v4: ported changes for computing variance in unit root models
git-svn-id: https://www.dynare.org/svn/dynare/dynare_v4@720 ac1d8469-bf42-47a9-8791-bf33cf982152time-shift
michel
parent
161753429a
commit
c85730c108
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@ -1,47 +1,61 @@
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% solves x-a*x*a'=b for b (and then x) symmetrical
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function x=lyapunov_symm(a,b)
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n = size(b,1);
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if n == 1
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x=b/(1-a*a);
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return
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end
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x=zeros(n,n);
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[u,t]=schur(a);
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b=u'*b*u;
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for i=n:-1:2
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if t(i,i-1) == 0
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if i == n
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c = zeros(n,1);
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else
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c = t(1:i,:)*(x(:,i+1:end)*t(i,i+1:end)')+...
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t(i,i)*t(1:i,i+1:end)*x(i+1:end,i);
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end
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x(1:i,i)=(eye(i)-t(1:i,1:i)*t(i,i))\(b(1:i,i)+c);
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x(i,1:i-1)=x(1:i-1,i)';
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else
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if i == n
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c = zeros(n,1);
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c1 = zeros(n,1);
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else
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c = t(1:i,:)*(x(:,i+1:end)*t(i,i+1:end)')+...
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t(i,i)*t(1:i,i+1:end)*x(i+1:end,i)+...
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t(i,i-1)*t(1:i,i+1:end)*x(i+1:end,i-1);
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c1 = t(1:i,:)*(x(:,i+1:end)*t(i-1,i+1:end)')+...
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t(i-1,i-1)*t(1:i,i+1:end)*x(i+1:end,i-1)+...
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t(i-1,i)*t(1:i,i+1:end)*x(i+1:end,i);
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end
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z = [eye(i)-t(1:i,1:i)*t(i,i) -t(1:i,1:i)*t(i,i-1);...
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-t(1:i,1:i)*t(i-1,i) eye(i)-t(1:i,1:i)*t(i-1,i-1)]...
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\[b(1:i,i)+c;b(1:i,i-1)+c1];
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x(1:i,i) = z(1:i);
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x(1:i,i-1) = z(i+1:end);
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x(i,1:i-1)=x(1:i-1,i)';
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x(i-1,1:i-2)=x(1:i-2,i-1)';
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i = i - 1;
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end
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end
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if i == 2
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c = t(1,:)*(x(:,2:end)*t(1,2:end)')+t(1,1)*t(1,2:end)*x(2:end,1);
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x(1,1)=(b(1,1)+c)/(1-t(1,1)*t(1,1));
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end
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x=u*x*u';
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% solves x-a*x*a'=b for b (and then x) symmetrical
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function [x,ns_var]=lyapunov_symm(a,b)
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global options_
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info = 0;
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if size(a,1) == 1
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x=b/(1-a*a);
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return
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end
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[u,t] = schur(a);
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if exist('ordeig','builtin')
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e1 = abs(ordeig(t)) > 2-options_.qz_criterium;
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else
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e1 = abs(my_ordeig(t)) > 2-options_.qz_criterium;
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end
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k = sum(e1);
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[u,t] = ordschur(u,t,e1);
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n = length(e1)-k;
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b=u(:,k+1:end)'*b*u(:,k+1:end);
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t = t(k+1:end,k+1:end);
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x=zeros(n,n);
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for i=n:-1:2
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if t(i,i-1) == 0
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if i == n
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c = zeros(n,1);
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else
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c = t(1:i,:)*(x(:,i+1:end)*t(i,i+1:end)')+...
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t(i,i)*t(1:i,i+1:end)*x(i+1:end,i);
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end
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q = eye(i)-t(1:i,1:i)*t(i,i);
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x(1:i,i) = q\(b(1:i,i)+c);
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x(i,1:i-1) = x(1:i-1,i)';
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else
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if i == n
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c = zeros(n,1);
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c1 = zeros(n,1);
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else
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c = t(1:i,:)*(x(:,i+1:end)*t(i,i+1:end)')+...
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t(i,i)*t(1:i,i+1:end)*x(i+1:end,i)+...
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t(i,i-1)*t(1:i,i+1:end)*x(i+1:end,i-1);
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c1 = t(1:i,:)*(x(:,i+1:end)*t(i-1,i+1:end)')+...
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t(i-1,i-1)*t(1:i,i+1:end)*x(i+1:end,i-1)+...
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t(i-1,i)*t(1:i,i+1:end)*x(i+1:end,i);
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end
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q = [eye(i)-t(1:i,1:i)*t(i,i) -t(1:i,1:i)*t(i,i-1);...
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-t(1:i,1:i)*t(i-1,i) eye(i)-t(1:i,1:i)*t(i-1,i-1)];
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z = q\[b(1:i,i)+c;b(1:i,i-1)+c1];
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x(1:i,i) = z(1:i);
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x(1:i,i-1) = z(i+1:end);
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x(i,1:i-1)=x(1:i-1,i)';
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x(i-1,1:i-2)=x(1:i-2,i-1)';
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i = i - 1;
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end
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end
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if i == 2
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c = t(1,:)*(x(:,2:end)*t(1,2:end)')+t(1,1)*t(1,2:end)*x(2:end,1);
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x(1,1)=(b(1,1)+c)/(1-t(1,1)*t(1,1));
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end
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x=u(:,k+1:end)*x*u(:,k+1:end)';
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ns_var = [];
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ns_var = find(any(abs(u(:,1:k)) > 1e-8,2));
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@ -0,0 +1,18 @@
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function eval = my_ordeig(t)
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n = size(t,2);
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eval = zeros(n,1);
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for i=1:n-1
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if t(i+1,i) == 0
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eval(i) = t(i,i);
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else
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k = i:i+1;
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eval(k) = eig(t(k,k));
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i = i+1;
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end
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end
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if i < n
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t(n) = t(n,n);
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end
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@ -1,196 +1,204 @@
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% Copyright (C) 2001 Michel Juillard
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%
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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 M_.Sigma_e
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% for a subset of variables ivar (indices in M_.endo_names)
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% Theoretical HP filtering is available as an option
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function gamma_y=th_autocovariances(dr,ivar)
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global M_ options_
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if sscanf(version('-release'),'%d') < 13
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warning off
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else
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eval('warning off MATLAB:dividebyzero')
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end
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nar = options_.ar;
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gamma_y = cell(nar+1,1);
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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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npred = dr.npred;
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nstatic = dr.nstatic;
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kstate = dr.kstate;
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order = dr.order_var;
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iv(order) = [1:length(order)];
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nx = size(ghx,2);
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ikx = [nstatic+1:nstatic+npred];
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A = zeros(nx,nx);
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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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n0 = length(i0);
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A(i0,:) = ghx(ikx,:);
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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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j2 = j1;
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for k1 = 1:n1
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j1(k1) = find(k0(1:n0,1)==k0(i1(k1),1));
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j2(k1) = find(k0(i0,1)==k0(i1(k1),1));
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end
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A(i1,i0(j2))=eye(n1);
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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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% order of variables with preset variances in ghx and ghu
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iky = iv(ivar);
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aa = ghx(iky,:);
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bb = ghu(iky,:);
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if options_.order == 2
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%save temp A AS b dr M_.Sigma_e ikx iky nx n0
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vx = lyapunov_symm(A,b);
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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} = AS(iky,:)*Ex+(dr.ghs2(iky)+dr.ghxx(iky,:)*vx(:)+dr.ghuu(iky,:)*M_.Sigma_e(:))/2;
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end
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if options_.hp_filter == 0
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if options_.order < 2
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vx = lyapunov_symm(A,b);
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end
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gamma_y{1} = aa*vx*aa'+ bb*M_.Sigma_e*bb';
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k = find(abs(gamma_y{1}) < 1e-12);
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gamma_y{1}(k) = 0;
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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 *sy';
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gamma_y{2} = 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} = aa*vxy./sy;
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end
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end
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% variance decomposition
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if M_.exo_nbr > 1
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gamma_y{nar+2} = zeros(nvar,M_.exo_nbr);
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SS(M_.exo_names_orig_ord,M_.exo_names_orig_ord)=M_.Sigma_e+1e-14*eye(M_.exo_nbr);
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cs = chol(SS)';
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b1(:,M_.exo_names_orig_ord) = ghu1;
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b1 = b1*cs;
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b2(:,M_.exo_names_orig_ord) = ghu(iky,:);
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b2 = b2*cs;
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vx = lyapunov_symm(A,b1*b1');
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vv = diag(aa*vx*aa'+b2*b2');
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for i=1:M_.exo_nbr
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vx1 = lyapunov_symm(A,b1(:,i)*b1(:,i)');
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gamma_y{nar+2}(:,i) = abs(diag(aa*vx1*aa'+b2(:,i)*b2(:,i)'))./vv;
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end
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end
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else
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lambda = options_.hp_filter;
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ngrid = options_.hp_ngrid;
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freqs = 0 : ((2*pi)/ngrid) : (2*pi*(1 - .5/ngrid));
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tpos = exp( sqrt(-1)*freqs);
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tneg = exp(-sqrt(-1)*freqs);
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hp1 = 4*lambda*(1 - cos(freqs)).^2 ./ (1 + 4*lambda*(1 - cos(freqs)).^2);
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mathp_col = [];
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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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f_omega =(1/(2*pi))*( [inv(IA-A*tneg(ig))*ghu1;IE]...
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*M_.Sigma_e*[ghu1'*inv(IA-A'*tpos(ig)) ...
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IE]); % state variables
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g_omega = [aa*tneg(ig) bb]*f_omega*[aa'*tpos(ig); bb']; % selected variables
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f_hp = hp1(ig)^2*g_omega; % spectral density of selected filtered series
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mathp_col = [mathp_col ; (f_hp(:))']; % store as matrix row
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% 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);
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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 M_.exo_nbr > 1
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gamma_y{nar+2} = zeros(nvar,M_.exo_nbr);
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SS(M_.exo_names_orig_ord,M_.exo_names_orig_ord)=M_.Sigma_e+1e-14*eye(M_.exo_nbr);
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cs = chol(SS)';
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SS = cs*cs';
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b1(:,M_.exo_names_orig_ord) = ghu1;
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b2(:,M_.exo_names_orig_ord) = ghu(iky,:);
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mathp_col = [];
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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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f_omega =(1/(2*pi))*( [inv(IA-A*tneg(ig))*b1;IE]...
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*SS*[b1'*inv(IA-A'*tpos(ig)) ...
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IE]); % state variables
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g_omega = [aa*tneg(ig) b2]*f_omega*[aa'*tpos(ig); b2']; % selected variables
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f_hp = hp1(ig)^2*g_omega; % spectral density of selected filtered series
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mathp_col = [mathp_col ; (f_hp(:))']; % store as matrix row
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% 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 = [];
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SSi = cs(:,i)*cs(:,i)';
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for ig = 1:ngrid
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f_omega =(1/(2*pi))*( [inv(IA-A*tneg(ig))*b1;IE]...
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*SSi*[b1'*inv(IA-A'*tpos(ig)) ...
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IE]); % state variables
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g_omega = [aa*tneg(ig) b2]*f_omega*[aa'*tpos(ig); b2']; % selected variables
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f_hp = hp1(ig)^2*g_omega; % spectral density of selected filtered series
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mathp_col = [mathp_col ; (f_hp(:))']; % store as matrix row
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% 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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if sscanf(version('-release'),'%d') < 13
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warning on
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else
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eval('warning on MATLAB:dividebyzero')
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end
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% 10/18/2002 MJ
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% 10/30/2002 added autocorrelations, gamma_y is now a cell array
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% 01/20/2003 MJ added variance decomposition
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% 02/18/2003 MJ added HP filtering (Thanks to Jean Chateau for the code)
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% 04/28/2003 MJ changed handling of options
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% 05/19/2003 MJ don't assume lags are in increasing order in building A
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% 05/21/2003 MJ added global i_exo_var_,
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% variance decomposition: test M_.exo_nbr > 1
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% 05/29/2003 MJ removed global i_exo_var_
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% 06/10/2003 MJ test release for warning syntax
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% Copyright (C) 2001 Michel Juillard
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%
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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
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function [Gamma_y,ivar]=th_autocovariances(dr,ivar)
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global M_ options_
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exo_names_orig_ord = M_.exo_names_orig_ord;
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if sscanf(version('-release'),'%d') < 13
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warning off
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else
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eval('warning off MATLAB:dividebyzero')
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end
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nar = options_.ar;
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Gamma_y = cell(nar+1,1);
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if isempty(ivar)
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ivar = [1:M_.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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npred = dr.npred;
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nstatic = dr.nstatic;
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kstate = dr.kstate;
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order = dr.order_var;
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iv(order) = [1:length(order)];
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nx = size(ghx,2);
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ikx = [nstatic+1:nstatic+npred];
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A = zeros(nx,nx);
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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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A(i0,:) = ghx(ikx,:);
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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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j2 = j1;
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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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j2(k1) = find(k0(i0,1)==k0(i1(k1),1));
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end
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AS(:,j1) = AS(:,j1)+ghx(:,i1);
|
||||
i0 = i1;
|
||||
end
|
||||
b = ghu1*M_.Sigma_e*ghu1';
|
||||
|
||||
|
||||
[A,B] = kalman_transition_matrix(dr);
|
||||
% index of predetermined variables in A
|
||||
i_pred = [nstatic+(1:npred) M_.endo_nbr+1:length(A)];
|
||||
A = A(i_pred,i_pred);
|
||||
|
||||
if options_.order == 2
|
||||
[vx,ns_var] = lyapunov_symm(A,b);
|
||||
i_ivar = find(~ismember(ivar,dr.order_var(ns_var+nstatic)));
|
||||
ivar = ivar(i_ivar);
|
||||
iky = iv(ivar);
|
||||
aa = ghx(iky,:);
|
||||
bb = ghu(iky,:);
|
||||
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
|
||||
if options_.hp_filter == 0
|
||||
if options_.order < 2
|
||||
[vx, ns_var] = lyapunov_symm(A,b);
|
||||
i_ivar = find(~ismember(ivar,dr.order_var(ns_var+nstatic)));
|
||||
ivar = ivar(i_ivar);
|
||||
iky = iv(ivar);
|
||||
aa = ghx(iky,:);
|
||||
bb = ghu(iky,:);
|
||||
end
|
||||
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
|
||||
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');
|
||||
vv = diag(aa*vx*aa'+b2*b2');
|
||||
for i=1:M_.exo_nbr
|
||||
vx1 = lyapunov_symm(A,b1(:,i)*b1(:,i)');
|
||||
Gamma_y{nar+2}(:,i) = abs(diag(aa*vx1*aa'+b2(:,i)*b2(:,i)'))./vv;
|
||||
end
|
||||
end
|
||||
else
|
||||
if options_.order < 2
|
||||
iky = iv(ivar);
|
||||
aa = ghx(iky,:);
|
||||
bb = ghu(iky,:);
|
||||
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);
|
||||
|
||||
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 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 sscanf(version('-release'),'%d') < 13
|
||||
warning on
|
||||
else
|
||||
eval('warning on MATLAB:dividebyzero')
|
||||
end
|
||||
|
||||
Loading…
Reference in New Issue