Quantile Estimation via Residual Simulation (QRS)

• The paper introduces QRS as a method that transforms one-step-ahead point forecasts into full conditional quantile forecasts without assuming a full parametric distribution.
• The approach uses simulated empirical residuals from log-transformed data to generate a robust predictive distribution while mitigating negative forecast issues.
• Empirical results in Bitcoin volatility forecasting show that QRS on linear models achieves low CRPS and avoids quantile-crossing, matching more complex stacking methods.

Quantile Estimation through Residual Simulation (QRS) is a probabilistic forecasting method for converting one-step-ahead point forecasts into conditional quantile forecasts without building a full parametric distribution of the target variable. In "Probabilistic Forecasting Cryptocurrencies Volatility: From Point to Quantile Forecasts" (Dudek et al., 21 Aug 2025), QRS is formulated for a univariate volatility series, specifically daily realized variance $y_t$, or its log-transformation $z_t=\ln y_t$. The method approximates the predictive distribution at forecast origin $\tau$ by combining the current point forecast with an empirical distribution of in-sample residuals, thereby producing a full set of quantile forecasts from a wide range of base models, including statistical and machine learning algorithms (Dudek et al., 21 Aug 2025).

1. Definition and theoretical basis

QRS addresses the quantile estimation problem for a future time $\tau$: estimate the conditional $q$-quantile

$$Q_{q,\tau}=F^{-1}_{Y_\tau\mid\mathcal I_{\tau-1}}(q)$$

without specifying a full parametric law for $Y_\tau$ (Dudek et al., 21 Aug 2025). The method starts from a point-forecasting model that produces one-step-ahead forecasts $\hat y_t$ or, on the log scale, $\hat z_t$.

Its residual-simulation framework assumes

$Y_t = \hat Y_t + \varepsilon_t,$

with in-sample residuals

$z_t=\ln y_t$0

These residuals are treated as an empirical approximation to the future forecast-error distribution. Defining the empirical residual set

$z_t=\ln y_t$1

QRS approximates the predictive distribution at time $z_t=\ln y_t$2 as

$z_t=\ln y_t$3

The method is therefore nonparametric at the error-distribution stage and model-agnostic at the point-forecast stage. A plausible implication is that its performance depends directly on the quality of the underlying point forecast and on how informative the historical residuals are about future forecast errors.

2. Algorithmic construction

The procedure described in (Dudek et al., 21 Aug 2025) converts a point forecast into a set of simulated future values and then extracts empirical quantiles from those simulations.

First, an optional preprocessing step applies the log-transformation $z_t=\ln y_t$4. This is used to stabilize variance and enforce positivity after back-transformation. Next, a base model is fitted on historical data $z_t=\ln y_t$5 or $z_t=\ln y_t$6 to generate in-sample one-step-ahead forecasts $z_t=\ln y_t$7 for $z_t=\ln y_t$8 and the next-day forecast $z_t=\ln y_t$9.

The in-sample residuals are then computed as

$\tau$0

From these residuals, the empirical residual set $\tau$1 is formed. The paper describes three variants for generating forecast-error draws: directly reusing each $\tau$2 as a simulated error, sampling with replacement $\tau$3 times from $\tau$4, or fitting a kernel-density estimator $\tau$5 on $\tau$6 using a Gaussian kernel with cross-validated bandwidth and drawing $\tau$7 samples from $\tau$8 (Dudek et al., 21 Aug 2025).

Each simulated residual $\tau$9 is added to the current point forecast: $\tau$0 If the model is estimated on the log scale, the simulated values are generated as

$\tau$1

followed by back-transformation

$\tau$2

The recommended number of simulations depends on the residual-generation strategy. If in-sample residuals are simply recycled, then $\tau$3 equals the training-window length $\tau$4. If residuals are sampled with replacement or drawn from a fitted kernel, the recommended range is $\tau$5–$\tau$6 in order to ensure stable quantile estimates (Dudek et al., 21 Aug 2025).

3. Quantile extraction and predictive distribution

Once the simulated ensemble $\tau$7 is available, QRS estimates the empirical $\tau$8-quantile as

$\tau$9

Operationally, one sorts the simulated values in ascending order and selects the $q$0-th order statistic. If a continuous CDF $q$1 has been fitted, the quantile may also be obtained by numerical inversion: $q$2

This extraction mechanism makes QRS a direct simulation-based route from a point forecast to a predictive quantile function. The method does not estimate quantiles through separate quantile-specific models at this stage; instead, all forecast quantiles are induced from a single predictive simulation ensemble. This suggests a structural reason why QRS avoids quantile-crossing when the simulated sample is sorted into a valid empirical distribution.

4. Base models, ensembles, and stacking

QRS is explicitly described as model-agnostic. In the reported application it can be applied to residuals from statistical models such as HAR, HAR-R, ARFIMA, and GARCH, and from machine learning models such as LASSO, RR, SVR-G, SVR-L, MLP, FNM, RF, and LSTM (Dudek et al., 21 Aug 2025). The same mechanism can be used on a single base model or on an ensemble point forecast, such as the cross-model mean

$q$3

or a cross-model median.

The paper contrasts QRS with a separate framework termed probabilistic stacking. In probabilistic stacking, the vector of base point forecasts

$q$4

is optionally augmented with additional predictors, including lagged daily, weekly, and monthly realized variance, and then passed to a quantile-regression meta-model $q$5 that directly outputs

$q$6

Two specific meta-models are highlighted. Quantile Linear Regression (QLR) is defined as

$q$7

with estimation by minimizing the pinball loss $q$8. Quantile Regression Forest (QRF) is described as an extension of Random Forest that retains the full empirical distribution of $q$9 in each leaf and estimates conditional quantiles as weighted sums of leaf-node responses (Dudek et al., 21 Aug 2025).

Within this comparison, QRS functions as a residual-based post-processing method, whereas probabilistic stacking is a quantile-level meta-modeling approach. The empirical discussion in (Dudek et al., 21 Aug 2025) reports that QRS on single strong base models, especially linear models on $$Q_{q,\tau}=F^{-1}_{Y_\tau\mid\mathcal I_{\tau-1}}(q)$$0-RV, often outperforms or matches these more sophisticated stacking solutions in overall CRPS, calibration, and interval-score metrics.

5. Empirical evaluation in cryptocurrency volatility forecasting

The reported application concerns Bitcoin realized-variance forecasting. The study states that it is, to the best of the authors’ knowledge, the first in the literature to propose and systematically evaluate probabilistic forecasts of variance in cryptocurrency markets based on predictions derived from multiple base models (Dudek et al., 21 Aug 2025).

The evaluation uses several probabilistic and point-forecast proxy metrics. Continuous Ranked Probability Score (CRPS) is computed as a sum of pinball losses over a dense grid of quantiles: $$Q_{q,\tau}=F^{-1}_{Y_\tau\mid\mathcal I_{\tau-1}}(q)$$1 Calibration is assessed through relative frequency (coverage),

$$Q_{q,\tau}=F^{-1}_{Y_\tau\mid\mathcal I_{\tau-1}}(q)$$2

with nominal target $$Q_{q,\tau}=F^{-1}_{Y_\tau\mid\mathcal I_{\tau-1}}(q)$$3, and through Mean Absolute ReFr Error (MARFE),

$$Q_{q,\tau}=F^{-1}_{Y_\tau\mid\mathcal I_{\tau-1}}(q)$$4

The evaluation also includes the Winkler Score (WS) for a $$Q_{q,\tau}=F^{-1}_{Y_\tau\mid\mathcal I_{\tau-1}}(q)$$5 interval $$Q_{q,\tau}=F^{-1}_{Y_\tau\mid\mathcal I_{\tau-1}}(q)$$6, and the point-forecast proxies $$Q_{q,\tau}=F^{-1}_{Y_\tau\mid\mathcal I_{\tau-1}}(q)$$7 and $$Q_{q,\tau}=F^{-1}_{Y_\tau\mid\mathcal I_{\tau-1}}(q)$$8, both based on $$Q_{q,\tau}=F^{-1}_{Y_\tau\mid\mathcal I_{\tau-1}}(q)$$9 (Dudek et al., 21 Aug 2025).

The principal empirical findings reported for Bitcoin are summarized below.

Finding	Reported result
QRS on $Y_\tau$0-RV linear models	Lowest mean CRPS $Y_\tau$1; MARFE $Y_\tau$2
Raw-RV QRS	Negative lower quantiles sometimes occurred; wider CRPS dispersion
Probabilistic stacking	QLR-l and QRF-l mean CRPS $Y_\tau$3
Quantile-crossing	QLR in $Y_\tau$4 of days; QRF and QRS never
90% interval score	Best values around $Y_\tau$5

More specifically, QRS applied to linear base models on $Y_\tau$6-RV, including HAR-l, HAR-R-l, ARFIMA-l, RR-l, and SVR-L-l, achieved the lowest mean CRPS and competitive calibration. Probabilistic stacking via QLR-l and QRF-l produced mean CRPS in the same general range and similar interval scores, but did not substantially outperform the simplest QRS-l specifications. The best 90% interval scores were around $Y_\tau$7, reached by QRF-l, QLR-l, and some QRS-l instantiations (Dudek et al., 21 Aug 2025).

These results support a narrow but important conclusion: in the reported Bitcoin setting, residual simulation around strong point forecasts—especially linear forecasts on logarithmized volatility—was sufficient to match or exceed more elaborate quantile-level meta-models.

6. Practical considerations, limitations, and interpretation

Several implementation considerations are explicit in the source description. Base-model choice matters: QRS is recommended for models with low point-forecast error and reasonably stable, near-IID residuals. In the reported study, the best QRS partners were simple linear autoregressions on $Y_\tau$8-RV (Dudek et al., 21 Aug 2025). Data transformation also matters. Applying $Y_\tau$9 to realized variance avoids negative simulated draws and mitigates outliers, whereas on raw RV the method may generate inadmissible negative lower quantiles.

The method is computationally light. The reported runtime is approximately $\hat y_t$0 per daily date for 99 quantiles, because QRS avoids retraining. By comparison, QLR takes approximately $\hat y_t$1, described as one LP per quantile, and QRF takes approximately $\hat y_t$2, described as a single forest (Dudek et al., 21 Aug 2025). This computational profile is consistent with the architecture of QRS: once a point forecast and residual history are available, quantiles are produced by simulation and sorting rather than repeated optimization.

The paper also identifies several pitfalls. QRS inherits any systematic bias in the base model and assumes stationarity of the residual distribution. Kernel-density fitting may fail to converge at extreme $\hat y_t$3, in which case interpolation may be required. The method does not adapt to time-varying heteroscedasticity in forecast errors (Dudek et al., 21 Aug 2025). These caveats limit the circumstances under which historical residuals can reliably approximate future forecast uncertainty.

A common misconception would be to treat QRS as a full generative model for volatility. The formulation in (Dudek et al., 21 Aug 2025) does not present it that way. It is a residual-based approximation scheme for conditional quantiles built on top of an existing point forecast. Another possible misconception is that more sophisticated quantile meta-models necessarily dominate residual simulation. The reported Bitcoin results do not support that generalization: QRS on well-specified linear models operating on $\hat y_t$4-transformed realized volatility often matched or exceeded QLR and QRF under CRPS, calibration, and interval-score criteria (Dudek et al., 21 Aug 2025).

7. Position within probabilistic volatility forecasting

Within the framework of cryptocurrency volatility forecasting, QRS occupies the transition from deterministic to probabilistic prediction. The paper introduces probabilistic forecasting methods that leverage point forecasts from a wide range of base models to estimate conditional quantiles of cryptocurrency realized variance, and QRS is one of the central mechanisms for doing so (Dudek et al., 21 Aug 2025). In this setting, the method supplies a distributional layer over existing forecasters rather than replacing them.

The broader significance claimed in the source is domain-specific: traditional deterministic forecasting methods are described as inadequate for capturing the full spectrum of potential volatility outcomes in cryptocurrency markets, while probabilistic methods provide more comprehensive insight into uncertainty and risk. The empirical outcome that QRS performed particularly well when paired with linear base models on log-transformed realized volatility suggests that, in at least this application, the fidelity of residual behavior and the effect of scale transformation may be more consequential than increasing model complexity.

In that sense, QRS can be understood as a parsimonious probabilistic wrapper around point forecasts. In the reported evidence, its simplicity, model-agnosticity, and low computational cost coincide with strong empirical performance in Bitcoin realized-variance forecasting, especially relative to more elaborate stacking-based alternatives (Dudek et al., 21 Aug 2025).