  When considering the problem of forecasting a continuous-time stochastic
process over an entire time-interval in terms of its recent past, the notion of
Autoregressive Hilbert space processes (ARH) arises. This model can be seen as
a generalization of the classical autoregressive processes to Hilbert space
valued random variables. Its estimation presents several challenges that were
addressed by many authors in recent years. In this paper, we propose an
extension based on this model by introducing a conditioning process on the arh.
In this way, we are aiming a double objective. First, the intrinsic linearity
of arh is overwhelm. Second, we allow the introduction of exogenous covariates
on this function- valued time series model. We begin defining a new kind of
processes that we call Conditional arh. We then propose estimators for the
infinite dimensional parameters associated to such processes. Using two classes
of predictors defined within the arh framework, we extend these to our case.
Consistency results are provided as well as a real data application related to
electricity load forecasting.
