The paper by George Tauchen and Robert Hussey develops a discrete state-space solution method for nonlinear rational expectations models. The method uses numerical quadrature rules to approximate the integral operators that arise in stochastic intertemporal models, particularly asset pricing models. The key component is a technique based on numerical quadrature to form a discrete approximation to the conditional density of exogenous variables. This technique calibrates a Markov chain whose probability distribution closely approximates the given time series. The method is particularly useful for approximating asset pricing models and has potential applications in other areas as well. An empirical application uses the method to study the relationship between the risk premium and the conditional variability of equity returns under an ARCH endowment process. The paper extends previous work by showing how to use quadrature to calibrate the Markov chain approximation to the conditional density, making it more efficient and applicable to larger problems with complex dynamics. The method is demonstrated through various examples and theoretical results, and its effectiveness is evaluated through Monte Carlo simulations and empirical applications.The paper by George Tauchen and Robert Hussey develops a discrete state-space solution method for nonlinear rational expectations models. The method uses numerical quadrature rules to approximate the integral operators that arise in stochastic intertemporal models, particularly asset pricing models. The key component is a technique based on numerical quadrature to form a discrete approximation to the conditional density of exogenous variables. This technique calibrates a Markov chain whose probability distribution closely approximates the given time series. The method is particularly useful for approximating asset pricing models and has potential applications in other areas as well. An empirical application uses the method to study the relationship between the risk premium and the conditional variability of equity returns under an ARCH endowment process. The paper extends previous work by showing how to use quadrature to calibrate the Markov chain approximation to the conditional density, making it more efficient and applicable to larger problems with complex dynamics. The method is demonstrated through various examples and theoretical results, and its effectiveness is evaluated through Monte Carlo simulations and empirical applications.