Stochastic discount factor (SDF) processes in dynamic economies admit a permanent-transitory decomposition in which the permanent component characterizes pricing over long investment horizons. This paper introduces an empirical framework to analyze the permanent-transitory decomposition of SDF processes. Specifically, we show how to estimate nonparametrically the solution to the Perron-Frobenius eigenfunction problem of Hansen and Scheinkman (2009). Our empirical framework allows researchers to (i) recover the time series of the estimated permanent and transitory components and (ii) estimate the yield and the change of measure which characterize pricing over long investment horizons. We also introduce nonparametric estimators of the continuation value function in a class of models with recursive preferences by reinterpreting the value function recursion as a nonlinear Perron-Frobenius problem. We establish consistency and convergence rates of the eigenfunction estimators and asymptotic normality of the eigenvalue estimator and estimators of related functionals. As an application, we study an economy where the representative agent is endowed with recursive preferences, allowing for general (nonlinear) consumption and earnings growth dynamics.

We propose a framework for analyzing the sensitivity of counterfactuals to parametric assumptions about the distribution of latent variables in structural models. In particular, we derive bounds on counterfactuals as the distribution of latent variables spans nonparametric neighborhoods of a given parametric specification while other “structural” features of the model are maintained. Our approach recasts the infinite‐dimensional problem of optimizing the counterfactual with respect to the distribution of latent variables (subject to model constraints) as a finite‐dimensional convex program. We also develop an MPEC version of our method to further simplify computation in models with endogenous parameters (e.g., value functions) defined by equilibrium constraints. We propose plug‐in estimators of the bounds and two methods for inference. We also show that our bounds converge to the sharp nonparametric bounds on counterfactuals as the neighborhood size becomes large. To illustrate the broad applicability of our procedure, we present empirical applications to matching models with transferable utility and dynamic discrete choice models.

It is generally difficult to know whether the parameters in nonlinear econometric models are point‐identified. We provide computationally attractive procedures to construct confidence sets (CSs) for identified sets of the full parameter vector and of subvectors in models defined through a likelihood or a vector of moment equalities or inequalities. The CSs are based on level sets of “optimal” criterion functions (such as likelihoods, optimally‐weighted or continuously‐updated GMM criterions). The level sets are constructed using cutoffs that are computed via Monte Carlo (MC) simulations from the quasi‐posterior distribution of the criterion. We establish new Bernstein–von Mises (or Bayesian Wilks) type theorems for the quasi‐posterior distributions of the quasi‐likelihood ratio (QLR) and profile QLR in partially‐identified models. These results imply that our MC CSs have exact asymptotic frequentist coverage for identified sets of full parameters and of subvectors in partially‐identified regular models, and have valid but potentially conservative coverage in models whose local tangent spaces are convex cones. Further, our MC CSs for identified sets of subvectors are shown to have exact asymptotic coverage in models with singularities. We provide local power properties and uniform validity of our CSs over classes of DGPs that include point‐ and partially‐identified models. Finally, we present two simulation experiments and two empirical examples: an airline entry game and a model of trade flows.