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
|
function [hessian_mat, gg, htol1, ihh, hh_mat0, hh1, hess_info] = mr_hessian(x,func,penalty,hflag,htol0,hess_info,varargin)
% function [hessian_mat, gg, htol1, ihh, hh_mat0, hh1, hess_info] = mr_hessian(x,func,penalty,hflag,htol0,hess_info,varargin)
% numerical gradient and Hessian, with 'automatic' check of numerical
% error
%
% adapted from Michel Juillard original routine hessian.m
%
% Inputs:
% - func function handle. The function must give two outputs:
% the log-likelihood AND the single contributions at times t=1,...,T
% of the log-likelihood to compute outer product gradient
% - x parameter values
% - penalty penalty due to error code
% - hflag 0: Hessian computed with outer product gradient, one point
% increments for partial derivatives in gradients
% 1: 'mixed' Hessian: diagonal elements computed with numerical second order derivatives
% with correlation structure as from outer product gradient;
% two point evaluation of derivatives for partial derivatives
% in gradients
% 2: full numerical Hessian, computes second order partial derivatives
% uses Abramowitz and Stegun (1965) formulas 25.3.24 and 25.3.27
% p. 884.
% - htol0 'precision' of increment of function values for numerical
% derivatives
% - hess_info structure storing the step sizes for
% computation of Hessian
% - varargin other inputs:
% varargin{1} --> DynareDataset
% varargin{2} --> DatasetInfo
% varargin{3} --> DynareOptions
% varargin{4} --> Model
% varargin{5} --> EstimatedParameters
% varargin{6} --> BayesInfo
% varargin{7} --> Bounds
% varargin{8} --> DynareResults
%
% Outputs
% - hessian_mat hessian
% - gg Jacobian
% - htol1 updated 'precision' of increment of function values for numerical
% derivatives
% - ihh inverse outer product with modified std's
% - hh_mat0 outer product hessian with modified std's
% - hh1 updated hess_info.h1
% - hess_info structure with updated step length
% Copyright (C) 2004-2017 Dynare Team
%
% This file is part of Dynare.
%
% Dynare is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% Dynare is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
n=size(x,1);
[f0,exit_flag, ff0]=penalty_objective_function(x,func,penalty,varargin{:});
h2=varargin{7}.ub-varargin{7}.lb;
hmax=varargin{7}.ub-x;
hmax=min(hmax,x-varargin{7}.lb);
if isempty(ff0)
outer_product_gradient=0;
else
outer_product_gradient=1;
end
hess_info.h1 = min(hess_info.h1,0.5.*hmax);
if htol0<hess_info.htol
hess_info.htol=htol0;
end
xh1=x;
f1=zeros(size(f0,1),n);
f_1=f1;
if outer_product_gradient
ff1=zeros(size(ff0));
ff_1=ff1;
ggh=zeros(size(ff0,1),n);
end
i=0;
hhtol=hess_info.htol*ones(n,1);
while i<n
i=i+1;
hess_info.htol=hhtol(i);
h10=hess_info.h1(i);
hcheck=0;
xh1(i)=x(i)+hess_info.h1(i);
try
[fx,exit_flag,ffx]=penalty_objective_function(xh1,func,penalty,varargin{:});
catch
fx=1.e8;
end
it=1;
dx=(fx-f0);
ic=0;
icount = 0;
h0=hess_info.h1(i);
while (abs(dx(it))<0.5*hess_info.htol || abs(dx(it))>(3*hess_info.htol)) && icount<10 && ic==0
icount=icount+1;
if abs(dx(it))<0.5*hess_info.htol
if abs(dx(it)) ~= 0
hess_info.h1(i)=min(max(1.e-10,0.3*abs(x(i))), 0.9*hess_info.htol/abs(dx(it))*hess_info.h1(i));
else
hess_info.h1(i)=2.1*hess_info.h1(i);
end
hess_info.h1(i) = min(hess_info.h1(i),0.5*hmax(i));
hess_info.h1(i) = max(hess_info.h1(i),1.e-10);
xh1(i)=x(i)+hess_info.h1(i);
try
[fx,exit_flag,ffx]=penalty_objective_function(xh1,func,penalty,varargin{:});
catch
fx=1.e8;
end
end
if abs(dx(it))>(3*hess_info.htol)
hess_info.h1(i)= hess_info.htol/abs(dx(it))*hess_info.h1(i);
hess_info.h1(i) = max(hess_info.h1(i),1e-10);
xh1(i)=x(i)+hess_info.h1(i);
try
[fx,exit_flag,ffx]=penalty_objective_function(xh1,func,penalty,varargin{:});
catch
fx=1.e8;
end
iter=0;
while (fx-f0)==0 && iter<50
hess_info.h1(i)= hess_info.h1(i)*2;
xh1(i)=x(i)+hess_info.h1(i);
[fx,exit_flag,ffx]=penalty_objective_function(xh1,func,penalty,varargin{:});
ic=1;
iter=iter+1;
end
end
it=it+1;
dx(it)=(fx-f0);
h0(it)=hess_info.h1(i);
if (hess_info.h1(i)<1.e-12*min(1,h2(i)) && hess_info.h1(i)<0.5*hmax(i))
ic=1;
hcheck=1;
end
end
f1(:,i)=fx;
if outer_product_gradient
if any(isnan(ffx)) || isempty(ffx)
ff1=ones(size(ff0)).*fx/length(ff0);
else
ff1=ffx;
end
end
xh1(i)=x(i)-hess_info.h1(i);
[fx,exit_flag,ffx]=penalty_objective_function(xh1,func,penalty,varargin{:});
f_1(:,i)=fx;
if outer_product_gradient
if any(isnan(ffx)) || isempty(ffx)
ff_1=ones(size(ff0)).*fx/length(ff0);
else
ff_1=ffx;
end
ggh(:,i)=(ff1-ff_1)./(2.*hess_info.h1(i));
end
xh1(i)=x(i);
if hcheck && hess_info.htol<1
hess_info.htol=min(1,max(min(abs(dx))*2,hess_info.htol*10));
hess_info.h1(i)=h10;
hhtol(i) = hess_info.htol;
i=i-1;
end
end
h_1=hess_info.h1;
xh1=x;
xh_1=xh1;
gg=(f1'-f_1')./(2.*hess_info.h1);
if outer_product_gradient
if hflag==2
gg=(f1'-f_1')./(2.*hess_info.h1);
hessian_mat = zeros(size(f0,1),n*n);
for i=1:n
if i > 1
k=[i:n:n*(i-1)];
hessian_mat(:,(i-1)*n+1:(i-1)*n+i-1)=hessian_mat(:,k);
end
hessian_mat(:,(i-1)*n+i)=(f1(:,i)+f_1(:,i)-2*f0)./(hess_info.h1(i)*h_1(i));
temp=f1+f_1-f0*ones(1,n);
for j=i+1:n
xh1(i)=x(i)+hess_info.h1(i);
xh1(j)=x(j)+h_1(j);
xh_1(i)=x(i)-hess_info.h1(i);
xh_1(j)=x(j)-h_1(j);
temp1 = penalty_objective_function(xh1,func,penalty,varargin{:});
temp2 = penalty_objective_function(xh_1,func,penalty,varargin{:});
hessian_mat(:,(i-1)*n+j)=-(-temp1 -temp2+temp(:,i)+temp(:,j))./(2*hess_info.h1(i)*h_1(j));
xh1(i)=x(i);
xh1(j)=x(j);
xh_1(i)=x(i);
xh_1(j)=x(j);
j=j+1;
end
i=i+1;
end
elseif hflag==1
hessian_mat = zeros(size(f0,1),n*n);
for i=1:n
dum = (f1(:,i)+f_1(:,i)-2*f0)./(hess_info.h1(i)*h_1(i));
if dum>eps
hessian_mat(:,(i-1)*n+i)=dum;
else
hessian_mat(:,(i-1)*n+i)=max(eps, gg(i)^2);
end
end
end
gga=ggh.*kron(ones(size(ff1)),2.*hess_info.h1'); % re-scaled gradient
hh_mat=gga'*gga; % rescaled outer product hessian
hh_mat0=ggh'*ggh; % outer product hessian
A=diag(2.*hess_info.h1); % rescaling matrix
% igg=inv(hh_mat); % inverted rescaled outer product hessian
ihh=A'*(hh_mat\A); % inverted outer product hessian
if hflag>0 && min(eig(reshape(hessian_mat,n,n)))>0
hh0 = A*reshape(hessian_mat,n,n)*A'; %rescaled second order derivatives
hh = reshape(hessian_mat,n,n); %rescaled second order derivatives
sd0=sqrt(diag(hh0)); %rescaled 'standard errors' using second order derivatives
sd=sqrt(diag(hh_mat)); %rescaled 'standard errors' using outer product
hh_mat=hh_mat./(sd*sd').*(sd0*sd0'); %rescaled inverse outer product with 'true' std's
igg=inv(hh_mat); % rescaled outer product hessian with 'true' std's
ihh=A'*(hh_mat\A); % inverted outer product hessian
hh_mat0=inv(A)'*hh_mat*inv(A); % outer product hessian with 'true' std's
sd=sqrt(diag(ihh)); %standard errors
sdh=sqrt(1./diag(hh)); %diagonal standard errors
for j=1:length(sd)
sd0(j,1)=min(varargin{6}.p2(j), sd(j)); %prior std
sd0(j,1)=10^(0.5*(log10(sd0(j,1))+log10(sdh(j,1))));
end
ihh=ihh./(sd*sd').*(sd0*sd0'); %inverse outer product with modified std's
igg=inv(A)'*ihh*inv(A); % inverted rescaled outer product hessian with modified std's
hh_mat=inv(igg); % outer product rescaled hessian with modified std's
hh_mat0=inv(A)'*hh_mat*inv(A); % outer product hessian with modified std's
% sd0=sqrt(1./diag(hh0)); %rescaled 'standard errors' using second order derivatives
% sd=sqrt(diag(igg)); %rescaled 'standard errors' using outer product
% igg=igg./(sd*sd').*(sd0*sd0'); %rescaled inverse outer product with 'true' std's
% hh_mat=inv(igg); % rescaled outer product hessian with 'true' std's
% ihh=A'*igg*A; % inverted outer product hessian
% hh_mat0=inv(A)'*hh_mat*inv(A); % outer product hessian with 'true' std's
end
if hflag<2
hessian_mat=hh_mat0(:);
end
if any(isnan(hessian_mat))
hh_mat0=eye(length(hh_mat0));
ihh=hh_mat0;
hessian_mat=hh_mat0(:);
end
hh1=hess_info.h1;
save hess.mat hessian_mat
else
hessian_mat=[];
ihh=[];
hh_mat0 = [];
hh1 = [];
end
htol1=hhtol;
|
