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

# The Granularity Paradox: How Temporal Disaggregation Inflates In-Sample Fit and Compounds Out-of-Sample Error
Source: [https://arxiv.org/html/2607.05450](https://arxiv.org/html/2607.05450)
###### Abstract

This paper explores the “Granularity Paradox” in time\-series forecasting, wherein finer temporal disaggregation \(e\.g\., Monthly to Weekly or Daily\) improves in\-sample model diagnostics and expands the dataset size \(NN\), but degrades out\-of\-sample multi\-step ahead forecasting accuracy due to recursive error compounding over longer prediction horizons \(HH\)\. Conversely, coarse aggregation \(Annual\) eliminates recursive error propagation but reduces the data available to estimators\. We formalize this trade\-off and benchmark 10 forecasting models—spanning naïve baselines, statistical, machine learning, and deep learning architectures—across six temporal granularities \(Annual, Monthly, Quarterly, Bi\-Weekly, Weekly, Daily\) 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 recursive forecasting, with Holt\-Winters reaching a TestR2R^\{2\}of−151\-151and TPFE of425\.85%425\.85\\%at the Daily grain, while the LSTM traces a U\-shaped error curve—worsening from Monthly \(19\.66%19\.66\\%\) through Bi\-Weekly \(35\.94%35\.94\\%\) before overcoming the error propagation penalty at Daily, achieving a TPFE of4\.35%4\.35\\%and an out\-of\-sampleR2R^\{2\}of0\.660\.66\. Linear Regression remains stable across all six granularities \(16\.116\.1–17\.0%17\.0\\%TPFE\), confirming that the paradox is driven by recursive feedback topology, not model complexity\. The results also 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—a finding with direct implications for forecasting practice\. We introduce a consensus\-dissensus diagnostic that compares the directional behaviour of pointwise metrics against cumulative TPFE across granularities, enabling the identification of models whose standard diagnostics mask systematic error propagation\.

###### keywords:

Forecasting; Temporal Aggregation; Error Propagation; Deep Learning; Granularity Paradox; Public Procurement

\{Frontmatter\}

\[1\]Hugo Moreira\\orcid0000\-0002\-4199\-6006 \[1\]\\orgdivCIES \- Centro de Investigação e Estudos de Sociologia,\\orgnameIscte \- Instituto Universitário de Lisboa,\\cityLisbon,\\countryPortugal

## 1Introduction

Temporal aggregation is a core design decision in forecasting workflows\. When modeling a fixed physical planning horizon \(such as a 1\-year budget cycle\), the researcher must choose the temporal grain of the target series: daily, weekly, monthly, quarterly, or annual\. This choice establishes a fundamental trade\-off\.

Finer granularities expand the sample size \(NN\), providing machine learning and neural architectures with the data volume necessary to estimate high\-dimensional parameter spaces\. However, disaggregation also inflates the forecasting horizon \(HH\) required to cover the planning period\. For example, a 1\-year planning horizon requiresHm​o​n​t​h​l​y=12H\_\{monthly\}=12steps,Hw​e​e​k​l​y=52H\_\{weekly\}=52steps, orHd​a​i​l​y=365H\_\{daily\}=365steps\. For state\-dependent or autoregressive models that generate multi\-step ahead forecasts recursively, this inflation ofHHtriggers exponential error propagation, as each intermediate prediction is fed back as an input for subsequent steps\.

This paper formalizes this phenomenon as theGranularity Paradox\. We demonstrate that in\-sample fit diagnostics \(such as trainingR2R^\{2\}\) are deceptive indicators of out\-of\-sample performance across granularities\. We benchmark 10 forecasting models — including naïve baselines as recursive\-error controls — across six frequencies to map the boundary where the benefit of sample size expansion is overtaken by the penalty of recursive error compounding\.

## 2Theoretical Foundations of Temporal Aggregation

Temporal aggregation—the process of consolidating high\-frequency observations \(e\.g\., daily\) into coarser low\-frequency intervals \(e\.g\., monthly or annual\)—is a standard operation in time\-series analysis\(Hamilton,[1994](https://arxiv.org/html/2607.05450#bib.bib8)\)\. It acts primarily as a low\-pass filter, smoothing out high\-frequency noise, localized volatility, and short\-term seasonal fluctuations to reveal long\-term trends\. However, this filtering comes at a measurable informational and statistical cost\(Silvestrini and Veredas,[2008](https://arxiv.org/html/2607.05450#bib.bib23)\)\.

### 2\.1Information Loss and Parameter Estimation

Temporal aggregation alters the stochastic properties of the underlying process, affecting the consistency and efficiency of parameter estimators\. Earlier work byWei \([2006](https://arxiv.org/html/2607.05450#bib.bib25)\)demonstrated that estimating parameters on aggregated distributed lag models leads to a systematic degradation of precision\. This degradation is driven by two statistical mechanisms: the amplification of the variance of the aggregated error terms, and the inflation of multicollinearity among the aggregated explanatory variables\. Furthermore,Silvestrini and Veredas \([2008](https://arxiv.org/html/2607.05450#bib.bib23)\)show that the loss of information is more pronounced when the original disaggregated series exhibits negative autocorrelation, a feature common in error\-correction systems or rapid inventory replenishment cycles\.

### 2\.2Causal Distortions and Cointegration

Beyond parameter estimation, temporal aggregation can distort the structural identification of causal relationships\. As shown byTiao and Wei \([1976](https://arxiv.org/html/2607.05450#bib.bib24)\), if the underlying high\-frequency process features a unidirectional dynamic causal relationship \(e\.g\.,X→YX\\to Ywithout feedback\), the temporal aggregation operator can artificially induce a bidirectional feedback loop in the aggregated data\. This spurious causality arises because lagging relationships at a fine resolution are compressed into contemporaneous associations at a coarser resolution, impeding the identification of the true causal direction\.

Additionally, aggregation modifies the integration and cointegration properties of the series\. Depending on the aggregation factor, a process containing a seasonal unit root may be transformed into one with a non\-seasonal unit root\(Silvestrini and Veredas,[2008](https://arxiv.org/html/2607.05450#bib.bib23)\)\. This modification changes the asymptotic distributions of unit root and cointegration tests, meaning that model specifications must be adjusted carefully to avoid spurious regressions\.

### 2\.3Multi\-Scale Modeling and Hierarchical Reconciliation

To resolve the conflict between high\-frequency detail \(which is statistically rich but noisy\) and low\-frequency stability \(which is clean but data\-scarce\), multi\-scale forecasting frameworks have been developed\. The Multiple Aggregation Prediction Algorithm \(MAPA\), proposed byKourentzes et al\. \([2014](https://arxiv.org/html/2607.05450#bib.bib13)\), models a time series across multiple non\-overlapping aggregation levelskk\. Letyty\_\{t\}be a series sampled at the base frequencyt=1,…,nt=1,\\dots,n\. The aggregated seriesYi\[k\]Y^\{\[k\]\}\_\{i\}at aggregation levelkkis defined as:

Yi\[k\]=∑t=1\+\(i−1\)​ki​kytfor​i=1,…,⌊n/k⌋Y^\{\[k\]\}\_\{i\}=\\sum\_\{t=1\+\(i\-1\)k\}^\{ik\}y\_\{t\}\\quad\\text\{for \}i=1,\\dots,\\lfloor n/k\\rfloor\(1\)A linear Holt exponential smoothing model is fitted at each individual aggregation levelkkto extract the level \(LtL\_\{t\}\) and trend \(TtT\_\{t\}\) components:

Lt=α​Yt\[k\]\+\(1−α\)​\(Lt−1\+Tt−1\)L\_\{t\}=\\alpha Y^\{\[k\]\}\_\{t\}\+\(1\-\\alpha\)\(L\_\{t\-1\}\+T\_\{t\-1\}\)\(2\)Tt=β​\(Lt−Lt−1\)\+\(1−β\)​Tt−1T\_\{t\}=\\beta\(L\_\{t\}\-L\_\{t\-1\}\)\+\(1\-\\beta\)T\_\{t\-1\}\(3\)These components are subsequently mapped back to the base frequency and averaged, allowing each scale to contribute the dynamic component it estimates most robustly:

f=1K​∑k=1KLk\+Tkkf=\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\frac\{L\_\{k\}\+T\_\{k\}\}\{k\}\(4\)
Similarly, temporal hierarchies organize these relationships into a coherent tree structure\(Athanasopoulos et al\.,[2017](https://arxiv.org/html/2607.05450#bib.bib1)\)\. This ensures that operational decisions taken at short\-term horizons align perfectly with strategic goals at long\-term horizons, eliminating inconsistencies\. The coherence of independent forecasts generated at different levels of the hierarchy is achieved via forecast reconciliation\. Lety^h\\hat\{y\}\_\{h\}be the vector of independent \(unreconciled\) forecasts across all levels of the hierarchy\. The reconciled, coherent forecastsy~h\\tilde\{y\}\_\{h\}are computed via a stable linear projection:

y~h=S​\(S′​W−1​S\)−1​S′​W−1​y^h\\tilde\{y\}\_\{h\}=S\(S^\{\\prime\}W^\{\-1\}S\)^\{\-1\}S^\{\\prime\}W^\{\-1\}\\hat\{y\}\_\{h\}\(5\)whereSSis the structural summing matrix mapping the base nodes to the aggregated nodes, andWWis the covariance matrix of the base forecast errors\(Athanasopoulos et al\.,[2024](https://arxiv.org/html/2607.05450#bib.bib2)\)\.

## 3Multi\-Step Ahead Forecasting and Error Propagation

Projecting time\-series trajectories over a long future planning horizonHHrequires a choice of how the model generates forecasts for multiple steps\. The forecasting literature divides these strategies into recursive, direct, and multiple\-output \(MIMO\) approaches\(Ben Taieb et al\.,[2012](https://arxiv.org/html/2607.05450#bib.bib3)\)\.

### 3\.1Recursive \(Iterated\) Strategy and the Granularity Paradox

The recursive strategy trains a single model for a single\-step\-ahead prediction \(h=1h=1\)\. For longer horizons, forecasts are generated sequentially, with the predictions of previous steps fed back as autoregressive inputs for the next step\. If the model is perfectly specified, the recursive strategy is asymptotically more efficient than its competitors because it preserves the full temporal dependence structure of the data\(Marcellino et al\.,[2006](https://arxiv.org/html/2607.05450#bib.bib17)\)\. However, in the presence of misspecification or parameter bias, this feedback loop compounds errors exponentially\.

Let us formalize the trade\-off\. For a first\-order autoregressive process,yt=ϕ​yt−1\+ϵty\_\{t\}=\\phi y\_\{t\-1\}\+\\epsilon\_\{t\}, the out\-of\-sample prediction at stephhis:

y^T\+h=ϕ^h​yT\\hat\{y\}\_\{T\+h\}=\\hat\{\\phi\}^\{h\}y\_\{T\}\(6\)If the estimated parameter has a biasδ\\deltasuch thatϕ^=ϕ\+δ\\hat\{\\phi\}=\\phi\+\\delta, the forecast error at stephhcompounds as:

eT\+h=yT\+h−y^T\+h=ϕh​yT−\(ϕ\+δ\)h​yTe\_\{T\+h\}=y\_\{T\+h\}\-\\hat\{y\}\_\{T\+h\}=\\phi^\{h\}y\_\{T\}\-\(\\phi\+\\delta\)^\{h\}y\_\{T\}\(7\)For a daily forecasting model \(H=365H=365\), even a minor parameter biasδ\\deltaleads to exponential divergence over the horizon, whereas a monthly model \(H=12H=12\) limits this error compounding\. TheGranularity Paradoxarises because daily estimation expands the sample sizeNN\(reducing the estimator variance ofϕ^\\hat\{\\phi\}\), but simultaneously inflates the forecasting horizonHH\(amplifying the propagation of the biasδ\\deltaover the horizon\)\.

### 3\.2Direct and Multiple\-Output \(MIMO\) Strategies

The direct strategy avoids recursive error propagation by trainingHHindependent models, each optimized for a specific forecasting steph∈\[1,H\]h\\in\[1,H\]\. While robust to short\-term model misspecification, this approach ignores the temporal correlations between successive forecasts, resulting in highly erratic prediction trajectories\. It also imposes a substantial computational burden sinceHHseparate models must be trained and maintained\(Marcellino et al\.,[2006](https://arxiv.org/html/2607.05450#bib.bib17)\)\.

The Multiple\-Input Multiple\-Output \(MIMO\) strategy resolves these issues by predicting the entire vector of future outputsYH=\[yt\+1,…,yt\+H\]′Y\_\{H\}=\[y\_\{t\+1\},\\dots,y\_\{t\+H\}\]^\{\\prime\}simultaneously with a single multi\-output model\. As shown byBen Taieb et al\. \([2012](https://arxiv.org/html/2607.05450#bib.bib3)\), MIMO models preserve the temporal covariance structure of the forecasting errors without the recursive instability of iterated methods or the computational overhead of direct methods\. Other non\-parametric methods, such as lazy learning\(Bontempi et al\.,[1999](https://arxiv.org/html/2607.05450#bib.bib5)\), defer parameter fitting to the moment of inference, fitting local weighted regressions on the nearest neighbors in the embedded state space\.

## 4Related Work

The interaction between temporal aggregation and forecasting accuracy has been investigated from multiple angles\.Nikolopoulos et al\. \([2011](https://arxiv.org/html/2607.05450#bib.bib18)\)introduced the Aggregate–Disaggregate Intermittent Demand Approach \(ADIDA\), establishing a formal framework in which a series is first aggregated to a coarser level, forecasted with a simple model, and then disaggregated back to the operational frequency\.Rostami\-Tabar et al\. \([2013](https://arxiv.org/html/2607.05450#bib.bib21)\)extended this line of inquiry by demonstrating empirically that temporal aggregation of non\-overlapping intervals can improve forecast accuracy for stationary demand processes, whilePetropoulos and Kourentzes \([2015](https://arxiv.org/html/2607.05450#bib.bib20)\)showed that combining forecasts across multiple aggregation levels outperforms single\-scale methods for intermittent series\. Our work departs from this literature by focusing not on demand intermittency but on the recursive error compounding mechanism that emerges when state\-dependent models are projected over increasingly long horizonsHH, a consequence of granularity choice that prior aggregation studies have not isolated\.

In the broader forecasting benchmarking literature, the M4 Competition\(Makridakis et al\.,[2020](https://arxiv.org/html/2607.05450#bib.bib16)\)established that hybrid statistical–neural methods can outperform pure approaches, but evaluated models at a fixed temporal resolution per series without systematically varying the granularity as a controlled experimental factor\.Makridakis et al\. \([2018](https://arxiv.org/html/2607.05450#bib.bib15)\)raised early concerns about the gap between statistical and machine learning methods, noting that ML models often underperform classical methods when sample sizes are limited — a finding that aligns with our observation of deep learning data hunger at coarse granularities\.

Recent advances in deep learning for time series have introduced architectures that address multi\-horizon forecasting through fundamentally different mechanisms\.Salinas et al\. \([2020](https://arxiv.org/html/2607.05450#bib.bib22)\)proposed DeepAR, an autoregressive recurrent network that generates probabilistic forecasts, explicitly modeling the uncertainty that compounds over recursive horizons\.Lim et al\. \([2021](https://arxiv.org/html/2607.05450#bib.bib14)\)introduced the Temporal Fusion Transformer \(TFT\), which uses attention mechanisms to weight temporal dependencies at multiple scales without recursive state propagation\. Notably,Zeng et al\. \([2023](https://arxiv.org/html/2607.05450#bib.bib26)\)demonstrated that simple linear models \(DLinear\) can match or exceed Transformer\-based architectures for long\-term forecasting, consistent with our finding that Linear Regression exhibits stability across all granularities\. Our benchmark complements this body of work by demonstrating that the choice of temporal grain — a design decision upstream of model selection — can exceed the effect of architectural sophistication on out\-of\-sample accuracy\.

## 5Algorithmic Paradigms and Architectures

Forecasting models span distinct computational paradigms, differing in their parameter complexity, linearity assumptions, and requirements for manual feature engineering\.

### 5\.1Classical Box\-Jenkins and Gradient Tree Boosting

The Box\-Jenkins methodology\(Box and Jenkins,[1976](https://arxiv.org/html/2607.05450#bib.bib6)\)remains the standard for linear statistical forecasting\. The Seasonal Autoregressive Integrated Moving Average with Exogenous Regressors \(SARIMAX\) model assumes that a seriesyty\_\{t\}can be explained by its own lags, lags of the stochastic errors, and exogenous variablesXtX\_\{t\}:

ϕp​\(L\)​ΦP​\(Ls\)​\(1−L\)d​\(1−Ls\)D​yt=β′​Xt\+θq​\(L\)​ΘQ​\(Ls\)​ϵt\\phi\_\{p\}\(L\)\\Phi\_\{P\}\(L^\{s\}\)\(1\-L\)^\{d\}\(1\-L^\{s\}\)^\{D\}y\_\{t\}=\\beta^\{\\prime\}X\_\{t\}\+\\theta\_\{q\}\(L\)\\Theta\_\{Q\}\(L^\{s\}\)\\epsilon\_\{t\}\(8\)whereLLis the lag operator,ssis the seasonal period, andϵt\\epsilon\_\{t\}is a white\-noise process\. While statistically efficient and parsimonious under linearity, SARIMAX’s performance degrades in the presence of complex non\-linear dynamics or structural breaks\.

In the machine\-learning domain, the Extreme Gradient Boosting \(XGBoost\) algorithm\(Chen and Guestrin,[2016](https://arxiv.org/html/2607.05450#bib.bib7)\)builds ensembles of additive decision trees to minimize a regularized objective function:

ℒ\(t\)=∑i=1nl​\(yi,y^i\(t−1\)\+ft​\(xi\)\)\+Ω​\(ft\)\\mathcal\{L\}^\{\(t\)\}=\\sum\_\{i=1\}^\{n\}l\\left\(y\_\{i\},\\hat\{y\}\_\{i\}^\{\(t\-1\)\}\+f\_\{t\}\(x\_\{i\}\)\\right\)\+\\Omega\(f\_\{t\}\)\(9\)where the regularization termΩ​\(ft\)=γ​T\+12​λ​∑j=1Twj2\\Omega\(f\_\{t\}\)=\\gamma T\+\\frac\{1\}\{2\}\\lambda\\sum\_\{j=1\}^\{T\}w\_\{j\}^\{2\}penalizes tree complexity to prevent overfitting\. Because XGBoost is inherently non\-temporal, applying it to time\-series forecasting requires the manual construction of lag variables \(tabularization\) and is sensitive to scaling and hyperparameter tuning\(Makridakis et al\.,[2018](https://arxiv.org/html/2607.05450#bib.bib15); Bergstra and Bengio,[2012](https://arxiv.org/html/2607.05450#bib.bib4)\)\.

### 5\.2Deep Learning Architectures: LSTM and N\-BEATS

Deep learning architectures automate feature extraction for complex non\-linear dynamics\. The Long Short\-Term Memory \(LSTM\) network\(Hochreiter and Schmidhuber,[1997](https://arxiv.org/html/2607.05450#bib.bib10)\)handles long\-term dependencies by introducing a cell stateCtC\_\{t\}regulated by forget \(ftf\_\{t\}\), input \(iti\_\{t\}\), and output \(oto\_\{t\}\) gates:

ft=σ​\(Wf​xt\+Uf​ht−1\+bf\)f\_\{t\}=\\sigma\\left\(W\_\{f\}x\_\{t\}\+U\_\{f\}h\_\{t\-1\}\+b\_\{f\}\\right\)\(10\)it=σ​\(Wi​xt\+Ui​ht−1\+bi\)i\_\{t\}=\\sigma\\left\(W\_\{i\}x\_\{t\}\+U\_\{i\}h\_\{t\-1\}\+b\_\{i\}\\right\)\(11\)ot=σ​\(Wo​xt\+Uo​ht−1\+bo\)o\_\{t\}=\\sigma\\left\(W\_\{o\}x\_\{t\}\+U\_\{o\}h\_\{t\-1\}\+b\_\{o\}\\right\)\(12\)C~t=tanh⁡\(Wc​xt\+Uc​ht−1\+bc\)\\tilde\{C\}\_\{t\}=\\tanh\\left\(W\_\{c\}x\_\{t\}\+U\_\{c\}h\_\{t\-1\}\+b\_\{c\}\\right\)\(13\)Ct=ft⊙Ct−1\+it⊙C~tC\_\{t\}=f\_\{t\}\\odot C\_\{t\-1\}\+i\_\{t\}\\odot\\tilde\{C\}\_\{t\}\(14\)ht=ot⊙tanh⁡\(Ct\)h\_\{t\}=o\_\{t\}\\odot\\tanh\(C\_\{t\}\)\(15\)where⊙\\odotdenotes element\-wise multiplication\. LSTMs are highly parameter\-dense and suffer from data hunger, requiring large sample sizesNNto achieve generalization out\-of\-sample\.

In contrast, the N\-BEATS architecture\(Oreshkin et al\.,[2020](https://arxiv.org/html/2607.05450#bib.bib19)\)is a deep feed\-forward neural network that avoids recurrent or convolutional operations\. It relies on a stack of fully\-connected blocks linked via forward and backward residual connections\. N\-BEATS enforces interpretability by decomposing forecasts into trend and seasonal components using dedicated basis functions\. The implementation of such models is facilitated by modern Python libraries like Darts\(Herzen et al\.,[2022](https://arxiv.org/html/2607.05450#bib.bib9)\)\.

## 6Mathematical Framework and Evaluation Metrics

Pointwise metrics, such as Mean Absolute Error \(M​A​EMAE\) and Root Mean Square Error \(R​M​S​ERMSE\), evaluate the accuracy of individual steps\(Hyndman and Koehler,[2006](https://arxiv.org/html/2607.05450#bib.bib12)\)\. However, for cumulative budgeting and volume planning, the net deviation over the entire planning horizon is the primary operational constraint\. Letyty\_\{t\}be the observed value andy^t\\hat\{y\}\_\{t\}be the forecasted value\. For a test periodTT, we define theCumulative Forecast Error \(CFE\)as:

C​F​E=∑t∈Ty^t−∑t∈TytCFE=\\sum\_\{t\\in T\}\\hat\{y\}\_\{t\}\-\\sum\_\{t\\in T\}y\_\{t\}\(16\)
To capture the absolute magnitude of this planning gap, we define theTotal Absolute Forecast Error \(TAFE\):

T​A​F​E=\|∑t∈Ty^t−∑t∈Tyt\|=\|C​F​E\|TAFE=\\left\|\\sum\_\{t\\in T\}\\hat\{y\}\_\{t\}\-\\sum\_\{t\\in T\}y\_\{t\}\\right\|=\|CFE\|\(17\)
To compare forecasting performance across different scales and temporal granularities, we define theTotal Percentage Forecast Error \(TPFE\):

T​P​F​E=\|∑t∈Ty^t−∑t∈Tyt\|∑t∈Tyt×100%TPFE=\\frac\{\\left\|\\sum\_\{t\\in T\}\\hat\{y\}\_\{t\}\-\\sum\_\{t\\in T\}y\_\{t\}\\right\|\}\{\\sum\_\{t\\in T\}y\_\{t\}\}\\times 100\\%\(18\)
We contrast these cumulative metrics with the classical coefficient of determination \(R2R^\{2\}\) evaluated out\-of\-sample:

R2=1−∑t∈T\(yt−y^t\)2∑t∈T\(yt−y¯\)2R^\{2\}=1\-\\frac\{\\sum\_\{t\\in T\}\(y\_\{t\}\-\\hat\{y\}\_\{t\}\)^\{2\}\}\{\\sum\_\{t\\in T\}\(y\_\{t\}\-\\bar\{y\}\)^\{2\}\}\(19\)
### 6\.1Consensus\-Dissensus Diagnostic

Pointwise metrics and cumulative metrics may respond differently to changes in temporal granularity\. For a given model, letΔpw\\Delta\_\{\\text\{pw\}\}denote the directional change of pointwise metrics \(RMSE, MAE\) when moving from a coarser to a finer granularity, and letΔTPFE\\Delta\_\{\\text\{TPFE\}\}denote the corresponding directional change of the cumulative TPFE\. We classify a model–granularity pair as exhibitingconsensuswhenΔpw\\Delta\_\{\\text\{pw\}\}andΔTPFE\\Delta\_\{\\text\{TPFE\}\}move in the same direction \(both improve or both degrade\), anddissensuswhen they diverge—specifically, when pointwise metrics improve or remain stable while cumulative TPFE degrades\. Dissensus signals that the model contains systematic forecasting bias that compounds over the planning horizonHH, a pattern that pointwise metrics, by averaging over individual steps, are unable to detect\. This diagnostic is applied post\-hoc to the benchmarking results and does not require additional model runs; it requires only that evaluation includes both pointwise and cumulative metrics across at least two granularities\.

## 7Empirical Evaluation

We evaluate 10 forecasting models using a 13\-year public procurement micro\-data dataset from the Portuguese contracts registry \(Portal BASE\), filtered by CPV Division 72 \(IT Services\), spanning 1 January 2012 to 31 December 2024\. The models are:

1. 1\.Naïve Baselines: Persistence, Drift, Rolling Mean — included as recursive and non\-recursive reference controls\.
2. 2\.Statistical/Autoregressive: Linear Regression, ARIMAX, SARIMAX, Holt\-Winters \(ARIMAX and SARIMAX incorporate overall public contract value as an exogenous covariate\)\.
3. 3\.Machine Learning: XGBoost \(with lag structures\)\.
4. 4\.Deep Learning: PyTorch LSTM, Darts N\-BEATS\.

The naïve baselines deserve explicit inclusion rather than relegation to sanity checks: Persistence is itself a first\-order recursive model \(y^t\+h=yT\\hat\{y\}\_\{t\+h\}=y\_\{T\}\), and as shown below, it exposes the most striking in\-sample/out\-of\-sample divergence of any model in the benchmark \(TrainR2=0\.78R^\{2\}=0\.78with Test TPFE of23\.89%23\.89\\%at the Daily grain\)\. They therefore serve as critical controls for the recursive\-error\-propagation thesis, not merely as lower bounds on accuracy\.

We resample the series to six granularities: Annual \(N=13N=13,H=1H=1\), Quarterly \(N=52N=52,H=4H=4\), Monthly \(N=156N=156,H=12H=12\), Bi\-Weekly \(N=341N=341,H=26H=26\), Weekly \(N=680N=680,H=52H=52\), and Daily \(N=4754N=4754,H=365H=365\)\. All parameterized models are re\-optimized per fold using an expanding\-window backtesting scheme over 8 folds\(see Hyndman and Athanasopoulos,[2021](https://arxiv.org/html/2607.05450#bib.bib11), ch\. 5 for backtesting methodology\)\. Reported metrics are means across the 8 folds\. Hyperparameters for the LSTM, N\-BEATS, and XGBoost models are selected by grid search on the first fold and held fixed across granularities to avoid confounding the granularity effect with per\-grain tuning\. Random seeds are fixed for reproducibility \(LSTM seed = 42; N\-BEATS uses Darts defaults\)\. Deep\-learning results are therefore single\-seed point estimates; the directional patterns are robust but the precise magnitudes \(notably the LSTM Daily TPFE\) should not be interpreted as optimised lower bounds\.

Several models exhibit negative TrainR2R^\{2\}at coarse granularities \(e\.g\., LSTM Annual−2\.28\-2\.28, Holt\-Winters Annual−6\.46\-6\.46, Persistence Monthly−0\.23\-0\.23\)\. These are not estimation artefacts: whenNNis too small to resolve a model’s parameter space, the fitted model can perform worse than the sample mean, yieldingR2<0R^\{2\}<0by definition\. Negative in\-sample fit at coarse grains is itself a diagnostic of model–data mismatch, and is reported without correction\.

## 8Results

The comparative backtesting results are reported in TableLABEL:tab:comparison\_results\.

Table 1:Comparative Benchmarking Results across Granularities \(8\-Fold Backtesting averages\)\. TAFE and TPFE are cumulative errors over the planning horizon; “N/A” denotes that the model could not be fit at the given granularity \(e\.g\., AnnualN=13N=13is insufficient to estimate seasonal/state components\) or that the metric is mathematically undefined \(R2R^\{2\}atH=1H=1\)\.Boldmarks the best TPFE, best TestR2R^\{2\}, and worst TestR2R^\{2\}across all model–granularity combinations\.ModelGranularityTrainR2R^\{2\}TestR2R^\{2\}Test RMSETest MAETest TAFETest TPFENaïve BaselinesPersistenceAnnual0\.2970N/A75\.84M75\.84M75\.84M18\.12%PersistenceMonthly\-0\.2322\-6\.026314\.50M12\.52M88\.38M26\.46%PersistenceQuarterly0\.1786\-4\.409825\.82M23\.43M70\.29M17\.16%PersistenceBi\-Weekly\-0\.3990\-3\.31899\.38M7\.88M159\.72M44\.68%PersistenceWeekly\-0\.5099\-1\.25275\.06M3\.59M83\.56M24\.45%PersistenceDaily0\.7841\-1\.1176720\.48K509\.18K82\.86M23\.89%DriftAnnual0\.3939N/A67\.13M67\.13M67\.13M15\.98%DriftMonthly\-0\.5152\-7\.580115\.26M13\.52M103\.54M30\.63%DriftQuarterly0\.1957\-4\.148025\.64M23\.40M69\.52M17\.34%DriftBi\-Weekly\-0\.7370\-4\.17949\.83M8\.43M174\.63M48\.91%DriftWeekly\-0\.1789\-1\.48745\.10M3\.74M78\.79M23\.37%DriftDaily\-0\.1843\-1\.3268726\.68K530\.10K78\.21M22\.83%Rolling MeanAnnual\-0\.9975N/A110\.39M110\.39M110\.39M25\.29%Rolling MeanMonthly0\.0175\-2\.610013\.20M10\.69M85\.06M22\.44%Rolling MeanQuarterly0\.3556\-5\.147226\.67M23\.89M76\.58M18\.09%Rolling MeanBi\-Weekly\-0\.0074\-1\.50357\.98M6\.23M107\.94M30\.98%Rolling MeanWeekly\-0\.1062\-1\.00075\.14M3\.72M113\.10M31\.29%Rolling MeanDaily0\.6220\-1\.2479759\.26K573\.44K118\.19M31\.08%Statistical / AutoregressiveLinear RegressionAnnual0\.6606N/A72\.43M72\.43M72\.43M16\.96%Linear RegressionMonthly0\.3380\-0\.286611\.55M8\.42M69\.63M16\.09%Linear RegressionQuarterly0\.5286\-3\.148624\.65M21\.55M70\.46M16\.32%Linear RegressionBi\-Weekly0\.2318\-0\.20476\.78M4\.67M70\.59M16\.36%Linear RegressionWeekly0\.1490\-0\.10194\.53M2\.87M72\.07M16\.78%Linear RegressionDaily0\.1491\-0\.1007647\.46K408\.68K70\.63M16\.32%ARIMAXAnnual0\.8620N/A52\.90M52\.90M52\.90M13\.55%ARIMAXMonthly0\.3077\-0\.665912\.56M9\.50M87\.13M20\.65%ARIMAXQuarterly0\.4645\-4\.292625\.32M22\.54M72\.69M18\.26%ARIMAXBi\-Weekly0\.1311\-0\.68027\.71M5\.31M120\.72M29\.85%ARIMAXWeekly0\.0057\-0\.38965\.00M3\.19M130\.88M32\.45%ARIMAXDaily0\.1332\-0\.3912715\.84K456\.11K131\.28M32\.63%SARIMAXAnnualN/AN/AN/AN/AN/AN/ASARIMAXMonthly0\.1897\-0\.639511\.95M9\.08M80\.26M19\.20%SARIMAXQuarterly0\.3606\-4\.319327\.11M23\.84M81\.40M18\.95%SARIMAXBi\-Weekly\-0\.2005\-0\.69317\.57M5\.78M103\.73M26\.46%SARIMAXWeekly\-0\.3541\-0\.43914\.80M3\.22M76\.31M24\.60%SARIMAXDaily0\.1094\-0\.4486689\.78K455\.80K84\.44M26\.05%Holt\-WintersAnnual\-6\.4565N/A126\.20M126\.20M126\.20M37\.83%Holt\-WintersMonthly0\.3455\-0\.668512\.62M9\.97M80\.13M19\.01%Holt\-WintersQuarterly0\.3645\-6\.294031\.45M27\.36M96\.73M20\.82%Holt\-WintersBi\-Weekly0\.1787\-0\.79108\.06M5\.77M93\.23M22\.59%Holt\-WintersWeekly\-0\.0835\-3\.93698\.69M6\.87M286\.19M65\.45%Holt\-WintersDaily0\.2648\-151\.00005\.34M4\.71M1706\.53M425\.85%Machine LearningXGBoostAnnual0\.5864N/A108\.29M108\.29M108\.29M26\.88%XGBoostMonthly0\.6627\-1\.069814\.13M11\.14M81\.60M20\.95%XGBoostQuarterly0\.6656\-7\.642734\.84M30\.82M116\.27M29\.04%XGBoostBi\-Weekly0\.5616\-0\.70277\.71M5\.43M96\.38M25\.30%XGBoostWeekly0\.4644\-0\.55085\.35M3\.61M100\.34M25\.83%XGBoostDaily0\.70370\.3005493\.26K271\.46K62\.03M17\.15%Deep LearningLSTM PyTorchAnnual\-2\.2790N/A170\.65M170\.65M170\.65M41\.50%LSTM PyTorchMonthly0\.2388\-0\.777512\.71M9\.68M76\.49M19\.66%LSTM PyTorchQuarterly0\.2173\-6\.273933\.82M29\.89M112\.92M27\.12%LSTM PyTorchBi\-Weekly\-0\.0657\-0\.93368\.51M6\.17M150\.11M35\.94%LSTM PyTorchWeekly0\.0395\-0\.44835\.19M3\.36M150\.73M34\.59%LSTM PyTorchDaily0\.67190\.6627339\.51K186\.35K17\.13M4\.35%N\-BEATSAnnualN/AN/AN/AN/AN/AN/AN\-BEATSMonthly0\.5472\-0\.694512\.04M9\.40M59\.05M16\.00%N\-BEATSQuarterly\-0\.2862\-2\.810038\.73M31\.94M94\.98M19\.80%N\-BEATSBi\-Weekly0\.5840\-0\.48057\.29M5\.68M45\.65M11\.30%N\-BEATSWeekly0\.5185\-0\.35295\.03M3\.63M56\.48M12\.17%N\-BEATSDaily0\.7677\-0\.2966691\.66K482\.20K41\.12M10\.77%Table 1:Comparative Benchmarking Results \(continued\)To visualize the performance trajectories and allow direct comparison of absolute performance magnitudes across different models, Table[2](https://arxiv.org/html/2607.05450#S8.T2)presents the trajectories normalized globally across all models and granularities\. Due to the massive amplitude of the raw metrics \(spanning several orders of magnitude\), a logarithmic scaling is applied: for errors \(RMSE, MAE, TPFE\), values arelog10\\log\_\{10\}\-scaled and mapped to\[0,1\]\[0,1\]where the global minimum corresponds to0\(far\-left, best, green circles\) and the global maximum corresponds to11\(far\-right, worst, red circles\)\. ForR2R^\{2\}, the log\-distance to 1 \(log10⁡\(1−R2\)\\log\_\{10\}\(1\-R^\{2\}\)\) is mapped so that the best observed global fit is at11\(far\-right, green\) and the worst at0\(far\-left, red\)\. This inter\-model dashboard visually highlights the absolute dominance of deep learning models at fine resolutions, as well as the magnitude of statistical model collapses\.

Table 2:Global Logarithmic Performance Trajectories \(Vertical sparklines normalized globally across all 10 models and 6 granularities on a logarithmic scale; TestR2R^\{2\}is clipped at a minimum of−1\.0\-1\.0, and TPFE is clipped at a maximum of100%100\\%; Annual→\\rightarrowQuarterly→\\rightarrowMonthly→\\rightarrowBi\-Weekly→\\rightarrowWeekly→\\rightarrowDaily from top to bottom; green circles indicate global best performance, red indicate global worst; for errors, left is best/lowest; forR2R^\{2\}, right is best/highest\)
## 9Discussion

Several key statistical insights emerge from the empirical benchmarking:

1. 1\.The In\-Sample Fit Illusion: Finer temporal resolution \(Daily\) yields high trainingR2R^\{2\}values that bear no relationship to out\-of\-sample accuracy\. A clear example is the naïve Persistence baseline: at the Daily grain it attains TrainR2=0\.7841R^\{2\}=0\.7841— comparable to the deep models — yet its TestR2R^\{2\}falls to−1\.12\-1\.12and its TPFE to23\.89%23\.89\\%\. The model captured the high\-frequency autocorrelation while acquiring no out\-of\-sample forecasting skill\. The effect is amplified for parameterized seasonal models: Holt\-Winters at the Daily grain produces a TestR2R^\{2\}of−151\-151and a TPFE of425\.85%425\.85\\%, the worst outcome in the benchmark, because its seasonal state is recursively re\-fed overH=365H=365steps\. This constitutes evidence that high in\-sample fit at fine grains is a structural predictor of out\-of\-sample degradation, not merely a heuristic warning\.
2. 2\.Autoregressive vs\. Trend Stability: Linear Regression remains stable across the granularity paradox, maintaining TPFE in the narrow band16\.0916\.09–16\.96%16\.96\\%across all*six*granularities \(Annual through Daily, a0\.870\.87pp spread\)\. This occurs because the model projects predictions directly as a function of time, without the recursive feedback loop characteristic of autoregressive and state\-updating models\. The six\-grain flatness — confirmed across two intermediate granularities \(Quarterly, Bi\-Weekly\) that were absent from prior work — constitutes empirical evidence that the paradox is driven by recursive feedback topology, not by model complexity or sample size\.
3. 3\.The Non\-Monotonic LSTM Threshold \(U\-Shaped Error Curve\): The extended six\-grain benchmark reveals that the LSTM’s “data hunger threshold” is not a monotonic boundary but a U\-shaped error curve\. The LSTM’s TPFE worsens from Monthly \(19\.66%19\.66\\%\) through Quarterly \(27\.12%27\.12\\%\) to Bi\-Weekly \(35\.94%35\.94\\%\), then partially recovers at Weekly \(34\.59%34\.59\\%\) before decreasing to4\.35%4\.35\\%at Daily\. The intermediate granularities form an adverse operating region:NNis larger than Monthly, but the recursive horizonHHgrows faster than the data can compensate, so error propagation dominates\. Only at Daily \(N=4754N=4754\) does the sample size finally resolve the LSTM’s high\-dimensional parameter space, yielding a TestR2R^\{2\}of0\.660\.66\. This non\-monotonicity means that practitioners who choose an intermediate grain \(Bi\-Weekly, Weekly\) thinking it “balances” data and horizon are in fact selecting the*worst*operating point for recursive deep models\.
4. 4\.N\-BEATS Robustness vs\. LSTM Sensitivity: N\-BEATS shows a flatter TPFE profile \(10\.7710\.77–19\.80%19\.80\\%across all fine grains\) and already achieves10\.77%10\.77\\%at Daily and11\.30%11\.30\\%at Bi\-Weekly — competitive with the LSTM’s Daily result, but without the intermediate\-grain degradation\. Its trend\-block inductive bias appears to resolve the multi\-step dependency without requiring the full Daily sample, whereas the LSTM’s recurrent hidden state accumulates error linearly withHHuntil the data volume is large enough to compensate\.
5. 5\.Metric Consensus vs\. Cumulative Dissensus: Because the objective of temporal aggregation in this context is cumulative planning and budgeting, TPFE serves as the step\-agnostic, goal\-dependent primary evaluation criterion\. Analyzing the consensus between pointwise metrics \(RMSE, MAE,R2R^\{2\}\) and TPFE reveals the nature of a model’s error propagation\. For non\-recursive models \(Linear Regression\), there is perfect consensus: both pointwise and cumulative metrics agree on stability\. In contrast, recursive autoregressive models \(e\.g\., SARIMAX, ARIMAX\) exhibit a clear dissensus: pointwise metrics \(RMSE, MAE\) improve or stabilize at finer grains \(due to sample size expansion reducing parameter variance\), yet cumulative TPFE degrades\. This dissensus exposes systematic forecasting bias that propagates recursively over the daily planning horizon \(H=365H=365\), which pointwise metrics hide\. Conversely, unbiased high\-variance models benefit from error cancellation over long planning periods, where positive and negative daily fluctuations cancel out, yielding a low TPFE despite high high\-frequency pointwise variance \(as seen in the daily LSTM\)\.

Table 3:Consensus and Dissensus Classifications between Pointwise Fit and Cumulative Budgeting Error \(Daily vs\. Monthly resolution\)
## 10Conclusion

The choice of temporal grain is not merely a matter of data representation, but a design decision that controls error dynamics\. We summarize three methodological conclusions that emerge from the empirical results\.

Pointwise metrics are insufficient for cross\-granularity evaluation\.RMSE and MAE decrease monotonically as temporal resolution increases—Holt\-Winters Daily records a Test RMSE of 5\.34M versus 12\.62M at Monthly—yet its cumulative TPFE rises from19\.01%19\.01\\%to425\.85%425\.85\\%\. This occurs because RMSE and MAE evaluate individual step accuracy within the observation window of each granularity, not the accumulated deviation over the planning horizon\. These metrics remain useful for diagnosing point\-by\-point prediction quality at a fixed granularity, but they are uninformative—and potentially misleading—when used to compare model adequacy across granularities or to assess cumulative planning accuracy\.

Forecasting without a goal\-dependent cumulative metric creates distorted evaluation\.Across the benchmark, models evaluated exclusively on in\-sampleR2R^\{2\}, RMSE, and MAE would rank Persistence at Daily as a competitive forecaster \(TrainR2=0\.78R^\{2\}=0\.78, low RMSE\)\. Only when evaluated against TPFE—a metric anchored to the actual planning objective—does the model’s inadequacy become visible\. This finding generalizes beyond our specific dataset: any forecasting exercise that lacks a metric tied to the downstream decision variable \(cumulative budget deviation, total demand, aggregate volume\) risks producing models that perform well on diagnostic criteria while failing the task they were built for\.

In\-sample\-only evaluation creates a perverse incentive structure\.In\-sample fit diagnostics systematically reward two choices: finer temporal disaggregation \(which inflatesNNand improves training statistics\) and greater model complexity \(which reduces training residuals\)\. Both choices increase the recursive forecasting horizonHHor the parameter space, amplifying out\-of\-sample error propagation\. A workflow that monitors only in\-sample metrics therefore incentivizes precisely the design decisions that degrade forecasting performance\. This is not a theoretical concern: the benchmark shows that the model–granularity combination with the highest trainingR2R^\{2\}\(Persistence at Daily,0\.780\.78\) is outperformed out\-of\-sample by Linear Regression \(16\.32%16\.32\\%TPFE vs\.23\.89%23\.89\\%\), a model whose trainingR2R^\{2\}\(0\.150\.15\) would appear inferior by in\-sample standards\.

From a practical standpoint, finer granularities should be selected for high\-capacity models \(such as LSTMs\) only when the resulting sample size crosses the threshold required to resolve the architecture’s parameter space\. For low\-data regimes, non\-recursive trend models or parsimonious monthly statistical models remain preferable\. In all cases, evaluation should include at least one goal\-dependent cumulative metric aligned with the planning objective\.

## 11Limitations and Reproducibility

The benchmark is restricted to a single series \(CPV Division 72, IT Services\), so the absolute TPFE magnitudes are dataset\-specific\. The directional findings — recursive degradation of autoregressive models at fine grains, U\-shaped LSTM error curve, Linear Regression scale\-invariance — are structural properties of the model–granularity interaction and should generalise, but confirming this across additional CPV divisions is left for future work\. Confidence intervals from multi\-seed runs and Diebold–Mariano significance tests between models are also deferred: the present results are single\-seed point estimates and should be read as a structural comparison rather than a ranking\.

The dataset is derived from the Portuguese public contracts registry \(Portal BASE\), which is publicly accessible at[https://www\.base\.gov\.pt](https://www.base.gov.pt/)\. The CPV Division 72 filter and the resampling protocol are described in Section[7](https://arxiv.org/html/2607.05450#S7)\(Empirical Evaluation\); full pre\-processing code, model configurations, and the backtesting pipeline are available from the author on request\. Software stack: Python 3\.11, PyTorch 2\.x, Darts, statsmodels, scikit\-learn, XGBoost\.

## References

- Athanasopoulos et al\. \[2017\]George Athanasopoulos, Rob J\. Hyndman, Nikolaos Kourentzes, and Fotios Petropoulos\.Forecasting with temporal hierarchies\.*European Journal of Operational Research*, 262\(1\):60–74, 2017\.
- Athanasopoulos et al\. \[2024\]George Athanasopoulos, Rob J\. Hyndman, Nikolaos Kourentzes, and Anastasios Panagiotelis\.Forecast reconciliation: A review\.*International Journal of Forecasting*, 40\(2\):430–456, 2024\.
- Ben Taieb et al\. \[2012\]Souhaib Ben Taieb, Gianluca Bontempi, Amir F\. Atiya, and Antti Sorjamaa\.A review and comparison of strategies for multi\-step ahead time series forecasting based on the NN5 forecasting competition\.*Expert Systems with Applications*, 39\(8\):7067–7083, 2012\.
- Bergstra and Bengio \[2012\]James Bergstra and Yoshua Bengio\.Random search for hyper\-parameter optimization\.*Journal of Machine Learning Research*, 13:281–305, 2012\.
- Bontempi et al\. \[1999\]Gianluca Bontempi, Mauro Birattari, and Hugues Bersini\.Lazy learning for local modelling and control design\.*International Journal of Control*, 72\(7\-8\):643–658, 1999\.
- Box and Jenkins \[1976\]George E\. P\. Box and Gwilym M\. Jenkins\.*Time Series Analysis: Forecasting and Control*\.Holden\-Day, San Francisco, revised edition, 1976\.
- Chen and Guestrin \[2016\]Tianqi Chen and Carlos Guestrin\.XGBoost: A scalable tree boosting system\.In*Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining*, pages 785–794, 2016\.
- Hamilton \[1994\]James D\. Hamilton\.*Time Series Analysis*\.Princeton University Press, 1994\.
- Herzen et al\. \[2022\]Julien Herzen, Francesco Lässig, Samuele Giuliano Piazzetta, Thomas Neuer, Léo Tafti, Guillaume Raille, Tomas Van Pottelbergh, Marek Pasieka, Andrzej Skrodzki, and Nicolas Huguenin\.Darts: User\-friendly modern machine learning for time series\.*Journal of Machine Learning Research*, 23:1–6, 2022\.
- Hochreiter and Schmidhuber \[1997\]Sepp Hochreiter and Jürgen Schmidhuber\.Long short\-term memory\.*Neural Computation*, 9\(8\):1735–1780, 1997\.
- Hyndman and Athanasopoulos \[2021\]Rob J\. Hyndman and George Athanasopoulos\.*Forecasting: Principles and Practice*\.OTexts, Melbourne, Australia, 3rd ed\. edition, 2021\.
- Hyndman and Koehler \[2006\]Rob J\. Hyndman and Anne B\. Koehler\.Another look at measures of forecast accuracy\.*International Journal of Forecasting*, 22\(4\):679–688, 2006\.
- Kourentzes et al\. \[2014\]Nikolaos Kourentzes, Fotios Petropoulos, and Juan R\. Trapero\.Improving forecasting via multiple temporal aggregation\.*International Journal of Forecasting*, 30\(3\):464–476, 2014\.
- Lim et al\. \[2021\]Bryan Lim, Sercan O\. Arık, Nicolas Loeff, and Tomas Pfister\.Temporal fusion transformers for interpretable multi\-horizon time series forecasting\.*International Journal of Forecasting*, 37\(4\):1748–1764, 2021\.
- Makridakis et al\. \[2018\]Spyros Makridakis, Evangelos Spiliotis, and Vassilios Assimakopoulos\.Statistical and Machine Learning forecasting methods: Concerns and ways forward\.*PLOS ONE*, 13\(3\):e0194889, 2018\.
- Makridakis et al\. \[2020\]Spyros Makridakis, Evangelos Spiliotis, and Vassilios Assimakopoulos\.The M4 competition: 100,000 time series and 61 forecasting methods\.*International Journal of Forecasting*, 36\(1\):54–74, 2020\.
- Marcellino et al\. \[2006\]Massimiliano Marcellino, James H\. Stock, and Mark W\. Watson\.A comparison of direct and iterated multistep AR methods for forecasting macroeconomic time series\.*Journal of Econometrics*, 135\(1\-2\):499–526, 2006\.
- Nikolopoulos et al\. \[2011\]Konstantinos Nikolopoulos, Aris A\. Syntetos, John E\. Boylan, Fotios Petropoulos, and Vassilios Assimakopoulos\.An aggregate–disaggregate intermittent demand approach \(ADIDA\) to forecasting\.*Journal of the Operational Research Society*, 62\(3\):544–554, 2011\.
- Oreshkin et al\. \[2020\]Boris N\. Oreshkin, Dmitri Carpov, Nicolas Chapados, and Yoshua Bengio\.N\-BEATS: Neural basis expansion analysis for interpretable time series forecasting\.In*Proceedings of the 8th International Conference on Learning Representations \(ICLR\)*, 2020\.
- Petropoulos and Kourentzes \[2015\]Fotios Petropoulos and Nikolaos Kourentzes\.Forecast combinations for intermittent demand\.*Journal of the Operational Research Society*, 66\(6\):914–924, 2015\.
- Rostami\-Tabar et al\. \[2013\]Bahman Rostami\-Tabar, M\. Zied Babai, Aris Syntetos, and Yves Ducq\.Demand forecasting by temporal aggregation\.*Naval Research Logistics*, 60\(6\):479–498, 2013\.
- Salinas et al\. \[2020\]David Salinas, Valentin Flunkert, Jan Gasthaus, and Tim Januschowski\.DeepAR: Probabilistic forecasting with autoregressive recurrent networks\.*International Journal of Forecasting*, 36\(3\):1181–1191, 2020\.
- Silvestrini and Veredas \[2008\]Andrea Silvestrini and David Veredas\.Temporal aggregation of univariate and multivariate time series models: a survey\.*Journal of Economic Surveys*, 22\(3\):458–497, 2008\.
- Tiao and Wei \[1976\]George C\. Tiao and William W\. S\. Wei\.Effect of temporal aggregation on the dynamic relationship of two time series variables\.*Biometrika*, 63\(3\):513–523, 1976\.
- Wei \[2006\]William W\. S\. Wei\.*Time Series Analysis: Univariate and Multivariate Methods*\.Pearson Addison Wesley, 2nd edition, 2006\.
- Zeng et al\. \[2023\]Ailing Zeng, Muxi Chen, Lei Zhang, and Qiang Xu\.Are transformers effective for time series forecasting?In*Proceedings of the AAAI Conference on Artificial Intelligence*, volume 37, pages 11121–11128, 2023\.