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LIN_DSGE.m
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LIN_DSGE.m
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%%==============================================================================
%% Solve linear DSGE model calibrated at weekly frequency
%%==============================================================================
function [AMAT,BMAT,SS,xeq,xvari,s3]=LIN_DSGE(w,gamma,alpha)
global delta eta c a sigma_a omega rho_a s z q f u varsigma uinv
global W_bar C_bar TH_bar N_bar H_bar U_bar UC_bar UF_bar Y_bar
global wpos cpos thpos npos hpos upos ucpos ufpos ypos apos a_pos
%% --- Set up positions of variables ---
xlead = 1; % Number of leads in system (recruiting)
xlag = 1 ; % Number of lags in system (employment)
xnames = ['TH';'W ';'C ';'N ';'H ';'U ';'Y ';'A ';'A_'];
xnum = size(xnames,1); % Number of variables in system
xeq = xnum ; % Number of equations (same)
xvari=xnum-1;
colzero = 0+xlag*xnum; % Position counter for start of contemp. coefs
collead = 0+xlag*xnum+xnum; % Position counter for start of lead coefs
collag = 0 ;% Position counter for start of lag coefs
wzero = colzero+wpos;
czero = colzero+cpos;
thzero = colzero+thpos;
nzero = colzero+npos;
hzero = colzero+hpos;
uzero = colzero+upos;
yzero = colzero+ypos;
azero = colzero+apos;
a_zero = colzero+a_pos;
wlead = collead+wpos;
clead = collead+cpos;
thlead = collead+thpos;
nlead = collead+npos;
hlead = collead+hpos;
ulead = collead+upos;
ylead = collead+ypos;
alead = collead+apos;
a_lead = collead+a_pos;
wlag = collag+wpos;
clag = collag+cpos;
thlag = collag+thpos;
nlag = collag+npos;
hlag = collag+hpos;
ulag = collag+upos;
ylag = collag+ypos;
alag = collag+apos;
a_lag = collag+a_pos;
% Now we have one vector with all of the leads, contemporaneous and
% lags stacked in one column
% Determine number of total posible coefficients per equation:
xcoef = xeq*(xlag+xlead+1);
% Initialize the coeffienct matrix, where each
% row is an equation of the model
cof = zeros(xeq,xcoef) ; % Coef matrix --- Each row is an equation
%% --- Steady-state shares ---
xx=alpha.*N_bar.^alpha;
sa= w./xx ;
sb= (c.*a./q(TH_bar))./xx;
sc=c.*a./q(TH_bar).*(1-s).*delta./xx ;
s2=(c.*a.*s)./(q(TH_bar).*N_bar.^(alpha-1));
s3=1./((1-s).*N_bar)-1;
s4=UC_bar./(1-UC_bar);
s5=UC_bar./U_bar;
%% --- equilibrium conditions: Setup coefficients vectors for each equation ---
% Euler equation
cof(1,azero) = 1-gamma.*sa-sb;
cof(1,nzero) = (alpha-1);
cof(1,thlead) = sc.*eta;
cof(1,thzero) =-eta.*sb;
cof(1,alead) = sc;
% Unemployment definition
cof(2,uzero) = s3;
cof(2,nlag) = 1;
% Definition of tightness
cof(3,thzero) = 1-eta;
cof(3,hzero) = -1;
cof(3,uzero) = 1;
% Employment fluctuations
cof(4,nzero) = 1;
cof(4,nlag) = -(1-s);
cof(4,hzero) = -s;
% Resource constraint
cof(5,yzero) = 1;
cof(5,azero) = -s2;
cof(5,czero) = s2-1;
cof(5,thzero) = -s2*eta;
cof(5,hzero) = -s2;
% Output
cof(6,yzero) =1;
cof(6,azero) =-1;
cof(6,nzero) =-alpha;
% Rigid wwage
cof(7,wzero)=1;
cof(7,azero)=-gamma;
% Correlated error term: technology
cof(8,azero) = -1;
cof(8,alag) = rho_a;
cof(8,a_zero) = 1;
% 0 = SHOCKS
cof(9,a_zero) = 1;
%% --- Use AIM procedure to solve model ---
uprbnd = 1+1e-8; % Tolerance values for AIM program
condn = 1e-8;
[cofb,rts,ia,nex,nnum,lgrts,mcode] = aim_eig(cof,xeq,xlag,xlead,condn,uprbnd);
scof = obstruct(cof,cofb,xeq,xlag,xlead);
s0 = scof(:,(xeq*xlag+1):xeq*(xlag+1)); %Contemp. coefs from obs.
%structure
amat=zeros(xeq*xlag,xeq*xlag); % Initialize A matrix
bmat=cofb(1:xeq,((xlag-1)*xeq+1):xlag*xeq); % Lag 1 coefficients
i=2;
while i<=xlag;
bmat=[bmat cofb(1:xeq,((xlag-i)*xeq+1):(xlag-i+1)*xeq)]; % Lag i coefs
i=i+1;
end;
amat(1:xeq,:)=bmat; % Coefs for equations
if xlag>1;
amat((length(cofb(:,1))+1):length(amat(:,1)),...
1:xeq*(xlag-1))=eye(xeq*(xlag-1));
end;
b = zeros(length(amat(:,1)),length(s0(1,:)));
b(1:length(s0(:,1)),1:length(s0(1,:))) = inv(s0); % Store coefs
% check unique/stable REE: mcode = 1
if mcode ~=1
'ERROR: NON_UNIQUE REE'
disp(['mcode = ' num2str(mcode)]) % unstable? non-unique?
end
AMAT=amat;
BMAT=b;
% to compute and plot impulse response use reduced form solution: x_t=amat*x_{t-1}+b*shock
SS=[];
SS(wpos,1)= W_bar;
SS(cpos,1)=C_bar;
SS(thpos,1)=TH_bar;
SS(npos,1)=N_bar;
SS(hpos,1)=H_bar;
SS(upos,1)=U_bar;
SS(ypos,1)=Y_bar;
SS(apos,1)=a;
SS(a_pos,1)=0;