% 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 © 2007-2013 Frank Smets and Raf Wouters
 * Copyright © 2013-15 Johannes Pfeifer
 * Copyright © 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 <https://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] = identification.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 > 3e-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
