System Reduction and Solution Algorithms for Singular Linear Difference Systems under Rational Expectations

Robert G. King; Mark W. Watson

doi:10.1023/A:1020576911923

System Reduction and Solution Algorithms for Singular Linear Difference Systems under Rational Expectations

Robert G. King, Mark W. Watson

Research output: Contribution to journal › Article › peer-review

55 Scopus citations

Abstract

A first-order linear difference system under rational expectations is, AEy_t+1{pipe}I_t=By_t+C(F)Ex_t{p ipe}I_t, where y_t is a vector of endogenous variables;x_t is a vector ofexogenous variables; Ey_t+1{pipe}I_t is the expectation ofy_t+1 givendate t information; and C(F)Ex_t{pipe}I_t =C₀x_t+C₁Ex_t+1{pipe}I_t+...+C_nEx_t+n{pipe}I_t. If the model issolvable, then y_tcan be decomposed into two sets of variables:dynamicvariables d_t that evolve according to Ed_t+1{pipe}I_t = Wd_t + Ψ_d(F)Ex_t{pipe}I_t and other variables thatobey the dynamicidentities f_t =-Kd_t-Ψ_f(F)Ex_t{pipe}I_t. We developan algorithm for carrying out this decomposition and for constructing theimplied dynamic system. We also provide algorithms for (i) computing perfectforesight solutions and Markov decision rules; and (ii) solutions to relatedmodels that involve informational subperiods.

Original language	English (US)
Pages (from-to)	57-86
Number of pages	30
Journal	Computational Economics
Volume	20
Issue number	1-2
DOIs	https://doi.org/10.1023/A:1020576911923
State	Published - Oct 2002

All Science Journal Classification (ASJC) codes

Economics, Econometrics and Finance (miscellaneous)
Computer Science Applications

Keywords

algorithm
in practice
models
solutions
system reduction

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10.1023/A:1020576911923

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title = "System Reduction and Solution Algorithms for Singular Linear Difference Systems under Rational Expectations",

abstract = "A first-order linear difference system under rational expectations is, AEyt+1{pipe}It=Byt+C(F)Ext{p ipe}It, where yt is a vector of endogenous variables;xt is a vector ofexogenous variables; Eyt+1{pipe}It is the expectation ofyt+1 givendate t information; and C(F)Ext{pipe}It =C0xt+C1Ext+1{pipe}It+...+CnExt+n{pipe}It. If the model issolvable, then ytcan be decomposed into two sets of variables:dynamicvariables dt that evolve according to Edt+1{pipe}It = Wdt + Ψd(F)Ext{pipe}It and other variables thatobey the dynamicidentities ft =-Kdt-Ψf(F)Ext{pipe}It. We developan algorithm for carrying out this decomposition and for constructing theimplied dynamic system. We also provide algorithms for (i) computing perfectforesight solutions and Markov decision rules; and (ii) solutions to relatedmodels that involve informational subperiods.",

keywords = "algorithm, in practice, models, solutions, system reduction",

author = "King, {Robert G.} and Watson, {Mark W.}",

note = "Funding Information: The authors are also affiliated with the National Bureau of Economic Research and the Federal Reserve Bank of Richmond. This article combines results from two of our earlier working papers (1995a, b). We have had useful support on this research project from many people. Examples of singularities in real business cycle models provided by Marianne Baxter, Mario Crucini, and Sergio Rebelo were one of the motivations for this research. Matthew Van Vlack and Pierre Sarte aided in the initial development of the MATLAB programs that tested our algorithms and Julie Thomas helped us refine them. Michael Kouparitsas and Steven Lange worked on the translations into GAUSS. Financial support was provided by the National Science Foundation through grants SES-91-22463 and SBR-94-09629.",

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N1 - Funding Information: The authors are also affiliated with the National Bureau of Economic Research and the Federal Reserve Bank of Richmond. This article combines results from two of our earlier working papers (1995a, b). We have had useful support on this research project from many people. Examples of singularities in real business cycle models provided by Marianne Baxter, Mario Crucini, and Sergio Rebelo were one of the motivations for this research. Matthew Van Vlack and Pierre Sarte aided in the initial development of the MATLAB programs that tested our algorithms and Julie Thomas helped us refine them. Michael Kouparitsas and Steven Lange worked on the translations into GAUSS. Financial support was provided by the National Science Foundation through grants SES-91-22463 and SBR-94-09629.

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N2 - A first-order linear difference system under rational expectations is, AEyt+1{pipe}It=Byt+C(F)Ext{p ipe}It, where yt is a vector of endogenous variables;xt is a vector ofexogenous variables; Eyt+1{pipe}It is the expectation ofyt+1 givendate t information; and C(F)Ext{pipe}It =C0xt+C1Ext+1{pipe}It+...+CnExt+n{pipe}It. If the model issolvable, then ytcan be decomposed into two sets of variables:dynamicvariables dt that evolve according to Edt+1{pipe}It = Wdt + Ψd(F)Ext{pipe}It and other variables thatobey the dynamicidentities ft =-Kdt-Ψf(F)Ext{pipe}It. We developan algorithm for carrying out this decomposition and for constructing theimplied dynamic system. We also provide algorithms for (i) computing perfectforesight solutions and Markov decision rules; and (ii) solutions to relatedmodels that involve informational subperiods.

AB - A first-order linear difference system under rational expectations is, AEyt+1{pipe}It=Byt+C(F)Ext{p ipe}It, where yt is a vector of endogenous variables;xt is a vector ofexogenous variables; Eyt+1{pipe}It is the expectation ofyt+1 givendate t information; and C(F)Ext{pipe}It =C0xt+C1Ext+1{pipe}It+...+CnExt+n{pipe}It. If the model issolvable, then ytcan be decomposed into two sets of variables:dynamicvariables dt that evolve according to Edt+1{pipe}It = Wdt + Ψd(F)Ext{pipe}It and other variables thatobey the dynamicidentities ft =-Kdt-Ψf(F)Ext{pipe}It. We developan algorithm for carrying out this decomposition and for constructing theimplied dynamic system. We also provide algorithms for (i) computing perfectforesight solutions and Markov decision rules; and (ii) solutions to relatedmodels that involve informational subperiods.

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KW - models

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System Reduction and Solution Algorithms for Singular Linear Difference Systems under Rational Expectations

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