105/replication_package/examples/rbc/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

%----------------------------------
% Make a vector of parameter values
%----------------------------------
BETA=.96; GAMMA=2; ALPHA=.3; RHO=.8; DELTA=.1; SIGMA=.02;
params=eval(symparams);

%----------------------------------------------------------------------
% Prepare an initial guess - in this case I use a perturbation solution
%----------------------------------------------------------------------

% Steady state values

kss=((1/BETA-1+DELTA)/ALPHA)^(1/(ALPHA-1));
zss=0;
css=kss^ALPHA-DELTA*kss;

nxss=[kss;zss];
nyss=css;

% Cross moments of the shocks

M=get_moments(nodes,weights,model.order(2));

% Compute the perturbation solution (keep the 4 outputs):

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

% Explanation of outputs:
% derivs=structure with the perturbation solution as explained in Levintal
% (2017): "Fifth-Order Perturbation Solution to DSGE Models".
% stoch_pert=the perturbation solution in the form of unique polynomial coefficients.
% nonstoch_pert=same as stoch_pert but without correction for the model volatility (i.e. this is a perturbation solution of a deterministic version of the model)

%-------------------------------------
% Solve the model by Taylor projection
%-------------------------------------

x0=nxss; % the approximation point (here we use the steady state, but it could be any arbitrary state)
c0=nxss; % the center of the initial guess

% tolerance parameters for the Newton solver
tolX=1e-6;
tolF=1e-6;
maxiter=10;

% model.jacobian='exact'; % this is the default
% model.jacobian='approximate'; % for large models try the approximate jacobian.

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

%------------------------------------------------------------------
% Compute the residual function and the model variables at point x0
%------------------------------------------------------------------

[R_fun0,g_fun0,Phi_fun0,auxvars0]=residual(coeffs,x0,params,c0,nodes,weights);

% R_fun0 is the residual function at x0.
% g_fun0 is the control variables at x0, namely, g(x0).
% Phi_fun0 is the function Phi at x0 and each future node, namely, Phi(x0,g(x0),epsp), for each node of the quadrature.
% auxvars0 is the auxiliary functions at x0 and each future node.

% compute the function g(x) at x0
y0=evalg(x0,coeffs,c0);

% compute the function Phi(x,y,epsp) at x0, y0 and epsp0
epsp0=.02;
xp0=evalPhi(x0,y0,epsp0,params);

%---------------------------------
% simulate the model for T periods
%---------------------------------
T=100;
shocks=randn(1,T+1); % draw shocks

% preallocate
x_simul=zeros(model.n_x,T+1);
y_simul=zeros(model.n_y,T);
R_simul=zeros(model.n_y,T);

x_simul(:,1)=x0;

% option=1; % compute only simulated variables
option=2; % compute model residuals

for t=1:T
    xt=x_simul(:,t);
    epsp=shocks(t+1);

    % Option 1 - compute only the simulated variables
    if option==1
        yt=evalg(xt,coeffs,c0);
        
        y_simul(:,t)=yt;
        x_simul(:,t+1)=evalPhi(xt,yt,epsp,params);
    else
    % Option 2 - compute also model residuals
        [Rt,yt]=residual(coeffs,xt,params,c0,nodes,weights);
        
        y_simul(:,t)=yt;
        x_simul(:,t+1)=evalPhi(xt,yt,epsp,params);
        R_simul(:,t)=Rt;
    end
end

%-------------------------------------------
% Solve the model again at a different state
%-------------------------------------------
% This is useful when the long run domain of the model is far from the
% initial state, so we need to approximate the solution at the long run state
% (e.g. the risky steady state or the mean of the ergodic distribution)
% rather than the steady state.

x1=x0*1.1; % take some arbitrary state
[coeffs1,model]=tpsolve(coeffs,x1,model,params,c0,nodes,weights,tolX,tolF,maxiter); % solve at x1

%-----------------------
% Use a different solver
%-----------------------

% The function tpsolve uses the Newton method for up to maxiter iterations. If it fails, it
% switches automatically to fsolve for another maxiter iterations. You can
% control the parameters of the second solver by optimoptions. The
% supported solvers are fsolve and lsqnonlin.

% For example, do one Newton iteration and switch to lsqnonlin:

x2=x1*1.1;
maxiter=1; % one Newton iteration
OPTIONS = optimoptions('lsqnonlin','TolX',tolX,'TolF',tolF,'MaxIter',10,'Display','iter-detailed'); % 10 more iterations by lsqnonlin

[coeffs2,model]=tpsolve(coeffs,x2,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS);


% or switch to fsolve:

maxiter=1; % one Newton iteration
OPTIONS = optimoptions('fsolve','TolX',tolX,'TolF',tolF,'MaxIter',10,'Display','iter-detailed'); % 10 more iterations by fsolve

[coeffs3,model]=tpsolve(coeffs,x2,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS);