  Forecast reconciliation is a post-forecasting process that involves
transforming a set of incoherent forecasts into coherent forecasts which
satisfy a given set of linear constraints for a multivariate time series. In
this paper we extend the current state-of-the-art cross-sectional probabilistic
forecast reconciliation approach to encompass a cross-temporal framework, where
temporal constraints are also applied. Our proposed methodology employs both
parametric Gaussian and non-parametric bootstrap approaches to draw samples
from an incoherent cross-temporal distribution. To improve the estimation of
the forecast error covariance matrix, we propose using multi-step residuals,
especially in the time dimension where the usual one-step residuals fail. To
address high-dimensionality issues, we present four alternatives for the
covariance matrix, where we exploit the two-fold nature (cross-sectional and
temporal) of the cross-temporal structure, and introduce the idea of
overlapping residuals. We assess the effectiveness of the proposed
cross-temporal reconciliation approaches through a simulation study that
investigates their theoretical and empirical properties and two forecasting
experiments, using the Australian GDP and the Australian Tourism Demand
datasets. For both applications, the optimal cross-temporal reconciliation
approaches significantly outperform the incoherent base forecasts in terms of
the Continuous Ranked Probability Score and the Energy Score. Overall, the
results highlight the potential of the proposed methods to improve the accuracy
of probabilistic forecasts and to address the challenge of integrating
disparate scenarios while coherently taking into account short-term
operational, medium-term tactical, and long-term strategic planning.
