Before revision:  We propose sieve estimators of the eigenvalue-eigenfunction pair which are used to decompose the SDF into its permanent and transitory components , as well as estimators of the long-run yield and the entropy of the permanent component of the SDF, allowing for a wide variety of empirically relevant setups.Consistency and convergence rates are established.The estimators of the eigenvalue, yield and entropy are shown to be asymptotically normal and semiparametrically efficient when the SDF is observable. We also introduce nonparametric estimators of the continuation value under Epstein-Zin preferences, thereby extending the scope of our estimators to an important class of recursive preferences. The estimators are simple to implement, perform favorably in simulations, and may be used to numerically compute the eigenfunction and its eigenvalue in fully specified models when analytical solutions are not available .
After revision:  We propose sieve estimators of the eigenvalue-eigenfunction pair which are used to decompose the SDF into its permanent and transitory components , as well as estimators of the long-run yield and the change of measure which characterize pricing over long investment horizons. The methodology is nonparametric, i.e., it does not impose any tight parametric restrictions on the dynamics of the state variables and the SDF process. We derive the large-sample properties of the estimators and illustrate favorable performance in simulations. The methodology is applied to study an economy where the representative agent is endowed with recursive preferences, allowing for general (nonlinear) consumption and earnings growth dynamics .