Quantile Regression Averaging (QRA) is a forecast combination approach to the computation of prediction intervals. It involves applying quantile regression to the point forecasts of a small number of individual forecasting models or experts. It has been introduced in 2014 by Jakub Nowotarski and Rafał Weron and originally used for probabilistic forecasting of electricity prices and loads. Despite its simplicity it has been found to perform extremely well in practice - the top two performing teams in the price track of the Global Energy Forecasting Competition (GEFCom2014) used variants of QRA.
Introduction The individual point forecasts are used as independent variables and the corresponding observed target variable as the dependent variable in a standard quantile regression setting. The Quantile Regression Averaging method yields an interval forecast of the target variable, but does not use the prediction intervals of the individual methods. One of the reasons for using point forecasts (and not interval forecasts) is their availability. For years, forecasters have focused on obtaining accurate point predictions. Computing probabilistic forecasts, on the other hand, is generally a much more complex task and has not been discussed in the literature nor developed by practitioners so extensively. Therefore, QRA may be found particularly attractive from a practical point of view as it allows to leverage existing development of point forecasting.
Computation
The quantile regression problem can be written as follows:
Q y ( q | X t ) = X t β q {\displaystyle Q_{y}(q|X_{t})=X_{t}\beta _{q}} , where Q y ( q | ⋅ ) {\displaystyle Q_{y}(q|\cdot )} is the conditional q-th quantile of the dependent variable ( y t {\displaystyle y_{t}} ), X t = [ 1 , y ^ 1 , t , . . . , y ^ m , t ] {\displaystyle X_{t}=[1,{\hat {y}}_{1,t},...,{\hat {y}}_{m,t}]} is a vector of point forecasts of m {\displaystyle m} individual models (i.e. independent variables) and βq is a vector of parameters (for quantile q). The parameters are estimated by minimizing the loss function for a particular q-th quantile:
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