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function [yhat,Estr,rcon,Rcon,u,v,d] = fn_fcstcnd(valuecon,stepcon,varcon,nstepsm,...
nconstr,eq_ms,nvar,lags,phil,Sband,yfore_h,imf3s_h,Bh_h,forep)
% ******* There are some BUG problems when calling this fucntion. ******* 3/15/2004.
% [yhat,Estr,rcon,Rcon,u,v,d] = fn_fcstcnd(valuecon,stepcon,varcon,nstepsm,...
% nconstr,eq_ms,nvar,lags,phil,Sband,yfore_h,imf3s_h,Bh_h,forep)
%
% Conditional forecasting in the identified model with or without error bands
% It handles conditions on average values as well, so "valuecon" must be
% expressed at average (NOT sum) levels (i.e., arithmetic averages of log(y) over
% the period of stepcon{i}. 5/22/01.
% Unconditional forecast when imf3s_h, etc is fixed and nconstr=0.
% Note that length(eq_ms)==1 implies one-one mapping between MS shocks and, say, FFR
% if nstepsm==nconstr. If this condition does not hold, this procedure is incorrect.
% I don't have time to fix it now (3/20/99). Meantime, consult or use the old code
% fidencond.m.
%
% 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
% nstepsm: maximum number of steps in all DLS constraints
% nconstr: number of DLS constraints
% eq_ms: Equation location of MS shocks. If [], all shocks.
% 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)
% Sband: 1: draws from random shocks E; 0: no random shocks For now (4/27/01), no option
% for Aband because I don't think it works best to do both Aband and Sband in one function.
% yfore_h: uncondtional forecasts: forep-by-nvar. Never used when nconstr=0.
% In this case, may set it to [];
% imf3s_h: 3-dimensional impulse responses matrix: impsteps-by-nvar shocks-by-nvar responses
% Never used when nconstr=0. In this case, may set it to [];
% Bh_h: 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
% forep: # of forecast periods (e.g., monthly for a monthly model)
% eq_Cms: equation location of MS shocks
% ------
% yhat: conditional forecasts: forep-by-nvar
% Estr: backed-out structural shocks (from constrained Gaussians)
% 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 and k>=q
%
% See the old code fidencond.m. I wond't use fn_fcstidcnd?.m, 5/22/01.
% Written by Tao Zha March 1998 .
% Revised November 1998, May 2001 (Delete A0_h as
% input arg so that previous programs may not be compatible).
%
% 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/>.
%
DLSIdShock = ~isempty(eq_ms); % if not empty, the MS shock is identified as in DLS
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
kts = nvar*nstepsm; % k -- ts: total shocks some of which are restricted and others
% are free.
%*** initializing
Rcon = zeros(kts,nconstr); % R: k-by-q
Econ = zeros(kts,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.
tcwc = nvar*lags; % total coefficients without constant
phi=phil;
%----------------------------------------------------
% Form rcon, Rcon, and Econ (the mean of structural shocks)
%----------------------------------------------------
A0in = reshape(imf3s_h(1,:,:),nvar,nvar); % <<>>1 nvar shocks-by-nvar responses
if nconstr % Conditional forecasts.
for i=1:nconstr
rcon(i)=length(stepcon{i})*valuecon(i) - sum(yfore_h(stepcon{i},varcon(i)),1);
%<<>>2 Automatically taking care of average conditions.
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 DLSIdShock % Assuming the Fed can't see all other shocks within a month
Rmat(1:stepcon{i}(j),eq_ms) = Rmat(1:stepcon{i}(j),eq_ms) + ...
imf3s_h(stepcon{i}(j):-1:1,eq_ms,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
%
[u d v]=svd(Rcon,0); %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; the mean of structural shocks
else % Unconditional forecasts.
Econ = zeros(kts,1); % the mean of shocks is zero under no variable condition
Rcon = NaN;
rcon = NaN;
u = NaN;
d = NaN;
v = NaN;
end
%---------------------------------------
% No uncertainty at all. In other words, no future shocks.
%---------------------------------------
if (~Sband) %| (nconstr & (length(eq_ms)==1))
% length(eq_ms)==1 implies one-one mapping between MS shocks and, say, FFR
% if nstepsm==nconstr. If this condition does not hold, this procedure
% is incorrect. I don't have time to fix it now (3/20/99). So I use
% this as a proximation
Estr = reshape(Econ,nvar,nstepsm);
Estr = Estr'; % transpose so that
% Estr: structural shocks. Row--steps, Column--n shocks
Estr = [Estr;zeros(forep-nstepsm,nvar)];
% Now, forep-by-nvar -- ready for forecasts
Ures = Estr*A0in; % <<>>1 nstepsm-by-nvar
% Ures: reduced-form residuals. Row--steps; Column--n shocks
% Note: We don't use /A0_h so as to eliminate small discrepancies to be
% completely compatible with the computation of Rmat and Estr, which uses A0in.
% ** 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
%---------------------------------------
% With random future shocks.
%---------------------------------------
else
if nconstr % Conditional forecasts.
%--------------
% Condition on variables with all shocks backed out. Straight DLS forecasts. No A random but S random.
%--------------
if ~DLSIdShock
Ome = eye(kts) - 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
Estr1 = Econ + Stdcon*randn(kts,1); % Draws of constrained (conditioned) shocks.
Estr2 = reshape(Estr1,nvar,nstepsm);
Estr2 = Estr2'; % transpose so that
% Estr2: structural shocks. Row--nstepsm, Column--n shocks
Estr = [Estr2;randn(forep-nstepsm,nvar)]; % Second concatenated part: draws of free shocks.
% Now, forep-by-nvar -- ready for forecasts
%--------------
% Condition on variables with identified MS shocks backed out, no A random but S random.
%--------------
else % other shocks are indepedent of the eq_ms shock
% 3/20/99 The following may be problematic because Osk should depend
% on u (A0_h and Bh_h) in general. I have NOT worked out any good version.
%/*
% Osk = randn(kts,1); % other shocks
% for j=1:nstepsm
% Osk(nvar*(j-1)+eq_ms)=0; % no shock to the MS or identified equation
% end
% Estr = Econ + Osk; % Econ is non zero only at position
% % eq_ms*j where j=1:nstepsm
% Estr = reshape(Estr,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
%
disp('DLS')
Ome = eye(kts) - 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
tmp1=zeros(nvar,nstepsm);
tmp1(eq_ms,:)=randn(1,nstepsm);
tmp2=tmp1(:);
%Estr1 = Econ + Stdcon*randn(kts,1);
%jnk = reshape(Stdcon*tmp2,nvar,nstepsm)
Estr1 = Econ + Stdcon*tmp2;
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
else % Unconditional forecasts.
Estr = randn(forep,nvar); % Unconditional draws.
end
Ures = Estr*A0in; % <<>>1 nstepsm-by-nvar
% Ures: reduced-form residuals. Row--steps; Column--n shocks
% Note: We don't use /A0_h so as to eliminate small discrepancies to be
% completely compatible with the computation of Rmat and Estr, which uses A0in.
% ** 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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