Dunare AIM Solver Subsystem

AIM=20 Solver Subsystem=20
APPENDIX=20 1: AIM System SPecification and Dynare Mapping=20
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 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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Dunare AIM Solver Subsystem

Contents

APPENDIX 1: AIM System SPecification and Dynare Mapping

APPENDIX 2: dynAIMsolver1 Function Specification