The Granularity Paradox: How Temporal Disaggregation Inflates In-Sample Fit and Compounds Out-of-Sample Error

Abstract: This paper explores the "Granularity Paradox" in time-series forecasting, wherein finer temporal disaggregation (e.g., Monthly to Weekly/Daily) improves in-sample diagnostics and dataset size (N), but degrades out-of-sample accuracy due to recursive error compounding over longer horizons (H). Conversely, coarse aggregation (Annual) eliminates recursive error propagation but reduces data available to estimators. We formalize this trade-off and benchmark 10 models - spanning naïve, statistical, machine learning, and deep learning architectures - across six granularities using a 13-year public procurement dataset. The empirical results reveal a non-monotonic threshold structure: recursive autoregressive and seasonal models degrade substantially under high-frequency forecasting (e.g., Holt-Winters reaches a Test R-squared of -151 and TPFE of 425.85% at the Daily grain), while the LSTM traces a U-shaped error curve, worsening from Monthly (19.66%) through Bi-Weekly (35.94%) before overcoming the error propagation penalty at Daily (TPFE of 4.35%, R-squared of 0.66). Linear Regression remains stable across all granularities (16.3-17.0% TPFE), confirming that the paradox is driven by recursive feedback topology, not model complexity. The results demonstrate that standard pointwise metrics (RMSE, MAE) systematically mask cumulative error propagation, and that evaluating forecasts without goal-dependent cumulative metrics produces misleading assessments of model adequacy. We introduce a consensus-dissensus diagnostic comparing the directional behaviour of pointwise metrics against cumulative TPFE across granularities, enabling the identification of models whose standard diagnostics mask systematic error propagation.

• The paper demonstrates that finer temporal granularity inflates in-sample fit while compounding out-of-sample forecast error in recursive models.
• It provides a rigorous theoretical and empirical analysis showing that pointwise metrics can mask cumulative biases in forecasting performance.
• Empirical benchmarks reveal that while non-recursive models remain robust, recursive architectures like LSTM exhibit U-shaped error dynamics across granularities.

The Granularity Paradox in Time-Series Forecasting: A Technical Analysis

Introduction

Temporal aggregation is a foundational design decision in time-series modeling with direct implications for both statistical efficiency and operational utility. "The Granularity Paradox: How Temporal Disaggregation Inflates In-Sample Fit and Compounds Out-of-Sample Error" (2607.05450) systematically characterizes the error dynamics induced by the choice of granularity. The paper provides a formal and empirical basis for the paradox where finer temporal granularity increases apparent in-sample fit while simultaneously worsening out-of-sample multi-step forecasting due to recursive error propagation. The findings challenge conventional reliance on pointwise metrics and in-sample diagnostics across forecasting pipelines.

Theoretical Formalization of Aggregation Effects

Temporal aggregation reduces high-frequency volatility but introduces fundamental tradeoffs. Coarse-grained aggregation reduces estimator variance but limits available data points, which particularly handicaps parameter-rich models. Finer disaggregation, conversely, provides larger $N$ for estimation but requires longer recursive prediction horizons ($H$) for practical planning. Notably, the stochastic properties of the underlying series, unit root behavior, and cointegration structure are all impacted by the aggregation operator, often in nontrivial ways. Crucially, the Granularity Paradox is not a simple function of model complexity or sample size, but emerges from the interaction of recursive feedback structure and horizon inflation.

Error Propagation in Multi-Step Forecasting

Recursive forecasting strategies, common across both classical (e.g., ARIMA, Holt-Winters) and modern (e.g., RNNs) time-series models, are disproportionately affected by horizon length. Even minor bias in parameter estimation is exponentially compounded as predictions are fed sequentially over large $H$—a phenomenon mathematically formalized for first-order autoregressive processes within the paper. Direct and multiple-output (MIMO) strategies avoid some of these limitations but increase variance or computational cost. The analysis reveals that error propagation should be a routine diagnostic consideration whenever forecasting horizon scales with granularity.

Empirical Benchmarking Across Granularities

The empirical component centers on a 13-year panel of public procurement micro-data, resampled to six granularities (Annual, Quarterly, Monthly, Bi-Weekly, Weekly, Daily), and includes 10 modeling paradigms ranging from classical statistical methods to deep neural architectures. Each model is backtested across 8 rolling folds, and cross-granular performance is evaluated along both pointwise (RMSE, MAE, $R^2$) and cumulative (Total Absolute Forecast Error—TAFE, Total Percentage Forecast Error—TPFE) axes.

Several salient results emerge:

• In-sample diagnostics such as training $R^2$ are systematically misleading at fine grains. Naive recursive baselines (e.g., Persistence) exhibit training $R^2 \approx 0.78$ at the daily scale, but their cumulative out-of-sample error is catastrophic (TPFE $>23\%$; Holt-Winters' TPFE surges to $425.85\%$ at Daily).
• Autoregressive and seasonal models collapse under high-frequency recursive forecasting, with the penalty outpacing any possible gain from larger $N$.
• Linear Regression is robust to granularity choice. Its TPFE remains essentially invariant ($16.1$–$H$0) across all granularities, exposing that the paradox is primarily due to recursive model architecture, not model sophistication or data volume.
• LSTM models exhibit a non-monotonic, U-shaped error curve across grains. Intermediate granularities (Bi-Weekly, Weekly) form an adverse region where expanded $H$1 amplifies error more than $H$2 can offset; only at fine ($H$3) resolutions does LSTM recover, achieving the best cumulative performance (TPFE $H$4, $H$5).
• N-BEATS demonstrates flatter performance over granularities, suggesting structural robustness to error propagation compared to pure recurrent architectures.
• Pointwise metrics (RMSE, MAE) systematically obscure cumulative bias, creating a consensus-dissensus gap between operational forecast quality (TPFE) and diagnostic metrics.

Diagnostic and Evaluation Advances

A consensus-dissensus diagnostic paradigm is proposed, comparing the directional trends of pointwise and cumulative metrics as granularity varies. For non-recursive models (e.g., Linear Regression), both metric classes move in lockstep. For recursive models, dissensus is observed: pointwise error improves or plateaus with finer grains, but cumulative error explodes. This dichotomy exposes systematic forecasting bias and underscores the necessity of cumulative, goal-dependent metrics in any evaluation protocol if the objective is aggregate planning or budgeting accuracy.

Practical and Theoretical Implications

The analysis compels several implications for both research and applied forecasting:

• Granularity selection interacts with model architecture to control error dynamics. For high-capacity models, only large sample sizes at fine granularities can exploit the model's potential without catastrophic error propagation.
• In-sample fit or pointwise accuracy should not guide model/grain selection where operational metrics are aggregate in nature. Pipelines that diagnose performance in-sample or by RMSE/MAE per step are incentivized toward design choices (finer grain, added complexity) that actively degrade out-of-sample planning accuracy.
• Recursive, state-dependent models are most vulnerable in intermediate-granularity regimes. Tradeoff regions where $H$6 grows more slowly than $H$7 offer the worst-case scenario for multi-step propagation error.
• Non-recursive trend models are preferable in low-data regimes or where aggregate accuracy is paramount.

Theoretically, these effects are not idiosyncratic but driven by structural properties of recursive state updating. As such, they generalize across datasets and modeling frameworks.

Limitations and Prospective Directions

The study is limited to a single domain (IT Services within public procurement), and results are provided as point estimates; further multi-seed benchmarks and expanded panel validation across additional sectors are natural extensions. The insight that the worst in-sample fit often heralds the best out-of-sample stability (for non-recursive trend models) suggests new directions for theoretical modeling of regularization and feedback topologies. Moreover, integrating cumulative metrics into automated hyperparameter optimization and model selection pipelines remains an open challenge.

Conclusion

This study demonstrates that temporal disaggregation inflates in-sample fit while inducing severe out-of-sample degradation for recursive forecasters, a structural paradox independent of model class or complexity. Linear trend models are robust to these effects; capacity-hungry architectures (e.g., LSTM) are rescued at fine grains only by sufficiently large $H$8. Pointwise error metrics are inadequate for cross-granularity comparisons; cumulative, goal-dependent metrics are a necessary inclusion in any pipeline concerned with planning accuracy. These findings have immediate implications for forecasting practice and point to the importance of diagnostic frameworks sensitive to error propagation and operational requirements.