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function [yhat,Estr,rcon,Rcon,u,v,d] = fidcnderr(valuecon,stepcon,varcon,nconstr,...
nstepsm,nvar,lags,phil,eq_iden,Aband,Sband,yforeml,imf3sml,Bhml,...
yfore_h,imf3s_h,Bh_h,Cms,TLindx,TLnumber,nCms,eq_Cms)
%function [yhat,Estr,rcon,Rcon,u,v,d] = fidcnderr(valuecon,stepcon,varcon,nconstr,...
% nstepsm,nvar,lags,phil,eq_iden,Aband,Sband,yforeml,imf3sml,Bhml,...
% yfore_h,imf3s_h,Bh_h,Cms,TLindx,TLnumber,nCms,eq_Cms)
% To be done (8/25/98): (1) yforeml, imf3sml, and Bhml should not be there.
% (2) MS stuff need to be generalized.
% (3) perhaps, imf3s_h needs to be re-examined.
% Conditional forecasting in the identified model with one draw for error bands
%
% valuecon: vector of values conditioned
% stepcon: sequence (cell) of steps conditioned; if length(stepcon{i}) > 1, the condition
% is then an arithmetic average of log(y) over the stepcon{i} period.
% varcon: vector of variables conditioned
% nconstr: number of constraints
% nstepsm: maximum number of steps in all constraints
% nvar: number of variables in the BVAR model
% lags: number of lags in the BVAR model
% phil: the 1-by-(nvar*lags+1) data matrix where k=nvar*lags+1
% (last period plus lags before the beginning of forecast)
% eq_iden: identified equation or shock (in terms of number). If equ_iden=[], then
% 'fidencond' is, similar to RATS, to compute forecasts with *all* shocks.
% Sband: 1: generate error bands from random shocks E; 0: no random shocks
% yfore: uncondtional forecasts: forep-by-nvar
% imf3s: 3-dimensional impulse responses matrix:
% impsteps-by-nvar shocks-by-nvar responses
% Bh: reduced-form parameter matrix: k-by-nvar, y(t) = X(t)*Bh+e(t)
% where X(t) is k-by-nvar and y(t) is 1-by-nvar
% ------
% yhat: conditional forecasts: forep-by-nvar
% Estr: backed-out structural shocks (from N(0,1))
% rcon: vector - the difference between valuecon and log(yfore) (unconditional forecasts)
% Rcon: k-by-q (q constranits and k=nvar*max(nsteps)) so that
% Rcon'*e = rcon where e is k-by-1
% [u,d,v]: svd(Rcon,0)
%
%% See Zha's note "Forecast (1)" p. 5, RATS manual (some errors in RATS), etc.
%
%% Some notations: y(t+1) = y(t)B1 + e(t+1)inv(A0). e(t+1) is 1-by-n.
%% Let r(t+1)=e(t+1)inv(A0) + e(t+2)C + .... where inv(A0) is impulse
%% response at t=1, C at t=2, etc. The row of inv(A0) or C is
%% all responses to one shock.
%% Let r be q-by-1 (such as r(1) = r(t+1)
%% = y(t+1) (constrained) - y(t+1) (forecast)).
%% Use impulse responses to find out R (k-by-q) where k=nvar*nsteps
%% where nsteps the largest constrained step. The key of the program
%% is to creat R using impulse responses
%% Optimal solution for shock e where R'*e=r and e is k-by-1 is
%% e = R*inv(R'*R)*r.
%
%
% Copyright (C) 1997-2012 Tao Zha
%
% This 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.
%
% It 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.
%
% If you did not received a copy of the GNU General Public License
% with this software, see <http://www.gnu.org/licenses/>.
%
IdenShock = ~isempty(eq_iden); % if not empty, the shock is identified
forep=size(yfore_h,1);
impsteps=size(imf3s_h,1);
if (forep<nstepsm) | (impsteps<nstepsm)
disp('Increase # of forecast or impulse steps!!')
disp('Or decrease # of constraints (nconstr) or constrained steps (stepcon(i))!!')
error('Maximum of conditional steps > # of forecast or impulse steps!!')
end
kfs = nvar*nstepsm; % k -- fs: free shocks
%*** initializing
Rcon = zeros(kfs,nconstr); % R: k-by-q
Econ = zeros(kfs,1); % E: k-by-1
rcon = zeros(nconstr,1); % r: q-by-1
%rcon=valuecon-diag(yfore(stepcon,varcon)); % another way is to use "loop" below.
A0in = reshape(imf3s_h(1,:,:),nvar,nvar); % nvar shocks-by-nvar responses
for i=1:nconstr
if IdenShock
rcon(i)=length(stepcon{i})*valuecon(i) - ...
sum(yforeml(stepcon{i},varcon(i)),1); % <<>>
else
rcon(i)=length(stepcon{i})*valuecon(i) - ...
sum(yfore_h(stepcon{i},varcon(i)),1); %<<>>
end
Rmat = zeros(nstepsm,nvar);
r2mat = zeros(nstepsm,1); % simply one identified equation
% Must be here inside the loop because it's matrix of one column of Rcon
for j=1:length(stepcon{i})
if IdenShock % Rcon only at ML; assuming the Fed can't see all other shocks
Rmat(1:stepcon{i}(j),eq_iden) = Rmat(1:stepcon{i}(j),eq_iden) + ...
imf3sml(stepcon{i}(j):-1:1,eq_iden,varcon(i));
% Rmat: row--nstepsm, column--nvar shocks (here all shocks except
% the identified one are set to zero) for a particular
% endogenous variable 'varcon(i)'. See Zha Forcast (1), pp.6-7
else % Rcon random with (A0,A+)
Rmat(1:stepcon{i}(j),:) = Rmat(1:stepcon{i}(j),:) + ...
imf3s_h(stepcon{i}(j):-1:1,:,varcon(i));
% Rmat: row--nstepsm, column--nvar shocks (here all shocks are
% *not* set to zero) for a particular endogenous
% variable 'varcon(i)'. See Zha Forcast (1), pp.6-7
end
end
Rmatt = Rmat'; % Now, nvar-by-nstepsm. I think here is where RATS has an error
% i.e. "OVERR" is not transposed when overlaid to "CAPR"
Rcon(:,i)=Rmatt(:); % Rcon: k-by-q where q=nconstr
end
if nconstr
[u d v]=svd(Rcon,0); %trial
% rtr = Rcon'*Rcon; %trial
% rtrinv = inv(Rcon'*Rcon); %trial
vd=v.*(ones(size(v,2),1)*diag(d)'); %trial
dinv = 1./diag(d); % inv(diag(d))
vdinv=v.*(ones(size(v,2),1)*dinv'); %trial
rtr=vd*vd'; % R'*R
rtrinv = vdinv*vdinv'; % inv(R'*R)
Econ = Rcon*rtrinv*rcon; % E = R*inv(R'R)*r; mean
else
Econ = zeros(kfs,1);
Rcon = NaN;
rcon = NaN;
u = NaN;
d = NaN;
v = NaN;
end
tcwc = nvar*lags; % total coefficients without constant
phi=phil;
phis=phil; % for exact backed out shocks
%
if (Sband==0) % no random or random only from (A0,A+)
Estr = reshape(Econ,nvar,nstepsm);
Estr = Estr'; % transpose so that
% Estr: structural shocks. Row--steps, Column--n shocks
Estr = [Estr;randn(forep-nstepsm,nvar)];
% Now, forep-by-nvar -- ready for forecasts
Ures = Estr*A0in; % nstepsm-by-nvar
% Ures: reduced-form residuals. Row--steps; Column--n shocks
% ** reconstruct x(t) for y(t+h) = x(t+h-1)*B
% ** where phi = x(t+h-1) with last column being constant
%
yhat = zeros(forep,nvar);
for k=1:forep
yhat(k,:) = phi*Bh_h + Ures(k,:);
phi(1,nvar+1:tcwc) = phi(1,1:tcwc-nvar);
phi(1,1:nvar) = yhat(k,:);
%
end
else % random from shocks E and possibly (A0,A+) depending on if imf3s is random
if nconstr
if IdenShock % other shocks are indepedent of the eq_iden shock
Osk = randn(kfs,1); % other shocks
for j=1:nstepsm
Osk(nvar*(j-1)+eq_iden)=0; % no shock to the MS or identified equation
end
Estr = Econ + Osk; % Econ is non zero only at position
% eq_iden*j where j=1:nstepsm
else
Ome = eye(kfs) - u*u'; % note, I-u*u' = I - R*inv(R'*R)*R'
%[u1 d1 v1] = svd(Ome); % too slow
[u1 d1] = eig(Ome);
Stdcon = u1*diag(sqrt(diag(abs(d1)))); % lower triagular chol of conditional variance
% see Zha's forecast (1), p.17
if Cms
if TLindx % tight
Estr1 = Econ + Stdcon*randn(kfs,1);
Estr2 = reshape(Estr1,nvar,nstepsm);
Estr2 = Estr2'; % transpose so that
% Estr2: structural shocks. Row--nstepsm, Column--n shocks
Estr = [Estr2;randn(forep-nstepsm,nvar)];
% Now, forep-by-nvar -- ready for forecasts
Sindx = find(Estr(1:nCms,eq_Cms)<TLnumber);
while ~isempty(Sindx)
Estr1 = Econ + Stdcon*randn(kfs,1);
Estr2 = reshape(Estr1,nvar,nstepsm);
Estr2 = Estr2'; % transpose so that
% Estr2: structural shocks. Row--nstepsm, Column--n shocks
Estr = [Estr2;randn(forep-nstepsm,nvar)];
% Now, forep-by-nvar -- ready for forecasts
Sindx = find(Estr(1:nCms,eq_Cms)<TLnumber);
end
else % loose
Estr1 = Econ + Stdcon*randn(kfs,1);
Estr2 = reshape(Estr1,nvar,nstepsm);
Estr2 = Estr2'; % transpose so that
% Estr2: structural shocks. Row--nstepsm, Column--n shocks
Estr = [Estr2;randn(forep-nstepsm,nvar)];
% Now, forep-by-nvar -- ready for forecasts
Sindx = find(Estr(1:nCms,eq_Cms)>TLnumber);
while ~isempty(Sindx)
Estr1 = Econ + Stdcon*randn(kfs,1);
Estr2 = reshape(Estr1,nvar,nstepsm);
Estr2 = Estr2'; % transpose so that
% Estr2: structural shocks. Row--nstepsm, Column--n shocks
Estr = [Estr2;randn(forep-nstepsm,nvar)];
% Now, forep-by-nvar -- ready for forecasts
Sindx = find(Estr(1:nCms,eq_Cms)>TLnumber);
end
end
else
Estr1 = Econ + Stdcon*randn(kfs,1);
Estr2 = reshape(Estr1,nvar,nstepsm);
Estr2 = Estr2'; % transpose so that
% Estr2: structural shocks. Row--nstepsm, Column--n shocks
Estr = [Estr2;randn(forep-nstepsm,nvar)];
% Now, forep-by-nvar -- ready for forecasts
end
end
else
if Cms
if TLindx % tight
Estr = randn(forep,nvar);
% Now, forep-by-nvar -- ready for forecasts
Sindx = find(Estr(1:nCms,eq_Cms)<TLnumber);
while ~isempty(Sindx)
Estr = randn(forep,nvar);
% Now, forep-by-nvar -- ready for forecasts
Sindx = find(Estr(1:nCms,eq_Cms)<TLnumber);
end
else % loose
Estr = randn(forep,nvar);
% Now, forep-by-nvar -- ready for forecasts
Sindx = find(Estr(1:nCms,eq_Cms)>TLnumber);
while ~isempty(Sindx)
Estr = randn(forep,nvar);
% Now, forep-by-nvar -- ready for forecasts
Sindx = find(Estr(1:nCms,eq_Cms)>TLnumber);
end
end
else
Estr = randn(forep,nvar);
% Now, forep-by-nvar -- ready for forecasts
end
end
%
%disp('HERE')
Ures = Estr*A0in; % nstepsm-by-nvar
% Ures: reduced-form residuals. Row--steps; Column--n shocks
% ** reconstruct x(t) for y(t+h) = x(t+h-1)*B
% ** where phi = x(t+h-1) with last column being constant
%
yhat = zeros(forep,nvar);
for k=1:forep
yhat(k,:) = phi*Bh_h + Ures(k,:);
phi(1,nvar+1:tcwc) = phi(1,1:tcwc-nvar);
phi(1,1:nvar) = yhat(k,:);
end
end
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