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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’s growth across theory and practice, alongside its many applications, calls for a diverse account of available approaches. This article reviews forecasting theory and practice across varied methods and contexts, and notes that theoretical results can lead to improved practice.
Problem
Forecasting’s growth across theory and practice and its many applications call for a broad account of available methods and their use.
Source: §1 · p. 7Method
The article reviews forecasting theory and practice, covering a variety of approaches and applications.
Source: §1 · p. 8Results
Theoretical results can lead to improved practice.
Source: §2.1 · p. 9Takeaways & Limitations
The review offers a point of reference for forecasting theory and practice.
Source: §1 · p. 8The review’s selection of approaches is subjective and not comprehensive.
Source: §2.7.9 · p. 64
Abstract
from arXiv · showhide
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 passage identifies a weather-forecasting section attributed to Claudio Carnevale, but provides no substantive discussion of its content.
3.5.2 Weather forecasting
- The weather-forecasting entry credits Claudio Carnevale and lists page 141.Source: §3.5.3 · p. 6
4 Forecasting: benefits, practices, value, and limitations (Spyros Makridakis) … 1. Introduction1
Forecasting has expanded in theory and practice, and this review surveys developments across a broad range of methods and applications. It highlights how forecasts support decisions under uncertainty and how established and newer methods can be used together.
1. Introduction1
- Tailored forecasts can inform decisions by managers, public officials, investors, and other decision makers, while visualizations can help communicate uncertainty.Examples include hurricane forecast cones and rapidly updated timelines of a team’s probability of winning.Source: §1 · pp. 7–8
- The review offers a broad view of forecasting theory and practice, using expert presentations to describe the state of the art.It covers a variety of developments but cannot encompass the field’s full breadth.Source: §1 · pp. 7–8
- Computing advances and larger, more complex data sets have expanded forecasting toolboxes, including neural networks, machine learning, Bayesian forecasting, and complex regression models.Judgmental forecasting research has also expanded, driven largely by the “wisdom of crowds” notion.Source: §1 · p. 7
- Top-performing entries in the M4 Competition combined forecasts from multiple methods, and aggregation has received increased attention and performed well.Combining forecasts can help when it is difficult to know which individual forecast will be most accurate.Source: §1 · p. 7
- ARIMA, exponential smoothing, and other simple methods remain valuable because they are robust and less prone to overfitting than more complex methods.They can also contribute to ensembles, which are more valuable when the component methods’ forecast errors are not highly correlated.Source: §1 · p. 7
- Probability forecasts quantify decision risks and are increasingly communicated to the public and used as decision inputs.They are generated through Bayesian and other approaches and used across applications including elections, medicine, science, sports, and economic measures.Source: §1 · p. 8
2. Theory
This section surveys forecasting theory and methods across model formulation, data preparation, uncertainty representation, forecast combination, judgment, and evaluation. It describes their approaches, practical uses, assumptions, limitations, and tradeoffs.
2.1. Introduction to forecasting theory2
- Forecasts need not be exact: for a future time period, they can be point forecasts, prediction intervals, percentiles, or entire prediction distributions.These possible outputs together may be considered “the forecast.”Source: §2.1 · p. 9
- Forecasting procedures should address practical problems: understanding their essential features can develop theory that then improves practice.Source: §2.1 · p. 9
- A forecasting method is a predetermined sequence of steps, while stochastic models add data-generating assumptions for intervals and distributions and may not be unique.Source: §2.1 · p. 9
- Data characteristics can determine method choice, including judgmental methods when observations are absent, and may require preprocessing such as treating seasonality.Source: §2.1 · pp. 9–10
2.2.1. Box-Cox transformations3
- Box-Cox transformations were motivated by normality and constant-variance assumptions; λ = 0 gives the log transform, λ = 1 gives no transformation up to an additive constant, and intermediate values bridge multiplicative and additive models.Source: §2.2.1 · p. 11
- Back-transforming forecast quantiles yields the original-scale quantiles, but intervals symmetric on the transformed scale will not be symmetric on the original scale.Source: §2.2.1 · p. 11
- Back-transforming the forecast mean can produce biased original-scale forecasts unless corrected, because the transformation is nonlinear.Source: §2.2.1 · p. 11
2.2.2. Time series decomposition4
- Decomposition represents a series through components, commonly additive or multiplicative; the simplest single-season additive form has trend, seasonality, and remainder, while logs enable multiplicative use of additive methods.Source: §2.2.2 · p. 11
- Model-based decompositions can yield confidence and prediction intervals; STR additionally accommodates missing values, outliers, multiple seasonal or cyclic components, exogenous influences, and complex schedules.Source: §2.2.2 · p. 12
2.2.3. Anomaly detection and time series forecasting5
- Forecast intervals are sensitive to contextual anomalies, with anomalies at the forecast origin having the greatest impact; anomalies may also signal critical events or data-quality problems.Source: §2.2.3 · p. 13
- Forecasting anomaly detection covers contextual anomalies within a series, anomalous subsequences, and anomalous series within a collection.Source: §2.2.3 · p. 13
- Model-based detection flags observations whose deviations from predictions exceed a threshold, whereas feature-based detection uses statistical features to identify atypical behavior.Source: §2.2.3 · p. 13
2.2.4. Robust handling of outliers in time series forecasting6
- A few aberrant observations can bias estimates and reduce ARIMA prediction efficiency, especially near the forecast origin, making preliminary analysis of extreme observations necessary.Source: §2.2.4 · p. 14
- Robust forecasting can distinguish influential outliers that affect parameter estimates, detect level shifts, and improve accuracy through timely detection or robust estimation without removing outliers.The section also describes robust parameter estimation as an alternative to correcting or removing outliers.Source: §2.2.4 · p. 14
2.2.5. Exogenous variables and feature engineering7
- Exogenous predictors may need temporal-resolution alignment; feature engineering can encode interactions, while dimension reduction compresses numerous predictors into a lower-dimensional representation.PCA is useful when candidate predictors outnumber observations, and regularisation or partial least squares can mitigate challenges from correlated predictors.Source: §2.2.5 · p. 15
2.3.1. Exponential smoothing models8
- Exponential smoothing weights past observations less as they recede, and the appropriate method depends on the series components, such as trend or seasonality.Simple Exponential Smoothing is adequate when no clear trend or seasonal pattern is present.Source: §2.3.1 · pp. 16–17
- State-space exponential smoothing models extend traditional point-forecast methods to provide future probability distributions, including prediction intervals.The models use a statistical framework based on a single source of error.Source: §2.3.1 · p. 16
- Exponential smoothing variants match level, trend, and seasonal patterns; state-space likelihood enables AIC-based automatic model selection, and additive or multiplicative seasonal forms are Holt-Winters (Winters, 1960).Source: §2.3.1 · p. 17
2.3.2. Time-series regression models9
- Time-series regression forecasts a target from predictor variables under a linear relationship, with coefficients estimating each predictor’s effect while accounting for the others.Examples include using past sales, prices, advertising, promotions, special days, and holidays to forecast product sales.Source: §2.3.2 · p. 17
- A high R2 can reflect overfitting rather than forecastability, so regression should be assessed with cross-validation or complexity-aware criteria such as AIC, AICc, BIC, and adjusted R2.Source: §2.3.2 · pp. 17–18
- Regression diagnostics examine residual correlation and autocorrelation; normality is rarely necessary for accuracy, while homoscedasticity matters chiefly for prediction intervals.Violations may indicate unexplained target variance and a need for other or additional regressors.Source: §2.3.2 · p. 18
- Regression can incorporate categorical indicators for events, holidays, and seasonality; Fourier terms and lagged variables can help with long series showing complex seasonal patterns.Source: §2.3.2 · p. 18
- When potential regressors are numerous relative to observations, stepwise regression, dimension reduction, or shrinkage can facilitate training and help avoid overfitting.Source: §2.3.2 · p. 18
2.3.3. Theta method and models10
- The theta method is a simple, enduring forecasting approach: Nikolopoulos and Thomakos (2019) show AR(1) forecasts are theta forecasts, and Hyndman and Billah (2003) show the same for SES forecasts with drift.In its simplest form, it combines the last observation with a trend function.Source: §2.3.3 · pp. 18–19
- The theta method combines simplicity, theoretical foundations, adaptability, and performance, and is recommended as a critical benchmark because it is simple to apply, freely implemented, scalable, and performs well.The discussion argues that complex methods should be judged against whether they can outperform the theta method.Source: §2.3.3 · pp. 19–20
2.3.4. Autoregressive integrated moving average (ARIMA) models11
- ARIMA(p, d, q) models are estimated by maximum likelihood, while ARIMA(p, d, 0) permits ordinary least squares; order selection uses autocorrelation diagnostics or information criteria.Empirical autocorrelation and partial autocorrelation help identify p and q for low orders, while AIC and BIC are used in practice otherwise.Source: §2.3.4 · p. 20
- ARIMA forecasts and prediction intervals are straightforward to compute, while the differencing order d is selected using unit-root tests whose power is limited.d = 0 models the series level, d = 1 models first differences, and d = 2 is rare; practitioners often find signals for d = 1. Seasonal ARIMA adds seasonal orders and periodicity, while ARIMAX incorporates external regressors.Source: §2.3.4 · pp. 20–21
2.3.5. Forecasting for multiple seasonal cycles12
- For multiple seasonal cycles, MLR is fast, flexible, and supports exogenous regressors but cannot decompose components that change over time; Prophet is easier to use and robust to missing values, structural changes, and outliers.MLR is a common approach, while Prophet is an automated Generalised Additive Model approach.Source: §2.3.5 · p. 21
- Multiple-seasonality methods trade flexibility and evolving components against computational cost: MSARIMA and SSARIMA support evolving terms or regressors but can be inflexible or expensive, while seasonal Holt-Winters variants model daily, weekly, or intraday cycles.SSARIMA is especially costly for high-frequency series; the seasonal Holt-Winters extensions include double and triple seasonal models and a parsimonious seasonal-state approach, whose implementation can be inflexible and computationally costly.Source: §2.3.5 · p. 21
- TBATS combines Fourier terms, exponential-smoothing state space, and Box-Cox transformation in an automated model, whereas FASSTER adds state switching to capture irregular multiple seasonality.TBATS allows terms to evolve and can produce accurate forecasts, but is inflexible, potentially slow, and does not allow covariates; FASSTER is described as fast, flexible, and supporting exogenous regressors.Source: §2.3.5 · p. 22
- Machine-learning approaches to multiple seasonal cycles show promising real-data results and can accommodate any exogenous regressor and nonlinear series, though interpretability may be problematic.Examples include MLP (Dudek, 2013; Zhang and Qi, 2005), RNN, GRNN (Dudek, 2015), and LSTM, with applications reported by Bandara et al. (2020a) and Xie and Ding (2020).Source: §2.3.5 · p. 22
2.3.6. State-space models13
- State-space systems formulate time-series models through typically unobserved but meaningful states, with transition equations governing state dynamics and observation equations linking the system to data.Typical states include trends, seasonal components, and time-varying parameters.Source: §2.3.6 · p. 22
- State-space systems estimate states and their covariance over time using Kalman filtering for online applications or smoothing over the full sample for offline applications.Unknown system-matrix parameters are normally estimated by maximum likelihood using prediction error decomposition.Source: §2.3.6 · p. 23
- Nonlinear and non-Gaussian state-space models are possible, but require more computationally burdensome recursive algorithms or simulation methods.Options include extended or unscented Kalman filters, particle filters, Monte Carlo, bootstrapping, and importance sampling.Source: §2.3.6 · p. 23
- State-space systems are flexible containers for univariate, multivariate, nonlinear, non-Gaussian, and other model families in a single framework.Examples include regression, ARIMAx, exponential smoothing, VARs, and stochastic-volatility models.Source: §2.3.6 · p. 23
2.3.7. Models for population processes14
- Cohort-component models use matrix algebra to project populations by age and other characteristics from assumptions about fertility, mortality, and migration.The population vector advances as Pt+1 = GPt, with G representing a growth matrix.Source: §2.3.7 · p. 24
- Among demographic drivers, mortality is most predictable, fertility is intermediate, and migration is least predictable; components may be projected deterministically or probabilistically.The components can be forecast separately or uncertainty can be based on past prediction errors.Source: §2.3.7 · pp. 24–25
- Demographic model extensions can represent multiple regions or states and geographic levels, but require detailed transition data.Examples include multiregional, multistate, and multilevel models.Source: §2.3.7 · p. 25
- Microsimulation offers a pragmatic alternative for forecasting population trajectories, using large-sample Monte Carlo simulations based on available transition rates.Current methodological developments mainly predict individual demographic components rather than describe the population renewal mechanism.Source: §2.3.7 · p. 25
2.3.8. Forecasting count time series16
- Predictive mass functions provide coherent count forecasts on discrete support and capture distributional, including tail, information.For low-count series with small support, point forecasts can mislead and continuous approximations such as Gaussian ones are particularly inappropriate.Source: §2.3.8 · pp. 25–26
2.3.9. Forecasting with many variables17
- In non-stationary settings with unit roots and structural breaks, simple forecast models can outperform structural models, especially at short horizons.VARs increase estimation uncertainty through greater parameterisation, and their curse of dimensionality has motivated factor-augmented variants.Source: §2.3.9 · p. 27
- Multivariate models require forecasting explanatory variables or specifying scenarios, while VARs use lagged system variables and can serve as baseline forecasts.Scenarios, such as setting explanatory variables to their means, can illustrate outcomes of different policy choices.Source: §2.3.9 · p. 27
- Panel time-series forecasting centers on dependence across units, with the Global VAR approach used at the country level.Source: §2.3.9 · pp. 27–28
- Panel forecasting must address common outliers, structural breaks, split trends, and unit-specific variation; Nymoen and Sparrman (2015) first applied indicator saturation in a panel context, but software limits applications.The gets R package can be used with panel data.Source: §2.3.9 · p. 28
2.3.10. Functional time series models18
- Functional forecasting often uses FPCA for dimension reduction; Aue et al. (2015) forecast principal-component scores via VAR, while many models’ stationarity requirement is often rejected.Source: §2.3.10 · pp. 28–29
2.3.11. ARCH/GARCH models19
- ARCH weights past squared forecast errors, while GARCH adds past squared conditional volatility and captures volatility clustering.Choosing the estimation period is contested: periods that are too long can be irrelevant to the forecast horizon, while periods that are too short add noise.Source: §2.3.11 · p. 29
- GARCH extensions address nonlinearity, asymmetry, and long memory, while multivariate VEC and BEKK models represent conditional covariance with different estimation and positivity tradeoffs.Source: §2.3.11 · pp. 29–30
2.3.12. Markov switching models20
- Markov-switching AR/VAR models allow latent regimes to change mean, autoregressive, and shock-variance parameters, capturing nonlinearities and regime-related variance, skewness, kurtosis, and persistence.Source: §2.3.12 · pp. 30–31
- Multivariate Markov-switching models risk over-parameterisation, while their regime-transition probabilities may vary over time in response to lagged endogenous, exogenous, or state values.Source: §2.3.12 · p. 31
- Maximum likelihood estimation commonly uses the EM algorithm, and Bayesian Monte Carlo Markov chain approaches have reported advantages for multivariate Markov-switching models.Source: §2.3.12 · p. 31
2.3.13. Threshold models21
- Threshold autoregressive models assign regimes according to a threshold variable’s position relative to K −1 levels; the variable may be exogenous or a lag of the series, yielding SETAR models.Source: §2.3.13 · p. 31
- Unlike TAR’s abrupt regime shifts at threshold crossings, STAR allows gradual changes among regimes.Source: §2.3.13 · p. 32
- Threshold models also address volatility, with GJR-GARCH of Glosten et al. (1993) interpretable as a special case, and include multivariate VAR-threshold and error-correction extensions.Source: §2.3.13 · p. 32
2.3.14. Low and high prices in volatility models22
- Daily low and high prices add information about intraday return variation and support range-based variance estimators, but these estimators ignore temporal dependence and cannot produce multi-period dynamic volatility forecasts.Source: §2.3.14 · p. 32
- The review groups low/high-price volatility models into approaches based on price range or range-based estimators, conditional variance, and conditional mean of the range, whose forecasts require scaling.Source: §2.3.14 · p. 33
- Low/high-price multivariate volatility models range from standard multivariate models using range-based variance estimators to models that use low/high prices directly for both variances and covariances.Most models in the first group omit covariance modeling; co-range models apply low/high prices to both variance and covariance construction.Source: §2.3.14 · p. 33
2.3.15. Forecasting with DSGE models23
- DSGE forecasting uses Bayesian estimation and a state-space representation, and evaluates both point and density forecasts.Point forecasts can be assessed with static and dynamic analyses; density forecasts draw on Bayesian uncertainty and are commonly evaluated using probability integral transforms and log predictive density scores.Source: §2.3.15 · pp. 33–34
2.3.16. Robust equilibrium-correction forecasting devices24
- Equilibrium-correction forecasts can fail after abrupt equilibrium changes, continuing to converge toward the previous equilibrium; flexible estimation windows can help adapt after structural breaks.The subsection notes that estimating parameters over smaller or more flexible subsamples can robustify forecasts, including through adaptive selection of estimation windows.Source: §2.3.16 · p. 35
- Targeted transformations of estimated equilibria can improve equilibrium-correction forecasts over naive forecasts at short and long horizons while retaining embedded economic theory.Differencing the equilibrium can remove components susceptible to shifts, but it also increases forecast-error variance; other proposed approaches include updating the equilibrium, modeling structural breaks, or smoothing transformed-equilibrium estimates.Source: §2.3.16 · p. 35
2.3.17. Forecasting with data subject to revision25
- Evaluation of forecasting with revised data should mimic real time; pseudo-real-time analysis uses data up to each forecast origin from the latest available vintage when the study is conducted.The passage identifies out-of-sample forecast performance as the gold standard and distinguishes pseudo-real-time analysis by its use of latest-available rather than origin-time vintages.Source: §2.3.17 · p. 35
- Using revised data instead of the vintage available at forecast time can overstate real-time performance; real-time-vintage estimation matches lightly revised training data to forecast-origin maturity, and accounting for revisions can improve accuracy.Source: §2.3.17 · p. 36
2.3.18. Innovation diffusion models27
- The Bass model represents adoption with market potential, innovation, and imitation parameters; its Generalised Bass extension adds an intervention function for shocks such as marketing, prices, or policy.Source: §2.3.18 · p. 37
- Guseo and Guidolin (2009)'s GGM makes market potential knowledge-dependent, separates information from adoption, and improves forecasts over the simpler BM, especially in early-series slowdowns.Source: §2.3.18 · p. 37
2.3.19. The natural law of growth in competition28
- Logistic S-curves describe growth in competition across biological and human systems, with nested S-curves and connections to Lotka–Volterra competition.Source: §2.3.19 · p. 38
- Each S-curve has a life cycle of seasons, with different behaviors suited to periods of stable conditions, disruption, learning, or preparation.Source: §2.3.19 · p. 38
- On a flattened growth curve, innovations may go unappreciated, while qualitative S-curves can still be accurate, informative, and insightful.During steep growth, the public rapidly assimilates innovations that exceed the curve; once growth flattens, innovators may receive no credit whether or not they innovate.Source: §2.3.19 · pp. 38–39
2.3.20. Synchronic and diachronic competition29
- Competition models capture interaction-driven life-cycle perturbations that univariate diffusion and other time-series models miss, especially for medium- and long-term prediction.ODE-based systems analysis distinguishes synchronic competition between co-present competitors from diachronic competition in which competitors arrive later; Lotka–Volterra models are a pioneering synchronic approach.Source: §2.3.20 · p. 40
- For Lotka–Volterra models without closed-form solutions, a stacked equation form supports first-stage nonlinear least-squares inference, followed by short-term SARMAX refinement.The first stage requires no strong assumptions about the stochastic error distribution; the estimated solution can serve as exogenous input for forecasts.Source: §2.3.20 · pp. 40–41
2.3.21. Estimation and representation of uncertainty30
- For multi-step forecasting, simultaneous forecast intervals account for temporal dependence in forecast errors, unlike marginal intervals.Marginal intervals can be derived from parametric or non-parametric distributions, quantile regression, bootstrapping, or quantile random forests. Stochastic optimization uses scenario representations such as random vectors or path forecasts, whereas robust optimization places uncertain parameters in deterministic uncertainty sets without specific distributional assumptions.Source: §2.3.21 · pp. 41–42
- Forecast uncertainty can be shaped as polyhedral or ellipsoidal regions for integration into optimization, but communicating uncertainty remains a major adoption bottleneck.The bottleneck is especially noted for multivariate time-series use cases and adverse weather events.Source: §2.3.21 · p. 42
2.3.22. Forecasting under fat tails31
- Fat tails occur across natural and human-related phenomena, making forecasting fat-tailed random variables pivotal in many fields of life and science; overlooking their implications can generate substantial forecast errors.Source: §2.3.22 · p. 42
- For fat-tailed variables, forecasting is unreliable when α ≤ 2.5, while forecasting can proceed for α > 2.5 and improves as α increases, according to Taleb (2020).For 1 < α ≤ 2, the law of large numbers can converge so slowly that an unavailable number of observations is needed for reliable forecasts; α ≤ 1 lacks a defined mean and is described as not predictable.Source: §2.3.22 · pp. 42–43
2.4.1. Foundations of Bayesian forecasting32
- Bayesian forecasting produces a probabilistic forecast by integrating uncertainty about unknown parameters, and can also average over model uncertainty using posterior model probabilities.The framework combines a predictive model, an observed-data model, and prior beliefs; practical predictive-density integration is usually infeasible analytically and requires computation.Source: §2.4.1 · pp. 43–44
2.4.2. Implementation of Bayesian forecasting33
- When the posterior is accessible, exact simulation yields predictive-density estimates that can be made arbitrarily accurate with enough posterior draws.When likelihoods are intractable or model unknowns are high-dimensional, approximation methods are required; approximate Bayesian computation produces predictives shown equivalent to exact predictives in large samples and numerically indistinguishable in investigated finite samples.Source: §2.4.2 · p. 44
- Approximating the posterior to compute p(yn+1|y) does not significantly reduce predictive accuracy.Bayesian prediction nevertheless assumes the model, or the models averaged, contain the data-generating process.Source: §2.4.2 · p. 45
- Alternatives combine predictive distributions using task-specific weights or calibration, or replace the posterior with one tailored to the predictive task and measure of interest.The review notes that these approaches do not require the true model to be spanned by the constituent set and cites proofs of theoretical validity.Source: §2.4.2 · p. 45
2.4.3. Bayesian forecasting with copulas34
- Bayesian copulas jointly model marginal distributions and copula parameters, provide probabilistic forecasts through the predictive density, and incorporate expert information through priors.Joint modelling is associated with improved forecasting efficiency.Source: §2.4.3 · p. 45
- Dynamic copula dependence can be modelled with autoregressive or GARCH forms, factor copulas, latent-variable stochastic autoregression, or covariate-dependent parameters.Factor copulas simplify computation, while covariate-dependent copulas improve interpretability and allow ARMA-like or GARCH-like tail dependence.Source: §2.4.3 · p. 46
- Unequal-length and mixed-frequency series can be handled by separating marginal and copula parameters, decomposing dependence into linear and nonlinear components, or using data augmentation.The cited mixed-frequency decomposition uses high- and low-frequency data for linear and nonlinear dependencies, respectively.Source: §2.4.3 · p. 46
- Traditional copula families support lower-dimensional forecasting, while higher-dimensional settings require computational approaches such as low-rank approximations or factor copulas with stochastic loadings.The review identifies d < 10 as a setting where traditional approaches are straightforward.Source: §2.4.3 · p. 46
2.5.1. Leading indicators and Granger causality35
- A leading indicator Granger-causes a forecast variable when including it in the information set lowers the mean square error of an optimal forecast at some horizon.Leading indicators support short-term forecasting and nowcasting of economic variables such as quarterly GDP, which is published with a lag.Source: §2.5.1 · pp. 46–47
- Granger causality is not sufficient to establish causation, and treating it as such risks the post hoc ergo propter hoc fallacy.Economic activity is rarely monocausal.Source: §2.5.1 · p. 47
- A bivariate Granger test asks whether past values of a leading indicator jointly predict the forecast variable beyond its own past values.The null is that the indicator does not Granger-cause the forecast variable; the maximum lag can be chosen using an information criterion such as AIC or BIC.Source: §2.5.1 · p. 47
2.5.2. Model complexity36
- Empirical studies provide little evidence that complex models forecast better than simple ones; the bias-variance trade-off makes forecast MSE U-shaped in complexity, with optimal complexity rising with sample size.Complexity includes both parameter count and the functional form of relationships; estimation variance grows with complexity and declines with sample size, while simple methods are more prone to misspecification bias.Source: §2.5.2 · p. 48
- Evidence on simplicity is mixed: broad support and M1/M2 results favor simpler models; complex nonlinear autoregressions were less accurate for most US monthly series, while M4 tended to favor more complex models.Source: §2.5.2 · p. 48
2.5.3. Variable selection37
- When many predictors are available, variable selection reduces dimensionality either by aggregating predictors into factors or shrinking coefficients, sometimes to zero.Frequentist options include LASSO and Complete Subset Regression, which averages forecasts from OLS models fitted to all regressor combinations; Bayesian selection instead averages across sparse models weighted by variable-inclusion probabilities and is effective in economic forecasting, including Bayesian VARs.Source: §2.5.3 · p. 49
2.5.4. Model selection38
- Forecast failures can arise from mis-measured forecast origins and unanticipated location shifts, so model selection must account jointly for observations and variables, including outliers and breaks.Autometrics uses impulse and step indicator saturation to detect outliers and breaks while selecting variables; because in-sample fit may not ensure success after crises or pandemics, robust variants can avoid systematic mis-forecasts.Source: §2.5.4 · pp. 49–50
- Forecasting-oriented selection uses loose significance thresholds, around AIC’s implied 10% to 16%, to retain relevant regressors; its costs are small relative to avoiding systematic failure from missed location shifts.Averaging undominated terminal models across paths can also provide a robust forecasting device. Castle et al. (2018a) report that the threshold works regardless of location shifts at or near the forecast origin.Source: §2.5.4 · p. 50
2.5.5. Cross-validation for time-series data39
- Time-series cross-validation uses rolling-origin evaluation, moving observations into training data and optionally refitting on sliding or expanding windows.Ordinary cross-validation is not straightforward for time series because of non-stationarity, serial dependence, withheld test values, and the problem of using future observations to predict the past; rolling-origin methods may still use data suboptimally, especially with small samples.Source: §2.5.5 · p. 51
- With purely autoregressive models, random-fold cross-validation can estimate generalisation error accurately when in-sample model errors are uncorrelated.Underfitting can be checked by testing residuals for serial correlation, for example with the Ljung-Box test.Source: §2.5.5 · p. 51
2.6.1. Forecast combination: a brief review of statistical approaches40
- Statistical forecast combinations use estimated weights to seek better accuracy than their component forecasts, with approaches ranging from error-variance minimisation and regression to Bayesian and time-varying weights.More recent approaches include Akaike weights, bootstrapped or cross-validated combinations, neural networks, and forecasts from multiple temporal levels.Source: §2.6.1 · pp. 51–52
- Evidence that statistical combinations improve accuracy is near unanimous, including in forecasting competitions and empirical studies (Elliott et al., 2005; Jose and Winkler, 2008; Andrawis et al., 2011; Kourentzes et al., 2019).The review points to combining statistical and machine-learning forecasts and including human judgment as future directions.Source: §2.6.1 · p. 52
- Statistical forecast-combination approaches generate combined probability forecasts, and their results need to be interpretable and suitable for decision-making.Source: §2.6.1 · p. 52
2.6.2. Density forecast combinations41
- Density forecasts represent complete forecast uncertainty, helping decision makers assess model-related risks; pooling can mitigate individual-density misspecification when the true density is non-normal.Combining normal densities produces a flexible mixture that can accommodate heavier tails, skewness, kurtosis, and approximate nonlinear specifications.Source: §2.6.2 · p. 53
- Density-combination methods range from simple averaging to time-varying schemes; linear opinion pools choose convex weights by minimizing Kullback-Leibler divergence from the true density.Extensions allow nonlinear transformations, value-dependent weights, time-varying weights, or Bayesian predictive synthesis; the state-space scheme with nonnegative weights summing to one requires particle filters for inference.Source: §2.6.2 · p. 53
2.6.3. Ensembles and predictive probability post processors42
- For decisions under unknown future events, predictive probability densities support expected-benefit or loss calculations, while combining multiple forecasts provides one predictive probability conditional on them.Bayesian inversion can derive predictive probabilities from model forecasts, and uncertainty post-processors include Bayesian model averaging, model output statistics, ensemble model output statistics, and model conditional processors.Source: §2.6.3 · p. 54
2.6.4. The wisdom of crowds43
- The simple average is more accurate than randomly selecting a crowd forecast, can outperform nearly all individuals, and often beats more complex point-aggregation schemes.Mannes et al. (2012) identify individual expertise and crowd diversity as factors in average-forecast quality; diversity can yield gains even when individuals differ substantially in accuracy. When two forecasts bracket the truth, their average beats random selection; without bracketing, it performs as well as the average expert.Source: §2.6.4 · p. 55
- When crowd expertise varies widely, standout forecasters may outperform the group, while a small diverse crowd with a minimum expertise level can be effective.Mannes et al. (2014) suggest that combining a small crowd can be especially powerful in practice.Source: §2.6.4 · p. 55
- Averaging probabilities is common, but crowd diversity can make probability forecasts underconfident; averaging quantiles or using trimmed opinion pools can yield sharper or better-calibrated forecasts.The linear opinion pool is described as “hard to beat in practice”.Source: §2.6.4 · p. 55
2.7.1. Forecasting with big data44
- Big-data methods use selection, shrinkage, variable combination, and machine learning, but their accuracy gains for macroeconomic forecasts remain uncertain and black-box models are mainly used for forecasting rather than policy analysis.Castle et al. (2020a) find no forecasting improvements from big data sets over small models.Source: §2.7.1 · p. 56
- Big data can improve nowcasts when events have already occurred, but future predictor values remain uncertain and non-stationarity undermines validation approaches that assume stable data.Models assuming stationary distributions, even after differencing, do not forecast well ex ante; forecasters must also guard against false positives, sampling bias, and structural breaks.Source: §2.7.1 · p. 56
2.7.2. Forecasting on distributed systems45
- Big-data time series pose storage, complexity, and computational-cost challenges, while distributed systems struggle with dependencies among observations and often lack advanced analysis tools.Scalable algorithms and additional processing considerations are therefore needed for distributed time-series forecasting.Source: §2.7.2 · p. 57
- Distributed forecasting approaches include Spark-based nearest-neighbour and multi-step methods, MapReduce estimation, and combining local estimators to approximate global models.Talavera-Llames et al. (2016) reported satisfactory performance for a Spark nearest-neighbours algorithm; Galicia et al. (2018) split multi-step forecasting into h sub-problems.Source: §2.7.2 · pp. 57–58
2.7.3. Agent-based models50
- Agent-based models explore complex systems from the bottom up, aggregating individual behavioral changes, and are useful with limited data, uncertain interventions, or poorly understood dynamics.They account for environmental- and individual-level factors that may influence the forecast variable.Source: §2.7.3 · p. 58
- Agent-based modeling must balance simplicity and realism, and developing a model requires subject expertise; the KISS principle is often recommended.Models are calibrated with empirical data and validated by testing whether their results differ significantly from observations.Source: §2.7.3 · pp. 58–59
2.7.4. Feature-based time series forecasting51
- Feature-based forecasting uses series characteristics to select an appropriate model or combine candidates because no single model performs best for every series.Features are chosen for the data and application context, and can support model selection or estimate combination weights.Source: §2.7.4 · pp. 59–60
- Feature-based approaches estimate model-combination weights using predicted errors, learned feature relationships, or extracted image features; features also inform interval forecasting.Examples include Kang et al. (2020a), who use 26 features and nonlinear regression for nine methods, and FFORMA (Montero-Manso et al., 2020), which uses 42 features and XGBoost for nine methods.Source: §2.7.4 · p. 60
2.7.5. Forecasting with bootstrap52
- The iid bootstrap is inefficient for dependent time-series data because resampling must preserve dependence; block methods do so by resampling data segments.Variants include nonoverlapping, moving, circular, and stationary block bootstraps.Source: §2.7.5 · pp. 60–61
- When a time series is driven by iid innovations, sieve bootstrap extends iid resampling to dependent data by first fitting parametric models (Bühlmann, 1997).Source: §2.7.5 · p. 61
- The sieve bootstrap filters residuals with an autoregressive model but is nonparametric in spirit, and its generated sample is not a subsample of the original data.Unlike earlier dependent-data bootstrap methods, its sample is conditionally stationary and does not preserve a dependence structure.Source: §2.7.5 · p. 61
2.7.6. Bagging for time series forecasting54
- Bagging for time series requires careful resampling to account for serial dependence and non-stationarity, and its framework comprises data treatment, resampling, forecasting, and aggregation.Petropoulos et al. (2018a) found that addressing model uncertainty alone could achieve superior performance, motivating Bootstrap Model Combination with different model forms fitted to the original data.Source: §2.7.6 · p. 62
- Bagged ETS outperformed the original benchmarks on the M3 data set.It applies moving block bootstrap to the remainder after STL decomposition and aggregates ETS forecasts using the median.Source: §2.7.6 · p. 62
- Meira et al. (2020) found that analysing ensemble prediction intervals before final aggregation is important for improving interval performance.Their treating and pruning strategies address prediction intervals for both model selection and forecast combinations, tested on real series from the M forecasting competitions.Source: §2.7.6 · p. 62
2.7.7. Multi-step ahead forecasting55
- Recursive forecasting iterates a one-step model, whereas direct forecasting fits separate models at each horizon; their relative performance depends on model specification and context.For linear models, recursion is more efficient when correctly specified, while direct forecasts are more robust to misspecification; practical comparisons are mixed and depend on factors including data, sample size, and horizon.Source: §2.7.7 · pp. 62–63
- Multi-step-ahead strategy selection is an empirical design choice because strategies trade off forecast bias and variance.The strategy should therefore be included in model selection.Source: §2.7.7 · p. 63
2.7.8. Neural networks56
- Neural networks model non-stationary, nonlinear, complex data through weighted layers, but the complexity needed for universal approximation can make training computationally infeasible.Networks transform weighted inputs through hidden-layer functions and train by adjusting weights against a loss; MLPs, RBFs, and RNNs are common topologies.Source: §2.7.8 · p. 64
2.7.9. Deep probabilistic forecasting models57
- Deep probabilistic neural forecasts can output distribution parameters, selected quantiles, or parameters of existing probabilistic models, but the approaches reviewed are a subjective selection.Open challenges include modeling dependence across series and obtaining reliable uncertainty estimates when only a few series are available.Source: §2.7.9 · pp. 64–65
2.7.10. Machine learning59
- Machine-learning methods learn data relationships without prescribing a generating process, can succeed on long high-frequency series, but require sufficient data and careful preprocessing, computation, and tuning.Short complex series may yield sub-optimal forecasts without preprocessing; implementation also requires resources and expertise to set hyperparameters.Source: §2.7.10 · p. 66
- Cross-learning lets one model forecast multiple series, reducing computation, and can transfer across similar datasets to support generalized forecasting methods.The review considers cross-learning the most effective way to apply machine learning to batch time-series forecasting.Source: §2.7.10 · p. 66
- Cross-learning can mitigate data limitations by capturing patterns shared across series, including seasonal cycles and special events.It can use hierarchical structure and categorical or explanatory inputs to identify common patterns among otherwise differing series.Source: §2.7.10 · p. 66
2.7.11. Machine learning with (very) noisy data60
- For very noisy data, LASSO and elastic net shrink coefficients to limit overfitting, while stringent BIC-based tuning can select the degree of shrinkage.Very low signal-to-noise ratios heighten overfitting risk, making adequate shrinkage important for uncovering predictive signal.Source: §2.7.11 · p. 67
- Neural networks model nonlinear relationships but can overfit noisy data; SGD regularization, weight penalties, and dropout are described as safeguards.This concern is especially salient for asset-return forecasts, where the review urges mindfulness about overfitting.Source: §2.7.11 · pp. 67–68
2.7.12. Clustering-based forecasting62
- Clustering-based forecasting uses similarities among data entries; one approach trains a dedicated forecaster for each cluster, and such systems are reported to yield lower errors.Clustering can divide training data into similar-pattern subsets before fitting cluster-specific forecasters.Source: §2.7.12 · p. 68
2.7.12. Clustering-based forecasting
- A second clustering approach forecasts from cluster centroids or label sequences, matching a predicted sequence to historical ones and averaging the following load curves.The review notes that clustering algorithm choice should account for both effectiveness and complexity.Source: §2.7.12 · p. 68
2.7.12. Clustering-based forecasting62
- The optimal number of clusters is selected by trial and error as the count producing the lowest forecasting error.Source: §2.7.12 · p. 69
2.7.13. Hybrid methods63
- Hybrid methods combine diverse techniques across forecasting, data preprocessing, and parameter identification, aiming to exploit complementary strengths and address individual-method limitations.Preprocessing hybrids may decompose series or signals or classify historical data; postprocessing hybrids forecast residual errors to correct base forecasts.Source: §2.7.13 · p. 69
- Hybrid methods can be robust to sudden parameter changes, but require understanding of base methods, careful selection for the problem, and may respond slowly to new data.Hybrids with meteorological inputs may underperform when conditions are unstable.Source: §2.7.13 · pp. 69–70
- Hybrid methods can outperform kNN, NN, and ARIMA-based models; Smyl’s hybrid achieved the best M4 competition performance, and physical–AI and weather-informed approaches can improve forecasts.The weather-informed approach combines historical meteorological data, an optimal learning algorithm, and weather classification; such hybrids may underperform under unstable meteorological conditions.Source: §2.7.13 · pp. 69–70
2.8.1. Parametric methods for intermittent demand forecasting64
- Croston’s method addresses SES’s issue-point bias by forecasting demand sizes and inter-demand intervals separately with SES, then taking their ratio for mean demand per period.Demand sizes represent positive demand amounts, while intervals indicate how intermittent demand is.Source: §2.8.1 · p. 70
- The Syntetos-Boylan Approximation (SBA) yields approximately unbiased estimates, has repeatedly improved empirical inventory forecasting, and serves as a benchmark for new intermittent-demand methods.Syntetos and Boylan (2001) identified inversion bias in Croston’s method; their 2005 modification addressed it.Source: §2.8.1 · p. 70
2.8.2. Non-parametric intermittent demand methods65
- Non-parametric intermittent-demand methods include bootstrapping, which resamples historical demand to construct an empirical distribution, and overlapping or non-overlapping block aggregation.In non-overlapping blocks, the series is divided into consecutive blocks; in overlapping blocks, the oldest observation is advanced at each period.Source: §2.8.2 · p. 71
- Willemain et al. (2004)’s Markov-chain bootstrap, which switches between no-demand and demand periods, outperformed SES and Croston’s method in reported comparisons.Zhou and Viswanathan (2011) separately resample demand intervals and sizes, with good reported performance for long lead-times.Source: §2.8.2 · p. 71
- The empirical-EVT improvement to OB better models lead-time-demand tails and yields higher achieved target cycle service levels than the original method; OB has a clear advantage over NOB unless demand history is short.Source: §2.8.2 · p. 71
2.8.3. Classification methods66
- Classification groups series so subsets can share forecasting methods, but general categorisation rules may not identify the best method; SBC classifies intermittent demand using average demand interval and demand-size variation.SBC rules assume iid demand and compare methods by expected mean square error.Source: §2.8.3 · p. 72
- Boylan et al. (2008) found SBC cut-off rules robust to inexact specification even for series not necessarily meeting iid assumptions.Combining standard intermittent-demand methods did not directly improve accuracy, while combinations across temporal frequencies using methods selected by extended SBC gave good results.Source: §2.8.3 · p. 72
2.8.4. Peak over the theshold67
- A peak-over-threshold decomposition separates baseline from extreme values to create intermittent series resembling those Croston (1972) analyzed, enabling forecasting of extremes but not their exact timing.The approach instead aims to provide a satisfactory cumulative forecast over a long period.Source: §2.8.4 · p. 73
2.9.1. Fuzzy logic68
- Fuzzy logic represents uncertainty and vagueness by allowing statements or variables to be partially true or false, with fuzzy-set membership grades from 0 to 1.This contrasts with Boolean logic's two values, true and false.Source: §2.9.1 · p. 74
- Fuzzy inference maps crisp inputs to graded memberships through membership functions and fuzzification, then applies fuzzy rules; Mamdani outputs membership functions, while Sugeno outputs a crisp value.Defuzzification converts membership values back to crisp values.Source: §2.9.1 · p. 74
2.9.2. Association rule mining69
- Association-rule quality is commonly evaluated with support, confidence, and lift; confidence represents implication strength as a conditional probability, while lift measures correlation between antecedent and consequent, and mining first computes frequent itemsets.Source: §2.9.2 · p. 75
- Association rule mining exhaustively extracts statistically relevant correlations from categorical transaction data, but can be computationally demanding; continuous variables may first be discretised.The mining process extracts frequent itemsets before deriving rules that meet minimum support and confidence thresholds.Source: §2.9.2 · p. 75
2.9.3. Forecasting with text information70
- Text can provide forecasting predictors after cleaning and numerical conversion through sentiment, bag-of-words, or embeddings; choosing the right feature-extraction technique is the key challenge.Sentiment analysis can encode polarity, while bag-of-words records word occurrence and Word2Vec represents words using surrounding-word context.Source: §2.9.3 · p. 76
2.10.1. Cross-sectional hierarchical forecasting71
- Cross-sectional hierarchical time series aggregate series by geographic or logical relationships; retail product sales can roll up through categories and families to shop or distribution-centre totals.Source: §2.10.1 · p. 77
- Coherent cross-sectional forecasts respect aggregation constraints and have improved accuracy in reported studies.Independent forecasts generally do not add up across hierarchical levels, motivating coherence for decisions at different levels.Source: §2.10.1 · p. 77
- Bottom-up aggregates bottom-level forecasts, while top-down disaggregates top-level forecasts; unbiased top-down forecasts are possible despite earlier bias concerns.In two-stage reconciliation, independently produced base forecasts are adjusted to satisfy hierarchy constraints; MinT minimizes the trace of reconciled forecast-error covariance (Wickramasuriya et al., 2019).Source: §2.10.1 · p. 77
- More complex reconciliation can increase forecast variability through covariance-matrix estimation or implicit combination weights, motivating approximations that balance uncertainty against finer tuning.For elliptical probabilistic base forecasts, analytical reconciliation is possible; otherwise, bootstrapping in-sample errors offers a non-parametric approach (Panagiotelis et al., 2020).Source: §2.10.1 · p. 78
2.10.2. Temporal aggregation72
- Temporal aggregation can reduce intermittence in slow-moving series and has improved forecasting performance for both slow- and fast-moving series.Higher aggregation may provide structural information that improves forecasts for intermittent data.Source: §2.10.2 · pp. 78–79
- Multiple temporal aggregation (MTA) avoids selecting a single level and addresses model uncertainty; it typically improves accuracy, though one level can perform better with strong seasonality (Kourentzes et al., 2017; Spiliotis et al., 2019b, 2020c).Different frequencies can reveal different patterns, enabling forecast combinations across temporal levels.Source: §2.10.2 · pp. 78–79
- Temporal hierarchies apply hierarchical reconciliation across time frequencies, and have also been used to obtain reconciled probabilistic forecasts.Source: §2.10.2 · p. 79
2.10.3. Cross-temporal hierarchies73
- Cross-sectional hierarchies vary the unit at a fixed period, while temporal hierarchies vary the period for a fixed unit; many organizational decisions require both dimensions (SKU example).In such cases, the hierarchies function more as statistical devices for accuracy through forecast combinations than as a way to satisfy the original rationale for hierarchical forecasting.Source: §2.10.3 · p. 79
- Cross-temporal hierarchies provide coherent forecasts across units and periods, a common outlook across decision-makers, and greater accuracy than temporal or cross-sectional hierarchies.Their construction remains challenging as hierarchy size grows; one heuristic compartmentalizes estimation but may be suboptimal in reconciliation error, while sequential reconciliation is beneficial but does not ensure coherency.Source: §2.10.3 · pp. 79–80
2.10.4. Ecological inference forecasting74
- Ecological inference forecasting predicts contingency-table inner cells from known margins, but underidentification requires similarity assumptions or covariates, and mathematical programming offers a suitable framework.The approach remains vulnerable to ecological fallacy and aggregation bias.Source: §2.10.4 · pp. 80–81
2.11.1. Judgmental forecasting75
- Judgmental forecasters use information-dependent heuristics, including anchoring on the last observation and adjusting for trend when forecasting time series.The heuristic used depends on available information.Source: §2.11.1 · pp. 81–82
2.11.2. Judgmental adjustments of computer-based forecasts76
- Judgmental adjustments can improve computer-based forecast accuracy by incorporating important extra information, but cognitive and motivational biases can prompt unwarranted adjustments that reduce accuracy.Source: §2.11.2 · p. 82
- Judgmental adjustments can harm accuracy through political interference, optimism bias, double counting, or attention to random movements over long-run patterns (Oliva and Watson, 2009; Fildes et al., 2009; Van den Broeke et al., 2019; Bolger and Harvey, 1993).Recent-outcome feedback may be noisy and encourage over-focus on recent events, whereas feedback on adjustment biases across periods may help; it is less useful when interventions are rare (Goodwin and Fildes, 1999; Petropoulos et al., 2016, 2017).Source: §2.11.2 · pp. 82–83
- Explaining statistical-forecast rationales improved adjustments for simple patterns, while guidance on when to adjust and restrictions on small adjustments were ineffective.Decomposing judgment into easier tasks and providing analogous past outcomes also improved adjustment accuracy (Webby et al., 2005; Lee et al., 2007).Source: §2.11.2 · p. 82
- The reviewed evidence on judgmental adjustment is largely based on laboratory studies, with forecasting support systems discussed as a way to improve adjustment effectiveness.The review directs readers to §3.7.3 for research on forecast adjustment in practice.Source: §2.11.2 · p. 83
2.11.3. Judgmental model selection77
- Judgmental selection had better overall accuracy than algorithmic selection in one comparison, although automatic selection was better at choosing the best models.Participants were also less adept than an FSS providing the optimal algorithm, and judgmental selection was better at avoiding the worst models.Source: §2.11.3 · p. 83
- People’s model choices outperformed averaging model outputs, especially when model quality differences were larger and data noise was lower (De Baets and Harvey, 2020).Participants also adjusted more toward advice from models they perceived as good quality than toward medium- or bad-quality models (De Baets and Harvey, 2020).Source: §2.11.3 · p. 83
2.11.4. Panels of experts78
- The Delphi method can produce more accurate forecasts than unstructured methods and avoid groupthink through anonymity, but multiple rounds add cost and may increase dropout.The method uses repeated anonymous surveys with summaries and feedback to support revisions toward consensus; its outcome is reported as more accurate in decision-making.Source: §2.11.4 · p. 84
- Interaction-group members have provided more accurate judgments than individuals, but evidence about group forecasting is mixed and arranging meetings makes the method less attractive.Panelists interact and debate toward a common forecast.Source: §2.11.4 · p. 84
- Structured analogies elicit experts’ recollections and descriptions of analogous cases, which are ranked for similarity and summarized, usually with similarity-weighted averages.Semi-structured analogies have also been proposed.Source: §2.11.4 · p. 84
- Nikolopoulos et al. (2015) supported that structured analytic and interaction methods could produce forecasts 50% more accurate than unstructured methods such as unaided judgment.A caveat is identifying genuine experts; experience and interaction among experts are also supported as important.Source: §2.11.4 · p. 85
2.11.5. Scenarios and judgmental forecasting79
- In mobile-phone demand forecasts, optimistic scenarios prompted larger adjustments and greater confidence, while pessimistic scenarios produced the largest downward predictions.Optimistic scenarios also moved lower interval bounds most toward the model-based centre, whereas pessimistic scenarios moved upper bounds most toward it.Source: §2.11.5 · p. 85
- Goodwin et al. (2019b) found resilient contrast effects when forecasters saw opposing scenarios, and such exposure increased confidence in initial predictions.Scenario benefits for performance were circumstantial: Goodwin et al. (2019a) found scenarios worsened forecast accuracy and moved production-order decisions farther from optimality, despite decision makers’ fondness for scenarios.Source: §2.11.5 · p. 86
2.11.6. Trusting model and expert forecasts80
- Users often distrust algorithms and prefer human advice, but feedback, understandable models, and opportunities to adjust forecasts can support algorithm trust.Expert forecasts also face distrust related to group, contextual, and motivational biases; credibility and prior experience with a source shape acceptance.Source: §2.11.6 · pp. 86–87
2.12.1. Benchmarking81
- Benchmarking can exaggerate a new method’s benefits through selective metrics, competing methods, or test data, with publication bias compounding the risk.Rigorous evaluation calls for multiple suitable benchmarks, diverse error metrics tailored to the forecast and problem, and statistical tests of method differences.Source: §2.12.1 · p. 87
- The use of hypothesis testing in forecasting remains debated, as it does more generally in statistics.Source: §2.12.1 · p. 87
- Rolling-window sample sizes should be chosen with reference to the latest literature on window choice.Source: §2.12.1 · p. 87
- Where possible, implement and benchmark new forecasting methods in open-source languages such as C, Python, and R to support replicability.Source: §2.12.1 · p. 87
- Methods should be evaluated on appropriate benchmark datasets.Examples include the U.S. macroeconomic dataset (Stock and Watson, 2012) and similar EU and Australian datasets.Source: §2.12.1 · pp. 87–88
- Energy forecasting benchmarks include GEFCom data (Hong et al., 2016) and the IEEE 118 Bus Test Case data (Pe˜na et al., 2018).The Makridakis et al. (2020c) data provide a benchmark dataset for general forecasting.Source: §2.12.1 · p. 88, M Forecasting competitions · p. 88
M Forecasting competitions
This section examines how forecasting methods are compared and evaluated, from competition findings and point-forecast metrics to scoring, calibration, and assessment of predictive distributions. It also considers decision-focused evaluation, how forecasts are communicated and used, and challenges in verification and in balancing human judgment with data.
M Forecasting competitions
- Forecast Hub and Business Forecast Lab exemplify websites demonstrating forecasting methods on real data in epidemiology, macroeconomics, and business.The paper identifies this publication trend as having future potential.Source: M Forecasting competitions · p. 88
2.12.2. Point, interval, and pHDR forecast error measures84
- Different point forecast error measures select different forecasts from the same density: MSE favors the expectation, while MAE and MASE favor the median.The section notes that point, interval, and predictive highest-density-region (pHDR) forecasts can be evaluated after the outcome is observed.Source: §2.12.2 · pp. 88–89
- MAPE favors the (−1)-median, sMAPE an unnamed numerically minimized functional, and pinball loss the corresponding quantile; no loss generally favors the mode.These examples show why a point forecast should be evaluated with a measure aligned to the functional it is meant to elicit.Source: §2.12.2 · p. 89
- For intermittent demand, MAPE is undefined when actuals are zero, and the MAE-minimizing median may be zero while the MSE-minimizing mean is nonzero.The authors recommend choosing the error measure—and tailoring the forecasting algorithm—to the desired functional of the possibly implicit density, rather than routinely using multiple point measures.Source: §2.12.2 · p. 89
- Equal-tailed intervals can be evaluated with the proper interval score, while quantile forecasts can also be scored separately using hinge loss.A shortest interval at specified coverage is not elicitable for practically relevant distribution classes; pHDRs are not elicitable even for unimodal densities, and proposed evaluation includes an ad hoc Winkler-score variation or a coverage test that ignores pHDR volume.Source: §2.12.2 · p. 89
- Shortest intervals and pHDRs have elicitation limitations, complicating their evaluation.A shortest interval with specified coverage is not elicitable for practically relevant distribution classes; pHDRs are not elicitable even for unimodal densities, and proposed pHDR evaluation includes an ad hoc Winkler-score variation or a coverage test that ignores pHDR volume.Source: §2.12.2 · p. 89
2.12.3. Scoring expert forecasts85
- Scoring forecasts against observed outcomes can assess expertise and guide improvement, while proper scoring rules incentivize honest reports and decision-aligned measures can target practical utility.For binary events the Brier score is used, and logarithmic scoring is available for discrete events with more than two outcomes.Source: §2.12.3 · p. 90
2.12.4. Evaluating probabilistic forecasts86
- Full predictive distributions may be reported as CDFs or densities, dense quantile grids for univariate forecasts, or ensembles of simulated paths for multivariate forecasts.Reporting full multivariate distributions can be impracticable, especially for non-parametric or copula-based methods.Source: §2.12.4 · p. 91
- Proper scoring rules are recommended because they address calibration and sharpness simultaneously.For univariate forecasts, the CRPS and log-score are popular strictly proper choices.Source: §2.12.4 · p. 91
- Multivariate evaluation is more difficult: density-based log scores may be impractical, while later studies found energy scores discriminate well with Diebold-Mariano tests (Ziel and Berk, 2019).An earlier study by Pinson and Tastu (2013) concluded that energy scores did not discriminate well between dependency structures.Source: §2.12.4 · p. 91
2.12.5. Assessing the reliability of probabilistic forecasts87
- A calibrated system has uniformly distributed PIT values, and PIT histograms reveal overdispersion, underdispersion, or systematic bias.For ensemble forecasts, the verification rank histogram is the discrete PIT equivalent; small test sets require care in choosing the number of bins (Thorarinsdottir and Schuhen, 2018; Heinrich, 2020).Source: §2.12.5 · p. 92
2.12.6. Statistical tests of forecast performance88
- The Diebold-Mariano test compares whether two forecasts have equal predictive ability under a loss function, with few assumptions and broad applicability (Diebold and Mariano, 1995).Its population-level setting accommodates multiple periods and many error types, including non-Gaussian and serially or contemporaneously correlated errors.Source: §2.12.6 · p. 93
- For multiple forecasts, tests include checks of superior predictive ability and multiple comparisons with a control or the best; these were used to analyze the M3 competition.The M3 analysis used model rankings rather than forecast accuracy scores as its primitives.Source: §2.12.6 · p. 93
- The comparison of the best was used in analyses of the subsequent M4 and M5 competitions (Makridakis et al., 2020c, 2021b).Source: §2.12.6 · p. 93
2.12.7. Forecasting competitions89
- The M and M3 competitions found that simple methods were at least as accurate as more complex models, and M3’s winning Theta method combined linear regression with exponential smoothing.Early competitions also showed the value of combining forecasts to improve predictive accuracy.Source: §2.12.7 · p. 94
- In M4, independently trained machine-learning methods underperformed statistical benchmarks, while cross-learning-based neural-network and machine-learning submissions were among the top performers; five of the top six used forecast combinations (Makridakis et al., 2020c; Smyl, 2020; Montero-Manso et al., 2020).The competition’s main lesson was that machine learning can improve forecasting performance when used properly.Source: §2.12.7 · p. 94
- M2 found limited benefits from domain expertise, tourism results found no added value from exogenous variables and strong yearly naïve forecasts, and NN3 found statistical methods superior as neural networks narrowed the gap.These findings illustrate how competition results can depend on application context and forecast frequency.Source: §2.12.7 · p. 94
- Energy competitions demonstrated practices for load, price, solar, wind, probabilistic, and hierarchical forecasting, while Kaggle reviews highlighted hierarchical information, cross-learning, feature engineering, and ensembles as routes to better performance (Hong et al., 2014, 2016, 2019; Bojer and Meldgaard, 2020).The Kaggle review reported these approaches outperforming traditional statistical methods; those insights preceded the M5 competition on hierarchical retail data (Makridakis et al., 2021b,c).Source: §2.12.7 · pp. 94–95
2.13. The future of forecasting theory90
- Future forecasting should balance data-driven methods with expert judgment, use probabilistic forecasts, and model forecasting and decision-making together when forecasts can affect outcomes.The section describes this feedback problem as self-negating or potentially self-fulfilling forecasts, also called the prophet dilemma.Source: §2.13 · p. 95
- Future methods must address growing data and distributed ownership through privacy-preserving learning, while causal-model identification and inference may affect forecasting theory.The section also notes that access to data and contributions to distributed learning could be monetized.Source: §2.13 · pp. 95–96
- Forecast verification should further examine forecast quality and value, link to psychology and behavioural economics, and clarify what makes a forecast good.The section identifies multivariate probabilistic forecasting as a pressing verification area.Source: §2.13 · p. 96
- Forecast evaluation should move beyond plugging forecasts into decision problems to determine which forecasts are fundamentally better for which classes of decisions.The passage calls for suitable theoretical frameworks to make this assessment.Source: §2.13 · p. 96
- Links with psychology and behavioural economics can improve appraisal of how forecasts are communicated, perceived, and acted upon.Source: §2.13 · p. 96
- The complementarity of theoretical developments and applications has advanced forecasting, and application areas’ needs and challenges will continue to fuel its theory.Source: §2.13 · p. 96
3. Practice
This section surveys how forecasting methods are applied across diverse practical domains, from supply chains, finance and energy to health, climate, transport and business. It describes how methods and forecasts support decisions while addressing uncertainty, data and implementation constraints, and challenges in evaluating performance.
3.1. Introduction to forecasting practice91
- Forecast Value Added (FVA) measures performance change attributable to a process step or participant, treating activities that fail to improve quality as process waste.FVA compares each step with the prior one, helping identify where review or adjustment may warrant correction or elimination.Source: §3.1 · pp. 97–98
- In Newell Rubbermaid, the statistical forecast improved on naive accuracy by five percentage points, while management review and adjustment added negative value; adjusted forecasts beat naive by only two points.Naive accuracy averaged 60% across products, and the findings motivate investigation of causes and possible process corrections.Source: §3.1 · pp. 97–98, Table 1 · p. 98
- Prediction intervals are uncommon, software intervals often underestimate uncertainty, and decision makers may reject even accurate intervals when they seem too wide to inform decisions.Practitioners may also leave intervals unused; poor calibration alone may not explain this reluctance.Source: §3.1 · p. 98
- In the M4 competition, a simple combination reduced overall weighted average (OWA) error by 17.9% versus naive, while the top six methods improved OWA by over 5% versus that benchmark.In contrast, one study of eight businesses found 52% of forecasts failed to beat naive.Source: §3.1 · p. 98
- Forecast accuracy may be limited by the forecasted behaviour, and even accurate forecasts matter little if management overrides them or they do not inform decisions.When desired accuracy is unattainable, alternatives include supply-chain re-engineering to respond to demand variation and demand smoothing through pricing or promotions.Source: §3.1 · p. 98
3.2.1. Demand management92
- Demand forecasts support supply-chain decisions including inventory, scheduling, workforce planning, distribution, and marketing, while challenges include intermittent demand, new products, short-life-cycle products, and hierarchical data.Better demand forecasts can improve inventory management and service levels.Source: §3.2.1 · p. 99
- Deep-learning global models can capture long memory, cross-series latent correlations, sparse or cold-start settings, exogenous covariates, and varied distributional assumptions (Montero-Manso and Hyndman, 2020; Smyl, 2020; Chen et al., 2020; Law et al., 2019; Salinas et al., 2019a).These capabilities address the challenges of forecasting many related series and incorporating external variables.Source: §3.2.1 · p. 100
3.2.2. Forecasting in the supply chain93
- Forecasts inform supply-chain decisions and commonly combine statistical methods with management judgment, while more research is needed to establish big-data analytics’ usefulness.Source: §3.2.2 · p. 100
- Cross-temporal reconciliation blends information from different sources and hierarchy levels into coherent forecasts.It is described as a data-driven way to break through organisational information silos.Source: §3.2.2 · p. 100
- Real-company studies report improved performance through collaboration, but case-study evidence is scarce, initiatives often stall at pilots, and barriers include organisational silos and lack of trust (Boone and Ganeshan, 2008; Hill et al., 2018; Eksoz et al., 2019; Syntetos et al., 2016a; Panahifar et al., 2015; Fliedner, 2003).Source: §3.2.2 · p. 101
3.2.3. Forecasting for inventories94
- When demand parameters are not common knowledge, sharing demand data has value; accuracy gains relate to inventory savings, while parameter uncertainty exacerbates the bullwhip effect.Source: §3.2.3 · p. 101
- For replenishment, aggregating data to the lead-time frequency can exploit greater stability at lower frequencies, with forecasts disaggregated when needed.Source: §3.2.3 · p. 102
- Multiplying one-step forecast-error variance by lead time is unsound even for independent, identically distributed demand; smoothing mean square errors over lead time is a more direct approach.Unbiased forecasts also do not necessarily achieve target cycle service levels or fill rates on average.Source: §3.2.3 · p. 102
3.2.4. Forecasting in retail95
- Retailers rely on accurate demand forecasts for planning, marketing, purchasing, distribution, and staffing; inventory must balance storage and labour costs against stock-outs and lost sales.Source: §3.2.4 · p. 102
- Retail demand forecasting spans supply-chain position, product-hierarchy level and time granularity.Source: §3.2.4 · p. 102
- For frequent zero sales at SKU × store level, Croston’s method, SBA and TSB are established intermittent-demand approaches but have not yet been evaluated in retail.Source: §3.2.4 · p. 103
- The review finds no clear evidence that nonlinear models or novel machine-learning methods improve retail forecast accuracy.Source: §3.2.4 · p. 103
- Online retail forecasting needs to respond to interaction-driven feedback and integrate more tightly with retailers’ interactional tactics and omnichannel strategy.Forecasts, stock positions and personalised recommendations can feed back into subsequent demand and forecasts.Source: §3.2.4 · p. 104
3.2.5. Promotional forecasting96
- Promotional models use plans, prices and related product sales to forecast sales, while modelling proportional effects and synergy and estimating predictor elasticities.Logarithmic transformation allows an additive formulation for these models.Source: §3.2.5 · p. 104
- For SKU-level promotions with limited history or no past examples, Trapero et al. (2015) propose jointly estimating local SKU dynamics and pooled promotion effects to estimate uplifts for new SKUs.Product-group information accommodates heterogeneity in promotional effects.Source: §3.2.5 · pp. 104–105
- In practice, promotions are often forecast using judgmental adjustments, which have inconsistent performance.Source: §3.2.5 · p. 105
3.2.6. New product forecasting97
- New-product demand forecasting is among managers’ most difficult tasks because past data are unavailable, despite its greater consequences for company well-being.Source: §3.2.6 · p. 105
- Among the three approaches to new-product forecasting, management judgment is the most used; successful prediction-market applications for new-product forecasting have also been reported.The three broad approaches are management judgment, consumer judgment, and diffusion/formal mathematical models.Source: §3.2.6 · p. 105
- Customer surveys of purchase likelihood are not very reliable, whereas choice-based conjoint models paired with Logit have succeeded in new-product applications (Morwitz, 1997; McFadden, 1977; Meeran et al., 2017).Choice-based conjoint asks customers to select their preferred product among available alternatives.Source: §3.2.6 · p. 105
- Diffusion curves address missing historical data by modelling generic product-class demand growth, with new-product parameters estimated from analogous products, market intelligence, or regression.The Bass model is the most extensively used growth-curve family for new-product forecasting and has been extensively extended.Source: §3.2.6 · p. 106
3.2.7. Spare parts forecasting98
- When an item's failure and fault arrival characteristics can be captured, the Delay Time (DT) model is recommended for forecasting spare-parts demand with higher forecast accuracy.If this information is unavailable, time-series forecasting methods are recommended.Source: §3.2.7 · p. 106
- The reported inventory had 60% excess spare parts, with 18% having no demand and a total value of $1.5 billion; TSB addresses linearly decreasing demand and sudden obsolescence.An empirical investigation using individual demand histories of 8,000 spare-part SKUs from the automotive industry and the UK Royal Air Force found high forecast accuracy and inventory performance for TSB.Source: §3.2.7 · p. 106
3.2.8. Predictive maintenance99
- Preventive maintenance can waste remaining component life, risk catastrophic failures through late action, and itself cause failures; predictive maintenance uses forecasts to determine when action is needed.For prognostics, a key target is remaining useful time—the useful life left on an asset at a particular operating time.Source: §3.2.8 · p. 107
- Physics-based models require strong application-physics skills, while knowledge-based models aid diagnosis but perform more limitedly in prognostics; data-driven models are gaining popularity.Data-driven methods use signals such as vibration, lubricant analysis, thermography, or ultrasound; reported techniques include VARI-MAX/GARCH and Kalman-filter state-space models.Source: §3.2.8 · p. 107
- A review of predictive-maintenance applications identifies Random Forest, Artificial Neural Networks, Support Vector Machines, and K-means as the most frequently used machine-learning methods.The passages note growing practitioner and academic attention to predictive-maintenance intelligence.Source: §3.2.8 · p. 107
3.2.9. Reverse logistics100
- Returns forecasts often depend on the lagged relationship with sales, so methods such as SES that assume independence may perform poorly; product-level and serial-number data support return-distribution forecasts.Serial-number information matches each unit’s issue and return, while product-level information combines sales and returns histories with time-to-return information; unreturned past sales can be extrapolated with a return-time distribution.Source: §3.2.9 · p. 108
- Goltsos et al. (2019) found the Beta distribution best fit times-to-return and that serialisation can benefit both low- and high-volume products and improve product-level forecasting.Their evidence also suggests serialisation is worthwhile for low-volume products, especially when expensive, and supports accurate characterisation of returns for the robust product-level method.Source: §3.2.9 · p. 108
3.3.1. Macroeconomic survey expectations101
- Forecast efficiency is tested by regressing outturns on a constant and forecasts, with an efficient forecast implying a zero constant and unit forecast slope; consensus regressions can show a slope below one under partial information.A slope other than one implies forecast errors are systematically related to information available at the forecast origin.Source: §3.3.1 · p. 109
- Broer and Kohlhas (2018) and Bordalo et al. (2018) find that agents over-react to new information, producing a negative correlation between forecast revisions and errors.Evidence is often equivocal and may depend on data vintage, allowance for instabilities, and the assumed squared-error loss.Source: §3.3.1 · p. 109
3.3.2. Forecasting GDP and inflation103
- DSGE forecasting horseraces assess performance against competing models using rolling or recursive windows, and several studies report better results when estimating with real-time rather than revised data.Competitors include random-walk and Bayesian VAR models.Source: §3.3.2 · p. 110
- DSGE models can outperform competitors for medium- and long-run GDP and inflation forecasts, except for hybrid models, and their accuracy depends on capturing low-frequency trends.Smets and Wouters (2007) showed DSGEs could compete with Bayesian VARs; Del Negro and Schorfheide (2013) discuss the role of low-frequency trends.Source: §3.3.2 · p. 110
- Hybrid DSGE-statistical models outperform DSGEs, while financial frictions can improve baseline Smets and Wouters business-cycle predictions, particularly during the Great Recession.Updating DSGE parameters once a year is reported as sufficient for accurate and efficient predictions of main macro variables.Source: §3.3.2 · pp. 110–111
3.3.3. Forecasting unemployment104
- Unemployment forecasts are difficult because of non-stationarity and abrupt distributional shifts; model performance varies by country, period, and horizon, while profitability-based equilibrium-correction forecasts perform well over the business cycle.The approach uses impulse and step indicator saturation to capture outliers, breaks, and regime shifts, and Taylor expansions to allow non-linearities.Source: §3.3.3 · pp. 111–112
3.3.4. Forecasting productivity107
- The common approach forecasts productivity by estimating trend growth from aggregate data; examples include average historical growth outside the business cycle and filtering with the HP filter.Source: §3.3.4 · p. 112
- Productivity forecasts’ RMSE is more than 75% smaller than OBR’s five-year-ahead forecasts on average, and 84% smaller at the longest horizon.Martinez et al. (2021) use a local average of the last four years of growth rates to adjust long-term trend forecasts to recent patterns.Source: §3.3.4 · p. 113
3.3.5. Fiscal forecasting for government budget surveillance109
- Fiscal-forecast accuracy is complicated by data and country-specific issues, while insider information held by official institutions can improve accuracy.Source: §3.3.5 · p. 114
- Fiscal forecasting has no generally dominant method; judgment matters amid data problems, while combinations tend to outperform individual methods.Source: §3.3.5 · p. 114
- Short-term monitoring combines annual and intra-annual fiscal data to update year-end forecasts and flag target deviations (Pedregal and P´erez, 2010; Pedregal et al., 2014).A comprehensive, transparent system producing consistent forecasts across countries remains lacking.Source: §3.3.5 · p. 114
3.3.6. Interest rate prediction110
- Regime-switching and threshold models capture changing interest-rate dynamics; multi-state models are favored over single-state VARMA, and two-regime autoregressions improve US T-bill predictions.Source: §3.3.6 · p. 115
- Threshold models use observable lagged short rates to define interpretable regimes, and TAR-GARCH captures persistence and conditional heteroskedasticity.Source: §3.3.6 · p. 115
- Interest-rate regimes condition expectations-hypothesis relationships: single-state VARs cannot generate consistent distributions, and the hypothesis holds only in some states.Source: §3.3.6 · pp. 115–116
- The random walk remains hard to beat for mean-rate forecasts, while Markov-switching models perform well for density forecasts and generate varied nominal and real term-structure shapes.TAR models also report reasonable forecasts of conditional higher-order moments.Source: §3.3.6 · p. 116
- Markov-switching models remain more popular than interpretable threshold models for interest-rate prediction because discontinuous threshold-parameter likelihoods complicate statistical inference.The threshold models’ likelihood function is discontinuous with respect to threshold parameters.Source: §3.3.6 · p. 116
3.3.7. House price forecasting111
- The UK House Price Uncertainty index predicted property-price inflation ahead of the 2008 Q3 collapse and during the bust, while reflecting recessions, Brexit uncertainty, and COVID-19 (Yusupova et al., 2019).The index was found important for predicting property-price inflation ahead of the collapse and during the bust.Source: §3.3.7 · p. 117
3.3.8. Exchange rate forecasting114
- The exchange-rate disconnect puzzle remains unresolved, though PPP-based adjustment and controls for forecast-estimation error show some evidence of beating the random-walk benchmark.Ca’ Zorzi and Rubaszek (2020) report that gradual adjustment toward PPP performs well at shorter and longer horizons; extending PPP with other BEER fundamentals does not help forecasting.Source: §3.3.8 · p. 118
3.3.9. Financial time series forecasting with range-based volatility models115
- Range-based estimators better capture return variances and covariances than closing-price alternatives, and range-based volatility forecasts can outperform squared-return and standard GARCH forecasts.The price range and co-range are more efficient estimators of volatility and covariance, respectively.Source: §3.3.9 · p. 118
- Daily low/high-price multivariate volatility models forecast return covariances more accurately than models based on closing prices (Chou et al., 2009; Harris and Yilmaz, 2010).Examples include the RB DCC and hybrid EWMA models.Source: §3.3.9 · pp. 118–119
- Range-based volatility models produce more precise value-at-risk forecasts than approaches using closing prices.The cited applications include threshold CAViaR and HVAR models.Source: §3.3.9 · p. 119
- Multivariate range-based models improve hedging efficiency and portfolio-construction economic value, and low/high-price forecasts can support profitable investment strategies (Chou and Liu, 2010; He et al., 2010).The cited work includes RB DCC, RB-copula, VECM, and CARRS models.Source: §3.3.9 · p. 119
3.3.10. Copula forecasting with multivariate dependent financial times series116
- Copulas combine marginal and dependence models for joint probabilistic forecasts, capturing asymmetric financial dependence such as stronger stock-return correlation in downturns.Point forecasts from marginal models are straightforward but ignore dependence information.Source: §3.3.10 · p. 119
- Covariate-dependent copulas improve probabilistic forecasting performance and interpretability, while wind-power copula applications report advantages and reliable, sharp intervals with limited observations.The wind-power applications include time-adaptive quantile copulas and vine copulas.Source: §3.3.10 · pp. 119–120
3.3.11. Financial forecasting with neural networks117
- Neural networks model nonlinear, non-stationary financial series and are used in trading, derivatives pricing, and risk management, but hyperparameter sensitivity can foster data-snooping and poor generalization.The many modeling choices can produce strong in-sample and out-of-sample results that do not generalize to other samples.Source: §3.3.11 · pp. 120–121
3.3.12. Forecasting returns to investment style119
- Investment-style portfolios use observable security characteristics thought to predict future returns, with explanations framed as compensation for risk or as other return drivers.A momentum portfolio, for example, takes long positions in stocks with positive trailing 12-month returns and short positions in those with negative trailing returns.Source: §3.3.12 · p. 121
- Investment-style timing is debated: returns can be persistently positive yet highly volatile, and timing errors can incur trading costs and forgo returns and diversification.Proposed signals include serial correlation, valuation spreads, and economic and financial conditions; evidence for value added by valuation timing is mixed, and it may compound value-factor exposure.Source: §3.3.12 · pp. 121–122
3.3.13. Forecasting stock returns121
- For stock returns, a few factors extracted from hundreds of predictors improved out-of-sample US market forecasts, while partial least squares built target-relevant factors to improve market forecasts.Ludvigson and Ng (2007) report the first result; Kelly and Pruitt (2013) and Huang et al. (2015b) use partial least squares with valuation ratios and sentiment measures.Source: §3.3.13 · p. 122
- Rapach et al. (2010) show that averaging forecasts from individual univariate predictive regressions significantly improves out-of-sample market forecasts and helps guard against overfitting.Their combination forecast exerts a strong shrinkage effect.Source: §3.3.13 · p. 123
- Machine-learning approaches, including LASSO, elastic net, random forests, and neural networks, address overfitting or allow nonlinear relationships when forecasting stock returns.The LASSO is reported to improve out-of-sample fit for one-minute-ahead individual-stock returns and monthly industry-return forecasts.Source: §3.3.13 · p. 123
3.3.14. Forecasting crashes in stock markets122
- Volatility can be forecast with ARCH variants or realized-volatility models, but exact crash timing may be impossible; self-exciting jump models and duration or Hawkes processes offer alternatives.The crash probability within a foreseeable future may be forecast; the proposed self-exciting models reflect herding, with jumps capable of triggering new jumps across assets or markets.Source: §3.3.14 · p. 123
- Gresnigt et al. (2017a,b) document spillovers across assets and markets; Hawkes processes model crash clustering and predict nearby-crash probabilities, including reliable predictions of crashes within five days.Source: §3.3.14 · p. 124
3.4.1. Building energy consumption forecasting and optimisation123
- Building energy systems depend on building characteristics, occupancy and behaviour, equipment, weather, and calendar effects, so accurate forecasts typically require tailored, building-specific methods.Source: §3.4.1 · p. 124
- No building-energy forecasting class is consistently best: findings depend on building type, dataset, horizon, and frequency, while combinations and hybrids have reported encouraging results.Source: §3.4.1 · p. 125
- Building forecasts support energy optimisation to reduce costs, improve network stability, and advance a carbon-free future; HVAC optimisation seeks lower energy use while maintaining comfort.Source: §3.4.1 · p. 125
3.4.2. Electricity price forecasting124
- Electricity-price forecasting must address market-specific autoregressive and nonstationary behavior, calendar effects, volatility, and positive or negative price spikes.Source: §3.4.2 · p. 126
- Electricity-price forecasts may also require predictions of external inputs, including demand, renewable generation, and relevant fuel prices.Source: §3.4.2 · p. 126
- Calibrated linear models can forecast day-ahead electricity prices accurately, and combining forecasts is considered appropriate when models differ in structure or calibration-window length.The combination findings cite Gaillard et al. (2016), Mirakyan et al. (2017), and Hubicka et al. (2018).Source: §3.4.2 · p. 128
- Intraday electricity prices are more volatile and have stronger spikes than day-ahead prices, making probabilistic forecasts especially relevant; European markets are close to weak-form efficiency, and naive point benchmarks perform remarkably well.Source: §3.4.2 · p. 128
3.4.3. Load forecasting125
- Short-term load forecasting supports unit commitment and optimal power flow and is vital for market entities such as retailers, aggregators, and prosumers.Applications for these entities include strategic bidding, portfolio optimisation, and tariff design.Source: §3.4.3 · pp. 129–130
- Ma (2021)'s ensemble on GEFCom2012 data outperformed benchmark forecasters including the Theta method, NN, and ARIMA.Source: §3.4.3 · p. 130
- Sideratos et al. (2020)'s radial-basis-network/CNN model had lower error than persistence, NN, and SVM, while Semero et al. (2020)'s approach outperformed four other methods.The Sideratos model uses an initial radial basis network forecast as input to a CNN for final forecasting; Semero et al. use EMD, ANFIS, and PSO.Source: §3.4.3 · p. 130
- The prosumer-microgrid hybrid system selected the best individual load and weather forecasts through optimisation and was more accurate than applying individual forecasters alone.Source: §3.4.3 · p. 130
3.4.4. Crude oil price forecasting126
- Crude-oil prices are volatile and driven by hard-to-predict external factors, and probabilistic forecasts remain underexplored despite their importance for risk-aware decisions.Source: §3.4.4 · pp. 130–131
- Recent crude-oil studies have beaten naive no-change forecasts, while econometric, machine-learning, and hybrid approaches target complex price dynamics.The review notes improved hybrid-model performance, but also computational-cost and overfitting challenges for machine-learning techniques.Source: §3.4.4 · p. 131
3.4.5. Forecasting renewable energy technologies127
- Renewable-technology diffusion forecasts inform energy planning but face short time series and uncertain deployment; relevant factors include policy, emissions, and economic development.Source: §3.4.5 · pp. 131–132
- Renewable-technology studies use bottom-up aggregation, Grey System methods for limited data, and growth-curve and innovation-diffusion models.Source: §3.4.5 · p. 132
3.4.6. Wind power forecasting128
- Wind forecasts support farm operations, energy markets, and power-system operations; short-term methods combine numerical weather predictions with turbine, farm, or regional power-curve models.Source: §3.4.6 · pp. 132–133
- Weather unpredictability and complex power curves challenge wind forecasting, while probabilistic forecasts are increasingly used in practice.Source: §3.4.6 · p. 133
- For very short-term wind forecasts, recent observations matter more than NWP, and spatially dependent time-series models, notably VAR variants, are state-of-the-art.Very short-term forecasts cover minutes to a few hours ahead.Source: §3.4.6 · p. 133
- Wind-power errors may not be normally distributed because output is bounded between zero and rated power, so transformations are recommended, with the choice depending on the series (Pinson, 2012).Source: §3.4.6 · p. 133
3.4.7. Wave forecasting129
- At one hour ahead, simulated wave-farm forecast errors are typically 5–7%, versus 17% for solar and 22% for wind.For wave energy, time-series models are more accurate in the first few forecast hours, while numerical wave models are widely used beyond those horizons.Source: §3.4.7 · pp. 133–134
- A bivariate VARMA-GARCH approach to probabilistic wave-energy-flux forecasting outperformed univariate flux modelling and bivariate flux-and-wind-speed modelling.In another approach, Gaussian-copula modelling of temporal dependence did not improve probabilistic forecasts of wave heights and periods.Source: §3.4.7 · p. 134
3.4.8. Solar power forecasting130
- PV forecasting methods are difficult to compare because performance depends on historical data, weather forecasts, horizon, resolution, and installation conditions.The cited review comparison was considered not very meaningful because studies used different conditions and metrics.Source: §3.4.8 · p. 134
- The PHANN hybrid method outperforms classical neural-network methods and shows good forecasting performance, especially on sunny days.The cited comparison covers measured and forecast hourly PV output on sunny and cloudy days.Source: §3.4.8 · p. 135, Figure 10 · p. 135
- With limited PV history, chronologically consecutive training samples suit day-ahead forecasting; randomly mixed samples appear more effective when more data are available.Independent-trial ensembles are generally most effective.Source: §3.4.8 · p. 136
3.4.9. Long-term simulation for large electrical power systems131
- Long-term scheduling of hydro, wind, and solar futures uses stochastic scenarios in dispatch optimisation under uncertainty, generally with time-series scenarios and multi-stage stochastic approaches.Source: §3.4.9 · p. 136
- For long-term renewable-power planning, especially hydropower, periodic autoregressive (PAR) models remain the streamflow-simulation benchmark, while portfolio simulation still faces challenges incorporating exogenous variables.PAR fits a model for each historical period and simulates residuals to generate future time series; exogenous factors include climate, meteorological, calendar, and economic variables.Source: §3.4.9 · p. 137
3.4.10. Collaborative forecasting in the energy sector132
- Collaborative forecasting can improve individual renewable-plant accuracy, while privacy-preserving analytics and data markets offer complementary ways to share data and address scarcity.A renewable-energy data-market model linked data prices to forecasting-related extra electricity-market revenue, and results showed higher revenue from combining electricity and data markets.Source: §3.4.10 · pp. 137–138
- Data owners may resist sharing over confidentiality, uncertainty about shareable data, or unclear benefits; privacy-preserving methods also require scrutiny for data-recovery risks, and heterogeneous data sources remain a research challenge.Combining numerical, textual, and categorical sources is described as a promising avenue for collaborative forecasting.Source: §3.4.10 · pp. 137–138
3.5.1. Forecasting two aspects of climate change133
- UK CO2 multi-step forecasts for 2012–2017 show an advantage from step-indicator saturation (SIS), with the selected model also identifying policy shifts and outliers.Saturation estimation detected four step shifts coinciding with major policy interventions, including the 2008 Climate Change Act, and numerous outliers, revealing a cointegrated relation.Source: §3.5.1 · p. 139, Figure 12 · p. 139
- A three-equation model of atmospheric CO2, Antarctic temperature, and ice volume uses orbital drivers to produce 100,000-year forecasts within past-data ranges ±2.2SE.The model covers 800,000 years of Ice Ages, and its forecasts begin 10,000 years ago because humanity has affected climate since then.Source: §3.5.1 · p. 139
- Because atmospheric CO2 already exceeds 400ppm, outside the Ice-Age range, the authors conditionally forecast the next 100,000 years under 400ppm and 560ppm scenarios.The dynamic forecasts show Ice and Temperature with ±2SE bands; orbital drivers continue to affect all three variables, while the exogenised-CO2 scenario switches off that relation.Source: §3.5.1 · p. 139
- In the climate-change example, Antarctic temperatures above 5°C are described as dangerous, with positive temperatures lasting thousands of years.Source: §3.5.1 · pp. 139–140
3.5.2. Weather forecasting134
- Numerical weather prediction combines physical principles and observations, propagates initial-state uncertainty with ensembles, and parameterizes small-scale processes that models cannot resolve directly.For fixed computational resources, increasing grid resolution competes with ensemble size, while a larger ensemble generally estimates prediction uncertainty better.Source: §3.5.2 · pp. 140–141
- Raw weather ensembles have systematic magnitude and spread errors, and statistical post-processing must correct them while preserving consistency across space, time, and variables.That consistency matters when weather forecasts feed further prediction models, such as hydrometeorological models.Source: §3.5.2 · p. 141
- Beyond two weeks, initial-condition weather noise grows large, so longer-range predictability is associated with slowly evolving Earth-system components that can inform user-relevant forecasts.The cited examples include ENSO, monsoon rains, the Madden Julian Oscillation, the Indian Ocean dipole, and the North Atlantic Oscillation.Source: §3.5.2 · p. 141
3.5.3. Air quality forecasting135
- In the initial air-quality configuration, correlations ranged from 0.6 for lazy learning to 0.75 for neural networks, and an ensemble beat each station’s best individual model.The systems used the last three available particulate-matter measurements as inputs.Source: §3.5.3 · p. 142
- Adding wind speed increased the best neural-network model’s median correlation from 0.75 to 0.82 and reduced RMSE from 15µg/m3.The reported RMSE passage gives the starting value but not the resulting value.Source: §3.5.3 · p. 142
3.5.4. Forecasting and decision making for floods and water resources management136
- Flood and reservoir decisions use probabilistic forecasts to balance expected losses, including short-term flood reduction against longer-term water-resource benefits.Reservoir operators decide how much water to release preventively, with the objective based on a linear combination of long- and short-term expected losses.Source: §3.5.4 · pp. 142–143
- Integrating the flood-damage function over a water-level forecast yielded expected damage of 6.59 million dollars, while using the forecast mean implied zero damage.In the example, water level is Normally distributed with mean 10 meters and standard deviation 5 meters, and the dike elevation is 10.5 meters; probabilistic forecasts support alert and evacuation decisions.Source: §3.5.4 · p. 143
- At Lake Como, Bayesian decisions based on predictive probability distributions increased management reliability by reducing wrong decisions (Todini, 2017, 2018).The approach was associated with an average 12% reduction in water deficit and a 3% increase in electricity production.Source: §3.5.4 · p. 143
3.6.1. Healthcare137
- Clinical forecasting supports risk screening, diagnosis, disease progression, prognosis, and treatment recommendations, using structured and unstructured data that make preprocessing crucial.Examples include time-series, regression, classification-tree, neural-network, Markov, and Bayesian-network methods.Source: §3.6.1 · pp. 143–144
- Forecasting supports healthcare operational, tactical, and strategic planning, but grouped and hierarchical health-supply-chain demand remains underinvestigated.Demand forecasts are used in emergency departments, ambulance services, and hospitals; supply-chain forecasting helps avoid shortages and excessive inventory.Source: §3.6.1 · p. 144
3.6.2. Epidemics and pandemics138
- Ioannidis et al. (2020) argued COVID-19 forecasting had largely failed and recommended predictive distributions and continuous evaluation, while Taleb et al. (2020) warned against point forecasts for fat-tailed risks.Taleb et al. also cautioned against naive empirical approaches to tail-risk management.Source: §3.6.2 · p. 145
- COVID-19 supply-chain forecasts found excess grocery and electronics demand and reduced automotive demand, with earlier lockdowns associated with greater grocery demand.The findings support securing key-product volumes before lockdowns and considering rationing when that is not possible.Source: §3.6.2 · p. 145
- Dengue forecasting has used laboratory-confirmed spatiotemporal cases and web data such as Google Trends and tweets; SARIMA models have been popular.The reviewed web-data example is de Almeida Marques-Toledo et al. (2017), and the spatiotemporal example is Naish et al. (2014).Source: §3.6.2 · pp. 145–146
- Dengue forecasts use climate covariates and count regressions, including overdispersion-aware models, alongside hybrid ARIMA-NNAR, wavelet, and Google Trends approaches.The review identifies spatio-temporal hierarchical forecasting as a future research direction.Source: §3.6.2 · p. 146
3.6.3. Forecasting mortality 139
- Mortality forecasting methods fall into expectation, explanation, and extrapolation approaches; time-series extrapolation yields probability distributions and forecast intervals.Explanatory methods connect mortality changes to related variables, while expectation methods produce subjective, scenario-based deterministic forecasts.Source: §3.6.3 · p. 147
3.6.4. Forecasting fertility140
- Fertility forecasts support population and service planning, but deficient birth registration and ongoing transitions from high to low fertility increase uncertainty in many low- and middle-income countries.Forecasts must address quantum and tempo, and methods include Lee-Carter extensions, cohort-ARIMA, parametric fertility curves, and Bayesian approaches.Source: §3.6.4 · p. 148
- Bohk-Ewald et al. (2018) found few fertility methods outperform naive freezing of age-specific rates, and the outperformers differ greatly in method.Comparative performance research across the broad range of fertility approaches remains limited.Source: §3.6.4 · p. 148
- European statistical-office fertility forecasts tend to be deterministic and use expert panels.Statistics Canada’s elicitation protocol requests a full probability distribution of the total fertility rate (TFR).Source: §3.6.4 · p. 148
- Fertility forecasting could incorporate birth order and survey data, while alternative data sources may help when vital statistics are unavailable.Google data have been used to predict COVID-19’s effect on US fertility; the section also identifies delayed motherhood and assisted reproduction as areas needing long-term study.Source: §3.6.4 · p. 148
3.6.5. Forecasting migration141
- Migration forecasting is shifting from prediction toward contingency planning amid high uncertainty and non-stationary processes.Work includes causal models, early warnings, nowcasting, and data-driven changepoint detection with digital traces; coherent uncertainty across horizons, especially long-range, remains a challenge.Source: §3.6.5 · p. 149
- Migration models can produce reasonable forecasts and errors up to a decade ahead, but predictability varies by flow, with family migration more predictable than asylum.Covariates endogenous to migration may themselves require prediction, risking exploding uncertainty unless structured models are used.Source: §3.6.5 · p. 149
3.6.6. Forecasting risk for violence and wars142
- Statistical analyses find no clear trend toward declining human belligerence or conflict severity, and long intervals between wars do not establish a lasting decline.War casualties have an extremely fat-tailed distribution, making outcomes erratic and difficult to foresee.Source: §3.6.6 · p. 150
- Extreme value theory can model war-casualty tail risk for prudential forecasts, while avoiding naive extrapolations.The section notes caveats in applying these methods.Source: §3.6.6 · p. 150
3.7.1. Support systems143
- Business forecasting support systems address complex tasks involving diverse series, multiple unstructured data sources, and human judgment.A typical system retrieves data, provides visualisations and summary statistics, supports preprocessing, and produces forecasts that users may adjust.Source: §3.7.1 · p. 151
- Forecasting support systems should combine automatic statistical methods with judgment, let users track and interact with the procedure, and adapt to company context.The stated requirements also include efficiency.Source: §3.7.1 · p. 151
- Many support systems lack state-of-the-art methods and do not guide judgmental forecasts or adjustments within a well-organised framework, limiting user options and bias-mitigation methods.The section also calls for simple but exhaustive interfaces adapted to company requirements and users’ strengths and weaknesses, and suggests web-based systems to improve accessibility, availability, and compatibility.Source: §3.7.1 · p. 151
3.7.2. Cloud resource capacity forecasting144
- Cloud providers face forecasting problems at operational, tactical, and strategic scales, from resource provisioning to decisions about data-centre locations.Cloud forecasting also involves demands for scale, rapid response, fine granularity, and automation; predictive scaling may require forecasts in seconds without a human operator controlling the response.Source: §3.7.2 · p. 152
3.7.3. Judgmental forecasting in practice145
- Judgment combined with statistical methods accounted for 55.6% of forecasts, compared with 15.6% made by judgment alone in the surveyed companies.The section covers unaided judgment, judgmental adjustments, and judgment in forecasting support systems.Source: §3.7.3 · p. 153
- People have difficulty learning cross-series correlations and may add noise when using one variable to forecast another (Harvey et al., 1994; Lim and O’Connor, 1996c,b; Gray et al., 1965).The section also describes optimism, added noise, and damped trends in judgmental extrapolation from a single series.Source: §3.7.3 · p. 153
- Forecasts from un-trended independent series may fall between the last observation and the mean, but this and damped trends should not automatically be treated as biases because real series may have modest autocorrelation and damped trends.The section relates these patterns to insufficient adjustment from an anchor at the last data point.Source: §3.7.3 · p. 153
- Combining statistical models with judgment can address external events, low-frequency events, and patchy or insufficient data that statistical models generally cannot account for.Judgmental adjustments to statistical model outputs are described as the most frequent form of judgmental forecasting in practice.Source: §3.7.3 · p. 153
- Negative and larger judgmental adjustments improved accuracy more than positive and smaller adjustments in studies of supply-chain companies.Positive adjustments may reflect wishful thinking or optimism, while large adjustments may be made when there is a highly valid reason; other studies confirmed the findings (Franses and Legerstee, 2009b; Syntetos et al., 2009).Source: §3.7.3 · pp. 153–154
- Practitioners adjust forecasts to add intuition, experience, event or insider information, control, expectations, and bias compensation, but only the first three reasons may improve accuracy; the others can harm it.Practitioners may refrain when they trust and understand the baseline, have explanations or supporting materials, or face policies against adjustment; these conditions make the baseline easier to accept and adjustments less frequent (Onkal and Gonul, 2005; Gonul et al., 2009).Source: §3.7.3 · p. 154
- A no-adjustment default and requiring explanations increased acceptance of statistical forecasts and improved accuracy, whereas adjustment guidance and restrictions did not improve accuracy and met forecaster resistance.Changing presentation can also affect accuracy: trends were easier to discern in graphs than tables, and line-versus-point graph choices altered accuracy (Harvey and Bolger, 1996; Theocharis et al., 2018).Source: §3.7.3 · pp. 154–155
- Automatic correction for judgmental biases by the forecast support system was more effective than automatically combining judgmental and statistical inputs with equal or varying weights; integrative judgment improved accuracy while preserving forecaster input.Source: §3.7.3 · p. 155
3.7.4. Trust in forecasts146
- Trust is essential for forecasts to be used in decisions, and practitioners associate it with support tools, forecaster competence and credibility, forecast combinations, and users’ knowledge of forecasting methods.Building trust requires communication clarity and perceptions of competence and integrity, while forecasters must engage users and demonstrate objectivity and expertise and users must communicate expectations and informational needs.Source: §3.7.4 · p. 155
3.7.5. Communicating forecast uncertainty147
- Effective uncertainty communication helps forecasters influence end-users to respond appropriately, and proposed frameworks decompose it into communicator, uncertainty object, expression format, audience, and effect.Organisations also face the challenge of investing in forecast management and resilient systems for collaborative forecasting.Source: §3.7.5 · p. 156, §3.7.4 · p. 156
- Forecasters use consistent terminology, yet recipients interpret even commonly used uncertainty terms inconsistently, and numeracy and cognitive load can impede comprehension (Budescu and Wallsten, 1985; Clark, 1990; ¨Ulk¨umen et al., 2016).Wording and verbal representation also affect uncertainty communication (Joslyn et al., 2009).Source: §3.7.5 · p. 156
- More uncertainty information increased perceived uncertainty but caused only a small decrease in trust, while uncertainty visualisation improved forecast comprehension in a homeland-security study.The review also notes that empirical evidence on effective uncertainty communication remains limited.Source: §3.7.5 · pp. 156–157
3.8.1. Tourism demand forecasting148
- Tourism demand forecasting uses statistical models, machine learning, and hybrids, while publication lags and low-frequency economic predictors can constrain forecasts.Leading indicators have been proposed for short-term forecasting and nowcasting (Chatziantoniou et al., 2016).Source: §3.8.1 · p. 157
- Web-based leading indicators have generally improved tourism-demand forecast accuracy, but not in every case, supporting the view that no single approach is best.Examples include Google Trends, Google Analytics, and Facebook ‘likes’.Source: §3.8.1 · p. 158
- During COVID-19 travel restrictions, scenario, hybrid, and judgmental forecasting played an important role, highlighting a limitation of quantitative tourism-demand forecasting as practiced.Source: §3.8.1 · p. 158
3.8.2. Forecasting for aviation149
- Aviation forecasts support congestion measurement, flight scheduling, and stand and gate assignment; random forests have been applied to predict air-traffic delays using temporal and network-delay states.Source: §3.8.2 · p. 158
- Carson et al. (2011) found that aggregating airport-specific forecasts could improve national forecasts, while Hopman et al. (2021) found itinerary-based XGBoost outperformed traditional time-series forecasts.Hopman et al. use ticket prices, social-media posts, and airline reviews to forecast itinerary-based bookings.Source: §3.8.2 · p. 159
- Guo et al. (2020) combine regression trees and copula simulations to forecast transfer connection times and passenger flows, generating accurate rolling forecasts from real-time information at Heathrow.Their model parameters do not update over time, leaving dynamic real-time parameter updating as a stated challenge.Source: §3.8.2 · pp. 159–160
3.8.3. Traffic flow forecasting150
- STR achieved test-set MSE 102.4, MAE 62.8, and MAPE 14.3%, outperforming Double-Seasonal Holt-Winters by 44% in MAPE.The example uses log-transformed A1 highway vehicle-flow data and temporal inputs; figures 15 and 16 show the decomposition, forecasts, and errors.Source: §3.8.3 · p. 160, Figure 15 · p. 161, Figure 16 · p. 161
3.8.4. Call arrival forecasting151
- Call arrivals can exhibit intraday, intraweek, and intrayear seasonality, interday and intraday dependence, heteroscedasticity, and overdispersion relative to a Poisson model.These properties vary with data aggregation and collection frequency and affect modelling and forecasting approaches.Source: §3.8.4 · pp. 161–162
- Call-arrival methods include distribution-free time-series approaches such as ARMA, exponential smoothing, Fourier transforms, and regression, followed by methods designed to capture multiple seasonality.The latter include double seasonal exponential smoothing and multiplicative double seasonal ARMA.Source: §3.8.4 · p. 162
- Time-inhomogeneous Poisson models use fixed or mixed approaches to account for overdispersed call data and sometimes interday and intraday dependence.Source: §3.8.4 · p. 162
- SVD reduces the dimensionality of high-frequency call data to produce interday volume forecasts and intraday updates.Source: §3.8.4 · p. 162
3.8.5. Elections forecasting152
- Election forecasts are highly public and can influence voters, political parties, campaign organizations, and international investors (Pavía et al., 2019).Source: §3.8.5 · p. 163
- Election forecasting uses polls, information markets, statistical and econometric models, and emerging social-media surveillance; econometric models can use structural information months ahead.Source: §3.8.5 · p. 163
- After adjustments, polls remained powerful and robust predictors across 86 countries and more than 500 elections, despite non-sampling errors.Source: §3.8.5 · p. 163
- Gallup forecast Roosevelt’s 1936 victory with a small representative sample, while a nearly 2.5-million-response Literacy Digest sample failed (Squire, 1988).Source: §3.8.5 · p. 163
3.8.6. Sports forecasting153
- In sport, bookmaker forecasts improved relative to expert forecasts as betting became more competitive, while prediction-market studies tended to find a lack of efficiency (Forrest et al., 2005; Gil and Levitt, 2007; Croxson and Reade, 2014; Angelini and De Angelis, 2019).Source: §3.8.6 · p. 164
- Sports forecasts shape attendance expectations and event-hosting decisions, but projected hosting impacts are often inflated for political reasons.Source: §3.8.6 · p. 164
- Sports examples span soccer and tennis, while growing data is beginning to bring machine and deep learning methods to dominate the field.Source: §3.8.6 · p. 165
3.8.7. Forecasting for megaprojects154
- Megaproject success is difficult to define and forecast across stakeholders and timeframes, while leading indicators could project stakeholder perceptions during execution.Source: §3.8.7 · p. 165
- Early megaproject decisions shape outcomes, and failing to account for unforeseen events frequently leads to cost overruns.Source: §3.8.7 · p. 165
- Pooling semi-structured analogies through interaction groups further amplified their forecasting-performance advantage over unaided judgment in a study with 69 semi-experts.The study compared unaided judgment, semi-structured analogies, and interaction groups.Source: §3.8.7 · p. 166
3.8.8. Competing products155
- UCRCD forecasting was described as more realistic and as avoiding long-term prediction mistakes, while the new entrant gained cross-brand and within-brand word-of-mouth effects (Guseo and Mortarino, 2014).After competition began, cimetidine experienced a negative word-of-mouth effect from its own adopters while benefiting from growth in category market potential.Source: §3.8.8 · p. 166
- LVch outperformed restricted and unrestricted UCRCD analyses where the residual market was not fully accessible, improving medium- and long-term interpretation and forecasting through word-of-mouth components.Source: §3.8.8 · p. 167
- Diachronic competition models face complex parameterization with many rivals, and forecasts conditional on exogenous controls may be affected by unknown future policies.The controls can represent policy regulations, innovations, and prices, and the models can simulate strategic interventions.Source: §3.8.8 · p. 167
3.8.9. Forecasting under data integrity attacks157
- Data-integrity attacks can evade detection by skilled, coordinated adversaries while causing over-forecasts, unnecessary upgrade and maintenance expenses, and poor planning or business decisions.The described attack templates include ramping, scaling, random attacks, pulse, and smooth-curve attacks.Source: §3.8.9 · p. 167
- Widely used load models fail under large integrity attacks; anomaly preprocessing evolved from fixed thresholds to data-driven thresholds.Source: §3.8.9 · p. 167
- New attack templates can fool anomaly-detection-enabled forecasters, and adversaries may manipulate related inputs without altering target series, motivating further research.Source: §3.8.9 · p. 168
3.8.10. The forecastability of agricultural time series158
- Common-factor models have seen little application to agricultural time series despite related commodities research, leaving open avenues for applied forecasting (including common-factor modeling).Source: §3.8.10 · p. 168
- Potential applications include identifying and using common factors and studying COVID-19 or climate-change effects on agricultural production.Source: §3.8.10 · p. 169
3.8.11. Forecasting in the food and beverage industry159
- Association-rule mining of categorical grind size, amount, and pressure finds that low pressure, high ground amount, and fine grinding forecast low-quality espresso.Source: §3.8.11 · p. 170
- Coffee forecasts illustrate that coarse grounds can imply excessive flow and low quality, even when dosage and pressure are optimal.Low coffee quantity, coarse grounds, and high pressure are also associated with a low-quality forecast due to excessive flow rate.Source: §3.8.11 · p. 170
3.8.12. Dealing with logistic forecasts in practice160
- Logistic forecasts require choosing cumulative or per-period counts based on whether a market niche will fill; US COVID-19 cases illustrated fits to daily rather than cumulative infections.Some early cumulative COVID-19 fits worked where diffusion behaved accordingly, but the US example was illuminated by fitting daily infections.Source: §3.8.12 · p. 170
- The final ceiling can be difficult to determine early, and forecasters often lack a theory for assigning quantitative uncertainty to S-curve parameters.A study of 35,000 noisy simulated S-curve fits produced lookup tables and graphs for parameter uncertainties conditional on data coverage, error per point, and required confidence; one claim is that ceiling estimates before the midpoint are untrustworthy.Source: §3.8.12 · pp. 170–171
- Logistic fitting programs may bias estimates toward low ceilings, so forecasters should compare justified fits with different data weights and favor the highest ceiling.Larger errors can accentuate the bias; heavier weighting of recent historical data often produces the higher ceiling.Source: §3.8.12 · p. 171
3.9. The future of forecasting practice161
- Business forecasting practice ranges from ad hoc judgment to sophisticated systems, while operational forecasting faces proliferating products, shorter horizons, and disrupted supply chains.The Group anticipated shorter histories and less viability for traditional models as demand chains restart, alongside demand for greater agility.Source: §3.9 · p. 172
- Practitioners anticipate greater AI/ML automation, more use of big data and social media for longer-horizon causal forecasts, and increased risk management, with cloud solutions displacing desktop tools.They also expect quantum computing to improve algorithms and change thinking about forecasting and uncertainty.Source: §3.9 · pp. 172–173
- Forecasting systems are expected to integrate planning across functions and connect more closely to resilience, alert, and root-cause analysis systems.The Group also anticipated expanding S&OP into finance and performance management and building supply-chain capacity, inventory, and redundancy.Source: §3.9 · p. 173
- The Group unanimously expected forecasters’ jobs to broaden and become more demanding, with technology redirecting some effort away from forecast production.The excerpt does not specify which tasks would receive that redirected effort.Source: §3.9 · p. 173
- Attention to disease models highlights their strengths and weaknesses, prompting forecasters to temper claims and resist exploitation of their models.Source: §3.9 · p. 173
- AI/ML will add to forecasters’ tools but will not automatically solve forecasting issues, including data hunger, brittle capacity, dubious inputs, user distrust, and model bias (Kolassa, 2020c).Source: §3.9 · p. 174
- Practices may remain similar to today’s because academic developments translate slowly, politics and gaming constrain adoption, and departments move between trends without improving accuracy.Source: §3.9 · p. 174
- Companies may treat ERP forecasting as good enough, while higher management neglects forecasting, especially when forecasts contradict its desired messages.Source: §3.9 · p. 174
- COVID-19 concerns about disrupted patterns and volatility point to greater use of judgmental scenarios, likely alongside statistical and ML methods.The pandemic has raised practitioner concerns about forecasting amid disruption and an increasingly volatile environment.Source: §3.9 · p. 174
4. Forecasting: benefits, practices, value, and limitations162 · Disclaimer · Appendix A. List of acronyms
This section presents forecasting as essential to planning despite uncertainty, describing systematic methods and offering an encyclopedic reference to forecasting theory and practice. It also includes a glossary of acronyms spanning forecasting models and methods, evaluation measures, and application contexts.
4. Forecasting: benefits, practices, value, and limitations162
- The article compiles more than 140 sections and subsections, over 2,100 references, and contributions by 80 forecasting researchers and practitioners as an easy-to-use reference.The authors aim to turn it into an online resource updated as new information becomes available.Source: §4 · p. 175
- Forecasting is unavoidable because planning and most decisions require predictions, and forecasts are uncertain even when some turn out wrong.The article says uncertainty should be expressed through prediction intervals or probability distributions around the most likely forecast.Source: §4 · p. 175
- Prediction intervals can unsettle executives seeking exact figures, yet empirical research shows they are often too narrow and underestimate uncertainty.The intervals are based on past fluctuations and may be viewed as too wide or uninformative for business decisions.Source: §4 · pp. 175–176
- Systematic forecasting improves accuracy and uncertainty assessment over ad hoc judgment by extrapolating established patterns, provided those patterns remain stable.The article attributes these benefits to extensive empirical research, including forecasting competitions.Source: §4 · p. 176
- Forecasts and uncertainty should be evaluated against simple benchmarks; today’s stock price beats most professional forecasters, while weather forecasts are compared with persistence.The weather benchmark predicts that tomorrow will be the same as today.Source: §4 · p. 176
- Systematic methods have no prophetic power: their objectivity and optimal extrapolation depend on stable patterns and uncertainty that is not fat-tailed.The article cautions against exaggerated expectations of what forecasting can achieve.Source: §4 · p. 176
Appendix A. List of acronyms
- Evaluation and forecast-uncertainty terms include CRPS, MAPE, MASE, PIT, and the quantile score Qα.Source: Appendix A · pp. 180, 182, 183
- The glossary includes model acronyms such as GARCH, LSTM, and HAR.Source: Appendix A · pp. 180, 181, 184
- Application-related entries include GDP, HVAC, IEA, NWP, and HMD.Source: Appendix A · pp. 181–182
- It defines statistical-method acronyms including LASSO, OLS, and PCA.Source: Appendix A · pp. 181–182
- RMSE is Root Mean Squared Error, while sMAPE is symmetric Mean Absolute Percentage Error.Source: Appendix A · p. 183
- RNN means Recurrent Neural Network, SVM means Support Vector Machine, and XGBoost means eXtreme Gradient Boosting.Source: Appendix A · pp. 183–185
- The glossary expands SBA as Syntetos-Boylan Approximation and TSB as Teunter-Syntetos-Babai method.Source: Appendix A · pp. 183–184
- STLF and VSTLF expand to Short-Term Load Forecasting and Very Short-Term Load Forecasting, respectively.Source: Appendix A · p. 184
- S&OP denotes Sales and Operations Planning, and RET denotes Renewable Energy Technology.Source: Appendix A · pp. 183–184
Appendix B. Software · §2.3.1. Exponential smoothing models · §2.3.2. Time-series regression models
Appendix B catalogs indicative free or open-source software linked to the article’s theory sections, including tools for exponential smoothing and time-series regression. The listed tools are provided without liability, and users are advised to consult their documentation and licences.
Appendix B. Software
- Table A.1 lists indicative free or open-source packages, libraries, and toolboxes linked to the article’s theory sections.The authors state that the list is not an endorsement or guarantee.Source: Table A.1 · p. 186
- Users are advised to read each listed software’s documentation and licences; the authors assume no liability for the software.Source: Appendix B · p. 186
- R’s forecast package provides the isat function for running impulse and step indicator saturation.Source: Appendix B · p. 187
§2.3.1. Exponential smoothing models
- R forecast provides ets, forecast.ets, and ses functions for simple and automatic exponential smoothing.Source: §2.3.1. Exponential smoothing models · p. 187
- R smooth supports automatic exponential smoothing modelling, while Gretl’s expsmpars minimizes the sum of squared errors for simple exponential smoothing.Source: §2.3.1. Exponential smoothing models · p. 187
- R stats provides lm for fitting linear regression models.Source: §2.3.2. Time-series regression models · p. 187
§2.3.2. Time-series regression models
- Time-series regression entries include R leaps for linear-model selection, relaimpo for regressor importance, and MASS stepAIC for stepwise AIC selection.Source: §2.3.2. Time-series regression models · p. 187
- The section also lists Gretl tools for OLS, LAD, and MIDAS regression, and R forecast’s thetaf for forecasts and prediction intervals.Source: §2.3.2. Time-series regression models · p. 187, §2.3.3. Theta method and models · p. 187
§2.3.3. Theta method and models · §2.3.5. Forecasting for multiple seasonal cycles
These sections catalog software implementations for Theta methods and models, ARIMA-related analysis, and forecasting series with multiple seasonal cycles. The listed tools include forecasting methods, model-fitting functions, and unit-root tests.
§2.3.3. Theta method and models
- R forecTheta, R tsutils, and Gretl provide functions for fitting or forecasting with Theta methods and models.The listed tools include forecTheta’s stheta, stm, otm, dstm, and dotm functions, tsutils’ theta estimator, and Gretl’s stheta.Source: §2.3.3. Theta method and models · p. 187
- ARIMA-related implementations include functions for fitting and forecasting ARIMA or SARIMAX models and selecting a best-fitting SARIMAX model.The listed implementations include R forecast and Gretl functions.Source: §2.3.3. Theta method and models · p. 187
- R smooth provides state-space and multiple-seasonality ARIMA implementations, including functions for forecasting data with multiple seasonal cycles.The listed functions include auto.msarima, auto.ssarima, msarima, and ssarima.Source: §2.3.3. Theta method and models · p. 187, §2.3.5. Forecasting for multiple… · p. 188
- R and Gretl provide unit-root tests, while R forecast functions estimate the numbers of regular and seasonal differences required.The listed tests include augmented Dickey-Fuller and KPSS tests, alongside Gretl’s adf, adf-gls, kpss, and levinlin functions.Source: §2.3.3. Theta method and models · p. 187
- Gretl implementations cover seasonal unit-root tests, a double-unit-root test, and a unit-root test that allows possible structural breaks.The listed packages include DHF, DP, GHegy, Kapetanios, and PPtest.Source: §2.3.3. Theta method and models · p. 188
§2.3.5. Forecasting for multiple seasonal cycles
- For multiple seasonal cycles, R fable-related packages list ARIMA, TSLM, forecast, fasster, and ETS models.The cited package entries include a tidy time-series forecasting package and a list of model names.Source: §2.3.5. Forecasting for multiple… · p. 188
- Prophet is listed as an automatic forecasting procedure, with a tidy R interface available through fable.prophet.Source: §2.3.5. Forecasting for multiple… · p. 188
- R forecast lists tbats and dshw functions for forecasting data with multiple seasonal cycles.Source: §2.3.5. Forecasting for multiple… · p. 188
§2.3.6. State-space models … §2.3.10. Functional time series models
The listed implementations span state-space analysis, population projection, multivariate forecasting, and functional time-series analysis. The tools range from MATLAB and R packages to Gretl software and a demographic spreadsheet system.
§2.3.6. State-space models
- MATLAB tools cover general state-space modelling, time-series analysis in state-space form, and the SSMMATLAB statistical-analysis programs.The listed descriptions include linear, non-linear, and non-Gaussian state-space systems.Source: §2.3.6. State-space models · p. 188
- R, Python, and Gretl listings include state-space tools for ARIMA, structural models, Bayesian inference, multivariate models, filtering, smoothing, and forecasting.The descriptions include statsmodels statespace, bssm, mssm, KFAS, ksimul, and Gretl structural time-series tools.Source: §2.3.6. State-space models · pp. 188–189
§2.3.7. Models for population processes
- Population-process software includes Bayesian demographic methods, life-table tools, and BayesPop projections for countries worldwide.BayesPop uses several probabilistic components, including total fertility rate and life expectancy.Source: §2.3.7. Models for population processes · p. 189
- bayesTFR and bayesLife provide Bayesian hierarchical probabilistic projections of total fertility rate and life expectancy for countries worldwide.The listed population tools also include DAPPS, a Census Bureau spreadsheet system for demographic estimation and projections.Source: §2.3.7. Models for population processes · p. 189
§2.3.9. Forecasting with many variables
- Forecasting with many variables is supported by R VAR routines and Gretl system-model fitting with forecasting functionality.Source: §2.3.9. Forecasting with many variables · p. 189
- Functional time-series listings include stationarity and R ftsa tools, including ftsm and farforecast.Source: §2.3.9. Forecasting with many variables · p. 189, §2.3.10. Functional time series models · p. 189
§2.3.10. Functional time series models
- The R tseries garch package fits GARCH models to time series.Source: §2.3.11. ARCH/GARCH models · p. 189
§2.3.11. ARCH/GARCH models … §2.3.18. Innovation diffusion models
The review catalogs software implementations for several forecasting model families, from ARCH/GARCH to innovation diffusion. The listed tools span R, Python, and Gretl.
§2.3.11. ARCH/GARCH models
- PyFlux and Gretl tools support fitting or estimating ARCH/GARCH models, with PyFlux also covering ARIMA methods.Gretl’s listed tools include arch/garch and gig estimate/var fcast.Source: §2.3.11. ARCH/GARCH models · pp. 189–190
§2.3.12. Markov switching models
- R implementations listed for Markov switching include MSwM and NHMSAR, for fitting Markov switching models and non-homogeneous Markov switching autoregressive models.Source: §2.3.12. Markov switching models · p. 190
§2.3.13. Threshold models
- Threshold-model implementations include R packages TAR and TSA and Gretl tools for Hansen’s panel threshold model and SETAR estimation.TSA includes tar and star functions for threshold models and general time-series analysis.Source: §2.3.13. Threshold models · p. 190
§2.3.15. Forecasting with DSGE models
- The R BMR package estimates and forecasts Bayesian VAR and DSGE models.Source: §2.3.15. Forecasting with DSGE models · p. 190
- Dynare is a software platform for solving, estimating, and forecasting with DSGE models.Source: §2.3.15. Forecasting with DSGE models · p. 190
§2.3.18. Innovation diffusion models
- R packages cover innovation-diffusion forecasting, including Bass, Gompertz, Gamma/Shifted Gompertz, and S-curve models.DIMORA estimates Bass, Generalised Bass, GGM, and UCRCD models; diffusion forecasts new-product growth.Source: §2.3.18. Innovation diffusion models · p. 190, §2.3.19. The natural law of growth in… · p. 190
§2.3.19. The natural law of growth in competition · §2.3.22. Forecasting under fat tails
The entries catalog methods and software for uncertainty and quantile forecasting, covariance forecasting, prediction intervals, fat-tail distributions, and vine-copula analysis.
§2.3.19. The natural law of growth in competition
- Conditional kernel density estimation is listed for producing marginal distributions.Source: §2.3.19. The natural law of growth in… · p. 190
- gamlss provides semi-parametric uncertainty forecasting models, while gamboostLSS provides semi-parametric component-wise gradient boosting models for uncertainty.Source: §2.3.19. The natural law of growth in… · p. 190
- Gradient boosting trees and random forests are listed for quantile forecasting, alongside scikit-learn’s GradientBoostingRegressor.Source: §2.3.19. The natural law of growth in… · p. 190
- The listed quantreg methods estimate and support inference for conditional quantiles, and EnvStats includes functions for pointwise and simultaneous prediction intervals.Source: §2.3.19. The natural law of growth in… · p. 190
- Multivariate GARCH models are listed for forecasting covariance matrices, with rmgarch providing related methods.Source: §2.3.19. The natural law of growth in… · p. 190
- FatTailsR provides functions for Kiener distributions and fat tails.Source: §2.3.22. Forecasting under fat tails · p. 190
§2.3.22. Forecasting under fat tails
- VineCopula is listed for statistical analysis of vine copula models.Source: §2.4.3. Bayesian forecasting with… · p. 190
§2.4.3. Bayesian forecasting with copulas … §2.5.4. Model Selection
The sections point to software implementations for copula forecasting, Granger-causality testing, variable selection, and model selection.
§2.4.3. Bayesian forecasting with copulas
- R’s cdcopula implements covariate-dependent copula models, while FactorCopula covers copula models for mixed continuous and discrete data.Source: §2.4.3. Bayesian forecasting with… · pp. 190–191
§2.5.1. Leading indicators and Granger causality
- Gretl provides standard and frequency-wise Granger-causality tests, including the Breitung–Candelon test.Source: §2.5.1. Leading indicators and Granger… · p. 191
- R’s glmnet provides generalized linear models with Lasso or elastic-net regularization.Source: §2.5.3. Variable Selection · p. 191
§2.5.3. Variable Selection
- Gretl supports sequential variable removal and addition through omit and addlist.Source: §2.5.3. Variable Selection · p. 191
- Gretl’s regls supports regularized least squares, including Ridge, Lasso, and Elastic-Net.Source: §2.5.3. Variable Selection · p. 191
- Gretl’s fsboost and fsreg provide forward-stagewise boosted regression estimates with forecasting functionality.Source: §2.5.3. Variable Selection · p. 191
§2.5.4. Model Selection
- R’s forecast package includes CVar k-fold cross-validation applied to an autoregressive model.Source: §2.5.5. Cross-validation for… · p. 191
§2.5.5. Cross-validation for time-series data … §2.7.12. Clustering-based forecasting
The review’s resource listings pair forecasting approaches with software implementations, spanning time-series validation, distributed systems, feature-based methods, neural networks, and machine learning.
§2.5.5. Cross-validation for time-series data
- Gretl fcast supports recursive-window forecasts, while R forecastHybrid provides functions for ensemble time-series forecasts.Source: §2.5.5. Cross-validation for… · p. 191
§2.7.2. Forecasting on distributed systems
- Distributed forecasting resources include the OpenTSDB and RRDtool time-series databases, Timely, and Spark implementations of distributed ARIMA models.Source: §2.7.2. Forecasting on distributed… · p. 191
§2.7.3. Agent-based models
- R tsfeatures provides methods for extracting various features from time-series data.Source: §2.7.4. Feature-based time series… · p. 191
§2.7.4. Feature-based time series forecasting
- Feature-based forecasting resources include tsfresh and hctsa for time-series characteristics, fforma for model averaging or selection, and gratis for generating series with diverse characteristics.fforma trains a classifier to select or combine forecast models.Source: §2.7.4. Feature-based time series… · pp. 191–192
§2.7.5. Forecasting with bootstrap; §2.7.6. Bagging for time series forecasting
- Bootstrap resources include uniFCboot for interval forecasts without Gaussian innovations, while tshacks provides baggedClusterETS and treated ETS forecasts.Source: §2.7.5. Forecasting with bootstrap · p. 192, §2.7.6. Bagging for time series… · p. 192
§2.7.8. Neural Networks
- Neural-network resources include nnfor, neural prophet, RNN forecasting code, nnetar, and GluonTS implementations of deep-learning models such as DeepAR, DeepState, and NBEATS.Source: §2.7.8. Neural Networks · p. 192
§2.7.10. Machine learning
- Machine-learning resources include tools for recursive partitioning, support vector machines, Gaussian processes, random forests, boosted regression, and scikit-learn’s broad model collection.Source: §2.7.10. Machine learning · p. 193
§2.7.12. Clustering-based forecasting
- The clustering-based forecasting listing includes ESRNN-GPU, a GPU-enabled version of the hybrid model used by the M4 competition winner.Source: §2.7.13. Hybrid methods · p. 193
§2.7.13. Hybrid methods
The section points to R implementations for intermittent-demand forecasting, including Croston-related methods and MAPA.
§2.7.13. Hybrid methods
- The R package tsintermittent lists crost and tsb for parametric forecasting of intermittent demand.Source: §2.7.13. Hybrid methods · p. 194
- The forecast package provides forecasts for intermittent demand using Croston’s method.Source: §2.7.13. Hybrid methods · p. 194
- The tsintermittent package provides imapa, described as MAPA for intermittent-demand data.Source: §2.7.13. Hybrid methods · p. 194
§2.8.3. Classification methods … §2.12.6. Statistical tests of forecast performance
These sections catalogue software for classification, text-informed and hierarchical forecasting, ecological inference, and forecast evaluation. The listed implementations include tools for scoring probabilistic forecasts and testing forecast performance.
§2.8.3. Classification methods
- R tsutils lists classification functions and routines, and Python NLTK is identified as the Natural Language Toolkit.Source: §2.8.3. Classification methods · p. 194, §2.9.3. Forecasting with text… · p. 194
§2.9.3. Forecasting with text information
- For text-informed forecasting, SpaCy is described as an open-source library for advanced natural language processing; hts provides functions and routines for hierarchical and grouped time series forecasting.Source: §2.9.3. Forecasting with text… · p. 194, §2.10.1. Cross-sectional hierarchical… · p. 194
§2.10.1. Cross-sectional hierarchical forecasting
- MAPA is available through R packages mapa and mapasimple, described as functions and wrappers for using the Multiple Aggregation Prediction Algorithm for time series forecasting.Source: §2.10.2. Temporal aggregation · p. 194
§2.10.2. Temporal aggregation
- Temporal hierarchical forecasting is represented by the R package thief, while tsintermittent provides imapa for intermittent-demand data with automatic model selection based on PK classification.Source: §2.10.2. Temporal aggregation · p. 194
§2.10.4. Ecological inference forecasting
- Ecological-inference implementations include tools for local and global forecasts of inner table cells, using Bayesian models, ecological regression, linear programming, or iterative methods.The listed methods include Multinomial Dirichlet models and iterative versions of 2×2 methods.Source: §2.10.4. Ecological inference… · p. 194
§2.12.4. Evaluating probabilistic forecasts
- For probabilistic forecast evaluation, scoringRules provides scoring rules for parametric and simulated distribution forecasts, while verification provides the continuous ranked probability score.The listed R package forecast accuracy provides accuracy measures for a forecast model.Source: §2.12.4. Evaluating probabilistic… · p. 194, §2.10.4. Ecological inference… · p. 194
§2.12.6. Statistical tests of forecast performance
- Forecast-performance testing tools include forecast's Diebold–Mariano test and tsutils' Nemenyi test, alongside Gretl FEP routines and DiebMar for statistical tests.The listed tests cover forecast unbiasedness, efficiency, asymmetric loss, directional changes, and Diebold–Mariano comparisons.Source: §2.12.4. Evaluating probabilistic… · pp. 194–195, §2.12.6. Statistical tests of forecast… · p. 195
Appendix C. Data sets … §3.4.10. Collaborative forecasting in the energy sector
The paper’s appendix and application sections list indicative publicly available datasets for forecasting across economic, financial, and energy applications.
Appendix C. Data sets
- Appendix C presents an indicative list of publicly available datasets, including a database for EU macroeconomic variables.The list is organized alongside application sections covering forecasting topics across the paper.Source: Table A.2 · p. 196, Appendix C · p. 197
§3.3.7. House price forecasting
- House-price forecasting datasets include Zillow Prize, Sberbank Russian Housing Market, and Western Australia Rental Prices.Source: §3.3.7. House price forecasting · p. 197
§3.3.12. Forecasting returns to investment style
- The Winton Stock Market Challenge is listed as a dataset for forecasting returns to investment style.Source: §3.3.13. Forecasting stock returns · p. 197
§3.3.13. Forecasting stock returns
- Stock-return and energy-related forecasting examples include the BattleFin Big Data Combine, ASHRAE Great Energy Predictor III, and Global Energy Forecasting Competition 2012 – Load.Source: §3.3.13. Forecasting stock returns · p. 197, load forecasting · p. 197
load forecasting
- The Global Energy Forecasting Competition 2012 – Wind is listed among load-forecasting datasets.Source: §3.4.6. Wind power forecasting · p. 197
§3.4.6. Wind power forecasting
- Wind-power forecasting resources include numerical weather predictions and related data.Source: §3.4.6. Wind power forecasting · p. 197
§3.4.8. Solar power forecasting
- Solar-power forecasting resources include the AMS 2013–2014 Solar Energy Prediction Contest and solar time series from small-scale PV systems.Source: §3.4.8. Solar power forecasting · p. 197, §3.4.10. Collaborative forecasting in… · p. 197
§3.4.10. Collaborative forecasting in the energy sector
- Collaborative energy forecasting datasets include domestic-consumption and electric-vehicle charging data, plus wind-plant data with numerical weather predictions.Source: §3.4.10. Collaborative forecasting in… · p. 197
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