Quadrature-Based Methods for Obtaining Approximate Solutions to Nonlinear Asset Pricing Models

Abstract

The paper develops a state space solution method for a class of nonlinear rational expectations models. The method works by using numerical quadrature rules to approximate the integral operators that arise in stochastic intertemporal models. The method is particularly useful for approximating asset pricing models and has potential applications in other problems as well. An empirical application uses the method to study the relationship between the risk premium and the conditional variability of the equity return under an ARCH endowment process. NONLINEAR DYNAMIC RATIONAL EXPECTATIONS MODELS rarely admit explicit solutions. Techniques like the method of undetermined coefficients or forward- looking expansions, which often work well for linear models, rarely provide explicit solutions for nonlinear models. The lack of explicit solutions compli- cates the tasks of analyzing the dynamic properties of such models and generat- ing simulated realizations for applied policy work and other purposes. This paper develops a state-space approximation method for a specific class of nonlinear rational expectations models. The class of models is distinguished by two features: First, the solution functions for the endogenous variables are functions of at most a finite number of lags of an exogenous stationary state vector. Second, the expectational equations of the model take the form of integral equations, or more precisely, Fredholm equations of the second type. The key component of the method is a technique, based on numerical quadrature, for forming a approximation to a general time series conditional density. More specifically, the technique provides a means for calibrating a Markov chain, with a state space, whose probability distribution closely approximates the distribution of a given time series. The quality of the approximation can be expected to get better as the state space is made sufficiently finer. The term discrete is used here in reference to the range space of the random variables and not to the time index; time is always in our analysis. The discretization technique is primarily useful for taking a approxi- mation to the conditional density of the strictly exogenous variables of a model. The specification of this conditional density could be based on a variety of 1Financial support under NSF Grants SES-8520244 and SES-8810357 is acknowledged. We thank the co-editor and referees of earlier drafts for many, many helpful comments that substantially improved the manuscript.