Functional Autoregression Without Truncation: A Continuous-Regularization Approach

TL;DR

This paper introduces a continuous-regularization approach for functional autoregression that eliminates the need for discrete truncation, leading to more stable and accurate forecasts across various regimes.

Contribution

It proposes a Tikhonov-regularized estimator that replaces truncation with a data-driven regularization parameter, improving forecast accuracy and stability.

Findings

The regularized estimator closely matches the oracle FPCA rule.

It outperforms traditional truncation methods in challenging regimes.

Application to real data shows a 9.7% forecast error reduction.

Abstract

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.