Constrained functional time series: Applications to the Italian gas market

Motivated by market dynamic modelling in the Italian Natural Gas Balancing Platform, we propose a model for analyzing time series of functions, subject to equality and inequality constraints at the two edges of the domain, respectively, such as daily demand and offer curves. Specifically, we provide the constrained functions with suitable pre-Hilbert structures, and introduce a useful isometric bijective map that associates each possible bounded and monotonic function to an unconstrained one. We introduce a functional-to-functional autoregressive model that is used to forecast future demand/offer functions, and estimate the model via the minimization of a penalized mean squared error of prediction, with a penalty term based on the Hilbert–Schmidt squared norm of autoregressive lagged operators. The approach is of general interest and could be generalized to any situation in which one has to deal with functions that are subject to the above constraints which evolve over time.