Summary: We give a brief introduction to deterministic chaos and a link between chaotic deterministic models and stochastic time series models. We argue that it is often natural to determine the embedding dimension in a noisy environment first in any systematic study of chaos. Setting the stochastic models within the framework of nonlinear autoregression, we introduce the notion of a generalized partial autocorrelation and an order. We approach the estimation of the embedding dimension via order determination of an unknown nonlinear autoregression by cross-validation, and give justification by proving its consistency under global boundedness.
As a by-product, we provide a theoretical justification of the final prediction error approach of B. Auestad and D. Tøstheim [Biometrika 77, 669-688 (1990)]. Some illustrations based on the Hénon map and several real data sets are given. The bias of the residual sum of squares as essentially a noise variance estimator is quantified.