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
|
function of = fn_a0sfreefun2(b,Uistar,Uibar,nvar,nStates,n0,n0cumsum,tvstate,tvtot,constot,...
tvinx,constinx,indxTV,indxConst,Tkave,Sgm0tldinvaveConst,Sgm0tldinvaveTV)
% Negative logPosterior function for regime-switching vectorized free A0 parameters,
% which are b's in the WZ notation. The case of no asymmetric prior and no lag restrictions.
% It improves fn_a0sfreefun2.m in several aspects:
% (a) allows some equations to have constant parameters.
% It differs from fn_a0cfreefun_tv.m in several aspects:
% (a) does not deal with equations with only structural variances time varying;
% (b) cannot deal with lag restrictions;
% (c) only deal with the marginal distribution of A0_s.
% Note: (1) columns correspond to equations; (2) s stands for state.
% See Time-Varying BVAR NOTE pp.34-34c,40.
%
% b: sum(n0)*nStates-by-1 vector of free A0 parameters, vectorized from the sum(n0)-by-nStates matrix.
% Uistar: cell(nvar,1). In each cell, nvar*nSates-by-qi orthonormal basis for the null of the ith
% equation contemporaneous restriction matrix where qi is the number of free parameters.
% With this transformation, we have ai = Ui*bi or Ui'*ai = bi where ai is a vector
% of total original parameters and bi is a vector of free parameters. When no
% restrictions are imposed, we have Ui = I. There must be at least one free
% parameter left for the ith equation. See p.33.
% Uibar: cell(nvar,1). In each cell, we have nvar*nStates-by-qi*nStates, rearranged
% from Uistar or Ui. See p.33.
% nvar: Number of endogeous variables.
% nStates: NUmber of states.
% n0: nvar-by-1, ith element represents the number of free A0 parameters in ith equation for each state.
% n0cumsum: Equal to [0;cumsum(n0)];
% tvstate: Number of time-varying parameters for all equations for each state.
% tvtot: Total number of time-varying parameters.
% constot: Total number of constant parameters.
% tvinx: Vectorized index to include time-varying parameters for all equations under *each* state.
% constinx: Vectorized index to include constant parameters for all equations.
% indxTV: Index of acending order for equations with time-varying parameters.
% indxConst: Index of ascending order for equations with constant parameters. When [], all equations
% are time varying; when [1:nvar], all equations have constant parameters.
% Tkave: nStates-by-1 of sample sizes (excluding lags but including dummies if provided) for different states k,
% averaged (ave) over E-step draws. For T_k. See p.40.
% Sgm0tldinvaveConst
% Sgm0tldinvaveTV
% Sgm0tldinvave: nvar*nStates-by-nvar*nStates. The matrix inv(\Sigma~_0) averaged (ave)
% over E-step draws. Same for all equations because of no asymmetric prior and no lag
% restrictions. Resembles old SpH in the exponent term in posterior of A0,
% but NOT divided by fss (T) yet. See p.40.
%----------------
% of: Objective function (negative logPosterior).
%
% This function is called by tveml*.m which is an old program compared with szeml*.m. Thus,
% use fn_a0cfreefun_tv.m if possible.
% Tao Zha, August 2001.
%
% Copyright (C) 1997-2012 Tao Zha
%
% 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/>.
%
n0=n0(:);
A0=zeros(nvar,nvar,nStates);
b_shat = zeros(n0cumsum(end),nStates);
b_shat(tvinx,:) = reshape(b(1:tvtot),tvstate,nStates); % Time-varying parameter matrix.
b_shat(constinx,:) = repmat(b(tvtot+1:end), [1 nStates]); % Constant parameter matrix.
tra = 0.0;
for kj = 1:nvar
lenbjs=length(n0cumsum(kj)+1:n0cumsum(kj+1));
bj = zeros(nStates*lenbjs,1);
a0j = zeros(nStates*nvar,1); % See p.33.
for si=1:nStates
bj((si-1)*lenbjs+1:si*lenbjs) = b_shat(n0cumsum(kj)+1:n0cumsum(kj+1),si); % bj(si). See p.34a.
A0(:,kj,si) = Uistar{kj}((si-1)*nvar+1:si*nvar,:)*b_shat(n0cumsum(kj)+1:n0cumsum(kj+1),si);
a0j((si-1)*nvar+1:si*nvar) = A0(:,kj,si); % a0j(si). See p.33.
end
%tra = tra + 0.5*a0j'*Sgm0tldinvave*a0j; % negative exponential term
if find(kj==indxTV) % For time-varying equations.
tra = tra+0.5*bj'*(Uibar{kj}'*Sgm0tldinvaveTV*Uibar{kj})*bj;
else % % For constant parameter equations.
tra = tra+0.5*bj'*(Uibar{kj}'*Sgm0tldinvaveConst*Uibar{kj})*bj;
end
end
ada=0.0;
for si=1:nStates % See p.34a.
[A0l,A0u] = lu(A0(:,:,si));
ada = ada - Tkave(si)*sum(log(abs(diag(A0u)))); % negative log determinant of A0 raised to power T
end
of = ada + tra;
|
