It is proved that nonparametric autoregression is asymptotically equivalent in the sense of Le Cam’s deficiency distance to nonparametric regression with random design as well as with regular nonrandom design.
"Asymptotic equivalence of nonparametric autoregression and nonparametric regression." Ann. Statist. 34 (4) 1701 - 1732, August 2006. https://doi.org/10.1214/009053606000000560