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dynare/matlab/th_autocovariances.m

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function [Gamma_y,stationary_vars] = 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] 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 [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+1} [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.
% stationary_vars [integer] Vector of indices of stationary variables (as a subset of 1:length(ivar))
%
% SPECIAL REQUIREMENTS
%
% Copyright (C) 2001-2012 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/>.
if nargin<5
nodecomposition = 0;
end
if options_.order >= 3
error('Theoretical moments not implemented above 2nd order')
end
endo_nbr = M_.endo_nbr;
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:endo_nbr]';
end
nvar = size(ivar,1);
ghx = dr.ghx;
ghu = dr.ghu;
nspred = M_.nspred;
nstatic = M_.nstatic;
nx = size(ghx,2);
if options_.block == 0
%order_var = dr.order_var;
inv_order_var = dr.inv_order_var;
kstate = dr.kstate;
ikx = [nstatic+1:nstatic+nspred];
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
else
ghu1 = zeros(nx,M_.exo_nbr);
trend = 1:M_.endo_nbr;
inv_order_var = trend(M_.block_structure.variable_reordered);
ghu1(1:length(dr.state_var),:) = ghu(dr.state_var,:);
end;
b = ghu1*M_.Sigma_e*ghu1';
if options_.block == 0
ipred = nstatic+(1:nspred)';
else
ipred = dr.state_var;
end;
% state space representation for state variables only
[A,B] = kalman_transition_matrix(dr,ipred,1:nx,M_.exo_nbr);
% Compute stationary variables (before HP filtering),
% and compute 2nd order mean correction on stationary variables (in case of
% HP filtering, this mean correction is computed *before* filtering)
if options_.order == 2 || options_.hp_filter == 0
[vx, u] = lyapunov_symm(A,B*M_.Sigma_e*B',options_.qz_criterium,options_.lyapunov_complex_threshold);
if options_.block == 0
iky = inv_order_var(ivar);
else
iky = ivar;
end;
stationary_vars = (1:length(ivar))';
if ~isempty(u)
x = abs(ghx*u);
iky = iky(find(all(x(iky,:) < options_.Schur_vec_tol,2)));
stationary_vars = find(all(x(inv_order_var(ivar(stationary_vars)),:) < options_.Schur_vec_tol,2));
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} = NaN*ones(nvar, 1);
Gamma_y{nar+3}(stationary_vars) = AS(iky,:)*Ex+(dr.ghs2(iky)+dr.ghxx(iky,:)*vx(:)+...
dr.ghuu(iky,:)*M_.Sigma_e(:))/2;
end
end
if options_.hp_filter == 0
v = NaN*ones(nvar,nvar);
v(stationary_vars,stationary_vars) = aa*vx*aa'+ bb*M_.Sigma_e*bb';
k = find(abs(v) < 1e-12);
v(k) = 0;
Gamma_y{1} = v;
% autocorrelations
if nar > 0
vxy = (A*vx*aa'+ghu1*M_.Sigma_e*bb');
sy = sqrt(diag(Gamma_y{1}));
sy = sy(stationary_vars);
sy = sy *sy';
Gamma_y{2} = NaN*ones(nvar,nvar);
Gamma_y{2}(stationary_vars,stationary_vars) = aa*vxy./sy;
for i=2:nar
vxy = A*vxy;
Gamma_y{i+1} = NaN*ones(nvar,nvar);
Gamma_y{i+1}(stationary_vars,stationary_vars) = aa*vxy./sy;
end
end
% variance decomposition
if ~nodecomposition && M_.exo_nbr > 1 && size(stationary_vars, 1) > 0
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)';
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,options_.lyapunov_complex_threshold,1);
vv = diag(aa*vx*aa'+b2*b2');
vv2 = 0;
for i=1:M_.exo_nbr
vx1 = lyapunov_symm(A,b1(:,i)*b1(:,i)',options_.qz_criterium,options_.lyapunov_complex_threshold,2);
vx2 = abs(diag(aa*vx1*aa'+b2(:,i)*b2(:,i)'));
Gamma_y{nar+2}(stationary_vars,i) = vx2;
vv2 = vv2 +vx2;
end
if max(abs(vv2-vv)./vv) > 1e-4
warning(['Aggregate variance and sum of variances by shocks ' ...
'differ by more than 0.01 %'])
end
for i=1:M_.exo_nbr
Gamma_y{nar+2}(stationary_vars,i) = Gamma_y{nar+ ...
2}(stationary_vars,i)./vv2;
end
end
else% ==> Theoretical HP filter.
% By construction, all variables are stationary when HP filtered
iky = inv_order_var(ivar);
stationary_vars = (1:length(ivar))';
aa = ghx(iky,:);
bb = ghu(iky,:);
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);
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
end
% Variance decomposition
if ~nodecomposition && 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;
imathp_col = real(ifft(mathp_col))*(2*pi);
Gamma_y{nar+2}(:,i) = abs(diag(reshape(imathp_col(1,:),nvar,nvar)))./vv;
end
end
end
if exist('OCTAVE_VERSION')
warning('on', 'Octave:divide-by-zero')
else
warning on MATLAB:dividebyzero
end
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