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Solving linear difference systems with lagged expectations by a method of undetermined coefficients

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Peng-fei Wang
Yi Wen

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Abstract

This paper proposes a solution method to solve linear difference models with lagged expectations. Variables with lagged expectations expand the model's state space greatly when N is large; and getting the system into a canonical form solvable by the traditional methods involves substantial manual work (e.g., arranging the state vector and the associated coefficient matrices to accommodate variables with lagged expectations), which is prone to human errors. Our method avoids the need of expanding the state space of the system and shifts the burden of analysis from the individual economist/model solver toward the computer. Hence it can be a very useful tool in practice, especially in testing and estimating economics models with a high order of lagged expectations. Examples are provided to demonstrate the usefulness of the method. We also discuss the implications of lagged expectations on the equilibrium properties of indeterminate DSGE models, such as the serial correlation properties of sunspots shocks in these models.

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Paper provided by Federal Reserve Bank of St. Louis in its series Working Papers with number 2006-003.

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Date of creation: 2006
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Handle: RePEc:fip:fedlwp:2006-003

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Keywords: Monetary policy Macroeconomics

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  1. N. Gregory Mankiw & Ricardo Reis, 2002. "Sticky Information Versus Sticky Prices: A Proposal To Replace The New Keynesian Phillips Curve," The Quarterly Journal of Economics, MIT Press, vol. 117(4), pages 1295-1328, November. [Downloadable!] (restricted)
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  2. Benhabib Jess & Farmer Roger E. A., 1994. "Indeterminacy and Increasing Returns," Journal of Economic Theory, Elsevier, vol. 63(1), pages 19-41, June. [Downloadable!] (restricted)
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  3. Klein, Paul, 2000. "Using the generalized Schur form to solve a multivariate linear rational expectations model," Journal of Economic Dynamics and Control, Elsevier, vol. 24(10), pages 1405-1423, September. [Downloadable!] (restricted)
  4. Mathias Trabandt, 2004. "Sticky Information vs. Sticky Prices: A Horse Race in a DSGE Framework," 2004 Meeting Papers 543, Society for Economic Dynamics. [Downloadable!]
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  5. Sims, Christopher A, 2002. "Solving Linear Rational Expectations Models," Computational Economics, Springer, vol. 20(1-2), pages 1-20, October. [Downloadable!]
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  6. King, Robert G & Watson, Mark W, 1998. "The Solution of Singular Linear Difference Systems under Rational Expectations," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(4), pages 1015-26, November.
  7. Burnside, Craig & Eichenbaum, Martin & Rebelo, Sergio, 1993. "Labor Hoarding and the Business Cycle," Journal of Political Economy, University of Chicago Press, vol. 101(2), pages 245-73, April. [Downloadable!] (restricted)
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  8. Andres, Javier & Lopez-Salido, J. David & Nelson, Edward, 2005. "Sticky-price models and the natural rate hypothesis," Journal of Monetary Economics, Elsevier, vol. 52(5), pages 1025-1053, July. [Downloadable!] (restricted)
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  9. Burnside, Craig & Eichenbaum, Martin, 1996. "Factor-Hoarding and the Propagation of Business-Cycle Shocks," American Economic Review, American Economic Association, vol. 86(5), pages 1154-74, December. [Downloadable!] (restricted)
  10. Lawrence J. Christiano & Martin Eichenbaum & Charles Evans, 2001. "Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy," NBER Working Papers 8403, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
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  11. Blanchard, Olivier Jean & Kahn, Charles M, 1980. "The Solution of Linear Difference Models under Rational Expectations," Econometrica, Econometric Society, vol. 48(5), pages 1305-11, July. [Downloadable!] (restricted)
  12. Lubik, Thomas A. & Schorfheide, Frank, 2003. "Computing sunspot equilibria in linear rational expectations models," Journal of Economic Dynamics and Control, Elsevier, vol. 28(2), pages 273-285, November. [Downloadable!] (restricted)
  13. Thomas A. Lubik & Frank Schorfheide, 2004. "Testing for Indeterminacy: An Application to U.S. Monetary Policy," American Economic Review, American Economic Association, vol. 94(1), pages 190-217, March. [Downloadable!] (restricted)
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  14. Wen, Yi, 1998. "Capacity Utilization under Increasing Returns to Scale," Journal of Economic Theory, Elsevier, vol. 81(1), pages 7-36, July. [Downloadable!] (restricted)
  15. Benjamin D. Keen, 2007. "Sticky Price And Sticky Information Price-Setting Models: What Is The Difference?," Economic Inquiry, Western Economic Association International, vol. 45(4), pages 770-786, October. [Downloadable!] (restricted)
  16. Lawrence J. Christiano, 1998. "Solving dynamic equilibrium models by a method of undetermined coefficients," Working Paper 9804, Federal Reserve Bank of Cleveland. [Downloadable!]
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  1. Alexander Meyer-Gohde, 2007. "Solving Linear Rational Expectations Models with Lagged Expectations Quickly and Easily," SFB 649 Discussion Papers SFB649DP2007-069, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany. [Downloadable!]
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  2. N. Gregory Mankiw & Ricardo Reis, 2006. "Sticky Information in General Equilibrium," NBER Working Papers 12605, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
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