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🏇 Better minimal state space handling and unit tests

time-shift
Willi Mutschler 2020-01-16 14:44:31 +01:00
parent 1aa3dda449
commit 5525a7c515
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GPG Key ID: 91E724BF17A73F6D
4 changed files with 773 additions and 115 deletions
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22
matlab/get_identification_jacobians.m
View File

@ -442,14 +442,16 @@ if ~no_identification_minimal
dMINIMAL = [];
else
% Derive and check minimal state vector of first-order
[CheckCO,minnx,minA,minB,minC,minD,dminA,dminB,dminC,dminD] = get_minimal_state_representation(oo.dr.ghx(oo.dr.pruned.indx,:),... %A
oo.dr.ghu(oo.dr.pruned.indx,:),... %B
oo.dr.ghx(oo.dr.pruned.indy,:),... %C
oo.dr.ghu(oo.dr.pruned.indy,:),... %D
oo.dr.derivs.dghx(oo.dr.pruned.indx,:,:),... %dA
oo.dr.derivs.dghu(oo.dr.pruned.indx,:,:),... %dB
oo.dr.derivs.dghx(oo.dr.pruned.indy,:,:),... %dC
oo.dr.derivs.dghu(oo.dr.pruned.indy,:,:)); %dD
SYS.A = oo.dr.ghx(oo.dr.pruned.indx,:);
SYS.dA = oo.dr.derivs.dghx(oo.dr.pruned.indx,:,:);
SYS.B = oo.dr.ghu(oo.dr.pruned.indx,:);
SYS.dB = oo.dr.derivs.dghu(oo.dr.pruned.indx,:,:);
SYS.C = oo.dr.ghx(oo.dr.pruned.indy,:);
SYS.dC = oo.dr.derivs.dghx(oo.dr.pruned.indy,:,:);
SYS.D = oo.dr.ghu(oo.dr.pruned.indy,:);
SYS.dD = oo.dr.derivs.dghu(oo.dr.pruned.indy,:,:);
[CheckCO,minnx,SYS] = get_minimal_state_representation(SYS,1);
if CheckCO == 0
warning_KomunjerNg = 'WARNING: Komunjer and Ng (2011) failed:\n';
warning_KomunjerNg = [warning_KomunjerNg ' Conditions for minimality are not fullfilled:\n'];
@ -457,6 +459,10 @@ if ~no_identification_minimal
fprintf(warning_KomunjerNg); %use sprintf to have line breaks
dMINIMAL = [];
else
minA = SYS.A; dminA = SYS.dA;
minB = SYS.B; dminB = SYS.dB;
minC = SYS.C; dminC = SYS.dC;
minD = SYS.D; dminD = SYS.dD;
%reshape into Magnus-Neudecker Jacobians, i.e. dvec(X)/dp
dminA = reshape(dminA,size(dminA,1)*size(dminA,2),size(dminA,3));
dminB = reshape(dminB,size(dminB,1)*size(dminB,2),size(dminB,3));

334
matlab/get_minimal_state_representation.m
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@ -1,40 +1,55 @@
function [CheckCO,minnx,minA,minB,minC,minD,dminA,dminB,dminC,dminD] = get_minimal_state_representation(A,B,C,D,dA,dB,dC,dD)
% [CheckCO,minnx,minA,minB,minC,minD,dminA,dminB,dminC,dminD] = get_minimal_state_representation(A,B,C,D,dA,dB,dC,dD)
% Derives and checks the minimal state representation of the ABCD
% representation of a state space model
function [CheckCO,minns,minSYS] = get_minimal_state_representation(SYS, derivs_flag)
% Derives and checks the minimal state representation
% Let x = A*x(-1) + B*u and y = C*x(-1) + D*u be a linear state space
% system, then this function computes the following representation
% xmin = minA*xmin(-1) + minB*u and and y=minC*xmin(-1) + minD*u
%
% -------------------------------------------------------------------------
% INPUTS
% A: [endo_nbr by endo_nbr] Transition matrix from Kalman filter
% for all endogenous declared variables, in DR order
% B: [endo_nbr by exo_nbr] Transition matrix from Kalman filter
% mapping shocks today to endogenous variables today, in DR order
% C: [obs_nbr by endo_nbr] Measurement matrix from Kalman filter
% linking control/observable variables to states, in DR order
% D: [obs_nbr by exo_nbr] Measurement matrix from Kalman filter
% mapping shocks today to controls/observables today, in DR order
% dA: [endo_nbr by endo_nbr by totparam_nbr] in DR order
% SYS [structure]
% with the following necessary fields:
% A: [nspred by nspred] in DR order
% Transition matrix for all state variables
% B: [nspred by exo_nbr] in DR order
% Transition matrix mapping shocks today to states today
% C: [varobs_nbr by nspred] in DR order
% Measurement matrix linking control/observable variables to states
% D: [varobs_nbr by exo_nbr] in DR order
% Measurement matrix mapping shocks today to controls/observables today
% and optional fields:
% dA: [nspred by nspred by totparam_nbr] in DR order
% Jacobian (wrt to all parameters) of transition matrix A
% dB: [endo_nbr by exo_nbr by totparam_nbr] in DR order
% dB: [nspred by exo_nbr by totparam_nbr] in DR order
% Jacobian (wrt to all parameters) of transition matrix B
% dC: [obs_nbr by endo_nbr by totparam_nbr] in DR order
% dC: [varobs_nbr by nspred by totparam_nbr] in DR order
% Jacobian (wrt to all parameters) of measurement matrix C
% dD: [obs_nbr by exo_nbr by totparam_nbr] in DR order
% dD: [varobs_nbr by exo_nbr by totparam_nbr] in DR order
% Jacobian (wrt to all parameters) of measurement matrix D
% derivs_flag [scalar]
% (optional) indicator whether to output parameter derivatives
% -------------------------------------------------------------------------
% OUTPUTS
% CheckCO: [scalar] indicator, equals to 1 if minimal state representation is found
% minnx: [scalar] length of minimal state vector
% minA: [minnx by minnx] Transition matrix A for evolution of minimal state vector
% minB: [minnx by exo_nbr] Transition matrix B for evolution of minimal state vector
% minC: [obs_nbr by minnx] Measurement matrix C for evolution of controls, depending on minimal state vector only
% minD: [obs_nbr by minnx] Measurement matrix D for evolution of controls, depending on minimal state vector only
% dminA: [minnx by minnx by totparam_nbr] in DR order
% CheckCO: [scalar]
% equals to 1 if minimal state representation is found
% minns: [scalar]
% length of minimal state vector
% SYS [structure]
% with the following fields:
% minA: [minns by minns] in DR-order
% transition matrix A for evolution of minimal state vector
% minB: [minns by exo_nbr] in DR-order
% transition matrix B for evolution of minimal state vector
% minC: [varobs_nbr by minns] in DR-order
% measurement matrix C for evolution of controls, depending on minimal state vector only
% minD: [varobs_nbr by minns] in DR-order
% measurement matrix D for evolution of controls, depending on minimal state vector only
% dminA: [minns by minns by totparam_nbr] in DR order
% Jacobian (wrt to all parameters) of transition matrix minA
% dminB: [minnx by exo_nbr by totparam_nbr] in DR order
% dminB: [minns by exo_nbr by totparam_nbr] in DR order
% Jacobian (wrt to all parameters) of transition matrix minB
% dminC: [obs_nbr by minnx by totparam_nbr] in DR order
% dminC: [varobs_nbr by minns by totparam_nbr] in DR order
% Jacobian (wrt to all parameters) of measurement matrix minC
% dminD: [obs_nbr by exo_nbr by totparam_nbr] in DR order
% dminD: [varobs_nbr by u_nbr by totparam_nbr] in DR order
% Jacobian (wrt to all parameters) of measurement matrix minD
% -------------------------------------------------------------------------
% This function is called by
@ -42,8 +57,9 @@ function [CheckCO,minnx,minA,minB,minC,minD,dminA,dminB,dminC,dminD] = get_minim
% -------------------------------------------------------------------------
% This function calls
% * check_minimality (embedded)
% * minrealold (embedded)
% =========================================================================
% Copyright (C) 2019 Dynare Team
% Copyright (C) 2019-2020 Dynare Team
%
% This file is part of Dynare.
%
@ -60,102 +76,123 @@ function [CheckCO,minnx,minA,minB,minC,minD,dminA,dminB,dminC,dminD] = get_minim
% You should have received a copy of the GNU General Public License
% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
% =========================================================================
nx = size(A,2);
ny = size(C,1);
nu = size(B,2);
if nargin == 1
derivs_flag = 0;
end
realsmall = 1e-7;
[nspred,exo_nbr] = size(SYS.B);
varobs_nbr = size(SYS.C,1);
% Check controllability and observability conditions for full state vector
CheckCO = check_minimality(A,B,C);
if CheckCO == 1 % If model is already minimal
minnx = nx;
minA = A;
minB = B;
minC = C;
minD = D;
if nargout > 6
dminA = dA;
dminB = dB;
dminC = dC;
dminD = dD;
end
CheckCO = check_minimality(SYS.A,SYS.B,SYS.C);
if CheckCO == 1 % If model is already minimal, we are finished
minns = nspred;
minSYS = SYS;
else
%Model is not minimal
realsmall = 1e-7;
% create indices for unnecessary states
endogstateindex = find(abs(sum(A,1))<realsmall);
exogstateindex = find(abs(sum(A,1))>realsmall);
minnx = length(exogstateindex);
% remove unnecessary states from solution matrices
A_1 = A(endogstateindex,exogstateindex);
A_2 = A(exogstateindex,exogstateindex);
B_2 = B(exogstateindex,:);
C_1 = C(:,endogstateindex);
C_2 = C(:,exogstateindex);
D = D;
ind_A1 = endogstateindex;
ind_A2 = exogstateindex;
% minimal representation
minA = A_2;
minB = B_2;
minC = C_2;
minD = D;
% Check controllability and observability conditions
CheckCO = check_minimality(minA,minB,minC);
if CheckCO ~=1
j=1;
while (CheckCO==0 && j<minnx)
combis = nchoosek(1:minnx,j);
i=1;
while i <= size(combis,1)
ind_A2 = exogstateindex;
ind_A1 = [endogstateindex ind_A2(combis(j,:))];
ind_A2(combis(j,:)) = [];
% remove unnecessary states from solution matrices
A_1 = A(ind_A1,ind_A2);
A_2 = A(ind_A2,ind_A2);
B_2 = B(ind_A2,:);
C_1 = C(:,ind_A1);
C_2 = C(:,ind_A2);
D = D;
% minimal representation
minA = A_2;
minB = B_2;
minC = C_2;
minD = D;
% Check controllability and observability conditions
CheckCO = check_minimality(minA,minB,minC);
if CheckCO == 1
minnx = length(ind_A2);
break;
%Model is not minimal
try
minreal_flag = 1;
% In future we will use SLICOT TB01PD.f mex file [to do @wmutschl], currently use workaround
[mSYS,U] = minrealold(SYS,realsmall);
minns = size(mSYS.A,1);
CheckCO = check_minimality(mSYS.A,mSYS.B,mSYS.C);
if CheckCO
minSYS.A = mSYS.A;
minSYS.B = mSYS.B;
minSYS.C = mSYS.C;
minSYS.D = mSYS.D;
if derivs_flag
totparam_nbr = size(SYS.dA,3);
minSYS.dA = zeros(minns,minns,totparam_nbr);
minSYS.dB = zeros(minns,exo_nbr,totparam_nbr);
minSYS.dC = zeros(varobs_nbr,minns,totparam_nbr);
% Note that orthogonal matrix U is such that (U*dA*U',U*dB,dC*U') is a Kalman decomposition of (dA,dB,dC) %
for jp=1:totparam_nbr
dA_tmp = U*SYS.dA(:,:,jp)*U';
dB_tmp = U*SYS.dB(:,:,jp);
dC_tmp = SYS.dC(:,:,jp)*U';
minSYS.dA(:,:,jp) = dA_tmp(1:minns,1:minns);
minSYS.dB(:,:,jp) = dB_tmp(1:minns,:);
minSYS.dC(:,:,jp) = dC_tmp(:,1:minns);
end
i=i+1;
minSYS.dD = SYS.dD;
end
j=j+1;
end
catch
minreal_flag = 0; % if something went wrong use below procedure
end
if nargout > 6
dminA = dA(ind_A2,ind_A2,:);
dminB = dB(ind_A2,:,:);
dminC = dC(:,ind_A2,:);
dminD = dD;
if minreal_flag == 0
fprintf('Use naive brute-force approach to find minimal state space system.\n These computations may be inaccurate/wrong as ''minreal'' is not available, please raise an issue on GitLab or the forum\n')
% create indices for unnecessary states
exogstateindex = find(abs(sum(SYS.A,1))>realsmall);
minns = length(exogstateindex);
% remove unnecessary states from solution matrices
A_2 = SYS.A(exogstateindex,exogstateindex);
B_2 = SYS.B(exogstateindex,:);
C_2 = SYS.C(:,exogstateindex);
D = SYS.D;
ind_A2 = exogstateindex;
% minimal representation
minSYS.A = A_2;
minSYS.B = B_2;
minSYS.C = C_2;
minSYS.D = D;
% Check controllability and observability conditions
CheckCO = check_minimality(minSYS.A,minSYS.B,minSYS.C);
if CheckCO ~=1
% do brute-force search
j=1;
while (CheckCO==0 && j<minns)
combis = nchoosek(1:minns,j);
i=1;
while i <= size(combis,1)
ind_A2 = exogstateindex;
ind_A2(combis(j,:)) = [];
% remove unnecessary states from solution matrices
A_2 = SYS.A(ind_A2,ind_A2);
B_2 = SYS.B(ind_A2,:);
C_2 = SYS.C(:,ind_A2);
D = SYS.D;
% minimal representation
minSYS.A = A_2;
minSYS.B = B_2;
minSYS.C = C_2;
minSYS.D = D;
% Check controllability and observability conditions
CheckCO = check_minimality(minSYS.A,minSYS.B,minSYS.C);
if CheckCO == 1
minns = length(ind_A2);
break;
end
i=i+1;
end
j=j+1;
end
end
if derivs_flag
minSYS.dA = SYS.dA(ind_A2,ind_A2,:);
minSYS.dB = SYS.dB(ind_A2,:,:);
minSYS.dC = SYS.dC(:,ind_A2,:);
minSYS.dD = SYS.dD;
end
end
end
function CheckCO = check_minimality(A,B,C)
function CheckCO = check_minimality(a,b,c)
%% This function computes the controllability and the observability matrices of the ABCD system and checks if the system is minimal
%
% Inputs: Solution matrices A,B,C of ABCD representation of a DSGE model
% Outputs: CheckCO: equals 1, if both rank conditions for observability and controllability are fullfilled
n = size(A,1);
CC = B; % Initialize controllability matrix
OO = C; % Initialize observability matrix
n = size(a,1);
CC = b; % Initialize controllability matrix
OO = c; % Initialize observability matrix
if n >= 2
for indn = 1:1:n-1
CC = [CC, (A^indn)*B]; % Set up controllability matrix
OO = [OO; C*(A^indn)]; % Set up observability matrix
CC = [CC, (a^indn)*b]; % Set up controllability matrix
OO = [OO; c*(a^indn)]; % Set up observability matrix
end
end
CheckC = (rank(full(CC))==n); % Check rank of controllability matrix
@ -163,4 +200,87 @@ CheckO = (rank(full(OO))==n); % Check rank of observability matrix
CheckCO = CheckC&CheckO; % equals 1 if minimal state
end % check_minimality end
function [mSYS,U] = minrealold(SYS,tol)
% This is a temporary replacement for minreal, will be replaced by a mex file from SLICOT TB01PD.f soon
a = SYS.A;
b = SYS.B;
c = SYS.C;
[ns,nu] = size(b);
[am,bm,cm,U,k] = ControllabilityStaircaseRosenbrock(a,b,c,tol);
kk = sum(k);
nu = ns - kk;
nn = nu;
am = am(nu+1:ns,nu+1:ns);
bm = bm(nu+1:ns,:);
cm = cm(:,nu+1:ns);
ns = ns - nu;
if ns
[am,bm,cm,t,k] = ObservabilityStaircaseRosenbrock(am,bm,cm,tol);
kk = sum(k);
nu = ns - kk;
nn = nn + nu;
am = am(nu+1:ns,nu+1:ns);
bm = bm(nu+1:ns,:);
cm = cm(:,nu+1:ns);
end
mSYS.A = am;
mSYS.B = bm;
mSYS.C = cm;
mSYS.D = SYS.D;
end
function [abar,bbar,cbar,t,k] = ObservabilityStaircaseRosenbrock(a,b,c,tol)
%Observability staircase form
[aa,bb,cc,t,k] = ControllabilityStaircaseRosenbrock(a',c',b',tol);
abar = aa'; bbar = cc'; cbar = bb';
end
function [abar,bbar,cbar,t,k] = ControllabilityStaircaseRosenbrock(a, b, c, tol)
% Controllability staircase algorithm of Rosenbrock, 1968
[ra,ca] = size(a);
[rb,cb] = size(b);
ptjn1 = eye(ra);
ajn1 = a;
bjn1 = b;
rojn1 = cb;
deltajn1 = 0;
sigmajn1 = ra;
k = zeros(1,ra);
if nargin == 3
tol = ra*norm(a,1)*eps;
end
for jj = 1 : ra
[uj,sj,vj] = svd(bjn1);
[rsj,csj] = size(sj);
p = flip(eye(rsj),2);
p = permute(p,[2 1 3:ndims(eye(rsj))]);
uj = uj*p;
bb = uj'*bjn1;
roj = rank(bb,tol);
[rbb,cbb] = size(bb);
sigmaj = rbb - roj;
sigmajn1 = sigmaj;
k(jj) = roj;
if roj == 0, break, end
if sigmaj == 0, break, end
abxy = uj' * ajn1 * uj;
aj = abxy(1:sigmaj,1:sigmaj);
bj = abxy(1:sigmaj,sigmaj+1:sigmaj+roj);
ajn1 = aj;
bjn1 = bj;
[ruj,cuj] = size(uj);
ptj = ptjn1 * ...
[uj zeros(ruj,deltajn1); ...
zeros(deltajn1,cuj) eye(deltajn1)];
ptjn1 = ptj;
deltaj = deltajn1 + roj;
deltajn1 = deltaj;
end
t = ptjn1';
abar = t * a * t';
bbar = t * b;
cbar = c * t';
end
end % Main function end

100
tests/minimal state space system/as2007_minimal.mod Normal file
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@ -0,0 +1,100 @@
% this is the Smets and Wouters (2007) model for which Komunjer and Ng (2011)
% derived the minimal state space system. In Dynare, however, we use more
% powerful minreal function
% created by Willi Mutschler (@wmutschl, 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 <http://www.gnu.org/licenses/>.
% =========================================================================
var y R g z c dy p YGR INFL INT;
varobs y R p c YGR INFL INT;
varexo e_r e_g e_z;
parameters tau phi psi1 psi2 rhor rhog rhoz rrst pist gamst nu cyst;
rrst = 1.0000;
pist = 3.2000;
gamst= 0.5500;
tau = 2.0000;
nu = 0.1000;
kap = 0.3300;
phi = tau*(1-nu)/nu/kap/exp(pist/400)^2;
cyst = 0.8500;
psi1 = 1.5000;
psi2 = 0.1250;
rhor = 0.7500;
rhog = 0.9500;
rhoz = 0.9000;
model;
#pist2 = exp(pist/400);
#rrst2 = exp(rrst/400);
#bet = 1/rrst2;
#gst = 1/cyst;
#cst = (1-nu)^(1/tau);
#yst = cst*gst;
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+dy(+1)+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;
INT = pist+rrst+4*gamst+400*R;
dy = y - y(-1);
end;
shocks;
var e_r; stderr 0.2/100;
var e_g; stderr 0.6/100;
var e_z; stderr 0.3/100;
end;
steady_state_model;
z=0; g=0; c=0; y=0; p=0; R=0; dy=0;
YGR=gamst; INFL=pist; INT=pist+rrst+4*gamst;
end;
stoch_simul(order=1,irf=0,periods=0);
options_.qz_criterium = 1;
indx = [M_.nstatic+(1:M_.nspred)]';
indy = 1:M_.endo_nbr';
SS.A = oo_.dr.ghx(indx,:);
SS.B = oo_.dr.ghu(indx,:);
SS.C = oo_.dr.ghx(indy,:);
SS.D = oo_.dr.ghu(indy,:);
[CheckCO,minnx,minSS] = get_minimal_state_representation(SS,0);
Sigmax_full = lyapunov_symm(SS.A, SS.B*M_.Sigma_e*SS.B', options_.lyapunov_fixed_point_tol, options_.qz_criterium, options_.lyapunov_complex_threshold, 1, options_.debug);
Sigmay_full = SS.C*Sigmax_full*SS.C' + SS.D*M_.Sigma_e*SS.D';
Sigmax_min = lyapunov_symm(minSS.A, minSS.B*M_.Sigma_e*minSS.B', options_.lyapunov_fixed_point_tol, options_.qz_criterium, options_.lyapunov_complex_threshold, 1, options_.debug);
Sigmay_min = minSS.C*Sigmax_min*minSS.C' + minSS.D*M_.Sigma_e*minSS.D';
([Sigmay_full(:) - Sigmay_min(:)]')
sqrt(([diag(Sigmay_full), diag(Sigmay_min)]'))
dx = norm( Sigmay_full - Sigmay_min, Inf);
if dx > 1e-12
error('something wrong with minimal state space computations')
else
fprintf('numerical error for moments computed from minimal state system is %d\n',dx)
end

432
tests/minimal state space system/sw_minimal.mod Normal file
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@ -0,0 +1,432 @@
% this is the Smets and Wouters (2007) model for which Komunjer and Ng (2011)
% derived the minimal state space system. In Dynare, however, we use more
% powerful minreal function
/*
* This file provides replication files for
* Smets, Frank and Wouters, Rafael (2007): "Shocks and Frictions in US Business Cycles: A Bayesian
* DSGE Approach", American Economic Review, 97(3), 586-606, that are compatible with Dynare 4.5 onwards
*
* To replicate the full results, you have to get back to the original replication files available at
* https://www.aeaweb.org/articles.php?doi=10.1257/aer.97.3.586 and include the respective estimation commands and mode-files.
*
* Notes:
* - The consumption Euler equation in the paper, equation (2), premultiplies the risk premium process \varepsilon_t^b,
* denoted by b in this code, by the coefficient c_3. In the code this prefactor is omitted by setting the
* coefficient to 1. As a consequence, b in this code actually is b:=c_3*\varepsilon_t^b. As a consequence, in
* the arbitrage equation for the value of capital in the paper, equation (4), the term 1*\varepsilon_t^b
* is replaced by 1/c_3*b, which is equal to \varepsilon_t^b given the above redefinition. This rescaling also explains why the
* standard deviation of the risk premium shock in the AR(1)-process for b has a different standard deviation than reported
* in the paper. However, the results are unaffected by this scaling factor (except for the fact that the posterior distribution
* reported in the paper cannot be directly translated to the present mod-file due to parameter correlation in the posterior.
* - As pointed out in Del Negro/Schorfheide (2012): "Notes on New-Keynesian Models"
* in the code implementation of equation (8) for both the flex price and the sticky price/wage economy,
* there is a (1/(1+cbetabar*cgamma)) missing in the i_2 in front of q_t (denoted qs in the code).
* Equation (8) in the paper reads:
* (1-(1-delta)/gamma)*(1+beta*gamma^(1-sigma))*gamma^2*varphi
* which translates to the code snippet:
* (1-(1-ctou)/cgamma)*(1+cbetabar*cgamma)*cgamma^2*csadjcost
* But the code implements
* (1-(1-ctou)/cgamma)*cgamma^2*csadjcost
* which corresponds to an equation reading
* (1-(1-delta)/gamma)*gamma^2*varphi
* - Chib/Ramamurthy (2010): "Tailored randomized block MCMC methods with application to DSGE models", Journal of Econometrics, 155, pp. 19-38
* have pointed out that the mode reported in the original Smets/Wouters (2007) paper is not actually the mode. \bar \pi (constepinf) is estimated lower
* while \bar \l (constelab) is higher.
* - Note that at the prior mean, [cmap,crhopinf] and [cmaw,crhow] are pairwise collinear. Thus, running identification at the prior
* mean will return a warning. But this is only a local issue. These parameters are only indistinguishable at the prior mean, but not
* at different points.
* - In the prior Table 1A in the paper, the
* - habit parameter $\lambda$ is erroneously labeled h
* - the fixed cost parameter $\phi_p$ is labeled $\Phi$
* - Table 1B claims that $\rho_{ga}$ follows a beta prior with B(0.5,0.2^2), but the code shows that it actually
* follows a normal distribution with N(0.5,0.25^2)
*
* This file was originally written by Frank Smets and Rafeal Wouters and has been updated by
* Johannes Pfeifer.
*
* Please note that the following copyright notice only applies to this Dynare
* implementation of the model
*/
/*
* Copyright (C) 2007-2013 Frank Smets and Raf Wouters
* Copyright (C) 2013-15 Johannes Pfeifer
* Copyright (C) 2020 Dynare Team
*
* This 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.
*
* This file 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 can receive a copy of the GNU General Public License
* at <http://www.gnu.org/licenses/>.
*/
var labobs ${lHOURS}$ (long_name='log hours worked')
robs ${FEDFUNDS}$ (long_name='Federal funds rate')
pinfobs ${dlP}$ (long_name='Inflation')
dy ${dlGDP}$ (long_name='Output growth rate')
dc ${dlCONS}$ (long_name='Consumption growth rate')
dinve ${dlINV}$ (long_name='Investment growth rate')
dw ${dlWAG}$ (long_name='Wage growth rate')
ewma ${\eta^{w,aux}}$ (long_name='Auxiliary wage markup moving average variable')
epinfma ${\eta^{p,aux}}$ (long_name='Auxiliary price markup moving average variable')
zcapf ${z^{flex}}$ (long_name='Capital utilization rate flex price economy')
rkf ${r^{k,flex}}$ (long_name='rental rate of capital flex price economy')
kf ${k^{s,flex}}$ (long_name='Capital services flex price economy')
pkf ${q^{flex}}$ (long_name='real value of existing capital stock flex price economy')
cf ${c^{flex}}$ (long_name='Consumption flex price economy')
invef ${i^{flex}}$ (long_name='Investment flex price economy')
yf ${y^{flex}}$ (long_name='Output flex price economy')
labf ${l^{flex}}$ (long_name='hours worked flex price economy')
wf ${w^{flex}}$ (long_name='real wage flex price economy')
rrf ${r^{flex}}$ (long_name='real interest rate flex price economy')
mc ${\mu_p}$ (long_name='gross price markup')
zcap ${z}$ (long_name='Capital utilization rate')
rk ${r^{k}}$ (long_name='rental rate of capital')
k ${k^{s}}$ (long_name='Capital services')
pk ${q}$ (long_name='real value of existing capital stock')
c ${c}$ (long_name='Consumption')
inve ${i}$ (long_name='Investment')
y ${y}$ (long_name='Output')
lab ${l}$ (long_name='hours worked')
pinf ${\pi}$ (long_name='Inflation')
w ${w}$ (long_name='real wage')
r ${r}$ (long_name='nominal interest rate')
a ${\varepsilon_a}$ (long_name='productivity process')
b ${c_2*\varepsilon_t^b}$ (long_name='Scaled risk premium shock')
g ${\varepsilon^g}$ (long_name='Exogenous spending')
qs ${\varepsilon^i}$ (long_name='Investment-specific technology')
ms ${\varepsilon^r}$ (long_name='Monetary policy shock process')
spinf ${\varepsilon^p}$ (long_name='Price markup shock process')
sw ${\varepsilon^w}$ (long_name='Wage markup shock process')
kpf ${k^{flex}}$ (long_name='Capital stock flex price economy')
kp ${k}$ (long_name='Capital stock')
;
varexo ea ${\eta^a}$ (long_name='productivity shock')
eb ${\eta^b}$ (long_name='Investment-specific technology shock')
eg ${\eta^g}$ (long_name='Spending shock')
eqs ${\eta^i}$ (long_name='Investment-specific technology shock')
em ${\eta^m}$ (long_name='Monetary policy shock')
epinf ${\eta^{p}}$ (long_name='Price markup shock')
ew ${\eta^{w}}$ (long_name='Wage markup shock')
;
parameters curvw ${\varepsilon_w}$ (long_name='Curvature Kimball aggregator wages')
cgy ${\rho_{ga}}$ (long_name='Feedback technology on exogenous spending')
curvp ${\varepsilon_p}$ (long_name='Curvature Kimball aggregator prices')
constelab ${\bar l}$ (long_name='steady state hours')
constepinf ${\bar \pi}$ (long_name='steady state inflation rate')
constebeta ${100(\beta^{-1}-1)}$ (long_name='time preference rate in percent')
cmaw ${\mu_w}$ (long_name='coefficient on MA term wage markup')
cmap ${\mu_p}$ (long_name='coefficient on MA term price markup')
calfa ${\alpha}$ (long_name='capital share')
czcap ${\psi}$ (long_name='capacity utilization cost')
csadjcost ${\varphi}$ (long_name='investment adjustment cost')
ctou ${\delta}$ (long_name='depreciation rate')
csigma ${\sigma_c}$ (long_name='risk aversion')
chabb ${\lambda}$ (long_name='external habit degree')
ccs ${d_4}$ (long_name='Unused parameter')
cinvs ${d_3}$ (long_name='Unused parameter')
cfc ${\phi_p}$ (long_name='fixed cost share')
cindw ${\iota_w}$ (long_name='Indexation to past wages')
cprobw ${\xi_w}$ (long_name='Calvo parameter wages')
cindp ${\iota_p}$ (long_name='Indexation to past prices')
cprobp ${\xi_p}$ (long_name='Calvo parameter prices')
csigl ${\sigma_l}$ (long_name='Frisch elasticity')
clandaw ${\phi_w}$ (long_name='Gross markup wages')
crdpi ${r_{\Delta \pi}}$ (long_name='Unused parameter')
crpi ${r_{\pi}}$ (long_name='Taylor rule inflation feedback')
crdy ${r_{\Delta y}}$ (long_name='Taylor rule output growth feedback')
cry ${r_{y}}$ (long_name='Taylor rule output level feedback')
crr ${\rho}$ (long_name='interest rate persistence')
crhoa ${\rho_a}$ (long_name='persistence productivity shock')
crhoas ${d_2}$ (long_name='Unused parameter')
crhob ${\rho_b}$ (long_name='persistence risk premium shock')
crhog ${\rho_g}$ (long_name='persistence spending shock')
crhols ${d_1}$ (long_name='Unused parameter')
crhoqs ${\rho_i}$ (long_name='persistence risk premium shock')
crhoms ${\rho_r}$ (long_name='persistence monetary policy shock')
crhopinf ${\rho_p}$ (long_name='persistence price markup shock')
crhow ${\rho_w}$ (long_name='persistence wage markup shock')
ctrend ${\bar \gamma}$ (long_name='net growth rate in percent')
cg ${\frac{\bar g}{\bar y}}$ (long_name='steady state exogenous spending share')
;
// fixed parameters
ctou=.025;
clandaw=1.5;
cg=0.18;
curvp=10;
curvw=10;
// estimated parameters initialisation
calfa=.24;
cbeta=.9995;
csigma=1.5;
cfc=1.5;
cgy=0.51;
csadjcost= 6.0144;
chabb= 0.6361;
cprobw= 0.8087;
csigl= 1.9423;
cprobp= 0.6;
cindw= 0.3243;
cindp= 0.47;
czcap= 0.2696;
crpi= 1.488;
crr= 0.8762;
cry= 0.0593;
crdy= 0.2347;
crhoa= 0.9977;
crhob= 0.5799;
crhog= 0.9957;
crhols= 0.9928;
crhoqs= 0.7165;
crhoas=1;
crhoms=0;
crhopinf=0;
crhow=0;
cmap = 0;
cmaw = 0;
constelab=0;
%% Added by JP to provide full calibration of model before estimation
constepinf=0.7;
constebeta=0.7420;
ctrend=0.3982;
model(linear);
//deal with parameter dependencies; taken from usmodel_stst.mod
#cpie=1+constepinf/100; %gross inflation rate
#cgamma=1+ctrend/100 ; %gross growth rate
#cbeta=1/(1+constebeta/100); %discount factor
#clandap=cfc; %fixed cost share/gross price markup
#cbetabar=cbeta*cgamma^(-csigma); %growth-adjusted discount factor in Euler equation
#cr=cpie/(cbeta*cgamma^(-csigma)); %steady state net real interest rate
#crk=(cbeta^(-1))*(cgamma^csigma) - (1-ctou); %R^k_{*}: steady state rental rate
#cw = (calfa^calfa*(1-calfa)^(1-calfa)/(clandap*crk^calfa))^(1/(1-calfa)); %steady state real wage
//cw = (calfa^calfa*(1-calfa)^(1-calfa)/(clandap*((cbeta^(-1))*(cgamma^csigma) - (1-ctou))^calfa))^(1/(1-calfa));
#cikbar=(1-(1-ctou)/cgamma); %k_1 in equation LOM capital, equation (8)
#cik=(1-(1-ctou)/cgamma)*cgamma; %i_k: investment-capital ratio
#clk=((1-calfa)/calfa)*(crk/cw); %labor to capital ratio
#cky=cfc*(clk)^(calfa-1); %k_y: steady state output ratio
#ciy=cik*cky; %consumption-investment ratio
#ccy=1-cg-cik*cky; %consumption-output ratio
#crkky=crk*cky; %z_y=R_{*}^k*k_y
#cwhlc=(1/clandaw)*(1-calfa)/calfa*crk*cky/ccy; %W^{h}_{*}*L_{*}/C_{*} used in c_2 in equation (2)
#cwly=1-crk*cky; %unused parameter
#conster=(cr-1)*100; %steady state federal funds rate ($\bar r$)
// flexible economy
[name='FOC labor with mpl expressed as function of rk and w, flex price economy']
0*(1-calfa)*a + 1*a = calfa*rkf+(1-calfa)*(wf) ;
[name='FOC capacity utilization, flex price economy']
zcapf = (1/(czcap/(1-czcap)))* rkf ;
[name='Firm FOC capital, flex price economy']
rkf = (wf)+labf-kf ;
[name='Definition capital services, flex price economy']
kf = kpf(-1)+zcapf ;
[name='Investment Euler Equation, flex price economy']
invef = (1/(1+cbetabar*cgamma))* ( invef(-1) + cbetabar*cgamma*invef(1)+(1/(cgamma^2*csadjcost))*pkf ) +qs ;
[name='Arbitrage equation value of capital, flex price economy']
pkf = -rrf-0*b+(1/((1-chabb/cgamma)/(csigma*(1+chabb/cgamma))))*b +(crk/(crk+(1-ctou)))*rkf(1) + ((1-ctou)/(crk+(1-ctou)))*pkf(1) ;
[name='Consumption Euler Equation, flex price economy']
cf = (chabb/cgamma)/(1+chabb/cgamma)*cf(-1) + (1/(1+chabb/cgamma))*cf(+1) +((csigma-1)*cwhlc/(csigma*(1+chabb/cgamma)))*(labf-labf(+1)) - (1-chabb/cgamma)/(csigma*(1+chabb/cgamma))*(rrf+0*b) + b ;
[name='Aggregate Resource Constraint, flex price economy']
yf = ccy*cf+ciy*invef+g + crkky*zcapf ;
[name='Aggregate Production Function, flex price economy']
yf = cfc*( calfa*kf+(1-calfa)*labf +a );
[name='Wage equation, flex price economy']
wf = csigl*labf +(1/(1-chabb/cgamma))*cf - (chabb/cgamma)/(1-chabb/cgamma)*cf(-1) ;
[name='Law of motion for capital, flex price economy (see header notes)']
kpf = (1-cikbar)*kpf(-1)+(cikbar)*invef + (cikbar)*(cgamma^2*csadjcost)*qs ;
// sticky price - wage economy
[name='FOC labor with mpl expressed as function of rk and w, SW Equation (9)']
mc = calfa*rk+(1-calfa)*(w) - 1*a - 0*(1-calfa)*a ;
[name='FOC capacity utilization, SW Equation (7)']
zcap = (1/(czcap/(1-czcap)))* rk ;
[name='Firm FOC capital, SW Equation (11)']
rk = w+lab-k ;
[name='Definition capital services, SW Equation (6)']
k = kp(-1)+zcap ;
[name='Investment Euler Equation, SW Equation (3)']
inve = (1/(1+cbetabar*cgamma))* (inve(-1) + cbetabar*cgamma*inve(1)+(1/(cgamma^2*csadjcost))*pk ) +qs ;
[name='Arbitrage equation value of capital, SW Equation (4)']
pk = -r+pinf(1)-0*b
+ (1/((1-chabb/cgamma)/(csigma*(1+chabb/cgamma))))*b
+ (crk/(crk+(1-ctou)))*rk(1)
+ ((1-ctou)/(crk+(1-ctou)))*pk(1) ;
[name='Consumption Euler Equation, SW Equation (2)']
c = (chabb/cgamma)/(1+chabb/cgamma)*c(-1)
+ (1/(1+chabb/cgamma))*c(+1)
+((csigma-1)*cwhlc/(csigma*(1+chabb/cgamma)))*(lab-lab(+1))
- (1-chabb/cgamma)/(csigma*(1+chabb/cgamma))*(r-pinf(+1) + 0*b) +b ;
[name='Aggregate Resource Constraint, SW Equation (1)']
y = ccy*c+ciy*inve+g + 1*crkky*zcap ;
[name='Aggregate Production Function, SW Equation (5)']
y = cfc*( calfa*k+(1-calfa)*lab +a );
[name='New Keynesian Phillips Curve, SW Equation (10)']
pinf = (1/(1+cbetabar*cgamma*cindp)) * ( cbetabar*cgamma*pinf(1) +cindp*pinf(-1)
+((1-cprobp)*(1-cbetabar*cgamma*cprobp)/cprobp)/((cfc-1)*curvp+1)*(mc) ) + spinf ;
[name='Wage Phillips Curve, SW Equation (13), with (12) plugged for mu_w']
w = (1/(1+cbetabar*cgamma))*w(-1)
+(cbetabar*cgamma/(1+cbetabar*cgamma))*w(1)
+(cindw/(1+cbetabar*cgamma))*pinf(-1)
-(1+cbetabar*cgamma*cindw)/(1+cbetabar*cgamma)*pinf
+(cbetabar*cgamma)/(1+cbetabar*cgamma)*pinf(1)
+(1-cprobw)*(1-cbetabar*cgamma*cprobw)/((1+cbetabar*cgamma)*cprobw)*(1/((clandaw-1)*curvw+1))*
(csigl*lab + (1/(1-chabb/cgamma))*c - ((chabb/cgamma)/(1-chabb/cgamma))*c(-1) -w)
+ 1*sw ;
[name='Taylor rule, SW Equation (14)']
r = crpi*(1-crr)*pinf
+cry*(1-crr)*(y-yf)
+crdy*(y-yf-y(-1)+yf(-1))
+crr*r(-1)
+ms ;
[name='Law of motion for productivity']
a = crhoa*a(-1) + ea;
[name='Law of motion for risk premium']
b = crhob*b(-1) + eb;
[name='Law of motion for spending process']
g = crhog*(g(-1)) + eg + cgy*ea;
[name='Law of motion for investment specific technology shock process']
qs = crhoqs*qs(-1) + eqs;
[name='Law of motion for monetary policy shock process']
ms = crhoms*ms(-1) + em;
[name='Law of motion for price markup shock process']
spinf = crhopinf*spinf(-1) + epinfma - cmap*epinfma(-1);
epinfma=epinf;
[name='Law of motion for wage markup shock process']
sw = crhow*sw(-1) + ewma - cmaw*ewma(-1) ;
ewma=ew;
[name='Law of motion for capital, SW Equation (8) (see header notes)']
kp = (1-cikbar)*kp(-1)+cikbar*inve + cikbar*cgamma^2*csadjcost*qs ;
// measurement equations
[name='Observation equation output']
dy=y-y(-1)+ctrend;
[name='Observation equation consumption']
dc=c-c(-1)+ctrend;
[name='Observation equation investment']
dinve=inve-inve(-1)+ctrend;
[name='Observation equation real wage']
dw=w-w(-1)+ctrend;
[name='Observation equation inflation']
pinfobs = 1*(pinf) + constepinf;
[name='Observation equation interest rate']
robs = 1*(r) + conster;
[name='Observation equation hours worked']
labobs = lab + constelab;
end;
steady_state_model;
dy=ctrend;
dc=ctrend;
dinve=ctrend;
dw=ctrend;
pinfobs = constepinf;
robs = (((1+constepinf/100)/((1/(1+constebeta/100))*(1+ctrend/100)^(-csigma)))-1)*100;
labobs = constelab;
end;
shocks;
var ea;
stderr 0.4618;
var eb;
stderr 1.8513;
var eg;
stderr 0.6090;
var eqs;
stderr 0.6017;
var em;
stderr 0.2397;
var epinf;
stderr 0.1455;
var ew;
stderr 0.2089;
end;
estimated_params;
// PARAM NAME, INITVAL, LB, UB, PRIOR_SHAPE, PRIOR_P1, PRIOR_P2, PRIOR_P3, PRIOR_P4, JSCALE
// PRIOR_SHAPE: BETA_PDF, GAMMA_PDF, NORMAL_PDF, INV_GAMMA_PDF
stderr ea,0.4618,0.01,3,INV_GAMMA_PDF,0.1,2;
stderr eb,0.1818513,0.025,5,INV_GAMMA_PDF,0.1,2;
stderr eg,0.6090,0.01,3,INV_GAMMA_PDF,0.1,2;
stderr eqs,0.46017,0.01,3,INV_GAMMA_PDF,0.1,2;
stderr em,0.2397,0.01,3,INV_GAMMA_PDF,0.1,2;
stderr epinf,0.1455,0.01,3,INV_GAMMA_PDF,0.1,2;
stderr ew,0.2089,0.01,3,INV_GAMMA_PDF,0.1,2;
crhoa,.9676 ,.01,.9999,BETA_PDF,0.5,0.20;
crhob,.2703,.01,.9999,BETA_PDF,0.5,0.20;
crhog,.9930,.01,.9999,BETA_PDF,0.5,0.20;
crhoqs,.5724,.01,.9999,BETA_PDF,0.5,0.20;
crhoms,.3,.01,.9999,BETA_PDF,0.5,0.20;
crhopinf,.8692,.01,.9999,BETA_PDF,0.5,0.20;
crhow,.9546,.001,.9999,BETA_PDF,0.5,0.20;
cmap,.7652,0.01,.9999,BETA_PDF,0.5,0.2;
cmaw,.8936,0.01,.9999,BETA_PDF,0.5,0.2;
csadjcost,6.3325,2,15,NORMAL_PDF,4,1.5;
csigma,1.2312,0.25,3,NORMAL_PDF,1.50,0.375;
chabb,0.7205,0.001,0.99,BETA_PDF,0.7,0.1;
cprobw,0.7937,0.3,0.95,BETA_PDF,0.5,0.1;
csigl,2.8401,0.25,10,NORMAL_PDF,2,0.75;
cprobp,0.7813,0.5,0.95,BETA_PDF,0.5,0.10;
cindw,0.4425,0.01,0.99,BETA_PDF,0.5,0.15;
cindp,0.3291,0.01,0.99,BETA_PDF,0.5,0.15;
czcap,0.2648,0.01,1,BETA_PDF,0.5,0.15;
cfc,1.4672,1.0,3,NORMAL_PDF,1.25,0.125;
crpi,1.7985,1.0,3,NORMAL_PDF,1.5,0.25;
crr,0.8258,0.5,0.975,BETA_PDF,0.75,0.10;
cry,0.0893,0.001,0.5,NORMAL_PDF,0.125,0.05;
crdy,0.2239,0.001,0.5,NORMAL_PDF,0.125,0.05;
constepinf,0.7,0.1,2.0,GAMMA_PDF,0.625,0.1;//20;
constebeta,0.7420,0.01,2.0,GAMMA_PDF,0.25,0.1;//0.20;
constelab,1.2918,-10.0,10.0,NORMAL_PDF,0.0,2.0;
ctrend,0.3982,0.1,0.8,NORMAL_PDF,0.4,0.10;
cgy,0.05,0.01,2.0,NORMAL_PDF,0.5,0.25;
calfa,0.24,0.01,1.0,NORMAL_PDF,0.3,0.05;
end;
stoch_simul(order=1,irf=0,periods=0);
options_.qz_criterium = 1;
indx = [M_.nstatic+(1:M_.nspred)]';
indy = 1:M_.endo_nbr';
SS.A = oo_.dr.ghx(indx,:);
SS.B = oo_.dr.ghu(indx,:);
SS.C = oo_.dr.ghx(indy,:);
SS.D = oo_.dr.ghu(indy,:);
[CheckCO,minnx,minSS] = get_minimal_state_representation(SS,0);
Sigmax_full = lyapunov_symm(SS.A, SS.B*M_.Sigma_e*SS.B', options_.lyapunov_fixed_point_tol, options_.qz_criterium, options_.lyapunov_complex_threshold, 1, options_.debug);
Sigmay_full = SS.C*Sigmax_full*SS.C' + SS.D*M_.Sigma_e*SS.D';
Sigmax_min = lyapunov_symm(minSS.A, minSS.B*M_.Sigma_e*minSS.B', options_.lyapunov_fixed_point_tol, options_.qz_criterium, options_.lyapunov_complex_threshold, 1, options_.debug);
Sigmay_min = minSS.C*Sigmax_min*minSS.C' + minSS.D*M_.Sigma_e*minSS.D';
([Sigmay_full(:) - Sigmay_min(:)]')
sqrt(([diag(Sigmay_full), diag(Sigmay_min)]'))
dx = norm( Sigmay_full - Sigmay_min, Inf);
if dx > 1e-8
error(sprintf('something wrong with minimal state space computations, as numerical error is %d',dx))
else
fprintf('numerical error for moments computed from minimal state system is %d\n',dx)
end