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<title>Description of th_autocovariances</title>
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<div><a href="../index.html">Home</a> > <a href="index.html">.</a> > th_autocovariances.m</div>
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<!--<table width="100%"><tr><td align="left"><a href="../index.html"><img alt="<" border="0" src="../left.png"> Master index</a></td>
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<td align="right"><a href="index.html">Index for . <img alt=">" border="0" src="../right.png"></a></td></tr></table>-->
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<h1>th_autocovariances
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</h1>
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<h2><a name="_name"></a>PURPOSE <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="box"><strong>Copyright (C) 2001 Michel Juillard</strong></div>
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<h2><a name="_synopsis"></a>SYNOPSIS <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="box"><strong>function [Gamma_y,ivar]=th_autocovariances(dr,ivar) </strong></div>
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<h2><a name="_description"></a>DESCRIPTION <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="fragment"><pre class="comment"> Copyright (C) 2001 Michel Juillard
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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 (indices in lgy_)
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Theoretical HP filtering is available as an option</pre></div>
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<!-- crossreference -->
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<h2><a name="_cross"></a>CROSS-REFERENCE INFORMATION <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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This function calls:
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<ul style="list-style-image:url(../matlabicon.gif)">
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<li><a href="kalman_transition_matrix.html" class="code" title="function [A,B] = kalman_transition_matrix(dr)">kalman_transition_matrix</a> makes transition matrices out of ghx and ghu for Kalman filter</li><li><a href="lyapunov_symm.html" class="code" title="function [x,ns_var]=lyapunov_symm(a,b)">lyapunov_symm</a> solves x-a*x*a'=b for b (and then x) symmetrical</li></ul>
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This function is called by:
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<ul style="list-style-image:url(../matlabicon.gif)">
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<li><a href="disp_th_moments.html" class="code" title="function disp_th_moments(dr,var_list)">disp_th_moments</a> Copyright (C) 2001 Michel Juillard</li></ul>
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<!-- crossreference -->
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<h2><a name="_source"></a>SOURCE CODE <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="fragment"><pre>0001 <span class="comment">% Copyright (C) 2001 Michel Juillard</span>
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0002 <span class="comment">%</span>
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0003 <span class="comment">% computes the theoretical auto-covariances, Gamma_y, for an AR(p) process</span>
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0004 <span class="comment">% with coefficients dr.ghx and dr.ghu and shock variances Sigma_e_</span>
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0005 <span class="comment">% for a subset of variables ivar (indices in lgy_)</span>
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0006 <span class="comment">% Theoretical HP filtering is available as an option</span>
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0007
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0008 <a name="_sub0" href="#_subfunctions" class="code">function [Gamma_y,ivar]=th_autocovariances(dr,ivar)</a>
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0009 <span class="keyword">global</span> M_ options_
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0010
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0011 exo_names_orig_ord = M_.exo_names_orig_ord;
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0012 <span class="keyword">if</span> sscanf(version(<span class="string">'-release'</span>),<span class="string">'%d'</span>) < 13
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0013 warning off
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0014 <span class="keyword">else</span>
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0015 eval(<span class="string">'warning off MATLAB:dividebyzero'</span>)
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0016 <span class="keyword">end</span>
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0017 nar = options_.ar;
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0018 Gamma_y = cell(nar+1,1);
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0019 <span class="keyword">if</span> isempty(ivar)
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0020 ivar = [1:M_.endo_nbr]';
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0021 <span class="keyword">end</span>
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0022 nvar = size(ivar,1);
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0023
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0024 ghx = dr.ghx;
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0025 ghu = dr.ghu;
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0026 npred = dr.npred;
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0027 nstatic = dr.nstatic;
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0028 kstate = dr.kstate;
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0029 order = dr.order_var;
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0030 iv(order) = [1:length(order)];
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0031 nx = size(ghx,2);
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0032
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0033 ikx = [nstatic+1:nstatic+npred];
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0034
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0035 A = zeros(nx,nx);
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0036 k0 = kstate(find(kstate(:,2) <= M_.maximum_lag+1),:);
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0037 i0 = find(k0(:,2) == M_.maximum_lag+1);
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0038 i00 = i0;
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0039 n0 = length(i0);
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0040 A(i0,:) = ghx(ikx,:);
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0041 AS = ghx(:,i0);
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0042 ghu1 = zeros(nx,M_.exo_nbr);
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0043 ghu1(i0,:) = ghu(ikx,:);
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0044 <span class="keyword">for</span> i=M_.maximum_lag:-1:2
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0045 i1 = find(k0(:,2) == i);
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0046 n1 = size(i1,1);
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0047 j1 = zeros(n1,1);
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0048 j2 = j1;
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0049 <span class="keyword">for</span> k1 = 1:n1
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0050 j1(k1) = find(k0(i00,1)==k0(i1(k1),1));
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0051 j2(k1) = find(k0(i0,1)==k0(i1(k1),1));
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0052 <span class="keyword">end</span>
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0053 AS(:,j1) = AS(:,j1)+ghx(:,i1);
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0054 i0 = i1;
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0055 <span class="keyword">end</span>
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0056 b = ghu1*M_.Sigma_e*ghu1';
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0057
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0058
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0059 [A,B] = <a href="kalman_transition_matrix.html" class="code" title="function [A,B] = kalman_transition_matrix(dr)">kalman_transition_matrix</a>(dr);
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0060 <span class="comment">% index of predetermined variables in A</span>
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0061 i_pred = [nstatic+(1:npred) M_.endo_nbr+1:length(A)];
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0062 A = A(i_pred,i_pred);
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0063
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0064 <span class="keyword">if</span> options_.order == 2
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0065 [vx,ns_var] = <a href="lyapunov_symm.html" class="code" title="function [x,ns_var]=lyapunov_symm(a,b)">lyapunov_symm</a>(A,b);
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0066 i_ivar = find(~ismember(ivar,dr.order_var(ns_var+nstatic)));
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0067 ivar = ivar(i_ivar);
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0068 iky = iv(ivar);
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0069 aa = ghx(iky,:);
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0070 bb = ghu(iky,:);
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0071 Ex = (dr.ghs2(ikx)+dr.ghxx(ikx,:)*vx(:)+dr.ghuu(ikx,:)*M_.Sigma_e(:))/2;
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0072 Ex = (eye(n0)-AS(ikx,:))\Ex;
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0073 Gamma_y{nar+3} = AS(iky,:)*Ex+(dr.ghs2(iky)+dr.ghxx(iky,:)*vx(:)+dr.ghuu(iky,:)*M_.Sigma_e(:))/2;
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0074 <span class="keyword">end</span>
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0075 <span class="keyword">if</span> options_.hp_filter == 0
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0076 <span class="keyword">if</span> options_.order < 2
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0077 [vx, ns_var] = <a href="lyapunov_symm.html" class="code" title="function [x,ns_var]=lyapunov_symm(a,b)">lyapunov_symm</a>(A,b);
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0078 i_ivar = find(~ismember(ivar,dr.order_var(ns_var+nstatic)));
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0079 ivar = ivar(i_ivar);
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0080 iky = iv(ivar);
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0081 aa = ghx(iky,:);
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0082 bb = ghu(iky,:);
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0083 <span class="keyword">end</span>
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0084 Gamma_y{1} = aa*vx*aa'+ bb*M_.Sigma_e*bb';
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0085 k = find(abs(Gamma_y{1}) < 1e-12);
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0086 Gamma_y{1}(k) = 0;
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0087
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0088 <span class="comment">% autocorrelations</span>
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0089 <span class="keyword">if</span> nar > 0
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0090 vxy = (A*vx*aa'+ghu1*M_.Sigma_e*bb');
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0091
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0092 sy = sqrt(diag(Gamma_y{1}));
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0093 sy = sy *sy';
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0094 Gamma_y{2} = aa*vxy./sy;
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0095
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0096 <span class="keyword">for</span> i=2:nar
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0097 vxy = A*vxy;
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0098 Gamma_y{i+1} = aa*vxy./sy;
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0099 <span class="keyword">end</span>
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0100 <span class="keyword">end</span>
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0101
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0102 <span class="comment">% variance decomposition</span>
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0103 <span class="keyword">if</span> M_.exo_nbr > 1
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0104 Gamma_y{nar+2} = zeros(length(ivar),M_.exo_nbr);
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0105 SS(exo_names_orig_ord,exo_names_orig_ord)=M_.Sigma_e+1e-14*eye(M_.exo_nbr);
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0106 cs = chol(SS)';
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0107 b1(:,exo_names_orig_ord) = ghu1;
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0108 b1 = b1*cs;
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0109 b2(:,exo_names_orig_ord) = ghu(iky,:);
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0110 b2 = b2*cs;
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0111 vx = <a href="lyapunov_symm.html" class="code" title="function [x,ns_var]=lyapunov_symm(a,b)">lyapunov_symm</a>(A,b1*b1');
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0112 vv = diag(aa*vx*aa'+b2*b2');
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0113 <span class="keyword">for</span> i=1:M_.exo_nbr
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0114 vx1 = <a href="lyapunov_symm.html" class="code" title="function [x,ns_var]=lyapunov_symm(a,b)">lyapunov_symm</a>(A,b1(:,i)*b1(:,i)');
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0115 Gamma_y{nar+2}(:,i) = abs(diag(aa*vx1*aa'+b2(:,i)*b2(:,i)'))./vv;
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0116 <span class="keyword">end</span>
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0117 <span class="keyword">end</span>
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0118 <span class="keyword">else</span>
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0119 <span class="keyword">if</span> options_.order < 2
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0120 iky = iv(ivar);
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0121 aa = ghx(iky,:);
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0122 bb = ghu(iky,:);
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0123 <span class="keyword">end</span>
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0124 lambda = options_.hp_filter;
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0125 ngrid = options_.hp_ngrid;
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0126 freqs = 0 : ((2*pi)/ngrid) : (2*pi*(1 - .5/ngrid));
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0127 tpos = exp( sqrt(-1)*freqs);
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0128 tneg = exp(-sqrt(-1)*freqs);
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0129 hp1 = 4*lambda*(1 - cos(freqs)).^2 ./ (1 + 4*lambda*(1 - cos(freqs)).^2);
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0130
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0131 mathp_col = [];
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0132 IA = eye(size(A,1));
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0133 IE = eye(M_.exo_nbr);
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0134 <span class="keyword">for</span> ig = 1:ngrid
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0135 f_omega =(1/(2*pi))*( [inv(IA-A*tneg(ig))*ghu1;IE]<span class="keyword">...</span>
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0136 *M_.Sigma_e*[ghu1'*inv(IA-A'*tpos(ig)) <span class="keyword">...</span>
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0137 IE]); <span class="comment">% state variables</span>
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0138 g_omega = [aa*tneg(ig) bb]*f_omega*[aa'*tpos(ig); bb']; <span class="comment">% selected variables</span>
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0139 f_hp = hp1(ig)^2*g_omega; <span class="comment">% spectral density of selected filtered series</span>
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0140 mathp_col = [mathp_col ; (f_hp(:))']; <span class="comment">% store as matrix row</span>
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0141 <span class="comment">% for ifft</span>
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0142 <span class="keyword">end</span>;
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0143
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0144 <span class="comment">% covariance of filtered series</span>
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0145 imathp_col = real(ifft(mathp_col))*(2*pi);
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0146
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0147 Gamma_y{1} = reshape(imathp_col(1,:),nvar,nvar);
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0148
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0149 <span class="comment">% autocorrelations</span>
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0150 <span class="keyword">if</span> nar > 0
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0151 sy = sqrt(diag(Gamma_y{1}));
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0152 sy = sy *sy';
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0153 <span class="keyword">for</span> i=1:nar
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0154 Gamma_y{i+1} = reshape(imathp_col(i+1,:),nvar,nvar)./sy;
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0155 <span class="keyword">end</span>
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0156 <span class="keyword">end</span>
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0157
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0158 <span class="comment">%variance decomposition</span>
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0159 <span class="keyword">if</span> M_.exo_nbr > 1
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0160 Gamma_y{nar+2} = zeros(nvar,M_.exo_nbr);
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0161 SS(exo_names_orig_ord,exo_names_orig_ord)=M_.Sigma_e+1e-14*eye(M_.exo_nbr);
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0162 cs = chol(SS)';
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0163 SS = cs*cs';
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0164 b1(:,exo_names_orig_ord) = ghu1;
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0165 b2(:,exo_names_orig_ord) = ghu(iky,:);
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0166 mathp_col = [];
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0167 IA = eye(size(A,1));
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0168 IE = eye(M_.exo_nbr);
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0169 <span class="keyword">for</span> ig = 1:ngrid
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0170 f_omega =(1/(2*pi))*( [inv(IA-A*tneg(ig))*b1;IE]<span class="keyword">...</span>
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0171 *SS*[b1'*inv(IA-A'*tpos(ig)) <span class="keyword">...</span>
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0172 IE]); <span class="comment">% state variables</span>
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0173 g_omega = [aa*tneg(ig) b2]*f_omega*[aa'*tpos(ig); b2']; <span class="comment">% selected variables</span>
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0174 f_hp = hp1(ig)^2*g_omega; <span class="comment">% spectral density of selected filtered series</span>
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0175 mathp_col = [mathp_col ; (f_hp(:))']; <span class="comment">% store as matrix row</span>
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0176 <span class="comment">% for ifft</span>
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0177 <span class="keyword">end</span>;
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0178
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0179 imathp_col = real(ifft(mathp_col))*(2*pi);
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0180 vv = diag(reshape(imathp_col(1,:),nvar,nvar));
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0181 <span class="keyword">for</span> i=1:M_.exo_nbr
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0182 mathp_col = [];
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0183 SSi = cs(:,i)*cs(:,i)';
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0184 <span class="keyword">for</span> ig = 1:ngrid
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0185 f_omega =(1/(2*pi))*( [inv(IA-A*tneg(ig))*b1;IE]<span class="keyword">...</span>
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0186 *SSi*[b1'*inv(IA-A'*tpos(ig)) <span class="keyword">...</span>
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0187 IE]); <span class="comment">% state variables</span>
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0188 g_omega = [aa*tneg(ig) b2]*f_omega*[aa'*tpos(ig); b2']; <span class="comment">% selected variables</span>
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0189 f_hp = hp1(ig)^2*g_omega; <span class="comment">% spectral density of selected filtered series</span>
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0190 mathp_col = [mathp_col ; (f_hp(:))']; <span class="comment">% store as matrix row</span>
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0191 <span class="comment">% for ifft</span>
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0192 <span class="keyword">end</span>;
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0193
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0194 imathp_col = real(ifft(mathp_col))*(2*pi);
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0195 Gamma_y{nar+2}(:,i) = abs(diag(reshape(imathp_col(1,:),nvar,nvar)))./vv;
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0196 <span class="keyword">end</span>
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0197 <span class="keyword">end</span>
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0198 <span class="keyword">end</span>
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0199 <span class="keyword">if</span> sscanf(version(<span class="string">'-release'</span>),<span class="string">'%d'</span>) < 13
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0200 warning on
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0201 <span class="keyword">else</span>
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0202 eval(<span class="string">'warning on MATLAB:dividebyzero'</span>)
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0203 <span class="keyword">end</span>
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0204</pre></div>
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