Research on Forecastability, Mutual Information, and the Limits of Prediction
Forecastability is a property of the data, not the model. This page presents research on measuring forecastability using auto-mutual information (AMI) and entropy. It addresses a prior problem, before model selection or benchmarking: determining how much of the future is knowable from the past.
Rather than benchmarking models against each other, these papers examine the structure of the data itself and the limits that structure imposes on achievable accuracy. Together, they develop forecastability as a measurable property of time series, show how it varies across forecast horizons, and link it directly to forecast error, model choice, and economic value.
Forecastability as an Information-Theoretic Limit on Prediction
Abstract: Forecasting is usually framed as a problem of model choice. This paper starts earlier, asking how much predictive information is available at each horizon. Under logarithmic loss, the answer is exact: the mutual information between the future observation and the declared information set equals the maximum achievable reduction in expected loss. This paper develops the consequences of that identity. Forecastability, defined as this mutual information evaluated across horizons, forms a profile whose shape reflects the dependence structure of the process and need not be monotone. Three structural properties are derived: compression of the information set can only reduce forecastability; the gap between the profile under a finite lag window and the full history gives an exact truncation error budget; and for processes with periodic dependence, the profile inherits the periodicity. Predictive loss decomposes into an irreducible component fixed by the information structure and an approximation component attributable to the method; their ratio defines the exploitation ratio, a normalised diagnostic for method adequacy. The exact equality is specific to log loss, but when forecastability is near zero, classical inequalities imply that no method under any loss can materially improve on the unconditional baseline. The framework provides a theoretical foundation for assessing, prior to any modelling, whether the declared information set contains sufficient predictive information at the horizon of interest.
APA
Catt, P. M. (2026). Forecastability as an Information-Theoretic Limit on Prediction. arXiv. https://doi.org/10.48550/arXiv.2603.27074
BibTeX
@article{catt2026forecastability,
author = {Catt, Peter M.},
title = {Forecastability as an Information-Theoretic Limit on Prediction},
year = {2026},
journal = {arXiv preprint arXiv:2603.27074},
doi = {10.48550/arXiv.2603.27074},
url = {https://doi.org/10.48550/arXiv.2603.27074}
}
An Information-Theoretic Diagnostic Analytics Framework for Mapping Past–Future Dependence in Horizon-Specific Forecastability
Abstract: In many social, business, economic, and physical systems, the true data-generating process is unknown, requiring forecasters to rely exclusively on observed time series. This study proposes a pre-modeling diagnostic analytics framework for horizon-specific forecastability assessment that evaluates forecastability before model selection begins, enabling informed decisions about whether additional modeling effort is likely to justify its cost. Forecastability is operationalized using auto-mutual information at lag h, which quantifies how much past observations reduce uncertainty about future values, and is estimated via a k-nearest-neighbor estimator computed strictly on training data to preserve out-of-sample validity. The diagnostic signal is validated against realized out-of-sample symmetric mean absolute percentage error across 42,355 time series spanning six temporal frequencies, using benchmark and higher-capacity probe models under a rolling-origin protocol. The results reveal a strong frequency-dependent relationship between measurable dependence and realized forecast error: for five of six frequencies, auto-mutual information exhibits a consistent negative rank association with realized error, supporting its use as a forecast triage signal for modeling investment decisions, whereas the daily series shows weaker discrimination despite measurable dependence. Across all frequencies, median forecast error declines monotonically from low to high forecastability terciles, demonstrating clear decision-relevant separation. Overall, the findings establish measurable past-future dependence as a practical screening tool for analytics-driven forecasting strategy, identifying when advanced models are likely to add value, when simple baselines suffice, and when attention should shift from accuracy improvement to robust decision design, thereby supporting a diagnostic-first approach to modeling effort and resource allocation in organizational forecasting contexts.
APA
Catt, P. M. (2026). An information-theoretic diagnostic analytics framework for mapping past–future dependence in horizon-specific forecastability (under review). SSRN. https://doi.org/10.2139/ssrn.6416626
BibTeX
@article{catt2026diagnostic,
author = {Catt, Peter M.},
title = {An Information-Theoretic Diagnostic Analytics Framework for Mapping Past--Future Dependence in Horizon-Specific Forecastability},
year = {2026},
journal = {SSRN Electronic Journal},
doi = {10.2139/ssrn.6416626},
note = {Manuscript under review}
}
Entropy as an A Priori Indicator of Forecastability
Abstract: The ability to accurately determine the a priori forecastability of a time series is an important endeavour for forecasting practitioners as it provides guidance on the potential for accurate forecasts and the associated degree of effort that is warranted. Measures of entropy, such as sample entropy, provide an assessment of the regularity or similarity within a time series. We posit that series with low in-sample entropy, i.e. high regularity, will positively correlate with low out-of-sample forecast error, as measured by the mean absolute scaled error (MASE). To assess this we adopt the 3003 time series used in the M3 forecasting competition spanning micro, industry, macro, finance, and demographic series. We calculate the in-sample sample entropy and out-of-sample MASE for all 3003 series using a common univariate forecasting method. We demonstrate that the a priori sample entropy is indeed a useful predictor of out-of-sample forecast performance, subject to the well-researched problem of structural breaks. We recommend that forecasting practitioners adopt such entropy measures, alongside well-established tests for seasonality and trend, to better understand the likelihood of successful forecasting outcomes.
APA
Catt, P. M. (2014). Entropy as an a priori indicator of forecastability. (SSRN Working Paper No. 6235738). SSRN. https://dx.doi.org/10.2139/ssrn.6235738
BibTeX
@unpublished{catt2014entropy,
author = {Catt, Peter M.},
title = {Entropy as an A Priori Indicator of Forecastability},
year = {2014},
month = {November},
note = {SSRN Working Paper No. 6235738},
url = {https://dx.doi.org/10.2139/ssrn.6235738}
}
Forecastability: Insights from Physics, Graphical Decomposition, and Information Theory
Abstract: This paper explores the concept of forecastability from multiple theoretical perspectives, drawing on physics, graphical decomposition methods, and information theory. Beginning with foundational concepts from classical determinism, quantum mechanics, and chaos theory, the work establishes a classification of data-generating processes along a continuum from deterministic to chaotic to complex to random. Six illustrative time series, spanning a pure sine wave, the Box-Jenkins airline passengers series, the Hénon map, polystyrene shipments, IBM stock price changes, and pseudo-random numbers, are analysed using time plots, classical decomposition, and lag plots to reveal their underlying structural components. The paper evaluates the coefficient of variation as a forecastability metric and identifies critical limitations, including its sensitivity to near-zero means and its inability to detect patterns beyond trend and seasonality. As an alternative, the paper proposes normalised approximate entropy (ApEn), an information-theoretic measure of series regularity that captures a broader range of deterministic structure. Empirical results across the six series demonstrate that ApEn provides a more reliable ordering of relative forecastability than the coefficient of variation, offering forecasting practitioners a principled basis for assessing how predictable a given time series is and, by extension, the degree to which investment in sophisticated modelling methods may be justified.
APA
Catt, P. M. (2009). Forecastability: Insights from physics, graphical decomposition, and information theory. Foresight: The International Journal of Applied Forecasting, 13, 24–33.
BibTeX
@article{catt2009forecastability,
author = {Catt, Peter M.},
title = {Forecastability: Insights from Physics, Graphical Decomposition, and Information Theory},
journal = {Foresight: The International Journal of Applied Forecasting},
year = {2009},
volume = {13},
pages = {24--33}
}