations (Collopy and Armstrong 1992). Testing by Vokurka,
Flores, and Pearce (1996) provided supporting evidence for the relative accuracy of RBF forecasts.
One of the 99 RBF rules, thecontrary series ruleis especially important, as well as simple and
inexpensive to apply. It states that one should not extrapolate a trend if the direction of a time series expected by
domain experts is contrary to the recent trend of the time series. The use of that rule alone yielded improvements
in extrapolating time-series data from five data sets. In particular, for longer-term (six-years ahead) forecasts, the
error reductions exceeded 40% (Armstrong and Collopy 1993).
12. Judgmental bootstrapping
This method was developed in the early 1900s to provide forecasts of the size of the upcoming corn
harvest in the U.S. In the 1940s, the method was used successfully for personnel selection (Meehl 1954) and has
been supported by subsequent research (e.g., Dawes and Corrigan 1974; Grove, Zaid, Lebow, Snitz, and Nelson
2000). The method uses regression analysis to estimate coefficients for the variables that experts use to make
judgmental forecasts. The dependent variable is not the outcome, but rather the experts’ predictions of the
outcome given the values of the causal variables. Among researchers in forecasting, the method has, in recent
decades, been called “judgmental bootstrapping.” In effect, it uses a quantitative model of the experts’ use of
causal information for forecasting to improve upon the expertsforecast accuracy.
In comparative studies to date, the bootstrap model’s forecasts were more accurate than those of the
experts whose judgments they were based on. The gain in accuracy arises from the quantitative model’s more
consistent application of the expert’s mental model. In addition, the model does not become distracted by
irrelevant information and variables, nor does it become tired or irritable.
The first step for developing a judgmental bootstrap model is to ask experts to identify causal variables
based on their domain knowledge. Then ask them to make predictions using data on the variables. For example,
they could be asked to forecast the likelihood of success of doctoral candidates.
Judgmental bootstrap models can be estimated from experts’ predictions made on the basis of
hypothetical data on the causal variables. Doing so allows the forecaster to ensure that the causal variables vary
11
substantially and independently of one another. That use of experimental design overcomes many of the
deficiencies of multiple regression. It also enables one to make forecasts for situations for which actual data are
not available. Once developed, the bootstrap model can provide forecasts at a low cost and for different
situationse.g., for a new product with different features.
Despite the discovery of the method and evidence on its usefulness, its early use was confined to
agricultural predictions. Social scientists rediscovered the method in the 1960s, and tested its predictive validity.
A review of those studies found that judgmental bootstrapping forecasts were more accurate than those from
unaided judgments in eight of 11 comparisons, with two tests finding no difference and one finding a small loss
in accuracy (Armstrong 2001a). The one failure occurred when the experts relied on an irrelevant variable that
was not excluded from the bootstrap model. The typical error reduction was about 6% relative to unaided
judgment.
Many universities taught the methods to their students, but we are aware of only one that adopted the
method, despite the fact that one of the earliest validation tests showed that it provided a more accurate and less
expensive way of predicting success in a PhD program (Dawes 1971).
In 2002, the Oakland Athletics baseball team adopted a version of judgmental bootstrapping. Attempts
were made to block the use of the method by the experts who traditionally used their judgment to make the
selection decisionsthe managers, owners, and scouts. But the new general manager persisted, and the team
performed well. Other professional sports teams subsequently adopted the method, improving both won-lost
ratios and profitability (Armstrong 2012).
13. Segmentation
Segmentation in forecasting involves structuring the problem in order to make best use of knowledge
and data about parts, or sub-populations, that are expected to behave differently. Appropriate methods are used to
make forecasts for each part, and the forecasts for the parts are then added to derive a forecast for the whole.
Segmentation attracted widespread attention when it was used to forecast the 1960 Kennedy-Nixon election
outcome (Pool, Abelson and Popkin 1965).
The Port of New York Authority used the method in 1955 to forecast air travel demand ten years hence.
Their analysts divided airline travelers into segments of 130 business traveler types and 160 personal traveler
types. The personal travelers were segmented by age, then by occupation, income, and education; and the
business travelers were segmented by occupation, then industry, and finally income. Data on each segment were
obtained from the census and from a survey on travel behavior. To derive the forecast, the official projected air
travel population for 1965 was allocated among the segments, and the number of travelers and trip frequency
were extrapolated using 1935 as the starting year with zero travelers. The resulting forecast of 90 million trips
was only 3% different from the actual 1965 figure (Armstrong 1985).
To use segmentation, identify important causal variables that can be used to define the segments, and
their priorities. Then determine cut-pointse.g., different age categories of peoplefor each variable. Use more
cut-points when there are non-linearities in the relationships and fewer cut points when the samples of data are
smaller. Next, forecast the population of each segment and the behavior of the population within each segment by
using the typical behavior. Finally, combine the population and behavior forecasts for each segment and sum
across segments. The method is most likely to be useful when much data are available.
Segmentation is suitable for situations in which variables are interrelated, the effects of variables are non-
linear, and prior causal knowledge is good. These conditions occurred, to a reasonable extent, in a study where
data from 2,717 gas stations were used to estimate a segmentation model for forecasting weekly gasoline sales
volumes. Data were available on nine binary variables and ten other variables including type of area, traffic
volumes, length of street frontage, presence of a canopy, and whether the station was open 24 hours a day. The
method was tested using a holdout sample of 3,000 stations. The segmentation model forecast errors (Mean
Absolute Percentage Errors) were 29% smaller than the errors of a multiple regression model estimated using the
same variables and data (Armstrong and Andress 1970).
A review of the literature on segmentation is provided by Armstrong (1985, Chapter 9). While the
evidence on predictive validity is not substantial, the method is sensible, as it is based on decomposition. Interest
12
in segmentation fell away after the 1970s, but we expect that it would be more useful now than ever before, given
the availability of large databases.
14. Simple regression
Simple regression analysis can be used to forecast the effect of changes in a single causal variable. The
method is conservative in that it reduces the effect size estimate toward the meanvia the calculation of a
constant termin response to variations in the relationship found in the estimation data. For a forecasting model
estimated using simple regression to be useful, one must be able to control or accurately forecast the causal
variable.
The traditional form of a simple regression model is y = a + bx, where yis the variable to be forecast
(dependent variable), ais the constant, b is the effect size, and x” is the causal variable. The method is
appropriate for forecasting problems that involve good prior knowledge about a strong causal relationship, along
with valid and reliable data on the dependent and causal variables. A basic assumption is that the forecaster must
be able to accurately control or forecast the causal variable.
Transform the data so that the simple regression model provides a realistic representation of the causal
relationship. For example, calculating logarithms of the causal and dependent variables before estimating the
model will result in an effect size estimate in the form of an elasticity. Elasticities are the percentage change in the
variable to be forecast that would result from a one percent change in the causal variable. A price elasticity of
demand of -1.2 for beef, for example, means that one would expect a price increase of 10% to result in a 12%
decrease in the quantity demanded, all else being equal. Other transformations to consider include expressing the
variables in per capita terms, and adjusting the data for the effect of currency inflation and seasonality.
The least squares method of estimating regression model coefficients has the effect of giving extreme
data values an excessive influence on the estimate of the effect size. To avoid that, adjust or remove outliers from
the estimation data. One way to do soknown as “winsorizing”is to set the outlier to the value of the most
extreme observation in which you have confidence (Tukey 1962). Forecasters should specify the rules for
determining outliers before doing any analysis in order to avoid the temptation to make adjustments to support a
preferred hypothesis. Another sensible approach is to estimate the regression model by minimizing the absolute
error (e.g., Dielman 1986; Dielman 1989).
Multiple regression
What if more than one causal variable is important? Multiple regression analysis (MRA) might seem to
be an obvious solution, but its use with non-experimental data leads to multicollinearity and interactions among
causal variables. In addition, data on the variables are typically subject to measurement errors and validity
concerns that make assessing the relative weight of each variable problematic. That complexity puts MRA at a
considerable disadvantage to simple regression as a method for estimating causal relationships: MRA fails
Occam’s razor.
To our knowledge, MRA was adopted without any testing of its predictive validity. The first
comparative test that we are aware of involved making ten-year ahead forecasts of the populations of 100
counties in North Carolina. A multiple regression model with six causal variables was used to make the forecasts.
For comparison, six simple regressions were estimated, one for each variable; their forecasts were then averaged
for each county. The Mean Absolute Percentage Error of forecasts from the MRA model was 64% higher than
that of the combined simple-regression model forecasts (Namboodiri and Lalu 1971).
Another test obtained forecasts for 20 data sets using MRA models with from 3 to 19 causal variables.
The data sets included problems such as predicting professors’ salaries and high school dropout rates. MRA was
compared with an equal weights model using the same variables, and also with the simple “take-the-best(causal
variable) approach based on the forecaster’s information. The MRA produced 1% fewer correct forecasts than
were obtained from equal weights models and 3% fewer than from the take-the-best approach (Table 5-4,
Czerlinski, Gigerenzer and Goldstein, 1999).
MRA forecasts of the popular vote for U.S. Presidential elections from MRA models were available
from eight leading political forecasters. Their accuracy was compared with those from a simple regression using
the “best” variable (typically the “economy”). Forecasts were made for each of the last 100 days of the ten U.S.
13
Presidential election years from 1972 to 2008 (a total of 1,000 forecasts). The MRA forecasts were less accurate
than the simple regression model forecasts with a Mean Absolute Error of 3.8% compared to 3.6% (Graefe and
Armstrong 2012).
Data models
Beginning in the 1960s, advances in technology made it feasible for analysts to use tests of statistical
significance to select multiple “predictor variables” and estimate relationships. We refer to the resulting models as
“data models.” The trend started in the mid-1900s with stepwise regression. It spawned procedures with names
such as big data, analytics, data mining, and neural nets. One claim is that objectivity is increased by letting the
data speak for themselves. As we show below, in practice these techniques have the opposite effect.
Einhorn (p. 367, 1972) was among the first to warn against data models. He concluded, “Access to
powerful new computers has encouraged routine use of highly complex analytic techniques, often in the absence
of any theory, hypotheses, or model to guide the researcher's expectations of results.” He likened the practice to
alchemy. For a further discussion of the deficiencies of regression analysis in practice, see Armstrong (2012b).
The only scientific way to identify relationships in complex situation is to conduct experiments to
identify the effects of proposed causal variables under different conditions. Data models ignore cumulative
scientific knowledge, and rely only on the data.
Despite the widespread understanding that correlation does not imply causation, data models are based
on statistically significant correlations: not on causal relationships but on “predictor variables.”. About 32% of the
182 regression papers published in the American Economic Review in the 1980s relied on statistical significance
for choosing predictor variables (Ziliak and McCloskey 2004). The situation was worse in the 1990s, as 74% of
137 such papers did so.
Statistical significance testing is detrimental to advances in science (Armstrong 2007a,b). A theoretical
analysis titled “Why most published research findings are false” demonstrated how using statistical significance
testing along with testing for a preferred hypothesis leads to the publication of incorrect research findings
(Ioannidis 2005). Data models can be, and are, used to support any desired conclusions through such dubious
practices as proposing hypotheses after analyzing the data, trying out variables in order to find ones that support a
preferred hypothesis, discarding observations that do not support the desired hypothesis, selecting unreasonable
null hypotheses, using large sample sizes to ensure statistical significance, and ignoring findings by other
researchers that do not support the desired hypothesis. These procedures are common tactics in advocacy
research. Armstrong and Green (2018) summarize evidence on the extent to which such questionable procedures
are used in scientific journals.
Our searches have been unable to find any experimental comparisons showing that MRA or other data
modeling techniques have out-of-sample predictive validity equal to that of the simple evidence-based methods
identified in Exhibit 1. To the contrary, the evidence that we have found shows that data models are unsuited to
forecasting.
A comprehensive analysis of the accuracy of data mining found that forecasts from data-mining models
had consistently lower out-of-sample predictive validity than simple alternative models. In one test, the authors of
the study asked a data-mining expert to make predictions using a set of data. The expert did so, and identified
many statistically significant relationships in the data. Unbeknownst to the data miner, the numbers were random
(Keogh and Kasetty 2003) In personal correspondence with us, Keogh stated, “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.”
15. Knowledge models
Some forecasting problems are characterized by knowledge of many important causal variables.
Consider, for example, predicting which players will do well in sports, who would be an effective company
executive, which countries will have the highest economic growth, or which applicants for immigration are most
likely to pose a security risk. Knowledge models are suitable for such problems.
Benjamin Franklin proposed a form of a knowledge model in a letter to his friend, Joseph Priestley, who
had written to Franklin about a “vexing decision” he was struggling to make. Franklin’s method was to list pros
14
and cons for each alternative giving each a subjective weight, then to sum the lists to determine which alternative
has the largest score in its favor. Franklin called his approach “prudential algebra.” 4
A similar approach, called “experience tables,” was used in the early 1900s for deciding which prisoners
should be given parole (Burgess 1936). Another version was called “configural analysis.” It came into limited use
in the mid-1900s. The approach was found to have predictive validity (e.g., see Babst, Gottfredson and Ballard,
1968). Yet another version was developed more recently under the term “index methodwhere there was
considerable testing as we describe below.
We propose the name “knowledge model” because the term is more descriptive than the previous terms.
Exhibit 2 provides a checklist for developing a knowledge model.
Exhibit 2: Knowledge Model Development Checklist
a.
Identify all important causal variables using domain knowledge and findings from experiments
¨
b.
Discard a causal variable if it cannot be controlled, or accurately forecast
¨
c.
Determine the directions of causal variables’ effects on the variable to be forecast
¨
d.
Determine the relative magnitudes of causal variables’ effects on the variable to be forecast if possible
¨
e.
Specify model as dependent variable score equals the sum of weighted causal variables
¨
f.
Estimate relationship between scores and dependent variable values by regression analysis if feasible
¨
a. Identify all important causal variables using domain knowledge and findings from experiments
Follow the scientific method by using prior knowledge to identify causal variables. With knowledge models,
causal variables can be as simple as binary; for example, “is taller than opponent” for an election forecasting
model. In some situations, causal variables are obvious from logical relationships. In cases where they are not,
consider surveying three to five domain experts. When the validity of a proposed causal variable is uncertain,
consult findings from experiments, especially meta-analyses of experiments, in order to determine whether there
is sufficient support for the use of the variable. Consider this example as an illustration of the importance of
relying on experimental evidence: evidence on the direction of the effect of each of 56 persuasion principles from
Armstrong (2010) was obtained from non-experimental data as well as from experimental data. The findings
from different experiments were in the same direction for each principle, but for only two-thirds of the principles
in the non-experimental data (Armstrong and Patnik 2009).
b. Discard a causal variable if it cannot be controlled, or accurately forecastIf a causal variable
cannot be forecast or controlled, including it in a model can only harm the accuracy of forecasts from the model.
c. Determine the directions of causal variables’ effects on the variable to be forecastThe directional
effects of some variables are obvious from logic or common knowledge about the domain. If the direction is not
obvious, refer to experimental studies. For example, opinions about the effects of gun regulations on crime vary
and opposing opinions among voters and politicians have led U.S. counties and states to change their laws to
either restrict or make gun ownership easier. These natural experiments provide a method to scientifically
determine which opinion is correct, as was done by Lott (2010) and Lott (2016). If there is neither obviousness
nor experimental evidence in its favor, discard the variable.
d. Determine the relative magnitudes of causal variables’ effects on the variable to be forecast if
possibleConsider whether there is sufficient evidence that changes in some causal variables have stronger
influences on the dependent variable than others. Consult experimental evidence and consider surveying domain
experts to determine differential weights. Vary weights from unity only if there is strong evidence of differences
in effect sizes among the causal variables. Avoid changing the a priori weights to improve in-sample fit.
e. Specify model as dependent variable score equals the sum of weighted causal variable values
Knowledge models simply calculate a score by summing the prooducts of the signed causal variable weights and
the variable values. The score is a forecast: a higher score means the outcome is more likely.
f. Estimate relationship between the score and the dependent variable values by regression analysis if
feasibleWhere sufficient historical data are available on the dependent variable, one can estimate the
4 The text of Franklin’s 1772 letter is available at onlinelibrary.wiley.com/doi/10.1002/9781118602188.app1/pdf
15
relationship between the knowledge model scores and a continuous dependent variable using simple regression
analysis. Quantitative forecasts can then be obtained by applying the regression-estimated parametersconstant
and score coefficientto the knowledge model score for a particular situation.
While we believe that Benjamin Franklin was correct when he suggested considering differential
weights, they should be used only when they are supported by strong evidence. For example, how much do
experts know about the causal relationships, and how much experimental data is available on the relationships?
For problems where domain knowledge and data are insufficient for estimating differential weights, use equal
weights.
The first empirical demonstration of the power of equal weights was by Schmidt (1971). That was
followed by Einhorn and Hogarth (1975) and Dana and Dawes (2004) who showed some of the conditions under
which equal weights models provide more accurate forecasts than regression weights.
Lichtman’s “Keys to the White Housemodel used 13 equally-weighted variables selected by an expert
to forecast the popular vote in U.S. Presidential elections. The model accurately predicted which candidate won
the popular vote for all elections from 1984 to date, except 2016 (Armstrong and Cuzán 2006.) Another equal-
weights election forecasting model included all of the 27 variables that had been used in nine independent
econometric (multiple regression) models. The ex ante average forecast error was 29% lower than the average
error of the most accurate of the ten original regression models (Graefe 2015). Graefe and Armstrong (2013)
reviewed empirical forecasting studies in psychology, biology, economics, elections, health, and personnel
selection. Knowledge models provided more accurate forecasts than did regression models for ten of the 13
studies.
Even when there is a strong case for differential weights, consider adjusting the weights toward equality.
Equalizing was tested in election forecasting using eight independent econometric election forecasting models
estimated from data that was standardized and positively correlated with the dependent variable. Where
equalizing coefficients by 100% amounts to using equal weights, equalizing by between 10% and 60% reduced
the absolute errors of the forecasts for all of the models. (Graefe, Armstrong, and Green 2014).
A study assessed the predictions from a knowledge model of the relative effectiveness of the advertising
in 96 pairs of advertisements used differential weights influenced by experimental evidence. With 195 potentially
relevant variables, regression was not feasible. Guessing would result in 50% correct predictions of which of each
pair was more effective. Judgmental predictions by novices were correct for 54% of the pairs; those with
experience in advertising made 55% correct predictions. Copy testing (e.g., showing ads to subjects and asking
them to assess their likelihood of purchase) yielded 59% correct predictions. In contrast, the knowledge model
forecasts were correct for 75% of the pairs of advertisementsan error reduction of 37% compared to copy
testing (Armstrong, Du, Green and Graefe 2016). In an extension of the study, the model was tested using
weights that were equalized across groups of variables. At 32%, the resulting error reduction was broadly similar
(Green, Armstrong, Du, and Graefe 2016).
16. and 17. Combining Forecasts
The last two methods listed in Exhibit 1 deal with combining forecasts. We regard them as the most
important methods to improve ex ante forecast accuracy.
The basic rules for combining within and across methods are: (1) obtain forecasts from variations of all
valid evidence-based methods that are the products of diverse experts, data, procedures, and implementations; (2)
for each component method, combine forecasts from the variations by calculating equally-weighted averages; (3)
combine the combined forecasts from the component methods by calculating an equally-weighted average across
the methods used. The rules for equal weighting should only be relaxed if there is strong evidence of differences
in forecast accuracy, in which case, the weights should be specified before making the forecasts.
For important problems, we suggest obtaining forecasts from at least two variations within each
component method, and from three different component methods. That is, combine across combined forecasts in
order to improve reliability and validity. For more details on combining forecasts, see Graefe, Armstrong, Jones,
and Cuzán (2014) and Graefe (2015).
16
The combining procedures described guarantee that the resulting forecast will not be the worst forecast,
and that it will perform at least as well as the typical component forecast. In addition, the absolute error of the
combined forecast will be smaller than the average of the component forecast errors when the components’ range
includes (brackets) the true value. Combined forecasts can be, and often are, more accurate than the most accurate
component forecast. Because bracketing is always possible, combining should always be used. Thus, when two
or more forecasts from evidence-based methods can be obtained, the method of combining forecasts should
always be used.
Combining is not intuitive. 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). In another experiment, a paid panel of U.S. adults were given data on five experts’
recent forecast errors in predicting attendance at film screenings. When asked to nominate which experts
forecasts they would combine for forecasting attendance at future screenings, only 5% of the 203 participants
chose to use forecasts from all five experts. The rest chose to combine the forecasts only of the experts whose
previous errors had been smallest (Mannes, Soll, and Larrick 2014).
With the same intuition, when New York City officials received two different forecasts for an impending
snowstorm in January 2015, they acted on the forecast that they believed would be the best. As it turned out, it
was the worst.
Much research remains to be done on combining forecasts. In particular, we need to learn more about (1)
how to combine forecasts in order to produce the greatest gains in forecast accuracy, (2) whether and under what
conditions some methods contribute more to increase the accuracy of a combined forecast than others, and (3) the
marginal effects on accuracy of adding more methods and of adding more method variations to a forecast
combination.
Combining forecasts from variations of a single method or from independent forecasters using the same
method helps to compensate for mistakes, errors in the data, and small sample sizes in any of the component
forecasts. In other words, combining within a single method is likely to be most useful for improving the
reliability of forecasts. However, forecasts from a single method are less likely to bracket the outcome than
forecasts from different methods because any one particular method might tend to produce forecasts that are
biased in the same direction.
One review identified 30 studies that compared combinations of forecasts mostly from a single method.
The unweighted arithmetic mean error of the combined forecasts was 12.5% smaller than the average error of the
typical forecast, with a range from 3% to 24% (Armstrong 2001c).
Another study compared the accuracy of the forecasts from eight independent multiple regression
models for forecasting the popular vote in U.S. Presidential elections with the accuracy of an average of their
forecasts. The combined forecasts reduced error compared to the typical individual model’s forecast by 36%
across the 15 elections in the study (Graefe, Armstrong, and Green 2014).
Different forecasting methods are likely to have different biases because they utilize different
assumptions, knowledge, and data. As a consequence, forecasts from diverse methods are more likely than those
from a single method to bracket the actual outcome. Moreover, by including more information about the
situation, combining forecasts across multiple methods is also likely to increase reliability. For example, one
study examined the effect of combining time-series extrapolations and intentions forecasts on accuracy. The
study found that combining forecast from the two methods reduced errors by one-third compared to extrapolation
forecasts alone (Armstrong, Morwitz, and Kumar 2000).
Consider also the case of combining the forecasts of economists who ascribe to different economic
theories. In one study, combinations of 12-month ahead real GNP growth forecasts from two economists with
similar theories reduced the Mean Square Errors by 11% on average, whereas combinations of forecasts from
two economists with dissimilar theories reduced errors by 23%. Combinations from pairs of economists who
used similar forecasting techniques reduced errors by 2%, while combinations from pairs who used dissimilar
techniques yielded a 21% error reduction (Table 2 in Batchelor and Dua 1995). The error-reduction advantage for
diversity in combinations was much larger for five of the six other comparisons in the study, in which economists
with similar/dissimilar theories/techniques forecast the GNP deflator, corporate profit growth, and the
unemployment rate.
17
The PollyVote.com election-forecasting project provided data for testing the accuracy of combining
forecasts across four to six different methods for predicting the popular vote in the seven U.S. Presidential
elections from 1992 to 2016. The individual method forecasts (e.g., election polls) were first combined.
Combined forecasts from several methods were then combined. Over the 100 days prior to the elections, the
Mean Absolute Error of the PollyVote forecast was, at 1.1 percentage points, smaller than the average errors of
each of the component combinations which ranged from 1.2 to 2.6 percentage points with a median of 1.8
(Graefe, Armstrong, Jones, and Cuzán 2017).
Combining across methods provided an error reduction of roughly 40% relative to the typical single
method combination. Taken together with the finding from the previously mentioned error reduction of 12.5%
for combining within a method (Armstrong 2001c), a crude estimate of the expected error reduction from
combining within methods then across methods is that it would be more than one-half.
FORECASTING PRINCIPLES: GOLDEN RULE AND OCCAM’S RAZOR
We turn our attention now from methods to principles. The forecasting methods listed in the Exhibit 1
checklist are consistent with forecasting principles, so following the Forecasting Methods Application Checklist
can help to ensure that the principles are adhered to. More importantly, however, forecasters who persist in using
methods other than those listed in Exhibit 1 can greatly improve the accuracy of their forecasts if they take steps
to comply with two overarching forecasting principles: the Golden Rule, and Occam’s Razor.
The Golden Rule and Simple Forecasting checklists described below provide guidance on how to
comply with the two principles. They differ from the previously published checklist of principlesthe
Forecasting Audit checklist, available at ForecastingPrinciples.comwhich is intended for forecasting
academics and practitioners. For example, we used the Forecasting Audit checklist to assess the forecasting
procedures used to produce the International Panel on Climate Change projections of global mean temperatures
(Green and Armstrong 2007b).
In contrast, the checklists that we present in this section are intended to empower all interested parties to
conduct audits of forecasting procedures. The two principles checklists apply to all types of forecasting problems,
and to all forecasting methods.
Golden Rule
The Golden Rule is to be conservative. More specifically, to be conservative by adhering to cumulative
knowledge about the situation and about forecasting methods. (Armstrong, Green, and Graefe 2015). The Golden
Rule of Forecasting is also an ethical principle, as it implies “forecast unto others as you would have them
forecast unto you.” The Rule is a useful reference when objectivity must be demonstrated, as is the case in legal
or public policy disputes (Green, Armstrong, and Graefe 2015).
Exhibit 3 is a revised version of Armstrong, Green, and Graefe’s Table 1 (2015). It includes 28
guidelines logically deduced from the Golden Rule of Forecasting. There are two key changes from the
previously published version. The first is that Guideline 5 now includes the injunction to “combine forecasts from
diverse methods.” The change is based on the evidence presented in the previous section.
The second change is that Guideline 6 originally suggested caution in using judgmental adjustments, but
is now a prohibition: one should “avoid adjusting forecasts.” The primary reason is that the use of diverse
methods leads to increased use of information about the situation, and hence a lower likelihood of bias arising due
to the omission of key information. Moreover, adjustments are liable to introduce intentional bias. For example, a
survey of nine divisions within a British multinational firm found that 64% of the 45 respondents agreed that
“forecasts are frequently politically modified” (Fildes and Hastings 1994). In another study, 29 Israeli political
surveys were classified according to the independence of the pollster from low to high, as “in-house”such as a
poll run by a political party“commissioned,” or “self-supporting.” The independent polls provided forecasts
that were more accurate than the in-house pollsters. For example, 71% of the most independent polls had
relatively high accuracy, whereas 60% of the most dependent polls had relatively low accuracy (Table 4, Shamir
1986).
18
Exhibit 3: Golden Rule of Forecasting Checklist: Version 2
Comparisons*
Guideline
N
Error
reduction
1.
Problem formulation
n
%
1.1
Use all important knowledge and information by…
1.1.1
¨
selecting evidence-based methods validated for the situation
7
3
18
1.1.2
¨
decomposing to best use knowledge, information, judgment
17
9
35
1.2
Avoid bias by…
1.2.1
¨
concealing the purpose of the forecast
1.2.2
¨
specifying multiple hypotheses and methods
1.2.3
¨
obtaining signed ethics statements before and after forecasting
1.3
¨
Provide full disclosure to enable audits, replications, extensions
1
2.
Judgmental methods
2.1
¨
Avoid unaided judgment
2
1
45
2.2
¨
Use alternative wording and pretest questions
2.3
¨
Ask judges to write reasons against the forecasts
2
1
8
2.4
¨
Use judgmental bootstrapping
11
1
6
2.5
¨
Use structured analogies
3
3
57
2.6
¨
Combine independent forecasts from many diverse judges
18
10
15
3.
Extrapolation methods
3.1
¨
Use the longest time series of valid and relevant data
3.2
¨
Decompose by causal forces
1
1
64
3.3
Modify trends to incorporate more knowledge if the…
3.3.1
¨
series is variable or unstable
8
8
12
3.3.2
¨
historical trend conflicts with causal forces
1
1
31
3.3.3
¨
forecast horizon is longer than the historical series
1
1
43
3.3.4
¨
short and long-term trend directions are inconsistent
3.4
Modify seasonal factors to reflect uncertainty if
3.4.1
¨
estimates vary substantially across years
2
2
4
3.4.2
¨
few years of data are available
3
2
15
3.4.3
¨
causal knowledge about seasonality is weak
3.5
¨
Combine forecasts from diverse alternative extrapolation methods
1
1
16
4.
Causal methods
4.1
¨
Use prior knowledge to specify variables, relationships, and effects
1
1
32
4.2
¨
Modify effect estimates to reflect uncertainty
1
1
5
4.3
¨
Use all important variables
5
4
45
4.4
¨
Combine forecasts from alternative causal models
5
5
22
5.
¨
Combine forecasts from diverse methods
6.
¨
Avoid adjusting forecasts
Totals and Unweighted Average for Guidelines 1 through 4
90
55
29
* N: Number of papers with findings on effect direction.
n: Number of papers with findings on effect size. %: Average effect size (geometric mean).
19
Meehl (1954) concluded that forecasters should not make subjective adjustments to forecasts made by
quantitative methods. Since then, research in psychology has continued to support Meehl’s findings (see Grove et
al. 2000). Research on adjusting forecasts from statistical models found that adjustments often increase errors
(e.g., Belvedere and Goodwin 2017; Fildes, Goodwin, Lawrence, and Nikolopoulos 2009) or have mixed results
(e.g., Franses 2014; Lin, Goodwin, and Song 2014).
Consider a problem that is often dealt with by judgmentally adjusting a statistical forecast: forecasting
sales of a product that is subject to periodic promotions (e.g., see Fildes and Goodwin 2007). The need for
adjustment could be avoided by decomposing the problem into sub-problems, separately forecasting the level, the
trend, and the effects of promotions. Trapero, Pedregal, Fildes, and Kourentzes (2013) provides support for that
approach, finding an average reduction of Mean Absolute Errors of about 20% compared to adjusted forecasts.
We have been unable to find any evidence that adjustments would reduce forecast errors relative to the
errors of forecasts derived in ways that were consistent with the guidance presented in this paper. In particular,
following Guideline 1.1.2to decompose the forecasting problem to make best use of knowledge, information,
and judgmentand the revised Guideline 5to combine forecasts from diverse methodshelps to ensure that
all relevant knowledge and information are included in the forecast, leaving no valid reason for adjusting
forecasts.
Our literature search found evidence on the effects of 18 of the guidelines on forecast accuracy. On
average, the violation of a typical guideline increased error by 40%, as detailed in Exhibit 3. Errors can be
expected to accumulate as more guidelines are violated. Although, we have no systematic information on the
extent that the Golden Rule is followed in practice, we expect that forecasting studies published in scientific
journals typically violate most of the Golden Rule Guidelines. For example, our audit concluded that the U.N.’s
International Panel on Climate Change ignored the Golden Rule in deriving their projections of dangerous
manmade global warming.
Any stakeholder can use the Golden Rule of Forecasting Checklist. Experts and non-experts can
complete the Golden Rule of Forecasting Checklist in less than an hour. Stakeholders do not need to be
forecasting experts to use the checklist because the onus is on forecasters to fully and clearly disclose their
methods (Guideline 1.3.) To help improve the reliability of the checklist ratings, stakeholders could ask at least
three people, each working independently, to rate the forecasting procedures and then average the ratings.
Simple Forecasting
The “simplicity principle” (Occam’s Razor) is the scientific principle that the simplest explanation of
evidence is the best. The principle was proposed by Aristotle and later named after 14th-century scholar William
of Ockham. The principle also applies to scientific forecasting: forecasters should use methods that are no more
complex than necessary to develop the simplest model that is consistent with knowledge about the situation.
Do forecasters ascribe to Occam’s razor? Apparently not: in 1978, when 21 of the world’s leading
experts in econometric forecasting were asked whether more complex econometric methods produced more
accurate forecasts than simple methods, 72% replied that they did. In that survey, “complexity” was defined as an
index reflecting the methods used to develop the forecasting model: (1) the use of coefficients other than 0 or 1;
(2) the number of variables; (3) the functional relationship; (4) the number of equations; and (5) the use of
simultaneous equations (Armstrong 1978).
Starting in the 1950s, researchers developed complex statistical models to extrapolate time-series data.
They derived models using mathematics, and reported on the ability of the models to fit data. The models were
popular and widely used by academics and practitioners, but their predictive validity was not tested against
alternative methods.
In the late 1970s, researchers were invited to enter their models in a competition to extrapolate 111
unidentified business and economic time series of monthly, quarterly, and annual data up to six years ahead. The
accuracies of the forecasts from the different methods were assessed against those of the relevant no-change
benchmark model forecasts. The simple naïve models performed well, with only minor differences in accuracy
compared with forecasts from the more complex models. The findings were published with commentary by 14
leading statisticians (Makridakis and Hibon 1979). Makridakis went on to conduct extensions of the
competitionswhich were referred to as the M-competitions (Makridakis et al. 1993, Makridakis and Hibon
20
2000)—that led to the conclusion that simple methods provide extrapolation forecasts that are competitive with
those from complex methods.
A series of tests from across different kinds of problemssuch as the forecasting of high school dropout
ratesfound that simple heuristics were typically at least as accurate as complex forecasting methods, and often
more accurate (Gigerenzer, Todd, et al. 1999).
We proposed a new operational definition of simplicity in forecasting, one that could be assessed by any
stakeholder. It consisted of a four-item checklist to rate simplicity in forecasting as the ease of understanding by a
potential forecast user. The checklist was created before any analysis was done and it was not changed as a result
of testing. Exhibit 4 provides an abridged version of the checklist provided on ForecastingPrinciples.com (Green
and Armstrong 2015).
Exhibit 4: “Simple Forecasting” Checklist: Occam’s Razor
Are the descriptions of the following aspects of the forecasting process
sufficiently uncomplicated as to be easily understood by decision makers?
Simplicity
rating
(010)
1.
method
[__]
2.
representation of cumulative knowledge
[__]
3.
relationships in models
[__]
4.
relationships among models, forecasts, and decisions
[__]
Simple Forecasting Average (out of 10)
[__]
Our search identified 32 published papers that allowed for a comparison of the accuracy of forecasts
from simple methods with those from complex methods. Four of those papers tested judgmental methods, 17
tested extrapolative methods, 8 tested causal methods, and 3 tested forecast combining methods. The findings
were consistent across the methods. On average, across each comparison, the more complex methods produced
ex ante forecast errors that were 27% larger than those from simpler evidence-based methods. The finding was
surprising because the papers had apparently proposed the more complex methods with the expectation that they
would provide more accurate forecasts. To our knowledge, complex methods have never been shown to provide
forecasts for complex situations that are as accurate as those from simple evidence-based methods. The late
Arnold Zellner, founder of the Journal of Econometrics, reached the same conclusion.5
ASSESSING FORECAST UNCERTAINTY
A forecast’s uncertainty affects its utility. For example, if demand for automobiles is forecast to increase
by 20% next year, manufacturers might consider hiring more employees and investing in more machinery. If the
forecast had a high level of uncertainty such that a decline in demand is also likely, however, expanding
operations might not be prudent.
This section first describes error measures for estimating prediction intervals. Currently, the estimates of
prediction intervals are typically much too narrow. We suggest doubling the width of statistically estimated 95%
confidence intervals to approximate the likely 95% prediction intervals. But use of the guidelines below, in
Exhibit 5, would be better.
Error measures
Earlier, we discussed error measures suitable for evaluating forecasting methods by comparing the
accuracy of their forecasts with those from alternative methods. Here, for the purpose of estimating prediction
intervals that are useful for managerial decisions we suggest the Mean Absolute Deviation (MAD) of forecasts
from actual values. The MAD is easy to calculate, and is easily understood by decision makers. On the other
5 García-Ferrer, A., “Professor Zellner: An Interview.” International Journal of Forecasting 14, 1998, 303-312.
21
hand, the commonly used Root Mean Square Error (RMSE) measure should be avoided as it cannot be related to
benefits.
For forecasting problems that are expected to involve asymmetric errorsi.e., negative errors are larger
than positive errors, or vice versacalculate the logarithms of the forecast and actual values and calculate the
errors using the logged values. Use those errors to estimate prediction intervals, and then convert the bounds of
the intervals back to actual values (p. 281, Armstrong and Collopy 2001).
Loss functions can also be asymmetric. For example, the losses due to a forecast that is too low by 50
units may differ from the losses if a forecast is too high by 50 units. Asymmetric errors are, however, a problem
for the planner, not the forecaster: the planner must assess the damages resulting from positive versus negative
errors.
Methods to forecast uncertainty
Exhibit 5 presents a checklist of methods to forecast uncertainty. The checklist includes four valid
methods to use, and two commonly used but invalid methods to avoid.
Exhibit 5: Methods to Forecast Uncertainty Checklist
1.
Use empirical prediction intervals or likelihoods estimated from out-of-sample tests
2.
Decompose errors by source in order to estimate the uncertainty of each
3.
Use structured judgment to estimate prediction intervals or likelihoods
4.
Combine alternative valid estimates of uncertainty
5.
Avoid using statistical fit with historical data to assess uncertainty
6.
Avoid using tests of statistical significance to assess uncertainty
1. Use empirical prediction intervals or likelihoods estimated from out-of-sample tests
Traditional statistical confidence intervals estimated from historical data are usually too narrow. One
study showed that the percentage of actual values that fell outside the 95% confidence intervals for extrapolation
forecasts was often greater than 50% (Makridakis, Hibon, Lusk, and Belhadjali 1987).
In order to provide forecast users with useful information on forecast uncertainty, there is no alternative
to estimating empirical prediction intervals based on out-of-sample forecast errors. To that end, simulate the
actual forecasting procedure as closely as possible and use the distribution of the errors of the resulting forecasts
to assess uncertainty. Tashman (2000) provides guidance on out-of-sample testing. For more on estimating
prediction intervals, see Chatfield (2001).
When analyzing time-series forecast errors, use successive updating to increase the number of
predictions. If sufficient validation data are not available, consider using data from analogous situations.
2. Decompose errors by source in order to estimate the uncertainty of each
Most forecasting problems are subject to several sources of forecast error. To help ensure that all
possible errors are accounted for, consider decomposing errors by source of error to estimate each, then combine
the estimates. For example, when polling to predict the outcomes of political elections, survey researchers report
only the error expected due to random variation based on the size of the sample. Response and non-response bias
errors are ignored. As a consequence, the 95% confidence intervals reported for polls are about half as large as
they should be (Buchanan 1986). In other words, decision-makers should double political pollsconfidence
intervals to obtain more realistic estimates of the prediction intervals.
When uncertainty is highsuch as with surveying citizens to forecast their behavior in response to
changes in government regulations—response error is likely to be particularly high due to survey respondents’
lack of self-knowledge about how they make decisions (see Nisbett and Wilson 1977). Non-response can also be
a large source of error because people who are most affected by the topic of the survey are more likely to
22
respond. While the latter error can be reduced to some extent by the “extrapolation-across-waves” method
(Armstrong and Overton 1977), forecasters still need to consider that source of error when assessing uncertainty.
As with analyses of survey responses, regression models’ diagnostic statistics ignore key sources of
uncertainty such as the omission of key variables, the difficulty in controlling or forecasting the causal variables,
inability to make accurate forecasts of the causal variables, and the difficulty of assessing the relative importance
of causal variables that are correlated with one another. These problems are magnified when analysts strive for a
close fit with historical data, and even more so when data-mining techniques are used to achieve a close fit.
3. Use structured judgment to estimate prediction intervals or likelihoods
One common judgmental approach to assessing uncertainty is to ask experts to express their confidence
in their own judgmental forecasts in the form of 95% prediction intervals. One concern with that approach is that
experts are typically overconfident about the accuracy of their forecasts. For example, an analysis of judgmental
confidence intervals for economic forecasts from 22 economists over 11 years found that outcomes were within
the range of their 95% confidence intervals only 57% of the time (McNees 1992). Another study tracked
members of a ten-year panel who provided 13,300 estimates of expected stock market returns by company; the
actual returns were within the executives’ 80% confidence intervals only 36% of the time (Ben-David, et al.
2013).
A number of structured approaches can improve the calibration of judgmental forecasts. Ensure that the
judgments are obtained from many experts and obtain independent anonymous estimates. The Delphi technique
can be used for that purpose. Ask experts to list all sources of uncertainty, and all reasons why their forecasts
might be wrong. That approach was shown to be effective by Arkes (2001).
Finally, to improve the calibration of forecasters’ estimates of uncertainty in the future, ensure that they
receive timely, accurate, frequent, and well-summarized information on what actually happened and reasons why
their forecasts were right or wrong. For example, weather forecasters use such procedures, and their forecasts are
well-calibrated for a few days ahead: When they say that there is a 40% chance of rain, rain falls 40% of the time
on average (Murphy and Winkler 1984).
4. Combine alternative valid estimates of uncertainty
The logic behind combining uncertainty estimates is the same as that for combining forecasts. Thus, the
estimates of uncertainty based on combined estimates can never be worse than the typical estimate, and the
combined estimate will always be better than the typical estimate as long as bracketing of the uncertainty
estimates occurs.
5. Avoid using statistical fit with historical data to assess uncertainty
In a study using data consisting of 31 observations on 30 variables, stepwise regression was used with a
rule that only variables with a t-statistic greater than 2.0 would be included in the model. The data were from a
book of random numbers. Despite that, the stepwise method delivered an eight-variable regression model with
good statistical fitan R2 of 0.85 adjusted for degrees of freedom (Armstrong 1970).
Measures of statistical fit do not provide useful information about out-of-sample predictive validity
(Armstrong 2001d). Experiments testing analysts’ interpretation of standard statistical fit information on
regression models found that 72% grossly underestimated the uncertainty of forecasts associated with changes to
the model’s causal (policy) variable (Soyer and Hogarth 2012). Further discussion on why forecasters should
avoid such measures as adjusted R2 is provided in Armstrong (2001e).
6. Avoid using tests of statistical significance to assess uncertainty
Statistical significance tests do not provide estimates of forecast uncertainty. Attempts to use them in that
way will likely lead to confusion and poor decision making. Experimental studies over more than half a century
support that conclusion (e.g., see Ziliak and McCloskey 2008; Hubbard 2016; and Armstrong and Green 2018).
One experiment presented leading researchers with a treatment difference between two drugs, as well as
a “p-value” for the difference, and asked them which of the drugs they would recommend to a potential patient.
When the difference in the effects of treatments was large but reported to be p > 0.05, nearly half responded that
23
they would advise that there was no difference between the two drugs. By contrast, when the difference between
the treatment effects was small but reported to be statistically significant (p < 0.05), 87% of the respondents
replied that they would advise taking the drug (McShane and Gal 2015). Many of those teaching statistics also
failed to draw logical conclusions as was shown in another experiment by McShane and Gal (2017).
Errors in interpretation of findings due to the provision of statistical significance information have led to
poor decisions. Hauer (2004) described the harm caused by decisions related to automobile traffic safety, such as
the “right-turn-on-red policy. Ziliak and McCloskey (2008) provide other examples.
To our knowledge, no scientific study has shown that statistical significance testing has led to better
forecasts, decisions, or scientific contributions. Schmidt (1996) offered this challenge: “Can you articulate even
one legitimate contribution that significance testing has made (or makes) to the research enterprise (i.e., any way
in which it contributes to the development of cumulative scientific knowledge)?” Schmidt and Hunter (1997)
stated that no such cases have been reported, and they repeated the challenge, as we have and hereby do again.
DISCUSSION
The accumulation of scientific knowledge about forecasting over the past century enables improvements
in forecast accuracy. Regrettably, that knowledge is often ignored, and forecasting practice appears to be in
decline. There are two related reasons: first, advocacy research has tended to replace objective forecasting; and
second, an unsupported faith in data models has resulted in forecasters ignoring cumulative knowledge about
causal relationships and validated forecasting methods (Armstrong and Green 2018).
What recourse do clients and citizens have when they make decisions on the basis of forecasts that turn
out to be inaccurate? The answer has traditionally been that there is none, because it has been impossible to
distinguish between forecasts that were wrong due to random or unpredictable changes in the situation and those
that were wrong due to the forecaster’s failure to follow evidence-based procedures. This paper follows in the
footsteps of medicine, engineering, and aviation by providing checklists that can be used to hold forecasters
including scientists who make forecasts, and public policy makersresponsible if they fail to follow evidence-
based forecasting procedures.
Forecasters can use the checklists to improve the accuracy of their forecasts and, by communicating that
they have followed the checklists, protect themselves from claims against them. Forecasters who follow the
checklists might alsoas do medical practitionersobtain protection against damage claims by arranging
insurance on the understanding that they follow the forecasting procedures required by the checklists.
Science requires that the predictive validities of hypotheses and new findings are tested. Milton
Friedman (1953) viewed out-of-sample predictive validity testing of competing hypotheses as an essential
element of economics as a social science. The checklists in this paper can help scientists to design such
tests.
CONCLUSIONS
Forecasting practice can be improved such that the accuracy of forecasts upon which decision makers in
business and public policy depend is greatly increased. The best way to achieve that objective is to require
forecasters to comply with evidence-based checklists.
Exhibit 1, the Forecasting Methods Application Checklist, lists 15 individual evidence-based forecasting
methods. The use of those methods substantially improves the ex ante accuracy of forecasts relative to forecasts
from commonly used methods, including experts’ unaided judgments. Error reductions range from
approximately 5%—for damped-trend extrapolation and decomposition by seasonalityto over 50%for
simulated interaction and knowledge models.
Rather than hoping to identify the one best model, forecasters should employ diverse models from
different evidence-based methods, and combine the forecasts from them. Doing so reduces the bias than can arise
from using a single method and improves reliability by incorporating more knowledge and information.
Combining also avoids the risk of making the worst forecast and guarantees that the combined forecast will be
more accurate than the typical forecast if, as is likely, any of the forecasts in the combination bracket the outcome.
24
Data models are not suitable for forecasting. In particular, multiple regression approaches violate
evidence-based forecasting principles and provide forecasts that are substantially less accurate that those from the
methods listed in the Exhibit 1 checklist. Data models can be and are being used to support clients’ and funders’
prior beliefs and preferences to the further detriment of forecast accuracy.
The Golden Rule and Simple Forecasting principles checklists (Exhibits 3 and 4) can help forecasters to
implement the evidence-based methods listed in Exhibit 1, and can help forecasters to improve currently used
methods in situations where it is not feasible to replace them with evidence-based methods. Following the Golden
Rule of Forecasting can help forecasters reduce forecast errors by over half, while ignoring Occam’s Razor is
likely to increase errors by around 27%. The principles checklists can also help clients, sponsors, and users to
assess whether proper procedures were followed.
Procedures currently used to assess forecast uncertainty mislead analysts, clients, and users into
excessive confidence. As a rule of thumb, they are half as uncertain as they should be. Prediction intervals should
only be estimated by using out-of-sample testing, as described in the Exhibit 5 checklist.
Clients and other funders who are interested in accurate forecasts should require forecasters to follow the
five evidence-based checklists provided in this paper, and should audit the forecasters’ procedures to ensure that
they did so. Clients and other forecast stakeholders can use the checklists to assess the worth of forecasts by
determining whether they were the product of scientific forecasting procedures.
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The PollyVote uses evidence-based techniques for forecasting the popular vote in presidential elections. The forecasts are derived by averaging existing forecasts generated by six different forecasting methods. In 2016, the PollyVote correctly predicted that Hillary Clinton would win the popular vote. The 1.9 percentage-point error across the last 100 days before the election was lower than the average error for the six component forecasts from which it was calculated (2.3 percentage points). The gains in forecast accuracy from combining are best demonstrated by comparing the error of PollyVote forecasts with the average error of the component methods across the seven elections from 1992 to 2016. The average errors for last 100 days prior to the election were: public opinion polls (2.6 percentage points), econometric models (2.4), betting markets (1.8), and citizens’ expectations (1.2); for expert opinions (1.6) and index models (1.8), data were only available since 2004 and 2008, respectively. The average error for PollyVote forecasts was 1.1 percentage points, lower than the error for even the most accurate component method.
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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.
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