Quantile Regression Averaging (QRA) Overview

• Quantile Regression Averaging (QRA) is a forecasting method that converts point forecasts into conditional quantile predictions using linear regression techniques.
• It generates prediction intervals and a full predictive distribution by combining multiple quantile levels through a model-averaging framework.
• Extensions such as LQRA, FQRA, and iQRA improve performance via regularization, dimensionality reduction, and smoothing, especially in applications like electricity price forecasting.

Quantile Regression Averaging (QRA) is a probabilistic forecasting framework that converts a collection of point forecasts into conditional quantile forecasts by treating those point forecasts as regressors in a quantile regression. For a quantile level $\tau\in(0,1)$, QRA models the conditional $\tau$-quantile as a linear or affine combination of forecast inputs, with coefficients estimated by minimizing the quantile check loss; repeating this across a grid of quantile levels yields quantile curves, prediction intervals, and an empirical representation of the predictive distribution (Janczura, 2024, Zakrzewski et al., 2024). In a broader model-averaging sense, the same logic has been extended from ensembles of point forecasts to ensembles of quantile models, factor-augmented quantile autoregressions, complete subsets of regressors, functional quantile models, and semiparametric marginal quantile components (Phella, 2020, Lee et al., 2020, Xu et al., 2023, Tu et al., 2018).

1. Formal definition and conceptual scope

In its classical forecasting form, QRA starts with $K$ individual point forecasting models producing $\hat{y}_t^{(1)},\dots,\hat{y}_t^{(K)}$. For each quantile level $\tau$, QRA specifies

$$Q_{y_t}(\tau \mid X_t) = \beta_0^{(\tau)} + \sum_{k=1}^{K} \beta_k^{(\tau)} \hat{y}_t^{(k)},$$

or, in vector form,

$Q_{y_t}(\tau \mid X_t) = X_t^\top \beta^{(\tau)},$

where $X_t=(1,\hat{y}_t^{(1)},\dots,\hat{y}_t^{(K)})$ and $\beta^{(\tau)}$ is quantile-specific (Zakrzewski et al., 2024). In electricity price notation, with hourly day-ahead prices $P_{h,t}$ and point-forecast vector $\tau$0, the same idea is written as

$\tau$1

with a separate weight vector $\tau$2 for each $\tau$3 (Janczura, 2024).

This classical definition treats point forecasts as explanatory variables and estimates quantile-specific combination weights. The literature also uses the same model-averaging logic in a broader sense. In factor-augmented quantile autoregressions, model-specific quantile forecasts $\tau$4 are combined as

$\tau$5

with $\tau$6 on the unit simplex (Phella, 2020). In semiparametric high-dimensional conditional quantile prediction, the target conditional quantile $\tau$7 is approximated by an affine combination of one-dimensional marginal quantile regression functions,

$\tau$8

so that the “base forecasters” are marginal quantile components rather than distinct forecasting models (Tu et al., 2018). This suggests that QRA is not limited to forecast ensembles in the narrow time-series sense; it is a general quantile model averaging principle.

2. Estimation, loss functions, and probabilistic output

The canonical estimation problem uses the quantile check loss

$\tau$9

and fits a separate quantile regression for each $K$0 by solving

$K$1

Equivalent formulations in the electricity-price literature write the same objective in piecewise form with asymmetric absolute deviations (Janczura, 2024, Zakrzewski et al., 2024). In baseline QRA, quantiles are therefore estimated independently: for each selected $K$2, there is a distinct coefficient vector $K$3.

Once the regressions are fitted, QRA produces a discrete set of conditional quantiles

$K$4

which define prediction intervals and an approximate predictive distribution. Common examples include a 50% interval from $K$5 and $K$6, a 70% interval from $K$7 and $K$8, and a 90% interval from $K$9 and $\hat{y}_t^{(1)},\dots,\hat{y}_t^{(K)}$0 (Zakrzewski et al., 2024). In electricity-price applications, running QRA on a dense grid such as $\hat{y}_t^{(1)},\dots,\hat{y}_t^{(K)}$1 is used to describe the full predictive distribution and derive risk measures such as Value-at-Risk (Janczura, 2024).

A known issue is quantile crossing: independently estimated quantiles may violate monotonicity, so that a lower nominal quantile exceeds a higher one. In one implementation of Smoothing Quantile Regression Averaging, after obtaining predictions of the 99 percentiles, the results are sorted independently for each day and hour (Uniejewski, 2023). This indicates that standard separate-quantile QRA does not itself guarantee a coherent quantile function. A related development is Bayesian joint quantile autoregression, which supplies a quantile function that avoids crossing of the regression across quantiles; although it is not presented as QRA, it addresses a core structural problem that also arises in QRA systems (Castillo-Mateo et al., 2023).

3. Variants, regularization, and structural modifications

The contemporary QRA literature contains a large family of variants that preserve the basic forecast-combination structure while modifying the regressors, the loss, or the regularization.

Method	Core modification	Source
QRM	Quantile Regression on Mean, using the row mean of point predictions as a single regressor	(Zakrzewski et al., 2024)
FQRA / FQRM	PCA-based factor versions of QRA and QRM	(Zakrzewski et al., 2024)
sFQRA / sFQRM	Standardized factor variants using row standardization before PCA	(Zakrzewski et al., 2024)
LQRA	L1-regularized QRA with $\hat{y}_t^{(1)},\dots,\hat{y}_t^{(K)}$2 penalty	(Zakrzewski et al., 2024)
SQRA / SQRM	Smoothed QRA variants using kernel density estimation in the loss	(Zakrzewski et al., 2024)
iQRA	Isotonic QRA with stochastic order constraints and nonnegative coefficients	(Lipiecki et al., 20 Jul 2025)
ERA	Expectile regression averaging, replacing quantile loss by asymmetric squared loss	(Janczura, 2024)

Baseline QRA is linear quantile regression on point forecasts. QRM reduces dimensionality by averaging the point forecasts first and then fitting a one-regressor quantile regression. FQRA and FQRM use Principal Component Analysis on the point-prediction matrix to extract factors, thereby addressing multicollinearity and reducing dimensionality; the standardized variants apply row-wise standardization before PCA to improve comparability and numerical stability (Zakrzewski et al., 2024). In a related but distinct large-panel setting, a hybrid Factor-QRA approach uses PCA on a panel of 673 point forecasts and then performs quantile regression on the extracted factors or on a factor-based point forecast, with the number of factors chosen by BIC (Maciejowska et al., 2023).

Regularization enters most directly through LQRA and iQRA. LQRA adds an L1 penalty,

$\hat{y}_t^{(1)},\dots,\hat{y}_t^{(K)}$3

to perform feature selection among base models (Zakrzewski et al., 2024). Isotonic Quantile Regression Averaging (iQRA) instead imposes stochastic order constraints through nonnegative coefficients, using isotonic regularization to improve forecast accuracy, reliability, and computational costs; the paper states that isotonic regularization decreases the complexity of the quantile regression problem and offers a hyperparameter-free approach to variable selection (Lipiecki et al., 20 Jul 2025). This suggests a structural alternative to explicit penalization.

Smoothing Quantile Regression Averaging replaces the non-smooth check loss by a kernel-smoothed objective,

$\hat{y}_t^{(1)},\dots,\hat{y}_t^{(K)}$4

with bandwidth $\hat{y}_t^{(1)},\dots,\hat{y}_t^{(K)}$5 selected by a rule of thumb (Uniejewski, 2023). A distinct line of work proposes Expectile Regression Averaging (ERA), which mirrors QRA’s forecast-averaging structure but replaces the quantile loss with asymmetric squared loss (Janczura, 2024).

4. High-dimensional, semiparametric, and functional generalizations

Beyond forecast ensembles, QRA has been generalized to high-dimensional and semiparametric settings in which the “candidate models” are not separate forecasting engines but structured quantile components. In “Semiparametric model averaging for high dimensional conditional quantile prediction,” the target conditional quantile is approximated by an affine combination of one-dimensional marginal quantile regression functions,

$\hat{y}_t^{(1)},\dots,\hat{y}_t^{(K)}$6

where each $\hat{y}_t^{(1)},\dots,\hat{y}_t^{(K)}$7 is estimated by local linear quantile regression and the weights $\hat{y}_t^{(1)},\dots,\hat{y}_t^{(K)}$8 are estimated by penalized quantile regression with SCAD (Tu et al., 2018). The procedure is explicitly two-stage: first estimate marginal quantile functions nonparametrically, then estimate sparse model weights by minimizing

$\hat{y}_t^{(1)},\dots,\hat{y}_t^{(K)}$9

This is mathematically a QRA structure in which the base quantile forecasters are marginal quantile functions rather than external point forecasts (Tu et al., 2018).

Functional data-driven quantile model averaging extends the same idea to functional linear quantile regression. Candidate models are indexed by the number of retained functional principal components $\tau$0, each candidate producing

$\tau$1

and the averaged predictor is

$\tau$2

with weights chosen by K-fold cross-validation under the quantile loss (Xu et al., 2023). The paper proves asymptotic optimality of the selected weights in terms of minimizing the excess final prediction error when all candidate models are misspecified, and consistency when the true regression relationship belongs to the candidate set (Xu et al., 2023).

Other model-averaging formulations broaden QRA still further. Complete Subset Averaging for quantile regressions considers all submodels of a fixed size $\tau$3, averages them with equal weights, and selects $\tau$4 by leave-one-out cross-validation; the resulting predictor is

$\tau$5

where $\tau$6 (Lee et al., 2020). Jackknife Model Averaging for Composite Quantile Regression instead averages across candidate models while pooling multiple quantile levels in a composite quantile objective; it uses delete-one cross-validation to estimate model weights and proves asymptotic optimality in minimizing out-of-sample composite final prediction error (Wang et al., 2019). In macroeconomic factor-augmented quantile autoregressions, model-specific quantile forecasts are averaged using equal weights, AIC, BIC, QRIC, or jackknife weights, with latent factors estimated by principal components from a large panel (Phella, 2020). These developments indicate that QRA has become a general frequentist model-averaging framework for quantile-based prediction.

5. Applications in electricity price forecasting and related domains

In electricity price forecasting, QRA is described as “the golden standard in probabilistic approaches to predict energy prices” (Zakrzewski et al., 2024). Its appeal is tied to the structure of electricity markets: prices are highly volatile, exhibit spikes and heavy tails, and are affected by seasonality and fundamentals, while probabilistic forecasts are required for risk management and trading (Zakrzewski et al., 2024, Janczura, 2024). QRA is therefore routinely used to combine ARX-type models, spike-transformed models, weekly-mean models, and variants based on different variance-stabilizing transformations (Janczura, 2024, Uniejewski, 2023).

Empirical studies show both the strengths and the limits of the method. In one comparison on German day-ahead electricity prices, with asinh transformation applied beforehand, QRA achieved an overall pinball score of $\tau$7, coverage at 5% of $\tau$8, and coverage at 95% of $\tau$9; the corresponding ERA results were $$Q_{y_t}(\tau \mid X_t) = \beta_0^{(\tau)} + \sum_{k=1}^{K} \beta_k^{(\tau)} \hat{y}_t^{(k)},$$0, $$Q_{y_t}(\tau \mid X_t) = \beta_0^{(\tau)} + \sum_{k=1}^{K} \beta_k^{(\tau)} \hat{y}_t^{(k)},$$1, and $$Q_{y_t}(\tau \mid X_t) = \beta_0^{(\tau)} + \sum_{k=1}^{K} \beta_k^{(\tau)} \hat{y}_t^{(k)},$$2, respectively (Janczura, 2024). The same study reports that quantile-based methods produce prediction intervals that are too narrow, with QRA tending to under-estimate tail risk relative to expectile-based approaches (Janczura, 2024). A smoothing-based refinement, Smoothing Quantile Regression Averaging, is evaluated on German and Spanish electricity markets using data covering the COVID-19 pandemic and the Russian invasion of Ukraine; it is reported to lead to profit increases of up to $$Q_{y_t}(\tau \mid X_t) = \beta_0^{(\tau)} + \sum_{k=1}^{K} \beta_k^{(\tau)} \hat{y}_t^{(k)},$$3 on average compared to a benchmark strategy based solely on point forecasts (Uniejewski, 2023).

Hybrid factor-based extensions have also been evaluated from both statistical and economic perspectives. A probabilistic forecasting study for German EPEX SPOT and the Polish Power Exchange proposes a hybrid Factor-QRA approach and finds that the newly proposed method outperforms literature benchmarks in terms of empirical coverage and the Christoffersen test for conditional coverage (Maciejowska et al., 2023). The same work evaluates forecasts through a trading strategy using probabilistic price predictions to optimize the operation of an energy storage system and concludes that there is a need for the economic evaluation of forecasts in addition to statistical measures (Maciejowska et al., 2023). More recently, Isotonic Quantile Regression Averaging is reported to consistently outperform state-of-the-art postprocessing methods in terms of both reliability and sharpness, while producing well-calibrated prediction intervals across multiple confidence levels and offering a hyperparameter-free approach to variable selection (Lipiecki et al., 20 Jul 2025).

Software support has become part of the QRA ecosystem. The Python package ReModels implements baseline QRA together with QRM, FQRA, FQRM, sFQRA, sFQRM, LQRA, SQRA, and SQRM, and also provides dataset acquisition, variance-stabilizing transformations, point-forecast computation, and probabilistic forecast evaluation (Zakrzewski et al., 2024). This suggests that QRA is no longer only a methodological idea but also a reproducible workflow for electricity-market forecasting research.

6. Theoretical properties, evaluation criteria, and limitations

A major strand of the literature studies QRA through asymptotic optimality, sparsity, and oracle-type results. In semiparametric high-dimensional conditional quantile prediction, the SCAD-penalized weight estimator is shown to be consistent as $$Q_{y_t}(\tau \mid X_t) = \beta_0^{(\tau)} + \sum_{k=1}^{K} \beta_k^{(\tau)} \hat{y}_t^{(k)},$$4, to recover the zero weights with probability approaching 1, and to satisfy an oracle-type asymptotic normality result for the active weights (Tu et al., 2018). In factor-augmented quantile autoregression, QRIC-based and jackknife weights are motivated by minimizing quantile-specific forecast loss, and for UK GDP growth these approaches outperform equal weights and AIC/BIC on coverage and final prediction error, whereas for CPI inflation a naive QAR(1) outperforms all model averaging methodologies (Phella, 2020). In complete subset averaging, the selected subset size is asymptotically optimal in the sense of Li (1987), even when all candidate models are misspecified (Lee et al., 2020). In functional quantile model averaging, K-fold cross-validation weights are asymptotically optimal for the excess final prediction error, and consistency is obtained when the true model belongs to the candidate set (Xu et al., 2023). In composite quantile regression, delete-one cross-validation weights yield asymptotic optimality in minimizing out-of-sample composite final prediction error (Wang et al., 2019).

The evaluation of QRA forecasts is correspondingly quantile-centric. Proper scoring rules include the pinball loss and its aggregated versions (Janczura, 2024, Zakrzewski et al., 2024). Interval calibration is assessed by coverage probabilities, Average Empirical Coverage (AEC), empirical coverage, and average coverage error, together with Kupiec and Christoffersen tests for conditional coverage (Zakrzewski et al., 2024, Maciejowska et al., 2023). Comparative predictive ability is often examined with Diebold–Mariano or related tests (Janczura, 2024). In model-averaging theory, final prediction error and excess final prediction error provide the formal risk criteria under which asymptotic optimality is established (Xu et al., 2023).

Several limitations recur across the literature. Separate quantile fitting can produce quantile crossing (Zakrzewski et al., 2024). QRA’s effectiveness is limited by the quality and diversity of the underlying point forecasts (Zakrzewski et al., 2024). Computational cost rises with the number of quantiles, rolling calibration windows, and regularized variants, especially in high-dimensional settings (Zakrzewski et al., 2024, Tu et al., 2018). Bandwidth choice matters in kernel-based smoothing and in semiparametric marginal quantile estimation (Uniejewski, 2023, Tu et al., 2018). In electricity markets, structural regime changes and evolving renewable penetration can weaken the informativeness of historical residual structures (Zakrzewski et al., 2024). More specialized extensions address additional complications: semiparametric model averaging for high-dimensional quantile regression with nonignorable missingness uses a two-phase scheme combining semiparametric likelihood-based estimation for the missing mechanism with a semiparametric weighting procedure for quantile model averaging, and proves asymptotic optimality under complex missingness conditions (Xiong et al., 30 Aug 2025).

Future directions suggested in the literature include ultra-high dimensional extensions with screening or dimension reduction before quantile averaging, alternative penalties such as MCP or adaptive SCAD/LASSO, time-series dependence and dynamic QRA, Quantile Regression Forest, Quantile Regression Neural Network, and broader benchmarking against distributional neural networks and conformal prediction intervals (Tu et al., 2018, Xiong et al., 30 Aug 2025, Zakrzewski et al., 2024). This suggests that QRA is best understood not as a single fixed algorithm, but as a family of quantile model averaging methods whose central object is the weighted combination of conditional quantile forecasts under quantile-consistent loss.