105/replication_package/examples/rbc_MS_parameters/solve_model.m

clear,clc

%---------------------------------------------------------
% Add folder 'files' to the search path and load the model
%---------------------------------------------------------
addpath('files'); 
load('model')

%----------------------------------------------------------------------------
% Provide nodes and weights for the quadrature that approximates expectations
%----------------------------------------------------------------------------
n_e=1; % number of shocks.
[n_nodes,nodes,weights] = Monomials_2(n_e,eye(n_e)); % this quadrature function was written by Judd, Maliar, Maliar and Valero (2014).
nodes=nodes'; % transpose to n_e-by-n_nodes



%--------------------------------------------

% parameter values (for the fixed parameters)

%--------------------------------------------

GAMMA=2; ALPHA=.3; RHO=.8; SIGMA=.02; 

params=eval(symparams);



%--------------------------------------------------------------------

% Start with a perturbation solution for the case of fixed parameters

%--------------------------------------------------------------------

BETA=.96; DELTA=.1;

% Steady state values

kss=((1/BETA-1+DELTA)/ALPHA)^(1/(ALPHA-1));

zss=0;

css=kss^ALPHA-DELTA*kss;



nxss=[kss;zss];

nyss=css;



pert_order=model.order(2);



M=get_moments(nodes,weights,pert_order);



% Get a perturbation solution for the case of fixed parameters. You need to

% provide the fixed value of the ms parameters (the variable chi_fixed)



chi_fixed=eval(chi);



[derivs,stoch_pert,nonstoch_pert,model]=get_pert(model,params,M,nxss,nyss,chi_fixed);



%---------------------------------------

% Proceed to Markov-switching parameters

%---------------------------------------



% The values of the parameters in each regime (there are 2 regimes)

msparams=[BETA-.02,BETA+.02;% BETA varies across regimes

         DELTA-.05,DELTA+.05]; % DELTA varies across regimes



transition=[.9,.1;

            .8,.2];



initial_guess=stoch_pert; % the initial guess has n_regimes columns (one for each regime).



x0=nxss; % evaluate the system at the steady state

c0=nxss; % the center of the initial guess is the steady state.



tolX=1e-6;

tolF=1e-6;

maxiter=10;

[coeffs,model]=tpsolve(initial_guess,x0,model,params,msparams,transition,c0,nodes,weights,tolX,tolF,maxiter);





% compute the model residuals



[R_funMS,g_funMS,Phi_funMS,aux_funMS]=residual(coeffs,x0,params,msparams,transition,c0,nodes,weights);



% R_funMS is a n_y-by-n_regimes matrix of the model residuals.

% g_funMS is a n_y-by-n_regimes matrix of the endogenous control variables for each current regime.

% Phi_funMS is a n_x-by-n_nodes-by-n_regimes-by-n_regimes array of next period state variables for each future node and current and future regimes.

% aux_funMS is a similar array of the auxiliary functions (for each future

% node and current/future regimes)



% To compute the model residuals only for a specific regime, add the

% required regime as the last argument



specific_regime=1;



[R_specific,g_specific,Phi_specific,auxvars_specific]=residual(coeffs,x0,params,msparams,transition,c0,nodes,weights,specific_regime);



% Compute the function g given the state x0 and the regime specific_regime

g_fun=evalg(x0,specific_regime,coeffs,c0);



% Compute the function Phi given the state x0, the control y0, the current

% and future regimes and the future shock epsp

y0=g_fun;

current_regime=1;

future_regime=2;

epsp=0;

Phi_fun=evalPhi(x0,y0,epsp,future_regime,current_regime,params,msparams);





%---------------------------------

% simulate the model for T periods

%---------------------------------



T=10000;

shocks=randn(1,T+1);

rshock=rand(1,T+1); % to determine the regime



x_simul=zeros(model.n_x,T+1);

regime_simul=zeros(1,T+1);

y_simul=zeros(model.n_y,T);

R_simul=zeros(model.n_y,T);



coeffs=reshape(coeffs,[],model.n_regimes);



x_simul(:,1)=x0;

regime_simul(1)=1;



% option=1; % compute only simulated variables

option=2; % compute model residuals



for t=1:T

  

    xt=x_simul(:,t); % current state

    regimet=regime_simul(t); % current regime

    

    epsp=shocks(t+1); % future shock

    % future regime

    regimet_next=1+model.n_regimes-sum(double(logical((rshock(t+1)-cumsum(transition(regimet,:)))<0))); 

    

    % Option 1 - compute only the simulated variables

    if option==1

        yt=evalg(xt,regimet,coeffs,c0);



        y_simul(:,t)=yt;

        x_simul(:,t+1)=evalPhi(xt,yt,epsp,regimet_next,regimet,params,msparams);

        regime_simul(t+1)=regimet_next;

    else

    % Option 2 - compute also model residuals

        [Rt,yt]=residual(coeffs,xt,params,msparams,transition,c0,nodes,weights,regimet);



        y_simul(:,t)=yt;

        x_simul(:,t+1)=evalPhi(xt,yt,epsp,regimet_next,regimet,params,msparams);

        regime_simul(t+1)=regimet_next;

        R_simul(:,t)=Rt;       

    end

end



% Note that the simulated capital level is considerably different than the

% initial approximation point that we used. To improve accuracy, we can solve the

% model at the mean of the ergodic distribution.



meank=mean(x_simul(1,:));



ergodic_x0=[meank;0]; % solve at the mean of the ergodic distribution



[coeffs,model]=tpsolve(coeffs,ergodic_x0,model,params,msparams,transition,c0,nodes,weights,tolX,tolF,maxiter);



% simulate again and store residuals in R_simul2



x_simul(:,1)=ergodic_x0;

regime_simul(1)=1;

R_simul2=R_simul;

for t=1:T

    xt=x_simul(:,t);

    regimet=regime_simul(t);

    

    epsp=shocks(t+1);

    % future regime

    regimet_next=1+model.n_regimes-sum(double(logical((rshock(t+1)-cumsum(transition(regimet,:)))<0)));

    

    % Option 1 - compute only the simulated variables

    if option==1

        yt=evalg(xt,regimet,coeffs,c0);



        y_simul(:,t)=yt;

        x_simul(:,t+1)=evalPhi(xt,yt,epsp,regimet_next,regimet,params,msparams);

        regime_simul(t+1)=regimet_next;

    else

    % Option 2 - compute also model residuals

        [Rt,yt]=residual(coeffs,xt,params,msparams,transition,c0,nodes,weights,regimet);



        y_simul(:,t)=yt;

        x_simul(:,t+1)=evalPhi(xt,yt,epsp,regimet_next,regimet,params,msparams);

        regime_simul(t+1)=regimet_next;

        R_simul2(:,t)=Rt;        

    end

end



% compare mean errors for the two simulations



mean(abs(R_simul))



mean(abs(R_simul2))



% the second simulation is more accurate