Andrew Foerster, Juan Rubio-Ramírez, Daniel F. Waggoner, and Tao Zha
Working Paper 2013-1
March 2013

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This paper develops a general perturbation methodology for constructing high-order approximations to the solutions of Markov-switching DSGE models. We introduce an important and practical idea of partitioning the Markov-switching parameter space so that a steady state is well defined. With this definition, we show that the problem of finding an approximation of any order can be reduced to solving a system of quadratic equations. We propose using the theory of Grobner bases in searching all the solutions to the quadratic system. This approach allows us to obtain all the approximations and ascertain how many of them are stable. Our methodology is applied to three models to illustrate its feasibility and practicality.

JEL classification: C6, E1

Key words: Markov-switching parameters, partition, higher order approximations, no certainty equivalence, quadratic system, Grobner bases

The authors thank Leonardo Melosi, Seonghoon Cho, Rhys Bidder, seminar participants at Duke University, the Federal Reserve Bank of St. Louis, the 2010 Society of Economic Dynamics meetings, the 2011 Federal Reserve System Committee on Business and Financial Analysis Conference, the 2012 Annual Meeting of the American Economic Association, the 8th Dynare Conference, and the 2012 National Bureau of Economic Research (NBER) Workshop on Methods and Applications for DSGE Models for helpful comments. This research is supported in part by the National Science Foundation Grant Nos. SES-1127665 and SES-1227397. The views expressed here are the authors' and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining errors are the authors' responsibility.

Please address questions regarding content to Andrew Foerster, Federal Reserve Bank of Kansas City, 1 Memorial Drive, Kansas City, MO 64198, 816-881-2751,; Juan Rubio-Ramírez, Duke University, Federal Reserve Bank of Atlanta, CEPR, FEDEA, and BBVA Research, P.O. Box 90097, Durham, NC 27708, 919-660-1865,; Daniel F. Waggoner, Federal Reserve Bank of Atlanta, 1000 Peachtree Street, N.E., Atlanta, GA 30309-4470,; or Tao Zha, Federal Reserve Bank of Atlanta, Emory University, and NBER, 1000 Peachtree Street, N.E., Atlanta, GA 30309-4470,

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