We create an analytical structure that reveals the long-run risk-return relationship for nonlinear continuous-time Markov environments. We do so by studying an eigenvalue problem associated with a positive eigenfunction for a conveniently chosen family of valuation operators. The members of this family are indexed by the elapsed time between payoff and valuation dates, and they are necessarily related via a mathematical structure called a semigroup. We represent the semigroup using a positive process with three components: an exponential term constructed from the eigenvalue, a martingale, and a transient eigenfunction term. The eigenvalue encodes the risk adjustment, the martingale alters the probability measure to capture long-run approximation, and the eigenfunction gives the long-run dependence on the Markov state. We discuss sufficient conditions for the existence and uniqueness of the relevant eigenvalue and eigenfunction. By showing how changes in the stochastic growth components of cash flows induce changes in the corresponding eigenvalues and eigenfunctions, we reveal a long-run risk-return trade-off.

Continuous-time Markov processes can be characterized conveniently by their infinitesimal generators. For such processes there exist forward and reverse-time generators. We show how to use these generators to construct moment conditions implied by stationary Markov processes. Generalized method of moments estimators and tests can be constructed using these moment conditions. The resulting econometric methods are designed to be applied to discrete-time data obtained by sampling continuous-time Markov processes.

The purpose of this paper is to investigate testable implications of equilibrium asset pricing models. We derive a general representation for asset prices that displays the role of conditioning information. This representation is then used to examine restrictions implied by asset pricing models on the unconditional moments of asset payoffs and prices. In particular, we analyze the effect of information omission on the mean-variance frontier of one-period returns on portfolios of securities. Also, we deduce an information extension of equilibrium pricing functions that is useful in deriving restrictions on the unconditional moments of payoffs and prices.

[This paper describes a method for estimating and testing nonlinear rational expectations models directly from stochastic Euler equations. The estimation procedure makes sample counterparts to the population orthogonality conditions implied by the economic model close to zero. An attractive feature of this method is that the parameters of the dynamic objective functions of economic agents can be estimated without explicitly solving for the stochastic equilibrium.]

The ability of quantile regression models to characterize the heterogeneous impact of variables on different points of an outcome distribution makes them appealing in many economic applications. However, in observational studies, the variables of interest (e.g., education, prices) are often endogenous, making conventional quantile regression inconsistent and hence inappropriate for recovering the causal effects of these variables on the quantiles of economic outcomes. In order to address this problem, we develop a model of quantile treatment effects (QTE) in the presence of endogeneity and obtain conditions for identification of the QTE without functional form assumptions. The principal feature of the model is the imposition of conditions that restrict the evolution of ranks across treatment states. This feature allows us to overcome the endogeneity problem and recover the true QTE through the use of instrumental variables. The proposed model can also be equivalently viewed as a structural simultaneous equation model with nonadditive errors, where QTE can be interpreted as the structural quantile effects (SQE).

We demonstrate the asymptotic equivalence between commonly used test statistics for out-of-sample forecasting performance and conventional Wald statistics. This equivalence greatly simplifies the computational burden of calculating recursive out-of-sample test statistics and their critical values. For the case with nested models, we show that the limit distribution, which has previously been expressed through stochastic integrals, has a simple representation in terms of -distributed random variables and we derive its density. We also generalize the limit theory to cover local alternatives and characterize the power properties of the test.

This paper reconsiders the aliasing problem of identifying the parameters of a continuous time stochastic process from discrete time data. It analyzes the extent to which restricting attention to processes with rational spectral density matrices reduces the number of observationally equivalent models. It focuses on rational specifications of spectral density matrices since rational parameterizations are commonly employed in the analysis of the time series data.

We develop a new approach for the identification of Pareto‐improving tax reforms. This approach yields necessary and sufficient conditions for the existence of Pareto‐improving reform directions. A main insight is that “Two brackets are enough”: When the system cannot be improved by altering tax rates in one or two income brackets, then there is no continuous reform direction that is Pareto‐improving. We also show how to check whether a given tax reform is Pareto‐improving. We use these tools to study the introduction of the Earned Income Tax Credit (EITC) in the United States in 1975. A robust finding is that, prior to the EITC, the U.S. tax‐transfer system was not Pareto‐efficient. Under plausible assumptions about behavioral responses, the 1975 reform was not Pareto‐improving. Qualitatively, though, it had the right properties: A similar reform with earnings subsidies made available to a broader range of incomes would have been Pareto‐improving.