1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
|
<html xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<!--
This HTML is auto-generated from an M-file.
To make changes, update the M-file and republish this document.
-->
<title>Dunare AIM Solver Subsystem</title>
<meta name="generator" content="MATLAB 7.1">
<meta name="date" content="2008-09-02">
<meta name="m-file" content="Dynare AIM use Doc"><style>
body {
background-color: white;
margin:10px;
}
h1 {
color: #990000;
font-size: x-large;
}
h2 {
color: #990000;
font-size: medium;
}
/* Make the text shrink to fit narrow windows, but not stretch too far in
wide windows. On Gecko-based browsers, the shrink-to-fit doesn't work. */
p,h1,h2,div.content div {
/* for MATLAB's browser */
width: 600px;
/* for Mozilla, but the "width" tag overrides it anyway */
max-width: 600px;
/* for IE */
width:expression(document.body.clientWidth > 620 ? "600px": "auto" );
}
pre.codeinput {
background: #EEEEEE;
padding: 10px;
}
span.keyword {color: #0000FF}
span.comment {color: #228B22}
span.string {color: #A020F0}
span.untermstring {color: #B20000}
span.syscmd {color: #B28C00}
pre.codeoutput {
color: #666666;
padding: 10px;
}
pre.error {
color: red;
}
p.footer {
text-align: right;
font-size: xx-small;
font-weight: lighter;
font-style: italic;
color: gray;
}
</style></head>
<body>
<div class="content">
<h1>Dunare AIM Solver Subsystem</h1>
<introduction></introduction>
<h2>Contents</h2>
<div>
<ul>
<li><a href="#1">AIM Solver Subsystem</a></li>
<li><a href="#2">APPENDIX 1: AIM System SPecification and Dynare Mapping</a></li>
<li><a href="#11">APPENDIX 2: dynAIMsolver1 Function Specification</a></li>
</ul>
</div>
<h2>AIM Solver Subsystem<a name="1"></a></h2>
<p>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 <a href="http://www.federalreserve.gov/Pubs/oss/oss4/aimindex.html">http://www.federalreserve.gov/Pubs/oss/oss4/aimindex.html</a> ).
</p>
<p>It cosists of:</p>
<div>
<ul>
<li>New Dynare function <b>dynAIMsolver1(jacobia_, M_, dr)</b> which is called from <b>dr1.m</b> and which maps Dynare system to the AIM package subsystem. It then derives the solution for gy=dr.hgx and gu=dr.hgu from
the AIM outputs. ("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 compute and solve system passed on and returned
by dynAIMsolver1 whose names start with SP.. of which <b>SPAmalg.m</b> is the main driver:
</li>
</ul>
</div>
<div>
<ul>
<li>SPAmalg.m</li>
<li>SPBuild_a.m</li>
<li>SPShiftright.m</li>
<li>SPExact_shift.m</li>
<li>SPNumeric_shift.m</li>
<li>SPObstruct.m</li>
<li>SPEigensystem.m</li>
<li>SPReduced_form</li>
<li>SPCopy_w.m</li>
<li>SPAimerr.m</li>
</ul>
</div>
<p>The path to the AIM directory, if exists, is added by <b>dynare_config.m</b> using addpath
</p>
<p><b>USE:</b></p>
<p>Dynare DR1.m tries to invoke AIM solver instead default mjdgges if options_.useAIM == 1 is set and, if not check only, and
if 1st order solution is needed, i.e.:
</p><pre> if (options_.useAIM == 1) && (task == 0) && (options_.order == 1)</pre><p>For a start, options_.useAIM = 0 is set by default in <b>global_initialization.m</b> so that system uses mjdgges by default.
</p>
<p>If AIM is to be used, options_.useAIM = 1 needs to be set either in the model <a href="modelname">modelname</a>.mod file, before invoking, estimate and/or stoch_simul, or by issuing appropriate 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 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.
</p>
<p>Hence, system may not coverge or the resluts may not be accurate if there were many messages like</p>
<div>
<ul>
<li>"Error in AIM: aimcode=4 : Aim: too few big roots", or</li>
<li>"Error in AIM: aimcode=3 : 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 than 5, the system resets options_.useAIM = 0 and
tries to use mjdgges instead.
</p>
<h2>APPENDIX 1: AIM System SPecification and Dynare Mapping<a name="2"></a></h2>
<p>AIM System for thau lags and theta leads, and:</p>
<p><img vspace="5" hspace="5" src="Dynare AIM use Doc_eq5664.png"> </p>
<p><img vspace="5" hspace="5" src="Dynare AIM use Doc_eq45937.png"> </p>
<p>where xt+i is system vectors at time t for all lag/lead t+i and zt is vector of exogenous shocks.</p>
<p>The AIM input is array of matrices <b>H</b>:
</p>
<p><img vspace="5" hspace="5" src="Dynare AIM use Doc_eq45249.png"> </p>
<p>and its solution given as:</p>
<p><img vspace="5" hspace="5" src="Dynare AIM use Doc_eq69844.png"> </p>
<p>where Xt is matrix of vectors of all current system variables and forward looking leads xi for i=t,...,t+theta:</p>
<p><img vspace="5" hspace="5" src="Dynare AIM use Doc_eq235275.png"> </p>
<p>and AIM output in the form of endogenous transition matrix <b>bb</b>:
</p>
<p><img vspace="5" hspace="5" src="Dynare AIM use Doc_eq25810.png"> </p>
<p>and, for simple case of one lag system, the matrix Phi derived as:</p>
<p><img vspace="5" hspace="5" src="Dynare AIM use Doc_eq13129.png"> </p>
<p>For more lags, the phi equation becomes more complicated (see documentation on G.Anderson's site above).</p>
<p><b>Dynare AIM Mapping - input</b></p>
<p>For Dynare jacobian = [fy'-tau... fy'i ... fy'+theta fu'] - where -tau and +theta are subscripts, we have that its subset
without exogenous term fu' and expanded with zero columns represents <b>H</b>, i.e.:
</p>
<p><img vspace="5" hspace="5" src="Dynare AIM use Doc_eq372007.png"> </p>
<p>and for exogenous shocks terms:</p>
<p><img vspace="5" hspace="5" src="Dynare AIM use Doc_eq3403.png"> </p>
<p><b>Output</b> Dynare solution output:
</p>
<p><img vspace="5" hspace="5" src="Dynare AIM use Doc_eq76947.png"> </p>
<p>where Xt is again matrix of vectors all current system variables and forward looking leads xi for i=t,..., 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 <b>bb</b> without zero columns, and,
</li>
<li>gu (or dr.ghu) is derived from reordered AIM <b>SPObstruct.m</b> output as phi in:
</li>
</ul>
</div>
<p><img vspace="5" hspace="5" src="Dynare AIM use Doc_eq33587.png"> </p>
<h2>APPENDIX 2: dynAIMsolver1 Function Specification<a name="11"></a></h2>
<p><b>function [dr,aimcode]=dynAIMsolver1(jacobia_,M_,dr)</b></p>
<p><b>INPUTS</b></p>
<div>
<ul>
<li>jacobia_ - [matrix] 1st order derivative of the model</li>
<li>dr - [matlab structure] Decision rules for stochastic simulations.</li>
<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.</li>
<li> aimcode [integer] status</li>
</ul>
</div><pre>aimcode status is resolved by calling AIMerr as</pre><div>
<ul>
<li> (c==1) e='Aim: unique solution.';</li>
<li> (c==2) e='Aim: roots not correctly computed by real_schur.';</li>
<li> (c==3) e='Aim: too many big roots.';</li>
<li> (c==35) e='Aim: too many big roots, and q(:,right) is singular.';</li>
<li> (c==4) e='Aim: too few big roots.';</li>
<li> (c==45) e='Aim: too few big roots, and q(:,right) is singular.';</li>
<li> (c==5) e='Aim: q(:,right) is singular.';</li>
<li> (c==61) e='Aim: too many exact shiftrights.';</li>
<li> (c==62) e='Aim: too many numeric shiftrights.';</li>
<li> else e='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 ={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.</pre><p>GP July 2008</p>
<p>part of Dynare, copyright Dynare Team (1996-2008) Gnu Public License.</p>
<p class="footer"><br>
Published with MATLAB® 7.1<br></p>
</div>
<!--
##### SOURCE BEGIN #####
%% Dunare AIM Solver Subsystem
%% 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 ).
%
% 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.
% It then derives the solution for gy=dr.hgx and
% gu=dr.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..
% 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 == 1 is set and, if not check only, and if 1st order solution is needed, i.e.:
%
% if (options_.useAIM == 1) && (task == 0) && (options_.order == 1)
%
% For a start, options_.useAIM = 0 is set by default in *global_initialization.m* so that system uses mjdgges by default.
%
% If AIM is to be used, options_.useAIM = 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.
%
% *RELEASE NOTES:*
%
% 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=4 : Aim: too few big roots", or
% * "Error in AIM: aimcode=3 : 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 = 0 and tries to use mjdgges instead.
%% APPENDIX 1: AIM System SPecification and Dynare Mapping
%%
% AIM System for thau lags and theta leads, and:
%
% $$i=-\tau...+\theta $$
%
%%
%
% $$ \sum_{i=-\tau}^\theta(H_i*x_{t+i})= \Psi*z_t$$
%
%%
% 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*:
%
% $$H=[H_{-\tau} \ ...\ H_i \ ...\ H_{+\theta}] $$
%
% and its solution given as:
%%
%
% $$ X_t=\sum_{i=-\tau}^{-1}(B_i*x_{t+i}) + \phi*\Psi*z_t$$
%
% where Xt is matrix of vectors of all current system variables and forward
% looking leads xi for i=t,...,t+theta:
%
% $$X_t= \left[\begin{array}{c} {x_{t+\theta} } \\ {...} \\ {x_{t} }
% \end{array}\right]$$
%
%%
% and AIM output in the form of endogenous transition matrix *bb*:
%%
%
% $$bb=[B_{-\tau}... B_i \ ...\ B_{-1}]$$
%
% and, for simple case of one lag system, the matrix Phi derived as:
%
% $$ \phi=(H_O+H_1*B_{-1})^{-1}$$
%
%%
% For more lags, the phi equation becomes more complicated (see documentation on G.Anderson's site above).
%
% *Dynare AIM Mapping - input*
%
% For Dynare jacobian = [fy'-tau... fy'i ... fy'+theta fu'] - where -tau and +theta are subscripts,
% we have that its subset without exogenous term fu' and expanded with zero columns represents *H*, i.e.:
%
%%
%
% $$ [f_{y,-\tau}' \ ...\ f_{y,i}' \ ...\ f_{y,+\theta}']=[H_{-\tau} \
% ...\ H_i \ ...\ H_{+\theta}] $$
%
% and for exogenous shocks terms:
%
% $$ f_u' = - \Psi$$
%
% *Output*
% Dynare solution output:
%
% $$ X_t = \sum_{i=-\tau}^{-1}(g_{y,t+i}*x_{t+i})+ g_u*z_t $$
%
% where Xt is again matrix of vectors all current system variables and forward looking leads
% xi for i=t,..., 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:
%
% $$dr.ghu=g_u= - \phi * \Psi= - \phi * f_u'$$
%
%% APPENDIX 2: dynAIMsolver1 Function Specification
%%
% *function [dr,aimcode]=dynAIMsolver1(jacobia_,M_,dr)*
%
% *INPUTS*
%
% * jacobia_ - [matrix] 1st order derivative of the model
% * dr - [matlab structure] Decision rules for stochastic simulations.
% * M_ - [matlab structure] Definition of the model.
%
% *OUTPUTS*
%%
% * dr [matlab structure] Decision rules for stochastic simulations.
% * aimcode [integer] status
%
% aimcode status is resolved by calling AIMerr as
%
% * (c==1) e='Aim: unique solution.';
% * (c==2) e='Aim: roots not correctly computed by real_schur.';
% * (c==3) e='Aim: too many big roots.';
% * (c==35) e='Aim: too many big roots, and q(:,right) is singular.';
% * (c==4) e='Aim: too few big roots.';
% * (c==45) e='Aim: too few big roots, and q(:,right) is singular.';
% * (c==5) e='Aim: q(:,right) is singular.';
% * (c==61) e='Aim: too many exact shiftrights.';
% * (c==62) e='Aim: too many numeric shiftrights.';
% * else e='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 ={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.
##### SOURCE END #####
-->
</body>
</html>
|
