% ========================================================================= % Copyright © 2019-2020 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 . % ========================================================================= /* Check the policy functions obtained by perturbation at a high approximation order, using the Burnside (1998, JEDC) model (for which the analytical form of the policy function is known). As shown by Burnside, the policy function for yₜ is: yₜ = βⁱ exp[aᵢ+bᵢ(xₜ−xₛₛ)] where: θ² ⎛ 2ρ 1−ρ²ⁱ⎞ — aᵢ = iθxₛₛ + σ² ─────── ⎢i − ────(1−ρⁱ) + ρ² ─────⎥ 2(1−ρ)² ⎝ 1−ρ 1−ρ² ⎠ θρ — bᵢ = ───(1−ρⁱ) 1−ρ — xₛₛ is the steady state of x — σ is the standard deviation of e. With some algebra, it can be shown that the derivative of yₜ at the deterministic steady state is equal to: ∂ᵐ⁺ⁿ⁺²ᵖ yₜ ∞ (2p)! ──────────────── = ∑ βⁱ bᵢᵐ⁺ⁿ ρᵐ ───── cᵢᵖ exp(iθxₛₛ) ∂ᵐxₜ₋₁ ∂ⁿeₜ ∂²ᵖs ⁱ⁼¹ p! where: — s is the stochastic scale factor θ² ⎛ 2ρ 1−ρ²ⁱ⎞ — cᵢ = ─────── ⎢i − ────(1−ρⁱ) + ρ² ─────⎥ 2(1−ρ)² ⎝ 1−ρ 1−ρ² ⎠ Note that derivatives with respect to an odd order for s (i.e. ∂²ᵖ⁺¹s) are always equal to zero. The policy function as returned in the oo_.dr.g_* matrices has the following properties: — its elements are pre-multiplied by the Taylor coefficients; — derivatives w.r.t. the stochastic scale factor have already been summed up; — symmetric elements are folded (and they are not pre-multiplied by the number of repetitions). As a consequence, the element gₘₙ corresponding to the m-th derivative w.r.t. to xₜ₋₁ and the n-th derivative w.r.t. to eₜ is given by: 1 ∞ cᵢᵖ gₘₙ = ────── ∑ ∑ βⁱ bᵢᵐ⁺ⁿ ρᵐ ──── exp(iθxₛₛ) (m+n)! 0≤2p≤k-m-n ⁱ⁼¹ p! where k is the order of approximation. */ @#define ORDER = 3 var y x; varobs y; varexo e; parameters beta theta rho xbar; xbar = 0.0179; rho = -0.139; theta = -1.5; theta = -10; beta = 0.95; model; y = beta*exp(theta*x(+1))*(1+y(+1)); x = (1-rho)*xbar + rho*x(-1)+e; end; shocks; var e; stderr 0.0348; end; steady_state_model; x = xbar; y = beta*exp(theta*xbar)/(1-beta*exp(theta*xbar)); end; estimated_params; stderr e, normal_pdf, 0.0348,0.01; beta, normal_pdf, 0.95, 0.01; theta, normal_pdf, -10, 0.01; rho, normal_pdf, -0.139, 0.01; xbar, normal_pdf, 0.0179, 0.01; end; steady;check;model_diagnostics; stoch_simul(order=@{ORDER},k_order_solver,irf=0,drop=0,periods=0,nograph); identification(order=@{ORDER},nograph,no_identification_strength); %make sure everything is computed at prior mean [xparam_prior, estim_params_]= set_prior(estim_params_,M_,options_); M_ = set_all_parameters(xparam_prior,estim_params_,M_); [oo_.dr,info,M_.params] = resol(0,M_, options_, oo_.dr, oo_.steady_state, oo_.exo_steady_state, oo_.exo_det_steady_state); indpmodel = estim_params_.param_vals(:,1); indpstderr = estim_params_.var_exo(:,1); indpcorr = estim_params_.corrx(:,1:2); totparam_nbr = length(indpmodel) + length(indpstderr) + size(indpcorr,1); %% Verify that the policy function coefficients are correct i = 1:800; SE_e=sqrt(M_.Sigma_e); aux1 = rho*(1-rho.^i)/(1-rho); aux2 = (1-rho.^(2*i))/(1-rho^2); aux3 = 1/((1-rho)^2); aux4 = (i-2*aux1+rho^2*aux2); aux5=aux3*aux4; b = theta*aux1; c = 1/2*theta^2*SE_e^2*aux5; %derivatives wrt to rho only daux1_drho = zeros(1,length(i)); daux2_drho = zeros(1,length(i)); daux3_drho = 2/((1-rho)^3); daux4_drho = zeros(1,length(i)); daux5_drho = zeros(1,length(i)); for ii = 1:length(i) if ii == 1 daux1_drho(ii) = 1; daux2_drho(ii) = 0; else daux1_drho(ii) = rho/(rho^2 - 2*rho + 1) - 1/(rho - 1) + ((ii+1)*rho^ii)/(rho - 1) - rho^(ii+1)/(rho^2 - 2*rho + 1); daux2_drho(ii) = (2*rho)/(rho^4 - 2*rho^2 + 1) + (2*ii*rho^(2*ii-1))/(rho^2 - 1) - (2*rho^(2*ii+1))/(rho^4 - 2*rho^2 + 1); end daux4_drho(ii) = -2*daux1_drho(ii) + 2*rho*aux2(ii) + rho^2*daux2_drho(ii); daux5_drho(ii) = daux3_drho*aux4(ii) + aux3*daux4_drho(ii); end %derivatives of b and c wrt to all parameters db = zeros(size(b,1),size(b,2),M_.exo_nbr+M_.param_nbr); db(:,:,3) = aux1;%wrt theta db(:,:,4) = theta*daux1_drho;%wrt rho dc = zeros(size(c,1),size(c,2),M_.exo_nbr+M_.param_nbr); dc(:,:,1) = theta^2*SE_e*aux3*aux4;%wrt SE_e dc(:,:,3) = theta*SE_e^2*aux3*aux4;%wrt theta dc(:,:,4) = 1/2*theta^2*SE_e^2*daux5_drho; %wrt rho d2flag=0; g_0 = 1/2*oo_.dr.ghs2; if ~isequal(g_0,oo_.dr.g_0); error('something wrong'); end g_1 = [oo_.dr.ghx oo_.dr.ghu] +3/6*[oo_.dr.ghxss oo_.dr.ghuss]; if ~isequal(g_1,oo_.dr.g_1); error('something wrong'); end g_2 = 1/2*[oo_.dr.ghxx oo_.dr.ghxu oo_.dr.ghuu]; if ~isequal(g_2,oo_.dr.g_2); error('something wrong'); end g_3 = 1/6*[oo_.dr.ghxxx oo_.dr.ghxxu oo_.dr.ghxuu oo_.dr.ghuuu]; if ~isequal(g_3,oo_.dr.g_3); error('something wrong'); end tols = [1e-4 1e-4 1e-12 1e-12]; KRONFLAGS = [-1 -2 0 1]; for k = 1:length(KRONFLAGS) fprintf('KRONFLAG=%d\n',KRONFLAGS(k)); options_.analytic_derivation_mode = KRONFLAGS(k); DERIVS = identification.get_perturbation_params_derivs(M_, options_, estim_params_, oo_.dr, oo_.steady_state, oo_.exo_steady_state, oo_.exo_det_steady_state, indpmodel, indpstderr, indpcorr, d2flag); oo_.dr.dg_0 = permute(1/2*DERIVS.dghs2,[1 3 2]); oo_.dr.dg_1 = cat(2,DERIVS.dghx,DERIVS.dghu) + 3/6*cat(2,DERIVS.dghxss,DERIVS.dghuss); oo_.dr.dg_2 = 1/2*cat(2,DERIVS.dghxx,DERIVS.dghxu,DERIVS.dghuu); oo_.dr.dg_3 = 1/6*[DERIVS.dghxxx DERIVS.dghxxu DERIVS.dghxuu DERIVS.dghuuu]; for ord = 0:@{ORDER} g = oo_.dr.(['g_' num2str(ord)])(2,:); % Retrieve computed policy function for variable y dg = oo_.dr.(['dg_' num2str(ord)])(2,:,:); for m = 0:ord % m is the derivation order with respect to x(-1) v = 0; dv = zeros(1,M_.exo_nbr + M_.param_nbr); for p = 0:floor((@{ORDER}-ord)/2) % 2p is the derivation order with respect to s if ord+2*p > 0 % Skip the deterministic steady state constant v = v + sum(beta.^i.*exp(theta*xbar*i).*b.^ord.*rho^m.*c.^p)/factorial(ord)/factorial(p); %derivatives dv(:,1) = dv(:,1) + sum( beta.^i.*exp(theta*xbar*i).*ord.*b.^(ord-1).*db(:,:,1).*rho^m.*c.^p... +beta.^i.*exp(theta*xbar*i).*b.^ord.*rho^m.*p.*c.^(p-1).*dc(:,:,1)... )/factorial(ord)/factorial(p);%wrt SE_E dv(:,2) = dv(:,2) + sum( i.*beta.^(i-1).*exp(theta*xbar*i).*b.^ord.*rho^m.*c.^p... +beta.^i.*exp(theta*xbar*i).*ord.*b.^(ord-1).*db(:,:,2).*rho^m.*c.^p... +beta.^i.*exp(theta*xbar*i).*b.^ord.*rho^m.*p.*c.^(p-1).*dc(:,:,2)... )/factorial(ord)/factorial(p);%wrt beta dv(:,3) = dv(:,3) + sum( beta.^i.*exp(theta*xbar*i).*xbar.*i.*b.^ord.*rho^m.*c.^p... +beta.^i.*exp(theta*xbar*i).*ord.*b.^(ord-1).*db(:,:,3).*rho^m.*c.^p... +beta.^i.*exp(theta*xbar*i).*b.^ord.*rho^m.*p.*c.^(p-1).*dc(:,:,3)... )/factorial(ord)/factorial(p);%wrt theta dv(:,4) = dv(:,4) + sum( beta.^i.*exp(theta*xbar*i).*b.^ord.*m.*rho^(m-1).*c.^p... +beta.^i.*exp(theta*xbar*i).*ord.*b.^(ord-1).*db(:,:,4).*rho^m.*c.^p... +beta.^i.*exp(theta*xbar*i).*b.^ord.*rho^m.*p.*c.^(p-1).*dc(:,:,4)... )/factorial(ord)/factorial(p);%wrt rho dv(:,5) = dv(:,5) + sum( beta.^i.*exp(theta*xbar*i).*theta.*i.*b.^ord.*rho^m.*c.^p... +beta.^i.*exp(theta*xbar*i).*ord.*b.^(ord-1).*db(:,:,5).*rho^m.*c.^p... +beta.^i.*exp(theta*xbar*i).*b.^ord.*rho^m.*p.*c.^(p-1).*dc(:,:,5)... )/factorial(ord)/factorial(p);%wrt xbar end end if abs(v-g(ord+1-m)) > 1e-14 error(['Error in matrix oo_.dr.g_' num2str(ord)]) end chk_dg = squeeze(dg(:,ord+1-m,:))'; if isempty(indpstderr) chk_dv = dv(:,M_.exo_nbr+indpmodel); elseif isempty(indpmodel) chk_dv = dv(:,1:M_.exo_nbr); else chk_dv = dv; end fprintf('Max absolute deviation for dg_%d(2,%d,:): %e\n',ord,ord+1-m,norm( chk_dv - chk_dg, Inf)); if norm( chk_dv - chk_dg, Inf) > tols(k) error(['Error in matrix dg_' num2str(ord)]) chk_dv chk_dg end end end fprintf('\n'); end