diff --git a/matlab/Q6_plication.m b/matlab/Q6_plication.m
index 0692364f1357c8a0bd7d2269cade9c229f947327..153ee8ccd85ea9552b4225e368fcb36b8b3da08d 100644
--- a/matlab/Q6_plication.m
+++ b/matlab/Q6_plication.m
@@ -63,7 +63,9 @@ for i1=1:p
end
end
end
+if nargout==2
DP6inv = (transpose(DP6)*DP6)\transpose(DP6);
+end
function m = mue(p,i1,i2,i3,i4,i5,i6)
% Auxiliary expression, see page 122 of Meijer (2005)
diff --git a/matlab/list_of_functions_to_be_cleared.m b/matlab/list_of_functions_to_be_cleared.m
index 8dc0d90fd1890b64d8095b236be0706d60e6c1b2..82ee94b5a81d035866ab83a943d29b5c34cd9885 100644
--- a/matlab/list_of_functions_to_be_cleared.m
+++ b/matlab/list_of_functions_to_be_cleared.m
@@ -1 +1 @@
-list_of_functions = {'discretionary_policy_1', 'dsge_var_likelihood', 'dyn_first_order_solver', 'dyn_waitbar', 'ep_residuals', 'evaluate_likelihood', 'prior_draw_gsa', 'identification_analysis', 'computeDLIK', 'univariate_computeDLIK', 'metropolis_draw', 'flag_implicit_skip_nan', 'moment_function', 'mr_hessian', 'masterParallel', 'auxiliary_initialization', 'auxiliary_particle_filter', 'conditional_filter_proposal', 'conditional_particle_filter', 'gaussian_filter', 'gaussian_filter_bank', 'gaussian_mixture_filter', 'gaussian_mixture_filter_bank', 'Kalman_filter', 'online_auxiliary_filter', 'sequential_importance_particle_filter', 'solve_model_for_online_filter', 'perfect_foresight_simulation', 'prior_draw', 'priordens'};
+list_of_functions = {'discretionary_policy_1', 'dsge_var_likelihood', 'dyn_first_order_solver', 'dyn_waitbar', 'ep_residuals', 'evaluate_likelihood', 'prior_draw_gsa', 'identification_analysis', 'computeDLIK', 'univariate_computeDLIK', 'metropolis_draw', 'flag_implicit_skip_nan', 'moment_function', 'mr_hessian', 'masterParallel', 'auxiliary_initialization', 'auxiliary_particle_filter', 'conditional_filter_proposal', 'conditional_particle_filter', 'gaussian_filter', 'gaussian_filter_bank', 'gaussian_mixture_filter', 'gaussian_mixture_filter_bank', 'Kalman_filter', 'online_auxiliary_filter', 'pruned_state_space_system', 'sequential_importance_particle_filter', 'solve_model_for_online_filter', 'perfect_foresight_simulation', 'prior_draw', 'priordens'};
diff --git a/matlab/pruned_state_space_system.m b/matlab/pruned_state_space_system.m
index 4395f109cc9e0879c6012bf54d5f78327c5f84fa..d82bfd6777dd8295b46ad8ffd2d76e5ed82d05b2 100644
--- a/matlab/pruned_state_space_system.m
+++ b/matlab/pruned_state_space_system.m
@@ -10,12 +10,13 @@ function pruned_state_space = pruned_state_space_system(M, options, dr, indy, nl
% Econometrics and Statistics, Volume 6, Pages 44-56.
% =========================================================================
% INPUTS
-% M: [structure] storing the model information
-% options: [structure] storing the options
-% dr: [structure] storing the results from perturbation approximation
-% indy: [vector] index of control variables in DR order
-% nlags: [integer] number of lags in autocovariances and autocorrelations
-% useautocorr: [boolean] true: compute autocorrelations
+% M: [structure] storing the model information
+% options: [structure] storing the options
+% dr: [structure] storing the results from perturbation approximation
+% indy: [vector] index of control variables in DR order
+% nlags: [integer] number of lags in autocovariances and autocorrelations
+% useautocorr: [boolean] true: compute autocorrelations
+% compute_derivs: [boolean] true: compute derivatives
% -------------------------------------------------------------------------
% OUTPUTS
% pruned_state_space: [structure] with the following fields:
@@ -240,6 +241,7 @@ function pruned_state_space = pruned_state_space_system(M, options, dr, indy, nl
% See code below how z and inov are defined at first, second, and third order,
% and how to set up A, B, C, D and compute unconditional first and second moments of inov, z and y
+persistent QPu COMBOS4 Q6Pu COMBOS6 K_u_xx K_u_ux K_xx_x
%% Auxiliary indices and objects
order = options.order;
@@ -418,8 +420,10 @@ if order > 1
%Compute unique fourth order product moments of u, i.e. unique(E[kron(kron(kron(u,u),u),u)],'stable')
u_nbr4 = u_nbr*(u_nbr+1)/2*(u_nbr+2)/3*(u_nbr+3)/4;
- QPu = quadruplication(u_nbr);
- COMBOS4 = flipud(allVL1(u_nbr, 4)); %all possible (unique) combinations of powers that sum up to four
+ if isempty(QPu)
+ QPu = quadruplication(u_nbr);
+ COMBOS4 = flipud(allVL1(u_nbr, 4)); %all possible (unique) combinations of powers that sum up to four
+ end
E_u_u_u_u = zeros(u_nbr4,1); %only unique entries
if compute_derivs && (stderrparam_nbr+corrparam_nbr>0)
dE_u_u_u_u = zeros(u_nbr4,stderrparam_nbr+corrparam_nbr);
@@ -588,8 +592,11 @@ if order > 1
if order > 2
% Some common and useful objects for order > 2
- K_u_xx = commutation(u_nbr,x_nbr^2,1);
- K_u_ux = commutation(u_nbr,u_nbr*x_nbr,1);
+ if isempty(K_u_xx)
+ K_u_xx = commutation(u_nbr,x_nbr^2,1);
+ K_u_ux = commutation(u_nbr,u_nbr*x_nbr,1);
+ K_xx_x = commutation(x_nbr^2,x_nbr);
+ end
hx_hss2 = kron(hx,1/2*hss);
hu_hss2 = kron(hu,1/2*hss);
hx_hxx2 = kron(hx,1/2*hxx);
@@ -658,9 +665,11 @@ if order > 1
end
% Compute unique sixth-order product moments of u, i.e. unique(E[kron(kron(kron(kron(kron(u,u),u),u),u),u)],'stable')
- u_nbr6 = u_nbr*(u_nbr+1)/2*(u_nbr+2)/3*(u_nbr+3)/4*(u_nbr+4)/5*(u_nbr+5)/6;
- Q6Pu = Q6_plication(u_nbr);
- COMBOS6 = flipud(allVL1(u_nbr, 6)); %all possible (unique) combinations of powers that sum up to six
+ u_nbr6 = u_nbr*(u_nbr+1)/2*(u_nbr+2)/3*(u_nbr+3)/4*(u_nbr+4)/5*(u_nbr+5)/6;
+ if isempty(Q6Pu)
+ Q6Pu = Q6_plication(u_nbr);
+ COMBOS6 = flipud(allVL1(u_nbr, 6)); %all possible (unique) combinations of powers that sum up to six
+ end
E_u_u_u_u_u_u = zeros(u_nbr6,1); %only unique entries
if compute_derivs && (stderrparam_nbr+corrparam_nbr>0)
dE_u_u_u_u_u_u = zeros(u_nbr6,stderrparam_nbr+corrparam_nbr);
@@ -798,7 +807,7 @@ if order > 1
E_inovzlag1 = zeros(inov_nbr,z_nbr); % Attention: E[inov*z(-1)'] is not equal to zero for a third-order approximation due to kron(kron(xf(-1),u),u)
E_inovzlag1(id_inov6_xf_u_u , id_z1_xf ) = kron(E_xfxf,E_uu(:));
E_inovzlag1(id_inov6_xf_u_u , id_z4_xrd ) = kron(E_xrdxf',E_uu(:));
- E_inovzlag1(id_inov6_xf_u_u , id_z5_xf_xs ) = kron(reshape(commutation(x_nbr^2,x_nbr)*vec(E_xsxf_xf),x_nbr,x_nbr^2),vec(E_uu)) ;
+ E_inovzlag1(id_inov6_xf_u_u , id_z5_xf_xs ) = kron(reshape(K_xx_x*vec(E_xsxf_xf),x_nbr,x_nbr^2),vec(E_uu)) ;
E_inovzlag1(id_inov6_xf_u_u , id_z6_xf_xf_xf ) = kron(reshape(E_xf_xfxf_xf,x_nbr,x_nbr^3),E_uu(:));
Binovzlag1A= B*E_inovzlag1*transpose(A);
@@ -980,7 +989,7 @@ if order > 1
dE_inovzlag1(id_inov6_xf_u_u , id_z1_xf , jp3) = kron(dE_xfxf_jp3,E_uu(:)) + kron(E_xfxf,dE_uu_jp3(:));
dE_inovzlag1(id_inov6_xf_u_u , id_z4_xrd , jp3) = kron(dE_xrdxf_jp3',E_uu(:)) + kron(E_xrdxf',dE_uu_jp3(:));
- dE_inovzlag1(id_inov6_xf_u_u , id_z5_xf_xs , jp3) = kron(reshape(commutation(x_nbr^2,x_nbr)*vec(dE_xsxf_xf_jp3),x_nbr,x_nbr^2),vec(E_uu)) + kron(reshape(commutation(x_nbr^2,x_nbr)*vec(E_xsxf_xf),x_nbr,x_nbr^2),vec(dE_uu_jp3)) ;
+ dE_inovzlag1(id_inov6_xf_u_u , id_z5_xf_xs , jp3) = kron(reshape(K_xx_x*vec(dE_xsxf_xf_jp3),x_nbr,x_nbr^2),vec(E_uu)) + kron(reshape(K_xx_x*vec(E_xsxf_xf),x_nbr,x_nbr^2),vec(dE_uu_jp3)) ;
dE_inovzlag1(id_inov6_xf_u_u , id_z6_xf_xf_xf , jp3) = kron(reshape(dE_xf_xfxf_xf_jp3,x_nbr,x_nbr^3),E_uu(:)) + kron(reshape(E_xf_xfxf_xf,x_nbr,x_nbr^3),dE_uu_jp3(:));
dBinovzlag1A_jp3 = dB(:,:,jp3)*E_inovzlag1*transpose(A) + B*dE_inovzlag1(:,:,jp3)*transpose(A) + B*E_inovzlag1*transpose(dA(:,:,jp3));
@@ -1029,8 +1038,8 @@ else
+ C(stationary_vars,:)*transpose(E_inovzlag1)*D(stationary_vars,:)'...
+ D(stationary_vars,:)*Varinov*D(stationary_vars,:)';
end
-indzeros = find(abs(Var_y) < 1e-12); %find values that are numerical zero
-Var_y(indzeros) = 0;
+
+Var_y(abs(Var_y) < 1e-12) = 0; %find values that are numerical zero
if useautocorr
sdy = sqrt(diag(Var_y)); %theoretical standard deviation
sdy = sdy(stationary_vars);
@@ -1056,8 +1065,7 @@ if compute_derivs
+ dC(stationary_vars,:,jpV)*transpose(E_inovzlag1)*D(stationary_vars,:)' + C(stationary_vars,:)*transpose(dE_inovzlag1(:,:,jpV))*D(stationary_vars,:)' + C(stationary_vars,:)*transpose(E_inovzlag1)*dD(stationary_vars,:,jpV)'...
+ dD(stationary_vars,:,jpV)*Varinov*D(stationary_vars,:)' + D(stationary_vars,:)*dVarinov(:,:,jpV)*D(stationary_vars,:)' + D(stationary_vars,:)*Varinov*dD(stationary_vars,:,jpV)';
end
- indzeros = find(abs(dVar_y_tmp) < 1e-12); %find values that are numerical zero
- dVar_y_tmp(indzeros) = 0;
+ dVar_y_tmp(abs(dVar_y_tmp) < 1e-12) = 0; %find values that are numerical zero
dVar_y(stationary_vars,stationary_vars,jpV) = dVar_y_tmp;
if useautocorr
dsy = 1/2./sdy.*diag(dVar_y(:,:,jpV));
@@ -1089,7 +1097,7 @@ for i = 1:nlags
E_inovzlagi = zeros(inov_nbr,z_nbr);
E_inovzlagi(id_inov6_xf_u_u , id_z1_xf ) = kron(hxi*E_xfxf,E_uu(:));
E_inovzlagi(id_inov6_xf_u_u , id_z4_xrd ) = kron(hxi*E_xrdxf',E_uu(:));
- E_inovzlagi(id_inov6_xf_u_u , id_z5_xf_xs ) = kron(hxi*reshape(commutation(x_nbr^2,x_nbr)*vec(E_xsxf_xf),x_nbr,x_nbr^2),vec(E_uu));
+ E_inovzlagi(id_inov6_xf_u_u , id_z5_xf_xs ) = kron(hxi*reshape(K_xx_x*vec(E_xsxf_xf),x_nbr,x_nbr^2),vec(E_uu));
E_inovzlagi(id_inov6_xf_u_u , id_z6_xf_xf_xf ) = kron(hxi*reshape(E_xf_xfxf_xf,x_nbr,x_nbr^3),E_uu(:));
Var_yi(stationary_vars,stationary_vars,i) = C(stationary_vars,:)*Var_zi*C(stationary_vars,:)' + C(stationary_vars,:)*Ai*tmp + D(stationary_vars,:)*E_inovzlagi*C(stationary_vars,:)';
end
@@ -1125,12 +1133,12 @@ if compute_derivs
E_inovzlagi = zeros(inov_nbr,z_nbr);
E_inovzlagi(id_inov6_xf_u_u , id_z1_xf ) = kron(hxi*E_xfxf,E_uu(:));
E_inovzlagi(id_inov6_xf_u_u , id_z4_xrd ) = kron(hxi*E_xrdxf',E_uu(:));
- E_inovzlagi(id_inov6_xf_u_u , id_z5_xf_xs ) = kron(hxi*reshape(commutation(x_nbr^2,x_nbr)*vec(E_xsxf_xf),x_nbr,x_nbr^2),vec(E_uu));
+ E_inovzlagi(id_inov6_xf_u_u , id_z5_xf_xs ) = kron(hxi*reshape(K_xx_x*vec(E_xsxf_xf),x_nbr,x_nbr^2),vec(E_uu));
E_inovzlagi(id_inov6_xf_u_u , id_z6_xf_xf_xf ) = kron(hxi*reshape(E_xf_xfxf_xf,x_nbr,x_nbr^3),E_uu(:));
dE_inovzlagi_jpVi = zeros(inov_nbr,z_nbr);
dE_inovzlagi_jpVi(id_inov6_xf_u_u , id_z1_xf ) = kron(dhxi_jpVi*E_xfxf,E_uu(:)) + kron(hxi*dE_xfxf(:,:,jpVi),E_uu(:)) + kron(hxi*E_xfxf,vec(dE_uu(:,:,jpVi)));
dE_inovzlagi_jpVi(id_inov6_xf_u_u , id_z4_xrd ) = kron(dhxi_jpVi*E_xrdxf',E_uu(:)) + kron(hxi*dE_xrdxf(:,:,jpVi)',E_uu(:)) + kron(hxi*E_xrdxf',vec(dE_uu(:,:,jpVi)));
- dE_inovzlagi_jpVi(id_inov6_xf_u_u , id_z5_xf_xs ) = kron(dhxi_jpVi*reshape(commutation(x_nbr^2,x_nbr)*vec(E_xsxf_xf),x_nbr,x_nbr^2),vec(E_uu)) + kron(hxi*reshape(commutation(x_nbr^2,x_nbr)*vec(dE_xsxf_xf(:,:,jpVi)),x_nbr,x_nbr^2),vec(E_uu)) + kron(hxi*reshape(commutation(x_nbr^2,x_nbr)*vec(E_xsxf_xf),x_nbr,x_nbr^2),vec(dE_uu(:,:,jpVi)));
+ dE_inovzlagi_jpVi(id_inov6_xf_u_u , id_z5_xf_xs ) = kron(dhxi_jpVi*reshape(K_xx_x*vec(E_xsxf_xf),x_nbr,x_nbr^2),vec(E_uu)) + kron(hxi*reshape(K_xx_x*vec(dE_xsxf_xf(:,:,jpVi)),x_nbr,x_nbr^2),vec(E_uu)) + kron(hxi*reshape(K_xx_x*vec(E_xsxf_xf),x_nbr,x_nbr^2),vec(dE_uu(:,:,jpVi)));
dE_inovzlagi_jpVi(id_inov6_xf_u_u , id_z6_xf_xf_xf ) = kron(dhxi_jpVi*reshape(E_xf_xfxf_xf,x_nbr,x_nbr^3),E_uu(:)) + kron(hxi*reshape(dE_xf_xfxf_xf(:,:,jpVi),x_nbr,x_nbr^3),E_uu(:)) + kron(hxi*reshape(E_xf_xfxf_xf,x_nbr,x_nbr^3),vec(dE_uu(:,:,jpVi)));
dVar_yi(stationary_vars,stationary_vars,i,jpVi) = dC(stationary_vars,:,jpVi)*Var_zi*C(stationary_vars,:)' + C(stationary_vars,:)*dVar_zi_jpVi*C(stationary_vars,:)' + C(stationary_vars,:)*Var_zi*dC(stationary_vars,:,jpVi)'...
+ dC(stationary_vars,:,jpVi)*Ai*tmp + C(stationary_vars,:)*dAi_jpVi*tmp + C(stationary_vars,:)*Ai*dtmp_jpVi...
@@ -1179,7 +1187,7 @@ if compute_derivs
end
end
end
-non_stationary_vars = setdiff(1:y_nbr,stationary_vars);
+non_stationary_vars = ~ismember((1:y_nbr)',stationary_vars);
E_y(non_stationary_vars) = NaN;
if compute_derivs
dE_y(non_stationary_vars,:) = NaN;
@@ -1195,7 +1203,7 @@ pruned_state_space.D = D;
pruned_state_space.c = c;
pruned_state_space.d = d;
pruned_state_space.Varinov = Varinov;
-pruned_state_space.Var_z = Var_z; %remove in future [@wmutschl]
+% pruned_state_space.Var_z = Var_z; %
pruned_state_space.Var_y = Var_y;
pruned_state_space.Var_yi = Var_yi;
if useautocorr
diff --git a/tests/pruning/AnSchorfheide_pruned_state_space.mod b/tests/pruning/AnSchorfheide_pruned_state_space.mod
index 9afd133b825973e6b69847a90519c44475c81c10..4858f73182499d0df44536d94b81dc085fb1fdb9 100644
--- a/tests/pruning/AnSchorfheide_pruned_state_space.mod
+++ b/tests/pruning/AnSchorfheide_pruned_state_space.mod
@@ -143,51 +143,51 @@ for iorder = 1:3
error('Something wrong with pruned_state_space.m compared to Andreasen et al 2018 Toolbox v2 at order %d.',iorder);
end
end
-skipline();
-fprintf('Note that at third order, there is an error in the computation of Var_z in Andreasen et al (2018)''s toolbox, @wmutschl is in contact to clarify this.\n');
-fprintf('EXAMPLE:\n')
-fprintf(' Consider Var[kron(kron(xf,xf),xf)] = E[kron(kron(kron(kron(kron(xf,xf),xf),xf),xf),xf)] - E[kron(kron(xf,xf),xf)]*E[kron(kron(xf,xf),xf)].''\n');
-fprintf(' Now note that xf=hx*xf(-1)+hu*u is Gaussian, that is E[kron(kron(xf,xf),xf)]=0, and Var[kron(kron(xf,xf),xf)] are the sixth-order product moments\n');
-fprintf(' which can be computed using the prodmom.m function by providing E[xf*xf''] as covariance matrix.\n');
-fprintf(' In order to replicate this you have to change UnconditionalMoments_3rd_Lyap.m to also output Var_z.\n')
-
-dynare_nx = M_.nspred;
-dynare_E_xf2 = pruned_state_space.order_3.Var_z(1:dynare_nx,1:dynare_nx);
-dynare_E_xf6 = pruned_state_space.order_3.Var_z((end-dynare_nx^3+1):end,(end-dynare_nx^3+1):end);
-dynare_E_xf6 = dynare_E_xf6(:);
-
-Andreasen_nx = M_.nspred+M_.exo_nbr;
-Andreasen_E_xf2 = outAndreasenetal.order_3.Var_z(1:Andreasen_nx,1:Andreasen_nx);
-Andreasen_E_xf6 = outAndreasenetal.order_3.Var_z((end-Andreasen_nx^3+1):end,(end-Andreasen_nx^3+1):end);
-Andreasen_E_xf6 = Andreasen_E_xf6(:);
-
-fprintf('Second-order product moments of xf and u are the same:\n')
-norm_E_xf2 = norm(dynare_E_xf2-Andreasen_E_xf2(1:M_.nspred,1:M_.nspred),Inf)
-norm_E_uu = norm(M_.Sigma_e-Andreasen_E_xf2(M_.nspred+(1:M_.exo_nbr),M_.nspred+(1:M_.exo_nbr)),Inf)
-
-% Compute unique sixth-order product moments of xf, i.e. unique(E[kron(kron(kron(kron(kron(xf,xf),xf),xf),xf),xf)],'stable')
-dynare_nx6 = dynare_nx*(dynare_nx+1)/2*(dynare_nx+2)/3*(dynare_nx+3)/4*(dynare_nx+4)/5*(dynare_nx+5)/6;
-dynare_Q6Px = Q6_plication(dynare_nx);
-dynare_COMBOS6 = flipud(allVL1(dynare_nx, 6)); %all possible (unique) combinations of powers that sum up to six
-dynare_true_E_xf6 = zeros(dynare_nx6,1); %only unique entries
-for j6 = 1:size(dynare_COMBOS6,1)
- dynare_true_E_xf6(j6) = prodmom(dynare_E_xf2, 1:dynare_nx, dynare_COMBOS6(j6,:));
-end
-dynare_true_E_xf6 = dynare_Q6Px*dynare_true_E_xf6; %add duplicate entries
-norm_dynare_E_xf6 = norm(dynare_true_E_xf6 - dynare_E_xf6, Inf);
-
-Andreasen_nx6 = Andreasen_nx*(Andreasen_nx+1)/2*(Andreasen_nx+2)/3*(Andreasen_nx+3)/4*(Andreasen_nx+4)/5*(Andreasen_nx+5)/6;
-Andreasen_Q6Px = Q6_plication(Andreasen_nx);
-Andreasen_COMBOS6 = flipud(allVL1(Andreasen_nx, 6)); %all possible (unique) combinations of powers that sum up to six
-Andreasen_true_E_xf6 = zeros(Andreasen_nx6,1); %only unique entries
-for j6 = 1:size(Andreasen_COMBOS6,1)
- Andreasen_true_E_xf6(j6) = prodmom(Andreasen_E_xf2, 1:Andreasen_nx, Andreasen_COMBOS6(j6,:));
-end
-Andreasen_true_E_xf6 = Andreasen_Q6Px*Andreasen_true_E_xf6; %add duplicate entries
-norm_Andreasen_E_xf6 = norm(Andreasen_true_E_xf6 - Andreasen_E_xf6, Inf);
-
-fprintf('Sixth-order product moments of xf and u are not the same!\n');
-fprintf(' Dynare maximum absolute deviations of sixth-order product moments of xf: %d\n',norm_dynare_E_xf6)
-fprintf(' Andreasen et al maximum absolute deviations of sixth-order product moments of xf: %d\n',norm_Andreasen_E_xf6)
-skipline();
-fprintf('Note that the standard deviations are set quite high to make the numerical differences more apparent.\n');
+% skipline();
+% fprintf('Note that at third order, there is an error in the computation of Var_z in Andreasen et al (2018)''s toolbox, @wmutschl is in contact to clarify this.\n');
+% fprintf('EXAMPLE:\n')
+% fprintf(' Consider Var[kron(kron(xf,xf),xf)] = E[kron(kron(kron(kron(kron(xf,xf),xf),xf),xf),xf)] - E[kron(kron(xf,xf),xf)]*E[kron(kron(xf,xf),xf)].''\n');
+% fprintf(' Now note that xf=hx*xf(-1)+hu*u is Gaussian, that is E[kron(kron(xf,xf),xf)]=0, and Var[kron(kron(xf,xf),xf)] are the sixth-order product moments\n');
+% fprintf(' which can be computed using the prodmom.m function by providing E[xf*xf''] as covariance matrix.\n');
+% fprintf(' In order to replicate this you have to change UnconditionalMoments_3rd_Lyap.m to also output Var_z.\n')
+%
+% dynare_nx = M_.nspred;
+% dynare_E_xf2 = pruned_state_space.order_3.Var_z(1:dynare_nx,1:dynare_nx);
+% dynare_E_xf6 = pruned_state_space.order_3.Var_z((end-dynare_nx^3+1):end,(end-dynare_nx^3+1):end);
+% dynare_E_xf6 = dynare_E_xf6(:);
+%
+% Andreasen_nx = M_.nspred+M_.exo_nbr;
+% Andreasen_E_xf2 = outAndreasenetal.order_3.Var_z(1:Andreasen_nx,1:Andreasen_nx);
+% Andreasen_E_xf6 = outAndreasenetal.order_3.Var_z((end-Andreasen_nx^3+1):end,(end-Andreasen_nx^3+1):end);
+% Andreasen_E_xf6 = Andreasen_E_xf6(:);
+%
+% fprintf('Second-order product moments of xf and u are the same:\n')
+% norm_E_xf2 = norm(dynare_E_xf2-Andreasen_E_xf2(1:M_.nspred,1:M_.nspred),Inf)
+% norm_E_uu = norm(M_.Sigma_e-Andreasen_E_xf2(M_.nspred+(1:M_.exo_nbr),M_.nspred+(1:M_.exo_nbr)),Inf)
+%
+% % Compute unique sixth-order product moments of xf, i.e. unique(E[kron(kron(kron(kron(kron(xf,xf),xf),xf),xf),xf)],'stable')
+% dynare_nx6 = dynare_nx*(dynare_nx+1)/2*(dynare_nx+2)/3*(dynare_nx+3)/4*(dynare_nx+4)/5*(dynare_nx+5)/6;
+% dynare_Q6Px = Q6_plication(dynare_nx);
+% dynare_COMBOS6 = flipud(allVL1(dynare_nx, 6)); %all possible (unique) combinations of powers that sum up to six
+% dynare_true_E_xf6 = zeros(dynare_nx6,1); %only unique entries
+% for j6 = 1:size(dynare_COMBOS6,1)
+% dynare_true_E_xf6(j6) = prodmom(dynare_E_xf2, 1:dynare_nx, dynare_COMBOS6(j6,:));
+% end
+% dynare_true_E_xf6 = dynare_Q6Px*dynare_true_E_xf6; %add duplicate entries
+% norm_dynare_E_xf6 = norm(dynare_true_E_xf6 - dynare_E_xf6, Inf);
+%
+% Andreasen_nx6 = Andreasen_nx*(Andreasen_nx+1)/2*(Andreasen_nx+2)/3*(Andreasen_nx+3)/4*(Andreasen_nx+4)/5*(Andreasen_nx+5)/6;
+% Andreasen_Q6Px = Q6_plication(Andreasen_nx);
+% Andreasen_COMBOS6 = flipud(allVL1(Andreasen_nx, 6)); %all possible (unique) combinations of powers that sum up to six
+% Andreasen_true_E_xf6 = zeros(Andreasen_nx6,1); %only unique entries
+% for j6 = 1:size(Andreasen_COMBOS6,1)
+% Andreasen_true_E_xf6(j6) = prodmom(Andreasen_E_xf2, 1:Andreasen_nx, Andreasen_COMBOS6(j6,:));
+% end
+% Andreasen_true_E_xf6 = Andreasen_Q6Px*Andreasen_true_E_xf6; %add duplicate entries
+% norm_Andreasen_E_xf6 = norm(Andreasen_true_E_xf6 - Andreasen_E_xf6, Inf);
+%
+% fprintf('Sixth-order product moments of xf and u are not the same!\n');
+% fprintf(' Dynare maximum absolute deviations of sixth-order product moments of xf: %d\n',norm_dynare_E_xf6)
+% fprintf(' Andreasen et al maximum absolute deviations of sixth-order product moments of xf: %d\n',norm_Andreasen_E_xf6)
+% skipline();
+% fprintf('Note that the standard deviations are set quite high to make the numerical differences more apparent.\n');