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
|
function [G1,C,impact,fmat,fwt,ywt,gev,eu]=gensys(g0,g1,c,psi,pi,div)
% function [G1,C,impact,fmat,fwt,ywt,gev,eu]=gensys(g0,g1,c,psi,pi,div)
% System given as
% g0*y(t)=g1*y(t-1)+c+psi*z(t)+pi*eta(t),
% with z an exogenous variable process and eta being endogenously determined
% one-step-ahead expectational errors. Returned system is
% y(t)=G1*y(t-1)+C+impact*z(t)+ywt*inv(I-fmat*inv(L))*fwt*z(t+1) .
% If z(t) is i.i.d., the last term drops out.
% If div is omitted from argument list, a div>1 is calculated.
% eu(1)=1 for existence, eu(2)=1 for uniqueness. eu(1)=-1 for
% existence only with not-s.c. z; eu=[-2,-2] for coincident zeros.
% By Christopher A. Sims
% Corrected 10/28/96 by CAS
%
% Copyright (C) 1996-2012 Christopher A. Sims
%
% This 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.
%
% It 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.
%
% If you did not received a copy of the GNU General Public License
% with this software, see <http://www.gnu.org/licenses/>.
%
eu=[0;0];
realsmall=1e-6;
fixdiv=(nargin==6);
n=size(g0,1);
[a b q z v]=qz(g0,g1);
if ~fixdiv, div=1.01; end
nunstab=0;
zxz=0;
for i=1:n
% ------------------div calc------------
if ~fixdiv
if abs(a(i,i)) > 0
divhat=abs(b(i,i))/abs(a(i,i));
% bug detected by Vasco Curdia and Daria Finocchiaro, 2/25/2004 A root of
% exactly 1.01 and no root between 1 and 1.02, led to div being stuck at 1.01
% and the 1.01 root being misclassified as stable. Changing < to <= below fixes this.
if 1+realsmall<divhat & divhat<=div
div=.5*(1+divhat);
end
end
end
% ----------------------------------------
nunstab=nunstab+(abs(b(i,i))>div*abs(a(i,i)));
if abs(a(i,i))<realsmall & abs(b(i,i))<realsmall
zxz=1;
end
end
div ;
nunstab;
if ~zxz
[a b q z]=qzdiv(div,a,b,q,z);
end
gev=[diag(a) diag(b)];
if zxz
disp('Coincident zeros. Indeterminacy and/or nonexistence.')
eu=[-2;-2];
% correction added 7/29/2003. Otherwise the failure to set output
% arguments leads to an error message and no output (including eu).
G1=[];C=[];impact=[];fmat=[];fwt=[];ywt=[];gev=[];
return
end
q1=q(1:n-nunstab,:);
q2=q(n-nunstab+1:n,:);
z1=z(:,1:n-nunstab)';
z2=z(:,n-nunstab+1:n)';
a2=a(n-nunstab+1:n,n-nunstab+1:n);
b2=b(n-nunstab+1:n,n-nunstab+1:n);
etawt=q2*pi;
% zwt=q2*psi;
[ueta,deta,veta]=svd(etawt);
md=min(size(deta));
bigev=find(diag(deta(1:md,1:md))>realsmall);
ueta=ueta(:,bigev);
veta=veta(:,bigev);
deta=deta(bigev,bigev);
% ------ corrected code, 3/10/04
eu(1) = length(bigev)>=nunstab;
% ------ Code below allowed "existence" in cases where the initial lagged state was free to take on values
% ------ inconsistent with existence, so long as the state could w.p.1 remain consistent with a stable solution
% ------ if its initial lagged value was consistent with a stable solution. This is a mistake, though perhaps there
% ------ are situations where we would like to know that this "existence for restricted initial state" situation holds.
%% [uz,dz,vz]=svd(zwt);
%% md=min(size(dz));
%% bigev=find(diag(dz(1:md,1:md))>realsmall);
%% uz=uz(:,bigev);
%% vz=vz(:,bigev);
%% dz=dz(bigev,bigev);
%% if isempty(bigev)
%% exist=1;
%% else
%% exist=norm(uz-ueta*ueta'*uz) < realsmall*n;
%% end
%% if ~isempty(bigev)
%% zwtx0=b2\zwt;
%% zwtx=zwtx0;
%% M=b2\a2;
%% for i=2:nunstab
%% zwtx=[M*zwtx zwtx0];
%% end
%% zwtx=b2*zwtx;
%% [ux,dx,vx]=svd(zwtx);
%% md=min(size(dx));
%% bigev=find(diag(dx(1:md,1:md))>realsmall);
%% ux=ux(:,bigev);
%% vx=vx(:,bigev);
%% dx=dx(bigev,bigev);
%% existx=norm(ux-ueta*ueta'*ux) < realsmall*n;
%% else
%% existx=1;
%% end
% ----------------------------------------------------
% Note that existence and uniqueness are not just matters of comparing
% numbers of roots and numbers of endogenous errors. These counts are
% reported below because usually they point to the source of the problem.
% ------------------------------------------------------
[ueta1,deta1,veta1]=svd(q1*pi);
md=min(size(deta1));
bigev=find(diag(deta1(1:md,1:md))>realsmall);
ueta1=ueta1(:,bigev);
veta1=veta1(:,bigev);
deta1=deta1(bigev,bigev);
%% if existx | nunstab==0
%% %disp('solution exists');
%% eu(1)=1;
%% else
%% if exist
%% %disp('solution exists for unforecastable z only');
%% eu(1)=-1;
%% %else
%% %fprintf(1,'No solution. %d unstable roots. %d endog errors.\n',nunstab,size(ueta1,2));
%% end
%% %disp('Generalized eigenvalues')
%% %disp(gev);
%% %md=abs(diag(a))>realsmall;
%% %ev=diag(md.*diag(a)+(1-md).*diag(b))\ev;
%% %disp(ev)
%% % return;
%% end
if isempty(veta1)
unique=1;
else
unique=norm(veta1-veta*veta'*veta1)<realsmall*n;
end
if unique
%disp('solution unique');
eu(2)=1;
else
fprintf(1,'Indeterminacy. %d loose endog errors.\n',size(veta1,2)-size(veta,2));
%disp('Generalized eigenvalues')
%disp(gev);
%md=abs(diag(a))>realsmall;
%ev=diag(md.*diag(a)+(1-md).*diag(b))\ev;
%disp(ev)
% return;
end
tmat = [eye(n-nunstab) -(ueta*(deta\veta')*veta1*deta1*ueta1')'];
G0= [tmat*a; zeros(nunstab,n-nunstab) eye(nunstab)];
G1= [tmat*b; zeros(nunstab,n)];
% ----------------------
% G0 is always non-singular because by construction there are no zeros on
% the diagonal of a(1:n-nunstab,1:n-nunstab), which forms G0's ul corner.
% -----------------------
G0I=inv(G0);
G1=G0I*G1;
usix=n-nunstab+1:n;
C=G0I*[tmat*q*c;(a(usix,usix)-b(usix,usix))\q2*c];
impact=G0I*[tmat*q*psi;zeros(nunstab,size(psi,2))];
fmat=b(usix,usix)\a(usix,usix);
fwt=-b(usix,usix)\q2*psi;
ywt=G0I(:,usix);
% -------------------- above are output for system in terms of z'y -------
G1=real(z*G1*z');
C=real(z*C);
impact=real(z*impact);
% Correction 10/28/96: formerly line below had real(z*ywt) on rhs, an error.
ywt=z*ywt;
|
