Empirical properties of forecasts with the functional autoregressive model

Abstract

We study the finite sample performance of predictors in the functional (Hilbertian) autoregressive model ${X_{n+1} = \Psi(X_n)+\varepsilon_n}$. Our extensive empirical study based on simulated and real data reveals that predictors of the form ${\hat\Psi(X_n)}$ are practically optimal in a sense that their prediction errors are comparable with those of the infeasible perfect predictor Ψ(X _n ). The predictions ${\hat\Psi(X_n)}$ cannot be improved by an improved estimation of Ψ, nor by a more refined prediction approach which uses predictive factors rather than the functional principal components. We also discuss the practical limits of predictions that are feasible using the functional autoregressive model. These findings have not been established by theoretical work currently available, and may serve as a practical reference to the properties of predictors of functional data.