diff --git a/matlab/Q6_plication.m b/matlab/Q6_plication.m index 0692364f1..153ee8ccd 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 8dc0d90fd..82ee94b5a 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 4395f109c..d82bfd677 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 9afd133b8..4858f7318 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');