Functional Autoregression Without Truncation: A Continuous-Regularization Approach

Functional autoregressive models of order one (FAR(1)) are predominantly estimated by projecting curves onto leading functional principal components and fitting a vector autoregression in score space, requiring a discrete truncation level $K$ chosen by an \emph{ad hoc} variance threshold. We demonstrate via Monte Carlo experiments that the truncation choice is both consequential and highly regime dependent: the optimal $K$ can differ by an order of magnitude across data-generating regimes, while commonly used high variance thresholds (95%, 99%) lead to substantial forecast deterioration, inflating error by up to $35\mathrm{\%}$ relative to an oracle benchmark. We propose a Tikhonov-regularized estimator ${\widehat{\mathrm{\Psi }}}_{\alpha }={\widehat{C}}_1({\widehat{C}}_0+\alpha I{)}^{-1}$ that replaces the discrete truncation choice with a continuous regularization parameter, selected in a data-driven manner. We establish the convergence rate $n^{-\beta \mathrm{/}(2(\beta +1))}$ under a source condition with smoothness parameter $\beta \in (0,1]$, achieving the saturation rate $n^{-1\mathrm{/}4}$ for smoother targets. Across three contrasting regimes and four sample sizes, the proposed estimator closely tracks the oracle-best FPCA rule and outperforms it in the most challenging wide-spectrum regime, without prior knowledge of the effective operator dimension. An application to 2{,}735 daily intraday PM10 curves from Vienna confirms a 9.7% reduction in mean forecast error relative to the popular 80% threshold and exhibits more stable parameter adaptation across 16 winter seasons.