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<DIV class=3Dcontent>
<H1>Dunare AIM Solver Subsystem</H1><INTRODUCTION></INTRODUCTION>
<H2>Contents</H2>
<DIV>
<UL>
  <LI><A=20
  =
href=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Aca=
demia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc.html#1"=
>AIM=20
  Solver Subsystem</A>=20
  <LI><A=20
  =
href=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Aca=
demia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc.html#2"=
>APPENDIX=20
  1: AIM System SPecification and Dynare Mapping</A>=20
  <LI><A=20
  =
href=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Aca=
demia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc.html#11=
">APPENDIX=20
  2: dynAIMsolver1 Function Specification</A> </LI></UL></DIV>
<H2>AIM Solver Subsystem<A name=3D1></A></H2>
<P>The AIM subsystem in the AIM subdirectory of the main Dynare matlab =
directory=20
contains MATLAB functions necessary for using Gary Anderson's AIM 1st =
order=20
solver as an alternative to Dynare's default mjdgges solver (see <A=20
href=3D"http://www.federalreserve.gov/Pubs/oss/oss4/aimindex.html">http:/=
/www.federalreserve.gov/Pubs/oss/oss4/aimindex.html</A>=20
). </P>
<P>It cosists of:</P>
<DIV>
<UL>
  <LI>New Dynare function <B>dynAIMsolver1(jacobia_, M_, dr)</B> which =
is called=20
  from <B>dr1.m</B> and which maps Dynare system to the AIM package =
subsystem.=20
  It then derives the solution for gy=3Ddr.hgx and gu=3Ddr.hgu from the =
AIM outputs.=20
  ("1" in the title is for 1st order solver). </LI></UL></DIV>
<DIV>
<UL>
  <LI>A subset of MATLAB routines from Gary Anderson's own AIM package =
needed to=20
  compute and solve system passed on and returned by dynAIMsolver1 whose =
names=20
  start with SP.. of which <B>SPAmalg.m</B> is the main driver: =
</LI></UL></DIV>
<DIV>
<UL>
  <LI>SPAmalg.m=20
  <LI>SPBuild_a.m=20
  <LI>SPShiftright.m=20
  <LI>SPExact_shift.m=20
  <LI>SPNumeric_shift.m=20
  <LI>SPObstruct.m=20
  <LI>SPEigensystem.m=20
  <LI>SPReduced_form=20
  <LI>SPCopy_w.m=20
  <LI>SPAimerr.m </LI></UL></DIV>
<P>The path to the AIM directory, if exists, is added by =
<B>dynare_config.m</B>=20
using addpath </P>
<P><B>USE:</B></P>
<P>Dynare DR1.m tries to invoke AIM solver instead default mjdgges if=20
options_.useAIM =3D=3D 1 is set and, if not check only, and if 1st order =
solution is=20
needed, i.e.: </P><PRE>  if (options_.useAIM =3D=3D 1) &amp;&amp; (task =
=3D=3D 0) &amp;&amp; (options_.order =3D=3D 1)</PRE>
<P>For a start, options_.useAIM =3D 0 is set by default in=20
<B>global_initialization.m</B> so that system uses mjdgges by default. =
</P>
<P>If AIM is to be used, options_.useAIM =3D 1 needs to be set either in =
the model=20
<A=20
href=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Aca=
demia/Economics/Dynare%20DSGE/V4/doc/AIM/modelname">modelname</A>.mod=20
file, before invoking, estimate and/or stoch_simul, or by issuing =
appropriate=20
command for estimate and/or stoch_simul. </P>
<P><B>RELEASE NOTES:</B></P>
<P>In the current implementation, as of July 2008, only first order =
solution is=20
supported and handling of exceptions is rather fundamental and, in =
particular,=20
when Blanchard and Kahn conditions are not met, only a large penalty =
value=20
1.0e+8 is being set. </P>
<P>Hence, system may not coverge or the resluts may not be accurate if =
there=20
were many messages like</P>
<DIV>
<UL>
  <LI>"Error in AIM: aimcode=3D4 : Aim: too few big roots", or=20
  <LI>"Error in AIM: aimcode=3D3 : Aim: too many big roots" =
</LI></UL></DIV>
<P>especially when issued close to the point of convergence.</P>
<P>However, if other exceptions occur and aimcode (see codes below) is =
higher=20
than 5, the system resets options_.useAIM =3D 0 and tries to use mjdgges =
instead.=20
</P>
<H2>APPENDIX 1: AIM System SPecification and Dynare Mapping<A =
name=3D2></A></H2>
<P>AIM System for thau lags and theta leads, and:</P>
<P><IMG hspace=3D5=20
src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq5664.p=
ng"=20
vspace=3D5> </P>
<P><IMG hspace=3D5=20
src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq45937.=
png"=20
vspace=3D5> </P>
<P>where xt+i is system vectors at time t for all lag/lead t+i and zt is =
vector=20
of exogenous shocks.</P>
<P>The AIM input is array of matrices <B>H</B>: </P>
<P><IMG hspace=3D5=20
src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq45249.=
png"=20
vspace=3D5> </P>
<P>and its solution given as:</P>
<P><IMG hspace=3D5=20
src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq69844.=
png"=20
vspace=3D5> </P>
<P>where Xt is matrix of vectors of all current system variables and =
forward=20
looking leads xi for i=3Dt,...,t+theta:</P>
<P><IMG hspace=3D5=20
src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq235275=
.png"=20
vspace=3D5> </P>
<P>and AIM output in the form of endogenous transition matrix <B>bb</B>: =
</P>
<P><IMG hspace=3D5=20
src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq25810.=
png"=20
vspace=3D5> </P>
<P>and, for simple case of one lag system, the matrix Phi derived =
as:</P>
<P><IMG hspace=3D5=20
src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq13129.=
png"=20
vspace=3D5> </P>
<P>For more lags, the phi equation becomes more complicated (see =
documentation=20
on G.Anderson's site above).</P>
<P><B>Dynare AIM Mapping - input</B></P>
<P>For Dynare jacobian =3D [fy'-tau... fy'i ... fy'+theta fu'] - where =
-tau and=20
+theta are subscripts, we have that its subset without exogenous term =
fu' and=20
expanded with zero columns represents <B>H</B>, i.e.: </P>
<P><IMG hspace=3D5=20
src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq372007=
.png"=20
vspace=3D5> </P>
<P>and for exogenous shocks terms:</P>
<P><IMG hspace=3D5=20
src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq3403.p=
ng"=20
vspace=3D5> </P>
<P><B>Output</B> Dynare solution output: </P>
<P><IMG hspace=3D5=20
src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq76947.=
png"=20
vspace=3D5> </P>
<P>where Xt is again matrix of vectors all current system variables and =
forward=20
looking leads xi for i=3Dt,..., t+theta, is mapped so that: </P>
<DIV>
<UL>
  <LI>gy (or dr.ghx) is a reordered subset of AIM <B>SPAmalg.m</B> =
output=20
  <B>bb</B> without zero columns, and,=20
  <LI>gu (or dr.ghu) is derived from reordered AIM <B>SPObstruct.m</B> =
output as=20
  phi in: </LI></UL></DIV>
<P><IMG hspace=3D5=20
src=3D"file:///C:/Documents%20and%20Settings/Sculptor/My%20Documents/Acad=
emia/Economics/Dynare%20DSGE/V4/doc/AIM/Dynare%20AIM%20use%20Doc_eq33587.=
png"=20
vspace=3D5> </P>
<H2>APPENDIX 2: dynAIMsolver1 Function Specification<A =
name=3D11></A></H2>
<P><B>function [dr,aimcode]=3DdynAIMsolver1(jacobia_,M_,dr)</B></P>
<P><B>INPUTS</B></P>
<DIV>
<UL>
  <LI>jacobia_ - [matrix] 1st order derivative of the model=20
  <LI>dr - [matlab structure] Decision rules for stochastic simulations. =

  <LI>M_ - [matlab structure] Definition of the model. </LI></UL></DIV>
<P><B>OUTPUTS</B></P>
<DIV>
<UL>
  <LI>dr [matlab structure] Decision rules for stochastic simulations.=20
  <LI>aimcode [integer] status </LI></UL></DIV><PRE>aimcode status is =
resolved by calling AIMerr as</PRE>
<DIV>
<UL>
  <LI>(c=3D=3D1) e=3D'Aim: unique solution.';=20
  <LI>(c=3D=3D2) e=3D'Aim: roots not correctly computed by real_schur.'; =

  <LI>(c=3D=3D3) e=3D'Aim: too many big roots.';=20
  <LI>(c=3D=3D35) e=3D'Aim: too many big roots, and q(:,right) is =
singular.';=20
  <LI>(c=3D=3D4) e=3D'Aim: too few big roots.';=20
  <LI>(c=3D=3D45) e=3D'Aim: too few big roots, and q(:,right) is =
singular.';=20
  <LI>(c=3D=3D5) e=3D'Aim: q(:,right) is singular.';=20
  <LI>(c=3D=3D61) e=3D'Aim: too many exact shiftrights.';=20
  <LI>(c=3D=3D62) e=3D'Aim: too many numeric shiftrights.';=20
  <LI>else e=3D'Aimerr: return code not properly specified'; =
</LI></UL></DIV>
<P><B>SPECIAL REQUIREMENTS</B></P>
<P>Dynare use:</P><PRE>     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
     (it does not depend on mjdgges output).</PRE><PRE>     2) for =
forward looking models, passing into dynAIMsolver aa =3D{Q'|1}*jacobia_
     can produce ~ one order closer results to the Dynare solutiion
     then when if plain jacobia_ is passed,
     i.e. diff &lt; e-14 for aa and diff &lt; *e-13 for jacobia_ if Q' =
is used.</PRE>
<P>GP July 2008</P>
<P>part of Dynare, copyright Dynare Team (1996-2008) Gnu Public =
License.</P>
<P class=3Dfooter><BR>Published with MATLAB=C2=AE 7.1<BR></P></DIV><!--=0A=
##### SOURCE BEGIN #####=0A=
%% Dunare AIM Solver Subsystem=20

%% AIM Solver Subsystem
% The AIM subsystem in the AIM subdirectory of the main Dynare matlab
% directory contains MATLAB functions necessary for using
% Gary Anderson's AIM 1st order solver as an alternative to Dynare's =
default mjdgges solver (see  =
http://www.federalreserve.gov/Pubs/oss/oss4/aimindex.html ).=20
%
% It cosists of:
%
% * New Dynare function *dynAIMsolver1(jacobia_, M_, dr)* which is =
called
% from *dr1.m* and which maps Dynare system to the AIM package =
subsystem.=20
% It then derives the solution for gy=3Ddr.hgx and
% gu=3Ddr.hgu from the AIM outputs. ("1" in the title is for 1st order
% solver).
%
% * A subset of MATLAB routines from Gary Anderson's own AIM package =
needed to compute
% and solve system passed on and returned by dynAIMsolver1 whose names =
start with SP..=20
% of which *SPAmalg.m* is the main driver:
%
% * SPAmalg.m
% * SPBuild_a.m
% * SPShiftright.m
% * SPExact_shift.m
% * SPNumeric_shift.m
% * SPObstruct.m
% * SPEigensystem.m
% * SPReduced_form
% * SPCopy_w.m
% * SPAimerr.m
%
% The path to the AIM directory, if exists, is added by =
*dynare_config.m* using addpath
%
% *USE:*
%
% Dynare DR1.m tries to invoke AIM solver instead default mjdgges if =
options_.useAIM =3D=3D 1 is set and, if not check only, and if 1st order =
solution is needed, i.e.:
%
%    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=
--></BODY></HTML>

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