Optimal Forecast Reconciliation with Uncertainty Quantification
arxiv(2024)
摘要
We propose to estimate the weight matrix used for forecast reconciliation as
parameters in a general linear model in order to quantify its uncertainty. This
implies that forecast reconciliation can be formulated as an orthogonal
projection from the space of base-forecast errors into a coherent linear
subspace. We use variance decomposition together with the Wishart distribution
to derive the central estimator for the forecast-error covariance matrix. In
addition, we prove that distance-reducing properties apply to the reconciled
forecasts at all levels of the hierarchy as well as to the forecast-error
covariance. A covariance matrix for the reconciliation weight matrix is
derived, which leads to improved estimates of the forecast-error covariance
matrix. We show how shrinkage can be introduced in the formulated model by
imposing specific priors on the weight matrix and the forecast-error covariance
matrix. The method is illustrated in a simulation study that shows consistent
improvements in the log-score. Finally, standard errors for the weight matrix
and the variance-separation formula are illustrated using a case study of
forecasting electricity load in Sweden.
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