In many prediction and decision problems, the relevant inputs are path-valued covariates rather than static feature vectors. This paper studies asymptotic theory and empirical applications for path regression using signatures. We first establish \(L^2\) approximation rates for truncated signature representations. We prove a minimax-optimal approximation rate over a class of smooth coefficient functionals of observable It\^{o} diffusions. Building on this approximation theory, we then develop asymptotic results for three signature-based learning procedures: Signature-OLS, Signature-LASSO, and Signature-Logistic. These results establish asymptotic normality for least-squares path regression, sparse recovery for high-dimensional signature regression, and latent-score consistency for binary-response classification. Extensive empirical studies cover three real-data applications: foreign-exchange realized-volatility forecasting from intraday price paths, battery end-of-life prediction from early HPPC pulse paths, and epileptic seizure detection from short EEG windows. The empirical results show that signatures provide informative representations of path-valued covariates relative to handcrafted features.
