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Forecasting: theory and practice

Fotios Petropoulos, Daniele Apiletti, Vassilios Assimakopoulos, Mohamed Zied Babai, Devon K. Barrow, Souhaib Ben Taieb, Christoph Bergmeir, Ricardo J. Bessa, Jakub Bijak, John E. Boylan, Jethro Browell, Claudio Carnevale, Jennifer L. Castle, Pasquale Cirillo, Michael P. Clements, Clara Cordeiro, Fernando Luiz Cyrino Oliveira, Shari De Baets, Alexander Dokumentov, Joanne Ellison, Piotr Fiszeder, Philip Hans Franses, David T. Frazier, Michael Gilliland, M. Sinan Gönül, Paul Goodwin, Luigi Grossi, Yael Grushka-Cockayne, Mariangela Guidolin, Massimo Guidolin, Ulrich Gunter, Xiaojia Guo, Renato Guseo, Nigel Harvey, David F. Hendry, Ross Hollyman, Tim Januschowski, Jooyoung Jeon, Victor Richmond R. Jose, Yanfei Kang, Anne B. Koehler, Stephan Kolassa, Nikolaos Kourentzes, Sonia Leva, Feng Li, Konstantia Litsiou, Spyros Makridakis, Gael M. Martin, Andrew B. Martinez, Sheik Meeran, Theodore Modis, Konstantinos Nikolopoulos, Dilek Önkal, Alessia Paccagnini, Anastasios Panagiotelis, Ioannis Panapakidis, Jose M. Pavía, Manuela Pedio, Diego J. Pedregal, Pierre Pinson, Patrícia Ramos, David E. Rapach, J. James Reade, Bahman Rostami-Tabar, Michał Rubaszek, Georgios Sermpinis, Han Lin Shang, Evangelos Spiliotis, Aris A. Syntetos, Priyanga Dilini Talagala, Thiyanga S. Talagala, Len Tashman, Dimitrios Thomakos, Thordis Thorarinsdottir, Ezio Todini, Juan Ramón Trapero Arenas, Xiaoqian Wang, Robert L. Winkler, Alisa Yusupova, Florian Ziel

arXiv:2012.03854v4stat.APcs.LGecon.EMstat.MLstat.OT

TL;DR

Forecasting addresses uncertainty in decision making through diverse methods and applications. This article offers an encyclopedic review of forecasting theory and practice, including uncertainty assessment and empirical findings on forecasting methods.

  • Problem

    Forecasting practice must address uncertainty across diverse applications, while improving on naive forecasts remains difficult.

  • Method

    The article provides an encyclopedic overview of theoretical models, methods, principles, uncertainty representation, and forecasting applications.

  • Results

    Empirical research reports more accurate forecasting and uncertainty assessment, with substantial benefits compared with ad-hoc judgments.

  • Takeaways & Limitations

    Forecasting theory and practice offer a broad and developing point of reference for preparing, producing, organising, and evaluating forecasts.

Abstract

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Forecasting has always been at the forefront of decision making and planning. The uncertainty that surrounds the future is both exciting and challenging, with individuals and organisations seeking to minimise risks and maximise utilities. The large number of forecasting applications calls for a diverse set of forecasting methods to tackle real-life challenges. This article provides a non-systematic review of the theory and the practice of forecasting. We provide an overview of a wide range of theoretical, state-of-the-art models, methods, principles, and approaches to prepare, produce, organise, and evaluate forecasts. We then demonstrate how such theoretical concepts are applied in a variety of real-life contexts. We do not claim that this review is an exhaustive list of methods and applications. However, we wish that our encyclopedic presentation will offer a point of reference for the rich work that has been undertaken over the last decades, with some key insights for the future of forecasting theory and practice. Given its encyclopedic nature, the intended mode of reading is non-linear. We offer cross-references to allow the readers to navigate through the various topics. We complement the theoretical concepts and applications covered by large lists of free or open-source software implementations and publicly-available databases.

3.5.2 Weather forecasting

The supplied material identifies a weather-forecasting subsection and lists nearby forecasting-application topics and contributors.

  • Weather forecasting is listed as subsection 3.5.2 and associated with Claudio Carnevale.
  • Floods and water-resources management is listed as a related forecasting application.
  • Social-good and demographic forecasting is listed among the surrounding application areas.

1. Introduction1

Forecasting is expanding through new methods, richer data, probabilistic communication, and broader applications. This review presents that growing field across theory and practice, emphasizing diverse methods, forecast combination, and uncertainty-aware decision support.

  • Forecasting now spans increasingly sophisticated methods, including neural networks, machine learning, Bayesian forecasting, and complex regression models.Rapid advances in computing have enabled analysis of larger and more complex data sets.
  • Combining forecasts can perform well because diverse methods may contribute less-correlated errors.The M4 Competition’s top-performing entries combined forecasts from multiple methods.
  • Older methods such as ARIMA and exponential smoothing remain valuable because they are robust and less prone to overfitting.They can also be included in ensembles with more sophisticated methods.
  • Probability forecasts quantify risk and are increasingly communicated to the public and used in decision making.Examples include precipitation, elections, medicine, science, sporting events, and economic measures.
  • This timely review covers forecasting theory and practice from highly theoretical to very applied perspectives.It uses short presentations by experts to portray the field’s current state of the art.

2. Theory

The theory of forecasting spans methods for producing, evaluating, and combining forecasts, with evidence favoring simplicity, calibration, combination, and context-sensitive model choice. It also addresses probabilistic prediction, fat-tailed uncertainty, machine learning, agent-based models, and hybrid approaches.

  • Foundations: Theory can improve forecasting practice when it begins with the essential features of the forecasting problem.The review links problem understanding to theoretical development and improved practice.
  • Forecasting frameworks: State-space systems provide a flexible framework for modelling and forecasting, including future probability distributions.They are presented as environments capable of handling many modelling and forecasting techniques.
  • Forecasting methods: The theta method is a simple, enduring approach and should be treated as a critical benchmark in forecasting.The review relates theta forecasts to AR(1) and SES-with-drift forecasts.
  • Uncertainty and predictability: Forecastability depends strongly on tail behavior: variables with α ≤1 are not predictable, whereas forecasting becomes possible for α > 2.5.For 1 < α ≤2, the law of large numbers may converge extremely slowly, making inference difficult even with many observations.
  • Model selection: Model complexity has a U-shaped relationship with MSE, and the optimal complexity increases with sample size; simple models often remain difficult to outperform.The random walk is described as a tough benchmark, while evidence on complexity is not uniform across settings.
  • Forecast combination: Combining forecasts generally improves accuracy, with evidence supporting simple averaging, diverse expertise, and hybrid statistical–machine-learning forecasts.The review also identifies human judgment and forecast combinations as promising directions.
  • Agent-based modelling: Agent-based modelling explores aggregate outcomes by representing individual agents and is useful when data are limited or uncertainty is high.Its framework includes setting up environments and agents, modelling, and calibration and validation.
  • Machine learning: Global machine-learning models can reduce computation, transfer across similar datasets, exploit shared patterns, and mitigate data limitations.The review describes global learning as effective for batch time-series forecasting.

M Forecasting competitions

Forecasting competitions and evaluation research show that forecast quality depends on matching methods and measures to the task, while simple methods and combinations remain strong benchmarks. The review also highlights a need to balance data-driven approaches with expert judgment and to develop more systematic probabilistic and decision-focused frameworks.

  • Forecast evaluation: Different point-error measures elicit different optimal forecasts from the same predictive density.MSE selects the expectation, while MAE and MASE select the median; quantile loss selects the corresponding quantile.
  • Forecast evaluation: MAPE is undefined with zero actuals, and intermittent demand can produce sharply different forecasts under different error measures.For example, the MAE-minimizing conditional median may be zero while the MSE-minimizing conditional mean is usually nonzero.
  • Competition evidence: Forecast competitions repeatedly find that simple statistical methods can match or outperform more complex approaches, while forecast combinations often improve accuracy.The M3 winner was the Theta method, and five of the six best M4 submissions used forecast combinations.
  • Competition evidence: The M4 competition found that properly used machine learning and cross-learning can increase forecasting performance, especially when combined across forecasts.The two best-performing M4 submissions used neural networks or machine learning, and five of the top six used different forecast-combination implementations.
  • Competition evidence: Other competitions report that domain expertise and exogenous variables may add limited value, while naive forecasts can perform very well at yearly frequency.The tourism competition specifically found no added value from exogenous variables and strong performance from yearly naive forecasts.
  • Future directions: Future forecasting research should connect probabilistic forecasting, verification, forecast value, behavioral science, causal modelling, and decision problems.The review also points toward modelling dynamic systems as wholes and developing frameworks for distributed learning and collaborative analytics.

3. Practice

Forecasting practice spans organisational, macroeconomic, financial, and operational settings, where model choice, information, human adjustment, and evaluation shape forecast performance. The reviewed evidence reports benefits from combinations, richer data, and specialised models, while also documenting biases, uncertainty miscalibration, and context dependence.

  • Forecast adjustment and value: Forecasts are shaped by organisational politics and personal agendas, so they may reflect aspirations rather than unbiased expectations.
  • Forecast adjustment and value: 60% accuracy was achieved by the naive random-walk benchmark across products, illustrating the importance of measuring forecast value against simple baselines.
  • Forecast adjustment and value: Management review can reduce value: one company’s statistical forecast improved accuracy by five percentage points, while subsequent adjustment delivered negative value.
  • Forecast uncertainty: Prediction intervals in software often underestimate uncertainty, whereas decision makers may reject intervals that are too wide to be informative.
  • Forecast adjustment and value: The M4 simple benchmark combination reduced overall weighted average error by 17.9% versus naive, while the top six methods reduced it by more than 5% further.
  • Forecast adjustment and value: Evidence across applications links collaboration, insider information, forecast combinations, and targeted adjustment with improved forecasting or operational outcomes.
  • Macroeconomic forecasting: Macroeconomic applications report that DSGE models can outperform competitors for medium- and long-run GDP and inflation forecasts, except against hybrid models.
  • Macroeconomic forecasting: Productivity forecasts are reported with RMSE more than 75% below OBR forecasts at five years and 84% below them at the longest horizon.

4. Forecasting: benefits, practices, value, and limitations162

Forecasting supports everyday planning and organisational decisions, but all forecasts are uncertain and require explicit risk assessment. The article presents systematic methods as more accurate and objective than ad-hoc judgment while emphasizing benchmark evaluation and important limits.

  • Article scope and purpose: The article offers an encyclopedic, single-source reference covering forecasting theory and practice, with more than 140 sections and contributions from 80 researchers and practitioners.It is intended as an easy-to-use resource and is designed to be regularly updated as new information becomes available.
  • The Myriad of Forecasts: Forecasting is pervasive because planning and most decisions require predictions about future conditions and their uncertainty.Applications range from commuting and education to production, pricing, advertising, and technology investment.
  • The Pervasiveness of Uncertainty: Forecasts outside some hard-science areas are uncertain and should include prediction intervals or probability distributions.These intervals estimate the likely range of future values or forecast errors, although users may find them too wide or uninformative.
  • More Accurate Ways of Forecasting and Assessing Uncertainty: Systematic forecasting approaches improve forecast accuracy and uncertainty assessment relative to ad-hoc judgment.They identify patterns and relationships mathematically and extrapolate them while reducing overoptimism and wishful thinking.
  • Using Benchmarks to Evaluate the Value of Forecasting: Forecast accuracy and uncertainty assessment should be evaluated against simple, readily available benchmarks rather than in isolation.Examples include using today’s stock price or today’s weather as forecasts for future periods.
  • Concluding remark: Systematic methods have no prophetic powers: accurate forecasts require patterns and relationships to remain fairly constant during the forecasting period.Their quantitative uncertainty assessment is also constrained when uncertainty is fat-tailed.

Appendix B. Software

Appendix B presents an indicative list of free or open-source forecasting software linked to the article’s theory sections. The list spans multiple forecasting tasks and implementations, but users are advised to consult each package’s documentation and licence.

  • Table A.1 links free or open-source packages, libraries, and toolboxes to the article’s theory sections.
  • The authors assume no liability for the listed software and strongly advise users to read the respective documentation and licences.
  • The listed software includes tools for impulse and step indicator saturation, outlier handling, exponential smoothing, regression, and Theta-method forecasting.
  • Implementations include R and Gretl functions for regression-model selection, relative regressor importance, stepwise AIC selection, and OLS, LAD, or MIDAS regression.

§2.3.3. Theta method and models

The listed implementations cover Theta-method forecasting, ARIMA-family models, unit-root testing, multiple seasonalities, tidy time-series modelling, Prophet, and state-space tools. The passages primarily provide software references rather than methodological exposition.

  • R and Gretl provide functions for univariate Theta-method forecasting and prediction intervals.
  • ARIMA-related implementations include automatic and direct ARIMA, SARIMAX, ARMAX, seasonal-unit-root, and multiple-seasonality models.
  • R packages and Gretl functions support augmented Dickey-Fuller, KPSS, Phillips-Perron, and other unit-root tests, as well as seasonal differencing estimation.
  • The software list also includes tidy time-series models, Prophet interfaces, machine-learning packages, and tools for multiple seasonal cycles.

§2.3.6. State-space models

The software references cover linear, nonlinear, non-Gaussian, Bayesian, multivariate, and structural state-space modelling. They also include demographic estimation and projection tools linked to forecasting applications.

  • MATLAB and R provide general state-space modelling tools for linear, nonlinear, non-Gaussian, and time-series applications.
  • Gretl, Python, and R tools support state-space analysis, forecasting, filtering, smoothing, structural time-series models, and exponential-family state-space models.
  • Available implementations include Bayesian dynamic linear models, non-Gaussian state-space models, multivariate models, and Kalman filtering and smoothing.
  • Demographic software covers life tables, population projections, fertility and life-expectancy projections, and demographic estimation.

§2.3.11. ARCH/GARCH models

The listed tools extend forecasting software coverage to functional time series, volatility models, regime-switching and threshold models, Bayesian macroeconometrics, DSGE forecasting, and diffusion models.

  • R, Python, and Gretl packages provide tools for autoregressive, ARCH, and GARCH time-series analysis and forecasting.
  • R and Gretl tools cover Markov-switching, non-homogeneous Markov-switching autoregressive, threshold, SETAR, and panel-threshold models.
  • Bayesian macroeconometric and Dynare platforms support Bayesian VAR, DSGE estimation, and forecasting.
  • Diffusion-model packages include Bass, Gompertz, and Gamma/Shifted Gompertz curves for forecasting new-product growth.

§2.3.19. The natural law of growth in competition

The listed resources cover uncertainty forecasting through conditional distributions, covariance models, fat-tailed distributions, semiparametric models, gradient boosting, quantile methods, and copula models.

  • Conditional kernel density estimation is listed for producing marginal distributions.
  • The resources include semiparametric uncertainty models, gradient boosting models, quantile estimation, prediction intervals, and copula analysis.
  • Multivariate GARCH models are listed for forecasting covariance matrices.

§2.4.3. Bayesian forecasting with copulas

The listed resources include covariate-dependent, factor, and vine copula tools alongside tests and regularization methods for forecasting-related model analysis and selection.

  • Covariate-dependent, factor, and vine copula models are represented by dedicated statistical-analysis resources.
  • The resources include standard and frequency-wise Granger-causality tests.
  • Regularized regression, sequential variable selection, boosted regression, and general-to-specific model selection are listed.
  • Additional entries cover cross-validation for autoregressive models and ensemble time-series forecasts.

§2.5.5. Cross-validation for time-series data

The resources span time-series databases, feature extraction, forecast combination and selection, intermittent-demand methods, neural networks, and machine-learning frameworks.

  • Time-series databases include InfluxDB, OpenTSDB, RRDtool, and Timely, with distributed ARIMA implementations also listed.
  • Feature-based forecasting resources extract, analyze, visualize, and use time-series characteristics for model selection or combination.
  • The listed software includes tools for generating time series with controllable characteristics, bootstrapping, forecast combination, and interval forecasting without Gaussian innovations.
  • Deep probabilistic and neural-network resources include DeepAR, DeepState, NBEATS, recurrent networks, feed-forward networks, and autoregressive neural networks.
  • Machine-learning resources cover regression, regularization, support vector machines, nearest neighbors, Gaussian processes, trees, ensembles, and multilayer perceptrons.

§2.7.13. Hybrid methods

The listed intermittent-demand resources include parametric and non-parametric methods, including Croston’s method, MAPA, and time-series categorisation.

  • Parametric intermittent-demand forecasting resources include Croston-related methods and the tsintermittent package.
  • MAPA is listed as a method for intermittent-demand data.
  • Time-series categorisation for intermittent demand is provided by the idclass resource.

§2.8.3. Classification methods

The section lists software implementations supporting forecasting classification, hierarchical and grouped time series, intermittent demand, contingency-table forecasts, accuracy assessment, scoring rules, and statistical performance tests.

  • Software implementations: R and Python packages provide classification, natural-language-processing, hierarchical, grouped, and multiple-aggregation forecasting routines.The listed implementations include tsutils, NLTK, SpaCy, hts, MAPA, thief, and tsintermittent.
  • Specialized forecasting methods: Specialized R packages forecast inner cells of 2 × 2 and R×C tables using Bayesian, ecological-regression, linear-programming, and iterative methods.The packages include ei, eiPack, lphom, and eiCompare.
  • Evaluation and testing: Forecast evaluation is supported by accuracy measures, scoring rules, continuous ranked probability score implementations, and predictive-accuracy tests.The listed tools include forecast, scoringRules, verification, and tsutils implementations.
  • Evaluation and testing: Additional statistical tests address forecast unbiasedness, efficiency, asymmetric loss, and directional changes.The passage also lists Gretl implementations of several forecast-performance tests, including Diebold-Mariano and Giacomini-White tests.

Appendix C. Data sets

The appendix catalogs indicative publicly available data sets for benchmarking forecasting methods across economic, energy, health, demographic, transport, tourism, electoral, sports, and other applications.

  • Overview: The appendix presents a list of indicative publicly available data sets for forecasting research.The table is identified as Table A.2 and is accompanied by a matching caption.
  • Economic and energy forecasting: Economic and energy examples include European macroeconomic variables, housing markets, electricity demand, wind, solar, and national generation data.Examples include Eurostat macroeconomic data, housing-market competitions, energy forecasting competitions, PV data, and Brazilian national electricity-system data.
  • Social good and demographic forecasting: Health and demographic collections cover influenza, West Nile virus, COVID-19, mortality, fertility, migration, and global population data.The listed sources include disease-prediction challenges, mortality databases, fertility databases, migration inventories, and UN population projections.
  • Other applications: Other application areas include tourism, aviation, traffic, elections, sports, weather, air quality, customer behavior, web traffic, and driver or property prediction.The appendix combines public databases and competition data spanning transport, environmental, commercial, electoral, and sports forecasting tasks.
  • Benchmarking resources: Some repositories contain large collections of time series intended for benchmarking forecasting methods across applications.One listed collection contains more than 490,000 time series, while another is described as a repository of data sets for benchmarking.
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