Is QRA just a simple forecast average?

No. It estimates quantile-specific combination relationships using observed outcomes.

Must the coefficients sum to one?

Not unless the chosen specification imposes that constraint.

Does a nominal 80% interval guarantee 80% realized coverage?

No. Coverage must be evaluated under the deployment conditions.

AI Glossary / Statistics and probability / Advanced

Quantile regression averaging

Definition

Quantile regression averaging, or QRA, combines forecasts by using them as inputs to a quantile regression model. In its original formulation, the inputs are point forecasts from several individual models and the response is the observed outcome. Separate regressions estimate selected conditional quantiles, which can be used to form prediction intervals. Despite the word averaging, QRA is not merely the arithmetic mean of forecasts or a direct average of existing interval endpoints.

What it means

Different forecasting models may capture different information about future outcomes. QRA learns how their predictions relate to a chosen outcome quantile. The combination coefficients can differ between lower, middle, and upper quantiles.

For instance, two demand models may have similar average accuracy but respond differently during high-demand periods. A quantile-based combination can use those differences when estimating an upper planning threshold. The method still depends on the quality and relevance of its inputs. A collection of forecasts does not automatically contain enough information to identify uncertainty, and combining poor forecasts does not guarantee a good result.

How it works

Generate historical base-model forecasts under conditions resembling actual deployment. For a time-series problem, each historical forecast should use only information available before its target outcome. Pair these forecasts with the subsequently observed outcomes to form a separate combination-training dataset.

For each desired quantile level, fit a quantile regression using the individual forecasts as explanatory variables. The objective is pinball loss, which weights underprediction and overprediction asymmetrically. State whether the specification includes an intercept, regularization, or constraints on the coefficients. Ordinary unconstrained regression coefficients need not be nonnegative or sum to one.

Apply the fitted combination to new base forecasts to obtain new quantile predictions. Check that quantiles are ordered and assess calibration, interval width, and pinball loss on suitably separated data. Rolling or expanding evaluation windows can match a sequential forecasting application. Correlated base forecasts can make estimated coefficients unstable, so model selection and regularization choices need their own validation rather than being judged solely by in-sample fit.

Three practical examples

An electricity forecasting team converts a pool of point forecasts into lower and upper price quantiles for planning.
A retailer evaluates a quantile combination of demand forecasts from seasonal, promotion-aware, and machine-learning models.
An AI operations team combines workload forecasts to estimate upper capacity requirements, while testing whether the estimated quantiles remain calibrated over time.

Think of it this way

Imagine several forecasters offering their best single-number estimates. A second analyst learns from their past forecasts and actual outcomes how to turn their opinions into low, middle, and high scenarios. The analyst is not simply taking the most pessimistic opinion or averaging the forecasters' own intervals. QRA formalizes this second-stage learning with a quantile-specific objective and a defined set of forecast inputs.

Worked example

In an original arithmetic illustration, two models forecast demand of 100 and 120 units. Suppose a previously fitted QRA specification gives the lower quantile as 0.4f1 + 0.3f2 and the upper quantile as 0.5f1 + 0.6f2. The coefficients are illustrative supplied values, not estimates claimed to come from a dataset.

The lower prediction is 0.4 × 100 + 0.3 × 120 = 76 units. The upper prediction is 0.5 × 100 + 0.6 × 120 = 122 units. If these are the 0.10 and 0.90 quantile models, they form a nominal central 80% prediction interval [76, 122], with width 46. The simple point-forecast average is 110, which is a different output.

If the realized demand is 130, the 0.90 pinball loss for the upper prediction is 0.90 × (130 - 122) = 7.2. If demand is 110, that loss is 0.10 × (122 - 110) = 1.2. These calculations explain combination and scoring; one or two outcomes cannot establish 80% coverage. The coefficient sums of 0.7 and 1.1 also show why unconstrained QRA need not be a convex average of the input forecasts.

Illustrative case study

Fictional case study: Elm Supply already maintains several point-demand models. It creates an archive of forecasts issued before each sales period and uses that archive to train lower and upper QRA models.

The team compares their interval scores with a simple baseline on later periods, checks performance during promotions, and tracks crossing quantiles. It treats the combined forecasts as estimates requiring ongoing assessment. This fictional case illustrates a reusable forecasting workflow, not a real empirical result or a promise that the ensemble will outperform every component.

Common misconceptions and mistakes

QRA does not require each base model to supply a full predictive distribution; the original approach uses point forecasts. It also should not be confused with simply averaging quantile predictions, which is a different combination method.

The word averaging does not guarantee positive coefficients or unit-sum weights unless constraints impose them. A nominal prediction interval is not automatically calibrated, and wider intervals are not always better. Training the second stage on unrealistically optimistic fitted values from the base models can also misrepresent deployment performance. Finally, prediction intervals describe future outcomes, not uncertainty intervals for the combination coefficients.

Business use

QRA can build uncertainty-aware forecasts from an existing investment in point-forecasting models. That can support decisions where overprediction and underprediction have different operational costs.

Maintain a reliable archive of forecast issue times, model versions, input availability, and realized outcomes. Compare useful baselines and include the cost of operating multiple base models. If forecasts become redundant or unstable, reconsider the combination design. Report both calibration and sharpness, and examine important periods separately. A good historical combination may need revision when demand mechanisms or base-model behavior change.

Check your understanding

In the worked example, does averaging the two point forecasts to obtain 110 reproduce the QRA interval?

Answer: No. The interval comes from two separately specified quantile combinations, producing endpoints 76 and 122.

Related terms

Quantile Regression
Pinball loss
Prediction interval
Calibration
forecast combination
ensemble forecasting
sharpness
rolling-origin evaluation
quantile crossing

Learn next

Learn point-forecast combination and quantile regression separately. Then construct honest historical forecast inputs and evaluate a second-stage combination using quantile scores and interval diagnostics.

Quantile Regression

Sources

Nowotarski and Weron's original article describes QRA using individual point forecasts, quantile-specific coefficients, and interval evaluation: https://link.springer.com/article/10.1007/s00180-014-0523-0

The core method section was read. The supplied coefficients, numerical calculations, operational extensions, and fictional case are original illustrations, not a reproduction of the paper's electricity-market experiment.

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