Computationally E cient Solution and Maximum Likelihood Estimation of Nonlinear Rational Expectations Models
نویسندگان
چکیده
This paper presents new, computationally e cient algorithms for solution and estimation of nonlinear dynamic rational expectations models. The innovations in the algorithms are as follows: (1) The entire solution path is obtained simultaneously by taking a small number of Newton steps, using analytic derivatives, over the entire path; (2) The terminal conditions for the solution path are derived from the uniqueness and stability conditions from the linearization of the model around the terminus of the solution path; (3) Unit roots are allowed in the model; (4) Very general models with expectational identities and singularities of the type handled by the King-Watson (1995a,b) linear algorithms are also allowed; and (5) Rank-de cient covariance matrices that arise owing to the presence of expectational identities are admissible. Reasonably complex models are solved in less than a second on a Sun Sparc20. This speed improvement makes derivative-based estimation methods feasible. Algorithms for maximum likelihood estimation and sample estimation problems are presented. (JEL E52, E43) Vice President and Research Assistant, respectively, Research Department, Federal Reserve Bank of Boston, Boston, MA, 02106. The views expressed in this paper are the authors' and do not necessarily re ect those of the Federal Reserve Bank of Boston or the Board of Governors of the Federal Reserve System. Please do not quote without permission of the authors. For a variety of reasons, nonlinear models for macroand microeconomics have grown in popularity in recent years. In macroeconomics, the recognition that most linear models cannot capture turning points in business cycles, the inherent nonlinearity in the consumer's budget constraint with time-varying interest rates, the presence of nonlinear adjustment costs in investment, and the nonlinearity of the convex production function all require some accommodation of nonlinearity. Researchers have employed a number of alternate strategies for computing the solutions to nonlinear models. Their approaches may be separated into three broad categories. 1. Linearize or log-linearize the system as in Kydland and Prescott (1982). In this case, one can apply the techniques developed for linear models. 2. Solve a reduced form version of the system by numerical integration and iteration using dynamic programming or the nite-element method, as in Christiano (1990) and McGrattan (1996), respectively. 3. Numerically solve for the model-consistent path of expectations (in the case of certainty equivalence) from an initial guess, as in Fair and Taylor (1983). Linearizing models involves approximations that can be evaluated only for simple, analytically tractable cases. Dynamic programming techniques generally require considerably more computing time, often several orders of magnitude greater than linear methods. This paper presents a method that provides a compromise between these two extremes, in the spirit of Fair and Taylor. The method directly solves the nonlinear functions that make up the model. However, it solves a perfect-foresight version of the functions, and thus does not fully incorporate the stochastic features of the model into the solution technique. The algorithm presented here uses Newton's method to jointly solve for the full time-path of nonlinear equations in the model. It utilizes the sparsity of the system to economize on computations (and storage). The method achieves a computational speed that makes derivative-based estimation methods feasible. The results discussed in section 2.3 suggest that, at least for
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تاریخ انتشار 1996