% This mod file compares the functionality of Dynare's pruned_state_space.m with the % external Dynare pruning toolbox of Andreasen, Fernández-Villaverde and Rubio-Ramírez (2018): % "The Pruned State-Space System for Non-Linear DSGE Models: Theory and Empirical Applications", % Review of Economic Studies, Volume 85, Issue 1, Pages 1–49. % The model under study is taken from An and Schorfheide (2007): "Bayesian Analysis of DSGE Models", % Econometric Reviews, Volume 26, Issue 2-4, Pages 113-172. % Note that we use version 2 of the toolbox, i.e. the one which is called % "Third-order GMM estimate package for DSGE models (version 2)" and can be % downloaded from https://sites.google.com/site/mandreasendk/home-1 % % Created by @wmutschl (Willi Mutschler, willi@mutschler.eu) % % ========================================================================= % Copyright (C) 2020 Dynare Team % % This file is part of Dynare. % % Dynare is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % % Dynare is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with Dynare. If not, see . % ========================================================================= % set this to 1 if you want to recompute using the Andreasen et al toolbox % otherwise the results are loaded from Andreasen_et_al_2018_Dynare44Pruning_v2.mat @#define Andreasen_et_al_toolbox = 0 var YGR INFL INT c p R g y z; %if ordering of var is changed comparison code below needs to be adapted varexo e_r e_g e_z e_junk; parameters tau nu kap cyst psi1 psi2 rhor rhog rhoz rrst pist gamst; tau = 2; nu = 0.1; kap = 0.33; cyst = 0.85; psi1 = 1.5; psi2 = 0.125; rhor = 0.75; rhog = 0.95; rhoz = 0.9; rrst = 1; pist = 3.2; gamst = 0.55; sig_r = .2; sig_g = .6; sig_z = .3; model; #pist2 = exp(pist/400); #rrst2 = exp(rrst/400); #bet = 1/rrst2; #phi = tau*(1-nu)/nu/kap/pist2^2; #gst = 1/cyst; #cst = (1-nu)^(1/tau); #yst = cst*gst; #dy = y-y(-1); 1 = exp(-tau*c(+1)+tau*c+R-z(+1)-p(+1)); (1-nu)/nu/phi/(pist2^2)*(exp(tau*c)-1) = (exp(p)-1)*((1-1/2/nu)*exp(p)+1/2/nu) - bet*(exp(p(+1))-1)*exp(-tau*c(+1)+tau*c+y(+1)-y+p(+1)); exp(c-y) = exp(-g) - phi*pist2^2*gst/2*(exp(p)-1)^2; R = rhor*R(-1) + (1-rhor)*psi1*p + (1-rhor)*psi2*(y-g) + e_r; g = rhog*g(-1) + e_g; z = rhoz*z(-1) + e_z; YGR = gamst+100*(dy+z); INFL = pist+400*p + e_junk; INT = pist+rrst+4*gamst+400*R; end; shocks; var e_r = sig_r^2; var e_g = sig_g^2; var e_z = sig_z^2; var e_junk = 0; end; steady_state_model; y = 0; R = 0; g = 0; z = 0; c = 0; p = 0; YGR = gamst; INFL = pist; INT = pist + rrst + 4*gamst; end; steady; check; model_diagnostics; @#for orderApp in [1, 2, 3] stoch_simul(order=@{orderApp},pruning,irf=0,periods=0); pruned_state_space.order_@{orderApp} = pruned_state_space_system(M_, options_, oo_.dr, [], options_.ar, 1, 0); @#if Andreasen_et_al_toolbox addpath('Dynare44Pruning_v2/simAndMoments3order'); %provide path to toolbox optPruning.orderApp = @{orderApp}; outAndreasenetal.order_@{orderApp} = RunDynarePruning(optPruning,oo_,M_,[oo_.dr.ghx oo_.dr.ghu]); rmpath('Dynare44Pruning_v2/simAndMoments3order'); close all; @#endif @#endfor @#if Andreasen_et_al_toolbox delete Andreasen_et_al_2018_Dynare44Pruning_v2.mat; pause(3); save('Andreasen_et_al_2018_Dynare44Pruning_v2.mat', 'outAndreasenetal') pause(3); @#endif load('Andreasen_et_al_2018_Dynare44Pruning_v2.mat') % Make comparisons only at orders 1 and 2 for iorder = 1:3 fprintf('ORDER %d:\n',iorder); pruned = pruned_state_space.(sprintf('order_%d',iorder)); outAndreasen = outAndreasenetal.(sprintf('order_%d',iorder)); %make sure variable ordering is correct if ~isequal(M_.endo_names,[outAndreasen.label_y; outAndreasen.label_v(1:M_.nspred)]) error('variable ordering is not the same, change declaration order'); end norm_E_yx = norm(pruned.E_y(oo_.dr.inv_order_var) - [outAndreasen.Mean_y; outAndreasen.Mean_v(1:M_.nspred)] , Inf); fprintf('max(sum(abs(E[y;x]''))): %d\n',norm_E_yx); norm_Var_y = norm(pruned.Var_y(oo_.dr.inv_order_var(1:(M_.endo_nbr-M_.nspred)),oo_.dr.inv_order_var(1:(M_.endo_nbr-M_.nspred))) - outAndreasen.Var_y , Inf); fprintf('max(sum(abs(Var[y]''))):: %d\n',norm_Var_y); norm_Var_x = norm(pruned.Var_y(M_.nstatic+(1:M_.nspred),M_.nstatic+(1:M_.nspred)) - outAndreasen.Var_v(1:M_.nspred,1:M_.nspred) , Inf); fprintf('max(sum(abs(Var[x]''))): %d\n',norm_Var_x); norm_Corr_yi1 = norm(pruned.Corr_yi(oo_.dr.inv_order_var(1:(M_.endo_nbr-M_.nspred)),oo_.dr.inv_order_var(1:(M_.endo_nbr-M_.nspred)),1) - outAndreasen.Corr_y(:,:,1) , Inf); fprintf('max(sum(abs(Corr[y,y(-1)]''))): %d\n',norm_Corr_yi1); norm_Corr_yi2 = norm(pruned.Corr_yi(oo_.dr.inv_order_var(1:(M_.endo_nbr-M_.nspred)),oo_.dr.inv_order_var(1:(M_.endo_nbr-M_.nspred)),2) - outAndreasen.Corr_y(:,:,2) , Inf); fprintf('max(sum(abs(Corr[y,y(-2)]''))): %d\n',norm_Corr_yi2); norm_Corr_xi1 = norm(pruned.Corr_yi(M_.nstatic+(1:M_.nspred),M_.nstatic+(1:M_.nspred),1) - outAndreasen.Corr_v(1:M_.nspred,1:M_.nspred,1) , Inf); fprintf('max(sum(abs(Corr[x,x(-1)]''))): %d\n',norm_Corr_xi1); norm_Corr_xi2 = norm(pruned.Corr_yi(M_.nstatic+(1:M_.nspred),M_.nstatic+(1:M_.nspred),2) - outAndreasen.Corr_v(1:M_.nspred,1:M_.nspred,2) , Inf); fprintf('max(sum(abs(Corr[x,x(-2)]''))): %d\n',norm_Corr_xi2); if iorder < 3 && any([norm_E_yx norm_Var_y norm_Var_x norm_Corr_yi1 norm_Corr_yi2 norm_Corr_xi1 norm_Corr_xi2] > 1e-5) error('Something wrong with pruned_state_space.m compared to Andreasen et al 2018 Toolbox v2 at order %d.',iorder); end if iorder==3 pruned_without_shock = load(['AnSchorfheide_pruned_state_space' filesep 'Output' filesep 'AnSchorfheide_pruned_state_space_results.mat']); pruned_without_shock = pruned_without_shock.oo_.pruned; norm_E_yx = norm(pruned.E_y - pruned_without_shock.E_y , Inf); fprintf('max(sum(abs(E[y;x]''))): %d\n',norm_E_yx); norm_Var_y = norm(pruned.Var_y - pruned_without_shock.Var_y, Inf); fprintf('max(sum(abs(Var[x]''))): %d\n',norm_Var_x); norm_Corr_yi1 = norm(pruned.Corr_yi(:,:,1) - pruned_without_shock.Corr_yi(:,:,1), Inf); fprintf('max(sum(abs(Corr[y,y(-1)]''))): %d\n',norm_Corr_yi1); norm_Corr_yi2 = norm(pruned.Corr_yi(:,:,2) - pruned_without_shock.Corr_yi(:,:,2), Inf); fprintf('max(sum(abs(Corr[y,y(-2)]''))): %d\n',norm_Corr_yi2); 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');