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Subject: Dunare AIM Solver Subsystem
Date: Tue, 2 Sep 2008 22:25:31 +0100
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TITLE>Dunare AIM Solver Subsystem

Dunare AIM Solver Subsystem
<INTRODUCTION></INTRODUCTION>

Contents

- <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=20
  
- <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=20
  
- <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 

AIM Solver Subsystem

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=20
). 

It cosists of:

- New Dynare function dynAIMsolver1(jacobia_, M_, dr) which =
is called=20
  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.=20
  ("1" in the title is for 1st order solver). 

- 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 SPAmalg.m is the main driver: =

- SPAmalg.m=20
  
- SPBuild_a.m=20
  
- SPShiftright.m=20
  
- SPExact_shift.m=20
  
- SPNumeric_shift.m=20
  
- SPObstruct.m=20
  
- SPEigensystem.m=20
  
- SPReduced_form=20
  
- SPCopy_w.m=20
  
- SPAimerr.m 

The path to the AIM directory, if exists, is added by =
dynare_config.m=20
using addpath 

USE:

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.: 

  if (options_.useAIM =3D=3D 1) && (task =
=3D=3D 0) && (options_.order =3D=3D 1)

For a start, options_.useAIM =3D 0 is set by default in=20
global_initialization.m so that system uses mjdgges by default. =

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.mod=20
file, before invoking, estimate and/or stoch_simul, or by issuing =
appropriate=20
command for estimate and/or stoch_simul. 

RELEASE NOTES:

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. 

Hence, system may not coverge or the resluts may not be accurate if =
there=20
were many messages like

- "Error in AIM: aimcode=3D4 : Aim: too few big roots", or=20
  
- "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=20
than 5, the system resets options_.useAIM =3D 0 and tries to use mjdgges =
instead.=20

APPENDIX 1: AIM System SPecification and Dynare Mapping

AIM System for thau lags and theta leads, and:

where xt+i is system vectors at time t for all lag/lead t+i and zt is =
vector=20
of exogenous shocks.

The AIM input is array of matrices H: 

and its solution given as:

where Xt is matrix of vectors of all current system variables and =
forward=20
looking leads xi for i=3Dt,...,t+theta:

and AIM output in the form of endogenous transition matrix bb: =

and, for simple case of one lag system, the matrix Phi derived =
as:

For more lags, the phi equation becomes more complicated (see =
documentation=20
on G.Anderson's site above).

Dynare AIM Mapping - input

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 H, i.e.: 

and for exogenous shocks terms:

Output Dynare solution output: 

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: 

- gy (or dr.ghx) is a reordered subset of AIM SPAmalg.m =
output=20
  bb without zero columns, and,=20
  
- gu (or dr.ghu) is derived from reordered AIM SPObstruct.m =
output as=20
  phi in: 

APPENDIX 2: dynAIMsolver1 Function Specification

function [dr,aimcode]=3DdynAIMsolver1(jacobia_,M_,dr)

INPUTS

- jacobia_ - [matrix] 1st order derivative of the model=20
  
- dr - [matlab structure] Decision rules for stochastic simulations. =

- M_ - [matlab structure] Definition of the model. 

OUTPUTS

- dr [matlab structure] Decision rules for stochastic simulations.=20
  
- aimcode [integer] status 

aimcode status is =
resolved by calling AIMerr as

- (c=3D=3D1) e=3D'Aim: unique solution.';=20
  
- (c=3D=3D2) e=3D'Aim: roots not correctly computed by real_schur.'; =

- (c=3D=3D3) e=3D'Aim: too many big roots.';=20
  
- (c=3D=3D35) e=3D'Aim: too many big roots, and q(:,right) is =
singular.';=20
  
- (c=3D=3D4) e=3D'Aim: too few big roots.';=20
  
- (c=3D=3D45) e=3D'Aim: too few big roots, and q(:,right) is =
singular.';=20
  
- (c=3D=3D5) e=3D'Aim: q(:,right) is singular.';=20
  
- (c=3D=3D61) e=3D'Aim: too many exact shiftrights.';=20
  
- (c=3D=3D62) e=3D'Aim: too many numeric shiftrights.';=20
  
- else e=3D'Aimerr: return code not properly specified'; =

SPECIAL REQUIREMENTS

Dynare use:

     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).

     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 < e-14 for aa and diff < *e-13 for jacobia_ if Q' =
is used.

GP July 2008

part of Dynare, copyright Dynare Team (1996-2008) Gnu Public =
License.

Published with MATLAB=C2=AE 7.1

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