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