Commit fe418672 authored by stepan's avatar stepan
Browse files

v4.1: Changes related to (posterior) 2nd order moments.

+ Efficiency changes:
++ Do not compute the variance decomposition if not needed.
++ Do not apply the Schur decomposition for each shock in case of a variance decomposition.
+ Bug corrections and cosmetic changes.
+ Corrections of some headers.


git-svn-id: https://www.dynare.org/svn/dynare/trunk@2411 ac1d8469-bf42-47a9-8791-bf33cf982152
parent 91264b69
......@@ -51,7 +51,7 @@ function oo_ = compute_moments_varendo(options_,M_,oo_,var_list_)
end
% VARIANCE DECOMPOSITION.
for i=1:NumberOfEndogenousVariables
for j=i:NumberOfExogenousVariables
for j=1:NumberOfExogenousVariables
oo_ = posterior_analysis('decomposition',var_list_(i,:),M_.exo_names(j,:),[],options_,M_,oo_);
end
end
\ No newline at end of file
......@@ -20,7 +20,7 @@ function [nvar,vartan,CorrFileNumber] = dsge_posterior_theoretical_correlation(S
% th_autocovariances.m
% posterior_moments.m
% Copyright (C) 2007-2008 Dynare Team
% Copyright (C) 2007-2009 Dynare Team
%
% This file is part of Dynare.
%
......@@ -38,6 +38,7 @@ function [nvar,vartan,CorrFileNumber] = dsge_posterior_theoretical_correlation(S
% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
type = 'posterior';
nodecomposition = 1;
% Set varlist (vartan)
[ivar,vartan] = set_stationary_variables_list;
......@@ -80,7 +81,7 @@ for file = 1:NumberOfDrawsFiles
set_parameters(pdraws{linee,1});
[dr,info] = resol(oo_.steady_state,0);
end
tmp = th_autocovariances(dr,ivar,M_,options_);
tmp = th_autocovariances(dr,ivar,M_,options_,nodecomposition);
for i=1:nar
Correlation_array(linea,:,:,i) = tmp{i+1};
end
......
......@@ -19,7 +19,7 @@ function [nvar,vartan,CovarFileNumber] = dsge_posterior_theoretical_covariance(S
% th_autocovariances.m
% posterior_moments.m
% Copyright (C) 2007-2008 Dynare Team
% Copyright (C) 2007-2009 Dynare Team
%
% This file is part of Dynare.
%
......@@ -37,7 +37,8 @@ function [nvar,vartan,CovarFileNumber] = dsge_posterior_theoretical_covariance(S
% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
type = 'posterior';
nodecomposition = 1;
% Set varlist (vartan)
[ivar,vartan] = set_stationary_variables_list;
nvar = length(ivar);
......@@ -79,7 +80,7 @@ for file = 1:NumberOfDrawsFiles
set_parameters(pdraws{linee,1});
[dr,info] = resol(oo_.steady_state,0);
end
tmp = th_autocovariances(dr,ivar,M_,options_);
tmp = th_autocovariances(dr,ivar,M_,options_,nodecomposition);
for i=1:nvar
for j=i:nvar
Covariance_matrix(linea,symmetric_matrix_index(i,j,nvar)) = tmp{1}(i,j);
......@@ -94,7 +95,7 @@ for file = 1:NumberOfDrawsFiles
Covariance_matrix = zeros(NumberOfLinesInTheLastCovarFile,nvar*(nvar+1)/2);
NumberOfCovarLines = NumberOfLinesInTheLastCovarFile;
CovarFileNumber = CovarFileNumber - 1;
elseif test<0;
elseif test<0
Covariance_matrix = zeros(MaXNumberOfCovarLines,nvar*(nvar+1)/2);
else
clear('Covariance_matrix');
......
......@@ -18,7 +18,7 @@ function [nvar,vartan,NumberOfDecompFiles] = ...
% th_autocovariances.m
% posterior_moments.m
% Copyright (C) 2007-2008 Dynare Team
% Copyright (C) 2007-2009 Dynare Team
%
% This file is part of Dynare.
%
......@@ -36,6 +36,7 @@ function [nvar,vartan,NumberOfDecompFiles] = ...
% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
type = 'posterior';% To be defined as a input argument later...
nodecomposition = 0;
% Set varlist (vartan)
[ivar,vartan] = set_stationary_variables_list;
......@@ -51,7 +52,6 @@ NumberOfDrawsFiles = length(DrawsFiles);
nexo = M_.exo_nbr;
NumberOfDrawsFiles = rows(DrawsFiles);
NumberOfSavedElementsPerSimulation = nvar*(nexo+1);
MaXNumberOfDecompLines = ceil(options_.MaxNumberOfBytes/NumberOfSavedElementsPerSimulation/8);
......@@ -84,7 +84,7 @@ for file = 1:NumberOfDrawsFiles
set_parameters(pdraws{linee,1});
[dr,info] = resol(oo_.steady_state,0);
end
tmp = th_autocovariances(dr,ivar,M_,options_);
tmp = th_autocovariances(dr,ivar,M_,options_,nodecomposition);
for i=1:nvar
for j=1:nexo
Decomposition_array(linea,(i-1)*nexo+j) = tmp{2}(i,j);
......
function [x,u]=lyapunov_symm(a,b,qz_criterium)
% Solves the Lyapunov equation x-a*x*a' = b, for b (and then x) symmetrical
% if a has some unit roots, the function computes only the solution of the stable subsystem
function [x,u] = lyapunov_symm(a,b,qz_criterium,method)
% Solves the Lyapunov equation x-a*x*a' = b, for b and x symmetric matrices.
% If a has some unit roots, the function computes only the solution of the stable subsystem.
%
% INPUTS:
% a: coefficient matrix (n x n)
% b: coefficient square matrix (n x n)
%
% a [double] n*n matrix.
% b [double] n*n matrix
% method [integer] Scalar, if method=0 [default] then U, T, n and k are not persistent.
% method=1 then U, T, n and k are declared as persistent variables and the schur decomposition is triggered.
% method=2 then U, T, n and k are declared as persistent variables and the schur decomposition is not performed.
% OUTPUTS
% x: solution matrix (m x m)
% ns_var: vector of indices of non-stationary variables (p x 1)
% (m + p = n)
% u: Schur vectors associated with unit roots
% x: [double] m*m solution matrix of the lyapunov equation, where m is the dimension of the stable subsystem.
% u: [double] Schur vectors associated with unit roots
%
% ALGORITHM
% uses reordered Schur decomposition
% Uses reordered Schur decomposition
%
% SPECIAL REQUIREMENTS
% needs Matlab >= 7.0.1 for ordeig function (otherwise uses my_ordeig)
% Copyright (C) 2006-2008 Dynare Team
% Copyright (C) 2006-2009 Dynare Team
%
% This file is part of Dynare.
%
......@@ -34,76 +34,90 @@ function [x,u]=lyapunov_symm(a,b,qz_criterium)
%
% You should have received a copy of the GNU General Public License
% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
ns_var = [];
u = [];
info = 0;
if size(a,1) == 1
x=b/(1-a*a);
return
end
[u,t] = schur(a);
if exist('OCTAVE_VERSION') || matlab_ver_less_than('7.0.1')
e1 = abs(my_ordeig(t)) > 2-qz_criterium;
else
e1 = abs(ordeig(t)) > 2-qz_criterium;
end
k = sum(e1);
if exist('ordschur','builtin')
% selects stable roots
[u,t] = ordschur(u,t,e1);
n = length(e1)-k;
b=u(:,k+1:end)'*b*u(:,k+1:end);
t = t(k+1:end,k+1:end);
elseif k > 0
% problem for Matlab version that don't have ordschur
error(['lyapunov_sym: you need a Matlab version > 6.5 to handle models' ...
' with unit roots'])
else
% no unit root
n = length(e1);
b=u'*b*u;
end
x=zeros(n,n);
i = n;
while i >= 2
if t(i,i-1) == 0
if i == n
c = zeros(n,1);
else
c = t(1:i,:)*(x(:,i+1:end)*t(i,i+1:end)')+...
t(i,i)*t(1:i,i+1:end)*x(i+1:end,i);
end
q = eye(i)-t(1:i,1:i)*t(i,i);
x(1:i,i) = q\(b(1:i,i)+c);
x(i,1:i-1) = x(1:i-1,i)';
i = i - 1;
if nargin<4
method = 0;
end
if method
persistent U T k n
else
if i == n
c = zeros(n,1);
c1 = zeros(n,1);
else
c = t(1:i,:)*(x(:,i+1:end)*t(i,i+1:end)')+...
t(i,i)*t(1:i,i+1:end)*x(i+1:end,i)+...
t(i,i-1)*t(1:i,i+1:end)*x(i+1:end,i-1);
c1 = t(1:i,:)*(x(:,i+1:end)*t(i-1,i+1:end)')+...
t(i-1,i-1)*t(1:i,i+1:end)*x(i+1:end,i-1)+...
t(i-1,i)*t(1:i,i+1:end)*x(i+1:end,i);
end
q = [eye(i)-t(1:i,1:i)*t(i,i) -t(1:i,1:i)*t(i,i-1);...
-t(1:i,1:i)*t(i-1,i) eye(i)-t(1:i,1:i)*t(i-1,i-1)];
z = q\[b(1:i,i)+c;b(1:i,i-1)+c1];
x(1:i,i) = z(1:i);
x(1:i,i-1) = z(i+1:end);
x(i,1:i-1)=x(1:i-1,i)';
x(i-1,1:i-2)=x(1:i-2,i-1)';
i = i - 2;
if exist('U','var')
clear('U','T','k','n')
end
end
u = [];
if size(a,1) == 1
x=b/(1-a*a);
return
end
if method<2
[U,T] = schur(a);
if exist('OCTAVE_VERSION') || matlab_ver_less_than('7.0.1')
e1 = abs(my_ordeig(T)) > 2-qz_criterium;
else
e1 = abs(ordeig(T)) > 2-qz_criterium;
end
k = sum(e1); % Number of unit roots.
n = length(e1)-k; % Number of stationary variables.
if exist('ordschur','builtin')
% Selects stable roots
[U,T] = ordschur(U,T,e1);
% B = U(:,k+1:end)'*b*U(:,k+1:end);
T = T(k+1:end,k+1:end);
elseif k > 0
% Problem for Matlab version that don't have ordschur
error(['lyapunov_sym: you need a Matlab version > 6.5 to handle models' ...
' with unit roots'])
%else
% No unit root
%n = length(e1);
%B=u'*b*u;
end
end
B = U(:,k+1:end)'*b*U(:,k+1:end);
x = zeros(n,n);
i = n;
while i >= 2
if T(i,i-1) == 0
if i == n
c = zeros(n,1);
else
c = T(1:i,:)*(x(:,i+1:end)*T(i,i+1:end)') + ...
T(i,i)*T(1:i,i+1:end)*x(i+1:end,i);
end
q = eye(i)-T(1:i,1:i)*T(i,i);
x(1:i,i) = q\(B(1:i,i)+c);
x(i,1:i-1) = x(1:i-1,i)';
i = i - 1;
else
if i == n
c = zeros(n,1);
c1 = zeros(n,1);
else
c = T(1:i,:)*(x(:,i+1:end)*T(i,i+1:end)') + ...
T(i,i)*T(1:i,i+1:end)*x(i+1:end,i) + ...
T(i,i-1)*T(1:i,i+1:end)*x(i+1:end,i-1);
c1 = T(1:i,:)*(x(:,i+1:end)*T(i-1,i+1:end)') + ...
T(i-1,i-1)*T(1:i,i+1:end)*x(i+1:end,i-1) + ...
T(i-1,i)*T(1:i,i+1:end)*x(i+1:end,i);
end
q = [eye(i)-T(1:i,1:i)*T(i,i) - T(1:i,1:i)*T(i,i-1) ; ...
-T(1:i,1:i)*T(i-1,i) eye(i) - T(1:i,1:i)*T(i-1,i-1) ];
z = q\[ B(1:i,i)+c ; B(1:i,i-1) + c1 ];
x(1:i,i) = z(1:i);
x(1:i,i-1) = z(i+1:end);
x(i,1:i-1) = x(1:i-1,i)';
x(i-1,1:i-2) = x(1:i-2,i-1)';
i = i - 2;
end
end
if i == 1
c = T(1,:)*(x(:,2:end)*T(1,2:end)') + T(1,1)*T(1,2:end)*x(2:end,1);
x(1,1) = (B(1,1)+c)/(1-T(1,1)*T(1,1));
end
end
if i == 1
c = t(1,:)*(x(:,2:end)*t(1,2:end)')+t(1,1)*t(1,2:end)*x(2:end,1);
x(1,1)=(b(1,1)+c)/(1-t(1,1)*t(1,1));
end
x = u(:,k+1:end)*x*u(:,k+1:end)';
u = u(:,1:k);
\ No newline at end of file
x = U(:,k+1:end)*x*U(:,k+1:end)';
u = U(:,1:k);
\ No newline at end of file
function [Gamma_y,ivar]=th_autocovariances(dr,ivar,M_,options_)
function [Gamma_y,ivar] = th_autocovariances(dr,ivar,M_,options_,nodecomposition)
% Computes the theoretical auto-covariances, Gamma_y, for an AR(p) process
% with coefficients dr.ghx and dr.ghu and shock variances Sigma_e_
% for a subset of variables ivar (indices in lgy_)
%
% Theoretical HPfiltering is available as an option
%
% INPUTS
% dr: structure of decisions rules for stochastic simulations
% ivar: subset of variables
% M_
% options_
% dr: [structure] Reduced form solution of the DSGE model (decisions rules)
% ivar: [integer] Vector of indices for a subset of variables.
% M_ [structure] Global dynare's structure, description of the DSGE model.
% options_ [structure] Global dynare's structure.
% nodecomposition [integer] Scalar, if different from zero the variance decomposition is not triggered.
%
% OUTPUTS
% Gamma_y: theoritical auto-covariances
% ivar: subset of variables
% Gamma_y [cell] Matlab cell of nar+3 (second order approximation) or nar+2 (first order approximation) arrays,
% where nar is the order of the autocorrelation function.
% Gamma_y{1} [double] Covariance matrix.
% Gamma_y{i} [double] Autocorrelation function (for i=1,...,options_.nar).
% Gamma_y{nar+2} [double] Variance decomposition.
% Gamma_y{nar+3} [double] Expectation of the endogenous variables associated with a second
% order approximation.
% ivar [integer] Vector of indices for a subset of variables.
%
% SPECIAL REQUIREMENTS
% Theoretical HP filtering is available as an option
%
% Copyright (C) 2001-2008 Dynare Team
% Copyright (C) 2001-2009 Dynare Team
%
% This file is part of Dynare.
%
......@@ -33,188 +41,181 @@ function [Gamma_y,ivar]=th_autocovariances(dr,ivar,M_,options_)
% You should have received a copy of the GNU General Public License
% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
exo_names_orig_ord = M_.exo_names_orig_ord;
if exist('OCTAVE_VERSION')
warning('off', 'Octave:divide-by-zero')
else
warning off MATLAB:dividebyzero
end
nar = options_.ar;
Gamma_y = cell(nar+1,1);
if isempty(ivar)
ivar = [1:M_.endo_nbr]';
end
nvar = size(ivar,1);
ghx = dr.ghx;
ghu = dr.ghu;
npred = dr.npred;
nstatic = dr.nstatic;
kstate = dr.kstate;
order = dr.order_var;
iv(order) = [1:length(order)];
nx = size(ghx,2);
ikx = [nstatic+1:nstatic+npred];
k0 = kstate(find(kstate(:,2) <= M_.maximum_lag+1),:);
i0 = find(k0(:,2) == M_.maximum_lag+1);
i00 = i0;
n0 = length(i0);
AS = ghx(:,i0);
ghu1 = zeros(nx,M_.exo_nbr);
ghu1(i0,:) = ghu(ikx,:);
for i=M_.maximum_lag:-1:2
i1 = find(k0(:,2) == i);
n1 = size(i1,1);
j1 = zeros(n1,1);
for k1 = 1:n1
j1(k1) = find(k0(i00,1)==k0(i1(k1),1));
end
AS(:,j1) = AS(:,j1)+ghx(:,i1);
i0 = i1;
end
b = ghu1*M_.Sigma_e*ghu1';
ipred = nstatic+(1:npred)';
% state space representation for state variables only
[A,B] = kalman_transition_matrix(dr,ipred,1:nx,dr.transition_auxiliary_variables,M_.exo_nbr);
if options_.order == 2 | options_.hp_filter == 0
[vx, u] = lyapunov_symm(A,B*M_.Sigma_e*B',options_.qz_criterium);
iky = iv(ivar);
if ~isempty(u)
iky = iky(find(all(abs(ghx(iky,:)*u) < options_.Schur_vec_tol,2)));
ivar = dr.order_var(iky);
end
aa = ghx(iky,:);
bb = ghu(iky,:);
if options_.order == 2 % mean correction for 2nd order
Ex = (dr.ghs2(ikx)+dr.ghxx(ikx,:)*vx(:)+dr.ghuu(ikx,:)*M_.Sigma_e(:))/2;
Ex = (eye(n0)-AS(ikx,:))\Ex;
Gamma_y{nar+3} = AS(iky,:)*Ex+(dr.ghs2(iky)+dr.ghxx(iky,:)*vx(:)+...
dr.ghuu(iky,:)*M_.Sigma_e(:))/2;
if nargin<5
nodecomposition = 0;
end
end
if options_.hp_filter == 0
Gamma_y{1} = aa*vx*aa'+ bb*M_.Sigma_e*bb';
k = find(abs(Gamma_y{1}) < 1e-12);
Gamma_y{1}(k) = 0;
% autocorrelations
if nar > 0
vxy = (A*vx*aa'+ghu1*M_.Sigma_e*bb');
sy = sqrt(diag(Gamma_y{1}));
sy = sy *sy';
Gamma_y{2} = aa*vxy./sy;
for i=2:nar
vxy = A*vxy;
Gamma_y{i+1} = aa*vxy./sy;
end
exo_names_orig_ord = M_.exo_names_orig_ord;
if exist('OCTAVE_VERSION')
warning('off', 'Octave:divide-by-zero')
else
warning off MATLAB:dividebyzero
end
% variance decomposition
if M_.exo_nbr > 1
Gamma_y{nar+2} = zeros(length(ivar),M_.exo_nbr);
SS(exo_names_orig_ord,exo_names_orig_ord)=M_.Sigma_e+1e-14*eye(M_.exo_nbr);
cs = chol(SS)';
b1(:,exo_names_orig_ord) = ghu1;
b1 = b1*cs;
b2(:,exo_names_orig_ord) = ghu(iky,:);
b2 = b2*cs;
vx = lyapunov_symm(A,b1*b1',options_.qz_criterium);
vv = diag(aa*vx*aa'+b2*b2');
for i=1:M_.exo_nbr
vx1 = lyapunov_symm(A,b1(:,i)*b1(:,i)',options_.qz_criterium);
Gamma_y{nar+2}(:,i) = abs(diag(aa*vx1*aa'+b2(:,i)*b2(:,i)'))./vv;
end
nar = options_.ar;
Gamma_y = cell(nar+1,1);
if isempty(ivar)
ivar = [1:M_.endo_nbr]';
end
else
if options_.order < 2
iky = iv(ivar);
aa = ghx(iky,:);
bb = ghu(iky,:);
nvar = size(ivar,1);
ghx = dr.ghx;
ghu = dr.ghu;
npred = dr.npred;
nstatic = dr.nstatic;
kstate = dr.kstate;
order = dr.order_var;
iv(order) = [1:length(order)];
nx = size(ghx,2);
ikx = [nstatic+1:nstatic+npred];
k0 = kstate(find(kstate(:,2) <= M_.maximum_lag+1),:);
i0 = find(k0(:,2) == M_.maximum_lag+1);
i00 = i0;
n0 = length(i0);
AS = ghx(:,i0);
ghu1 = zeros(nx,M_.exo_nbr);
ghu1(i0,:) = ghu(ikx,:);
for i=M_.maximum_lag:-1:2
i1 = find(k0(:,2) == i);
n1 = size(i1,1);
j1 = zeros(n1,1);
for k1 = 1:n1
j1(k1) = find(k0(i00,1)==k0(i1(k1),1));
end
AS(:,j1) = AS(:,j1)+ghx(:,i1);
i0 = i1;
end
lambda = options_.hp_filter;
ngrid = options_.hp_ngrid;
freqs = 0 : ((2*pi)/ngrid) : (2*pi*(1 - .5/ngrid));
tpos = exp( sqrt(-1)*freqs);
tneg = exp(-sqrt(-1)*freqs);
hp1 = 4*lambda*(1 - cos(freqs)).^2 ./ (1 + 4*lambda*(1 - cos(freqs)).^2);
b = ghu1*M_.Sigma_e*ghu1';
mathp_col = [];
IA = eye(size(A,1));
IE = eye(M_.exo_nbr);
for ig = 1:ngrid
f_omega =(1/(2*pi))*( [inv(IA-A*tneg(ig))*ghu1;IE]...
*M_.Sigma_e*[ghu1'*inv(IA-A'*tpos(ig)) ...
IE]); % state variables
g_omega = [aa*tneg(ig) bb]*f_omega*[aa'*tpos(ig); bb']; % selected variables
f_hp = hp1(ig)^2*g_omega; % spectral density of selected filtered series
mathp_col = [mathp_col ; (f_hp(:))']; % store as matrix row
% for ifft
end;
% covariance of filtered series
imathp_col = real(ifft(mathp_col))*(2*pi);
Gamma_y{1} = reshape(imathp_col(1,:),nvar,nvar);
% autocorrelations
if nar > 0
sy = sqrt(diag(Gamma_y{1}));
sy = sy *sy';
for i=1:nar
Gamma_y{i+1} = reshape(imathp_col(i+1,:),nvar,nvar)./sy;
end
ipred = nstatic+(1:npred)';
% state space representation for state variables only
[A,B] = kalman_transition_matrix(dr,ipred,1:nx,dr.transition_auxiliary_variables,M_.exo_nbr);
if options_.order == 2 | options_.hp_filter == 0
[vx, u] = lyapunov_symm(A,B*M_.Sigma_e*B',options_.qz_criterium);
iky = iv(ivar);
if ~isempty(u)
iky = iky(find(all(abs(ghx(iky,:)*u) < options_.Schur_vec_tol,2)));
ivar = dr.order_var(iky);
end
aa = ghx(iky,:);
bb = ghu(iky,:);
if options_.order == 2 % mean correction for 2nd order
Ex = (dr.ghs2(ikx)+dr.ghxx(ikx,:)*vx(:)+dr.ghuu(ikx,:)*M_.Sigma_e(:))/2;
Ex = (eye(n0)-AS(ikx,:))\Ex;
Gamma_y{nar+3} = AS(iky,:)*Ex+(dr.ghs2(iky)+dr.ghxx(iky,:)*vx(:)+...
dr.ghuu(iky,:)*M_.Sigma_e(:))/2;
end
end
%variance decomposition
if M_.exo_nbr > 1
Gamma_y{nar+2} = zeros(nvar,M_.exo_nbr);
SS(exo_names_orig_ord,exo_names_orig_ord) = M_.Sigma_e+1e-14*eye(M_.exo_nbr);
cs = chol(SS)';
SS = cs*cs';
b1(:,exo_names_orig_ord) = ghu1;
b2(:,exo_names_orig_ord) = ghu(iky,:);
mathp_col = [];
IA = eye(size(A,1));
IE = eye(M_.exo_nbr);
for ig = 1:ngrid
f_omega =(1/(2*pi))*( [inv(IA-A*tneg(ig))*b1;IE]...
*SS*[b1'*inv(IA-A'*tpos(ig)) ...
IE]); % state variables
g_omega = [aa*tneg(ig) b2]*f_omega*[aa'*tpos(ig); b2']; % selected variables
f_hp = hp1(ig)^2*g_omega; % spectral density of selected filtered series
mathp_col = [mathp_col ; (f_hp(:))']; % store as matrix row
% for ifft
end;
imathp_col = real(ifft(mathp_col))*(2*pi);
vv = diag(reshape(imathp_col(1,:),nvar,nvar));
for i=1:M_.exo_nbr
mathp_col = [];
SSi = cs(:,i)*cs(:,i)';
for ig = 1:ngrid
f_omega =(1/(2*pi))*( [inv(IA-A*tneg(ig))*b1;IE]...
*SSi*[b1'*inv(IA-A'*tpos(ig)) ...
IE]); % state variables
g_omega = [aa*tneg(ig) b2]*f_omega*[aa'*tpos(ig); b2']; % selected variables
f_hp = hp1(ig)^2*g_omega; % spectral density of selected filtered series
mathp_col = [mathp_col ; (f_hp(:))']; % store as matrix row
% for ifft
end;