Draft:Forecastability

The deterministic–random continuum

Forecastability has been discussed in relation to a broader classification of data-generating processes (DGPs) along a continuum from fully deterministic to purely random.^[12] Chaos theory, following Edward Lorenz's work on atmospheric convection in 1963, demonstrated that some deterministic nonlinear systems exhibit sensitive dependence on initial conditions, making long-horizon prediction practically impossible even when the DGP is known. Eckmann and Ruelle (1985) argued that this implies a fundamental bound on the predictability horizon in chaotic systems, independent of any forecasting method applied.^[12]

Data-generating processes are commonly classified along the following continuum:^[13][12]

• Deterministic: The DGP and its parameters are fully known; the series can in principle be forecast without error at any horizon.
• Chaotic: The DGP is known but sensitive to initial conditions. Theoretically deterministic but practically bounded in forecastability, as characterised by positive Lyapunov exponents.^[12]
• Complex: The DGP cannot be fully characterised, as in many economic and social systems where emergent behaviour arises from interactions among many agents.
• Random: The DGP has no exploitable pattern; future values are statistically independent of past values.

Catt (2009) situated typical business and economic time series within this continuum, arguing that their position between the deterministic and random extremes determines an upper bound on forecast accuracy achievable by any method.^[14]

Information-theoretic foundations

Information theory, originating with Claude Shannon's foundational 1948 paper, provides a mathematical framework for quantifying the reduction in uncertainty achievable by observing the past.^[15] The relevant quantity for time series forecastability is mutual information between past observations and future values:

${\displaystyle I({\text{Past}};{\text{Future}}_{h})=H({\text{Future}}_{h})-H({\text{Future}}_{h}\mid {\text{Past}})}$

where ${\displaystyle H(\cdot )}$ denotes Shannon entropy. High mutual information indicates that the past substantially reduces uncertainty about the future at horizon ${\displaystyle h}$; a value near zero indicates weak or absent past–future dependence.

Jaynes (1957) extended this framework through the maximum entropy principle, establishing information-theoretic reasoning as a foundation for statistical inference under uncertainty.^[16] Bialek, Nemenman and Tishby (2001) formalised the concept of predictive information as the mutual information between the past and future of a stochastic process, arguing that this quantity characterises how much of the future is knowable from the past, independently of any particular method.^[17] Related constructs in computational mechanics include excess entropy and statistical complexity, developed by Grassberger (1986) and Crutchfield and Feldman (2003).^[18][19]

Coefficient of variation

The coefficient of variation (CV), defined as the ratio of the standard deviation to the mean of a detrended and deseasonalised series, has been proposed as a practical numerical indicator of forecastability.^[9] Lower residual variability after removing trend and seasonality is taken to indicate greater forecastability.

Gilliland (2010) noted that the CV assumes no further exploitable structure remains after detrending and deseasonalisation, an assumption that may not hold for chaotic or complex series, and that the measure becomes unreliable when the series mean is near zero.^[9] Tiao and Tsay (1994) demonstrated that nonlinear structure can persist in series that appear low-variance under classical decomposition, motivating approaches capable of detecting non-linear dependence.^[20]

Entropy-based measures

Information-theoretic entropy measures assess series regularity without assuming that only trend and seasonality constitute exploitable structure.

Approximate entropy (ApEn), introduced by Pincus (1991), quantifies the likelihood that similar short patterns in a series remain similar at the next comparison step; small values indicate high regularity.^[21] Sample entropy (SampEn), developed by Richman and Moorman (2000), reduces the self-matching bias of ApEn in shorter series.^[22] Both produce a single scalar measure of overall regularity that does not vary by forecast horizon.

Goerg (2013) introduced forecastable component analysis (ForeCA), which identifies the most forecastable linear combination of a multivariate time series by minimising spectral entropy (the entropy of the normalised spectral density), and extends principal component analysis by optimising for forecastability rather than variance explained.^[23]

Lyapunov exponents

In dynamical systems, Lyapunov exponents quantify the rate of divergence of nearby trajectories. Eckmann and Ruelle (1985) established that the largest Lyapunov exponent determines the timescale over which prediction remains feasible in chaotic systems.^[12] Wang, Klee and Roos (2025) noted that reliable estimation requires long, high-quality series and tends to perform poorly on the short or sparse series common in business and economic forecasting.^[5]

Model-based and benchmark approaches

Gilliland (2010) described Forecast Value Added (FVA) analysis as a method for evaluating forecasting performance relative to a simple naïve baseline; a method that cannot consistently outperform a naïve seasonal repeat is taken to indicate low forecastability or poor model specification.^[9] Stock and Watson (2003) discussed variance decomposition and predictive R² as approaches to assessing predictability in macroeconomic forecasting contexts.^[24]

Auto-mutual information

Recent research has explored horizon-specific measures of forecastability based on the information-theoretic dependence between past and future observations. Auto-mutual information (AMI) at lag ${\displaystyle h}$ is the mutual information between a time series and its own values ${\displaystyle h}$ steps later:

${\displaystyle {\text{AMI}}(h)=I(X_{t};\,X_{t+h})}$

Fraser and Swinney (1986) introduced the measurement of mutual information between a time series and its lagged self in the context of chaotic systems analysis.^[25] Kantz and Schreiber (2004) noted that unlike the autocorrelation function, AMI captures both linear and nonlinear temporal dependencies without distributional assumptions, and is explicitly indexed to the lag ${\displaystyle h}$, allowing horizon-specific assessment.^[13]

Catt (2026) reported associations between AMI and realised forecast accuracy across benchmark series from the M4 competition.^[26] Wang, Klee and Roos (2025) provided a comparative review of time series forecastability measures, including AMI alongside entropy-based and model-based alternatives.^[5]