Computationally Efficient Solution and Maximum Likelihood Estimation of Nonlinear Rational Expectations Models
Jeffrey C. Fuhrer and C. Hoyt Bleakley - Federal Reserve Bank of Boston
This paper presents new, computationally efficient algorithms for solution
and estimation of nonlinear dynamic rational expectations models. The
algorithms incorporate the following features: (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
implied by 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-deficient 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 Sparc2O. This speed improvement makes
derivative-based estimation methods feasible. Algorithms for stochastic
simulation and maximum likelihood estimation are presented and implemented
for sample problems. (JEL E52, E43)
Nonlinear models in macro- and micro- economics have grown in popularity in recent years. In macroeconomics, this is the result of a recognition that most linear models may not adequately capture turning points in business cycles, the inherent nonlinearity in the consumer's budget constraint with time-varying interest rates, the possibility of nonlinear adjustment costs in investment, and the nonlinearity of the production function.
Researchers have employed a number of alternate strategies for computing the solutions to nonlinear models. Their approaches may be separated into three broad categories.
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 some canonical nonlinear models, the omission of the stochastic features of the model is of second- or third-order importance in solution accuracy. How good an approximation the perfect-foresight nonlinear method is for other models remains a question for further research; we discuss some methods for improving the accuracy of the approximation.
The rest of the paper is organized as follows. Section 1 presents the solution algorithm. In section 2, we apply the solution technique to a nonlinear sticky-price model and also to the stochastic growth model. We also compare our solution of the stochastic growth model with results from dynamic programming, and Fair-Taylor. Section 3 describes the maximum likelihood estimation procedure using our method, and includes an example. Section 4 presents an algorithm for stochastic simulation, and provides an example. Section 5 suggests several avenues of future research. Section 6 concludes.
Scheduled for Session 2.4 Rational Expectations Analysis - I