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 $35 \%$ relative to an oracle benchmark. We propose a Tikhonov-regularized estimator $\widehat{\Psi}_\alpha = \widehat{C}_1(\widehat{C}_0 + \alpha I)^{-1}$ that replaces the discrete truncation choice with a continuous regularization parameter, selected in a data-driven manner. We establish the convergence rate $n^{-\beta/(2(\beta+1))}$ under a source condition with smoothness parameter $\beta \in (0, 1]$, achieving the saturation rate $n^{-1/4}$ 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.