297 lines
13 KiB
Matlab
297 lines
13 KiB
Matlab
function [Gamma_y,stationary_vars] = th_autocovariances(dr,ivar,M_,options_,nodecomposition)
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% Computes the theoretical auto-covariances, Gamma_y, for an AR(p) process
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% with coefficients dr.ghx and dr.ghu and shock variances Sigma_e
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% for a subset of variables ivar.
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% Theoretical HP-filtering and band-pass filtering is available as an option
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%
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% INPUTS
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% dr: [structure] Reduced form solution of the DSGE model (decisions rules)
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% ivar: [integer] Vector of indices for a subset of variables.
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% M_ [structure] Global dynare's structure, description of the DSGE model.
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% options_ [structure] Global dynare's structure.
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% nodecomposition [integer] Scalar, if different from zero the variance decomposition is not triggered.
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%
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% OUTPUTS
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% Gamma_y [cell] Matlab cell of nar+3 (second order approximation) or nar+2 (first order approximation) arrays,
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% where nar is the order of the autocorrelation function.
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% Gamma_y{1} [double] Covariance matrix.
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% Gamma_y{i+1} [double] Autocorrelation function (for i=1,...,options_.nar).
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% Gamma_y{nar+2} [double] Variance decomposition.
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% Gamma_y{nar+3} [double] Expectation of the endogenous variables associated with a second
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% order approximation.
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% stationary_vars [integer] Vector of indices of stationary variables (as a subset of 1:length(ivar))
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%
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% SPECIAL REQUIREMENTS
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%
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% Algorithms
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% The means at order=2 are based on the pruned state space as
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% in Kim, Kim, Schaumburg, Sims (2008): Calculating and using second-order accurate
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% solutions of discrete time dynamic equilibrium models.
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% The solution at second order can be written as:
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% \[
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% \hat x_t = g_x \hat x_{t - 1} + g_u u_t + \frac{1}{2}\left( g_{\sigma\sigma} \sigma^2 + g_{xx}\hat x_t^2 + g_{uu} u_t^2 \right)
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% \]
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% Taking expectations on both sides requires to compute E(x^2)=Var(x), which
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% can be obtained up to second order from the first order solution
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% \[
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% \hat x_t = g_x \hat x_{t - 1} + g_u u_t
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% \]
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% by solving the corresponding Lyapunov equation.
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% Given Var(x), the above equation can be solved for E(x_t) as
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% \[
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% E(x_t) = (I - {g_x}\right)^{- 1} 0.5\left( g_{\sigma\sigma} \sigma^2 + g_{xx} Var(\hat x_t) + g_{uu} Var(u_t) \right)
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% \]
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%
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% Copyright © 2001-2023 Dynare Team
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%
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% This file is part of Dynare.
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%
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% Dynare is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% Dynare is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <https://www.gnu.org/licenses/>.
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if nargin<5
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nodecomposition = 0;
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end
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if options_.order >= 3
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error('Theoretical moments not implemented above 2nd order')
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end
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local_order = options_.order;
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if local_order~=1 && M_.hessian_eq_zero
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local_order = 1;
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end
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endo_nbr = M_.endo_nbr;
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if isoctave
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warning('off', 'Octave:divide-by-zero')
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else
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warning off MATLAB:dividebyzero
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end
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nar = options_.ar;
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Gamma_y = cell(nar+2,1);
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if isempty(ivar)
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ivar = [1:endo_nbr]';
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end
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nvar = size(ivar,1);
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ghx = dr.ghx;
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ghu = dr.ghu;
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nspred = M_.nspred;
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nstatic = M_.nstatic;
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nx = size(ghx,2);
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inv_order_var = dr.inv_order_var;
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kstate = dr.kstate;
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ikx = [nstatic+1:nstatic+nspred];
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k0 = kstate(find(kstate(:,2) <= M_.maximum_lag+1),:);
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i0 = find(k0(:,2) == M_.maximum_lag+1);
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i00 = i0;
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n0 = length(i0);
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AS = ghx(:,i0);
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ghu1 = zeros(nx,M_.exo_nbr);
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ghu1(i0,:) = ghu(ikx,:);
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for i=M_.maximum_lag:-1:2
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i1 = find(k0(:,2) == i);
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n1 = size(i1,1);
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j1 = zeros(n1,1);
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for k1 = 1:n1
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j1(k1) = find(k0(i00,1)==k0(i1(k1),1));
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end
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AS(:,j1) = AS(:,j1)+ghx(:,i1);
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i0 = i1;
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end
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b = ghu1*M_.Sigma_e*ghu1';
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ipred = nstatic+(1:nspred)';
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% state space representation for state variables only
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[A,B] = kalman_transition_matrix(dr,ipred,1:nx,M_.exo_nbr);
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% Compute stationary variables (before HP filtering),
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% and compute 2nd order mean correction on stationary variables (in case of
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% HP filtering, this mean correction is computed *before* filtering)
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if local_order == 2 || options_.hp_filter == 0
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[vx, u] = lyapunov_symm(A,B*M_.Sigma_e*B',options_.lyapunov_fixed_point_tol,options_.qz_criterium,options_.lyapunov_complex_threshold,[],options_.debug);
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iky = inv_order_var(ivar);
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stationary_vars = (1:length(ivar))';
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if ~isempty(u)
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x = abs(ghx*u);
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iky = iky(find(all(x(iky,:) < options_.schur_vec_tol,2)));
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stationary_vars = find(all(x(inv_order_var(ivar(stationary_vars)),:) < options_.schur_vec_tol,2));
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end
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aa = ghx(iky,:);
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bb = ghu(iky,:);
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if local_order == 2 % mean correction for 2nd order
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if ~isempty(ikx)
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Ex = (dr.ghs2(ikx)+dr.ghxx(ikx,:)*vx(:)+dr.ghuu(ikx,:)*M_.Sigma_e(:))/2;
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Ex = (eye(n0)-AS(ikx,:))\Ex;
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Gamma_y{nar+3} = NaN*ones(nvar, 1);
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Gamma_y{nar+3}(stationary_vars) = AS(iky,:)*Ex+(dr.ghs2(iky)+dr.ghxx(iky,:)*vx(:)+...
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dr.ghuu(iky,:)*M_.Sigma_e(:))/2;
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else %no static and no predetermined
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Gamma_y{nar+3} = NaN*ones(nvar, 1);
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Gamma_y{nar+3}(stationary_vars) = (dr.ghs2(iky)+ dr.ghuu(iky,:)*M_.Sigma_e(:))/2;
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end
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end
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end
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if options_.hp_filter == 0 && ~options_.bandpass.indicator
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v = NaN*ones(nvar,nvar);
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v(stationary_vars,stationary_vars) = aa*vx*aa'+ bb*M_.Sigma_e*bb';
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k = find(abs(v) < 1e-12);
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v(k) = 0;
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Gamma_y{1} = v;
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% autocorrelations
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if nar > 0
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vxy = (A*vx*aa'+ghu1*M_.Sigma_e*bb');
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sy = sqrt(diag(Gamma_y{1}));
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sy = sy(stationary_vars);
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sy = sy *sy';
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Gamma_y{2} = NaN*ones(nvar,nvar);
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Gamma_y{2}(stationary_vars,stationary_vars) = aa*vxy./sy;
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for i=2:nar
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vxy = A*vxy;
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Gamma_y{i+1} = NaN*ones(nvar,nvar);
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Gamma_y{i+1}(stationary_vars,stationary_vars) = aa*vxy./sy;
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end
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end
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% variance decomposition
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if ~nodecomposition && M_.exo_nbr > 0 && size(stationary_vars, 1) > 0
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if M_.exo_nbr == 1
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Gamma_y{nar+2} = ones(nvar,1);
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else
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Gamma_y{nar+2} = NaN(nvar,M_.exo_nbr);
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cs = get_lower_cholesky_covariance(M_.Sigma_e,options_.add_tiny_number_to_cholesky);
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b1 = ghu1;
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b1 = b1*cs;
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b2 = ghu(iky,:);
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b2 = b2*cs;
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vx = lyapunov_symm(A,b1*b1',options_.lyapunov_fixed_point_tol,options_.qz_criterium,options_.lyapunov_complex_threshold,1,options_.debug);
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vv = diag(aa*vx*aa'+b2*b2');
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vv2 = 0;
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for i=1:M_.exo_nbr
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vx1 = lyapunov_symm(A,b1(:,i)*b1(:,i)',options_.lyapunov_fixed_point_tol,options_.qz_criterium,options_.lyapunov_complex_threshold,2,options_.debug);
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vx2 = abs(diag(aa*vx1*aa'+b2(:,i)*b2(:,i)'));
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Gamma_y{nar+2}(stationary_vars,i) = vx2;
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vv2 = vv2 +vx2;
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end
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if max(abs(vv2-vv)./vv) > 1e-4
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warning(['Aggregate variance and sum of variances by shocks ' ...
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'differ by more than 0.01 %'])
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end
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for i=1:M_.exo_nbr
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Gamma_y{nar+2}(stationary_vars,i) = Gamma_y{nar+ ...
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2}(stationary_vars,i)./vv2;
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end
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end
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end
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else% ==> Theoretical filters.
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% By construction, all variables are stationary when HP filtered
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iky = inv_order_var(ivar);
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stationary_vars = (1:length(ivar))';
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aa = ghx(iky,:); %R in Uhlig (2001)
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bb = ghu(iky,:); %S in Uhlig (2001)
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lambda = options_.hp_filter;
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ngrid = options_.filtered_theoretical_moments_grid;
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freqs = 0 : ((2*pi)/ngrid) : (2*pi*(1 - .5/ngrid)); %[0,2*pi)
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tpos = exp( sqrt(-1)*freqs); %positive frequencies
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tneg = exp(-sqrt(-1)*freqs); %negative frequencies
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if options_.bandpass.indicator
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filter_gain = zeros(1,ngrid);
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lowest_periodicity=options_.bandpass.passband(2);
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highest_periodicity=options_.bandpass.passband(1);
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highest_periodicity=max(2,highest_periodicity); % restrict to upper bound of pi
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filter_gain(freqs>=2*pi/lowest_periodicity & freqs<=2*pi/highest_periodicity)=1;
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filter_gain(freqs<=-2*pi/lowest_periodicity+2*pi & freqs>=-2*pi/highest_periodicity+2*pi)=1;
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else
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filter_gain = 4*lambda*(1 - cos(freqs)).^2 ./ (1 + 4*lambda*(1 - cos(freqs)).^2); %HP transfer function
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end
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mathp_col = NaN(ngrid,length(ivar)^2);
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IA = eye(size(A,1));
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IE = eye(M_.exo_nbr);
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for ig = 1:ngrid
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if filter_gain(ig)==0
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f_hp = zeros(length(ivar),length(ivar));
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else
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f_omega =(1/(2*pi))*([(IA-A*tneg(ig))\ghu1;IE]...
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*M_.Sigma_e*[ghu1'/(IA-A'*tpos(ig)) IE]); % spectral density of state variables; top formula Uhlig (2001), p. 20 with N=0
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g_omega = [aa*tneg(ig) bb]*f_omega*[aa'*tpos(ig); bb']; % spectral density of selected variables; middle formula Uhlig (2001), p. 20; only middle block, i.e. y_t'
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f_hp = filter_gain(ig)^2*g_omega; % spectral density of selected filtered series; top formula Uhlig (2001), p. 21;
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end
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mathp_col(ig,:) = (f_hp(:))'; % store as matrix row for ifft
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end
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% Covariance of filtered series
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imathp_col = real(ifft(mathp_col))*(2*pi); % Inverse Fast Fourier Transformation; middle formula Uhlig (2001), p. 21;
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Gamma_y{1} = reshape(imathp_col(1,:),nvar,nvar);
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% Autocorrelations
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if nar > 0
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sy = sqrt(diag(Gamma_y{1}));
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sy = sy *sy';
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for i=1:nar
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Gamma_y{i+1} = reshape(imathp_col(i+1,:),nvar,nvar)./sy;
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end
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end
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% Variance decomposition
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if ~nodecomposition && M_.exo_nbr > 0
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if M_.exo_nbr == 1
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Gamma_y{nar+2} = ones(nvar,1);
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else
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Gamma_y{nar+2} = zeros(nvar,M_.exo_nbr);
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cs = get_lower_cholesky_covariance(M_.Sigma_e); %make sure Covariance matrix is positive definite
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SS = cs*cs';
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b1 = ghu1;
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b2 = ghu(iky,:);
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mathp_col = NaN(ngrid,length(ivar)^2);
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IA = eye(size(A,1));
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IE = eye(M_.exo_nbr);
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for ig = 1:ngrid
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if filter_gain(ig)==0
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f_hp = zeros(length(ivar),length(ivar));
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else
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f_omega =(1/(2*pi))*( [(IA-A*tneg(ig))\b1;IE]...
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*SS*[b1'/(IA-A'*tpos(ig)) IE]); % spectral density of state variables; top formula Uhlig (2001), p. 20 with N=0
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g_omega = [aa*tneg(ig) b2]*f_omega*[aa'*tpos(ig); b2']; % spectral density of selected variables; middle formula Uhlig (2001), p. 20; only middle block, i.e. y_t'
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f_hp = filter_gain(ig)^2*g_omega; % spectral density of selected filtered series; top formula Uhlig (2001), p. 21;
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end
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mathp_col(ig,:) = (f_hp(:))'; % store as matrix row for ifft
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end
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imathp_col = real(ifft(mathp_col))*(2*pi);
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vv = diag(reshape(imathp_col(1,:),nvar,nvar));
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for i=1:M_.exo_nbr
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mathp_col = NaN(ngrid,length(ivar)^2);
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SSi = cs(:,i)*cs(:,i)';
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for ig = 1:ngrid
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if filter_gain(ig)==0
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f_hp = zeros(length(ivar),length(ivar));
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else
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f_omega =(1/(2*pi))*( [(IA-A*tneg(ig))\b1;IE]...
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*SSi*[b1'/(IA-A'*tpos(ig)) IE]); % spectral density of state variables; top formula Uhlig (2001), p. 20 with N=0
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g_omega = [aa*tneg(ig) b2]*f_omega*[aa'*tpos(ig); b2']; % spectral density of selected variables; middle formula Uhlig (2001), p. 20; only middle block, i.e. y_t'
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f_hp = filter_gain(ig)^2*g_omega; % spectral density of selected filtered series; top formula Uhlig (2001), p. 21;
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end
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mathp_col(ig,:) = (f_hp(:))'; % store as matrix row for ifft
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end
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imathp_col = real(ifft(mathp_col))*(2*pi);
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Gamma_y{nar+2}(:,i) = abs(diag(reshape(imathp_col(1,:),nvar,nvar)))./vv;
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end
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end
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end
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end
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if isoctave
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warning('on', 'Octave:divide-by-zero')
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else
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warning on MATLAB:dividebyzero
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end
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