Quantifying prediction uncertainty for functional-and-scalar to functional autoregressive models under shape constraints

Motivated by demand and supply curve forecasting in energy markets, we discuss an autoregressive functional modeling framework that preserves curve constraints, includes exogenous scalar information, and provides prediction uncertainty quantification. The model is a functional autoregressive model that relies on a non-concurrent functional autoregressive model in a non-standard pre-Hilbert space in order to satisfy the curve constraints. Prediction uncertainty is quantified by means of a novel bootstrap approach for dependent functional data where the predictive bootstrap trajectories are represented alongside the prediction to show how forecasting confidence varies in the domain. Computational and numerical details are discussed in order to replicate the model estimation process an adequate number of times during the bootstrap phase. The method is applied to Italian natural gas market data.