Andrew Foerster, Juan Rubio-Ramírez, Daniel F. Waggoner, and Tao Zha
Working Paper 2014-16
August 2014

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Markov-switching DSGE (MSDSGE) modeling has become a growing body of literature on economic and policy issues related to structural shifts. This paper develops a general perturbation methodology for constructing high-order approximations to the solutions of MSDSGE models. Our new method, called "the partition perturbation method," partitions the Markov-switching parameter space to keep a maximum number of time-varying parameters from perturbation. For this method to work in practice, we show how to reduce the potentially intractable problem of solving MSDSGE models to the manageable problem of solving a system of quadratic polynomial equations. We propose to use the theory of Gröbner bases for solving such a quadratic system. This approach allows us to first obtain all the solutions and then determine how many of them are stable. We illustrate the tractability of our methodology through two examples.

JEL classification: C6, E3, G1

Key words: partition principle, naive perturbation, uncertainty, Taylor series, high-order expansion, time-varying coefficients, nonlinearity, Gröbner bases

For helpful comments, the authors thank Rhys Bidder, Han Chen, Seonghoon Cho, Lars Hansen, Giovanni Lombardo, Leonardo Melosi, Harald Uhlig, as well as 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's Workshop on Methods and Applications for DSGE Models. Zhao Li and Tong Xu provided excellent research assistance. This research is supported in part by National Science Foundation grants SES-1127665 and SES-1227397. The views expressed here are the authors' and not necessarily those of the Federal Reserve Banks of Atlanta and Kansas City, the Federal Reserve System, or the National Bureau of Economic Research. Any remaining errors are the authors' responsibility.

Please address questions regarding content to Andrew Foerster, Research Department, Federal Reserve Bank of Kansas City, 1 Memorial Drive, Kansas City, MO 64198, 816-881-2751,; Juan Rubio-Ramírez, Department of Economics, Duke University, PO Box 90097, Durham, NC 27708, 919-660-1865,; Daniel F. Waggoner, Federal Reserve Bank of Atlanta, 1000 Peachtree Street NE, Atlanta, GA 30309-4470, 404-498-8278,; or Tao Zha, Federal Reserve Bank of Atlanta and Emory University, 1000 Peachtree Street NE, Atlanta, GA 30309-4470, 404-498-8353,

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