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Subject: Dunare AIM Solver Subsystem
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Date: Tue, 2 Sep 2008 22:25:31 +0100
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<BODY>
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<DIV class=3Dcontent>
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<H1>Dunare AIM Solver Subsystem</H1><INTRODUCTION></INTRODUCTION>
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<H2>Contents</H2>
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<DIV>
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<UL>
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<LI><A=20
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=
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href=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Aca=
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demia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc.html#1"=
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>AIM=20
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Solver Subsystem</A>=20
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<LI><A=20
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=
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href=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Aca=
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demia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc.html#2"=
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>APPENDIX=20
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1: AIM System SPecification and Dynare Mapping</A>=20
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<LI><A=20
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=
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href=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Aca=
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demia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc.html#11=
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">APPENDIX=20
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2: dynAIMsolver1 Function Specification</A> </LI></UL></DIV>
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<H2>AIM Solver Subsystem<A name=3D1></A></H2>
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<P>The AIM subsystem in the AIM subdirectory of the main Dynare matlab =
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directory=20
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contains MATLAB functions necessary for using Gary Anderson's AIM 1st =
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order=20
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solver as an alternative to Dynare's default mjdgges solver (see <A=20
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href=3D"http://www.federalreserve.gov/Pubs/oss/oss4/aimindex.html">http:/=
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/www.federalreserve.gov/Pubs/oss/oss4/aimindex.html</A>=20
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). </P>
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<P>It cosists of:</P>
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<DIV>
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<UL>
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<LI>New Dynare function <B>dynAIMsolver1(jacobia_, M_, dr)</B> which =
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is called=20
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from <B>dr1.m</B> and which maps Dynare system to the AIM package =
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subsystem.=20
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It then derives the solution for gy=3Ddr.hgx and gu=3Ddr.hgu from the =
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AIM outputs.=20
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("1" in the title is for 1st order solver). </LI></UL></DIV>
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<DIV>
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<UL>
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<LI>A subset of MATLAB routines from Gary Anderson's own AIM package =
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needed to=20
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compute and solve system passed on and returned by dynAIMsolver1 whose =
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names=20
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start with SP.. of which <B>SPAmalg.m</B> is the main driver: =
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</LI></UL></DIV>
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<DIV>
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<UL>
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<LI>SPAmalg.m=20
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<LI>SPBuild_a.m=20
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<LI>SPShiftright.m=20
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<LI>SPExact_shift.m=20
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<LI>SPNumeric_shift.m=20
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<LI>SPObstruct.m=20
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<LI>SPEigensystem.m=20
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<LI>SPReduced_form=20
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<LI>SPCopy_w.m=20
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<LI>SPAimerr.m </LI></UL></DIV>
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<P>The path to the AIM directory, if exists, is added by =
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<B>dynare_config.m</B>=20
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using addpath </P>
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<P><B>USE:</B></P>
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<P>Dynare DR1.m tries to invoke AIM solver instead default mjdgges if=20
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options_.useAIM =3D=3D 1 is set and, if not check only, and if 1st order =
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solution is=20
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needed, i.e.: </P><PRE> if (options_.useAIM =3D=3D 1) && (task =
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=3D=3D 0) && (options_.order =3D=3D 1)</PRE>
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<P>For a start, options_.useAIM =3D 0 is set by default in=20
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<B>global_initialization.m</B> so that system uses mjdgges by default. =
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</P>
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<P>If AIM is to be used, options_.useAIM =3D 1 needs to be set either in =
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the model=20
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<A=20
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href=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Aca=
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demia/Economics/Dynare%20DSGE/V4/doc/AIM/modelname">modelname</A>.mod=20
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file, before invoking, estimate and/or stoch_simul, or by issuing =
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appropriate=20
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command for estimate and/or stoch_simul. </P>
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<P><B>RELEASE NOTES:</B></P>
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<P>In the current implementation, as of July 2008, only first order =
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solution is=20
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supported and handling of exceptions is rather fundamental and, in =
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particular,=20
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when Blanchard and Kahn conditions are not met, only a large penalty =
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value=20
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1.0e+8 is being set. </P>
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<P>Hence, system may not coverge or the resluts may not be accurate if =
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there=20
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were many messages like</P>
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<DIV>
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<UL>
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<LI>"Error in AIM: aimcode=3D4 : Aim: too few big roots", or=20
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<LI>"Error in AIM: aimcode=3D3 : Aim: too many big roots" =
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</LI></UL></DIV>
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<P>especially when issued close to the point of convergence.</P>
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<P>However, if other exceptions occur and aimcode (see codes below) is =
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higher=20
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than 5, the system resets options_.useAIM =3D 0 and tries to use mjdgges =
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instead.=20
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</P>
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<H2>APPENDIX 1: AIM System SPecification and Dynare Mapping<A =
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name=3D2></A></H2>
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<P>AIM System for thau lags and theta leads, and:</P>
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<P><IMG hspace=3D5=20
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src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
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emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq5664.p=
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ng"=20
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vspace=3D5> </P>
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<P><IMG hspace=3D5=20
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src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
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emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq45937.=
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png"=20
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vspace=3D5> </P>
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<P>where xt+i is system vectors at time t for all lag/lead t+i and zt is =
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vector=20
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of exogenous shocks.</P>
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<P>The AIM input is array of matrices <B>H</B>: </P>
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<P><IMG hspace=3D5=20
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src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
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emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq45249.=
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png"=20
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vspace=3D5> </P>
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<P>and its solution given as:</P>
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<P><IMG hspace=3D5=20
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src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
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emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq69844.=
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png"=20
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vspace=3D5> </P>
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<P>where Xt is matrix of vectors of all current system variables and =
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forward=20
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looking leads xi for i=3Dt,...,t+theta:</P>
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<P><IMG hspace=3D5=20
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src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
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emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq235275=
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.png"=20
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vspace=3D5> </P>
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<P>and AIM output in the form of endogenous transition matrix <B>bb</B>: =
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</P>
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<P><IMG hspace=3D5=20
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src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
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emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq25810.=
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png"=20
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vspace=3D5> </P>
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<P>and, for simple case of one lag system, the matrix Phi derived =
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as:</P>
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<P><IMG hspace=3D5=20
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src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
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emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq13129.=
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png"=20
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vspace=3D5> </P>
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<P>For more lags, the phi equation becomes more complicated (see =
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documentation=20
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on G.Anderson's site above).</P>
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<P><B>Dynare AIM Mapping - input</B></P>
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<P>For Dynare jacobian =3D [fy'-tau... fy'i ... fy'+theta fu'] - where =
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-tau and=20
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+theta are subscripts, we have that its subset without exogenous term =
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fu' and=20
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expanded with zero columns represents <B>H</B>, i.e.: </P>
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<P><IMG hspace=3D5=20
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src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
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emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq372007=
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.png"=20
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vspace=3D5> </P>
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<P>and for exogenous shocks terms:</P>
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<P><IMG hspace=3D5=20
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src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
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emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq3403.p=
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ng"=20
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vspace=3D5> </P>
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<P><B>Output</B> Dynare solution output: </P>
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<P><IMG hspace=3D5=20
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src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
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emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq76947.=
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png"=20
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vspace=3D5> </P>
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<P>where Xt is again matrix of vectors all current system variables and =
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forward=20
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looking leads xi for i=3Dt,..., t+theta, is mapped so that: </P>
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<DIV>
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<UL>
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<LI>gy (or dr.ghx) is a reordered subset of AIM <B>SPAmalg.m</B> =
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output=20
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<B>bb</B> without zero columns, and,=20
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<LI>gu (or dr.ghu) is derived from reordered AIM <B>SPObstruct.m</B> =
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output as=20
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phi in: </LI></UL></DIV>
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<P><IMG hspace=3D5=20
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src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
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emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq33587.=
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png"=20
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vspace=3D5> </P>
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<H2>APPENDIX 2: dynAIMsolver1 Function Specification<A =
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name=3D11></A></H2>
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<P><B>function [dr,aimcode]=3DdynAIMsolver1(jacobia_,M_,dr)</B></P>
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<P><B>INPUTS</B></P>
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<DIV>
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<UL>
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<LI>jacobia_ - [matrix] 1st order derivative of the model=20
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<LI>dr - [matlab structure] Decision rules for stochastic simulations. =
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<LI>M_ - [matlab structure] Definition of the model. </LI></UL></DIV>
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<P><B>OUTPUTS</B></P>
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<DIV>
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<UL>
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<LI>dr [matlab structure] Decision rules for stochastic simulations.=20
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<LI>aimcode [integer] status </LI></UL></DIV><PRE>aimcode status is =
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resolved by calling AIMerr as</PRE>
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<DIV>
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<UL>
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<LI>(c=3D=3D1) e=3D'Aim: unique solution.';=20
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<LI>(c=3D=3D2) e=3D'Aim: roots not correctly computed by real_schur.'; =
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<LI>(c=3D=3D3) e=3D'Aim: too many big roots.';=20
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<LI>(c=3D=3D35) e=3D'Aim: too many big roots, and q(:,right) is =
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singular.';=20
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<LI>(c=3D=3D4) e=3D'Aim: too few big roots.';=20
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<LI>(c=3D=3D45) e=3D'Aim: too few big roots, and q(:,right) is =
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singular.';=20
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<LI>(c=3D=3D5) e=3D'Aim: q(:,right) is singular.';=20
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<LI>(c=3D=3D61) e=3D'Aim: too many exact shiftrights.';=20
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<LI>(c=3D=3D62) e=3D'Aim: too many numeric shiftrights.';=20
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<LI>else e=3D'Aimerr: return code not properly specified'; =
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</LI></UL></DIV>
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<P><B>SPECIAL REQUIREMENTS</B></P>
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<P>Dynare use:</P><PRE> 1) the lognormal block in DR1 is being =
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invoked for some models and changing
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values of ghx and ghy. We need to return the AIM output
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values before that block and run the block with the current =
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returned values
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of gy (i.e. dr.ghx) and gu (dr.ghu) if it is needed even when the =
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AIM is used
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(it does not depend on mjdgges output).</PRE><PRE> 2) for =
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forward looking models, passing into dynAIMsolver aa =3D{Q'|1}*jacobia_
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can produce ~ one order closer results to the Dynare solutiion
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then when if plain jacobia_ is passed,
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i.e. diff < e-14 for aa and diff < *e-13 for jacobia_ if Q' =
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is used.</PRE>
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<P>GP July 2008</P>
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<P>part of Dynare, copyright Dynare Team (1996-2008) Gnu Public =
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License.</P>
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<P class=3Dfooter><BR>Published with MATLAB=C2=AE 7.1<BR></P></DIV><!--=0A=
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##### SOURCE BEGIN #####=0A=
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%% Dunare AIM Solver Subsystem=20
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%% AIM Solver Subsystem
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% The AIM subsystem in the AIM subdirectory of the main Dynare matlab
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% directory contains MATLAB functions necessary for using
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% Gary Anderson's AIM 1st order solver as an alternative to Dynare's =
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default mjdgges solver (see =
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http://www.federalreserve.gov/Pubs/oss/oss4/aimindex.html ).=20
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%
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% It cosists of:
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%
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% * New Dynare function *dynAIMsolver1(jacobia_, M_, dr)* which is =
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called
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% from *dr1.m* and which maps Dynare system to the AIM package =
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subsystem.=20
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% It then derives the solution for gy=3Ddr.hgx and
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% gu=3Ddr.hgu from the AIM outputs. ("1" in the title is for 1st order
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% solver).
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%
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% * A subset of MATLAB routines from Gary Anderson's own AIM package =
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needed to compute
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% and solve system passed on and returned by dynAIMsolver1 whose names =
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start with SP..=20
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% of which *SPAmalg.m* is the main driver:
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%
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% * SPAmalg.m
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% * SPBuild_a.m
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% * SPShiftright.m
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% * SPExact_shift.m
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% * SPNumeric_shift.m
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% * SPObstruct.m
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% * SPEigensystem.m
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% * SPReduced_form
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% * SPCopy_w.m
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% * SPAimerr.m
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%
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% The path to the AIM directory, if exists, is added by =
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*dynare_config.m* using addpath
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%
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% *USE:*
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%
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% Dynare DR1.m tries to invoke AIM solver instead default mjdgges if =
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options_.useAIM =3D=3D 1 is set and, if not check only, and if 1st order =
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solution is needed, i.e.:
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%
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|
% if (options_.useAIM =3D=3D 1) && (task =3D=3D 0) && (options_.order =
|
|
=3D=3D 1)=20
|
|
%
|
|
% For a start, options_.useAIM =3D 0 is set by default in =
|
|
*global_initialization.m* so that system uses mjdgges by default.=20
|
|
%
|
|
% If AIM is to be used, options_.useAIM =3D 1 needs to be set either in =
|
|
the model <modelname>.mod file, before invoking, estimate and/or =
|
|
stoch_simul, or by issuing appropriate command for estimate and/or =
|
|
stoch_simul.=20
|
|
%
|
|
% *RELEASE NOTES:*=20
|
|
%
|
|
% In the current implementation, as of July 2008, only first order =
|
|
solution is supported and handling of exceptions is rather fundamental =
|
|
and, in particular, when Blanchard and Kahn conditions are not met, only =
|
|
a large penalty value 1.0e+8 is being set.
|
|
%
|
|
% Hence, system may not coverge or the resluts may not be accurate if =
|
|
there were many messages like
|
|
%
|
|
% * "Error in AIM: aimcode=3D4 : Aim: too few big roots", or
|
|
% * "Error in AIM: aimcode=3D3 : Aim: too many big roots"
|
|
%
|
|
% especially when issued close to the point of convergence.
|
|
%
|
|
% However, if other exceptions occur and aimcode (see codes below) is =
|
|
higher than 5, the system resets options_.useAIM =3D 0 and tries to use =
|
|
mjdgges instead.
|
|
|
|
|
|
%% APPENDIX 1: AIM System SPecification and Dynare Mapping
|
|
%%
|
|
% AIM System for thau lags and theta leads, and:
|
|
%=20
|
|
% $$i=3D-\tau...+\theta $$=20
|
|
%=20
|
|
%%
|
|
%=20
|
|
% $$ \sum_{i=3D-\tau}^\theta(H_i*x_{t+i})=3D \Psi*z_t$$
|
|
%
|
|
%%
|
|
% where xt+i is system vectors at time t for all lag/lead t+i and zt is =
|
|
vector
|
|
% of exogenous shocks.
|
|
%
|
|
% The AIM input is array of matrices *H*:
|
|
%
|
|
% $$H=3D[H_{-\tau} \ ...\ H_i \ ...\ H_{+\theta}] $$
|
|
%
|
|
% and its solution given as:
|
|
%%
|
|
%=20
|
|
% $$ X_t=3D\sum_{i=3D-\tau}^{-1}(B_i*x_{t+i}) + \phi*\Psi*z_t$$
|
|
%
|
|
% where Xt is matrix of vectors of all current system variables and =
|
|
forward
|
|
% looking leads xi for i=3Dt,...,t+theta:
|
|
%
|
|
% $$X_t=3D \left[\begin{array}{c} {x_{t+\theta} } \\ {...} \\ {x_{t} }
|
|
% \end{array}\right]$$
|
|
%
|
|
%%
|
|
% and AIM output in the form of endogenous transition matrix *bb*:
|
|
%%
|
|
%
|
|
% $$bb=3D[B_{-\tau}... B_i \ ...\ B_{-1}]$$=20
|
|
%
|
|
% and, for simple case of one lag system, the matrix Phi derived as:
|
|
%
|
|
% $$ \phi=3D(H_O+H_1*B_{-1})^{-1}$$
|
|
%=20
|
|
%%
|
|
% For more lags, the phi equation becomes more complicated (see =
|
|
documentation on G.Anderson's site above).
|
|
%
|
|
% *Dynare AIM Mapping - input*
|
|
%
|
|
% For Dynare jacobian =3D [fy'-tau... fy'i ... fy'+theta fu'] - where =
|
|
-tau and +theta are subscripts,=20
|
|
% we have that its subset without exogenous term fu' and expanded with =
|
|
zero columns represents *H*, i.e.:
|
|
%
|
|
%%
|
|
%=20
|
|
% $$ [f_{y,-\tau}' \ ...\ f_{y,i}' \ ...\ f_{y,+\theta}']=3D[H_{-\tau} =
|
|
\
|
|
% ...\ H_i \ ...\ H_{+\theta}] $$
|
|
%
|
|
% and for exogenous shocks terms:=20
|
|
%
|
|
% $$ f_u' =3D - \Psi$$
|
|
%=20
|
|
% *Output*
|
|
% Dynare solution output:
|
|
%=20
|
|
% $$ X_t =3D \sum_{i=3D-\tau}^{-1}(g_{y,t+i}*x_{t+i})+ g_u*z_t $$
|
|
%
|
|
% where Xt is again matrix of vectors all current system variables and =
|
|
forward looking leads
|
|
% xi for i=3Dt,..., t+theta, is mapped so that:
|
|
%
|
|
% * gy (or dr.ghx) is a reordered subset of AIM *SPAmalg.m* output *bb*
|
|
% without zero columns, and,
|
|
% * gu (or dr.ghu) is derived from reordered AIM *SPObstruct.m* output =
|
|
as phi in:
|
|
%
|
|
% $$dr.ghu=3Dg_u=3D - \phi * \Psi=3D - \phi * f_u'$$
|
|
%
|
|
|
|
%% APPENDIX 2: dynAIMsolver1 Function Specification
|
|
|
|
%%
|
|
% *function [dr,aimcode]=3DdynAIMsolver1(jacobia_,M_,dr)*=20
|
|
%
|
|
% *INPUTS* =20
|
|
%=20
|
|
% * jacobia_ - [matrix] 1st order derivative of the model
|
|
% * dr - [matlab structure] Decision rules for stochastic =
|
|
simulations.
|
|
% * M_ - [matlab structure] Definition of the model.
|
|
% =20
|
|
% *OUTPUTS*
|
|
%%
|
|
% * dr [matlab structure] Decision rules for stochastic =
|
|
simulations.
|
|
% * aimcode [integer] status
|
|
%
|
|
% aimcode status is resolved by calling AIMerr as
|
|
%=20
|
|
% * (c=3D=3D1) e=3D'Aim: unique solution.';
|
|
% * (c=3D=3D2) e=3D'Aim: roots not correctly computed by =
|
|
real_schur.';
|
|
% * (c=3D=3D3) e=3D'Aim: too many big roots.';
|
|
% * (c=3D=3D35) e=3D'Aim: too many big roots, and q(:,right) is =
|
|
singular.';
|
|
% * (c=3D=3D4) e=3D'Aim: too few big roots.';
|
|
% * (c=3D=3D45) e=3D'Aim: too few big roots, and q(:,right) is =
|
|
singular.';
|
|
% * (c=3D=3D5) e=3D'Aim: q(:,right) is singular.';
|
|
% * (c=3D=3D61) e=3D'Aim: too many exact shiftrights.';
|
|
% * (c=3D=3D62) e=3D'Aim: too many numeric shiftrights.';
|
|
% * else e=3D'Aimerr: return code not properly specified';
|
|
% =20
|
|
% *SPECIAL REQUIREMENTS*
|
|
%
|
|
% Dynare use:=20
|
|
%%
|
|
% 1) the lognormal block in DR1 is being invoked for some models =
|
|
and changing
|
|
% values of ghx and ghy. We need to return the AIM output
|
|
% values before that block and run the block with the current =
|
|
returned values
|
|
% of gy (i.e. dr.ghx) and gu (dr.ghu) if it is needed even when =
|
|
the AIM is used =20
|
|
% (it does not depend on mjdgges output).
|
|
% =20
|
|
% 2) for forward looking models, passing into dynAIMsolver aa =
|
|
=3D{Q'|1}*jacobia_=20
|
|
% can produce ~ one order closer results to the Dynare solutiion =
|
|
|
|
% then when if plain jacobia_ is passed,=20
|
|
% i.e. diff < e-14 for aa and diff < *e-13 for jacobia_ if Q' is =
|
|
used. =20
|
|
%
|
|
% GP July 2008 =20
|
|
%
|
|
% part of Dynare, copyright Dynare Team (1996-2008)
|
|
% Gnu Public License.
|
|
=0A=
|
|
##### SOURCE END #####=0A=
|
|
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